Taxation
and International Capital Asset Pricing
Theory
Inauguraldissertation
zur Erlangung des akademischen Grades
�Doktor der Wirtschaftswissenschaften�
(Dr. rer. pol.)
eingereicht an der
Wirtschaftswissenschaftlichen Fakultät
der Europa-Universität Viadrina
in Frankfurt (Oder)
15. Januar 2011
von
Riad Nourallah
Simferopol
c© 2010
Riad Nourallah
All Rights Reserved
Abstract
Adler and Dumas (1983) laid the foundation for pricing international assets under de-
viation from Relative Purchasing Power Parity (PPP). Only Lally (1996) regards the
spectrum of international taxation but in his model - he disregards the tremendous im-
pact of exchange gains taxation in International Capital Asset Pricing Theory (IntCAPT).
Furthermore, the consensus in economic literature that exchange rates show evidence of
a non-linear behavior as elaborated by Dumas (1992), Grauwe (1993) and Serçu and
Uppal (1995) and that monetary policy ultimately determines in�ation as determined
by McCallum (1990) is ignored. In addition to this, a realistic version of the Tax In-
ternational Capital Asset Pricing Model (Tax � IntCAPM) should incorporate the fact
that dividends are stochastic, as developed in the Tax Capital Asset Pricing Model (Tax
� CAPM) of Lally (1998), Wiese (2006b) and Mai (2006a).
This dissertation develops a theory of taxation in pricing international assets. To under-
stand this theory, in the �rst part we introduce and discuss the research question and the
conceptual procedure of the dissertation. The review of the status of research provides
an extensive overview on research pertaining to taxation in IntCAPT. In the second part,
the framework of international taxation is introduced, and by introducing the features
of exchange gains taxation a new income type in IntCAPT is presented. The analysis
of the international tax system with the features of exchange gains taxation leads to the
new result that under the hypothesis of Relative PPP certain constellations of interna-
tional taxation lead to a Tax-IntCAPM that would be equal to the Tax-CAPM. With
the features of exchange gains taxation and the modeling of deviation from Relative PPP
by non-linear behavior of exchange rate and in�ation determined by monetary policy,
an extended model of taxation in IntCAPT � the Tax-IntCAPM � is developed and in-
terpreted. The new result is that the integration of exchange gains taxation into the
Tax-IntCAPM leads to an international pricing relationship composed of the risky asset's
excess return and its world risk premium, which is adapted by exchange gains tax factors.
The non-linear deterministic behavior of exchange rates and the determination of in�ation
by monetary policy lead to the integration of the market equilibrium exchange and in-
�ation rate into the Tax-IntCAPM. International tax arbitrage opportunities lead to the
derivation of the Tax-IntCAPM under short sale and borrowing restrictions. To imple-
ment this new international capital market model, the Tax-IntCAPM with homogeneous
I
expectations is derived and interpreted. The third part concludes the dissertation with an
extensive critique elaborating the boundaries of the models and a conclusion summarizing
the main results and analyzing the implications of the �ndings.
Keywords: Tax International Capital Asset Pricing Model - Empirical Tax Inter-
national Capital Asset Pricing Model - Tax International Capital Asset
Pricing Model under Restrictions - Exchange gains Taxation - Non-linear
Behavior of Exchange Rates - International Taxation - Purchasing Power
Parity - International Tax Arbitrage
II
Style manual used was the MLA Handbook for Writers of Research Papers, 7th Edition.
Computer software used was LaTeX2e and Microsoft R© Excel 2007.
Contents
List of Figures VII
List of Tables VIII
List of Symbols IX
List of Abbreviations XVIII
1 Introduction 1
2 Research Question and Conceptual Procedure 4
3 Review of the Status of Research 9
3.1 Tax Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Tax Capital Asset Pricing Model under Di�erent Tax Regimes . . . 11
3.1.2 Tax Capital Asset Pricing Model with Stochastic Dividends . . . . 12
3.1.3 Tax Capital Asset Pricing Model under Restrictions . . . . . . . . . 13
3.2 International Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . 15
3.2.1 Purchasing Power Parity . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.2 International Capital Asset Pricing Model under Purchasing Power
Parity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Taxation and International Capital Asset Pricing Theory . . . . . . . . . . 22
3.4 Exchange Rate Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Quantity Theory of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
IV
4 International Taxation 34
4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Corporate and Dividend Tax System . . . . . . . . . . . . . . . . . . . . . 39
4.3 Capital Gains and Interest Tax System . . . . . . . . . . . . . . . . . . . . 46
4.4 Exchange Gains Tax System . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Methods of Double Taxation Reduction . . . . . . . . . . . . . . . . . . . . 51
5 Tax International Capital Asset Pricing Model 58
5.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Model Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.1 Domestic Investor's Rate of Return . . . . . . . . . . . . . . . . . . 65
5.2.2 Deviation from Purchasing Power Parity and Real Exchange Rate . 67
5.2.3 Non-linear Behavior of Exchange Rates . . . . . . . . . . . . . . . . 69
5.2.4 Foreign Investor's Rate of Return . . . . . . . . . . . . . . . . . . . 76
5.2.5 Implementation of Foreign Investor's Rate of Return . . . . . . . . 81
5.3 Versions of Tax International Capital Asset Pricing Model . . . . . . . . . 85
5.4 Model Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4.1 Individual Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.2 Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.5 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5.1 Deterministic Dividends . . . . . . . . . . . . . . . . . . . . . . . . 112
5.5.2 Portfolio Composition . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Tax International Capital Asset Pricing Model under Restrictions 117
6.1 Tax Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Tax International Capital Asset Pricing Model under Short Sale and Bor-
rowing Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.1 Individual Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.2 Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7 Tax International Capital Asset Pricing Model with Homogeneous Expec-
tations 128
7.1 Model Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
V
Contents
7.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8 Critique 140
9 Conclusions 145
Appendix 147
A Taylor Series Approximation of the Utility Function 148
B Stein's Lemma 151
C Derivation of Tax International Capital Asset Pricing Model 154
C.1 Non-liner Behavior of Exchange Rate . . . . . . . . . . . . . . . . . . . . . 154
C.2 Expected Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
C.2.1 Stochastic Dividends . . . . . . . . . . . . . . . . . . . . . . . . . . 156
C.2.2 Deterministic Dividends . . . . . . . . . . . . . . . . . . . . . . . . 159
C.3 Risk Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
C.3.1 Domestic Investor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
C.3.2 Foreign Investor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
C.3.3 aggregate Covariance Term . . . . . . . . . . . . . . . . . . . . . . . 166
C.3.4 Pricing Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . 168
D Tax System International Capital Asset Pricing Model with Homogeneous
Expectations 171
Bibliography 175
VI
List of Figures
2.1 Structure of Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Structure of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Security Market Line of IntCAPM under the Assumption of Relative PPP 20
4.1 Corporate and Dividend Tax Systems . . . . . . . . . . . . . . . . . . . . . 40
4.2 Tax Rate Depending on the Relief and Imputation Factors in the Corporate
Tax System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Tax Rate Depending on the Relief and Imputation Factors in the Dividend
Tax System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Exchange Gains Tax System . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Methods of Avoiding and Reducing Double Taxation . . . . . . . . . . . . 52
5.1 Structure of the Tax-IntCAPM . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Deviation from Relative PPP . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Currency Exchange Market . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Time Series of Exchange Rate with α = 5, δ = 5 and γ = 25 . . . . . . . . 76
5.5 Model Solution of Tax-IntCAPM . . . . . . . . . . . . . . . . . . . . . . . 88
7.1 Tax-IntCAPM with Homogeneous Expectations . . . . . . . . . . . . . . . 136
C.1 Time Series of Exchange Rate with α = 4 . . . . . . . . . . . . . . . . . . . 155
List of Tables
4.1 Relief and Imputation Factors Depending on the Corporate and Dividend
Tax Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1 Taxing Right Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.1 Calculation of Components of βhE . . . . . . . . . . . . . . . . . . . . . . . 138
C.1 Value Table of Time Series of Exchange Rate . . . . . . . . . . . . . . . . . 154
C.2 Value Table of Time Series of Exchange Rate with α = 4 . . . . . . . . . . 155
List of Symbols
Notation Description Page
List
APPP Absolute Purchasing Power Parity 17
ADπD domestic investor's risk aversion factor under in�a-
tion
99
AFπF foreign investor's risk aversion factor under in�ation 99
AWπ world aggregate risk aversion factor under in�ation 100
AWdt world aggregate risk aversion factor under determin-
istic dividends
159
A expected coe�cient of world risk aversion 94
Cov covariance 20
DK+S dividends of the K+S AG stock 82
DSd−1demand for foreign currency on the previous day 72
DSd demand for foreign currency on the present day 70
D domestic state 35
E[Sd+1
]exchange rate of the next day 70
EI,P,t exchange gains after tax on interest and principal 51
E expected value 20
F foreign state 35
It interest after tax 48
I interest 36
MSD0 money supply of the domestic state at the beginning
of period
67
MSD money supply of the domestic state at the end of
period
67
List of Symbols
Notation Description Page
List
MSF0 money supply of the foreign state at the beginning of
period
81
MSF money supply of the foreign state at the end of period 81
M0 nominal value of the world market portfolio at the
beginning of period
101
Max Maximum 90
Min Minimum 54
PD0 price of domestic consumption bundle at the begin-
ning of period
18
PD price of domestic consumption bundle (at the end of
period)
17
P F0 price of foreign consumption bundle at the beginning
of period
18
P F price of foreign consumption bundle (at the end of
period)
17
RPPP Relative Purchasing Power Parity 18
R0 real exchange rate at the beginning of period 68
R real exchange rate (at the end of period) 67
SBaggr aggregated short sale and borrowing restriction factor 125
SB short sale and borrowing restriction factor 122
S0 nominal exchange rate at the beginning of period 18
Sd−1 exchange rate of the previous day 71
Sd exchange rate of the present day 70
Sm−1 exchange rate at the beginning of month m 83
Sm exchange rate at the end of month m 83
S nominal exchange rate (at the end of period) 17
TW weighted average over all Ti, where the weights are
the relative dividends
24
Td trade balance of the present day 71
Ti weighted average oftjd,j−tg,j1−tg,j for asset i over all in-
vestors j
24
X
List of Symbols
Notation Description Page
List
T weighted average of tI,j−tCG,j1−tCG,j
over all investors 24
U investor's utility function 90
V0,0 price of risk-free asset at the beginning of period 36
V0 price of risk-free asset at the end of period 36
VK+S,m−1 value of the K+S AG stock at the end of month m 82
VK+S,m value of the K+S AG stock at the beginning of month
m
82
Vi,0 price of asset i at the beginning of period 90
V ar variance 20
W0 initial real wealth 62
α sensitivity factor with α ≥ 0 70
βhE beta-factor of the Tax-IntCAPM with homogeneous
expectations
134
βrirw beta factor, sensitivity factor of risky asset i's and
world market return
20
δ sensitivity factor with δ > 0 71
εSt,CG domestic state's taxing right factor on capital gains 77
εSt,D domestic state's taxing right factor on dividends 78
εSt,I domestic state's taxing right factor on interest 80
ηCG,P,e foreign state's relief exchange gains taxing right fac-
tor on capital gains and principal
78
ηCG foreign state's relief taxing right factor on capital
gains
77
ηD,e foreign state's relief exchange gains taxing right fac-
tor on dividends
79
ηD foreign state's relief taxing right factor on dividends 78
ηI,P,e foreign state's exchange gains relief taxing right fac-
tor on interest
80
ηI foreign state's relief taxing right factor on interest 80
γ sensitivity factor with γ > 0 71
ι tax remission factor 55
XI
List of Symbols
Notation Description Page
List
λ1 lagrange variable 121
λ2 lagrange variable 121
µ expectation value of normal distribution 149
ω0 investor's value portion of risk-free assets in the port-
folio
90
ωi investor's value portion of assets in the portfolio 90
ω0,min investor-speci�c value portion 120
φ relief factor of dividend tax with 0 < φ < 1 41
πD in�ation rate of the domestic state 18
πF in�ation rate of the foreign state 18
ψ credit factor of corporate tax with 0 < ψ < 1 42
σ variance of normal distribution 149
% �at tax factor 56
CGi nominal capital gains of the risky asset i 36
CGF
i,t foreign investor's capital gains after international tax 77
CGi,t nominal capital gains after tax 48
DFi,t foreign investor's dividends after international tax 78
DCSi nominal dividends after tax in the classical system 41
DDEi nominal dividends after tax in the dividend exemp-
tion system
44
DDTSi nominal dividends after tax in the dividend tax sys-
tems
44
DISi nominal dividends after tax in the imputation system 42
DSRi nominal dividends after tax in the shareholder relief
system
41
Di nominal dividends of the risky asset i 36
EFD,t foreign investor's exchange gains on dividends after
tax
79
ECG,P,t exchange gains after tax on capital gains and princi-
pal
50
ED,t exchange gains on dividends after tax 51
XII
List of Symbols
Notation Description Page
List
M nominal value of the world market portfolio at the
end of period
102
PCSi nominal pro�t after tax in the classical system 41
PDEi nominal pro�t after tax in the dividend exemption
system
44
PDTSi nominal pro�t after tax in the corporate tax systems 44
P ISi nominal pro�t after tax in the imputation system 42
P SRi nominal pro�t after tax in the shareholder relief sys-
tem
41
Pi nominal pro�t of risky asset i 40
Vi price of risky asset i at the end of period 36
Wt investor's end of period real wealth after tax 90
rDCGi domestic investor's real capital gains rate of return
after tax
65
rDDi domestic investor's real dividend rate of return after
tax
65
rFCGi foreign investor's real capital gains rate of return after
tax
79
rFDi foreign investor's real dividend rate of return after
tax
79
rCG,CGt,aggr �rst aggregated capital gains rate of return 105
rCG,Dt,aggr second aggregated capital gains rate of return 166
rD,CGt,aggr �rst aggregated dividend rate of return 167
rD,Dt,aggr second aggregated dividend rate of return 168
ri real risky asset i's rate of return 20
rw real world market rate of return 20
rCGi,n nominal capital gains rate of return 65
rCGm,n nominal world market capital gains rate of return 102
rDi,n nominal dividend rate of return 65
rDm,n nominal world market dividend rate of return 102
ri,n nominal risky asset i's rate of return 24
XIII
List of Symbols
Notation Description Page
List
ζCG,P,e foreign state's exchange gains taxing right factor on
capital gains and principal
78
ζCG foreign state's taxing right factor on capital gains 77
ζD,e foreign state's exchange gains taxing right factor on
dividends
79
ζD foreign state's taxing right factor on dividends 78
ζI,P,e foreign state's exchange gains taxing right factor on
interest
80
ζI foreign state's taxing right factor on interest 80
d domestic investor 35
eRUB−EUR appreciation of Ruble-Euro exchange rate 83
e appreciation of nominal exchange rate 18
f foreign investor 35
n0i(k) exogenous number of risky asset i(k) 61
nDi demanded number of asset i by the domestic investor 61
nFi demanded number of asset i by the foreign investor 61
n0 investor's number of the risk-free asset in the portfo-
lio
90
rDI domestic investor's real interest rate of return after
tax
66
rFCGK+Sreal capital gains rate of return of the K+S AG stock
after tax
85
rFDK+Sreal capital gains rate of return of the K+S AG stock
after tax
85
rFI foreign investor's real interest rate of return after tax 81
rf real risk-free rate 20
rC,n nominal deterministic cash yield 24
rCGK+S,nnominal capital gains rate of return of the K+S AG
stock
82
rCw,n nominal dividend rate of return on the world equity
portfolio
24
XIV
List of Symbols
Notation Description Page
List
rDK+S,nnominal dividend rate of return of the K+S AG stock 82
rI,n nominal interest rate of return 66
rf,n nominal risk-free rate of return in asset i's economy 24
rfw,n world risk-free rate, weighted average of risk-free rate
of return, where the weights are the relative market
values of risky assets in each economy
25
rw,n nominal return on the world equity portfolio exclu-
sive of exchange rate changes
24
tDI domestic investor's tax rate on interest 66
tDSt,CG domestic investor's (source) capital gains tax rate 77
tDSt,D domestic investor's (source) dividend tax rate 78
tDSt,I domestic investor's (source) interest tax rate 80
tDSt domestic investor's source tax rate 54
tFCG,P,e foreign investor's exchange gains tax rate on capital
gains and principal
50
tFCG foreign investor's capital gains tax rate 77
tFD,e foreign investor's exchange gains tax rate on divi-
dends
51
tFD foreign investor's dividend tax rate 78
tFFCm foreign investor's �ctive credit tax rate 55
tFFt foreign investor's �at tax rate 56
tFI,P,e foreign investor's exchange gains tax rate on interest
and on principal
51
tFInt,CG,P,e foreign investor's exchange gains relief tax rate on
capital gains and principal
78
tFInt,CG foreign investor's capital gains relief tax rate 77
tFInt,D,e foreign investor's exchange gains relief tax rate on
dividends
79
tFInt,D foreign investor's relief tax rate on dividends 78
tFInt,I foreign investor's tax relief on interest 80
tFInt,e foreign investor's relief tax rate on exchange gains 57
XV
List of Symbols
Notation Description Page
List
tFInt foreign investor's relief tax rate 56
tFI foreign investor's interest tax rate 80
tFeff,CG,P,e foreign investor's e�ective exchange gains tax rate on
capital gains and principal
78
tFeff,CG foreign investor's e�ective capital gains tax rate 77
tFeff,D,e foreign investor's e�ective exchange gains tax rate on
dividends
79
tFeff,D foreign investor's e�ective dividend tax rate 79
tFeff,I,e foreign investor's e�ective exchange gains tax rate on
interest
80
tFeff,I foreign investor's e�ective interest tax rate 80
tFe foreign investor's exchange gains tax rate 57
tFtF ,Cml foreign investor's tax rate after application of the lim-
ited credit method
54
tFtF ,Cm foreign investor's tax rate after application of the un-
limited credit method
54
tFtF ,Dm foreign investor's tax rate after application of the de-
duction method
56
tFtF ,Em foreign investor's tax rate after application of the ex-
emption method
53
tFtF ,FCm foreign investor's tax rate after application of the �c-
tive credit method
55
tFtF ,F t foreign investor's tax rate after application of the ex-
emption method
56
tFtF ,Int,e foreign investor's tax rate on exchange gains 57
tFtF ,Int foreign investor's tax rate after application of the re-
lief methods
56
tFtF ,Rm foreign investor's tax rate after application of the re-
duction method
55
tF foreign investor's tax rate 53
XVI
List of Symbols
Notation Description Page
List
tjC,j tax rate of investor j on risky assets comprising the
impact of dividend imputation
24
tD/FI interest tax rate 48
tD/FCG capital gains tax rate 48
tD/FD integrated dividend tax rate with 0 ≤ t
D/FD < 1 46
tDCG domestic investor's capital gains tax rate 65
tDD domestic investor's dividend tax rate 66
tC corporate tax rate with 0 < tC < 1 40
tD dividend tax rate 41
tCG,j tax rate of investor j on capital gains 24
tI,j tax rate of investor j on interest 24
t point of time 36
v1 slack variable 120
v2 slack variable 120
XVII
List of Abbreviations
Notation Description Page
List
ADF Augmented Dickey � Fuller 21
APT Arbitrage Pricing Theory 17
AR(1) First � order Autoregressive Process 21
CAPM Capital Asset Pricing Model 4
CCAPM Consumption Capital Asset Pricing Model 16
CPI Consumer Price Index 28
CPP Commodity Price Parity 17
DTC Double Taxation Convention 83
EMH E�cient Market Hypothesis 60
EUR Euro 19
GARCH Generalized Autoregressive Conditional Het-
eroskedasticity
30
GMM Generalized Methods of Moments 30
HEM Heterogeneous Expectation Model 60
IAPM International Arbitrage Pricing Model 3
IAPT International Arbitrage Pricing Theory 16
ICAPM Intertemporal Capital Asset Pricing Model 3
List of Abbreviations
Notation Description Page
List
IntCAPT International Capital Asset Pricing Theory I
IntCCAPM International Consumption Capital Asset Pricing
Model
16
JLR Johansen Likelihood Ratio 21
LOOP Law Of One Price 17
MSCI ACWI Morgan Stanley Capital International All Country
World Index
138
OECD Organization for Economic Co-operation and Devel-
opment
2
OLS Ordinary Least Squares 151
PPP Purchasing Power Parity I
RUB Russian Ruble 19
SHS Schanz � Haigh � Simmons 2
STAR Smooth Transition Autoregression 28
SWARCH Switching Autoregressive Conditional Heteroskedas-
ticity
30
TAR Threshold Autoregression 28
Tax � CAPM Tax Capital Asset Pricing Model I
Tax � IntCAPM Tax International Capital Asset Pricing Model I
TSPM Time State Preference Model 10
UN United Nations 2
XIX
1 Introduction
�I am certain there is too much certainty in the world.�
Michael Crichton, US author and screenwriter (1942 � 2008)
International commerce is not a new phenomenon, it dates back into history beyond the
Phoenicians around 1200 B.C. (Kotabe (1999)). International capital market integration
commenced with the �ow of capital from Britain to the United States, Latin America
and the British colonies after 1700 (Neal (1990) and Zevin (1992)). In the last and in
future decades the dynamic process of globalization initialized by global externalizations
of economic, political and social developments enhanced and will enhance the impact of
international investments on the world economy.
The process of globalization has led to the integration of national capital markets, yet
nationhoods in our world economy are delineated along barriers in the form of taxation of
international investment and di�erent in�ation and exchange rates generating a form of
heterogeneity in international investors and consequently leading to the partial segmenta-
tion of markets.
The imposition of taxes extends back into history beyond the Egyptian Pharaohs; the tax
collectors � known as scribes � imposed a tax on cooking oil, whereas the Athenians intro-
duced international taxation by imposing a monthly poll tax on foreigners, people who
1
1 Introduction
were not born to an Athenian mother and father (Adams (1999)).1 The tax systems that
emerged in the 19th and 20th centuries were constructed for (relatively) closed economies.
Globalization created a new situation leading to the abolition of nation state borders for
the movement of capital. Multinational institutions like the United Nations (UN) and
Organization for Economic Co-operation and Development (OECD) exert worldwide pres-
sure to remove �scal barriers and construct model agreements of international taxation
(Wahl (2005)). Human beings of di�erent countries have totally di�erent tastes and
consume di�erent baskets of goods. The baskets of goods on which they consume the
income from their investments face di�erent processes of price. In such a setting, further
dimensions of internationality resulting from deviation from PPP arise. In the interna-
tional conference of Bretton Woods in New Hampshire in July 1944 a �xed exchange rate
regime was established pursuing the concept that the expansion of international trade
should be framed by the international �xation of exchange rates. In the beginning of
the 70s the imbalance of currency deposits led to the suspension of the �xed exchange
rate system of Bretton Woods and to free �oating exchange rates (James (1996) and
Samuelson and Nordhaus (2005)).2 The passing to �oating exchange rates symbolized
the approach to a more liberal policy of economy and distance from the theories of Keynes
pursuing the idea of a more intervening role of the state (James (1996) and Samuelson
and Nordhaus (2005)).
The delineation of national groups of investors by international taxation and deviation
1The theoretical framework for income taxation was established by Schanz (1896), Haig (1921) andSimmons (1962) (Schanz � Haigh � Simmons (SHS)). The income tax has a comprehensive characterin order to secure the universality of taxation and a synthetic nature, since di�erent incomes aresummed up to one tax unit. The income tax pursues the notion that income from all sources iscombined to determine the taxable income. A further income conception is the schedular systemor the dual income tax; under this hybrid conception capital income is taxed separately from otherincome. Capital income is taxed at relatively low rates. The reason for the low tax rates is theincreasing mobility of capital income; the rate is aligned to the bottom rate of the progressive taxon other income (Head (1997)). Further theoretical conceptions are the consumption or expenditureincome while the tax basis is the di�erence between income and savings (Kaldor (1955) and Andrews(1974)).
2The exact beginning of the suspension of the �xed exchange rate system of Bretton Woods is contro-versial in literature. James (1996) is of the opinion that the suspension of the system of BrettonWoods began in August 1971 with Nixon's announcement of the New Economic Program proclaiming�exible exchange rates and by ending the convertibility of the U.S. dollar to gold whereas Emminger(1986) pursues the opinion that the suspension of the system of Bretton Woods was in March 1973since the Smithsonian Agreement in December 1971 maintained the system of �exible exchange ratesuntil March 1973.
2
1 Introduction
from Relative PPP cause them to evaluate di�erently the returns from the same asset and
consequently lead to partial segmentation of markets (Adler and Dumas (1982)). The
segmentation of markets prevents investors from holding international assets and results
in their imperfection, leading to an equilibrium deviating from the perfect market outcome
(Dumas (1994a)).3
The compelling theoretical and empirical arguments show evidence of the existence of
home bias (French and Poterba (1991), Cooper and Kaplanis (1994) and Tesar and
Werner (1995)). The delineation of national groups of investors by deviation from Rela-
tive PPP and international taxation are explanations for the international market puzzle
why domestic assets can o�er superior risk and return relationship to foreign assets.
Theoretical models were developed to describe the features of international asset pricing.
However, the heterogeneity in investment opportunities for investors from di�erent coun-
tries in the form of international taxation changes the classic IntCAPM and desires a
model of international asset pricing integrating the features of international taxation.4
3In contrast to segmentation, incompleteness leads to the impossibility of trading assets (Dumas(1994a)).
4The terms Intertemporal Capital Asset Pricing Model (ICAPM) and International Arbitrage PricingModel (IAPM) are used for the International Capital Asset Pricing Model, however models of Intertem-poral Capital Asset Pricing Theory and of International Arbitrage Pricing Theory are summarized bythese terms (Dumas (1994a) and Zimmermann (2003)).
3
2 Research Question and
Conceptual Procedure
�If we knew what it was we were doing, it would not be called research, would it?�
Albert Einstein, US (German � born) physicist (1879 � 1955)
Various universal models of valuation of international assets were developed, but we are
ignorant of the in�uence of international taxation on the pricing of international assets.
Referred to the tension between knowledge and ignorance, the problem of pricing interna-
tional assets under barriers in the form of taxation arises. We discover a contradiction in
our supposed knowledge, since the IntCAPMs are not applicable for international a�airs.
IntCAPT draws the picture of homogenous investors pursuing the conception of interna-
tional investment motivated by the bene�t of diversi�cation (Grubel (1968) and Levy and
Sarnat (1970)). This picture is contradicted by barriers in the form of taxation and the
delineation of the world across countries along deviation from Relative PPP leading to het-
erogeneous market participants. The Capital Asset Pricing Model (CAPM) is considered
to be the fundamental model of integration of taxes into asset pricing theory.5 Although
the model is often criticized, the Capital Asset Pricing Model remains the best illustra-
tion of long-term tradeo�s between risk and return in the �nancial markets (Cousins
(2010)). Although very few investors actually use the CAPM without modi�cation, its
5The literature on the pricing of risky assets under an integrated tax system is based, either explicitlyor implicitly, on the seminal paper of Brennan (1970) on equilibrium asset pricing (Handley and
Maheswaran (2005)).
4
2 Research Question and Conceptual Procedure
principles are very valuable (Cousins (2010)). It is fully accepted that the in�uence of
international taxation in international investments is not denied, but international �nance
theory does not extensively focus on the problem of integrating taxation in international
market equilibrium models. It is a severe limit of international �nance theory that inter-
national taxation is not su�ciently considered in IntCAPT, since the relevance of taxes
leads to special kinds of market equilibrium.
The dissertation analyzes as research object the in�uence of taxation on IntCAPT.6 The
dissertation aims to describe taxation in IntCAPT through the construction and interpre-
tation of models. The central research question is: �What Are the Impacts of Taxation
on the International Capital Asset Pricing Model?" The aim of the dissertation is to de-
rive IntCAPMs under integration of international taxation. In order to �nd answers on
the research question, a research plan was developed which is illustrated by �gure 2.1 on
page 6.
The Tax-IntCAPM extends the IntCAPM through the integration of investor's interna-
tional income taxation. The aim of this extension is the derivation and interpretation of
a pricing relationship under consideration of an investor's international income taxation
in equilibrium.
Based on the research plan the structure and methodology of the dissertation is derived
and is illustrated by �gure 2.2 on page 7. After the introduction in the �rst chapter aiming
to put the dissertation in a broader context, the research problem leads to the elaboration
of the research question and the conceptual procedure in chapter two. An extensive
literature review on research pertaining to taxation in IntCAPT is given in chapter three.
The discussion of the IntCAPM leads to the hypothesis that the integration of taxation
in IntCAPT leads to a new model � the Tax-IntCAPM �.
6The theme of the doctorate is rooted in diverse contexts and has multifarious aspects. Furthermoremany issues are interconnected; the issues cannot be addressed within the con�nes of a particulardiscipline, but require an interdisciplinary approach. The author examines the topic through the eyesof specialists in several academic �elds and makes use of their research knowledge and tools. Heintends to produce transcendent insights and to commit to contextualism with a reaching for thegeneral. He aims to extend the boundaries of academic knowledge by eclecticism (Salter and Hearn
(1996), Starkey and Madan (2001) and Lamb (2005)).
5
2 Research Question and Conceptual Procedure
Research Problem
Pricing of International Assets under Barriers in the Form of Taxation
Research Question
What Are the Impacts of Taxation on the International Capital Asset Pricing Model?
Hypothesis
The Integration of Taxation in International Capital Asset Pricing Theory Leads to
aNew Model: theTax International Capital Asset Pricing Model � Tax-IntCAPM �
Research Object
Derivation and Interpretation of the Tax International Capital Asset Pricing Model
Figure 2.1: Structure of Research Plan
In chapter four the conception of international taxation is introduced.7 Hereby the con-
ception of taxation of the relevant income types in IntCAPT, the dividends, the capital
gains, interest and exchange gains are presented.8 The features of international taxation
� the assignment of right of taxation and the methods to avoid double taxation � are
extensively discussed in this chapter.
The research object � the derivation and interpretation of the Tax-IntCAPM � starts
in chapter �ve.9 In order to guarantee a deeper understanding of the models they are
illustrated by �gures and examples.10 In chapter �ve the Tax-IntCAPM is developed
7The conception of integration international taxation in IntCAPM makes use of the three-phase scheme(Wagner and Dirrigl (1980), Schult (1983) and Schult (2002)); in order to assess the tax burdenthe e�ective tax calculation according to Horst (1971) and Rose (1973) is applied. S. also Montag
(1977).8For the sake of readability we use solely the term gains instead of gains and losses.9The models are constructed according to the constructive and the contemplative system approach(Graumann (2004)). In the process of constructing models, the method of decreasing abstractionis used; the complexity of interdependencies is reduced by assumptions and in the course of a moreprofound analysis the assumptions are gradually dropped.
10The examples are illustrated by the German and Russian economies.
6
2 Research Question and Conceptual Procedure
Structure of Dissertation
Critique and Conclusions
Derivation and Interpretation
Introduction Chapter 1
Introduction
Chapter 3
Review of the State of
Art
Chapter 4
International Taxation
Chapter 9
Conclusions
Chapter 5
Tax International
Capital Asset Pricing
Model
Research
Problem
Hypothesis
Evaluation
Research Object
Chapter 8
Critique
Chapter 2
Research Question
and Conceptual
Proceeding
Chapter 6
Tax International
Capital Asset Pricing
Model under
Restrictions
Chapter 7
Tax International
Capital Asset Pricing
Model with
Homogeneous
Expectations
Figure 2.2: Structure of Dissertation
under integration of international taxation. The Tax-IntCAPM and the implication of
exclusion of international arbitrage opportunities due to di�erential taxation of assets
is discussed in chapter six, leading to a Tax-IntCAPM under short sale and borrowing
restrictions.
In chapter seven, the Tax-IntCAPM with homogeneous expectations is derived. We derive
hypotheses and the model is illustrated by an example.
The critique in chapter eight intends to give a deeper insight into the interpretation and
the limits of taxation in IntCAPT. The main results are summarized in chapter nine,
which �nishes by an outlook into the future. The proofs and additional information are
7
2 Research Question and Conceptual Procedure
given in the appendix.
8
3 Review of the Status of Research
“Reviewing has one advantage over suicide: in suicide you take it out on yourself;
in reviewing you take it out on other people.�
George Bernard Shaw, Irish dramatist and socialist (1856 � 1950)
This chapter provides a literature review on research pertaining to taxation in IntCAPT.
The literature review forms the basis for the dissertation. The review of the status of
research can be described as an alternate form of presentation of the contributions and
of critique. The critique is presented by elaborating the extensions of the successor and
by a �nal conclusion. This chapter is divided into seven sections. In the �rst section the
presentation of literature of Tax-CAPM is elaborated, followed by an extensive review of
IntCAPMs in the second section. In the third section taxation in IntCAPM is reviewed.
In section four we discuss exchange rate theories and in section �ve the quantity theory
of money. The chapter �nishes with an integrative survey of empirical evidence and a
conclusion.
3.1 Tax Capital Asset Pricing Model
The CAPM marks the birth of asset pricing theory (Fama and French (2004)).11 It forms
the basis for the integration of taxes in asset pricing theory.12 There exist further models
11For an overview, cf. Dimson and Mussavian (1999).12S. footnote 5.
9
3 Review of the Status of Research
such as the Time State Preference Model (TSPM) developed by Myers (1968), Kraus
and Litzenberger (1975) and Rubinstein (1976) and the Arbitrage Pricing Theory (APT)
- developed by Ross (1976).13 The TSPM is criticized since the derived model cannot
be represented as a handy empirical tool (Myers (1968)). The CAPM can be restated
in terms of the APT (Ross and Walsh (1983)). Due to its very general concept APT
does not develop clear statements about patterns of expected returns in an international
setting and is confronted with severe theoretic criticism (Gokey (1991) and Kruschwitz
and Lö�er (1997)).
The CAPM constitutes a theory for the structure of asset prices in national capital mar-
kets and was introduced independently by Sharpe (1963), Lintner (1965b), Mossin
(1966) and Treynor (1999).14 The model is based on the portfolio theory of Markowitz
(1959) which analyzes an investor's composition of a portfolio of risky and risk-free as-
sets. In Markowitz's model, an investor selects a portfolio at the beginning of the period
that produces an insecure return at the end of the period. The model assumes that in-
vestors are risk-averse and, when choosing among portfolios, they care only about the
mean and variance of their one-period investment return. As a result, investors choose
portfolios which minimize the variance of portfolio return (maximize expected return),
given expected return (variance); thus, the Markowitz approach is often called a �mean-
variance model� (Fama and French (2004)). The portfolio model provides a condition on
asset weights in mean-variance e�cient portfolios. The CAPM turns this statement into a
testable prediction about the relationship between risk and expected return by identifying
an investor's optimal portfolio composition under market equilibrium (Fama and French
(2004)).
Taxes are irrelevant only if the entire income types of CAPT are taxed at the same rate
or if there is a special linear relationship between dividend, interest and total return
(Mai (2008)).15 However, both constellations transpire to be implausible: On the one
hand under a stylized tax systems the equivalence of all tax rates for all income types
is unrealistic (Richter (2004)) and on the other hand � the linear relationship between
13The framework of the TSPM was laid by Arrow (1964) and has been expanded and expounded byDebreu (1959) and Hirshleifer (1964) and Hirshleifer (1966) (Myers (1968)).
14Cf. Sharpe (1964), Lintner (1965a), Lintner (1970), Merton (1972) and Sharpe (1999). S. Jensen(1972) for a review of these models.
15Cf. Long (1977), König (1990), Wiese (2006b).
10
3 Review of the Status of Research
dividend and total return was empirically rejected.16 Hence, taxation must be considered
in CAPT, otherwise, the pricing relationship would lead to suboptimal decisions (Long
(1977) and Wiese (2006b)).
Brennan (1970) relaxed the strong assumption of irrelevance of taxation that underpins
the original CAPM and laid the foundation for integration taxation in CAPT.17 There
are di�erent versions of the Tax-CAPM which vary in respect of the tax system, the
stochastic of return and restrictions to avoid tax arbitrage. The following review focuses
on these aspects of Tax-CAPM variants.
3.1.1 Tax Capital Asset Pricing Model under Di�erent Tax
Regimes
Brennan (1970) developed the basic version of the Tax-CAPM assuming linear, investor-
speci�c tax rates for interest and deterministic dividends which are unequal to the linear
and investor-speci�c tax rates for capital gains. Two strands of Tax-CAPM under di�erent
tax regimes can be di�erentiated between: The �rst strand includes models that have been
developed to allow for dividends being taxed at a di�erent rate to that of interest, and
some of these also allow for capital gains being taxed at a third rate. The second strand
ignores the linearity assumption of tax rates. All of these models contain a dividend yield
term that is certain.
Ashton (1989), Cli�e and Marsden (1992) and Lally and van Zijl (2003) among many
others18 integrate the imputation system into the Tax-CAPM. Lally and van Zijl (2003)
integrate the imputation system into the Tax-CAPM under the assumption of determin-
istic dividends and equal tax rates for dividends and capital gains which are unequal to
the tax rates of interest income.
16The linear relationship between dividend and total return is empirically rejected (Friend and Puckett
(1964), Litzenberger and Ramaswamy (1979) and Kalay (1982)). Only does Blume (1980) �ndempirical evidence for the linear relationship but solely for the small time horizon between 1947 and1956 in the United States.
17S. footnote 5.18Cf. Auerbach (1983), König (1990), Lally (1992), Monkhouse (1993), O�cer (1994), Okunev and
Tahir (1994), Brailsford and Davis (1995), Dempsey (1996), Wood (1997), Fa� (2000), Lally(2000), Lally (2002) and Hathaway and O�cer (2004).
11
3 Review of the Status of Research
To admit the greatest generality, the model should allow for each of the income types in
CAPT - dividends, capital gains and interest - to be di�erently taxed for any investor.
Jonas (2004), Wiese (2004), Wiese (2006b), and Dempsey and Partington (2008)
extend the models by di�erentiating tax rates for dividends, capital gains and interest.
Lally (1992) and Handley and Maheswaran (2005) develop a Tax-CAPM which is free
of all of these restrictions by abstracting from particular tax systems and present an asset
pricing model with integrated tax systems.19 Litzenberger and Ramaswamy (1979) and
König (1990) develop versions of the Tax-CAPM under the assumption of progressive
equal tax rates for deterministic dividends and interest; capital gains is assumed to be
tax-free.20
3.1.2 Tax Capital Asset Pricing Model with Stochastic
Dividends
A more realistic version of Tax-CAPM should incorporate the fact that dividends are
stochastic, since the future is insecure.21
Schwetzler and Piehler (2004) construct a simple model incorporating stochastic divi-
dends but ignore the stochastic relationship of dividends and capital gains. Lally (1998),
Wiese (2006b), Mai (2006a) and Mai (2008) extend the Tax-CAPM with stochastic
dividends and linear investor-speci�c varying tax rates for the entire income types in
CAPT.
Mai (2006a) and Mai (2008) analyze the Tax-CAPM with stochastic dividends and
consider constellations where the model incorporates a similar structure to the Tax-CAPM
19Wiese (2006b) integrates tax-free allowance into the model by equalizing the tax rate to zero. Wiese
(2006b) derives a version of the Tax-CAPM with stochastic capital gains but ignores the aspect thatthe tax-free allowance must also be stochastic (Mai (2008)).
20Wiese (2006b) extends this version of Tax-CAPM by di�erentiating between progressive tax rates fordeterministic dividends and interest. The tax rate for capital gains is assumed to be linear. Since taxrates are not stochastic, there is no fundamental di�erence to the Tax-CAPM with linear tax rates.The linear tax rates are to be replaced by investor-speci�c marginal tax rates in the equilibriumrelationship (Litzenberger and Ramaswamy (1979), König (1990) and Mai (2008)).
21In an empirical review Schulz (2006) reviews critically the assumption that dividends are determinis-tic. Wiese (2006b) comes to the conclusion that the extension of the Tax-CAPM to a multiperiodframework requires stochastic dividends, otherwise stocks would be risk-free.
12
3 Review of the Status of Research
with stochastic dividends and homogeneous expectations, a model where the tax rates are
equal for all investors. However, Mai (2006a) and Mai (2008) admit that simplifying
transformations of the β-factor are required which are formally not proved.
3.1.3 Tax Capital Asset Pricing Model under Restrictions
The versions of Tax-CAPM were derived in the framework of equilibrium, however it is
uncertain if this equilibrium exists due to the existence of tax arbitrage. The exclusion
of arbitrage is the central theorem in CAPT. Tax rate di�erentials across income classes
can generate tax arbitrage (McDonald (2001)); since investors are di�erently taxed, they
might use opportunities for tax arbitrage (Schäfer (1982), Dammon and Green (1987)
and Ross (1987)). Arbitrage is based on the concept of replication. The market is free
of arbitrage, if an asset can be replicated by a portfolio and the asset's and portfolio's
price are equal (value additivity) (Kruschwitz (2007)). In the case of integration of taxes,
di�erent replication portfolios can exist for an asset with deviating prices.22
Schäfer (1982), Dybvig and Ross (1986) and Ross (1987) among many others23 discuss
the relationship of exclusion of arbitrage with and without taxes.24 Market equilibrium
can be implemented by the introduction of short sale and borrowing restrictions. These
restrictions do not exclude arbitrage, but limit the use of arbitrage opportunities (Schäfer
(1982), Auerbach (1983), Dammon and Green (1987), Dammon (1988), Ashton (1989)
and Jones and Milne (1992)). The Tax-CAPM under restrictions can be di�erentiated
by short sale and borrowing restrictions.
Ross (1977) introduced short sale restrictions in CAPT. Litzenberger and Ramaswamy
(1980), Ashton (1989), König (1990), Wiese (2006b) and Mai (2008) analyze the Tax-
CAPM under the assumption of exclusion of short sales. Litzenberger and Ramaswamy
22Tax arbitrage can be avoided by special construction of tax systems and reduced by progressive taxrates which is not further analyzed since tax systems and rates are assumed to be given. As far asthe analysis of tax arbitrage under the assumption of progressive tax rates is concerned, cf. Schäfer(1982), Dybvig and Ross (1986), Dammon and Green (1987), Dammon (1988), Jones and Milne
(1992) and Basak and Croitoru (2001).23These include McDonald (2001), Gallmeyer and Srivastava (2003), Gallmeyer (2006) and Huang
(2008).24Lö�er and Schneider (2003) and Wilhelm and Schosser (2007) generalize the model of Ross (1987)
in a multiperiod context.
13
3 Review of the Status of Research
(1980) and König (1990) assume that capital gains is untaxed, whereas Ashton (1989)
develops an equilibrium pricing equation in the case of the dividend imputation system.
If a certain asset is held by one group of investors, then the Tax-CAPM under short sale
restrictions can be interpreted as a tax clientele CAPM and assets are distributed among
investors depending on their tax rates. If the tax rate on capital gains is less than on
dividends, the tax clienteles are formed based on the amount of tax rates and on dividend
return: Investors with the highest dividend tax rate will include assets with the lowest
dividend return into their portfolio (Litzenberger and Ramaswamy (1980), König (1990),
Wiese (2006b) and Mai (2008)).
The integration of short sale restrictions into the Tax-CAPM in the form that the investor
is only allowed to hold a positive amount of shares leads to an equilibrium pricing rela-
tionship that is applicable in special cases. It assumes that dividends are equally taxed
to capital gains and not only that investor's covariance term of total return and tangen-
tial portfolio is proportional to the beta factor but also that the risk premiums of all
investors are equal (Mai (2008)). Hence, conditions were introduced in order to derive a
market equilibrium which lacks economic interpretation (Litzenberger and Ramaswamy
(1980)).
Black (1972a) introduced borrowing restrictions in CAPT. Litzenberger and Ramaswamy
(1979), König (1990) andWiese (2006a) assume the existence of two kinds of borrowing
restrictions, the �rst one pursuing the conception that the borrowed amount is less than
an investor-speci�c investment in risky assets and the second one pursuing the idea that
interest shall not exceed dividends, which is assumed to be deterministic.
On account of the assumption of stochastic dividends, Mai (2008) incorporates short
sale restrictions on a limited amount and borrowing restrictions into the Tax-CAPM. The
entire amount of money invested in risky assets is restricted from acquiring a negative
value, implying that the short sale restrictions are not binding on every single risky asset
but on an investor's portfolio of risky assets. The borrowing restrictions incorporate
that the borrowed amount shall not exceed an investor-speci�c amount. He derives an
equilibrium pricing relationship which in comparison to the Tax-CAPM is adapted by a
term incorporating the market value of the short sale and borrowing restrictions.
14
3 Review of the Status of Research
3.2 International Capital Asset Pricing Model
In this section a brief discussion on the fundamental problems of asset pricing in the
international framework is provided. The issues shall give a �rst insight of di�erences
between national and international asset pricing theory.
An investor's motivation for international acquisition lies in the bene�t of diversi�cation
which was expounded in the models of Tobin (1958) and Markowitz (1959). These
insights were extended to an international framework by Grubel (1968) and solidi�ed for
international portfolio diversi�cation by Levy and Sarnat (1970) and Solnik (1974b)
among many others25. Through international diversi�cation investors could attain a
higher return (lower variances) for given variances (returns) on international portfolios
in comparison to domestic portfolios, which is due to the low correlations between na-
tional capital markets.26
In which manner is it necessary to extend the CAPM to render feasible an application on
an international setting?
1. In�ation rates di�er across states,
2. investors residing in di�erent states consume di�erent baskets of goods,
3. investors face di�erent investment opportunity sets.27
IntCAPMs can be categorized according to the following criteria (Stulz (1995) and Zim-
mermann (2003)):
1. Models with identical consumption and investment opportunity sets,
2. models with di�erent consumption and investment opportunity sets,
3. models with barriers to international investment.
25These include Lessard (1976), Levy and Sarnat (1993), Santis and Gerard (1997) and Gehrig and
Zimmermann (1999).26The low correlation is partly related to the states' specialization in particular industries (Roll (1992)).27Cf. Stulz (1984).
15
3 Review of the Status of Research
The investors' consumption opportunity set comprises the consumable commodities and
their prices, his investment opportunity set is described by the future distributions of
wealth (Stulz (1981a)) and (Stulz (1995)). Di�erent consumption and investment op-
portunity sets arise from the deviation from Relative PPP, which results from varying
in�ation and exchange rates and implies that asset returns are heterogeneously perceived.
Consequently, these di�erences a�ect an investor's portfolio choice and expected returns
(Stulz (1995)). Barriers to international investment occur through the integration of
taxation in IntCAPM.28
Solnik (1974a), Grauer (1976), Serçu (1980) and Adler and Dumas (1983) followed by
others29 extend the CAPM of Sharpe, Lintner and Mossin and the ICAPM of Merton
(1973) to an international context and develop International Capital Asset Pricing Models
(IntCAPMs).
Two strands of IntCAPMs can be identi�ed: Grauer (1976) among others30 develop
IntCAPMs under the postulate of PPP. The other strand of valuation concept in the
international setting comprises more general equilibrium models which were introduced
by Solnik (1974a), Serçu (1980) and Adler and Dumas (1983) and which develop
IntCAPMs without considering PPP. Like Solnik (1974a), Serçu (1980) and Adler and
Dumas (1983) assume that exchange rates are stochastic, but in contrast to him they do
not assume that exchange rates are independent from asset returns.
Further models of International Asset Pricing Theory comprise the International Con-
sumption Capital Asset Pricing Model (IntCCAPM) developed by Stulz (1981a) and the
International Arbitrage Pricing Theory (IAPT) originated by Solnik (1983) and elabo-
rated by Levine (1989) and Ikeda (1991).31 Stulz extends the Consumption Capital
Asset Pricing Model (CCAPM) of Breeden (1979) to an international setting,32 whereas
28Barriers in the form of taxation are explicitly discussed in chapter 3.3.29Further extensions of the IntCAPM were developed by Senbet (1979), Ross and Walsh (1983), Zapatero
(1995), Pavlova and Rigobon (2003), Chang (2005) and Soumaré (2006).30These include Kouri (1977), Frankel (1979), Fama and Farber (1979), Hodrick (1981) and Lucas
(1982).31Cf. also Stulz (1984) and Stulz (1995).32A severe limitation of Stulz's model is the empirical implementation since it requires the observation
of an investor's risk tolerance and time-varying local price indices; cf. Gokey (1991). Furthermore,the consumption rates are endogenous variables that are solved by investors when they optimize theirexpected lifetime expected utility (Stulz (1984)). For a critical review of consumption-based assetpricing, cf. Cornell (1981), Hansen (1982) and Wheatley (1983).
16
3 Review of the Status of Research
Solnik (1983) and the other authors transfer the Arbitrage Pricing Theory (APT) of Ross
(1976) to an international context and develop the IAPT.33
3.2.1 Purchasing Power Parity
In the �rst class of international valuation theories, apart from the general assumptions
of CAPT, it is assumed that markets are integrated and PPP is satis�ed (Stulz (1995)).
The origins of the PPP concept can be traced back to Spanish Salamanca School in the
16th century (Oxelheim and Wihloborg (2008)); the concept of PPP was introduced
in economic theory by Cassel (1916).34 Since then, the concept of PPP has become
embedded in the theories of many international economists (Frankel (1981), Taylor
(1995), Frankel and Rose (1995), Rogo� (1996)).35
According to the PPP hypothesis the determination of exchange rates depends on prices
of commodities. The PPP is based on the concept of Commodity Price Parity (CPP),
which implies in an arbitrage-free setting the same price of a commodity in every country
after translation into a common currency (Law Of One Price (LOOP)). The Absolute
PPP is de�ned in terms of the equality of the price of a representative domestic basket of
goods to the price of a foreign representative basket of goods (Serçu and Uppal (1995)).
The basket of goods contains the goods used for consumption in the domestic and foreign
countries.
APPP ≡ PDS = P F
with APPP Absolute Purchasing Power Parity
PD price of domestic consumption bundle
S nominal exchange rate
P F price of foreign consumption bundle
33As far as the critique of APT is concerned, c. chapter 3.1.34Cf. Cassel (1918) and Cassel (1925). After leaving the gold standard in 1914 countries experienced
signi�cantly di�erent rates of in�ation; Cassel proposed PPP as a means of adjusting exchange rates.35As far as an explicit overview of PPP is concerned, cf. Lothian and Taylor (1996), Sarno and Taylor
(2002a), Sarno and Taylor (2002b), Taylor and Taylor (2004) and Dornbusch (2008).
17
3 Review of the Status of Research
The ratio of the Absolute PPP at two di�erent instants leads to the dynamic version of
the Absolute PPP relationship, the Relative PPP (Serçu and Uppal (1995)).
RPPP ≡ S
S0
=PF
PD
PF0PD0
(3.1)
with RPPP Relative Purchasing Power Parity
S nominal exchange rate at the end of period
S0 nominal exchange rate at the beginning of period
P F price of foreign consumption bundle at the end of period
PD price of domestic consumption bundle at the end of period
P F0 price of foreign consumption bundle at the beginning of period
PD0 price of domestic consumption bundle at the beginning of period
The rate of in�ation is de�ned in terms of the appreciation of consumer goods. We can
insert the in�ation rate in the above equation.
PD
PD0
≡ 1 + πDP F
P F0
≡ 1 + πF (3.2)
with πD in�ation rate of the domestic state
πF in�ation rate of the foreign state
We de�ne the appreciation of nominal exchange rates as the quotient of end of period
nominal exchange rate and exchange rate in the beginning of period.
S
S0
≡ 1 + e (3.3)
with e appreciation of nominal exchange rate
18
3 Review of the Status of Research
So, we have
RPPP ≡ 1 + e =1 + πF
1 + πD⇒ RPPP ≡ (1 + e)
1 + πD
1 + πF= 1. (3.4)
Example 1 Relative Purchasing Power Parity
Prices in Germany rose by 0.4% and prices in Russia by 8.8% in 2009; the nominal
exchange rate Russian Ruble (RUB)/Euro (EUR) should appreciate by 8.36%.36
According to the hypothesis of Relative PPP, domestic and foreign in�ation rates are
related to exchange rate changes to the extent that di�erences in the development of price
levels are set o� by the appreciation of the nominal exchange rate and the appreciation of
the nominal exchange rate is determined by the relation of foreign to domestic in�ation.
The use of the Relative PPP allows for a di�erent basket of goods and varying weights to
be applied towards the goods within the consumption bundle. This relation is necessary
but not su�cient for the validity of Absolute PPP, but not vice versa, so the Relative
PPP is a weaker version of the Absolute PPP.
3.2.2 International Capital Asset Pricing Model under Purchasing
Power Parity?
Under the assumption of the prevalence of Relative PPP, the IntCAPM can be derived
(Grauer (1976), Stulz (1995), Zimmermann (2003) and Armitage (2005)).37 The
assumption of prevalence of Relative PPP leads to the conclusion of full integration of
national capital and goods markets; the real returns on international assets are the same
across countries.
E[ri] = rf +Cov[ri, rw]
V ar[rw]︸ ︷︷ ︸βrirw
(E[rw]− rf ) (3.5)
36As far as the data for the German and Russian in�ation rate is concerned, s. www.destatis.de andwww.cbr.ru/eng/analytics/macro/.
37S. chapter 5.3.
19
3 Review of the Status of Research
with E expected value
ri real risky asset i's rate of return
rf international real risk-free rate of return
Cov covariance
V ar variance
βrirw beta factor, sensitivity factor of risky asset i's and world market return
rw real world market rate of return
0 0.5 1 1.5
-
6
Expected return
βrirw
World market
risk premium
Risk-free
rate
Worldmarketreturn
.............................
.............................
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
...
......
......
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......
......
......
......
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......
......
......
......
........................................................................................................................................................
Figure 3.1: Security Market Line of IntCAPM under the Assumption of Relative PPP
Figure 3.1 represents eq. (3.5). Any investor determines the expected return of an asset
by the international real risk-free rate and a risk premium, composed of the product of
the beta-factor and the real expected return on the world market portfolio in excess of
the international real risk-free rate.38 As a result the expected return of a risky asset is
invariant to investors residing in di�erent states.
38The world market portfolio comprises the entire securities of the world in proportion to their capital-ization relative to the world wealth by use of consumption goods as the numeraire to measure worldwealth and each security's capitalization (Stulz (1995)). Under the assumptions of the existence ofan asset with a risk-free return as well as a zero world market beta in real terms and a zero covariancebetween in�ation rate and nominal asset returns, an adequate asset pricing equation in nominal termscan be derived (Zimmermann (2003)).
20
3 Review of the Status of Research
Heckscher (1916) proposed the idea of deviation from (Relative) PPP. Siegel (1972)
�rst raised the question of asset pricing when deviations from PPP are the source among
market participants. The main reasons for (Relative) PPP deviation lie in
1. the presence of non-traded goods and services,
2. the existence of signi�cant transaction costs,
3. di�erences in relative prices of goods, the composition of national consumption
baskets (heterogeneous preferences),
4. monetary policy.39
In empirical studies, the Relative PPP hypothesis was clearly rejected for short-period
considerations (Kravis and Lipsey (1978), Levich (1983), Glen (1992), Froot and Rogo�
(1995) and among others40). In most recent studies Taylor (2002) and Murray and
Papell (2005) among others41 are inconclusive as to whether the hypothesis of Relative
PPP holds as far as long periods - three to �ve years - are concerned.42 By use of the
Augmented Dickey � Fuller (ADF) unit root test and the multivariate Johansen Likelihood
Ratio (JLR) test, Taylor (2002) extends the long-run analysis to a set of 20 countries
over the 1870 - 1996 period and �nds support for the long run Relative PPP.43 Using
Andrews (1993) median-unbiased estimation method for the First � order Autoregressive
Process (AR(1)) speci�cation in the ADF regression for the dollar-sterling real exchange
rate (1791�1990) and the franc-sterling real exchange rate (1803�1990),Murray and Papell
(2005) are inconclusive about empirical evidence for the long run Relative PPP.44
Since Relative PPP does not hold to the extent that the exchange rate does not set o�
di�erences in the development of price levels, investors' portfolio composition di�ers and
39Cf. Stulz (1984), Oertmann (1997), Zimmermann (2003), Copeland (2005) and Armitage (2005).40Cf. Kravis (1975), Roll (1979) and Frankel (1981) and Taylor and Taylor (2004) .41Cf. Aliber and Stickney (1975), Gailliot (1970), Roll (1979), Adler and Dumas (1983), Frankel
(1986), Edison (1987), Huizinga (1987), Taylor (1988), Mark (1990), Abuaf and Jorion (1990),Glen (1992),Lothian and Taylor (1996), Rogo� (1996), Taylor (2001), Chen (2004), and Taylor
and Taylor (2004).42For an overview, cf. Sarno (2003).43For a description of the ADF unit root test, s. Greene (2003). For a description of the JLR test, s.
Johansen (1988) and Johansen (1991). The chief merit of this test may be the clean speci�cationof null and alternative hypotheses.
44For a description of AR(1) speci�cation, cf. Mills (1991).
21
3 Review of the Status of Research
real returns are di�erently measured. The fact that Relative PPP may fail introduces a
new dimension of internationality in international valuation which does not exist in the
framework of domestic pricing models.
3.3 Taxation and International Capital Asset Pricing
Theory
The general form of IntCAPM ignores the existence of taxation; however, rational in-
vestors judge the return and risk of assets after taxes. Two strands of Tax-IntCAPM
can be di�erentiated. The �rst models ignored the existence of exchange rates and the
question of deviation from PPP, the other strand implemented deviation of PPP in their
models.
Stapleton and Subrahmanyam (1977) raised the question of the e�ect of taxation in
IntCAPT under irrelevance of exchange rates. In Stapleton and Subrahmanyam (1977)
numerical examples of the e�ects of taxes on foreign assets are analyzed, but no pricing
relationship is presented. The �rst approach of modeling the segmentation of markets in
the form of taxation based on the concept of capital market equilibrium was initiated by
Black (1974). In a two country one period CAPM, a proportional tax is imposed on an
investor's long position of foreign risky assets, whereas the investor with a short position
of foreign risky assets receives a subsidy. Thus, in the Black model taxes are paid in
proportion to the net holdings of risky foreign assets. The tax may di�er from asset to
asset or from country to country. It may be assessed by the investor's own country or
by the country in which the asset is located. The tax is intended to represent a barrier
to international investment on the value of an individual's holdings of assets in foreign
countries. He states that the direction of deviations from CAPM is consistent with the
results of empirical studies.45 In comparison with the CAPM Black's model implicates for
the equilibrium pricing relationships that assets with a low (high) systematic risk have a
higher (lower) expected return; the capital market line adopts a �atter line. The optimal
portfolio choice implies a mixture of the international market portfolio, the minimum
45Empirical studies show that high β securities with expected returns are lower than predicted by theCAPM and vice versa (Jensen (1972) and Black (1972b)).
22
3 Review of the Status of Research
variance zero-beta portfolio of risky assets and a minimum variance country-portfolio
subject to a given level of tax being held. It is possible for the risk-free asset to be
non-traded (Stulz (1981)). The model can only be applied in a frame when taxes are
assumed to be modest in amount,46 since an increase in the level of taxation does not lead
to abstention from holding foreign assets; in contrast, it would lead to an enlargement of
holding foreign assets short since they are subsidized. Thus the model does not su�ciently
explain the e�ects of barriers in the form of taxation on IntCAPT.
Stulz (1981) extended Black's framework by constructing a model of asset pricing in
which di�erent taxes on long and on short positions of foreign risky securities are im-
posed.47 Thus, in contrast to Black's model, short positions of foreign assets are not
subsidized and the wealth tax is imposed on an investor's absolute holdings of foreign
assets. In contrast to Black's model the world market portfolio is ine�cient on account of
the existence of nontraded foreign assets, which fall below a critical beta and consequently
do not provide expected returns large enough to equalize the costs of holding them. The
Capital Market Line is replaced by a Security Market Band. Taxed long (short) posi-
tions of foreign assets plot on a security market line lying above (below) parallel to the
CAPM security market line for risky domestic assets, nontraded foreign risky assets plot
between these ones. Byun and Chen (1997) explicitly regard Stulz's model in the form
of di�erential taxation for the domestic and foreign investor. Tax positions from short
and long holdings are asymmetrically taxed. Wood (1997) develops Stulz's model under
incorporation of the imputation system. In the model's framework the exchange rate is
regarded as irrelevant, since the introduction of the exchange rate would not change the
results; the e�ect of barriers to international investment would be the same so long as
those barriers to international investment were of a type which in the limit can produce
complete segmentation (Stulz (1981) and Byun and Chen (1997)).
Wheatley (1983) and Stulz (1984) integrate the taxation of end of period price of a risky
asset in the CCAPM and derive identical solutions as in the above mentioned model. In
Sellin (1989) and Sellin (1990), a two country asset pricing model is developed where
the domestic investor's holding is subject to a variable tax. The asset pricing equation
consists of two components, the asset's covariance with the world market portfolio and
46The question of how to quantify the term modest in amount is raised.47Cf. Stulz (1995).
23
3 Review of the Status of Research
the covariance of the asset with the tax rate divided by the variance of the world market
portfolio. In equilibrium the market participant's holding consists of the risk-free rate, the
world market portfolio and a portfolio designed to hedge against changes in the stochastic
tax rate.
Lally (1996) was the �rst to raise the question of taxation in IntCAPT under consideration
of exchange rates which can deviate from Relative PPP. In Lally (1996) and Lally (1998)
an IntCAPM is developed incorporating the dividend imputation system. Based on the
model of Solnik (1974a) he assumes that exchange rate movements are stochastic but
independent from assets returns and ignores in�ation.48 Lally (1996) assumes unrestricted
international capital �ows which in combination with the assumption of homogeneous
expectations implies that home bias is solely dictated by personal taxation and exchange
rate uncertainty. Personal taxes vary across investors and for stochastic capital gains and
deterministic interest and cash yield (dividends etc.). Given these assumptions then, the
pre-tax expected rate of return on asset i is
E[ri,n] = rf,n (1− T ) + rC,nTi +Cov[ri,n, rw,n]
V ar[rw,n](E[rw,n]− rCw,nTW − rfw,n (1− T )). (3.6)
with ri,n nominal risky rate of return of asset i
rf,n nominal risk-free rate of return in asset i's economy
T weighted average of tI,j−tCG,j1−tCG,j
over all investors world wide
tI,j tax rate of investor j on interest
tCG,j e�ective tax rate of investor j on capital gains
(allowing for deferral opportunities)
rC,n nominal deterministic cash yield on asset i
Ti weighted average oftjC,j−tCG,j1−tCG,j
for asset i over all investors j world wide
tjC,j tax rate of investor j on risky asset i's cash yield
rw,n nominal return on the world portfolio of risky assets
exclusive of exchange rates
rCw,n nominal cash yield (dividends etc.) of world market portfolio
TW weighted average over all Ti, where the weights are the relative
48Since exchange rate movements are stochastic and unequal to unity, the hypothesis of Relative PPP isrejected, s. chapter 5.2.2 and eq. 5.11.
24
3 Review of the Status of Research
cash payouts (dividends etc.) of the risky assets
rfw,n world risk-free rate, weighted average of risk-free rate of return,
where the weights are the relative market values of risky assets
in each economy
Equivalent to the IntCAPM under PPP in chapter 3.2.2, the market risk premium involves
terms such as E[rw,n] and rfw,n which re�ect the extension to the world market. Since
deterministic cash yield is considered, the equilibrium pricing relationship is adapted by
the terms rC,n and rCw,n. The tax parameters T , Ti and TW involve averaging over all
investors in the world and TW relates to the world market portfolio. Imputation in any
country reduces Ti and hence the expected return. The extent of this e�ect will depend
upon what proportion of investors can bene�t from the credits and the value weight of
these investors.49 Furthermore, imputation in any state reduces Ti and hence TW with a
�ow-on e�ect to the expected return.
According to Lally (1996) the only manifestation of stochastic exchange rates is in the
world risk-free rate.50 Under the assumption of �xed exchange rates and no restrictions
on capital �ow, only one risk-free rate can prevail throughout the world.
The model of Lally (1996) is the most elaborated Tax-IntCAPM since it assumes varying
tax rates for capital gains, deterministic interest and cash yield under the assumption of
deviation from Relative PPP.
3.4 Exchange Rate Theories
The currency market is the largest �nancial market whose transactions are made up
of short term, intra-daily transactions and results from the interaction of traders with
49In most cases, only domestic investors bene�t from imputation credits; unless these investors have asigni�cant value weight in the world, the impact of imputation on Ti and hence on the expected returnis trivial (Lally (1996)).
50This statement cannot be concluded from the derivation of the model in the appendix, since the worldrisk-free rate is derived by weighting the risk-free rate of return which is derived in the absenceof exchange rates. In the implementation of the model, Lally (1996) ignores the consideration ofexchange rates for the world risk-free rate as well.
25
3 Review of the Status of Research
di�erent time-horizons, risk-pro�les or regulatory constraints (Guillaume (1997)). The
price of a foreign currency is determined by the supply and demand of the currency which
in turn is related to the trade between countries (Vélez-Pareja (2003)). However, currency
markets are characterized by abrupt changes traceable to central bank interventions and
attempts to control the value of currencies contrary to the natural market forces (Vaga
(1994) and Peters (1994)). Exchange rate markets are di�erent from capital markets
since their intention is not to raise capital but to create the ability to trade in stocks and
bonds; as pure trading markets they are a zero sum game (Peters (1994)). In contrast
to the stock market, whose asset values rise and fall with the economy, currencies have
no stable relationship with the economy (Peters (1994)).
The �rst approach of exchange rate theories can be categorized into macroeconomic ratio-
nal expectations news and structural models.51 The macroeconomic rational expectations
news and the structural model are confronted with severe theoretic criticism, since these
models contain an in�nity of possible information which reveal themselves to be unstable
and imply the selection of one particular solution.52 Furthermore, they face severe empir-
ical criticism. A �rst empirical puzzle is that the volatility of the exchange rate by far
exceeds the volatility of the underlying economic variables.53 The asset market approach
together with the rational expectation assumption imply that there is a long-run equilib-
rium relationship between the exchange rate and its fundamentals, which is rejected by
the results of tests performed by Baillie and Selover (1987), Abel and Mishkin (1983),
51For an overview of exchange rate regime classi�cation, cf. Baillie and McMahon (1989), McDonald
(1989), Mussa (1991), Grauwe (1993), Grauwe and Grimaldi (2002) and Bhatti and Moosa
(2010). According to the rational expectations news model the exchange rate is determined by thepresent value of the expected future path of the fundamental variable driving the exchange rate whilestructural models specify the economic structure of the fundamental variable (Grauwe (1993)). Thestructural models can be categorized into monetary models with �exible prices developed by Mussa
(1976), Frankel (1976), Bilson (1978a), Bilson (1978b) and Mussa (1979), monetary models withsticky prices developed by Dornbusch (1976b), Dornbusch (1976a) and extended by Frankel (1979)and the portfolio balance model which was developed by McKinnon and Oates (1966), Branson(1969), Branson (1975), Branson (1977), Allen and Kenen (1978), Isard (1980) and Dornbusch
and Fischer (1980). For an explicit overview, cf. Bhatti and Moosa (2010).52This selection is based on information not contained in the model; ad hoc assumptions � information
outside the model � must be introduced (Grauwe (1993) and Hofstadter (1999)).53This is a contradiction to the rational news expectations models, which predicted that the volatility
of the exchange rate can only increase when the variability of the underlying fundamental variablesincreases (Baxter and Stockman (1989), Flood and Rose (1995) and Grauwe and Grimaldi (2002)).Goodhart (1989) and Goodhart and Figliuoli (1991) found that exchange rate movements appear tooccur in the absence of observable news.
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3 Review of the Status of Research
and Boothe and Glassman (1987) among others54. A third puzzle relates to PPP and is
closely related to the previous one. Dumas (1992) has stressed the long time needed to
adjust to PPP.55 The transaction cost hypothesis implies a non-linearity in the adjustment
process. This hypothesis has been con�rmed by the empirical evidence based on Michael
(1997).
The theoretical and empirical criticism of the macroeconomic rational expectations news
and structural exchange rate models has led to new attempts to model the exchange
rate (Bhatti and Moosa (2010)). These attempts have led to three di�erent modeling
approaches.
Microeconomic models characterize explicitly the process of trading in the foreign ex-
change market (Grauwe and Grimaldi (2002) and Bhatti and Moosa (2010)). A number
of authors, e.g. Grauwe and Grimaldi (2002), Evans and Lyons (2002), Jeanne and
Rose (2002) and Kilian and Taylor (2003) among many others56, endogenize the market
structure and explicitly model the behavior of di�erent types of traders such as chartists
and fundamentalists. Central issues like the sources and persistence of heterogeneous
beliefs, excess volatility and exchange rate determination have to be further researched
(Frankel and Rose (1995)).
In Harvey (1999) and Harvey (2006) a Post-Keynesian approach is used to explain the
determination of exchange rates. The argument is that exchange rates are determined by
international investors' demand for currency as they act to adjust their portfolios with
an emphasis on psychological and institutional factors. In the Post-Keynesian model,
exchange rates are determined by the international supply and demand for each currency.
Demand comes from imports, foreign direct investment, portfolio investment, and o�cial
reserve management.
Finally, a third approach recognizes that heterogeneous agents have di�erent beliefs about
the behavior of the exchange rate. These di�erent beliefs introduce non-linear features
in the dynamics of the exchange rate.57 The appeal of non-linear dynamics lies in the
54Cutler (1989) report signi�cant autocorrelation at short horizons of a month but negative signi�cantautocorrelation at lower frequencies.
55For an overview of the PPP, s. chapter 3.2.1.56These include Grauwe (1993), Lyons (1995), Frenkel (1997), Hau (1998) and Obstfeld and Rogo�
(2000).57For an overview of non-linear exchange rate models, cf. Sarno (2003).
27
3 Review of the Status of Research
plethora of behaviors they allow, in the sense that the simplest non-linear deterministic
system is capable of generating complex dynamics (Ellis (1992)). A variable can converge
to a �xed point, to a cycle, where it oscillates between points, explode, or lead to chaotic
systems (Ellis (1992) and Grauwe (1993)). This is motivated by the notion that the
real exchange rate follows a deterministic nonlinear process which generates output that
mimics the output of stochastic systems. In other words, it is possible for the real exchange
rate to appear random but not to be really random (Assaf (2006)).
Empirical research has convincingly demonstrated the importance of non-linearities in
exchange rates. Dumas (1992) and Serçu and Uppal (1995) among many others58 suggest
that exchange rates adjust towards PPP equilibrium in a nonlinear fashion. Obstfeld and
Taylor (1997) among many others59 provide evidence using Threshold Autoregression
(TAR) models in favor of nonlinearity in real exchange rates.60 Obstfeld and Taylor (1997)
consider the Consumer Price Index (CPI) of 24 countries at monthly frequencies from 1980
to 1995. Meanwhile, Michael (1997) employs Smooth Transition Autoregression (STAR)
models. Their interwar data set comprises monthly observations on wholesale price indices
for the United Kingdom, United States, France, and Germany, and spot exchange rates
for sterling against the U.S. dollar, French franc, and German mark. The sample period
covers January 1921 to May 1925. The low-frequency data consist of annual observations
on exchange rates together with the wholesale price indices from 1791 to 1992 Michael
(1997) also found evidence in support of nonlinearity in PPP.61
3.5 Quantity Theory of Money
There is a wide consensus in economic literature that, in the long run, in�ation is a mone-
tary phenomenon; monetary policy ultimately determines in�ation, by in�uencing directly
or indirectly the rate at which the monetary side of the economy expands (McCallum
58These include Grauwe (1993), Granger and Teräsvirta (1993), Peel and Speight (1996), Taylor(2001), Liew (2003), Liew (2003), Liew (2004a), Liew (2004b) and Liew (2004c).
59These include O'Connell (1998) and Enders and Falk (1998).60For a description of TAR models, cf. Yadav (2006).61For a description of STAR models, cf. Dijk (2002).
28
3 Review of the Status of Research
(1990) and Borio and Filardo (2007)).62
According to the quantity theory of money, traded goods valued in money are equivalent
to transferred money. The idea of constant velocity in money and number in transactions
leads to the conception of quantity of money as the only variable of consumption goods
prices and in�ation rate (Friedman (1987)).63 Central banks adopt the apparatus of
monetary regime to in�ation targeting (Masson (2000), Mishkin (2000) and Ball and
Reyes (2008)).64
Hillinger and Süssmuth (2008) test the empirical distribution against a normal distribu-
tion with unity mean and �nd strong evidence for the quantity theory of money. Their
study comprised 164 countries for a period of 44 years from 1960 to 2003. By use of
the augmented Dickey�Fuller (ADF) test Ajuzie (2008) also �nds strong evidence of the
quantity theory of money for the period from January 1993 through June 2007 for the
US.
Mumtaz and Surico (2006) among many others65 investigate the impact of global factors
on in�ation. The panel of the study of Mumtaz and Surico (2006) includes 164 quar-
terly series of prices for 13 countries: United Kingdom, United States, Sweden, Spain,
Netherlands, New Zealand, Japan, Italy, Germany, France, Finland, Canada and Aus-
tralia. The full sample covers the period from 1961 to 2004 and they use the Bayesian
methods described by Kim and Nelson (2000).
They �nd evidence that in�ation appears to have become driven by global factors such
as labor costs and tradable and non-tradable goods and services.66 Borio and Filardo
(2007) argue that the robustness of the �ndings should be assessed further and argue that
62Even �scal policies imply di�erent ongoing in�ation rates only if they result in di�erent money stockgrowth rates (McCallum (1990)).
63The velocity equation can be expressed by a money demand function, cf. Friedman (1956) andGordon and Leeper (2002). Friedman (1987) assumes that money demand behavior is reasonablystable (McCallum (1990)). For empirical veri�cation, cf. Holtemöller (2004), Brand and Cassola
(2004) and Brüggemann and Lütkepohl (2006).64Empirical evidence thus far supports the presence of a unit root in in�ation rates, the results indicate
that most shocks to in�ation rates are temporary and that the targeting of in�ation rates is the resultof profound policies of central banks, cf. MacDonald and Murphy (1989), Moazzami (1991), Kingand Watson (1992), Lai (1997) and Loayza and Soto (2002).
65These include Morimoto (2003),Ciccarelli and Mojon (2005) and Borio and Filardo (2007).66For a contrary study, cf. Tootell (1998).
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3 Review of the Status of Research
studies about the in�uence of global factors need better information, better measures of
the contestability of markets and more micro-data.
Furthermore, the empirical study of Adler and Dumas (1983) shows that in�ation risk
represents a negligible factor in IntCAPT.67
3.6 Empirical Evidence
Most empirical investigations test the segmentation or integration of asset markets intend-
ing to �nd or reject barriers to international investments. In general, neither evidence
for segmentation nor integration can be found at signi�cant levels. Empirical studies as-
suming that markets are integrated against the hypothesis that markets are segmented
by barriers do not appear to present a very powerful framework; the (abnormal) returns
implied by barriers to international investment are not supported in empirical studies
(Stulz (1995)). The lack of power of asset pricing tests with unspeci�ed null hypotheses
has lead to a new approach of empirical tests. Models that explicitly quantify the impact
of barriers to international investment have been developed and tested (Stulz (1995)).
In their empirical test of the conditional version of the IntCAPM Dumas and Solnik
(1995) use the Generalized Methods of Moments (GMM) of Hansen (1982). Four coun-
tries are taken into account: Germany, the United Kingdom, Japan, and the United
States and the data covers the period from March 1970 to December 1991. Santis (1998)
tests the conditional version of an IntCAPM using a multivariate Generalized Autore-
gressive Conditional Heteroskedasticity (GARCH) process. They include ten countries
(Austria, Belgium, France, Germany, Italy, Japan, The Netherlands, Spain, UK and US)
and the sampling period covers 288 observations from January 1974 through December
1997. Ramchand and Susmel (1998) use a speci�cation of the Switching Autoregressive
Conditional Heteroskedasticity (SWARCH) model. The data cover the period from Jan-
uary 1980 through April 1996 and includes the countries Australia, Hong Kong, Japan,
France, Germany, Sweden, Switzerland, UK, US and Canada. Wu (2008) employs the
67Other studies reveal that the empirical evidence on the pricing of in�ation risk is inconclusive. Moerman
and van Dijk (2010) �nds empirical support for the pricing of in�ation risk whereas Javid and Ahmad(2009) produces mixed results.
30
3 Review of the Status of Research
Fama and MacBeth (1973) two-stage regression methodology. The data cover the pe-
riod from January 1977 to December 2006. Dumas and Solnik (1995), Santis (1998),
Ramchand and Susmel (1998) and Wu (2008) �nd strong support for the IntCAPM.
Cappiello and Fearnley (2000) test the conditional version of the IntCAPM by application
of a multivariate GARCH process. In Cappiello and Fearnley (2000), the observations
cover the period from February 1986 to December 1998; the USA, Europe and Japan are
included in the sample. Fearnley (2002) tests the conditional version of the IntCAPM
by application of a multivariate GARCH process. The sample data used for estimation
cover the period January 1993 to October 2001. US, Japanese and European stocks and
government bonds, two Euro currency deposits (Yen and a European currency basket),
and the world market portfolio of stocks and government bonds are estimated. Dahlquist
and Sallstrom (2002) use the GMM of Hansen (1982). The sample period is July 1973
to June 1999 and the set of assets consists of total market returns in 20 developed equity
markets.68
The tests of Cappiello (1998), Cappiello and Fearnley (2000), Fearnley (2002) and
Dahlquist and Sallstrom (2002) led to a rejection of the IntCAPM.
Cooper and Kaplanis (1994) use the GMM, the study comprising the time period from
January 1978 to December 1987 includes France, Italy, Japan, Spain, Sweden, UK, USA
and Germany. The study indicates that deviations from Relative PPP alone are insuf-
�cient to account for the degree of home bias observed. Fearnley (2002) comes to the
same conclusion by rejecting the IntCAPM of Adler and Dumas (1983) and suggests that
additional, unidenti�ed pricing factors contribute to return expectations.
3.7 Conclusions
The failure of the state of art is based on the discrepancy of theoretical and empirical
results and models of taxation in IntCAPT.
68The countries included are Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany,Hong Kong, Ireland, Italy, Japan, Netherlands, Norway, Singapore, Spain, Sweden, Switzerland, theU.K., and the U.S.
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3 Review of the Status of Research
The introduction of taxation was intended to represent a general kind of barrier to inter-
national investment, but taxation adopts a much more complex form than considered in
the theoretical models. Only Lally (1996) regards the international taxation spectrum
by di�erential taxation of capital gains, dividends and interest. Nevertheless, he assumes
exchange gains to be equivalently taxed to the underlying nature of income. In his model,
capital gains and exchange gains on capital, dividends and exchange gains on dividends
as well as interest and exchange gains on interest are regarded as only three income types.
Exchange gains are taxed by characterization of the recognition, character, nature, source
and hedging features, so, capital gains, exchange gains on capital, dividends, exchange
gains on dividends, interest and exchange gains on interest must be regarded as six di�er-
ent income types which are di�erently taxed.69 The characteristic features of the exchange
gains tax system � namely, the rise of exchange gains taxes depending on the character-
ization of the its recognition, character, nature, source and hedging features are ignored
in his model.
Furthermore, Lally (1996) admits that the return and cash-yield return on world portfolio
and weighted average of risk-free rate of return may vary substantially due to in�ation
rates. Hence, the consensus in economic literature that in the long run, monetary policy
ultimately determines in�ation, by in�uencing directly or indirectly the rate at which the
monetary side of the economy expands is neglected.
As Solnik (1974a) he assumes exchange rates to be stochastic but independent from
assets returns and ignores the extensions by Serçu (1980) and Adler and Dumas (1983)
who do not assume that exchange rates are independent from assets returns. Furthermore,
he ignores the exchange rate e�ects on the β-factor. In contrast to the derived models of
Solnik (1974a), Serçu (1980) and Adler and Dumas (1983) exchange rates reveal not
to be stochastiCf.70 Empirical studies show evidence of a non-linear behavior of exchange
rates, suggesting that they are not random.71
In contrast to Lally (1998), Wiese (2006b), Mai (2006a) and Mai (2008) he does
not assume stochastic dividends. However, a realistic version of Tax-IntCAPM should
69S. chapter 4.4.70S. chapter 3.4.71S. chapter 3.2.
32
3 Review of the Status of Research
incorporate the fact that dividends are stochastic, because the future is insecure.72
These assumptions reveal to be unrealistic.73 The theoretical signs indicate that further
components have to be integrated and the divergent empirical results may be attributed
to a severe theoretical lack of the IntCAPM - the ignorance of the framework of exchange
gains taxation and non-linear behavior of exchange rates in the IntCAPM -. The theo-
retical and empirical studies give the ball back to theorists, we need to know more about
the framework of taxation of exchange gains and the integration of non-linear behavior of
exchange rates in IntCAPT.
72S. chapter 3.1.2.73S. chapter 3.1.2, 3.4 and 3.5.
33
4 International Taxation
�The hardest thing in the world to understand is the income tax.�
Albert Einstein, US (German-born) physicist (1879 � 1955)
The aim of integration taxation in IntCAPM is the derivation of a pricing relationship.
The assumption that taxes are identical in their treatment of the entire income types in
IntCAPT - dividends, capital and exchange gains and interest - reveals itself to be invalid
under the prevailing tax systems. In most countries these parts of income are di�erently
taxed.
How to quantify the features of international taxation
The tax systems generally di�er from country to country and are subject to constant
changes. The valuation approaches typically used do not account for such di�erences or
changes - a general approach for deriving the equilibrium pricing relationship is needed
which can be adjusted to any situation depending on the characteristics of taxation in the
given country. In this chapter, we abstract from the mechanics of particular international
tax systems and present a model for a reasonably general integrated international tax
framework. The characteristic feature of international taxation is foreign investors' tax
treatment. The right of taxation of each income type in IntCAPT - dividends, capital
and exchange gains and interest - can be separately assigned to the source state or to
the state of residence.74 The right of taxation of dividends, capital gains and interest can
be assigned to both states. If the right of taxation is assigned to both states, two �scal
74For the sake of readability we use solely the term capital and exchange gains.
34
4 International Taxation
jurisdictions clash and the foreign investor can be subject to double taxation. In order to
avoid this (negative) characteristic, international tax law incorporates methods to avoid
double taxation.
By deriving a model integrating the features of international taxation, we have the possibil-
ity of quantifying and analyzing the features of international taxation. After a discussion
of the assumptions we introduce the national tax system that the domestic and the for-
eign investor is subject to. We present the corporate and dividend, the capital gains and
interest tax system. After introducing the national tax systems we focus on the taxation
of exchange gains. We conclude this chapter by presenting the methods to avoid and
reduce double taxation.
4.1 Assumptions
Our world is too complex to be understood, so models are constructed based on hypotheses
assuming away the complexities which are thought to have a minor impact on our world.
Hence, the assumptions of international taxation are presented which correspond to the
necessities of taxation in IntCAPT.
Assumption 1 World
The world consists of a domestic D and foreign F state, the states are delineated along
di�erent tax systems.75
Assumption 2 Investors
Each state is populated by one representative investor d and f which is an arti�cial agent
75Cf. Black (1974), Stulz (1981) and Byun and Chen (1997). The model is applied in a binationalframework, since international taxation is based on national and binational tax agreements. Conse-quently, the two country IntCAPM forms the basis in gainsing new scienti�c insight into taxation inIntCAPT. Factors of heterogeneity such as transaction costs, legal restrictions of foreign access suchas tari�s, the control of capital and exchange rates, limitation of foreigners' ownership, political risk,control of import and export of capital, reserve requirements on bank deposits and other assets andlimited information are excluded. For a discussion of these barriers, cf. Ross (1978), Garman (1981),Sellin (1990), Dermody and Prisman (1993), Jouini and Kallal (1995), Ardalan (1999), Zhang(2002) and Bodnar (2003). The tax rate can be regarded as a surrogate for the certainty-equivalentcost of the above barriers (Stulz (1984)).
35
4 International Taxation
who behaves the same as the aggregate ones (local representative).76
Assumption 3 Time
There is a period of one year, in the beginning of the period t=0 the decisions are made
and the outcome occurs at the end of period t =1, the insecure future.77
Assumption 4 Investment opportunities
As far as the investment opportunities are concerned we consider only equity �nanced
corporations.78 The investor has the opportunity to invest in risky and risk-free assets.
We assume that the return of the risky securities is composed of insecure dividends Di and
capital gains CGi - the di�erence between the value of the risky asset at the end of period Vi
and at the beginning of period Vi,0 - and the principal Vi,0.79 The value of the risky security
at the end of period and the dividends are assumed to be normally distributed.80 The return
of the risk-free security consists of nominal interest I - the di�erence between the value
of the risk-free asset at the end of period V0 and at the beginning of period V0,0 - and the
principal V0,0.81 In contrast to the domestic investor the foreign investor realizes exchange
gains. Exchange gains on the risky asset can be di�erentiated by the exchange gains on
the principal, on capital gains and on dividends (S − S0)Vi,0 + (S − S0)(Vi − Vi,0
)+
76Cf. Lengwiler (2006). Investors' behavior is di�erent within each state. However, in order to gains newscienti�c insight in the form of international delineation the concept of local representative revealsitself to be su�cient.
77For a discussion of a multiperiod model, s. chapter 8. No assumptions are made as far as the numberand the structure of states are concerned; the states can be �nite or in�nite (Jonas (2004)).
78This assumption is generally used in Tax-CAPT, cf. Mai (2006a), Wiese (2006b) and Mai (2008).The required return on debt and on partnerships follows a di�erent version of the CAPM because thetax treatment of debt and equity as well as of corporations and partnerships di�ers. For a discussionof CAPM for risky debt, cf. Sick (1990), Benninga and Sarig (2003) and Cooper and Davydenko
(2007). For a discussion of CAPM for partnerships, cf. Selim (2006).79This assumption is restrictive but customary, cf. Mai (2006a), Wiese (2006b) and Mai (2008). In
a one period model the entire return should consist of dividends, cf. Blaufus (2002). If the end ofperiod value decreased by the entire dividends, arbitrage opportunities would arise, cf. Elton and
Gruber (1970) and Elton (2005).80Cf. Lally (1998), Mai (2006a),Wiese (2006b) andMai (2008). The normal distribution seems not to
be descriptive for capital asset pricing because of limited liability and because empirical distributionsof returns have tails that are leptokurtic (Balvers (2001)). As far as the critique of the normaldistribution assumption is concerned, see chapter 8.
81This is an assumption generally used in CAPT, cf. Sharpe (1963), Lintner (1965b), Mossin (1966)and Treynor (1999). In reality, assets cannot be risk-free. Black (1972a) developed the CAPMwithout the assumption of risk-free assets.
36
4 International Taxation
(S − S0) Di which can be summarized as follows (S − S0) Vi+(S − S0) Di and in the case
of the risk-free asset by (S − S0)V0.82
The market approach requires the typi�cation of tax. We ignore specializations and in-
stead regard stylized income tax systems of a representative domestic and foreign investor
which incorporate the pricing relevant aspects of tax without considering every detail of
tax law (Ollmann and Richter (1999) and Mai (2008)). In determining the tax burden,
three elements are critical: the de�nition of income, the applicable tax rate and the rules
for netting the income with other sources (Desai and Gentry (2003)). Commission Of
The European Communities (1992), Kluge (2000), Rohatgi (2002), Lorié (2006) and Ja-
cobs (2007) among many others83 provide a good overview of the features of international
taxation.
Assumption 5 Principle of taxation
It is assumed that the domestic and foreign state raises income taxes. According to the
sovereignty principle and general international rules, states can raise taxes in the case of
a concrete point of contact on their territory.84 The point of contact of persons is the
principle of residence and the principle of nationality.85 In case of residence or nation-
ality, the person is unlimited tax liable with his entire income (world income principle,
universality principle), otherwise, the person is limited tax liable with his national income
(territoriality principle).86 Possibilities for tax evasion are excluded.87
Assumption 6 Tax base
Income is regarded as a net size, the expenses to yield gross income - the investments -
82Cf. Musgrave (1969).83These include Cnossen (1996), PricewaterhouseCoopers (1999), Buijink (1999), Spengel (1999),
European Commission (2001), Spengel (2001) and Kuipers (2004).84Cf. Kluge (2000), Rohatgi (2002) and Jacobs (2007).85Residence means the existence of a person's domicile and that the person is living for a certain period
in the country. The principle of nationality takes e�ect in the case of citizenship according to nationallaw (Rohatgi (2002) and Jacobs (2007)).
86Cf. Kluge (2000), Rohatgi (2002) and Jacobs (2007).87Foreign jurisdictions - tax havens - o�er potential for tax evasion as regards tax rates, legal concepts,
standards of administration, reporting and enforcement and governmental attitudes towards the lib-erty and privacy of taxpayers and the con�dentiality of �nancial and business transactions (Wang
(1995) and Niemann (2007)). Tax evasion shall solely have a tax rate e�ect and can be implementedinto the Tax-IntCAPM by a supplementary factor.
37
4 International Taxation
are deducted from gross income (objective net presentation principle).88 We ignore non-
deductible expenses and tax-free income (Bareis (2000)). It will be assumed that the tax
is raised at the end of period - the �ow of income to the investors at the end of period
results in immediate taxation and that portfolio switching at the beginning of the period
is not taxable.89 Expenses resulting from tax compliance are regarded as irrelevant.90
Redistribution e�ects of taxes to investors are ignored.91 Furthermore, we assume that
gains and losses are symmetrically taxed (full loss o�set).92 Equivalently, symmetry in
the tax treatment of long and short positions is assumed (Lally (1996)).93
Assumption 7 Tax scale
The tax scale is assumed to be linear and deterministic. Since taxes are proportional, the
average and the marginal tax-rate are identical.94 Stochastic returns lead to a stochastic
tax basis, which implies that average tax rates and marginal tax rates are stochastic in the
case of progressive tax; we assume linear tax rates to exclude stochastic tax rates (Mai
(2008)).95
88Cf. Kluge (2000), Rohatgi (2002) and Jacobs (2007).89This is an assumption generally used in Tax-CAPT, cf. e.g. Brennan (1970), Wiese (2006a) and
Mai (2008). We assumed a time horizon of one year, but in reality investors can determine bythemselves the point of time of acquisition and selling of assets. In a multiperiod option pricingmodel Constantinides (1983) develops a model to determine the optimal point of time of realization.
90Cf. Evans (2003) for an overview.91This assumption is restrictive but customary. For the integration of redistribution e�ects of taxes to
investors, cf. Eikseth and Lindset (2009) and Kruschwitz and Lö�er (2009).92Cf. Long (1977), Mai (2006b) and Dierkes (2007). This assumption implies that gains and losses
can be netted for tax purposes (Dierkes (2007)). This assumption also implies that the di�erentialtax treatment of debt and credit interest is neglected. If the tax on interest income di�ers in respectof debt and credit interest, then the tax payment depends on whether the investor borrows or lends.For the integration of di�erential taxation of debt and credit interest, cf. Elton (2007) and Hüper
(2009).93In the CAPM of Wood (1997) the symmetry assumption of short and long positions is abandoned.
He assumes that if an investor shorts an asset o�ering imputation credits, then the long holder iscompensated. Wood (1997) shows that the expected return on an asset o�ering imputation creditsdepends upon whether that economy's domestic investor has net long, net short or net zero positionsin that asset. However, this requires that the portfolio holdings of all investors must be observed(Lally (1996)).
94We assumed the existence of one representative investor in each state. Tax rates can vary accordingto the personal situation, so the after tax returns could be di�erent for every national investor. Theconcept of a representative investor in each state reveals itself to be fully su�cient to gains newscienti�c insight into taxation in IntCAPT, so varying tax rates for national investors are neglected.For a discussion of the Tax-CAPM with heterogeneous national investors, cf. chapter 3.1.
95For a discussion of progressive and stochastic tax rates, s. chapter 8.
38
4 International Taxation
Assumption 8 Tax e�ects
The return is una�ected by tax.96
The principles of taxation lay the foundation for an integrated tax framework. It is
assumed that the tax rates are di�erent in respect of the income classes, so all relevant
income types in the IntCAPM - dividends, capital and exchange gains and interest - are
di�erently taxed.97
In the following the dividends, capital gains, interest, exchange gains tax system and the
methods of double taxation reduction are presented.
4.2 Corporate and Dividend Tax System
Corporations are taxed according to the deferral principle, corporations are taxed on
the corporate level and investors on the personal level (Rohatgi (2002), Jacobs (2007),
Sche�er (2007) and Schreiber (2008)).98 In an integrated tax system, the corporate
and personal tax systems interact. We present the corporate tax system and derive the
dividend tax system. The corporate tax system consists of the classical, shareholder
relief, (full) imputation and dividend exemption system.99 The dividend tax system can
be categorized into a full tax, tax-reducing and tax-free system. Measures to reduce and
to avoid double taxation can be considered on the corporate and on the personal level.
The fundamental feature common to all double taxation reducing or avoiding systems
is the provision of a valuable tax bene�t to shareholders on dividend income (Handley
and Maheswaran (2005)). Figure 4.1 on page 40 illustrates the corporate and dividend
tax systems.100 Pro�ts (gross dividends) are the tax base for the corporate tax, whereas
96Cf. Mai (2008). This is an assumption generally accepted in Tax-CAPT. For a discussion, cf. Jonesand Lamont (2001) and Wiese (2004).
97Other types of taxes are neglected. This assumption is restrictive but customary in CAPT, cf. Brennan(1970), König (1990), Lally and van Zijl (2003) and Mai (2008). As far as the integration of thewealth tax in IntCAPT is concerned, cf. Black (1974), Stulz (1981) and Byun and Chen (1997) inchapter 3.3.
98For a discussion of taxing capital income on the corporate or personal level, cf. OECD (2007).99For an overview of corporate tax systems, cf. Giovannini (1992), Smith (1993), Bareis (2000), Hoek
(2003) and Jacobs (2007).100Cf. Jacobs (2007) for a comparable illustration.
39
4 International Taxation
Corporate tax systems
Classical
systemShareholder relief system
Imputation
systemDividend
exemption
method
Full imputation
system
tD = tC
Full imputation
system
tD > tC
Full dividend tax
systemDividend tax-reducing system Dividend tax-free system
Dividend tax systems
Tax rate
Income
Income
Figure 4.1: Corporate and Dividend Tax Systems
dividends are de�ned as the pro�t after corporate tax.
Di ≡ Pi (1− tC) (4.1)
with Di nominal dividends
Pi nominal pro�t
tC corporate tax rate with 0 < tC < 1
In the case of the classical system corporate tax is levied on pro�ts and dividend tax is
levied on dividends.101
101The term classical system was introduced by Tempel (1970).
40
4 International Taxation
PCSi = Pi − (tC + tD − tCtD) Pi
DCSi = Di − tDDi
(4.2)
with PCSi nominal pro�t after tax in the classical system
tD dividend tax rate with 0< tD < 1
DCSi nominal dividends after tax in the classical system
The classical system can be categorized as a full dividend tax system since neither tax
reducing nor avoiding measures are installed.
In the case of the shareholder relief system (classical system with tax scale) a reduced
dividend tax rate is applied on dividends.
P SRi = Pi − tCPi − (1− φ) tD
(Pi − tCPi
)DSRi = Di − (1− φ) tDDi
(4.3)
with P SRi nominal pro�t after tax in the shareholder relief system
DSRi nominal dividends after tax in the shareholder relief system
φ relief factor of dividend tax with 0< φ < 1
A further root of the shareholder relief system is the possibility of partial exemption from
dividend tax.
P SRi = Pi − tCPi − tD
(Pi − tCPi − φ
(Pi − tCPi
))DSRi = Di − tD
(Di − φDi
) (4.4)
The equations (4.3) and (4.4) are equal. The shareholder relief system can be classi�ed
as a dividend tax-reducing system.
41
4 International Taxation
The (full) imputation system is designed to reduce the total tax burden on corporate
income by compensating shareholders for tax paid by corporations. The compensation
takes the form of a tax imputation attached to dividend payments (Rohatgi (2002)). In
the case of the imputation method the tax base of the dividend tax is increased by part
of the corporate tax imposed on dividends and this part encumbering on dividends is
credited. Depending on the full imputation system (conduit system, dividend imputation
system or credit system), the tax base of the dividend tax is increased by the total corpo-
rate tax imposed on the dividends (grossing-up) and the corporate tax encumbering on
the dividends is credited.102
P ISi =Pi − tCPi − tD
(Pi (1− tC) + ψ
tC1− tC
Pi (1− tC))+ ψ
tC1− tC
Pi (1− tC)
=Pi − tCPi − tDPi + Pi (tCtD − ψtCtD + ψtC)
DISi =Di − tD
(Di + ψ
tC1− tC
Di
)+ ψ
tC1− tC
Di = Di − tDDi + ψ (1− tD)tC
1− tCDi
(4.5)
with P ISi nominal pro�t after tax in the imputation system
DISi nominal dividends after tax in the imputation system
ψ imputation factor of corporate tax with 0 < ψ ≤ 1
In the case of the imputation system, the imputation factor of corporate tax is in the
interval (0;1).
The classi�cation of the full imputation system ψ = 1 depends on the relationship of
corporate and dividend tax rate. If the dividend tax rate is higher than the corporate tax
rate the full imputation system reveals itself to be a dividend tax-reducing system. In the
case of equality of the tax rates the full imputation system is a dividend tax-free system.
If the dividend tax rate is lower than the corporate tax rate, the voluntary announcement
of dividend income is to be expected, since the shareholder would stand to gains to the
amount of the di�erence between corporate and dividend tax (alignment method) (Head
102The imputation system is subject to various restrictions. Smith (1993) and Rohatgi (2002) providea comprehensive overview of full imputation systems to be found internationally. A comparablestructure has the shareholder tax (Engels and Stützel (1968)).
42
4 International Taxation
(1997)). The measures of reducing the total tax burden are executed on the level of the
dividend tax. In that way only the dividend tax is levied on the pro�t. The corporate tax
is an advanced payment by the company on the shareholders' dividend tax and it takes a
withholding role (Mintz (1995) and Lally (2000)). The full imputation system transfers
the comprehensive income conception, that taxes are to be imposed on individuals. In
comparison with the imputation method the imputation factor of corporate tax of the full
imputation method is one, since the corporate tax imposed on dividends is fully credited.
The main rationale for a dividend tax imputation lies in the impossibility to tax unrealized
capital gains. Corporations and shareholders have the choice between the distribution of
dividends and the sale of shares. By eliminating the taxation of unrealized capital gains or
a corporate level tax, tax�payments could be deferred inde�nitely if the corporation were
to retain all earnings. A corporate tax eliminates the possibility of deferring the tax; the
integration is achieved by rebating the corporate tax to shareholders when they pay tax on
corporate distributions (McDonald (2001)). Another explanation for levying a corporate
tax is that it captures part of the bene�ts of public expenditures on goods and services, of
public expenditures on education and training and of the legal provisions that are o�ered
to the corporation (Mintz (1995) and OECD (2007)).103 The corporate tax also acts
as a withholding tax on equity income earned by non-resident shareholders, which would
otherwise escape taxation in the source country in the absence of a withholding tax on
dividends distributed to foreign shareholders (Mintz (1995) and OECD (2007)).104 If the
right of taxation is assigned to the source state, the foreign investor may not be able to
(fully) utilize imputation credits. The (full) imputation system leaves the foreign investor
liable to full taxation on dividends and hence the classical system (Treisch (2004) and
Jacobs (2007)).105 Assuming that the foreign investor cannot capture the value associated
with the imputation credit, the foreign investor is disadvantaged in relation to the domestic
investor.
In the case of the dividend exemption method, dividends are exempted from the tax base
of the dividend tax. The principal advantage of the dividend exclusion prototype is its
simplicity and relative ease of implementation.
103For a discussion, s. footnote 91.104For further reasons for levying the corporate tax, cf. OECD (2007).105Imputation is internationally not installed on account of evading imputation systems, administrative
and �scal reasons (Jacobs (2007)).
43
4 International Taxation
PDEi = Pi − tCPi − tD (0) (4.6)
DDEi = Di − tD (0) = Di (4.7)
with PDEi nominal pro�t after tax in the dividend exemption system
DDEi nominal dividends after tax in the dividend exemption system
The dividend exemption method can be classi�ed as a dividend tax-free system.
The di�erent corporate and dividend tax systems are modeled by assigning the relief fac-
tor of dividend tax and the imputation factor of corporate tax numbers from the interval
from [0; 1] (Bareis (2000) and Sureth and Niemann (2005)). Table 4.1 on page 44 illus-
trates the incorporation of the factors.
PDTSi = Pi (1− tC)− (1− φ) tDPi (1− tC) + ψ (1− tD) tCPi (4.8)
DDTSi = Di − (1− φ) tDDi + ψ (1− tD) tC
1−tCDi (4.9)
with PDTSi nominal pro�t after tax in the corporate tax systems
DDTSi nominal dividends after tax in the dividend tax systems
Corporate and dividend tax systems Relief factor of
dividends tax
φ
Imputation fac-
tor of corporate
tax ψ
Classical system 0 0
Shareholder relief method (0; 1) 0
Imputation system 0 (0; 1)
Full imputation system 0 1
Dividend exemption system 1 0
Table 4.1: Relief and Imputation Factors Depending on the Corporate and Dividend TaxSystems
44
4 International Taxation
The classical system is modeled by assigning the factors the number zero. In the case of
the shareholder relief method the imputation factor of the corporate tax is zero and the
relief factor of the dividend tax is assigned a number from the interval (0; 1). In the event
of the imputation system it is reversed. The full imputation system is characterized by
an imputation factor of the corporate tax of one and an imputation factor of dividend tax
of zero, whereas in the case of the dividend exemption method it is reversed.
Example 2 Corporate and Dividend Taxation
The dividend tax rate amounts to 35%, the corporate tax rate is 15%.
0 0.5 1
51015202530354045
-
6
Tax rate in (%)
Classical
system
Dividendexemptionsystem
Full imputationsystem
Imputationsystem
Shareholderrelief system
Imputation andrelief factor
ψ
φ
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................................
Figure 4.2: Tax Rate Depending on the Relief and Imputation Factors in the CorporateTax System
The shareholder relief system can be classi�ed between the classical system and the div-
idend exemption system. With the decrease of the relief factor the shareholder relief
system approaches the classical system, while with the increase of the relief factor the
shareholder relief system approaches the dividend exemption method.106 With the in-
crease of the imputation factor of corporate tax the imputation system approaches the
full imputation system, whereas a decrease of the imputation factor means the approach
to the classical system.
106Motivated by administrative complexities the trend of a shift from imputation tax based systems todividend exclusion based systems can be observed (Hoek (2003)).
45
4 International Taxation
0 0.5 1
510152025303540
-
6
Tax rate in (%)
Classical
system
Dividendexemptionsystem
Full imputationsystem
Imputationsystem
Shareholderrelief system
Imputation andrelief factor
ψ
φ
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................................
Figure 4.3: Tax Rate Depending on the Relief and Imputation Factors in the DividendTax System
The di�erent dividend tax systems are due to a tax rate e�ect, hence, the term (1− φ) tD−ψ (1− tD) tC
1−tCcan be replaced by tD/FD , so dividends after tax are expressed as follows:
DDTSi = Di − tD/FD Di. (4.10)
with DDTSi nominal dividends after tax in the dividend tax system
tD/FD integrated tax rate on dividends with 0 ≤ t
D/FD < 1
4.3 Capital Gains and Interest Tax System
After describing the dividend tax systems, the capital gains and interest tax system are
presented.107
There are a number of considerations in the design of capital gains tax rules. The design
dimensions can be categorized according to the following criteria (OECD (2006)).
107For an overview, cf. Giovannini (1992).
46
4 International Taxation
1. Taxation on a realization basis: Most countries apply taxation on a realization basis
rather than accrual basis.108
2. Setting of the statutory tax rate: Individuals can face lower tax rates on capital gains
income than on personal income or capital gains can be exempted from taxation
(Rohatgi (2002), Jacobs (2007) and Keen (2008)).
3. Treatment of losses: Ring-fencing rules restricting capital loss claims can be imple-
mented to protect the tax base. Certain assets can be restricted from capital loss
deductions.109
4. Rollover provision: Taxpayers are allowed to defer payment of capital gains tax that
might be triggered otherwise.110
5. Treatment of gains on a taxpayer's personal residence: Some countries provide a
full exemption for capital gains on homes.
6. Treatment of the in�ation component of (nominal) capital gains: Only real capital
gains can be included in the tax base. Nominal capital gains is adjusted to net out
the in�ation component.111
7. Treatment of non-residents. Non-residents can face lower tax rates on capital gains
income or they can be exempted from taxation.112
8. Transitional considerations: Capital gains tax can be introduced on an entirely
prospective basis. The tax is applied only to gains on assets acquired after the
108The realization-based system recognizes that accrual taxation poses signi�cant di�culties by requiringperiodic valuation of assets. Realization-based taxation can create liquidity problems for taxpayerswho do not have, su�cient cash to cover tax on accrued but unrealized gains, requiring them to borrowor sell assets to pay their tax liability, cf. OECD (2006). In our model a realization-based approachis adopted, s. assumption 6.
109In our model gains and losses are symmetrically taxed, s. assumption 6.110If rollover relief is provided valuation and cash-�ow problems can occur. For a description of the
di�erent types of rollovers, cf. OECD (2006). In our model a realization-based approach is adopted,s. assumption 6.
111Most countries do not attempt to adjust nominal capital gains to net out the in�ation component, sincein�ationary gains are not prevalent and because of the complexity of calculation, cf. OECD (2006).
112Certain countries have special realization rules that apply to taxable accrued gains where a residentbecomes a non-resident, while others apply tax at the time of realization, cf. OECD (2006).
47
4 International Taxation
commencement of the tax.113
Capital gains after tax is modeled by the following formula:
CGi,t =(Vi − Vi,0
)− tD/FCG
(Vi − Vi,0
)= CGi − tD/FCG CGi. (4.11)
with CGi,t nominal capital gains after tax
tD/FCG capital gains tax rate with 0 ≤ t
D/FCG < 1
Interest is subject to interest income taxation. In schedular systems individuals face �at
tax rates for interest income (Hess-Ingrassia (1995), Gordon (2003) and Benshalom
(2008)).114 On account of the linearity of tax rates the comprehensive income as well
as the schedular system are comprised by the formula. Interest after tax is modeled as
follows:
It =(V0 − V0,0
)− tD/FI
(V0 − V0,0
)= I − tD/FI I. (4.12)
with It nominal interest after tax
I nominal interest
tD/FI interest tax rate with 0 ≤ t
D/FI < 1
4.4 Exchange Gains Tax System
Foreign exchange gains and loss arise on the foreign investor's cross-border transactions.
Foreign exchange gains taxation can be e�ective in calming exchange rate volatility and
avoiding currency crises (Bird and Rajan (1999)). However, the taxation of exchange
gains and losses varies widely; the tax treatment of exchange gains and losses are, in
most countries, neither simple nor rational (Chown (1990)). These rules di�er on how
113On account of the assumption of a one period model, s. assumption 3, we regard this aspect asirrelevant.
114For an overview, cf. Giovannini (1992).
48
4 International Taxation
exchange gains are calculated, the timing when they are taxable, and their characterization
for tax purposes (Rohatgi (2002)). The key exchange gains conversion issues for the
foreign investor can be characterized as follows (L'Association Fiscale Internationale
(1986), International Bureau Of Fiscal Documentation (1988), OECD Committee On
Fiscal A�airs (1988) and Rohatgi (2002) among many others115):
1. Recognition: Exchange gains are recognized at the realization of exchange gains or
losses.116 The exchange rate is based on the transaction date.117
2. Character: Exchange gains can be taxed as ordinary income, as capital gains, as a
separate type of income or as outside the scope of taxation. If they are taxed as
ordinary income, they can be taxed at di�erent rates.118
3. Nature: Exchange gains can be a�ected by the nature of the underlying transaction.
4. Source: Exchange gains can be regarded as of either foreign or domestic source. If
exchange gains are regarded as income from the source state it can be non-taxable.119
5. Hedging: Hedged transactions can have both exchange gains and losses at di�erent
times with di�ering tax treatment.120
Exchange gains are only taxable in the state of residence. The overview 4.4 on page 50
illustrates the exchange gains tax system in various countries.121 In many countries such
as Russia, Austria and Canada exchange gains are taxed according to the underlying
115These include Gibbons (1985), Burge and Farber (1986),McGarry (1988), Chown (1990), Giovannini(1992), Suermann (1999) and Jacobs (2007).
116In a one-period model, the aspect that year-end losses can be recognized in the current year while gainscan be carried forward is regarded as irrelevant.
117We neglect the fact that certain states apply for taxation exchange rates which deviate from the marketrates.
118The distinction might depend on the nature and source of the underlying transaction. The tax treat-ment may di�er between trading and non-trading items, capital and revenue items, and monetary andnon-monetary items, s. International Bureau Of Fiscal Documentation (1988) and Rohatgi (2002).
119Exchange gains and losses and assets and liabilities can be asymmetrically taxed. Exchange gains areregarded as taxable whereas exchange loss is regarded as out of the scope of taxation. The asymmetrictax treatment of gains and loss is neglected on account of assumption 6.
120The exchange gains conversion issue of hedging with the asymmetric treatment of exchange gains andlosses is neglected, on account of the assumption of equal treatment of gains and losses, s. assump-tion 6.
121S. I would like to thank ECOVIS AG, and especially Prof. Dr. Peter Lüdemann who provided mewith his survey of exchange gains taxation. S. also www.kpmg.com/Global/en/IssuesAndInsights/
ArticlesPublications/TIES/Pages/Taxation-of-international-executives-2010.aspx.
49
4 International Taxation
Separate income type
Exchange gains tax system
Taxation according to the
underlying transaction
The Netherlands
Turkey
Spain
Romania
Argentina
USA
Australia
Italy
Russia
Austria
Canada
Ireland
South Africa
United Kingdom
Japan
tFD,e = t
FD
tFCG,P,e = t
FCG
tFI,P,e = t
FI
tFD,e ≠ t
FD ≠ 0
tFCG,P,e ≠ t
FCG ≠ 0
tFI,P,e ≠ t
FI ≠ 0
Anguilla
Bahamas
Cayman
Island
Egypt
Gibraltar
tFD,e = t
FD = 0
tFCG,P,e = t
FCG= 0
tFI,P,e = t
FI= 0
tFD,e ≠ t
FD = 0
tFCG,P,e ≠ t
FCG= 0
tFI,P,e ≠ t
FI= 0
Exchange gains
are taxable.
Exchange gains
are not taxable.
Belgium
China
Georgia
Greece
Hungary
Exchange gains
are taxable.
Exchange gains
are not taxable.
Kuwait
Qatar
United
Arab
Emirates
Finland
Angola
Bulgaria
Cambodia
Canada
Germany
Sweden
Figure 4.4: Exchange Gains Tax System
transaction, so it is equivalently taxed to dividend, capital gains and interest. In countries
where taxes seem not to be the primary source of revenue, exchange gains as well as
the underlying transaction are not subject to taxation. Other states as e.g. the USA,
Australia, Italy, the Netherlands and Belgium regard exchange gains as a separate income
type. In contrast to the USA, Australia and Italy, in the Netherlands, Belgium and other
countries exchange gains are not taxable. The exchange gains on capital gains and on the
principal after tax can be expressed by the following formula:
ECG,P,t = (S − S0) Vi − tFCG,P,e (S − S0) Vi. (4.13)
with ECG,P,t exchange gains after tax on capital gains and on principal
with tFCG,P,e exchange gains tax rate on capital gains and on principal
with 0 ≤ tFCG,P,e < 1
50
4 International Taxation
Exchange gains on dividends after tax are expressed by the following formula:
ED,t = (S − S0) Di − tFD,e (S − S0) Di. (4.14)
with ED,t exchange gains on dividends after tax
tFD,e tax rate on dividend exchange gains with 0 ≤ tFD,e < 1
For the exchange gains after tax on interest and on the principal we can derive the
following expression:
EI,P,t = (S − S0)V0 − tFI,P,e (S − S0)V0. (4.15)
with EI,P,t exchange gains after tax on interest and principal
tFI,P,e exchange gains tax rate on interest and on the principal
with 0 ≤ tFI,P,e < 1
4.5 Methods of Double Taxation Reduction
The foreign investor's subjection to the domestic and foreign state can lead to interna-
tional double taxation, since the world income principle and the territoriality principle
clash together. The domestic state (source state) and the foreign state (residence state)
impose taxes on the income of the foreign investor. Economic double (multiple) taxation
is de�ned as the taxation of an economic size by two (multiple) identical types of taxes
by two �scal sovereignties in the same period (Siegel and Bareis (2004)).122 Double taxa-
tion can be avoided by the conception of incorporating identical criteria of allocation and
assigning the right of taxation either to the domestic state or to the foreign state.123 Al-
ternatively, the domestic state can impose a withholding tax on the income of the foreign
122Economic undertaxation can be de�ned as the lower international in comparison to national taxation(Jacobs (2007)).
123These identical de�nitions of tax principles can be incarnated if the point of contact is solely the taxobject or the tax subject (Henselmann (1996)).
51
4 International Taxation
investor. Bilateral methods are incarnated in order to avoid or reduce double taxation.
Bilateral and national tax laws incorporate di�erent methods in order to avoid or reduce
double taxation (Rohatgi (2002), Schult (2002) and Jacobs (2007)). The methods of
avoiding or reducing double taxation are illustrated in �gure 4.5 on page 52.124 The ex-
Methods of avoiding and
reducing
double taxation
Reduction
of double taxation
Avoidence
of double taxation
CreditExemption Remission Flat taxDeduction
Without progressionWith progression Unlimited credit Limited credit
Per country limitation
Per basket limitation
Combination of both
Fictive credit Indirect credit
Figure 4.5: Methods of Avoiding and Reducing Double Taxation
emption and credit method can lead to the elimination of double taxation, whereas the
deduction, remission and �at tax lead to a reduction of double taxation. The exemption
method can be di�erentiated according to with progression (relative exemption) and with-
out progression.125 The credit method can be di�erentiated by unlimited, �ctive, indirect
124Cf. Henselmann (1996) and Bareis and Siegel (2004) for a comparable illustration.125The di�erentiation by with and without progression is irrelevant in our model on account of the
assumption of linear tax rates, s. assumption 7.
52
4 International Taxation
and limited credit.126 The limited credit method can be di�erentiated by per country
limitation, per basket limitation or a combination of both methods.127
Assumption 9 Per basket limitation
The per basket limitation is incorporated, capital gains, exchange gains on capital gains,
dividends, exchange gains on dividends, interest and exchange gains on interest are re-
garded as separate income types.
In the following we analyze the e�ects of the methods of avoidance and reducing double
taxation.
According to the exemption method the residence state does not take into consideration
foreign investors' income yielded in the source state.128 The exemption method is equiva-
lent to the assignment of the taxation right to the domestic state.
tFtF ,Em =tF (0) = 0
=tF − tF = 0(4.16)
with tFtF ,Em tax rate of the foreign state after application of the exemption method
tF foreign state's tax rate
As far as the exemption method is concerned the location of the investment is the de-
termining factor, so the concept of capital import neutrality (territoriality principle) is
pursued (Sche�er (2009)).
In the case of the credit method, the source tax of the domestic state is credited to the
income tax of the foreign state. A credit is allowed for domestic income taxes paid on
domestic source income where such income is also subject to foreign tax. The base point
of the credit method is the tax amount, the foreign state deduces the source tax from the
income tax.126The indirect credit can be regarded as irrelevant, since we do not consider a�liates in our model, s.
assumption 4.127The per country limitation can be regarded as irrelevant, since only two countries are regarded, s.
assumption 1.128Cf. Gordon and Hines (2002) and Sche�er (2009).
53
4 International Taxation
In the case of unlimited credit, the entire tax of the source (domestic) state is deducted
from the tax of the foreign state.129
tFtF ,Cm = tF − tDSt (4.17)
with tFtF ,Cm tax rate of the foreign state after application of the unlimited credit
method
tDSt domestic state's source tax
The conception of unlimited credit leads constantly to the tax burden of the state of
residence, the foreign state. The concept of capital export neutrality is pursued (Siegel
and Bareis (2004) and Sche�er (2009)). If the tax rate of the source state is higher than
the tax rate of the residence state, the residence state refunds the excess tax payment and
yields lower tax income than in comparison with a national investment.
In the case of limited credit the deduction shall not exceed the part of the income tax, as
computed before the deduction is given.130 The maximum credit may be computed either
by apportioning the total tax on total income according to the ratio between the income
for which the credit is to be given and the total income, or by applying the tax rate for
total income to the income for which credit is to be given. On account of the assumption
of linear tax rates, the computation of the maximum credit leads to the same result.
tFtF ,Cml = tF −Min(tDSt; t
F)
(4.18)
with tFtF ,Cml tax rate of the foreign state after application of the limited credit
method
Min minimum
129Cf. Gordon and Hines (2002) and Sche�er (2009).130Cf. Gordon and Hines (2002) and Sche�er (2009).
54
4 International Taxation
If the income tax of the residence state exceeds the source tax, the principle of capital
export neutrality is pursued and in the opposite case the principle of capital import
neutrality is pursued. The total tax burden of the foreign income measured by the highest
tax demands of the involving states corresponds at least to the level of tax of the residence
state. As a result, source tax reductions of the domestic state are not for the bene�t of
the shareholder. Creditable foreign taxes, exceeding the income tax of the residence state,
are not taken into account (credit overhang).131 The conception of the (unlimited) credit
consists in raising the (lower) tax level of the domestic state to the tax level of the foreign
state.
In order to make e�ective tax allowances, a �ctive credit can be given. In the case of �c-
tive credit, it is not the e�ective tax of the source state, but a �ctive (higher) tax amount
that is credited to tax in the foreign state.132
tFtF ,FCm = tF − tFFCm (4.19)
with tFtF ,FCm tax rate of the foreign state after application of the �ctive credit
method
tFFCm tax rate of �ctive credit method
In the case of the remission method a reduced tax rate on foreign income is applied.133
The remission method leads to a reduction of double taxation.
tFtF ,Rm = tF − ιtF (4.20)
with tFtF ,Rm tax rate of the foreign state after application of the remission method
ι tax remission factor
131A credit overhang is found, if the source tax of the domestic state exceeds the income tax of the foreignstate. The credit overhang is calculated by deducting the source tax from the income tax.
132Cf. Sche�er (2009).133Cf. Sche�er (2009).
55
4 International Taxation
The concept of the �at tax is that the foreign state raises a (lower) �at tax.134 The �at
tax method leads to the reduction of double taxation.
tFtF ,F t =tFFt
=tF − %tF(4.21)
with tFtF ,F t tax rate of the foreign state after application of the �at tax method
tFFt �at tax rate
% �at tax factor
In the case of the deduction method the tax of the source state is reduced from the tax
base of the residence state.135 A deduction is allowed for domestic income taxes paid on
domestic source income where such income is also subject to foreign tax.
tFtF ,Dm = tF − tF tDSt (4.22)
with tFtF ,Dm tax rate of the foreign state after application of the deduction method
The deduction method leads to a reduction of double taxation and it limits the double
tax relief to the foreign tax paid as multiplied by the tax rate.
The di�erent methods of double taxation reduction are due to a tax rate e�ect, so we
de�ne them by the following equation:
tFtF ,Int = tF − tFInt. (4.23)
with tFtF ,Int tax rate of the foreign state after application of the relief methods
tFInt relief tax rate
134Cf. Sche�er (2009).135Cf. Sche�er (2009).
56
4 International Taxation
Exchange gains a�ect only one jurisdiction and hence does not lead to double taxation
(Rohatgi (2002)). Since exchange gains are not taxed in the source state, the exemption,
the credit and the deduction method are not applicable for the taxation of exchange gains.
The di�erent methods of double taxation reduction for exchange gains taxation are due
to a tax rate e�ect and can be summarized by the following equation:
tFtF ,Int,e = tFe − tFInt,e. (4.24)
with tFtF ,Int,e tax rate on exchange gains of the foreign state after application
of the relief methods
tFe tax rate on exchange gains of the foreign state
tFInt,e relief tax rate
In our model, the exemption and the credit method in the case of equivalence of source
and residence tax rate lead to an equivalent tax burden of zero in the foreign state; foreign
investors' income is solely subject to tax in the source state.
57
5 Tax International Capital Asset
Pricing Model
“The government's view of the economy could be summed up in few short phra-
ses: If it moves, tax it. If it keeps moving, regulate it. And if it stops moving,
subsidize it.”
Ronald Reagan, 40th president of US (1911 � 2004)
Most IntCAPMs consider either di�erent consumption and investment opportunity sets
- deviations from PPP - or barriers to international investment - taxes on foreign invest-
ment - when PPP is assumed to hold. Rational investors judge the real return after taxes,
which implies that a more realistic model should incorporate both, deviations from PPP
and barriers in the form of taxes on international investment. The incorporation of inter-
national tax systems has the important bene�t of not only being descriptive on the macro
level but also of better describing the micro behavior of the international investor. Tax
e�ects are regarded as a very important factor in determining the investor's international
asset return and it is evident that deviation from PPP exists in the short run.
How is international asset pricing a�ected by international tax systems?
We analyze the general market equilibrium with stochastic dividend and tax rates that are
investor-speci�c for all income types in IntCAPM, dividend, interest, capital and exchange
gains. By integrating international taxation into the IntCAPM, the e�ect of international
58
5 Tax International Capital Asset Pricing Model
tax systems on individual investor behavior and on equilibrium returns can be studied by
derivation of the Tax-IntCAPM.136
The construction of the Tax-IntCAPM will allow the determination of the relevant mea-
sure of risk for any asset and the relationship between expected return and risk for assets
under integration of di�erent taxes and deviation from Relative PPP when the capital
market is in equilibrium. In order to analyze the e�ects of taxes in pricing international
assets, a theoretically sound model of international market valuation under international
taxation is required to develop a normative theory of taxation in IntCAPT.
Based on the concepts of Adler and Dumas (1983) andMai (2008) a binational version of
the Tax-IntCAPM is presented. In contrast to the model of Adler and Dumas (1983) the
Tax-IntCAPM is formulated in real terms in order to incorporate deviation from Relative
PPP. The framework of a discrete time model is applied, since the framework of the non-
linear exchange rate behavior with the option to integrate chaotic models is incorporated,
and which is founded on the basis of discrete time.
After discussion of the assumptions and set-up, we analyze di�erent versions of the Tax-
IntCAPM. The solution of the model derives the equilibrium pricing equation and an
intensive interpretation elaborates the economic concepts and implications.
5.1 Assumptions
Here, the assumptions of the Tax-IntCAPM are introduced. On the one hand, they build
on the assumptions of international taxation and on the other hand, they correspond to
the necessities of IntCAPT.
The economies in the domestic and foreign state can be comprised by making assump-
tions for the capital, goods and currency market, describing the investor's investment and
consumption opportunities and the barriers in the form of tax on international invest-
ment. The assumptions are rounded o� with the description of investors' behavior and a
conclusion.
136Furthermore, the derived Tax-IntCAPM should lay the basis for the derivation of a Tax-IntCAPM ina multiperiod context where dividends obviously are stochastic (Lally and van Zijl (2003), Jonas(2004) and Wiese (2006b)).
59
5 Tax International Capital Asset Pricing Model
Assumption 10 Exchange gains and in�ation e�ects
The nominal return is una�ected by exchange gains and in�ation.137
Assumption 11 Capital market
In the domestic state, there is a perfectly competitive capital market. A perfect capital
market implies that there are no arbitrage opportunities.138
Assumption 12 E�cient market
The E�cient Market Hypothesis (EMH) is pursued and the prices on assets re�ect imme-
diately and completely the entire relevant information.139
Assumption 13 Heterogeneous expectation
The concept of the Heterogeneous Expectation Model (HEM) is pursued.140 Market par-
ticipants di�er in degrees of international taxation and impact of deviation from Relative
PPP and hence in their risk aversion.
Assumption 14 Capital market equilibrium
The capital market is characterized by equilibrium in a world with taxes and deviation
from Relative PPP. Market participants are in a state of optimum and prices for all risky
137C. Mai (2008). This is an assumption generally accepted in Tax-CAPT. For a discussion, c. Jones
and Lamont (2001) and Wiese (2004).138No-arbitrage is a generally accepted condition in �nance. In general, if there is any arbitrage opportu-
nity, the market force would act as an invisible hand to drive the prices change and bring the marketback to an arbitrage-free setting. In reality, however, �nancial markets are never short of friction(Garman (1981)). The exclusion of tax arbitrage in the Tax-IntCAPM is discussed in chapter 6.
139C. Fama (1970a). The EMH is associated with the idea of a random walk, which is a term to character-ize a price series where all subsequent price changes represent random departures from previous prices.The logic of the random walk idea is that if the �ow of information is unimpeded and informationis immediately re�ected in stock prices, then tomorrow's price change will re�ect only tomorrow'snews and will be independent of the price changes today. But news is by de�nition unpredictableand, thus, resulting price changes must be unpredictable and random. As a result, prices fully re�ectall known information, and even uninformed investors buying a diversi�ed portfolio at the tableau ofprices given by the market will obtain a rate of return as generous as that achieved by the experts(Malkiel (2003)). Empirical evidence has been mixed, but has generally not supported strong formsof the EMH, c. Ross (1978), Garman (1981) and Sellin (1990). As far as the theoretical critiqueof the EMH is concerned, c. chapter 8.
140The extensive body of empirical evidence shows that investors do not behave the way homogeneousexpectation model predicts they should. The HEM assumes that individual investors hold di�eringbeliefs about an asset's future prospects (Miller (1977)).
60
5 Tax International Capital Asset Pricing Model
assets are assumed to move in such a way that an exogenous quantity of each risky asset
n0i(k) equals the aggregate demand for the risky asset nDi +n
Fi and the capital market is
cleared.141 World wealth is taken to be exogenous (endowment economy) (Lucas (1978)
and Pennacchi (2007)).
Assumption 15 Goods market
In each state there is a goods market which is characterized by bilateral trade. The price
process of the consumption bundle P is based on the quantity theory of money. The
in�ation rates adopt di�erent price processes in the states which are due to the state's
di�erent policies of money quantity providence. The states announce their policy of money
quantity providence at the beginning of the period.142
Assumption 16 Deviation from Relative PPP
The economic concept of nationhood is characterized by deviations from Relative PPP
(Adler and Dumas (1983)). At the beginning of the period PPP is assumed to hold but
the heterogeneous process of the in�ation and exchange rate leads to the irrelevance of
PPP at the end of period and hence the irrelevance of Relative PPP. Consequently, this
leads to the separation of the goods markets.143
Assumption 17 Currency market
The exchange rate process is characterized by a market approach in which demand for
foreign currency equals the supply. The non-linear behavior of exchange rates is assumed
to be determined by the interaction of speculators and traders.144
In the vein of the new paradigm we treat the capital, goods and currency market as com-
plex, interdependent systems which are characterized by the coexistence of randomness
and determination (Peters (1996)).
141The assumption of capital market equilibrium is a central theorem in CAPT. For a discussion ofdisequilibrium in capital markets, c. chapter 8.
142For a discussion of pricing in�ation risk and integration of global factors, c. chapters 3.5 and 8.143The good markets are considered to be separated if goods are di�erently evaluated.144S. chapter 3.4. The assumptions of the currency market are given in chapter 5.2.3 in detail.
61
5 Tax International Capital Asset Pricing Model
Assumption 18 Investment opportunity set
The representative investors face an equal investment opportunity set in the sense that
the available assets are the same for both. The capital market contains I+1 assets, i=0 is
a riskless asset, whereas i=1...I are risky assets. The capital market comprises all assets
except currencies.145 The investors have the opportunity of borrowing and lending at the
risk-free rate and short-selling is admitted.146
Assumption 19 Wealth
The investors possess initial real wealthW0 which equals unity in terms of the consumption
basket of the domestic state.147 Initial wealth is used in order to invest in exogenously given,
perfectly divisible and non-redundant risky stocks and risk-free bonds.148
Assumption 20 Consumption opportunity set
The investors consume the consumption bundle of their home state.
Assumption 21 Investors
Capital market investors behave as homo economicus,149 they optimize the well-being for
145It is assumed that all assets are marketable and so, the world capital market portfolio shall consistof all types of capital (except currencies), including real estate, privately held capital, publicly heldcapital (roads, parks, etc.), human capital which transpires not to be marketable. Balvers (2001)derives a CAPM integrating non-marketable assets.
146Short sales are necessary for e�ciently functioning markets. E�ective short sale constraints likelydecrease informational e�ciency and fundamental value e�ciency, and may lead to increased volatilityand market bubbles. Short sellers also increase liquidity, facilitate market making, and help marketsto identify corporate fraud (Jones and Lamont (2001), Francis (2005)). On the other hand shortsales can lead to arbitrage opportunities where investors and securities are subject to di�erentialtaxation and deviation from Relative PPP. In reality, investors may be charged a higher interestrate for borrowing money than the interest rate for lending money (Zhang (2004)). Brennan
(1971) examined the case where the borrowing rate di�ered from the lending rate and these rateswere di�erent for each investor. The Tax-IntCAPM under short sale and borrowing restrictions isdiscussed in chapter 6.
147Real wealth is nominal wealth divided by the price of the consumed commodity (Stulz (1984)).148The market participant's decision problem is regarded as investment based. The portfolio decision
problem is equivalent to the consumption-investment decision problem. For derivation, c. Lengwiler(2006). For empirical justi�cation, c. Siwik (2002).
149The term homo economicus is a model of the human being. The model refers to the concept of EconomicMan introduced by Mill (1909).
62
5 Tax International Capital Asset Pricing Model
themselves in the sense that they maximize their expected utility.150 The di�erent utility
concepts of the individuals include future consumption as their sole argument. The argu-
ment is assumed to be normally distributed. Investors' preferences display homotheticity,
non-satiation and variance-aversion.151 Investors are considered to be price takers.152
Investors' decisions are based on homogeneous availability and interpretation of informa-
tion, leading to homogenous estimation of state probabilities and hence homogeneous
expectations as far as the expectation value, variances and covariances are concerned for
pre-tax returns. However on account of investor-speci�c taxes, di�erent in�ation and ex-
change rates asset returns are heterogeneous for the domestic and foreign investor (Balvers
(2001)).
The structure of the Tax-IntCAPM is illustrated in �gure 5.1 on page 64. The capital,
goods and currency market are separated from each other, implying that the prices of
assets, goods and currencies are separately determined. We ignore the question of unique-
ness of equilibrium (Wiese (2004) and Mai (2008)).153 Since the nominal before tax
return is una�ected by tax, in�ation and exchange rates, it is possible to determine the
expected return on any risky asset endogenously, whereas the risk-free rate, the in�ation
and exchange rate is exogenously determined. The investor's consideration of one period
implies, that end of period consumption is equal to end of period real wealth, since at the
end of period the entire real wealth is used for consumption (Balvers (2001)). So, the
expected utility depends solely on the end of period wealth.
150The concept of expected utility (Neumann-Morgenstern utility) is the best instrument to make ratio-nal decisions in the state of insecurity (Kruschwitz (2007)). The concept of expected utility wasintroduced by Bernouilli (1738) and justi�ed axiomatically by Neumann and Morgenstern (1944).The concept of expected utility says that the utility of the investment corresponds to the expectedvalue of utility. Under the assumption of normal distribution of wealth the expected utility dependssolely on expected return and variance irrespective of the type of utility function (Balvers (2001) andKruschwitz (2007)), for proof s. appendix A.
151This implies that the utility function is continuous, twice di�erentiable, increasing and concave (Huangand Litzenberger (1988)). Epstein (1985) shows that mean-variance utility functions are implied bya set of decreasing absolute risk aversion postulates. The assumption of homotheticity implies that aproportional increase in consumption of all goods yields a proportional increase in utility. For givenprices the same consumption mix is optimal regardless of income. A homothetic utility function canbe represented by monotonic transformation. As far as asset pricing models under the assumption ofnon-homotheticity are concerned, c. Breeden (1979) and Balvers (2001).
152For a discussion of non-price-taking behavior, c. Basak (1994).153The existence of the uniqueness of equilibrium is regarded in Hens (2002). Botazzi (1998) shows that
the CAPM can have several equilibria.
63
5 Tax International Capital Asset Pricing Model
N
S
EW
Investor f
Currency markets
Capital market
Exchange
rates
S
S0
Exchange
rate gain/
loss
e
Deterministic chaos
Domestic and foreign
currency
Deviation of Purchasing Power Parity
Risky
± assets
Riskfree
i = 1...I
i = 0
Equilibrium
Bundle
of
goods
Bundle
of
goods
P D
P D
0
Inflation
πF
Inflation
πD
Domestic state D Foreign state F
Investor d time
P F
P F
0
Investment
Investor d
Money
Money
Trade
Nominal return rn
Taxation of sta
te FR
eal return after international tax
Quantity theory of money
Real return after tax
Quantity theory of money
World
Goods markets
Taxation of state D
Nominal return after
tax
Nominal return after
international tax
Nominal return after currency gain/loss
Investm
ent
Figure 5.1: Structure of the Tax-IntCAPM
64
5 Tax International Capital Asset Pricing Model
5.2 Model Set-up
The set-up of the model allows us to construct the foundations of the Tax-IntCAPM.
After the formulation of domestic investor's real rate of return, the concept of deviation
from Relative PPP, the integration in the real exchange rate, the non-linear behavior of
exchange rate is derived. The set-up of the model is concluded with the derivation and
implementation of foreign investor's rate of return.
5.2.1 Domestic Investor's Rate of Return
The stock return consists of capital gains and dividend. The nominal capital gains and
dividend rate of return are de�ned as:
rCGi,n ≡CGi
Vi,0rDi,n ≡
Di
Vi,0. (5.1)
with rCGi,n nominal capital gains rate of return
rDi,n nominal dividend rate of return
As far as the domestic investor is concerned, we formulate the stock rate of return after
integration of the national tax system comprising the dividend and the capital gains tax
system.154
1 + rDCGi + rDDi ≡Vi,0 + CGi
(1− tDCG
)+ Di
(1− tDD
)Vi,0 (1 + πD)
≡ rCGi,n1− tDCG1 + πD
+ rDi,n1− tDD1 + πD
+1
1 + πD
(5.2)
with rDCGi domestic investor's real capital gains rate of return after tax
rDDi domestic investor's real dividend rate of return after tax
tDCG domestic investor's tax rate on capital gains
154S. chapter 4.2 and chapter 4.3. The rate of returns are derived according to the concept of internalrate of return (Walker and Bos (2009)).
65
5 Tax International Capital Asset Pricing Model
tDD domestic investor's tax rate on dividend
The nominal interest rate of return is de�ned as:
rI,n ≡I
V0,0. (5.3)
with rI,n nominal interest rate of return
The domestic investor's risk-free borrowing and lending is subject to the interest tax sys-
tem of the domestic state.155
1 + rDI ≡V0,0 + I
(1− tDI
)V0,0 (1 + πD)
≡ rI,n1− tDI1 + πD
+1
1 + πD
(5.4)
with rDI domestic investor's real interest rate of return
tDI domestic investor's tax rate on interest
The in�ation rate is determined according to the quantity theory of money.156 We assume
a constant velocity in money and constant number in transactions.157 The price level of
consumption goods and hence the domestic in�ation rate is determined by a proportionate
increase of money supply of the domestic state. We can calculate the domestic in�ation
rate, which adopts numbers from the interval (-1;∞). Negative rates imply de�ation.
MSD
MSD0≡ PD
PD0
= 1 + πD (5.5)
155C. chapter 4.3.156C. chapter 3.5.157The velocity in money depends on payment habits such as salaries and taxes. The number in transac-
tions depends on income. These factors change solely in the long run, so it is reasonable to assumeconstant velocity in money and constant number in transactions, c. Gordon (2009).
66
5 Tax International Capital Asset Pricing Model
with MSD money supply of the domestic state at the end of period
MSD0 money supply of the domestic state at the beginning of the period
5.2.2 Deviation from Purchasing Power Parity and Real Exchange
Rate
The real exchange rate is de�ned in terms of the consumption of goods and is the relative
price of the foreign consumption basket to the domestic one. It is the nominal spot ex-
change rate multiplied by the ratio of domestic to foreign prices (Serçu and Uppal (1995)
and Chinn (2008)).
R ≡ SPD
P F(5.6)
with R real exchange rate (at the end of period)
The real exchange rate measures the value of the domestic state's goods against those ones
of the foreign state at the prevailing nominal exchange rate and is a measure to compare
living standards across countries. If the real exchange rate is equal to unity the goods in
the domestic country have the same value as the goods in the foreign country and PPP
is assumed to hold. According to PPP if the real exchange rate is unequal to unity, there
will be pressure on the nominal exchange rate to adjust, because the same good can be
purchased more cheaply in one country than in the other one.
Example 3 The Big Mac
The nominal Euro�Russian Ruble exchange rate is S = 42.1691RUBEUR
as of January 1,
2009.158 In Germany the Big Mac costs 2.99 EUR and in Russia the Big Mac costs 100
RUB.159 The real exchange rate amounts to R ≡ S PD
PF= 42.1691RUB
EUR2.99EUR100RUB
= 1.2609,
158S. www.oanda.com/convert/fxhistory.159McDonalds is a franchise enterprise, so the prices di�er from town to town. We chose as representative
Big Mac prices the ones in Berlin and Kazan.
67
5 Tax International Capital Asset Pricing Model
so the Big Mac in Germany is 26.09% more expensive than in Russia. If the Big Mac in
Germany costs 2.37 EUR, the real exchange rate would be equal to unity and the Big Mac
in Germany would cost the same as in Russia.
The appreciation of the real exchange rate is the quotient of the real exchange rate at the
end of period and at the beginning of the period (Balvers (2001)).
R
R0
≡SPD
PF
S0PD0PF0
(5.7)
with R0 real exchange rate at the beginning of period
The appreciation of the real exchange rate measures the value of the foreign state's goods
against those of the domestic state at the prevailing nominal exchange rate at two dif-
ferent points of time and expresses its appreciation. We can insert the de�nition of the
in�ation rate (eq. (3.2)) in the above equation.
R
R0
≡S(1 + πD
)S0 (1 + πF )
(5.8)
We insert the rate of nominal exchange appreciation in the above equation. The appreci-
ation of the real exchange rate is the appreciation of the nominal exchange rate and the
relation of domestic and foreign in�ation.
R
R0
≡ (1 + e)1 + πD
1 + πF(5.9)
Under the assumption of prevalence of Relative PPP the rate of appreciation of the real
exchange rate would equal unity as can be derived from the de�nition of Relative PPP
(eq. (3.4)).
RPPP ≡ (1 + e)1 + πD
1 + πF= 1 (5.10)
68
5 Tax International Capital Asset Pricing Model
The hypothesis of Relative PPP says that domestic and foreign in�ation rates are related
to exchange rate changes to the extent that di�erences in the development of price levels
are balanced by the appreciation of the exchange rate.
For theoretical and empirical reasons we reject the hypothesis of Relative PPP.160
RPPP ≡ (1 + e)1 + πD
1 + πF6= 1 (5.11)
Deviation from Relative PPP indicates a change of consumers' living standards and in
the relative competitiveness of countries (Serçu and Uppal (1995)). Figure 5.2 on page 70
illustrates the concept of deviation from Relative PPP.
5.2.3 Non-linear Behavior of Exchange Rates
The price process of the exchange rate is described by non-linearity and is not determined
by Relative PPP. We follow the model of Ellis (1992), since it retains the salient features
of the market.161 The exchange rate is determined by the interaction of speculators and
traders and is characterized by a market approach in which the change of demand for
foreign currency equals the change of supply.
Assumption 22 Demand for foreign currency
The speculator's demand for foreign currency is determined by the percentage deviation
of the expected future exchange rate from the current exchange rate.162
DSd = α
E[Sd+1
]Sd
− 1
(5.12)
160C. chapter 3.2.1.161Cf. also Karaguler (2000) and Vlad (2010). S. Dornbusch and Krugman (1976), Grauwe and
Vansanten (1990), Grauwe (1993), Lux (1998), Brock and Hommes (1998), Da Silva (2001),Federici and Gandolfo (2002) and Grauwe and Grimaldi (2006) for other non-linear models.
162Ellis (1992) does not specify how the expected exchange rate is determined. The domestic (foreign)speculators' demand for foreign (domestic) currency generates chaotic dynamics for some parametervalues (Ellis (1992), Karaguler (2000) and Vlad (2010)).
69
5 Tax International Capital Asset Pricing Model
Investor d
Irrelevance of RPPP
End of the period
Beginning of the period
Domestic state D Foreign state F
Investor d
Consumption bundle PD
0
Exchange rate S0
=
Investor f
Consumption bundle PD
Investor f
Consumption bundle PF≠SP
D
Exchange rate S
≠
Consumption bundle PF
0=S0PD
0
Inflation rate of
domestic state πD
Inflation rate of
foreign state πF
Exchange rate appreciation e
Appreciation of real exchange rate R ≠ 1
Figure 5.2: Deviation from Relative PPP
with DSd demand for foreign currency on the present day
α sensitivity factor with α ≥ 0
E[Sd+1
]expected exchange rate of the next day
Sd exchange rate of the present day
70
5 Tax International Capital Asset Pricing Model
The demand for foreign (domestic) currency generates non-linearity. If the expected
exchange rate is higher than the spot exchange rate and the sensitivity factor α is unequal
to zero, this will result in a speculators' demand.163 If the sensitivity factor α is equal to
zero or the expected future exchange rate is equal to the spot exchange rate, there is an
absence of speculative demand for the currency. α is a term generating non-linear demand.
If α 6= 1 the demand is non-linear in expected change of exchange rate. For α > 1 the
demands are proportionally larger. For 0 < α < 1 the demands are proportionally smaller
if the expected exchange rate is higher than the spot exchange rate.164
Assumption 23 Trade balance
It is assumed that the trade balance is determined by spot and previous day exchange rates,
the corresponding expected values and sensitivity factors.165
According to Ellis (1992) the presence of contract arrangements could justify the use of
this speci�cation.
Td = δ(Sd − E
[Sd
])+ γ
(Sd−1 − E
[Sd
])(5.13)
with Td trade balance of the present day
δ sensitivity factor with δ > 0
γ sensitivity factor with γ > 0
Sd−1 previous day's exchange rate
163Ellis (1992) leaves the question open if a supply of foreign currency will be generated in the case of ahigher value of the spot exchange rate than of the expected future exchange rate. According to Ellis(1992) if α =∞, the in�nite demand will be eliminated instantaneously.
164In Ellis (1992) the presence of transaction costs is given as an explanation for this type of behavior.165The trade balance must be interpreted as the trade surplus, which is the di�erence between the mone-
tary value of imports and exports in the domestic or foreign economy over the period of one day. Theterm balance is misleading since the values are �ow �gures. The domestic (foreign) nation's tradesurplus is equal to the supply or de�cit of domestic (foreign) currency which is provided by traders.For a discussion of the assumption, s. chapter 8.
71
5 Tax International Capital Asset Pricing Model
The trade balance is not only in�uenced by the present exchange rate but also by the
previous day's exchange rate. The previous day's exchange rate may have a greater e�ect
on the trade balance than the current exchange rate (Ellis (1992)). If the expected ex-
change rate is equal to the spot or to the previous day's exchange rate,
Sd−1 = Sd = E[Sd
](5.14)
the exchange rate is said to be at its steady state value. In this state, speculators do not
intend to borrow or lend, and the expected exchange rate is determined by exogenous
fundamentals and the trade balance becomes zero.166
The time frame of the model is short run, weekly at most, because the fundamentals are
not expected to change in the planning horizon.
Assumption 24 Currency market equilibrium
The foreign exchange market is cleared at each period, so that the deviation of demand is
equal to the trade surplus.167
DSd −DSd−1= Td (5.15)
with DSd−1demand for foreign currency on the previous day
Figure 5.3 on page 73 illustrates the non-linear behavior of exchange rates. We solve this
166Ellis (1992) mentions that fundamentals like the interest rate di�erential or relative money supplies
can determine the expected exchange rate. For α > 0, it follows that Sd−1 = Sd = E[Sd
]= E
[Sd+1
]and for α = 0, E
[Sd+1
]can adopt every value.
167The assumption that the expected exchange rate is determined by Relative PPP leads to constantdeviation of demand is found to be implausible.
72
5 Tax International Capital Asset Pricing Model
Net demand for
domestic and foreign
currency on the
actual day DSd
Chaoss
Net demand for currency on the
actual day DSdSpeculators
Sd-1
Expectation Expectation
Trade balance
Imports
- Exports
Net demand for
domestic and foreign
currency on the
actual day DSd
Net demand for currency on the
previous day DSd-1
SdTrade
surplusTraders
E[Sd] E[Sd-1]
Sd Sd-1α γ δ
~ ~
Figure 5.3: Currency Exchange Market
equation for the present exchange rates.
α
E[Sd+1
]Sd
− 1
− αE
[Sd
]Sd−1
− 1
= δ(Sd − E
[Sd
])+ γ
(Sd−1 − E
[Sd
])(5.16)
We place all terms on the left side and multiply by the current and previous exchange
rates to prepare the resolution of the quotients.
73
5 Tax International Capital Asset Pricing Model
α
E[Sd+1
]Sd
− 1
− αE
[Sd
]Sd−1
− 1
− δ (Sd − E [Sd])− γ (Sd−1 − E [Sd]) = 0 (5.17)
α
E[Sd+1
]Sd
− 1
SdSd−1 − α
E[Sd
]Sd−1
− 1
SdSd−1
− δ(Sd − E
[Sd
])SdSd−1 − γ
(Sd−1 − E
[Sd
])SdSd−1 = 0
(5.18)
The multiplication of the terms in parenthesis with the spot and previous day exchange
rates leads to the resolution of the quotients.
α(E[Sd+1
]Sd−1 − SdSd−1
)− α
(E[Sd
]Sd − SdSd−1
)− δ(Sd − E
[Sd
])SdSd−1 − γ
(Sd−1 − E
[Sd
])SdSd−1 = 0
(5.19)
We expand the terms in parenthesis and rearrange in respect of the future exchange rate.
αE[Sd+1
]Sd−1 − αE
[Sd
]Sd − δS2
dSd−1
+ δE[Sd
]SdSd−1 − γS2
d−1Sd + γE[Sd
]SdSd−1 = 0
(5.20)
δS2dSd−1 − δE
[Sd
]Sd−1Sd − γE
[Sd
]Sd−1Sd
+ γS2d−1Sd + αE
[Sd
]Sd − αE
[Sd+1
]Sd−1 = 0
(5.21)
δS2dSd−1 −
((δ + γ)E
[Sd
]Sd−1 − γS2
d−1 − αE[Sd
])Sd
−αE[Sd+1
]Sd−1 = 0
(5.22)
There are two roots for Sd; since the exchange rate cannot be negative the positive root
is chosen (Ellis (1992)).168
168For the correct derivation, cf. Vlad (2010).
74
5 Tax International Capital Asset Pricing Model
Sd =(δ + γ)E
[Sd
]Sd−1 − γS2
d−1 − αE[Sd
]2δSd−1
+
√((δ + γ)E
[Sd
]Sd−1 − γS2
d−1 − αE[Sd
])2+ 4αδE
[Sd+1
]S2d−1
2δSd−1
(5.23)
The non-linear process is formulated by a recursive function. In the model, if the ex-
change rate is known, the next day's exchange rate can be determined. By iterating the
present exchange rate we receive the process of the exchange rate and can calculate the
appreciation of the exchange rate.169
Example 4 Exchange rate process
As in Ellis (1992) we assume that the parameters adopt values of α = 5, δ = 5, and
γ = 25 and for reasons of tractability that the expected exchange rates E[Sd+1
]and
E[Sd
]are equal to unity.170 The starting point for the exchange rate is 0.8. We insert
the values into the formula leading to:
Sd =30Sd−1 − 25S2
d−1 − 5
10Sd−1
+
√(30Sd−1 − 25S2
d−1 − 5)2
+ 100S2d−1
10Sd−1
(5.24)
The time variation of the exchange rate for 25 days is shown in �gure 5.4. It displays non-
linear behavior. The maximum value is 2.0220, whereas the minimum value is 0.2078 for
the �rst 25 days. The value table of exchange rate time series is given in the appendix C.1
on page 154. The return of the nominal exchange rate is -0.7370%.171 The non-linear
process assumes a totally di�erent process in the case of a change in starting parameters.
169In the case of much larger values of the sensitivity factor γ than of the sensitivity factor δ the modelcan exhibit chaotic behavior (Ellis (1992)).
170To apply this model the parameters must be estimated.171S. value table of exchange rate time series in the appendix C.1 on page 154. The time frame of the
model is short run, weekly at most. Since we did not estimate the fundamentals, we refrain fromchanging them in the planning horizon.
75
5 Tax International Capital Asset Pricing Model
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2.0
-
6
Exchange rate
Day
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Figure 5.4: Time Series of Exchange Rate with α = 5, δ = 5 and γ = 25
If α adopts the value of 4, the exchange rate adopts a totally di�erent path, as shown in
appendix C.1 on page 155.
5.2.4 Foreign Investor's Rate of Return
The features of international taxation, comprising the dividend, interest, capital and
exchange gains tax system and the methods of double taxation reduction, have to be
considered in order to formulate a foreign investor's rate of return.172
By integrating the taxing right factors the di�erent constellations of assignment of the
taxing right of all income types are considered. Foreign investor's taxation is modeled
by assigning the right of taxation factors the numbers 0 and 1. Figure 5.1 illustrates the
taxation right.
172S. chapter 4.
76
5 Tax International Capital Asset Pricing Model
Right of taxation factors ε ζ η
Domestic state 1 0 0
Foreign state 0 1 (0/1)
Domestic and foreign state 1 1 (0/1)
Table 5.1: Taxing Right Factors
By assigning ε the value one and the other factors the value zero, the rate of return under
the assumption of assignment of the taxing right to the domestic state is modeled. By
assigning ζ the value one and ε the value zero, the assignment of the taxing right to the
foreign state is modeled. By assignment of unity to the factors ε and ζ the taxation in
the domestic and foreign state is modeled. If η is equal to one the methods of avoiding
and reducing double taxation are considered. Capital gains after tax is modeled by the
following formula:
CGF
i,t = CGi −(εSt,CGt
DSt,CG + ζCGt
FCG − ηCGtFInt,CG
)CGi. (5.25)
with CGF
i,t foreign investor's capital gains after international taxation
εSt,CG domestic state's taxing right factor on capital gains
tDSt,CG foreign investor's (source) capital gains tax rate
ζCG foreign state's assignment taxing right factor on capital gains
tFCG foreign investor's capital gains tax rate
ηCG foreign state's taxing right factor on capital gains relief
tFInt,CG foreign investor's capital gains relief tax rate
The taxation in the domestic and foreign state is due to a tax rate e�ect, so we can
summarize the above equation as follows:
CGF
i,t = CGi − tFeff,CGCGi. (5.26)
with tFeff,CG foreign investor's e�ective capital gains tax rate
77
5 Tax International Capital Asset Pricing Model
The taxation of the exchange gains on capital gains and on the principal can be modeled
by the following equation:
EFCG,P,t = (S − S0) Vi −
(ζCG,P,et
FCG,P,e − ηCG,P,etFInt,CG,P,e
)(S − S0) Vi (5.27)
= (S − S0) Vi − tFeff,CG,P,e (S − S0) Vi.
with EFCG,P,t foreign investor's exchange gains on capital gains and principal
after national tax
ζCG,P,e foreign state's exchange gains taxing right factor on capital
gains and principal
tFCG,P,e exchange gains tax rate on capital gains and principal
ηCG,P,e foreign state's relief exchange gains taxing right factor on capital
gains and principal
tFInt,CG,P,e foreign investor's relief exchange gains tax rate on capital
gains and principal
tFeff,CG,P,e foreign investor's e�ective exchange gains tax rate on capital gains
and principal
The taxation of foreign investor's dividend can be comprised by the following formula:
DFi,t = Di −
(εSt,Dt
DSt,D + ζDt
FD − ηDtFInt,D
)Di. (5.28)
with DFi,t foreign investor's dividend after international taxation
εSt,D domestic state's taxing right factor on dividend
tDSt,D foreign investor's (source) dividend tax rate
ζD foreign state's taxing right factor on dividend
tFD foreign investor's dividend tax rate
ηD foreign state's taxing right factor on dividend relief
tFInt,D foreign investor's relief tax rate on dividend
78
5 Tax International Capital Asset Pricing Model
The taxation in the domestic and foreign state is due to a tax rate e�ect, so we can
summarize the above equation as follows:
Di,t = Di − tFeff,DDi. (5.29)
with tFeff,D foreign investor's dividend e�ective tax rate
The taxation of the exchange gains on dividend can be modeled by the following equa-
tion:
EFD,t = (S − S0) Di −
(ζD,et
FD,e − ηD,etFInt,D,e
)(S − S0) Di (5.30)
= (S − S0) Di − tFeff,D,e (S − S0) Di.
with EFD,t foreign investor's exchange gains on dividend after tax
ζD,e foreign state's taxing right factor on exchange gains on dividend
tFD,e foreign investor's exchange gains tax rate on dividend
ηD,e foreign state's taxing right factor on exchange gains on dividend relief
tFInt,D,e foreign investor's exchange gains relief tax rate on dividend
tFeff,D,e foreign investor's dividend e�ective tax rate
Foreign investor's stock return is summarized by the following equation:
1 + rFCGi + rFDi ≡ rCGi,n1− tFeff,CG1 + πF
+ erCGi,n1− tFeff,CG,P,e
1 + πF+
1 + e
1 + πF
− etFeff,CG,P,e1 + πF
+ rDi,n1− tFeff,D1 + πF
+ erDi,n1− tFeff,D,e1 + πF
.
(5.31)
with rFCGi foreign investor's capital gains rate of return after tax
rFDi foreign investor's dividend rate of return after tax
Interest after tax is modeled by the following formula:
It = I −(εSt,It
DSt,I + ζIt
FI − ηItFInt,I
)I. (5.32)
79
5 Tax International Capital Asset Pricing Model
with εSt,I domestic state's taxing right factor on interest
tDSt,I foreign investor's (source) interest tax rate
ζI foreign state's taxing right factor on interest
tFI foreign investor's interest tax rate
ηI foreign state's taxing right factor on interest relief
tFInt,I foreign investor's interest relief tax rate
The taxation in the domestic and foreign state is due to a tax rate e�ect, so we can
summarize the above equation as follows:
It = I − tFeff,II. (5.33)
with tFeff,I foreign investor's interest e�ective tax rate
The taxation of the exchange gains on capital gains and on the principal can be modeled
by the following equation:
EI,P,t = (S − S0)V0 −(ζI,P,et
FI,P,e − ηI,P,etFInt,I,P,e
)(S − S0)V0 (5.34)
= (S − S0)V0 − tFeff,I,P,e (S − S0)V0.
with EI,P,t foreign investor's exchange gains after tax on interest and principal
ζI,P,e foreign state's taxing right factor on exchange gains on capital
gains and principal
tFI,P,e foreign investor's exchange gains tax rate on interest and on the principal
ηI,P,e foreign state's taxing right factor on exchange gains on capital
gains and principal relief
tFInt,I,P,e foreign investor's relief exchange gains tax rate on interest
and on the principal
tFeff,I,e foreign investor's e�ective exchange gains tax rate on capital gains
and on the principal
80
5 Tax International Capital Asset Pricing Model
The integration of the taxing right factors into the risk-free asset is equivalent.
1 + rFI ≡ rI,n1− tDeff,I1 + πF
+ rI,ne1− tDeff,I,P,e
1 + πF+
1 + e
1 + πF− e
tFeff,I,P,e1 + πF
(5.35)
with rFI foreign investor's interest rate of return after tax
The �rst term of foreign investor's rate of return incorporates the taxation of capital gains
(interest). In the second term the taxation of exchange gains on capital gains (interest)
is incarnated. In contrast to the dividend rate of return the third and fourth part of the
capital gains and interest rate of return incorporate the fact that the principal is solely
subject to exchange gains taxation.
The in�ation rate is determined according the quantity theory of money.173 The foreign
investor's in�ation rate is determined by the money supply of the foreign state. Equiva-
lently to the domestic in�ation rate the foreign in�ation rate is assumed to adopt numbers
from the interval (-1;∞).
MSF
MSF0≡ P F
P F0
= 1 + πF (5.36)
with MSF money supply of the foreign state at the end of period
MSF0 money supply of the foreign state at the beginning of the period
Since the tax, in�ation and exchange rates are di�erent in the economies, asset rate of
returns are heterogeneous for the investors.
5.2.5 Implementation of Foreign Investor's Rate of Return
We implement foreign investor's rate of return. As the nominal risk-free asset we consider
the eonia.174 As the risky asset we consider the K+S AG stock (WKN 716200) from
173S. chapter 3.5.174S. www.de.global-rates.com/zinssatze/eonia/2008.aspx.
81
5 Tax International Capital Asset Pricing Model
Xetra.175 On September 14, 2007, 6.75 % of the K+S AG stock were acquired by MCC
Holding Ltd. (Linea Ltd) whose holding manages assets of the Russian investor Andrey
Melnichenko. The share of K+S AG was increased to 15.001% until November 28, 2008.
The data comprises the period from November 1, 2007 to December 31, 2009.176 In order
to determine the risk-free rate we consider the average of the �rst month eonia.177 In order
to estimate the capital gains, we follow the methodology of Schmid and Trede (2006)
and apply the logarithm to use the additivity characteristic of continuous returns.178 We
correct the capital gains rate of return by the in�uences of dividend and stock-split and
annualize the return.179
rCGK+S,n= 12
∑Mm=1 ln
(1 +
VK+S,m−VK+S,m−1
VK+S,m−1
)M
(5.37)
with rCGK+S,nnominal capital gains rate of return of the K+S AG stock
VK+S,m value of the K+S AG stock at the end of month m
VK+S,m−1 value of the K+S AG stock at the beginning of month m
Dividends are distributed yearly. Equivalent to the capital gains rate of return of the
K+S AG stock we apply the logarithm.
rDK+S,n=
∑Tt=1 ln
(1 + DK+S
VK+S,t−1
)T
(5.38)
with rDK+S,nnominal dividend rate of return of the K+S AG stock
DK+S dividend of the K+S AG stock in the month
175S. www.deutsche-boerse.de. For a description of Xetra, cf. Schmid and Trede (2006).176Restricting the period of investigation to these years results from the consideration that the Tax-
IntCAPM shall re�ect the implications of the international tax system (Dausend and Schmitt (2007)).In November 2007 the K+S AG stock became part of the MSCI ACWI, s. �nancial report 2007 ofthe K+S AG. In order to build on the implementation of of foreign investor's rate of return the datastarts from November 1, 2007.
177S. www.de.global-rates.com/zinssatze/eonia/2008.aspx.178Cf. Stehle (1999) and Schulz (2006).179S. Schmid and Trede (2006).
82
5 Tax International Capital Asset Pricing Model
The appreciation of nominal exchange rate is determined on a monthly basis.180 Equiv-
alent to the determination of the capital gains rate of return of the K+S AG stock we
apply the logarithm and annualize the return.
eRUB−EUR = 12
∑Mm ln
(1 + Sm−Sm−1
Sm−1
)M
(5.39)
with eRUB−EUR appreciation of the Ruble-Euro exchange rate
Sm exchange rate at the end of month m
Sm−1 exchange rate at the beginning of month m
According to �2 German Fiscal Code respectively Art.7 Tax Code of the Russian Fed-
eration the Double Taxation Convention (DTC) Federal Republic of Germany - Russian
Federation takes precedence over German and Russian tax law. According to Art. 1 DTC
the convention is valid for the German limited liability corporation and the Russian in-
vestor. The German Russian DTC assigns the right of capital gains and interest taxation
to the Russian Federation (Art. 13 No. 4 and Art. 11 No.1 DTC). Dividends are subject
to a source tax tSt of 15% (Art. 10 I No. 2).181 According to Art. 208 Item 3 I in connec-
tion with Art. 214 Item 1 and Art. 224 Item 4 RITL the tax rate on foreign dividends
tFD amounts to 6%. According to Art. 224 Item 1 and 2 in connection with 214.2 RITL
the tax rate on interest amounts to 13%, since the nominal interest rate of return falls
below the critical margin of 9%. The tax rate on capital gains amounts to 13% according
to Art. 224 Item I. According to Art. 214 Item 1 RITL foreign taxes are credited, if
this is allowed in the DTC. Taxes imposed in Germany are to be credited on Russian
taxes (Art. 23 I DTC). According to Art. 23 Item I DTC the limited credit method is
applied. Incomes and expenses accepted for deduction of the taxpayer expressed (nomi-
nated) in foreign currency are converted into rubles at the rate of the Central Bank of
180We refrain from determining the exchange rate according to the model of Ellis (1992), since itsapplication requires the determination of several sensitivity parameters for the EUR-RUB exchangerate.
181In contrast to the OECD model convention Art. 10 I DTC does not explicitly incorporate the con�r-mation of the right of taxation to the state of residence. However, there is no material deviation fromthe model convention, since Art. 10 I Model Convention does not contain an independent materialregulatory content, cf. Tischbirek (2003).
83
5 Tax International Capital Asset Pricing Model
the Russian Federation established on the date of actual receipt of the incomes (Art. 210
Item 5 RITL).182 The tax base income (loss) under a securities purchase/sale deal shall
be determined as a di�erence between the sums received from the sale of the securities
and the expenses towards acquiring the securities (Art. 214.1 Item 3 RITL).
Example 5 Foreign Investor's Rate of Interest Return after Tax
The eonia and hence the nominal risk-free rate of return ri,n amounts to 2.52%. The
expected nominal capital gains rate of return of the K+S AG stock rCGK+S,nis 47.31%
whereas the expected nominal dividend rate of return of the K+S AG stock amounts to
rDK+S,n1.77%. The appreciation of the Ruble-Euro exchange rate eRUB−EUR amounts to -
8.99% whereas the in�ation rate for the Russian economy is 8.8%.183 Since the taxing right
on capital gains and interest is assigned to the Russian Federation and no double taxation
relief or avoiding methods are implemented Germany's taxing right factor on capital gains
εSt,CG and on interest εSt,I as well as Russia's taxing right factor on capital gains ηCG and
interest relief ηI are zero whereas the Russian state's taxing right factor on capital gains
ζCG and on interest ζI are equal to unity. The e�ective tax rate on capital gains tFeff,CGand e�ective tax rate on interest tFeff,I amount to 13%. The taxing right on dividend is
assigned to Germany and Russia, hence Germany's taxing right factor on dividend εSt,D,
Russia's taxing right factor on dividend ζD as well as its taxing right factor on dividend
relief ηCG are equal to unity. The Russian state applies the limited credit method, hence
the e�ective tax rate on dividend tFeff,D is 15%. Incomes and expenses in foreign currency
are converted into rubles at the exchange rate. The e�ective exchange gains tax rate on
capital gains and on the principal tFeff,CG,P,e and the e�ective exchange gains tax rate on
interest and on the principal tFeff,I,P,e amount to 13%, whereas the e�ective exchange gains
tax rate on dividend tFeff,D,e is 15%.
182The o�cial exchange rates of foreign currencies against the ruble are set by the Central Bank of theRussian Federation without assuming any liability to buy or sell foreign currency at the above rate.www.cbr.ru/eng/currency_base/monthly.aspx. Hence we can assume that the exchange rate fortax purposes is the market rate.
183S. example 1.
84
5 Tax International Capital Asset Pricing Model
We can calculate foreign investor's rate of risky and risk-free return.
rFCGK+S+ rFDK+S
≡ 0.47311− 0.13
1 + 0.088+ 0.4731(−0.0899) 1− 0.13
1 + 0.088
+1 + (−0.0899)
1 + 0.088− (−0.0899) 0.13
1 + 0.088+ 0.0177
1− 0.15
1 + 0.088
+0.0177(−0.0899) 1− 0.15
1 + 0.088− 1 = 0.2041
(5.40)
rFI ≡ 0.02521− 0.13
1 + 0.088+ 0.0252(−0.0899) 1− 0.13
1 + 0.088+
1 + (−0.0899)1 + 0.088
− (−0.0899) 0.13
1 + 0.088− 1 = −0.1344
(5.41)
with rFCGK+Sreal capital gains rate of return of the K+S AG stock after tax
rFDK+Sreal dividend rate of return of the K+S AG stock after tax
Foreign investor's real K+S stock rate of return amounts to 20.41% whereas the real risk-
free rate of return after tax is -13.44%.
5.3 Versions of Tax International Capital Asset
Pricing Model
The Tax-IntCAPM incorporates special constellations leading to di�erent versions of the
model.184 We equalize domestic and foreign investor's real rate of return after tax in
order to �nd the implications for the taxation of the di�erent income types - capital and
exchange gains, dividend and interest - in IntCAPT which can lead to di�erent versions
of the Tax-IntCAPM.
1 + rDCGi + rDDi = 1 + rFCGi + rFDi (5.42)
184Hereby we ignore constellations implying the irrelevance of tax, exchange and in�ation rates whichlead to the CAPM, IntCAPM and (international) Tax-CAPM. The international version of the Tax-CAPM is characterized by domestic and foreign investors' residence in the same monetary zone andsubjection to di�erent income classes.
85
5 Tax International Capital Asset Pricing Model
We apply this concept in order to compare domestic and foreign investor's rate of return
and to �nd out the implications for the tax system.185
rCGi,n1− tDCG1 + πD
+1
1 + πD+ rDi,n
1− tDD1 + πD
= rCGi,n1− tFeff,CG1 + πF
+ erCGi,n1− tFeff,CG,P,e
1 + πF+
1 + e
1 + πF
− etFeff,CG,P,e1 + πF
+ rDi,n1− tFeff,D1 + πF
+ erDi,n1− tFeff,D,e1 + πF
(5.43)
We regard the central theorem in IntCAPT - deviation from Relative PPP - as irrelevant
and assume that the hypothesis of Relative PPP holds. This approach enables us to
concentrate solely on tax e�ects.
In a �rst step, we multiply domestic and foreign investor's capital gains rate of return
with the domestic investor's in�ation rate 1+πD in order to incorporate the appreciation
of the real exchange rate and the concept of Relative PPP.
rCGi,n(1− tDCG
)+ 1 + rDi,n
(1− tDD
)= rCGi,n
(1− tFeff,CG
) (1 + πD
)1 + πF
+ erCGi,n
(1− tFeff,CG,P,e
) (1 + πD
)1 + πF
+(1 + e)
(1 + πD
)1 + πF
− etFeff,CG,P,e
(1 + πD
)1 + πF
+ rDi,n
(1− tFeff,D
) (1 + πD
)1 + πF
+ erDi,n
(1− tFeff,D,e
) (1 + πD
)1 + πF
(5.44)
Since we assume the existence of Relative PPP, which implies that the appreciation of
185Since the tax structure of dividends and interest rate of return is comprised by the tax structure of thestock rate of return, it is su�cient to regard the stock rate of return.
86
5 Tax International Capital Asset Pricing Model
the real exchange rate is equal to unity, we can simplify the above equation.
rCGi,n(1− tDCG
)+ rDi,n
(1− tDD
)= rCGi,n
(1− tFeff,CG
) (1 + πD
)1 + πF
+ erCGi,n
(1− tFeff,CG,P,e
) (1 + πD
)1 + πF
− etFeff,CG,P,e
(1 + πD
)1 + πF
+ rDi,n
(1− tFeff,D
) (1 + πD
)1 + πF
+ erDi,n
(1− tFeff,D,e
) (1 + πD
)1 + πF
(5.45)
The equivalence of domestic and foreign investors' rate of return depends decisively on the
features of international taxation. Under the assumption of Relative PPP and a certain
constellation of the international tax system the decision problem would be equivalent
to the one of the Tax-CAPM with investor-speci�c tax rates. If capital gains and the
principal are not taxable in either state and the tax rate on dividends is equal to the
exchange gains tax rate on dividends in the foreign state, then the decision problem
would be equivalent to the one of the Tax-CAPM with investor-speci�c tax rates.
rCGi,n + rDi,n(1− tDD
)= rCGi,n
(1 + πD
)(1 + e)
1 + πF+ rDi,n
(1− tFeff,D
) (1 + πD
)(1 + e)
1 + πF
(5.46)
rCGi,n + rDi,n(1− tDD
)= rCGi,n + rDi,n
(1− tFeff,D
)(5.47)
Under the assumption of Relative PPP certain tax constellations lead to the equivalence
of Tax-IntCAPM and the Tax-CAPM with investor-speci�c tax rates.
5.4 Model Solution
The solution of the Tax-IntCAPM will allow us to determine the relationship for interna-
tional assets between expected return and risk. The CAPM can alternatively be derived
by aggregating investors' individual optimum or by theoretical concepts of portfolio theory.
In general, the Tax-IntCAPM cannot be derived based on theoretical concepts of portfo-
lio theory, since the tangential portfolio is investor-speci�c on account of the assumption
87
5 Tax International Capital Asset Pricing Model
Derivation of Tax- IntCAPM
Market Equilibrium
Individual Optim
um
Maximization of
expected utility
Constraints
· Value shares sum to
unity
· Terminal wealth is set up
of the portfolio return
Integration of the
constraints into the
maximization
problem
5.48 – 5.52
5.53 – 5.54
Differentiation in
respect of the risky
asset value
5.55 – 5.56
Transformation
· Application of the definition of the
covariance
· Application of Stein’s
Lemma
5.57 – 5.60 5.61 5.69 5.70
Individual
optimum
5.81 – 5.84
Definition of the world
market return
· The world market port-
folio return is equal to
the value weighted return
of risky assets
Tax - IntCAPMSeparation of the
Market equilibrium
· Aggregation
· Supply is equal to
aggregated demand
· tax
· inflation
· appreciation of real exchange
rate
Integration of
international factors
Application of the
world market portfolio
5.75 – 5.805.85 – 5.985.71 – 5.74
Figure 5.5: Model Solution of Tax-IntCAPM
88
5 Tax International Capital Asset Pricing Model
of investor-speci�c tax, in�ation and appreciation of exchange rate.186 The process of
the derivation of the Tax-IntCAPM is illustrated in �gure 5.5 on page 88. The decision
problem of maximization expected utility of end of period real wealth after international
tax with the constraints that the value shares sum to unity and end of period wealth is
set up of the portfolio return is di�erentiated in respect of the value weighted shares of
the risky assets. After transformation of the derivation by application of the de�nition of
the covariance and Stein's Lemma we derive the individual optimum of the domestic and
foreign investor. We de�ne the world market return and derive the market equilibrium
by aggregation and equalization of supply and demand. By integration of international
factors and application of the world market portfolio we derive the equilibrium pricing
relationship of the Tax-IntCAPM.
The solution of the model will allow us to determine the relationship for international
assets between expected return and risk under the framework of international tax system.
We hypothesize that the integration of exchange gains taxation, deterministic in�ation
and non-linear exchange rates in IntCAPT leads to the integration of new factors whereas
the integration of the dividend, capital gains and interest tax system and the methods of
avoiding and reducing double taxation have solely a tax rate e�ect.
The derivation of investor's individual optimum forms the basis for the Tax-IntCAPM.
The domestic and foreign consumer maximize their expected utility of end of period real
wealth after national and international tax and hence, they achieve individual optimum.
5.4.1 Individual Optimum
The representative consumers intend to maximize their expected utility of end of period
real wealth after tax and hence, they achieve individual optimum. The formulas are con-
structed in such a way that they are valid for the domestic as well as for the foreign
investor.
Max{ωi}Ii=0
E[U(Wt
)](5.48)
186Cf. (Mai (2008)).
89
5 Tax International Capital Asset Pricing Model
with ωi investor's value portion of assets in the portfolio
Max Maximum
U investor's utility function
Wt investor's end of period real wealth after tax
The decision problem is subject to constraints. At the beginning of the period each
investor makes the decision about the composition of the portfolio under the budget con-
straint that the real value of the portfolio equals initial real wealth which is assumed to be
unity. PPP is assumed to hold at the beginning of period, so the real price of asset i is the
same as for the domestic investor and the foreign investor's portfolio at the beginning of
the period di�ers from the one of the domestic investor solely in respect of the demanded
number of each asset i.
I∑i=0
nDiVi,0PD0
= 1I∑i=0
nFiVi,0PD0
=I∑i=0
nFiVi,0S0P F
0
= 1 (5.49)
with nDi demanded number of asset i by the domestic investor
Vi,0 price of asset i at the beginning of period
PD0 price of domestic consumption bundle at the beginning of period
nFi demanded number of asset i by the foreign investor
The value weighted shares of the portfolio are made of the product of demanded number
and real price of the risky respectively the risk-free asset.
ωi =ni
Vi,0PD0∑I
i=0 niVi,0PD0
= niVi,0PD0
ω0 =n0
V0,0PD0∑I
i=0 niVi,0PD0
= n0V0,0PD0
(5.50)
with ω0 investor's value weighted share of the risk-free asset in the portfolio
n0 investor's number of the risk-free asset in the portfolio
V0,0 price of the risk-free asset at the beginning of period
90
5 Tax International Capital Asset Pricing Model
Initial real wealth equals one, so the value shares sum to unity. Since the entire wealth
is invested in risky and risk-free assets, the price of the portfolio equals initial wealth.
Short-selling is permitted, so the shares of the portfolio can also be negative.
s.t.I∑i=0
ωi = 1 (5.51)
Terminal real wealth after taxes is set up of the value weighted rate of return after tax of
the portfolio, consisting of risky and risk-free assets.187
s.t. Wt =I∑i=1
ωi (1 + rCGi + rDi) + ω0 (1 + rI) (5.52)
One part of the initial real wealth is invested in risky assets, the other part is invested into
risk-free assets (Huang and Litzenberger (1988), Balvers (2001) and Pennacchi (2007)).
The constraints, that the value shares sum to unity (eq. (5.51)) and that terminal real
wealth after taxes is set up of the return after tax of the portfolio, consisting of risky and
risk-free assets (eq. (5.52)) are integrated into the maximization problem.
Max{ωi}Ii=1
E
[U
((1−
I∑i=1
ωi) (1 + rI) +I∑i=1
ωi (1 + rCGi + rDi)
)](5.53)
The expansion and rearrangement of the above equation to express the excess rate of
return lead to the following result:
Max{ωi}Ii=1
E
[U
((1 + rI) +
I∑i=1
ωi (rCGi + rDi − rI)
)]. (5.54)
Investor's aim consists in determining the optimal share of each asset for the portfolio in
order to maximize the expected value of terminal real wealth after tax. We di�erentiate
partially in respect of the risky assets' shares by application of the chain rule and set the
187According to the theorem of linear transformation of normally distributed variables, the end-of-periodwealth is normally distributed, since the returns are assumed to be normally distributed (Balvers(2001)). Consider in constraint (eq. (5.52)) that initial wealth equals unity and can be omitted.
91
5 Tax International Capital Asset Pricing Model
derivative equal to zero.188
∂E[U((1 + rI) +
∑Ii=1 ωi (rCGi + rDi − rI)
)]∂ωi
= 0
E
∂U((1 + rI) +
∑Ii=1 ωi (rCGi + rDi − rI)
)∂ωi
(rCGi + rDi − rI)
= 0
for iε 1..I.
(5.55)
According to constraint (5.52) we can replace the variable of the utility function by ter-
minal real wealth after tax.
E
∂U(Wt
)∂ωi
(rCGi + rDi − rI)
= 0 (5.56)
The expected marginal utility of real wealth after tax has to be zero, what implies that
the change of the value shares would not lead to further expected utility and that the
investor achieved individual optimum. Since the utility function is concave, the �rst order
necessary condition leads to the maximization of investor's expected utility. The concavity
of the utility function implies that further increase of wealth leads to less enhancement of
investor's utility.189 The investors undertake risky investments only if the rate of return
on at least one risky asset exceeds the risk-free rate.190
By using the de�nition of covariance and the result of the �rst order condition that the
expected marginal utility of real wealth after taxes has to be zero (eq. (5.56)) we obtain
188We do not di�erentiate in respect of the risk-free asset share. The optimal share of the risk-free assetcan be determined by deduction of the sum of shares of the risky assets from unity.
189In mathematical terms this is expressed by the negativity of the second derivation which is necessaryfor a maximum, cf. Huang and Litzenberger (1988) and Pennacchi (2007).
190For proof and for derivation of the minimum risk premium for entire investment in the risky asset, cf.Huang and Litzenberger (1988).
92
5 Tax International Capital Asset Pricing Model
Cov
∂U(Wt
)∂ωi
, rCGi + rDi − rI
=
E
∂U(Wt
)∂ωi
(rCGi + rDi − rI)
−E
∂U(Wt
)∂ωi
E [rCGi + rDi − rI ]
(5.57)
− Cov
∂U(Wt
)∂ωi
, rCGi + rDi − rI
= E
[∂U(Wt)∂ωi
]E [rCGi + rDi − rI ] . (5.58)
By virtue of Stein's Lemma we can reformulate the negative of the covariance. Stein's
Lemma is a linearization result for covariances under the assumption of normal distri-
bution when one argument is a (possible) non-linear function of a normally distributed
variable. Stein's Lemma equates the covariance of a function of normal random variable
to the underlying covariance times the expectation number of the derivative function.
The gains from using Stein's Lemma lies in the separation of the utility risk components
from the covariances and hence, to derive tractable, interpretable results (Gron (2004),
Söderlind (2006a) and Söderlind (2006b)).191
− Cov
∂U(Wt
)∂ωi
, (rCGi + rDi − rI)
= −E
∂2U(Wt
)∂ωi∂Wt
Cov [Wt, rCGi + rDi
](5.59)
We replace the left hand side of equation (5.58) by the right hand side of equation (5.59)
where we applied Stein's Lemma.
E[rCGi + rDi − rI ]E
∂U(Wt
)∂ωi
= −E
∂2U(Wt
)∂ωi∂Wt
Cov [Wt, rCGi + rDi
]
191For proof of Stein's Lemma, s. appendix B.
93
5 Tax International Capital Asset Pricing Model
and de�ne the risk factor.192
A = −E
[∂2U(Wt)∂ωi∂Wt
]E
[∂U(Wt)∂ωi
] (5.60)
with A expected coe�cient of world risk aversion
We can derive the pricing relationship for the investors in individual optimum.
E[rCGi + rDi ] = rI + ACov[rCGi + rDi , Wt
](5.61)
The above equation describes uniquely investor's expected rate of return for each asset.
It says that the investor's expected rate of return of an asset depends on two components
in individual optimum: The risk-free rate of return and the risk premium.
The individual optimum of the Tax-IntCAPM di�ers from the IntCAPM only in respect
of use of real returns after tax.
The real risk-free rate of return after tax represents investor's price of time or the rate of
return that is required for delaying consumption (Lengwiler (2006) and Elton (2007)).
The risk premium expresses investor's marginal rate of substitution between expected
excess return and return variance. The investor expects a risk premium since he is risk
averse. The risk premium depends on the covariance of wealth and the real capital gains
and dividend rate of return after tax of the asset. Asset i's contribution to the risk of the
portfolio is expressed in the form of the covariance. The variance and hence the risk of
192According to Balvers (2001) the term A is similar to the coe�cient of world absolute risk aversionwhich expresses the subjective degree of risk aversion. The term is not equal due to the expectationsthat are taken. The coe�cient expresses investor's risk tolerance and serves as a measure to plausibleutility functions. For a description of the concept of absolute risk aversion, s. Pratt (1964), Zeckhauserand Keeler (1970), Arrow (1970), Huang and Litzenberger (1988) and Kruschwitz (2007).
94
5 Tax International Capital Asset Pricing Model
asset i is irrelevant, asset i's e�ect on the variance of the portfolio is decisive and in�uences
if the variance of the portfolio decreases or increases (Elton (2007)). If the real capital
gains and dividend rate of return after tax of the asset is positively correlated with end
of period wealth, the expected rate of return has to be higher than the risk-free rate of
return to compensate for the risk. Adversely, the expected rate of return of the asset has
to be lower than the risk-free rate, if the rate of return is negatively correlated with end
of period real wealth after tax. This can be interpreted, that the investor accepts a lower
rate of return for assets whose rate of return is negatively correlated with end of period
wealth, since these assets have high pay-o�s when the economy is in a relatively poor
state (Elton (2007)).
For the domestic investor we can express the following individual optimum relationship.
E[rDCGi + rDDi
]− rDI = AD Cov
[WDt , r
DCGi
+ rDDi
](5.62)
We consider solely the covariance term of the domestic investor. We decompose the co-
variance term by application of the linearity property.
Cov[WDt , r
DCGi
]+ Cov
[WDt , r
DDi
](5.63)
We consider the �rst covariance term. By replacing investor's terminal real wealth after
taxes by the de�nition of the portfolio return (eq. (5.52)), the following equation can be
derived.
Cov
[I∑i=1
ωDi(1 + rDCGi + rDDi
)+ ωD0
(1 + rDI
), rDCGi
](5.64)
Since the covariance with the risk-free asset is zero, we can derive the following term.
Cov
[I∑i=1
ωDi(1 + rDCGi + rDDi
), rDCGi
](5.65)
We decompose the covariance of the domestic investor by application of the linearity prop-
95
5 Tax International Capital Asset Pricing Model
erty:
Cov
[I∑i=1
ωDi rDCGi
, rDCGi
]+ Cov
[I∑i=1
ωDi rDDi, rDCGi
]. (5.66)
The �rst covariance describes the value weighted real capital gains rate of return with the
real capital gains rate of return after tax, the other one the value weighted real dividend
rate of return with the real capital gains rate of return after tax. By considering domes-
tic investor's nominal capital gains and dividend rate of return, we can separate the tax
system and in�ation rate from the risky real rate of return from the covariance.
Cov
[I∑i=1
rCGi,nωDi
1− tDCG1 + πD
, rCGi,n1− tDCG1 + πD
]
+Cov
[I∑i=1
rDi,nωDi
1− tDD1 + πD
, rCGi,n1− tDCG1 + πD
] (5.67)
Equivalently, we derive the covariance term with the dividend rate of return (See ap-
pendix C.3.1).
Cov
[I∑i=1
rCGi,nωDi
1− tDCG1 + πD
, rDi,n1− tDD1 + πD
]
+Cov
[I∑i=1
rDi,nωDi
1− tDD1 + πD
, rDi,n1− tDD1 + πD
] (5.68)
We aggregate the tax system and separate in�ation rates in order to derive an expression
depending solely on the nominal capital gains and dividend return. The left hand side of
96
5 Tax International Capital Asset Pricing Model
equation 5.62 and the above terms lead to domestic investor's individual optimum.(1 + πD
)2AD
(E[rDCGi + rDDi
]− rDI
)= Cov
[I∑i=1
rCGi,nωDi
(1− tDCG
)2, rCGi,n
]
+Cov
[I∑i=1
rDi,nωDi
(1− tDD
) (1− tDCG
), rCGi,n
]
Cov
[I∑i=1
rCGi,nωDi
(1− tDCG
) (1− tDD
), rDi,n
]
+Cov
[I∑i=1
rDi,nωDi
(1− tDD
)2, rDi,n
]
(5.69)
After deriving the pricing relationship term of the domestic investor, we derive analogously
the equilibrium pricing relationship for the foreign investor (See appendix C.3.2).
(1 + πF
)2AF
(E[rFCGi + rFDi ]− r
FI
)= Cov
[I∑i=1
rCGi,nωFi
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2, rCGi,n
]
+Cov
[I∑i=1
rDi,nωFi
(1− tFeff,D + e
(1− tFeff,D,e
))(1− tFeff,CG + e
(1− tFeff,CG,P,e
)), rCGi,n
]Cov
[I∑i=1
rCGi,nωFi
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))(1− tFeff,D + e
(1− tFeff,D,e
)), rDi,n
]+Cov
[I∑i=1
rDi,nωFi
(1− tFeff,D + e
(1− tFeff,D,e
))2, rDi,n
]
(5.70)
97
5 Tax International Capital Asset Pricing Model
5.4.2 Market Equilibrium
The capital market is in a state of equilibrium what implies that market participants
maximize expected utility - to achieve individual optimum - and that the endogenous price
process for all risky assets is assumed to develop such a way that an exogenous quantity
of each asset equals the aggregate demand for the asset (Nielsen (1990)). By aggregation
we develop an equilibrium relationship which is e�ective for the domestic as well as for
the foreign investor. According to the �rst welfare theorem the equilibrium allocation
leads principally to Pareto e�ciency and solves a speci�c Social Welfare Function, where
the individual's weights are determined by their shadow prices of wealth (competitive
Social Welfare Function) (Lengwiler (2006)).193 The concept of Walras' law is behind
the idea of equilibrium, Walras' law is associated with the concept of Walrasian auction.
The Walrasian auction is a form of auction where each participant calculates his demand
for the good at every possible price and submits it to an auctioneer. Then a price is
determined through tatonnement process equalizing the entire demand and supply across
all agents of the good. According to the concept of Walras it is su�cient to regard the
equilibrium of risky assets. An equilibrium of risky assets will lead to an equilibrium of
the riskless asset (Kruschwitz and Lö�er (2009)).
According to equation (5.49) the demanded numbers for asset i can be expressed by the
following equations.
nDi = ωDiPD0
Vi,0nFi = ωFi
PD0
Vi,0(5.71)
The aggregate demanded number of each risky asset has to correspond to the exogenously
given number (Brennan (1970), Kruschwitz (2007) and Albrecht and Maurer (2008)).
193According to the �rst welfare theorem competitive equilibrium (Walrasian equilibrium) leads to Paretoe�ciency what implies that no other allocation of (risky) assets would be preferred by the domesticand foreign investor. The shadow price is the change in expected utility of investor's individualoptimum by relaxing the constraints by one unit, cf. Stiglitz (1991), Sen (1993), Barr (2004) andKanbur (2008).
98
5 Tax International Capital Asset Pricing Model
nDi + nFi =
(ωDi + ωFi
)PD0
Vi,0= n0
i (5.72)(nDi + nFi
) Vi,0PD0
= ωDi + ωFi = n0i
Vi,0PD0
(5.73)
with n0i exogenous number of asset i
As long as investors can raise their expected utility by acquiring or selling assets, the
market is characterized by excess demand or supply. The price changes in such a way
that demand of both investors is equal to the supply of each asset and none of the in-
vestors is able to enhance expected utility. We reformulate equation (5.61) to express the
excess rate of return and sum the individual optimum of domestic and foreign investors.194
E[rDCGi + rDDi
]− rDI + E
[rFCGi + rFDi
]− rFI
=AD Cov[rDCGi + rDDi , W
Dt
]+ AF Cov
[rFCGi + rFDi , W
Ft
] (5.74)
Equations (5.72) and (5.74) express the market equilibrium in a general manner. The left
hand side of equation (5.74) contains the real excess rate of return of the domestic and
foreign investor. The right hand side is determined by the covariances of the investors
which are multiplied by the risk aversion parameter A. Based on the aggregation of
individual equilibrium we derive a pricing relationship which contains complex factors.
The market equilibrium depends on the expectations and risk preferences of investors.
We derive separately the international excess return and the world equity risk premium.
We de�ne the domestic and foreign risk aversion factor under in�ation.
ADπD =1 + πD
ADAFπF =
1 + πF
AF(5.75)
with ADπD domestic investor's risk aversion factor under in�ation
AFπF foreign investor's risk aversion factor under in�ation
194Equivalently, it is possible to aggregate based on the end of period values, cf. Brennan (1970), Wiese
(2004) and Wiese (2006a).
99
5 Tax International Capital Asset Pricing Model
and the world aggregate risk aversion factor under in�ation AWπ .
AWπ = ADπD + AFπF (5.76)
with AWπ world aggregate risk aversion factor under in�ation
After aggregation of the left hand side of the individual optimum of the domestic and for-
eign investor, we separate the national and international tax, in�ation and exchange rates
of the domestic and foreign investor. By de�ning the weighted tax factors, the weighted
foreign risk aversion factor and the foreign exchange gains tax factors we derive the excess
return of the Tax-IntCAPM. We de�ne the weighted tax factor on capital gains, dividend
and interest.
TCG =tDCGA
DπD + tFeff,CGA
FπF
AWπTD =
tDDADπD + tFeff,DA
FπF
AWπTI =
tDI ADπD + tFeff,IA
FπF
AWπ(5.77)
Under internationalization, the tax parameters re�ect the tax rates of all investors world-
wide. The value of the factors depends decisively on the features of international taxation
and the taxing right factors.195 If the domestic investor's tax rate is equal to foreign
investor's e�ective tax rate then the tax factors become tax rates.196 We de�ne the
weighted foreign risk aversion factor as
AFW =AFπF
AWπ. (5.78)
We de�ne the foreign tax factor on capital gains, dividend and interest exchange gains.
ET FCG =tFeff,CG,P,eA
FπF
AWπET FD =
tFeff,D,eAFπF
AWπET FI =
tFeff,I,P,eAFπF
AWπ(5.79)
195S. �gure 5.1.196Cf. Mai (2008).
100
5 Tax International Capital Asset Pricing Model
After some mathematical transformation the following formulation for the excess return
can be derived (See appendix C.2.1).
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
(5.80)
The excess return consists of an expression of capital gains, exchange gains on capital
gains and principal, dividend, exchange gains on dividend, interest and exchange gains
on interest and principal return.
The capital market is in a state of equilibrium. Both market participants maximize
expected utility to achieve individual optimum and an exogenous quantity of each asset
equals the aggregate demand for the asset. In order to derive the equilibrium pricing
equation further capital market relationships are to be explained. We de�ne the world
market equity portfolio in the absence of tax, in�ation and appreciation of exchange rate
(Ibbotson and Sinque�eld (1976), Lally (1996)). The nominal value of the world market
portfolio at the beginning of the period is given by
M0 ≡I∑i=1
n0iVi,0. (5.81)
with M0 nominal value of the world market portfolio at the beginning of period
The world market portfolio comprises the entire risky securities. The nominal value of
the world market equity portfolio at the end of period is the sum of capital gains and
dividend return of all risky assets.
101
5 Tax International Capital Asset Pricing Model
M ≡I∑i=1
n0i
(CGi + Di
)(5.82)
with M nominal value of the world market equity portfolio at the end of period
Based on equation (5.81), our de�nition of the world market equity portfolio at the be-
ginning of the period and equation (5.82) and our de�nition of the world market equity
portfolio at the end of period, we de�ne the nominal capital gains and dividend rate of
return of the world market equity portfolio.
rCGm,n + rDm,n ≡M
M0
− 1 =
∑Ii=1 n
0i
(CGi + Di
)∑I
i=1 n0iVi,0
− 1 (5.83)
with rCGm,n nominal world market capital gains rate of return
rDm,n nominal world market dividend rate of return
The idea consists in expressing the nominal rate of return of the world market equity
portfolio by the sum of dividend and capital gains rate of return. Hence, we reformulate
equation (5.83) by considering the de�nition of the nominal world market equity portfolio
(eq. (5.81)) and the nominal rate of return of risky assets (eq. (5.1)).
rCGm,n + rDm,n =
∑Ii=1 n
0i
(CGi + Di
)∑I
i=1 n0iVi,0
− 1
=I∑i=1
n0i
(CGi + Di
)n0iVi,0
− 1
n0iVi,0M0
=
I∑i=1
(rCGi,n + rDi,n)n0iVi,0M0
(5.84)
The term n0i Vi,0M0
expresses the share of each asset in the nominal world market equity
102
5 Tax International Capital Asset Pricing Model
portfolio. The world market equity portfolio rate of return is the sum of value weighted
capital gains and dividend rate of return of each risky asset.
After deriving the excess return, we derive the covariance terms in equilibrium. We extend
the right hand side of expression 5.69 with the market equity portfolio and consider that
the value weighted shares of the portfolio are made of the product of demanded number
and real price of the risky asset (eq. 5.50), so we have
M0
PD0
Cov
[I∑i=1
rCGi,nnDi
Vi,0M0
(1− tDCG
)2, rCGi,n
]︸ ︷︷ ︸
Domestic investor′s first capital gains covariance term
+M0
PD0
Cov
[I∑i=1
rDi,nnDi
Vi,0M0
(1− tDD
) (1− tDCG
), rCGi,n
]︸ ︷︷ ︸
Domestic investor′s second capital gains covariance term
.
(5.85)
The above covariances incorporate stochastic dividends and capital gains. Equivalently,
we can derive the following expressions for the covariance term with the dividend (eq. 5.69).
M0
PD0
Cov
[I∑i=1
rCGi,nnDi
Vi,0M0
(1− tDCG
) (1− tDD
), rDi,n
]︸ ︷︷ ︸
Domestic investor′s first dividend covariance term
+M0
PD0
Cov
[I∑i=1
rDi,nnDi
Vi,0M0
(1− tDD
)2, rDi,n
]︸ ︷︷ ︸Domestic investor′s second dividend covariance term
(5.86)
For the foreign investor we can equivalently derive the covariance of nominal capital gains
103
5 Tax International Capital Asset Pricing Model
and dividend rate of return (5.70). For the covariance with the capital gains, we have
M0
PD0
Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2, rCGi,n
]︸ ︷︷ ︸
Foreign investor′s first capital gains covariance term
+M0
PD0
Cov
[I∑i=1
rDi,nnFi
Vi,0M0
(1− tFeff,D + e
(1− tFeff,D,e
))(1− tFeff,CG + e
(1− tFeff,CG,P,e
)), rCGi,n
]︸ ︷︷ ︸Foreign investor′s second capital gains covariance term
(5.87)
whereas the covariance with dividend is as follows:
M0
PD0
Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))(1− tFeff,D + e
(1− tFeff,D,e
)), rDi,n
]︸ ︷︷ ︸Foreign investor′s first dividend covariance term
+M0
PD0
Cov
[I∑i=1
rDi,nnFi
Vi,0M0
(1− tFeff,D + e
(1− tFeff,D,e
))2, rDi,n
]︸ ︷︷ ︸
Foreign investor′s second dividend covariance term
.
(5.88)
We aggregate the �rst covariance term of capital gains of the domestic investor (5.85) and
of the foreign investor (5.87), so we have
M0
PD0
Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
(1− tDCG
)2, rCGi,n
]
+M0
PD0
Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2, rCGi,n
].
(5.89)
In contrast to the derivation of Wiese (2006b) it is not possible to aggregate domestic
and foreign investors' value weighted nominal capital gains rate of return in order to
derive an expression of capital gains world market equity return (Mai (2006a) and Mai
(2008)). By regarding the above equation we can state that it is not possible to derive
104
5 Tax International Capital Asset Pricing Model
an expression of
M0
PD0
Cov
[I∑i=1
rCGi,nnDi
Vi,0M0
(1− tDCG
)2, rCGi,n
]
+M0
PD0
Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2, rCGi,n
]6= MCov [rCGm,n, rCGi,n]X
(5.90)
since after separation of the tax and appreciation of exchange rate
(1− tDCG
)2 M0
PD0
Cov
[I∑i=1
rCGi,nnDi
Vi,0M0
, rCGi,n
]
+(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2 M0
PD0
Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
, rCGi,n
] (5.91)
we have an expression of
ACov[X, Z
]+B Cov
[Y , Z
](5.92)
which cannot be aggregated. The tax rate and the appreciation of the exchange rate as
well as the demanded numbers of risky asset are investor-speci�c and cannot be aggre-
gated in order to derive an expression of capital gains world market equity return. We
can derive the same result for the other terms. Nevertheless, it is possible to derive an
equilibrium pricing equation. We de�ne the aggregate nominal capital gains and dividend
rate of return for the four combinations. We de�ne the �rst aggregate capital gains rate
of return as follows:
rCG,CGt,aggr ≡
I∑i=1
rCGi,nnDi
Vi,0M0
(1− tDCG
)2+
I∑i=1
rCGi,nnFi
Vi,0M0
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2.
(5.93)
105
5 Tax International Capital Asset Pricing Model
We aggregate the �rst covariance term of capital gains of the domestic investor (5.85) and
of the foreign investor (5.87), so we have
M0
PD0
Cov
[I∑i=1
rCGi,nnDi
Vi,0M0
(1− tDCG
)2, rCGi,n
]
+M0
PD0
Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2, rCGi,n
] (5.94)
and sum them to one covariance
M0
P0
Cov
[I∑i=1
rCGi,nnDi
Vi,0M0
(1− tDCG
)2+
I∑i=1
rCGi,nnFi
Vi,0M0
((1− tFeff,CG
)+ e
(1− tFeff,CG,P,e
))2, rCGi,n
].
(5.95)
Having aggregated the �rst capital gains covariance term (eq. (5.93)), we insert the de-
�ned return into the above term.
M0
PD0
Cov[rCG,CGt,aggr , rCGi,n
](5.96)
Equivalently, we aggregate the other covariance terms and insert the de�ned returns into
the above terms (See appendix C.3.3). After that, we sum the covariances of nominal
capital gains and dividend rate of return.
M0
PD0
Cov[rCG,CGt,aggr , rCGi,n
]+M0
PD0
Cov[rCG,Dt,aggr , rCGi,n
]M0
PD0
Cov[rD,CGt,aggr , rDi,n
]+M0
PD0
Cov[rD,Dt,aggr, rDi,n
] (5.97)
We can sum the covariance terms, so we have
M0
PD0
Cov[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n
] M0
PD0
Cov[rCG,Dt,aggr + rD,Dt,aggr, rDi,n
]. (5.98)
106
5 Tax International Capital Asset Pricing Model
We put the derived formulations of the excess return (5.80) and the covariance terms (5.98)
together and consider that we divided the excess return by the world aggregate risk aver-
sion factor under in�ation AWπ (See appendix C.2.1) and hence, adapt the covariance
terms.
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
=M0
PD0 A
Wπ
(Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDi,n
]) (5.99)
In view of the fact that the covariance is taken with respect to the world market portfolio,
the world market portfolio with expected real rate of return E [rw,n] is taken as the bench-
mark in order to identify the world market risk (Serçu and Uppal (1995) and Fernandez
(2005)). CAPT prices every risky asset, so the world market equity portfolio can be
applied. This benchmark is perfectly correlated with the world market equity return. We
multiply the above equation with the share of each asset in the market portfolio n0kVk,0M0
.
We aggregate over all risky assets and recall that the world market equity portfolio rate
of return is the aggregated sum of value weighted capital and dividend rate of return of
each risky asset (eq. (5.107)) and consider the de�nition of the nominal value of the world
market equity portfolio at the beginning of period (eq. (5.81)) and that the aggregated
quotient∑I
i=1
n0kVk,0M0
equals unity. Finally, we eliminate the world aggregate risk aversion
factor under in�ation (S. appendix C.3.4). After these transformations we can derive the
Tax-IntCAPM.
107
5 Tax International Capital Asset Pricing Model
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
=(E[rCGm,n] (1− TCG) + E[rCGm,n]e
(AFW − ET FCG
)− eET FCG
+E[rDm,n] (1− TD) + E[rDm,n]e(AFW − ET FD
)− rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ rI,neET
FI,e
)Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDi,n
]Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGm,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDm,n
]
Under the framework of international taxation, deviation from Relative PPP and stochas-
tic dividends we developed the Tax International Capital Asset Pricing Model.
5.5 Interpretation
We derived the Tax-IntCAPM under the postulate of international tax and violation of
Relative PPP leading to heterogeneous equilibrium pricing relationships for international
market participants. The equilibrium depends on the preferences of the investors, since
investor-speci�c terminal wealth results from investor-speci�c tax, in�ation and appre-
ciation of exchange rates. Hence, the Tax-IntCAPM incorporates investor-speci�c ex-
pectations which lead to an equilibrium pricing equation depending on the preferences of
investors.197 The derived Tax-IntCAPM is a totally new model revealing the incorporation
of a fundamentally di�erent structure to the Tax-CAPM and the IntCAPM. We derived
a Tax-IntCAPM whose equilibrium is expressed by the above equation and the capital
market equilibrium pricing relationship that aggregated demand has to correspond to the
197The CAPM was derived for investor-speci�c preferences, cf. Lintner (1969) and Sharpe (1999). Theinvestor-speci�c CAPM is based on the assumption that investors estimate di�erently the returns, sothe expected excess returns are investor-speci�c.
108
5 Tax International Capital Asset Pricing Model
supply of risky assets.198 Nevertheless, the securities market line of the Tax-IntCAPM is
comparable to the concept of CAPM which says that the expected return depends on two
components: the risk-free rate of return and the risk premium.199
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+ E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)︸ ︷︷ ︸stock′s expected rate of return
= rI,n (1− TI) + rI,ne(AFW − ET FI
)− eET FI︸ ︷︷ ︸
risk−free rate of return(E [rCGm,n] (1− TCG) + E[rCG,m]e
(AFW − ET FCG
)− ET FCG
+ E[rDm,n] (1− TD) + E[rD,m]e(AFW − ET FD
)− rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
)︸ ︷︷ ︸worldmarket equity risk premium
Cov[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDi,n
]Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGm,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDm,n
]︸ ︷︷ ︸
β − factor
(5.100)
The above equation describes an investor's expected rate of return for each asset. Due
to the assumption of di�erential taxation of all income types in IntCAPT, the expected
return is decomposed into the expected return of capital gains, exchange gains on capital
gains, dividend and exchange gains on dividend.200 The investor's risk-free rate of return
represents the investor's price of time or the rate of return that is required for delaying
consumption. In contrast to Lally (1996) and as in Mai (2006a) and Mai (2008) due to
stochastic dividend the systematic risk is determined by a complex term incorporating the
stochastic relationship of dividend and capital gains (market) return. Furthermore the
198Due to mathematical reasons, s. eq. (5.90) to eq. (5.92) it was impossible to integrate the marketequilibrium theorem of CAPT so that the exogenous quantity of each asset has to equal the aggregatedemand for the asset into the asset pricing formula.
199S. also investors' individual optimum condition (eq. (5.61)).200Since capital gains and dividend are di�erently taxed, dividend policy proves not to be irrelevant
(Hamada and Scholes (1985) and Wiese (2006a)). For empirical evidence, cf. Holmen (2008).
109
5 Tax International Capital Asset Pricing Model
beta factor is dependent on tax and exchange rates. It is impossible to separate the tax
and appreciation of exchange rate from the covariances of the beta-factor, leading to the
conclusion that stochastic capital gains and dividends in the case of di�erential taxation
of dividend and capital gains lead to a beta factor whose covariances are in�uenced by
tax and exchange rates.201
The concept of asset pricing can be explained by comparing assets with equal mean
payo�s. Those assets which pay o� most when ex-post wealth is highest are , of course,
the assets that co-vary strongly with the market. However high ex-post wealth means low
marginal utility. Thus, those assets pay o� most when the payo� is least useful (and least
when the payo� is most useful). Those assets are considered riskier. The relationship
between beta and its expected rate of return is linear. The higher beta is for an asset,
the higher is supposed to be its equilibrium rate of return. The investor is rewarded for
bearing systematic risk that cannot be diversi�ed away, whereas the unsystematic risk is
not priced, since it can be averaged away in a diversi�ed portfolio (Elton (2007)). The
term
E[rCG,m] (1− TCG) + E[rCG,m]e(AFW − ET FCG
)− eET FCG
+E[rD,m] (1− TD) + E[rD,m]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
(5.101)
is the risk premium of the world equity market, the excess rate of return of the world
market equity portfolio. For the risk the investor takes, he is compensated by the risk
premium of the world equity market per unit of risk. E[rCG,m] and E[rD,m] are expected
to be smaller than their counterparts in the national Tax-CAPM, since the world market
implies greater diversi�cation and hence lower portfolio variance (Lally (1996)).
Equivalently to the Tax-IntCAPM of Lally (1996), the excess return and the world market
equity risk premium are adapted by factors but they adopt a more complicated structure
on account of the features of international taxation. The excess rate of return and the risk
premium are adapted by the weighted tax factor on capital gains, dividend and interest,
the foreign risk aversion and the foreign tax factor on capital gains, dividend and interest
exchange gains. In contrast to Lally (1996) di�erential in�ation rates are considered
201S. eq. 5.90 to eq. 5.92 (Mai (2008)).
110
5 Tax International Capital Asset Pricing Model
and are integrated into the risk aversion factors. The domestic tax rates are weighted by
the domestic investor's risk aversion factor under in�ation and the foreign tax rates are
weighted by the foreign investor's risk aversion factor under in�ation. The factors can be
interpreted as the equilibrium market value of a unit of domestic and foreign average tax
from security i.
The exchange rate is determined by the interaction of speculators and traders and is
characterized by a market approach in which the demand for foreign currency equals the
supply. The price process of the exchange rate is described by non-linearity and is not
determined by Relative PPP. Equivalently to the tax factors, the exchange gains rate
multiplied by the foreign risk aversion factor can be interpreted as the equilibrium value
of a unit exchange gains from security i.
In contrast to the Tax-IntCAPM of Lally (1996) it was found to be important to consider
the features of exchange gains taxation and the characteristic features of the exchange
gains tax system: The increase exchange gains taxes in dependence of the recognition,
character, nature, source and hedging.202 Foreign investors consider exchange gains tax-
ation of dividend, capital gains and interest in the pricing of international assets, which
leads to the integration of the foreign tax factor on the exchange gains of capital gains,
dividend and interest in the excess return and the market risk premium.
ET FCG =tFeff,CG,P,eA
FπF
AWπET FD =
tFeff,D,eAFπF
AWπET FI =
tFeff,I,P,eAFπF
AWπ(5.102)
The principal is subject to exchange gains taxation, so the capital gains (interest) rate of
return is adapted by the exchange gains factor on capital gains and interest.
The e�ect of exchange gains taxation into the Tax-IntCAPM cannot be limited by the
integration of the foreign tax factor on exchange gains. However, exchange gains taxation
also has an e�ect on the weighted tax factors, the foreign risk aversion factor and the
beta-factor. Exchange gains taxation is integrated into the weighted tax factors and the
foreign risk aversion factor via the utility function in the expected coe�cient of world risk
aversion. In the beta factor we can �nd exchange gains taxation in the aggregate return.
It was found to be impossible to separate exchange gains tax from these factors to analyze
202S. chapter 4.4.
111
5 Tax International Capital Asset Pricing Model
further the e�ects of exchange gains taxation, so we need to derive more simpli�ed models
which enable the separation of exchange gains e�ects to pursue further analysis.
5.5.1 Deterministic Dividends
We derived the Tax-IntCAPM under the assumption of stochastic dividend. The question
of how far the Tax-IntCAPM changes under the assumption of deterministic dividends is
raised. If we assume deterministic dividends, we can derive the following equation for the
domestic and foreign investors' individual optimum.203
E [rCGi ] + rD = rI + ACov[Wt, rCGi
](5.103)
We consider domestic and foreign investors' individual optimum and separate the risk
aversion factor A from the covariance. In contrast to the Tax-IntCAPM under stochastic
dividend apart from the in�ation rates we can separate the tax and appreciation of ex-
change rate from the covariance term, so the sum over the domestic and foreign investor
leads to the following equation.204(1 + πD
)2AD (1− tDCG)
2
(E[rDCGi
]+ rDDi − r
DI
)+
(1 + πF
)2AF(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2 (E [rFCGi]+ rFDi − rFI
)=
K∑k=1
Cov [rCGi,n, rCGk,n]ωDk +
K∑k=1
Cov [rCGi,n, rCGk,n]ωFk
(5.104)
Having transformed the equation and de�ned the factors we can derive the excess return
under deterministic dividend. Compare appendix C.2 for the process of transformation
203Cf. Mai (2008).204 For mathematical purposes we rede�ne the index.
112
5 Tax International Capital Asset Pricing Model
and appendix C.2.2 for the de�nition of the factors.
E[rCGi,n] (1− TCG,dt) + E[rCGi,n]e(AFW,dt − ET FCG,dt
)− eET FCG,dt
+rDi,n (1− TD,dt) + rDi,ne(AFW,dt − ET FD,dt
)−rI,n (1− TI,dt)− rI,ne
(AFW,dt − TI,dt
)+ eET FI,dt
(5.105)
The excess return is similar to the one of the Tax-IntCAPM under stochastic dividend.
Apart from the deterministic dividends the Tax-IntCAPM di�ers in respect of the factors
which are weighted by di�erent risk aversion terms.
In contrast to the Tax-IntCAPM under stochastic dividend it is possible to derive an asset
pricing relationship under integration of the market equilibrium postulate that an exoge-
nous quantity of each asset equals the aggregate demand for the asset. We consider the
condition for market equilibrium that apart from individual equilibrium the endogenous
price process for all risky assets is assumed to develop in such a way that an exogenous
quantity of each asset equals the aggregate demand for the asset.205
(nDi + nFi
) Vi,0PD0
= ωDi + ωFi = n0k
Vi,0PD0
(5.106)
We de�ne the capital gains world market return (eq. 5.84)
rCGw,n ≡∑K
k=1 n0kCGk,n
M0
− 1 =K∑k=1
((n0kCGk,n
n0kVk,0
− 1
)n0kVk,0M0
)
=K∑k=1
rCGk,nn0kVk,0M0
(5.107)
and aggregate the covariance terms of the left hand side of (eq. 5.104).
= Cov
[rCGi,n,
K∑k=1
rCGk,n(ωDk + ωFk
)]. (5.108)
According to the concept of market equilibrium we equalize demand and supply by consid-
ering that the value weighted shares of the portfolio are made of the product of demanded
205S. eq. 5.72.
113
5 Tax International Capital Asset Pricing Model
number and real price of the risky asset i.
Cov
[rCGi,n,
K∑k=1
rCGk,n(ωDk + ωFk
)]= Cov
[rCGi,n,
K∑k=1
rCGk,nn0k
Vi,0P0
](5.109)
The expansion of the right hand side of the above equation by the de�nition of the world
market portfolio (eq. 5.81) and the consideration of the de�nition of the capital gains
world market return (eq. 5.107) leads to a covariance term incorporating solely the capi-
tal gains (world market) rate of return.
Cov [rCGi,n, rCGw,n]M0
PD0
(5.110)
The application of the world market capital gains portfolio leads to the following Tax-
IntCAPM under deterministic dividend.
E[rCGi,n] (1− TCG,dt) + E[rCGi,n]e(AFW,dt − ET FCG,dt
)− eET FCG,dt
+rDi,n (1− TD,dt) + rDi,ne(AFW,dt − ET FD,dt
)−rI,n (1− TI,dt)− rI,ne
(AFW,dt − ET FI,dt
)+ eET FI,dt
=(E[rCGm,n] (1− TCG,dt) + E[rCGm,n]e
(AFW,dt − ET FCG,dt
)− eET FCG,dt + rDi,n (1− TD,dt)
+rDi,ne(AFW,dt − ET FD,dt
)− rI,n (1− TI,dt)
−rI,ne(AFW − ET FI,dt
)+ eET FI,dt
) Cov [rCGi,n, rCGm,n]V ar [rCGm,n]
(5.111)
Apart from the fact that the return of the dividend is secure, the beta factor is independent
from tax and exchange rates and determined by the relationship of covariance of capital
gains of the asset, world market capital gains return and variance of world market capital
gains return.
114
5 Tax International Capital Asset Pricing Model
5.5.2 Portfolio Composition
The Tax-IntCAPM is derived from a model of how the investors construct their portfolios,
so the model itself has major implications for the construction of portfolios. Our model
is in�uenced by tax and deviations from Relative PPP, so, domestic and foreign investors
have a di�erent composition of portfolios on account of heterogeneous preferences (Long
(1977) and Singer (1979)). By recalling investors' individual optimum expected rate of
return for assets (eq. (5.61)) and replacing investors' terminal real wealth after taxes by
the de�nition of the portfolio (eq. (5.52)), an investor's optimal asset allocation can be
derived. For the domestic investor we have
1
AD(E[rDCGi ] + rDDi − r
DI
)=Cov
[I∑i=1
ωDi(1 + rDCGi + rDDi
)+ ωD0
(1 + rDI
), rDCGi
] (5.112)
and for the foreign investor the following equation is valid.
1
AF(E[rFCGi ] + rFDi − r
FI
)= Cov
[I∑i=1
ωFi(1 + rFCGi + rFDi
)+ ωF0
(1 + rFI
), rFCGi
](5.113)
We separate the tax, in�ation and appreciation of exchange rate from the excess return
and the covariance, which leads to the following equation for the domestic investor
K∑k=1
ωDi Cov [rCGi,n, rCGk,n]
=1 + πD
AD(1− tDCG
)2 (E [rCGi,n](1− tDCG,D
)+ rD,n
(1− tDD
)− rI,n
(1− tDI
)) (5.114)
and to the following equation for the foreign investor.
K∑k=1
ωFi Cov
[rCGi,n
, rCGk,n
]
=
(1 + πF
)AF
((1− tF
eff,CG
)+ e
(1− tF
eff,CG,P,e
))2 (E [rCGi,n
] (1− tFeff,CG
)+ E
[rCGi,n
]e(1− tFeff,CG,P,e
)
−etFeff,CG,P,e + rDi,n
(1− tFeff,D
)+ rDi,n
e(1− tFeff,D,e
)− rI,n
(1− tFeff,I
)− rI,ne
(1− tFeff,I,P,e
)+ et
Feff,I,P,e
)(5.115)
115
5 Tax International Capital Asset Pricing Model
If we assume the existence of one asset, the well-known Samuelson theorem can be derived,
that the optimal portfolio weight is separable in risk aversion and market price of risk
(Samuelson (1969), Merton (1973) and Gron (2004)). For the domestic investor we have
ωDi
=1 + πD
AD(1− tDCG
)2V ar [rCGi,n]
(E [rCGi,n]
(1− tDCG
)+ rDi,n
(1− tDD
)− rI,n
(1− tDI
)) (5.116)
whereas for the foreign investor we have:
ωFi
=
(1 + πF
)AF
((1− tF
eff,CG
)+ e
(1− tF
eff,CG,P,e
))2 (E [rCGi,n
] (1− tFeff,CG
)+ E
[rCGi,n
]e(1− tFeff,CG,P,e
)
−etFeff,CG,P,e + rDi,n
(1− tFeff,D
)+ rDi,n
e(1− tFeff,D,e
)− rI,n
(1− tFeff,I
)− rI,ne
(1− tFeff,I,P,e
)+ et
Feff,I,P,e
).
(5.117)
The risk aversion coe�cient is a scaling constant, consequently an investor's relative hold-
ing of the risky assets is independent from the utility function (Brennan (1970)).
Every investor compiles his optimal portfolio. Based on the separation theorem of Tobin
the optimal portfolio consists of the risk-free asset and the tangential portfolio. On
account of investor-speci�c tax, in�ation and appreciation of exchange rate, the tangential
portfolios are investor-speci�c (Long (1977), Wiese (2006b) and Mai (2008)). The
tangential portfolio of the investors is unequal to the market portfolio (Sharpe (1999)).
116
6 Tax International Capital Asset
Pricing Model under Restrictions
�Things which restrict the common are to be interpreted rigidly�
Latin Proverb
Taxes are largely a source of embarrassment to �nancial economists, since taxes are re-
garded as signi�cant, but the equilibrium e�ect of taxes on international asset pricing
is unknown. We derived the Tax-IntCAPM in the framework of equilibrium. It is un-
certain whether this equilibrium exists due to the existence of arbitrage. The exclusion
of arbitrage is the central theorem in CAPT. Tax rate di�erentials across income classes
in an international framework can generate arbitrage (McDonald (2001)). In order to
build a relationship between an arbitrage-free international capital market we analyze
constellations leading to the exclusion of global arbitrage.
In models where investors and securities are subject to di�erential taxation in an inter-
national framework, there may be no set of prices that rule out in�nite gains to trade -
international tax arbitrage -. So the question arises
What are the implications of excluding global arbitrage opportunities in the
Tax-IntCAPM?
We discuss the implications of the exclusion of tax arbitrage. Having characterized the
restrictions that preclude international tax arbitrage, we develop the Tax-IntCAPM under
short sale and borrowing restriction concluding with an interpretation of the model.
117
6 Tax International Capital Asset Pricing Model under Restrictions
6.1 Tax Arbitrage
The Tax-IntCAPM is founded on the central theorem of equilibrium, but it is uncertain
whether this equilibrium exists due to the existence of tax arbitrage. Tax arbitrage is
found where an arbitrage portfolio after tax leads to a semi-positive pay-o� (arbitrage op-
portunity type 1) or where a positive pay-o� leads to a non-negative repayment (arbitrage
opportunity type 2) (Raab (1993)).206
The di�erential tax treatment of the domestic and foreign investor can lead to varying
marginal rates of substitution between investors. Hence, the central assumption of equilib-
rium is not satis�ed, since investors can realize unlimited gains through the substitution
of assets. The assumption of short-sale and borrowing restrictions limits these gains (local
arbitrage) and satis�es the central theorem of equilibrium despite varying marginal rates
of substitution.207 Schäfer (1982) and Mai (2008) among many others come to the
conclusion that in a model with investor-speci�c tax rates we have to include short sale
restrictions and borrowing restrictions in order to exclude unlimited arbitrage opportuni-
ties.208
6.2 Tax International Capital Asset Pricing Model
under Short Sale and Borrowing Restrictions
The Tax-IntCAPM was developed under the assumption that short sales are not excluded
and that investors are unlimited in their ability to borrow. We follow the model of Mai
(2008) and derive the Tax-IntCAPM under short sale restriction and the restrictions
206Semi-positive pay-o� implies a positive pay-o� in one state.207Cf. Raab (1993).208Other possibilities are the formation of an appropriate tax system and identical (marginal) taxation of
investors (Schäfer (1982) and Dammon and Green (1987)). Both variants are regarded as irrelevantsince taxes are assumed to be given and identical (marginal) taxation of investors is unrealistic (Wiese
(2006a)). We ignore the approach of Dammon and Green (1987) who derive the exclusion of taxarbitrage for incomplete markets without short-selling restriction, since incompleteness per se doesnot guarantee the absence of tax arbitrage when tax schedules are su�ciently heterogeneous. For acritique of the approach of Dammon and Green (1987), cf. Ross (1987).
118
6 Tax International Capital Asset Pricing Model under Restrictions
that the total amount invested in risky assets is nonnegative and implement a borrowing
restriction.209
The decision problem of the maximization of the expected utility of end of period real
wealth after international tax is extended by the constraint that the total amount invested
in risky assets must be positive and that the investor is limited in his ability to borrow.
The optimality condition is solved under application of the Lagrange conception. After
transformation by application of the de�nition of the covariance and Stein's Lemma we
derive the individual optimum of the domestic and foreign investor. We aggregate do-
mestic and foreign investors' demand and de�ne the weighted tax factor on capital gains,
dividend and interest, the weighted foreign risk aversion factor and the foreign tax factor
on capital gains, dividend and interest exchange gains. We equalize an investor's aggre-
gated demand to the supply of risky assets and derive the equilibrium pricing relationship
of the Tax-IntCAPM under short sale and borrowing restrictions.210
6.2.1 Individual Optimum
The representative consumers maximize their expected utility of end of period real wealth
after tax and achieve individual optimum. The formulas are constructed in such a way
that they are valid for the domestic as well as for the foreign investor (S. chapter 5.4.1).
Max{ωi}Ii=0
E[U(Wt
)](6.1)
s.t.I∑i=0
ωi = 1 (6.2)
s.t. Wt =I∑i=1
ωi (1 + rCGi + rDi) + ω0 (1 + rI) (6.3)
In order to implement short sale and borrowing restrictions into the Tax-IntCAPM we
integrate further constraints into the framework.
209S. chapter 3.1.3.210S. chapter 5.4 for derivation.
119
6 Tax International Capital Asset Pricing Model under Restrictions
s.t.I∑i=1
ωi ≥ 0 (6.4)
s.t. ω0 ≥ −ω0,min (6.5)
The constraints 6.4 and 6.5 are the decisive di�erences to the prior models. The �rst
constraint 6.4 excludes short sales. The entire amount of money invested in risky stocks
is restricted from adopting a negative value; an investor's portfolio is restricted from short
sale.211 The second constraint 6.5 contains the borrowing restriction, incorporating that
the borrowed share shall not fall below the investor-speci�c share ω0,min which is assumed
to be negative (Auerbach (1983)).212 Hence, the foreign investor can maximally borrow
the investor-speci�c amount. The inequalities are transformed into equalities by the in-
troduction of the slack variables v1 and v2 (Litzenberger and Ramaswamy (1979), König
(1990), Wiese (2006b) and Mai (2008)).
I∑i=1
ωi ≥ 0I∑i=1
ωi − v1 = 0 (6.6)
ω0 ≥ −ω0,min ω0 − v2 = −ω0,min (6.7)
The variables are positive if the restrictions are binding; otherwise they are zero (Litzen-
berger and Ramaswamy (1979) and Mai (2008)). If v1 is zero, net entire investment in
risky stocks is equal to zero and initial wealth is totally invested into the risk-free bond.
If v2 is zero, the investor borrows exactly the investor-speci�c amount. In order to inte-
grate the value share of risky stocks as the only variable in our maximization problem we
remember that the share of the risk-free asset is the opposite of the sum of shares of risky
assets (S. chapter 5.4.1).
211Single assets are not restricted from short-sale, cf. Auerbach (1983).212Litzenberger and Ramaswamy (1979), König (1990) and Wiese (2006b) integrate two di�erent kinds
of borrowing restrictions into the Tax-CAPM. The �rst restriction incorporates that the borrowedamount shall not exceed a speci�c share invested in risky assets, the other one that the debt interestshall not exceed deterministic dividend. Since we assume stochastic dividend the second restrictionsreveals not to be plausible to be incorporated in our model (Mai (2008)).
120
6 Tax International Capital Asset Pricing Model under Restrictions
ω0 = 1−I∑i=1
ωi (6.8)
We integrate the above expression into constraint 6.7, so the reformulated constraint be-
comes
1−I∑i=1
ωi − v2 + ω0,min = 0. (6.9)
Apart from the constraints, that the value shares sum to unity (eq. 6.2) and that ter-
minal real wealth after tax is set up of the return after tax of the portfolio, consisting
of risky and risk-free assets (eq. 6.3), the short-sale and borrowing restrictions (eq. (6.6)
and eq. (6.9)) are integrated into the investor's maximization problem. We expand and
rearrange the optimization equation and formulate the Lagrange equation (Litzenberger
and Ramaswamy (1979), Auerbach (1983), König (1990), Wiese (2006b) and Mai
(2008)).
∂L
∂ωi= E
[U
((1 + rI) +
I∑i=1
ωi (rCGi + rDi − rI)
)]−λ1
(I∑i=1
ωi − v1
)
−λ2
(1−
I∑i=1
ωi − v2 + ω0,min
)(6.10)
The Lagrangean multipliers λ represent the shadow prices of the short-selling and borrow-
ing restrictions. The Lagrangean multiplier λ1 (λ2) is the shadow price for the regrouping
of the entire invested amount from risky (risk-free) assets to the risk-free (risky) asset,
(Mai (2008)). We di�erentiate partially in respect of the shares by application of the
chain rule and set the derivative equal to zero.
E
∂U((1 + rI) +
∑Ii=1 ωi (rCGi + rDi − rI)
)∂ωi
(rCGi + rDi − rI)
− λ1 + λ2 = 0
for iε 1..I.
(6.11)
121
6 Tax International Capital Asset Pricing Model under Restrictions
We replace the variable of the utility function by terminal real wealth after tax.
E
∂U(Wt
)∂ωi
(rCGi + rDi − rI)
− λ1 + λ2 = 0 (6.12)
The application of the de�nition of the covariance leads to the following expression.
E
∂U(Wt
)∂ωi
E [rCGi + rDi − rI ] + Cov
∂U(Wt
)∂ωi
, rCGi + rDi − rI
− λ1 + λ2 = 0 (6.13)
After application of Stein's Lemma we have
E
∂U(Wt
)∂ωi
E [rCGi + rDi − rI ] + E
∂U(Wt
)∂Wt∂ωi
Cov [Wt, rCGi + rDi
]− λ1 + λ2 = 0.(6.14)
We divide by the term E
[∂U(Wt)∂ωi
]and consider the de�nition of the expected coe�cient
of global risk aversion A = −E
[∂2U(Wt)∂ωi∂Wt
]
E
[∂U(Wt)∂ωi
] (S. eq. 5.60 in chapter 5.4.1), so we have the
following equation.
E [rCGi + rDi − rI ]− ACov[Wt, rCGi + rDi
]− λ1 + λ2
E
[∂U(Wt)∂ωi
] = 0(6.15)
We de�ne the term λ1+λ2
E
[∂U(Wt)∂ωi
] as SB - the short sale and borrowing restriction factor -
and insert it into the above equation.
122
6 Tax International Capital Asset Pricing Model under Restrictions
E [rCGi + rDi ] = rI + ACov[Wt, rCGi + rDi
]+ SB (6.16)
Equation 6.16 represents an investor's individual optimum under short sale and borrowing
restrictions. If the investor does not short sale his portfolio and the borrowed amount
is less than the borrowing restriction, the short sale and borrowing restriction factor is
equal to zero. If the investor could enhance utility through further short selling, we have
λ1 > 0, λ2 = 0 or through further borrowing, we have λ1 = 0, λ2 > 0 and the short sale
and borrowing restriction factor is positive.213 The multipliers can be interpreted as a
lowest extra dividend that, when added to the expected return, would have eliminated
the incentive to short sale the portfolio or to exceed the allowed borrowing share (Serçu
(2001)).
The domestic and foreign investors' individual optimum is reformulated to express the
excess returns. We separate the risk aversion factor A and the in�ation rates from the
covariance. We aggregate the tax system in order to derive an expression depending solely
on the nominal capital gains and dividend return by application of the linearity property
of the covariance, which leads to domestic investor's individual optimum (S. eq. (5.69)).(1 + πD
)2AD
(E[rDCGi + rDDi
]− rDI
)−(1 + πD
)2AD
SBD
= Cov
[I∑i=1
rCGi,nωDi
(1− tDCG
)2, rCGi,n
]
+Cov
[I∑i=1
rDi,nωDi
(1− tDD
) (1− tDCG
), rCGi,n
]
Cov
[I∑i=1
rCGi,nωDi
(1− tDCG
) (1− tDD
), rDi,n
]
+Cov
[I∑i=1
rDi,nωDi
(1− tDD
)2, rDi,n
]
(6.17)
213Cf. Mai (2008).
123
6 Tax International Capital Asset Pricing Model under Restrictions
Having derived the pricing relationship term of the domestic investor, we derive analo-
gously the equilibrium pricing relationship for the foreign investor (See appendix C.3.2
and eq (5.70)).
(1 + πF
)2AF
(E[rFCG,i + rFD,i]− rFI
)−(1 + πF
)2AF
SBF
= Cov
[I∑i=1
rCGi,nωFi
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2, rCGi,n
]
+Cov
[I∑i=1
rDi,nωFi
(1− tFeff,D + e
(1− tFeff,D,e
))(1− tFeff,CG + e
(1− tFeff,CG,P,e
)), rCGi,n
]Cov
[I∑i=1
rCGi,nωFi
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))(1− tFeff,D + e
(1− tFeff,D,e
)), rDi,n
]+Cov
[I∑i=1
rDi,nωFi
(1− tFeff,D + e
(1− tFeff,D,e
))2, rDi,n
]
(6.18)
6.2.2 Market Equilibrium
By aggregating domestic and foreign investors' demand in a state of individual optimum
and equalization to the supply of risky assets, the state of equilibrium of the capital mar-
ket is expressed. Equivalently to the derivation of the Tax-IntCAPM, we �rst consider
the left hand side and decompose the return of the asset to formulate an expression of
excess return comprising the return of capital gains, exchange gains on capital gains and
principal, dividend, exchange gains on dividend, interest and exchange gains on interest
and principal after international tax. After some mathematical transformation and the
de�nition of domestic and foreign risk aversion factor under in�ation, the world aggregate
risk aversion factor under in�ation, the weighted tax factor on capital gains, dividend and
interest, weighted foreign risk aversion factor and the foreign exchange gains tax factor
on capital gains, dividend and interest , the following formulation for the excess return
124
6 Tax International Capital Asset Pricing Model under Restrictions
can be derived (S. chapter 5.4.2 and appendix C.2.1).
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI − SBaggr
(6.19)
We de�ne the aggregate short sale and borrowing restriction factor as
(1+πD)2
ADSBD +
(1+πF )2
AFSBF
AWπ= SBaggr. (6.20)
Having derived the excess return, we consider the right hand side and can derive the
following formulation (See chapter 5.4.2).
M0
PD0 A
Wπ
(Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDi,n
])(6.21)
If we put the derived formulations of the left hand side 6.19 and right hand side 6.21
together, we have
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI − SBaggr
=M0
PD0 A
Wπ
(Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDi,n
]).
(6.22)
As CAPT prices every risky asset, the world market portfolio can be applied. This bench-
mark is perfectly correlated with the world market return. Equivalently to the derivation
of the Tax-IntCAPM, we multiply the above equation with the share of each asset in the
market portfolio n0kVk,0M0
and extend it with the market equity portfolio.
125
6 Tax International Capital Asset Pricing Model under Restrictions
E[rCGm,n] (1− TCG) + E[rCGm,n]e(AFW − ET FCG
)− eET FCG
+E[rDm,n] (1− TD) + E[rDm,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI − SBaggr
=M0
P0AWπ
(Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGm,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDm,n
]).
(6.23)
We divide equation 6.22 by equation 6.23, so we have the Tax-IntCAPM under short sale
and borrowing restrictions.
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI − SBaggr
=(E[rCGm,n]
(1− TDCG
)+ E[rCGm,n]e
(AFW − ET FCG
)− eET FCG
+E[rDm,n](1− TDD
)+ E[rDm,n]e
(AFW − ET FD
)− rI,n
(1− TDI
)− rI,ne
(AFW − ETDI
)+ eET FI − SBaggr
)Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCG,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDi,n
]Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGm,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDm,n
]
6.2.3 Interpretation
The derivation of the Tax-IntCAPM under short sale and borrowing restrictions leads to
the integration of the aggregate short sale and borrowing factor. The structure of the equi-
librium pricing relationship with adapted borrowing and portfolio short sale restrictions is
comparable to the one without restrictions, since it is solely adapted by the term SBaggr.
The factor is reduced from the excess return and can be interpreted as the world market
(extra) payment of short sale and borrowing restrictions. In contrast to the Tax-CAPM of
Mai (2008) the factor is in�uenced by international tax, in�ation and exchange rates.
126
6 Tax International Capital Asset Pricing Model under Restrictions
In the case of the Tax-IntCAPM under deterministic dividend the tax factors can be elim-
inated from the covariance term and the equilibrium pricing relationship is equivalently
adapted by the term SBaggr,dt which is de�ned as
SBaggr,dt =
(1+πD)2
AD(1−tDCG)2SBD +
(1+πF )2
AF (1−tFeff,CG+e(1−tFeff,CG,P,e))2SBF
AWdt. (6.24)
Finally we analyze constellations leading to the irrelevance of SBaggr = 0.214 This is the
case if the restrictions are binding for neither the domestic investor nor for the foreign
investor. Furthermore it could be possible that the restrictions are binding for the domes-
tic and foreign investor but neutralized in aggregate. This can be the case if the shadow
prices for short sale and borrowing are neutralized in aggregate.215
214Cf. Mai (2008).215Under the assumption that the investor's tangential portfolio is equal to the market portfolio, Mai
(2008) draws further conclusions from the Tax-CAPM under short sale and borrowing restrictions.We disregard this aspect, since Mai (2006a) and Mai (2008) admit that simplifying transformationsof the β-factor are required which are formally not proved, s. chapter 3.1.2.
127
7 Tax International Capital Asset
Pricing Model with Homogeneous
Expectations
�But in this world nothing can be said to be certain, except death and taxes.�
Benjamin Franklin, author, diplomat, inventor, physicist, politician and printer
(1706 � 1790)
The Tax-IntCAPM under di�erential taxation of stochastic capital gains and dividends
leads to a CAPM which is di�cult to implement on account of unknown risk parameters
and the complicated structure of the beta-factor.216 In order to enable the implementation
of the model, it is necessary to introduce an additional restriction. The model assumes
simplistically that there is solely a foreign representative investor. So the question arises
How is the Tax-IntCAPM a�ected under the assumption of a representative
foreign investor?
216Domestic and foreign investors' risk aversion factor under in�ation ADπD and AFπF are empirically notobservable, as for that purpose a simultaneous determination of in�ation rates and risk tolerancesof all market participants would be required. The beta-factor is determined by a complex termincorporating the stochastic relationship of dividend and capital gains (exchange gains) (market)return. It is dependent on national and international taxation and exchange rates as well as the valueweighted share of each asset and the world market portfolio in the beginning of the period,whichmakes empirical studies di�cult (Wiese (2006a), Wiese (2006a) and Dausend and Schmitt (2007)).
128
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
The solution of the model will allow us to determine the pricing relationship under mar-
ket equilibrium for international assets between expected return and risk. An intensive
interpretation and example elaborates the economic conceptions and implications.
7.1 Model Solution
The derivation of an investor's individual optimum forms the basis for the Tax-IntCAPM.
The foreign consumer maximizes his expected utility of end of period real wealth after
tax and hence, achieves individual optimum (S. eq. (5.61)).
E[rFCGi + rFDi
]= rFI + AF Cov
[rFCGi + rFDi , W
Ft
](7.1)
The capital market is in a state of equilibrium - the foreign investor maximizes expected
utility and an exogenous quantity of each asset equals the foreign investor's demand for
the asset (S. eq. (5.72)).217
nFi =ωFi P
F0
Vi,0= n0
k (7.2)
nFiVi,0P F0
= ωFi = n0k
Vk,0P F0
(7.3)
The world market capital gains and dividend rate of return can be expressed by the
following equation (S. eq. (5.84)).
rCGm,n + rDm,n =
∑Kk=1 n
0k
(CGk + Dk
)M0
− 1
=K∑k=1
n0k
(CGk + Dk
)n0kVk,0
− 1
n0kVk,0M0
=
K∑k=1
(rCGk,n + rDk,n)n0kVk,0M0
(7.4)
217For mathematical purposes we rede�ne the index, s. footnote 204.
129
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
In order to derive the pricing relationship under market equilibrium, equation (7.1) has
to be formulated in nominal terms and separated by the in�uences of tax, in�ation and
appreciation of exchange rate. By considering the foreign investor's real capital gains and
dividend rate of return (eq. (5.31)) and replacing the investor's terminal real wealth after
tax by the de�nition of the portfolio (eq. (5.52)), we can derive the equilibrium pricing
relationship for the foreign investor (See appendix C.3.2).
(1 + πF
)2AF
(E[rFCGi + rFDi
]− rFI
)= Cov
[I∑i=1
rCGi,nωFi
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2, rCGi,n
]
+Cov
[I∑i=1
rDi,nωFi
(1− tFeff,D + e
(1− tFeff,D,e
))(1− tFeff,CG + e
(1− tFeff,CG,P,e
)), rCGi,n
]Cov
[I∑i=1
rCGi,nωFi
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))(1− tFeff,D + e
(1− tFeff,D,e
)), rDi,n
]+Cov
[I∑i=1
rDi,nωFi
(1− tFeff,D + e
(1− tFeff,D,e
))2, rDi,n
].
(7.5)
We de�ne the terms
tseFCG ≡ tFeff,CG + e(1− tFeff,CG,P,e
)tseFD ≡ tFeff,D + e
(1− tFeff,D,e
)(7.6)
130
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
and separate the tax and exchange rates from the covariance terms.(1 + πF
)2AF
(E[rFCGi + rFDi ]− r
FI
)=
(1− tseFCG
)2Cov
[I∑i=1
rCGi,nωFi , rCGi,n
]
+(1− tseFD
) (1− tseFCG
)Cov
[I∑i=1
rDi,nωFi , rCGi,n
]
+(1− tseFD
) (1− tseFCG
)Cov
[I∑i=1
rCGi,nωFi , rDi,n
]
+(1− tseFD
)2Cov
[I∑i=1
rDi,nωFi , rDi,n
]
(7.7)
We consider the de�nition of the foreign investor's value weighted share and expand the
above equation by the de�nition of the world market portfolio (eq. (5.81)).
(1 + πF
)2AF
M0
M0
(E[rFCGi + rFDi
]− rFI
)=
(1− tseFCG
)2Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
, rCGi,n
]M0
PF0
+(1− tseFD
) (1− tseFCG
)Cov
[I∑i=1
rDi,nnFi
Vi,0M0
, rCGi,n
]M0
PF0
+(1− tseFD
) (1− tseFCG
)Cov
[I∑i=1
rCGi,nnFi
Vi,0M0
, rDi,n
]M0
PF0
+(1− tseFD
)2Cov
[I∑i=1
rDi,nnFi
Vi,0M0
, rDi,n
]M0
PF0
(7.8)
131
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
Market equilibrium implies that the demand equals an exogenous quantity of each asset
(eq. (7.2)). (1 + πF
)2AF
M0
M0
(E[rFCGi + rFDi
]− rFI
)=
(1− tseFCG
)2Cov
[K∑k=1
rCGi,nn0k
Vk,0M0
, rCGi,n
]M0
PF0
+(1− tseFD
) (1− tseFCG
)Cov
[K∑k=1
rDi,nn0k
Vk,0M0
, rCGi,n
]M0
PF0
+(1− tseFD
) (1− tseFCG
)Cov
[K∑k=1
rCGi,nn0k
Vk,0M0
, rDi,n
]M0
PF0
+(1− tseFD
)2Cov
[K∑k=1
rDi,nn0k
Vk,0M0
, rDi,n
]M0
PF0
(7.9)
We can insert the world capital gains and dividend rate of return (eq. (7.4)).(1 + πF
)2AF
(E[rFCGi + rFDi
]− rFI
)=
((1− tseFCG
)2Cov [rCGi,n, rCGm,n] +
(1− tseFD
) (1− tseFCG
)Cov [rCGi,n, rDm,n]
+(1− tseFD
) (1− tseFCG
)Cov [rDi,n, rCGm,n] +
(1− tseFD
)2Cov [rDi,n, rDm,n]
)M0
PF0
(7.10)
Equivalently, we separate the investor's tax, in�ation and appreciation of exchange rate
from the excess rate of return in order to install the nominal rate of return before tax
by considering the de�nition of foreign real asset rate of returns after tax (eq. (5.31) and
eq. (5.35)). We reduce the in�ation rate and sum the terms in the bracket, so we can
derive the following equation.
132
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
1 + πF
AF(E[rCGi,n]
(1− tFeff,CG
)+ E[rCGi,n]e
(1− tFeff,CG,P,e
)− etFeff,D,e
+ E[rDi,n](1− tFeff,D
)+ E[rDi,n]e
(1− tFeff,D,e
)−rI,n
(1− tFeff,I
)− rI,ne
(1− tFeff,I,P,e
))=
((1− tseFCG
)2Cov [rCGi,n, rCGm,n] +
(1− tseFD
) (1− tseFCG
)Cov [rCGi,n, rDm,n]
+(1− tseFD
) (1− tseFCG
)Cov [rDi,n, rCGm,n] +
(1− tseFD
)2Cov [rDi,n, rDm,n]
)M0
PF0
(7.11)
CAPT prices every risky asset allowing the world market portfolio to be applied. This
benchmark is perfectly correlated with the world market capital gains and dividend rate
of return. The expected capital gains and dividend rate of return on the world market
portfolio is endogenously determined, since they are weighted averages of the expected
rate of returns of the individual assets. We can prove the application of the world market
portfolio to equation (7.11) by multiplying with the share of each asset in the nominal
world market portfolio n0kVk,0M0
and aggregating over all risky assets (König (1990) and Mai
(2008)). We recall that the world market portfolio capital gains and dividend return rate
of is the sum of value weighted rate of return of each risky asset (eq. (7.4)) and consider
that the aggregated quotient∑K
k=1
n0kVk,0M0
equals unity. The elimination of the term 1+πF
AF
and the real value of the world market portfolio M0
PF0produces the following equilibrium
pricing relationship (S. appendix D).
133
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
E[rCGi,n](1− tFeff,CG
)+ E[rCGi,n]e
(1− tFeff,CG,P,e
)− etFeff,CG,P,e
+ E[rDi,n](1− tFeff,D
)+ E[rDi,n]e
(1− tFeff,D,e
)−rI,n
(1− tFeff,I
)− rI,ne
(1− tFeff,I,P,e
)+ etFeff,I,P,e
=(E[rCGm,n]
(1− tFeff,CG
)+ E[rCGm,n]e
(1− tFeff,CG,P,e
)− etFeff,CG,P,e
+E[rDm,n]((1− tFeff,D
)+ E[rDm,n]e
(1− tFeff,D,e
)− rI,n
(1− tFeff,I
)− rI,ne
(1− tFeff,I,P,e
)+ etFeff,I,P,e
)βhE
βhE
=
(1−tseFCG
)2Cov
[rCGi,n
,rCGm,n
]+(1−tseFD
)(1−tseFCG
)(Cov
[rCGi,n
,rDm,n
]+Cov
[rDi,n
,rCGm,n
])+(1−tseFD
)2Cov
[rDi,n
,rDm,n
](1−tseF
CG
)2V ar
[rCGm,n
]+2(1−tseF
D
)(1−tseF
CG
)Cov
[rDm,n,rCGm,n
]+(1−tseF
D
)2V ar
[rDm,n
]
Based on equilibrium on the capital market we developed the Tax International Capital
Asset Pricing Model with homogeneous expectations.218
7.2 Interpretation
The derived version of the Tax-IntCAPM is independent from investors' preferences, since
all investors reside in the foreign state and have homogeneous expectations. The investors
have homogenous expectations and hold the same shares of assets in their tangential port-
folio, the tangential portfolio is the market portfolio.219 In contrast to the Tax-IntCAPM,
the excess rate of return and the risk premium are not in�uenced by domestic and foreign
investors' tax factors but adapted to foreign investor's tax and exchange rates. Equiva-
lently to the Tax-IntCAPM, the βhE factor is in�uenced by tax and exchange rates which
is due to di�erential taxation of stochastic dividend and capital gains.220 Since the foreign
218The derived Tax-IntCAPM is plausible. By ignoring the domestic investor in the Tax-IntCAPM thetax factors turn to tax rates and the foreign risk aversion factor becomes one. The β-factor of theTax-IntCAPM is equal to βhE . By ignoring tax rates and the appreciation of nominal exchange rate,we have the nominal IntCAPM.
219S. chapter 5.3.220Cf. Mai (2008).
134
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
investor resides in the foreign state, in�ation is irrelevant. An empirical version of the
Tax-IntCAPM with homogenous expectations can be developed to test the explanatory
power of the Tax-IntCAPM.221
Figure 7.1 represents the Tax-IntCAPM with homogenous expectations.222 Any investor
determines the expected return of an asset by the nominal risk-free rate and a world market
risk premium, composed of the product of the beta-factor and the nominal expected return
on the world market portfolio in excess of the nominal risk-free rate.
The above securities market line represents the IntCAPM under exchange gains. The
expected return of the world market portfolio, which is composed of world market dividend,
world market exchange gains on dividend, world market capital gains and world market
exchange gains on capital gains before tax, is the sum of interest, exchange gains on
interest and world market risk premium before tax. The middle securities market line is
equivalent to the IntCAPM under PPP as derived in chapter 3.2.2. The lower securities
market line represents the Tax-IntCAPM with homogeneous expectations. The expected
return of the world market portfolio, which is composed of world market dividend, world
market exchange gains on dividend, world market capital gains and world market exchange
gains on capital gains after tax, is the sum of interest, exchange gains on interest and world
market risk premium after tax.
The e�ects of an increase in the e�ective exchange gains tax rate on interest and principal
depend on the exchange gains on interest and principal and on the beta-factor. If the
beta-factor is higher than unity, an increase in the e�ective exchange gains tax rate on
interest and principal will lead to a higher expected stock return and vice versa.223 The
e�ects of a change in the e�ective exchange gains tax rate on capital gains and principal
221Due to the complicated structure of the βhE , Mai (2006b) considers an equivalent version of theTax-CAPM as impracticable. Every variable of the Tax-IntCAPM is empirically observable, so thereare no objections to implementing and testing the derived empirical version of Tax-IntCAPM. Roll(1977) believes that the CAPM cannot be tested empirically at all due to the non-observability of themarket portfolio; nevertheless, countless empirical studies exist (Dausend and Schmitt (2007)). Thetesting of an empirical version of the Tax-IntCAPM faces certain obstacles: Dividends are distributedannually, so there is a certain tension between the amount of data, the prevalence of the tax systemand the nature of the stock.
222Cf. Institut Der Wirtschaftsprüfer In Deutschland E.V. (2008) for a comparable illustration.223The equivalent conclusion we can take for the e�ective exchange gains tax rate on capital gains and
principal and dividend for deterministic dividend.
135
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
0 1
-
6
Expected return
β
βhE
Interestafter tax
Interest tax
Interest
Exchange gainon interest
Exchange gain tax on interestExchange gain on interestafter tax
World market risk premium after tax
World market
risk premium
World market
dividend
World marketexchange gainon dividend
World marketcapital gain
World market
exchange gainon capital gain
Dividendafter tax
Exchange gain on dividend after tax
Capital gainafter tax
Exchange gain on capitalgain after tax
Dividend tax
Exchange gain tax on dividend
Capital gain tax
Exchange gain taxon capital gain
Capital gain tax,Exchange gain taxon capital gain,
Dividend tax,
Exchange gain
tax on dividend
Tax−
IntCAP
M
IntCAPMunder exchange gain
IntCAP
Munder PPP
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Figure 7.1: Tax-IntCAPM with Homogeneous Expectations
as well as on dividends are hardly interpretable since these tax rates are an integral part
of the beta-factor.
The Tax-IntCAPM with homogeneous expectations allows us to derive hypotheses for
136
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
international stock returns.224
Hypothesis 1 International taxation of dividend, capital and exchange gains and inter-
est
In international stocks the source state's tax and the residence state's tax incorporating
the methods of reducing and avoiding double taxation on dividend, capital and exchange
gains and interest are priced.
The tax-clientele theory suggests that higher (lower) tax-rate investors should, ceteris
paribus, concentrate their portfolios in tax-favored assets (Forbes and Beranek (1988)
and Collins and Murphy (1995)).
Hypothesis 2 International tax clientele
The di�erential taxation of dividend, capital and exchange gains as well as interest lead
to international tax clienteles.
Hypothesis 3 Exchange gains
In international stocks' the non-linear deterministic behavior of exchange gains is priced.
Hypothesis 4 Exchange gains taxation
In the pricing of international stocks exchange gains taxation can lead to home bias.
7.3 Implementation
Based on chapter 5.2.5 we implement the Tax-IntCAPM with homogeneous expectations.
The beta-factor βhE involves the estimation of the nominal world market capital gains and
dividend rate of return, the market shall comprise all equities in the world (Lally (1996)).
As the risky asset we consider agains the K+S AG stock and as the world market portfolio
224These hypotheses need to be tested in an empirical version of the Tax-IntCAPM with homogeneousexpectations.
137
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
we consider the Morgan Stanley Capital International All Country World Index (MSCI
ACWI).225 In November 2007 the K+S AG stock became part of the MSCI ACWI.226
Example 6 Tax-IntCAPM
The expected nominal world capital gains rate of return E [rCGm,n] is -16.06% whereas
the expected nominal world dividend rate of return E [rDm,n] is 0.5%. Equivalently to the
determination of the rate of return of the K+S AG stock we applied the logarithm and
annualized the return. As in Schmid and Trede (2006) we annualize the variances and
covariances. The calculation beta-factor of the Tax-IntCAPM with homogeneous expecta-
tions consists of several components, these are presented in the following table.
Components of βhE Value
tseFCG 0.0518
tseFD 0.0736
Cov[rCGK+S ,n, rCGm,n
]0.0479
Cov[rCGK+S ,n, rDm,n
]-0.0473
Cov[rDK+S ,n, rCGm,n
]-0.0048
Cov[rDK+S ,n, rDm,n
]-0.0046
Cov [rCGm,n, rDm,n] 0.0447
V ar [rCGm,n] 0.0457
V ar [rDm,n] 0.0478
Table 7.1: Calculation of Components of βhE
−0.0412 = (1−0.0518)20.0479+(1−0.0518)(1−0.0736)(−0.0473+(−0.0048))+(1−0.0736)2(−0.0046)(1−0.0518)20.0457+2(1−0.0518)(1−0.0736)0.0447+(1−0.0736)20.0478
The low beta-factor might be due to the low empirical data of dividend. Under determin-
istic dividend the beta-factor would be 1.0481.
225The MSCI ACWI is a market capitalization weighted index that is designed to measure the equitymarket performance of developed and emerging markets. As of May 2010, the index comprised45 country indices, including Germany and Russia. The index measures the market performance,including both price performance and income from (gross) dividend payments, s. www.mscibarra.
com/products/indices/equity/definitions. The MSCI ACWI FM comprises 71 countries, butthe starting day was November 30, 2007 and hence the index does not include enough data.
226S. �nancial report 2007 of the K+S AG.
138
7 Tax International Capital Asset Pricing Model with Homogeneous Expectations
E[rCGi,n] (1− 0.13) + E[rCGi,n](−0.0899) (1− 0.13)− (−0.0899)0.13
+ E[rDi,n] (1− 0.15) + E[rDi,n](−0.0899) (1− 0.15)
−0.0252 (1− 0.13)− 0.0252(−0.0899) (1− 0.13) + (−0.0899)0.13
= (−0.1606 (1− 0.13) + (−0.1606)(−0.0899) (1− 0.13)− (−0.0899)0.13
+0.005 (1− 0.15) + 0.005(−0.0899) (1− 0.15)
− 0.0252 (1− 0.13)− 0.0252(−0.0899) (1− 0.13) + (−0.0899)0.13) (−0.0412)
Foreign investor's expected nominal capital gains rate of return of the K+S stock amounts
to 1.54% if the expected nominal dividend rate of return is 1.77%. If we assume that the
nominal capital gains rate of return is 47.31%, than the expected dividend rate of return
would be -45.08%.
We compare the Tax-IntCAPM with homogenous expectations with its counterpart where
exchange gains are not subject to taxation (S. appendix D).227 Under the assumption of
irrelevance of exchange gains taxation the expected nominal capital gains rate of return
would be 1.55% if the nominal dividend rate of return is 1.77%. If we assume that the
nominal capital gains rate of return is 47.31%, than the expected dividend rate of return
would be -45.19% under irrelevance of exchange gains taxation.
The derivation of the Tax-IntCAPM gives the ball back to the empiricists, we need to con-
duct empirical investigations to study the explanatory power of this new Tax-IntCAPM
and its hypotheses.
227Cf. Lally (1996).
139
8 Critique
�Criticism is prejudice made plausible."
Henry Louis Mencken, US editor (1880 � 1956)
It is evident that certain concepts do not seem to apply in the real world. Certain
assumptions do not coincide with the conditions of reality, and as a consequence certain
phenomena in reality are unexplainable by the model. The assumptions underlying the
Tax-IntCAPM are very restrictive. The restrictions of unlimited short-sale and borrowing
have been lifted and integrated into the Tax-IntCAPM. Nevertheless, further concepts are
to be reviewed. In this critique, we address the answers to the questions
How far is reality distorted by making certain assumptions and to what kind
of conclusions do they lead?
On account of the fact that the author's aim is to integrate taxation into IntCAPT,
we regard critically explicitly those concepts that impact the framework of international
taxation. We study the distortion of the framework of international taxation of these
assumptions and work out the results of an approach to reality of these assumptions.
We used formal models which are based on de�nitions and simplifying assumptions deemed
appropriate. On this basis we used logical operations to draw conclusions whose validity
can be checked by an expert third person at any time. Assuming that we did not make
mistakes during the logical operations, our results can only be legitimately criticized
by referring to a possible use of inappropriate assumptions (Kruschwitz and Lö�er
(2010)).
140
8 Critique
The critique of our models can be categorized into four dimensions: Incomplete interna-
tional taxation, incomplete exchange rate modeling, missing risk factors and the question
of investor's rationality.
Assumption 6 implies that gains and losses are equally taxed. Clark (2007) derives a
CAPM with asymmetric taxation of gains and losses. In general, capital and exchange
gains and losses are asymmetrically taxed and tax systems incorporate limited loss o�set.
Gains can be taxed at statutory rates while losses can be carried back to obtain refunds
of taxes paid in prior years or carried forward to be o�set against future taxes payable
(Long (1977) and Shevlin (1990)). Such a conception would imply the derivation of
the Tax-IntCAPT to a multiperiod context. Fama (1970b) discusses conditions for the
development of the single period CAPM to a multiperiod context. Wiese (2006a) assumes
that the extension to a multiperiod framework leads to a CAPM where investors hedge
against the risks of insecure dividends.
In assumption 7 we excluded the existence of progressive tax rates. If the tax takes a
progressive curve further income in�uences the tax payments (Mai (2008)). Litzenberger
and Ramaswamy (1979), Litzenberger and Ramaswamy (1980), König (1990) andWiese
(2006b) integrated progressive taxes for deterministic income in Tax-CAPT while Lai
(1989) and Wang (1995) apply progressive tax rates for stochastic income in the frame-
work of the CCAPM.228 Progressive tax rates are implemented in many countries and
they furthermore reduce arbitrage opportunities.229 Progressive tax rates imply stochas-
tic tax rates, which is in the vein of tax policy resulting from tax reforms and di�erent
interpretations of tax laws by investors, �scal authorities and tax courts (Singer (1979),
Niemann (2004) and Niemann (2007)).
In assumption 17 and in chapter 5.2.3 we lay the foundation for the non-linear model of
exchange rates.
We assumed that the trade balance is determined by exchange rates (assumption 23).
Other factors such as the cost of production in the exporting economy in relation to those
in the importing economy and taxes or restrictions on trade can also in�uence the trade
balance (Ostry and Rose (1992) and Siebert (2006)). In order to assess the domestic costs
228As far as the critique of the CCAPM is concerned, s. chapter 3.2.229S. footnote 22.
141
8 Critique
of production in relation to those of the foreign state, the concept of real exchange rates
should be applied. However, the empirical evidence of the relationship between exchange
rates and trade balance remains mixed. While the view that devaluation improves the
trade balance is empirically supported by Gylfason and Risager (1984), Bahmani-Oskooee
(1985) and Shirvani and Wilbratte (1997), Rose and Yellen (1989) and Rose (1991)
concluded that there is no signi�cant relationship between the trade balance and the real
exchange rate. Ostry and Rose (1992) conclude that there is no clear conclusion about
the e�ect of a tari� change on the trade balance. However, empirical studies support the
argument that exchange rates are likely to have a strong in�uence on the trade balance
(Prasad and Gable (1997)).
The model requires the determination of several sensitivity parameters. These parameters
must be determined in an estimation model and are found to have a highly unstable impact
on the exchange rate in the case of a slight change.
Our models explain the expected returns only by a single variable � the risk of an asset
relative to the market. It is reasonable to assume that there exist other dimensions of risk
in the world �nancial market that are su�ciently systematic to receive a non-zero reward.
Bodnar (2003) among many others230 introduce with country, industry, political and
liquidity further dimensions of international risk. Dimensions of risk must be recognized
in such a way that there are traded securities capable of indicating the price of each
risk dimension. From an empirical point of view it seems to be necessary to perform
some audacious transposition, so that a particular venture can be priced, even if only
approximately.
We assumed the normal distribution for our model (assumption 4). The normal distribu-
tion is not descriptive for capital asset pricing because of limited liability: In the context
of the prevailing legal system, all stock holders are protected by limited liability so their
rates of return are bounded below by -100% and empirical distributions of returns have
tails that are leptokurtic (Balvers (2001)). Furthermore, many commonly used utility
functions are not de�ned at negative values and hence cannot be applied in the framework
of normal distribution (Balvers (2001)). The CAPM should apply to all asset classes.
However, under the assumption of normal return distributions the pricing of options can
230These include Abuaf (1997), Chordia (1998), Chordia (2001) and Acharya and Pedersen (2004).
142
8 Critique
be excluded, since option returns are non-normally distributed. However, the multivari-
ate normality is solely a su�cient and not a necessary condition for investors to choose
mean-variance e�cient portfolios (Balvers (2001)). Elliptical distributions including nor-
mal distributions better describe asset return behavior (Balvers (2001) and Hamada and
Valdez (2004)).231 Owen and Rabinovitch (1983) and Ingersoll (1987) integrate ellip-
tical distributions into the CAPM. Hamada and Valdez (2004) derive call option prices
when the underlying is elliptically distributed.
For the Tax-IntCAPM we pursued the EMH (assumption 12). The validity of the EMH
has been questioned by critics. Many �nancial economists and statisticians began to
believe that stock prices are at least partially predictable. The assumption of capital mar-
ket equilibrium is a central theorem in CAPT (assumption 14). Disequilibrium theory
provides a sound theoretical reason why price trends are possible as markets adjust to
information shocks. The main thrust of this theory is that prices do not quickly adjust
to information shocks and therefore markets are in short-run disequilibrium (Beja and
Goldman (1980)). The adjustment period to a new equilibrium is slowed by factors such
as transaction costs and the cost of acquiring and evaluating information (Lukac (1988)).
In the vein of the neoclassical �nance paradigm, we assume that investors are rational (as-
sumption 21). They should act in an unbiased fashion and make decisions by maximizing
their self-interests. In essence, the economic concept of rationality means that economic
agents make the best choices possible for themselves. Although appealing, this concept
entails strong and unrealistic assumptions about human behavior and the functioning of
�nancial markets. We assumed that economic agents process new information correctly
and make decisions that are normatively acceptable (Barberis and Thaler (2002)) and
that they are capable of integrating and considering many di�erent pieces of information
and must fully understand the future consequences of all their actions. Moreover, �nan-
cial markets must be frictionless, such that security prices re�ect their fundamental value
and the in�uence of irrational market participants is corrected by rational traders. Unfor-
tunately, we as human beings do not possess all of these capabilities and characteristics.
We fail to update beliefs correctly, have limitations in the processing of information, and
only have certain capabilities to solve complex problems (Baltussen (2009)).
231For a description of elliptical distributions, c. Hamada and Valdez (2004).
143
8 Critique
Behavioral economists attribute the imperfections in �nancial markets to a combination of
cognitive biases such as overcon�dence, overreaction, representative bias, information bias,
and various other predictable human errors in reasoning and information processing. The
new breed of economists emphasized psychological and behavioral elements of stock-price
determination, and came to believe that future stock prices are somewhat predictable on
the basis of past stock price patterns as well as certain �fundamental� valuation metrics.
Moreover, many of these economists were even making the far more controversial claim
that these predictable patterns enable investors to earn excess risk-adjusted rates of return
(Hawawini and Keim (1994) and Malkiel (2003)).
The �eld of behavioral �nance argues that behavior that di�ers from that of the rational
investor can cause prices in �nancial markets to deviate from their fundamental value. By
applying insights from behavioral sciences, the �eld of behavioral �nance tries to improve
our understanding of �nancial decisions and their a�ect on market prices.
Behavioral tax research addresses the topics of tax compliance, tax knowledge acquisi-
tion and transmission, tax professionals' judgement and decision making, and behavioral
changes induced by tax incentives (Outslay (1995) and Kirchler (2007)). The impact
of taxes on international asset pricing by use of experimental economics as initiated by
Rolfe and White (1997), Collins and Murphy (1995) and Anderson and Butler (1997)
opens a new �eld of tax research in �nance.
The thesis intends to outline and inspire future research on taxation in IntCAPT. When
one considers all these extension possibilities of the model, it is obvious that the model
analyzed here is an elementary one. It provided some insights into the e�ects of taxation on
international asset pricing under varying market structures and is expected to build up the
means of moving to a more general and thus realistic, setting for a better understanding of
taxation in international asset pricing, exchange rate modeling, the integration of further
risk components and market participants' behavior.
144
9 Conclusions
�People do not like to think. If one thinks, one must reach conclusions.�
Helen Keller, US blind and deaf educator (1880 � 1968)
The dissertation is inspired by the compelling theoretical and empirical arguments that
the bene�t of international diversi�cation is distorted due to the existence of home bias.
The delineation of national groups of investors by deviation from Relative PPP and inter-
national taxation is evidence of the international market puzzle of why domestic assets
can o�er a superior risk and return relationship to foreign assets. The research problem of
the dissertation was the pricing of international assets under barriers in the form of taxes.
We raised the research question �What are the Impacts of Taxation on the International
Capital Asset Pricing Model?� and hypothesized that the integration of international tax-
ation leads to new models of IntCAPT. By deriving and interpreting the Tax-IntCAPM
we gain new insights into the framework of taxation in IntCAPT.
By modeling the dividend, capital gains, interest and exchange gains tax system as well
as the methods of double taxation reduction in chapter 4, we presented a model for a
general integrated international tax framework which lays the foundation for the theoret-
ical exposition in order to derive the Tax-IntCAPM in chapter 5. The heterogeneity of
international market participants was elaborated on by integrating a tax system di�ering
between the domestic and foreign states and the entire income classes in IntCAPT (capi-
tal gains, dividends, interest and exchange gains), the concept of deviation from Relative
PPP and the integration of non-linear behavior of exchange rates.
145
9 Conclusions
We analyzed certain tax constellations in which the decision problem of the Tax-IntCAPM
would be equivalent to the one of the Tax-CAPM.
Through the integration of a non-linear deterministic exchange rate process, characterized
by a market approach in which demand for foreign currency equals the supply, we pursued
the new approach of Peters (1996) to regard the capital and currency markets as complex,
interdependent systems which are characterized by the coexistence of randomness and de-
termination. The equilibrium pricing relationship of the Tax-IntCAPM is in line with the
concept of IntCAPT, which says that the expected international return is composed of the
risk-free rate and a world risk premium. The equilibrium pricing relationship incorporates
the taxation of all income types in international asset pricing - dividends, interest, capital
and exchange gains - leading to factors which are the equilibrium market gains (costs) of
international taxes and deviation from Relative PPP. In contrast to Lally (1996), di�er-
ential in�ation rates are considered. The value of the factors depends decisively on the
features of international taxation and the assignment of the right of taxation.
In contrast to the Tax-IntCAPM of Lally (1996), it was found to be important to consider
the features of exchange gains taxation and the characteristic features of the exchange
gains and losses tax system: The raise of exchange gains taxes in dependence of the
recognition, character, nature, source and hedging. Exchange gains are di�erently taxed
from the other income types in IntCAPT which leads to the integration of the foreign tax
factor on exchange gains of capital gains, dividends and interest in the excess return.
In chapter 6 we realize that tax rate di�erentials across income classes in an interna-
tional framework can generate arbitrage. In order to exclude international unlimited
arbitrage opportunities we have to include short sale and borrowing restrictions into the
Tax-IntCAPM. The imposition of an amounted short sale and borrowing restriction leads
to a Tax-IntCAPM which is adapted by the aggregated short sale and borrowing factor,
which can be interpreted as the market value of short sale and borrowing restrictions and
in�uenced by international taxation and exchange gains.
We operationalize the Tax-IntCAPM in chapter 7 through the assumption of homogeneous
foreign market participants and the Tax-IntCAPM with homogeneous expectations which
is not in�uenced by domestic and foreign investors' tax and exchange gains factors but
adapted by foreign investor's tax and exchange gains rates.
146
9 Conclusions
The critique in chapter 8 presents the boundaries of the model in order to elaborate on
the phenomena of reality which are not able to be explained by the model. We come
to the conclusion that stochastic tax rates, an advanced modeling of exchange rates, the
integration of further risk components, and experiments studying market participants'
behavior are ideas which should be considered in new approaches of Tax-IntCAPM.
The research in this thesis has been conducted with the hope that it will shed light
on a deeper understanding of the integration of taxation in IntCAPM. In the light of the
preceding chapters, the dissertation has been successful in answering the research question
�What are the Impacts of Taxation on the International Capital Asset Pricing Model?� by
deriving and providing perspectives on novel IntCAPMs incorporating the framework of
international taxation.
147
A Taylor Series Approximation of
the Utility Function
The Taylor series is a series expansion of a function about a point; it is a representation
of a function as an in�nite sum of terms calculated from the values of its derivatives at a
single point. The value of the utility function can be approximated by terms related to the
expected value of the random input under the assumption that the random input is close
to its means. Thus, the expected utility of wealth can be presented as the desirability of
the expected return plus the rate of change in the utility returns times the variance of
returns plus some smaller size terms (Huang and Litzenberger (1988) and Kruschwitz
(2007)).
U(W)= U
(E[W])
+∂U(E[W])
∂W
(W − E
[W])
1!
+∂2U
(E[W])
∂W
(W − E
[W])2
2!
+∂3U
(E[W])
∂W
(W − E
[W])3
3!+ ...
(A.1)
The utility function U(W)is monotically increasing and concave in its argument W
(implying desirability, non-satiation and variance aversion) and the wealth W represents
stochastic end-of-period wealth with mean of E[W]. Taking expectations produces:
148
A Taylor Series Approximation of the Utility Function
E[U(W)]
= U(E[W])
+∂U(E[W])
∂W
E[(W − E
[W])]
1!
+∂2U
(E[W])
∂W
E
[(W − E
[W])2]
2!
+N∑n=3
∂nU(E[W])
∂W
E[(W − E
[W])n]
n!.
(A.2)
The above equation indicates a preference for expected wealth and an aversion to variance
of wealth for investors with increasing and strictly concave utility functions. We cannot
de�ne the expected utility over the expected value and the variance of wealth for arbitrary
distributions and preferences, since the last term of the above equation involves higher
order moments. We assume that the returns are multivariate normally distributed; the
normal distribution can be completely described by its mean and variance. Normal dis-
tribution also has the advantage of being stable; e.g. the return of a portfolio consisting
of the sum of multivariate normally distributed assets is also normally distributed. The
assumption of normal distribution of wealth leads to:
E
[(W − µ
)k]=
0 ,if k is impair
1 ∗ 3 ∗ 5... ∗ (k − 1)σ2 ,if k is pair.(A.3)
with µ expected value of normal distribution
σ variance of normal distribution
Due to this rule every impair row member is void. The row can be expressed by a term
of mean and variance.
E[U(W)]
= U (µ) +∂2U (µ)
∂W
E
[(W − µ
)2]2!
+∂4U (µ)
∂W
E
[(W − µ
)4]4!
+ ... (A.4)
149
A Taylor Series Approximation of the Utility Function
E[U(W)]
= U (µ) +∂2U (µ)
∂W
1 ∗ σ2
2!+∂4U (µ)
∂W
1 ∗ 3 ∗ σ4
4!+ ... (A.5)
Under the assumption of normal distribution of wealth the expected utility depends solely
on the parameters µ and σ irrespective of the type of utility function (Balvers (2001)
and Kruschwitz (2007)).
150
B Stein's Lemma
Stein's Lemma provides a linearization result for covariances of normally distributed vari-
ables of which one argument is a (possible) non-linear, di�erentiable function.232 It says
that the covariance of a normally distributed variable and the di�erentiable function of a
normally distributed variable is equal to the expected function of the normally distributed
variable and the covariance of the variables.
Cov[X, h
(Y)]
= E
∂h(Y)
∂Y
Cov [X, Y ] (B.1)
with X normally distributed variable
Y normally distributed variable
h(Y)
di�erentiable function
Ingersoll's proof of Stein's Lemma is given below. The standard decomposition for any
two random variables according to Ordinary Least Squares (OLS) regression is used:
X = α +Cov
[X, Y
]V ar
[Y] Y + ε. (B.2)
with α parameter
ε error variable
232For proof of Stein's Lemma, s. Ingersoll (1987), Balvers (2001). Rubinstein (1974) and Huang and
Litzenberger (1988) illustrate the use of Stein's Lemma for portfolio choice problems and Constan-
tinides (1989) provides a further discussion.
151
B Stein's Lemma
It is assumed that the expected error variable is equal to zero.
E [ε] = E[εY]= 0 (B.3)
We insert the standard composition into the left hand side of equation (B.1) which leads to
Cov
α +Cov
[X, Y
]V ar
[Y] Y + ε, h
(Y) . (B.4)
From the linearity property of covariances
Cov[aX1 + bX2, Y
]= aCov
[X1, Y
]+ bCov
[X2, Y
](B.5)
follows for normal distributions.
Cov[X, Y
]V ar
[Y] Cov
[Y , h
(Y)]
+ Cov[ε, h(Y)]. (B.6)
On account of the independence of ε and Y the covariance between these arguments is
zero, which simpli�es our equation to:
Cov[X, Y
]V ar
[Y] Cov
[Y , h
(Y)]. (B.7)
Given Cov[X, Y
]= E
[(X − µx
)(Y − µy
)]for a continuous distribution and since
we know, that E[(Y − µY
)E[h(Y)]]
is equal to zero, the covariance between Y and
h(Y)can be written as follows:
Cov[Y , h
(Y)]
=
∫ ∞−∞
(Y − µY
)h(Y)f(Y)dY (B.8)
f(Y)is a normal density function given as
152
B Stein's Lemma
f(Y)=
1√V ar
[Y]2π
e
([− (Y−µY )2
2V ar[Y ]
]). (B.9)
The di�erentiation of the normal density function yields
∂f(Y)
∂Y= − Y − µY
V ar[Y]f (Y ) . (B.10)
Substituting the above term into equation
Cov[Y , h
(Y)]
= −V ar[Y] ∫ ∞−∞
h(Y)df(Y). (B.11)
The right hand side of the above equation is integrated by parts.
Cov[Y , h
(Y)]
= −V ar[Y](∫ ∞
−∞h(Y)d(Y)+ h
(Y)d(Y)∫ ∞−∞
)(B.12)
The last term vanishes at both limits, since the normal density converges to zero at the
limits, as long as h(Y)does not go to in�nity too quickly; or, in other words, as long as
h(Y)= 0e[Y
2]. Thus, we have
Cov[X, h
(Y)]
=Cov
[X, Y
]V ar
[Y] Cov
[Y , h
(Y)]
= E
∂h(Y)
∂Y
Cov [X, Y ] . (B.13)
153
C Derivation of Tax International
Capital Asset Pricing Model
C.1 Non-liner Behavior of Exchange Rate
The value table of example 4 is given. It contains the time series of the non-linear behavior
of exchange rate with α = 5.
Day Exchange rate Day Exchange rate Day Exchange rate
1 0.8 11 0.2144 21 0.24752 1.4430 12 1.1406 22 1.42433 0.4281 13 0.7512 23 0.44154 2.0189 14 1.5556 24 2.02205 0.2084 15 0.3597 25 0.20786 1.0830 16 1.9375 365 0.21047 0.8450 17 0.22588 1.3387 18 1.24519 0.5121 19 0.610210 1.9894 20 1.8505
Table C.1: Value Table of Time Series of Exchange Rate
The graph of the non-linear behavior of an exchange rate with α equal to 4 is given. If the
parameter α is 4, the exchange rate adopts a totally di�erent path. The maximum value
is 2.2634, whereas the minimum value is 0.1377 for the �rst 25 days. The value table of
example C.1 is given. It contains the time series of the non-linear behavior of exchange
rate with α = 4. The return of the nominal exchange rate is -0.5935%.
154
C Derivation of Tax International Capital Asset Pricing Model
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2.0
-
6
Exchange rate
Day
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.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.
Figure C.1: Time Series of Exchange Rate with α = 4
Day Exchange rate Day Exchange rate Day Exchange rate
1 0.8 11 2.1935 21 1.70972 1.5247 12 0.1460 22 0.24533 0.3236 13 0.7963 23 1.92714 2.2634 14 1.5349 24 0.18875 0.1377 15 0.3182 25 1.39206 0.6799 16 2.2504 365 0.32527 1.8552 17 0.13928 0.2045 18 0.70019 1.5740 19 1.801110 0.2988 20 0.2181
Table C.2: Value Table of Time Series of Exchange Rate with α = 4
C.2 Expected Return
In the following, we derive the expected return of the Tax-IntCAPM. First of all, we
derive the expected return under stochastic dividend. The derivation of the expected
return under deterministic dividend is equivalent, so we present the di�ering factors.
155
C Derivation of Tax International Capital Asset Pricing Model
C.2.1 Stochastic Dividends
We consider solely the excess return of the individual optimum and sum them over the
domestic and foreign investors.(1 + πD
)2AD
E[rDCGi + rDDi
]− rDI +
(1 + πF
)2AF
E[rFCGi + rFDi
]− rFI (C.1)
We decompose the return of the asset to formulate the nominal return of capital gains,
dividend and interest. We consider the return of the risky and risk-free asset of the
domestic investor (eq. (5.2) and eq. (5.4)) and of the foreign investor (eq. (5.31) and
eq. (5.35)). The investor's rate of returns is inserted into the excess return and we reduce
the in�ation rate.
1 + πD
ADE[rCGi,n]
(1− tDCG
)+
1 + πD
AD+
1 + πD
ADE[rDi,n]
(1− tDD
)−1 + πD
ADrI,n
(1− tDI
)− 1 + πD
AD
+1 + πF
AFE[rCGi,n]
(1− tFeff,CG
)+
1 + πF
AFE[rCGi,n]e
(1− tFeff,CG,P,e
)+1 + πF
AF(1 + e)− 1 + πF
AFetFeff,CG,P,e
+1 + πF
AFE[rDi,n]
(1− tFeff,D
)+
1 + πF
AFE[rDi,n]e
(1− tFeff,D,e
)−1 + πF
AFrI,n
(1− tFeff,I
)− 1 + πF
AFrI,ne
(1− tFeff,I,P,e
)−1 + πF
AF(1 + e) +
1 + πF
AFetFeff,I,P,e
(C.2)
The excess return consists of an expression of expected capital gains, expected dividend,
exchange gains and risk-free return. Each income type is di�erently taxed. We de�ne the
domestic and foreign risk aversion factor under in�ation.
ADπD =1 + πD
ADAFπF =
1 + πF
AF(C.3)
with ADπD domestic investor's risk aversion factor under in�ation
AFπF foreign investor's risk aversion factor under in�ation
156
C Derivation of Tax International Capital Asset Pricing Model
Having de�ned the terms we insert them into the expression (C.2) and summarize it.
ADπDE[rCGi,n](1− tDCG
)+ ADπDE[rDi,n]
(1− tDD
)− ADπDrI,n
(1− tDI
)+AFπFE[rCGi,n]
(1− tFeff,CG
)+ AFπFE[rCGi,n]e
(1− tFeff,CG,P,e
)− AFπF et
FCG,e
+AFπFE[rDi,n](1− tFeff,D
)+ AFπFE[rDi,n]e
(1− tFeff,D,e
)−AFπF rI,n
(1− tFeff,I
)− AFπF rI,ne
(1− tFeff,I,P,e
)+ AFπF et
Feff,I,P,e
(C.4)
We multiply the bracket by the de�ned terms ADπD and AFπF , so we have
E[rCGi,n](ADπD − t
DCGA
DπD
)+ E[rDi,n]
(ADπD − t
DDA
DπD
)− rI,n
(ADπD − t
DI A
DπD
)+E[rCGi,n]
(AFπF − t
Feff,CGA
FπF
)+ E[rCGi,n]e
(AFπF − t
Feff,CG,P,eA
FπF
)− AFπF et
FCG,e
+E[rDi,n](AFπF − t
Feff,DA
FπF
)+ E[rDi,n]e
(AFπF − t
Feff,D,eA
FπF
)−rI,n
(AFπF − t
Feff,IA
FπF
)− rI,ne
(AFπF − t
Feff,I,P,eA
FπF
)+ AFπF et
Feff,I,P,e
(C.5)
We de�ne the world aggregate risk aversion factor under in�ation AWπ .
AWπ = ADπD + AFπF (C.6)
with AWπ world aggregate risk aversion factor under in�ation
and sum foreign and domestic investors' nominal returns of capital and exchange gains,
dividends and interest.
E[rCGi,n](AWπ − tDCGADπD − t
Feff,CGA
FπF
)+ E[rCGi,n]e
(AFπF − t
Feff,CG,P,eA
FπF
)−etFeff,CG,P,eAFπF + E[rDi,n]
(AWπ − tDDADπD − t
Feff,DA
FπF
)+E[rDi,n]e
(AFπF − t
Feff,CG,P,eA
FπF
)−rI,n
(AWπ − tDI ADπD − t
Feff,IA
FπF
)−rI,ne
(AFπF − t
Feff,I,P,eA
FπF
)+ etFeff,I,P,eA
FπF
(C.7)
157
C Derivation of Tax International Capital Asset Pricing Model
We divide by the world aggregate risk aversion factor under in�ation.
E[rCGi,n]
(1−
tDCGADπD
AWπ−tFeff,CGA
FπF
AWπ
)+ E[rCGi,n]e
(AFπF
AWπ−tFeff,CG,P,eA
FπF
AWπ
)
−etFeff,CG,P,eA
FπF
AWπ
+E[rDi,n]
(1−
tDDADπD
AWπ−tFeff,DA
FπF
AWπ
)+ E[rDi,n]e
(AFπF
AWπ−tFeff,D,eAWπ
)
−rI,n
(1−
tDI ADπD
AWπ−tFeff,IA
FπF
AWπ
)− rI,ne
(AFπF
AWπ−tFeff,I,P,eA
FπF
AWπ
)+ e
tFeff,I,P,eAFπF
AWπ
(C.8)
We de�ne the weighted tax factor on capital gains, dividends and interest.
TCG =tDCGA
DπD + tFeff,CGA
FπF
AWπTD =
tDDADπD + tFeff,DA
FπF
AWπTI =
tDI ADπD + tFeff,IA
FπF
AWπ(C.9)
We de�ne the weighted foreign risk aversion factor as
AFW =AFπF
AWπ. (C.10)
We de�ne the foreign tax factor on capital gains, dividends and interest exchange gains.
ET FCG =tFeff,CG,P,eA
FπF
AWπET FD =
tFeff,D,eAFπF
AWπET FI =
tFeff,I,P,eAFπF
AWπ(C.11)
Finally, we insert the de�ned factors into equation (C.8); the derived excess return for
the Tax-IntCAPM is:
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
(C.12)
158
C Derivation of Tax International Capital Asset Pricing Model
C.2.2 Deterministic Dividends
The derivation of the excess return is equivalent to the one of the Tax-IntCAPM under
stochastic dividend. Apart from the deterministic dividends; the Tax-IntCAPM under
deterministic dividend di�ers in respect of the factors which are weighted by di�erent risk
aversion terms.
We de�ne the domestic and foreign risk aversion terms as follows:
1 + πD
AD (1− tDCG)2 = ADπD,tDCG
1 + πF
AF((1− tFeff,CG
)+ e
(1− tFeff,CG,P,e
))2 = AFπF ,tseFCG. (C.13)
We de�ne the world aggregate risk aversion factor under deterministic dividend as
AWdt = ADπD,tDCG+ AFπF ,tseFCG
. (C.14)
We de�ne the average tax factor on capital gains, dividends and interest under determin-
istic dividend.
TCG,dt =tDCGA
DπD,tDCG
+ tFeff,CGAFπF ,tseFCG
AWdtTD,dt =
tDDADπD,tDCG
+ tFeff,DAFπF ,tseFCG
AWdt
TI,dt =tDI A
DπD,tDCG
+ tFeff,IAFπF ,tseFCG
AWdt
(C.15)
We de�ne the weighted foreign risk aversion factor under deterministic dividend as
AFW,dt =AFπF ,tseFCG
AWdt. (C.16)
We de�ne the foreign exchange gains tax factor on capital gains, dividends and interest
159
C Derivation of Tax International Capital Asset Pricing Model
ET FCG,dt =tFeff,CG,P,eA
FπF ,tseFCG
AWdtET FD,dt =
tFeff,D,eAFπF ,tseFCG
AWdt
ET FI,dt =tFeff,I,P,eA
FπF ,tseFCG
AWdt.
(C.17)
Finally, we derive excess return for the Tax-IntCAPM under deterministic dividend:
E[rCGi,n] (1− TCG,dt) + E[rCGi,n]e(AFW,dt − ET FCG,dt
)− eET FCG,dt
+rD,n (1− TD,dt) + rD,ne(AFW,dt − ET FD,dt
)−rI,n (1− TI,dt)− rI,ne
(AFW,dt − TI,dt
)+ eET FI,dt
(C.18)
C.3 Risk Premium
Having derived the excess return, we derive the covariance terms of the risk premiums.
First of all we regard only the covariance of the domestic investor. Having derived the
domestic investor's covariance of the risk premium, we derive analogously the foreign
investor's covariance of the risk premium. Finally, the aggregate covariance terms are
derived.
C.3.1 Domestic Investor
The covariance with the capital gains rate of return was derived in chapter 5.4.1. In the
following we derive the covariance with the dividend rate of return.
Cov[WDt , r
DD
](C.19)
By replacing the investor's terminal real wealth after taxes by the de�nition of the port-
folio, the following expression can be derived.
160
C Derivation of Tax International Capital Asset Pricing Model
Cov
[I∑i=1
ωDi(1 + rDCGi + rDDi
)+ ωD0
(1 + rDI
), rDD
](C.20)
Since the covariance with the risk-free asset is zero, we can derive the following term.
Cov
[I∑i=1
ωDi(1 + rDCGi + rDDi
), rDD
](C.21)
We apply the linearity property of covariances.
Cov
[I∑i=1
ωDi rDCG, r
DD
]+ Cov
[I∑i=1
ωDi rDDi, rDD
]. (C.22)
The �rst covariance describes the value weighted real capital gains rate of return with
the real dividend rate of return after national tax, and the other one the value weighted
real dividend rate of return with the real dividend rate of return after national tax. By
considering the de�nition of nominal capital gains and dividend rate of return (eq. (5.1)),
we can separate the tax and in�ation rate from the risky real returns of the covariance.
Cov
[I∑i=1
rCGi,nωDi
1− tDCG1 + πD
, rDi,n1− tDD1 + πD
]
+Cov
[I∑i=1
rDi,nωDi
1− tDD1 + πD
, rDi,n1− tDD1 + πD
] (C.23)
161
C Derivation of Tax International Capital Asset Pricing Model
Now, we can derive the domestic investor's individual optimum.(1 + πD
)2AD
(E[rDCGi + rDDi
]− rDI
)= Cov
[I∑i=1
rCGi,nωDi
(1− tDCG
)2, rCGi,n
]
+Cov
[I∑i=1
rDi,nωDi
(1− tDD
) (1− tDCG
), rCGi,n
]
Cov
[I∑i=1
rCGi,nωDi
(1− tDCG
) (1− tDD
), rDi,n
]
+Cov
[I∑i=1
rDi,nωDi
(1− tDD
)2, rDi,n
]
(C.24)
C.3.2 Foreign Investor
For the foreign investor we can express the following individual optimum relationship.
E[rFCGi + rFDi
]− rFI = AF Cov
[W Ft , r
FCGi
+ rFDi
](C.25)
Having derived the domestic investor's term in chapter 5.4.1 and in appendix C.3.1 we
derive analogously the foreign investor's covariance pricing relationship term. We decom-
pose the covariance term by application of the linearity property.
Cov[W Ft , r
FCGi
]+ Cov
[W Ft , r
FDi
](C.26)
We consider the �rst covariance term. By replacing the foreign investor's terminal real
wealth after tax by the de�nition of the portfolio, the following expression can be derived.
162
C Derivation of Tax International Capital Asset Pricing Model
Cov
[I∑i=1
ωFi(1 + rFCGi + rFDi
)+ ωF0
(1 + rFI
), rFCG
](C.27)
Since the covariance with the risk-free asset is zero, we can derive the following term
Cov
[I∑i=1
ωFi(1 + rFCGi + rFDi
), rFCG
](C.28)
and decompose the covariance of the foreign investor into two terms:
Cov
[I∑i=1
ωFi rFCG, r
FCG
]+ Cov
[I∑i=1
ωFi rFDi, rFCG
]. (C.29)
One covariance describes the value weighted real rate of capital gains return with the real
capital gains rate of return, and the other one the value weighted real dividend rate of
return with the real capital gains rate of return. By considering the foreign investor's
capital gains and dividend rate of return after tax (eq. (5.31)), we can separate the tax,
in�ation and appreciation of exchange rates from the real capital gains and dividend rate
of return of the covariance.
Cov
[I∑i=1
rCGi,nωFi
1− tFeff,CG + e(1− tFeff,CG,P,e
)1 + πF
,
rCGi,n1− tFeff,CG + e
(1− tFeff,CG,P,e
)1 + πF
]
+Cov
[I∑i=1
rDi,nωFi
1− tFeff,D + e(1− tFeff,D,e
)1 + πF
,
rCGi,n1− tFeff,CG + e
(1− tFeff,CG,P,e
)1 + πF
](C.30)
163
C Derivation of Tax International Capital Asset Pricing Model
The covariance terms incorporate the stochastic relationship of capital gains and divi-
dends. Having derived the covariance with the capital gains rate of return we consider
the covariance with the dividend rate of return.
Cov[W Ft , r
FDi
](C.31)
By replacing the foreign investor's terminal real wealth after tax with the de�nition of
the portfolio, the following expression can be derived.
Cov
[I∑i=1
ωFi(1 + rFCGi + rFDi
)+ ωF0
(1 + rFI
), rFD
](C.32)
Since the covariance with the risk-free asset is zero, we can derive the following term.
Cov
[I∑i=1
ωFi(1 + rFCGi + rFDi
), rFD
](C.33)
We apply the linearity property of covariances:
Cov
[I∑i=1
ωFi rFCG, r
FD
]+ Cov
[I∑i=1
ωFi rFDi, rFD
]. (C.34)
One covariance describes the value weighted real capital gains rate of return with the real
dividend rate of return, and the other one the value weighted real dividend rate of return
with the real dividend rate of return. By considering the foreign investor's real return
after tax (eq. (5.31)), we can separate the tax, in�ation and appreciation of the exchange
rate.
164
C Derivation of Tax International Capital Asset Pricing Model
Cov
I∑i=1
rCGi,nωFi
1− tFeff,CG + e(1− tFeff,CG,P,e
)1 + πF
,
rDi,n1− tFeff,D + e
(1− tFeff,D,e
)1 + πF
+Cov
I∑i=1
rDi,nωFi
1− tFeff,D + e(1− tFeff,D,e
)1 + πF
,
rFD,n
1− tFeff,D + e(1− tFeff,D,e
)1 + πF
(C.35)
By dividing by the foreign risk aversion factor and multiplying by the in�ation rate, the
left hand side of equation C.25 and sum of the terms C.30 and C.35 lead to the foreign
investor's individual optimum.
(1 + πF
)2AF
(E[rFCGi + rFDi ]− r
FI
)= Cov
[I∑i=1
rCGi,nωFi
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))2, rCGi,n
]
+Cov
[I∑i=1
rDi,nωFi
(1− tFeff,D + e
(1− tFeff,D,e
))(1− tFeff,CG + e
(1− tFeff,CG,P,e
)), rCGi,n
]Cov
[I∑i=1
rCGi,nωFi
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))(1− tFeff,D + e
(1− tFeff,D,e
)), rDi,n
]+Cov
[I∑i=1
rDi,nωFi
(1− tFeff,D + e
(1− tFeff,D,e
))2, rDi,n
]
(C.36)
165
C Derivation of Tax International Capital Asset Pricing Model
C.3.3 aggregate Covariance Term
We aggregate the other covariance terms. The aggregate covariance of foreign and domes-
tic investor's second capital gains covariance term is de�ned as follows (Eq. (5.85) and
(5.87)):
M0
PD0
Cov
[I∑i=1
rDi,nnDi
Vi,0M0
(1− tDD
) (1− tDCG
)+
I∑i=1
rDi,nnFi
Vi,0M0
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))(1− tFeff,D + e
(1− tFeff,D,e
)), rCGi,n
].
(C.37)
The de�nition for the second aggregate capital gains rate of return is:
rCG,Dt,aggr ≡
I∑i=1
rDi,nnDi
Vi,0M0
(1− tDD
) (1− tDCG
)+
I∑i=1
rDi,nnFi
Vi,0M0
(1− tFeff,CG + e
(1− tFeff,CG,P,e
))(1− tFeff,D + e
(1− tFeff,D,e
))(C.38)
so we have
M0
PD0
Cov[rCG,Dt,aggr , rCGi,n
]. (C.39)
The aggregation of the investors' �rst dividend covariance term leads to the following
expression (Eq. (5.86) and (5.88)).
166
C Derivation of Tax International Capital Asset Pricing Model
M0
PD0
Cov
[I∑i=1
rCGi,nnDi
Vi,0M0
(1− tDD
) (1− tDCG
)+
I∑i=1
rCGi,nnFi
Vi,0M0
((1− tFeff,CG
)+ e
(1− tFeff,CG,P,e
))((1− tFeff,D
)+ e
(1− tFeff,D,e
)), rDi,n
](C.40)
The de�nition for the �rst aggregate dividend rate of return is:
rD,CGt,aggr ≡
I∑i=1
rCGi,nnDi
Vi,0M0
(1− tDD
) (1− tDCG
)+
I∑i=1
rCGi,nnFi
Vi,0M0
((1− tFeff,D
)+ e
(1− tFeff,D,e
))((1− tFeff,CG
)+ e
(1− tFeff,CG,P,e
))(C.41)
what leads to the following covariance term.
M0
PD0
Cov[rD,CGt,aggr , rDi,n
]. (C.42)
We de�ne the second covariance term of the domestic and foreign investor as follows
(Eq. (5.86) and (5.88)):
M0
PD0
Cov
[I∑i=1
rDi,nnDi
Vi,0M0
(1− tDD
)2+
I∑i=1
rDi,nnFi
Vi,0M0
((1− tFeff,D
)+ e
(1− tFeff,D,e
))2, rDi,n
].
(C.43)
167
C Derivation of Tax International Capital Asset Pricing Model
The second aggregate dividend rate of return is de�ned as
rD,Dt,aggr ≡
I∑i=1
rD,nnDi
Vi,0M0
(1− tDD
)2+
I∑i=1
rD,nnFi
Vi,0M0
((1− tFeff,D
)+ e
(1− tFeff,D,e
))2.
(C.44)
so, our covariance term is
M0
PD0
Cov[rD,Dt,aggr, rDi,n
]. (C.45)
C.3.4 Pricing Relationship
We put the derived formulations of the excess return (5.80) and covariance terms (5.98)
together and consider that we divided the excess return by the world aggregate risk aver-
sion factor under in�ation AWπ (See appendix C.2.1).
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+ E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)− rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
=M0
PD0 A
Wπ
(Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rD,n
]) (C.46)
CAPT prices every risky asset, consequently the world market portfolio can be applied.
This benchmark is perfectly correlated with the world market return. Equivalently to the
procedure in the Tax-IntCAPM, we multiply the above equation with the share of each
asset in the market portfolio n0i Vi,0M0
.
168
C Derivation of Tax International Capital Asset Pricing Model
E
[rCGi,n
n0iVi,0M0
](1− TCG) + E
[rCGi,n
n0iVi,0M0
]e(AFW,πF − ETFCG
)− n0iVi,0
M0eETFCG
+ E
[rDi,n
n0iVi,0M0
](1− TD) + E
[rDi,n
n0iVi,0M0
]e(AFW,πF − ETFD
)− rI,n
n0iVi,0M0
(1− TI)− rI,nen0iVi,0M0
(AFW,πF − ETFI
)+n0iVi,0M0
eETFI,e
=M0
PD0 AWπ
(Cov
[rCG,CGt,aggr + rD,CGt,aggr, rCGi,n
n0iVi,0M0
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rD,n
n0iVi,0M0
])(C.47)
We aggregate over all risky assets which leads to:
E
[I∑i=1
rCGi,nn0i Vi,0
M0
](1− TCG) + E
[I∑i=1
rCGi,nn0i Vi,0
M0
]e(AFW,πF − ETFCG
)−
I∑i=1
n0i Vi,0
M0eETFCG
+ E
[I∑i=1
rDi,nn0i Vi,0
M0
](1− TD) + E
[I∑i=1
rDi,nn0i Vi,0
M0
]e(AFW,πF − ETFD
)
−I∑i=1
rI,nn0i Vi,0
M0(1− TI)−
I∑i=1
rI,nen0i Vi,0
M0
(AFW,πF − ETFI
)+
I∑i=1
n0i Vi,0
M0eETFI
=M0
PD0 AWπ
(Cov
[rCG,CGt,aggr + rD,CGt,aggr,
I∑i=1
rCGi,nn0i Vi,0
M0
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr,
I∑i=1
rD,nn0i Vi,0
M0
])(C.48)
We recall that the world market portfolio rate of return is the aggregate sum of value
weighted capital and dividend rate of return of each risky asset (eq. (5.84)) and consider
that the aggregate quotient∑I
i=1n0i Vi,0M0
equals unity. So we have
E[rCG,m] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+ E[rD,m] (1− TD) + E[rDi,n]e(AFW − ET FD
)− rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
=M0
P0AWπ
(Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGm,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDm,n
]).
(C.49)
We divide eq. (C.47) by eq. (C.49) to eliminate the world aggregate risk aversion factor
under in�ation. Under the framework of international taxation, deviation from Relative
PPP and stochastic dividends we developed the Tax International Capital Asset Pricing
169
C Derivation of Tax International Capital Asset Pricing Model
Model.
E[rCGi,n] (1− TCG) + E[rCGi,n]e(AFW − ET FCG
)− eET FCG
+E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD
)−rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
=(E[rCG,m] (1− TCG) + E[rCGi,n]e
(AFW − ET FCG
)− eET FCG
+E[rD,m] (1− TD) + E[rDi,n]e(AFW − ET FD
)− rI,n (1− TI)− rI,ne
(AFW − ET FI
)+ eET FI
)Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rD,n
]Cov
[rCG,CGt,aggr + rD,CGt,aggr , rCGm,n
]+ Cov
[rCG,Dt,aggr + rD,Dt,aggr, rDm,n
]
(C.50)
170
D Tax System International Capital
Asset Pricing Model with
Homogeneous Expectations
CAPT prices every risky asset, consequently the world market portfolio can be applied.
This benchmark is perfectly correlated with the world market capital gains and dividend
rate of return. The expected capital gains and dividend rate of return on the world market
portfolio is endogenously determined, since they are weighted averages of the expected
rate of returns of the individual assets. We can prove the application of the world market
portfolio to equation (7.11) by multiplying with the share of each asset in the nominal
world market portfolio n0kVk,0M0
.
1 + πF
AF
(E
[rCGk,n
n0kVk,0
M0
](1− tFeff,CG
)+E
[rCGk,n
n0kVk,0
M0
]e(1− tFeff,CG,e
)− etFeff,CG,P,e
+ E
[rDk,n
n0kVk,0
M0
](1− tFeff,D
)+ E
[rDk,n
n0kVk,0
M0
]e(1− tFeff,D,e
)− rI,n
n0kVk,0
M0
(1− tFeff,I
)− rI,n
n0kVk,0
M0e(1− tFeff,I,P,e
)+n0kVk,0
M0etFeff,I,P,e
)
=(1− tseFCG
)2Cov
[rCGk,n
n0kVk,0
M0, rCGm,n
]+(1− tseFD
)(1− tseFCG
)Cov
[rCGk,n
n0kVk,0
M0, rDm,n
]
+(1− tseFD
)(1− tseFCG
)Cov
[rDk,n
n0kVk,0
M0, rCGm,n
]+(1− tseFD
)2Cov
[rDk,n
n0kVk,0
M0, rDm,n
]
(D.1)
171
D Tax System International Capital Asset Pricing Model with Homogeneous
Expectations
and aggregating over all risky assets (König (1990) and Mai (2008)).
1 + πF
AF
(E
[K∑k=1
rCGk,nn0kVk,0
M0
](1− tFeff,CG
)
+E
[K∑k=1
rCGk,nn0kVk,0
M0
]e(1− tFeff,CG,e
)− etFeff,CG,P,e
+ E
[K∑k=1
rDk,nn0kVk,0
M0
](1− tFeff,D
)+ E
[K∑k=1
rDk,nn0kVk,0
M0
]e(1− tFeff,D,e
)
−K∑k=1
rI,nn0kVk,0
M0
(1− tFeff,I
)−
K∑k=1
rI,nn0kVk,0
M0e(1− tFeff,I,P,e
)+
K∑k=1
n0kVk,0
M0etFI,e
)
=
((1− tseFCG
)2Cov
[K∑k=1
rCG,nn0kVk,0
M0, rCGm,n
]
+(1− tseFD
)(1− tseFCG
)Cov
[K∑k=1
rCG,nn0kVk,0
M0, rDm,n
]
+(1− tseFD
)(1− tseFCG
)Cov
[K∑k=1
rD,nn0kVk,0
M0, rCGm,n
]
+(1− tseFD
)2Cov
[K∑k=1
rD,nn0kVk,0
M0, rDm,n
])M0
PF0
(D.2)
We recall that the world market portfolio capital gains and dividend rate of return is
the sum of value weighted rate of return of each risky asset (eq. (7.4)) and consider the
de�nition of the nominal value of the world market portfolio in the beginning of period
(eq. (5.81)) and that the aggregated quotient∑K
k=1
n0kVk,0M0
equals unity.
1 + πF
AF
(E[rCG,m
] (1− tFeff,CG,e
)+ E
[rCG,m
]e(1− tFeff,CG,e
)− etFeff,CG,P,e
+ E[rD,m
] (1− tFeff,D
)+ E
[rD,m
]e(1− tFeff,D,e
)− rI,n
(1− tFeff,I
)− rI,ne
(1− tFeff,I,P,e
)+ etFeff,I,P,e
)=
((1− tseFCG
)2V ar
[rCGm,n
]+(1− tseFD
)(1− tseFCG
)Cov
[rCGm,n, rDm,n
]+(1− tseFD
)(1− tseFCG
)Cov
[rDm,n, rCGm,n
]+(1− tseFD
)2V ar
[rDm,n, rDm,n
]) M0
PF0
(D.3)
Dividing equation (7.11) by equation (D.3) to eliminate the term 1+πF
AWand the real value
172
D Tax System International Capital Asset Pricing Model with Homogeneous
Expectations
of the world market portfolio MPF0
produces:
E [rCGi,n](1− tFeff,CG
)+ E [rCGi,n] e
(1− tFeff,CG,e
)− etFeff,CG,P,e
+E [rDi,n](1− tFeff,D
)+ E [rDi,n] e
(1− tFeff,D,e
)−rI,n
(1− tFeff,I
)− rI,ne
(1− tFeff,I,P,e
)+ etFeff,I,P,e
=(E[rCGm,n]
(1− tFeff,CG
)+E [rCGm,n] e
(1− tFeff,CG,e
)− etFeff,CG,P,e
+E [rDm,n](1− tFeff,D
)+ E [rDm,n] e
(1− tFeff,D,e
)− rI,n
(1− tFeff,I
)− rI,ne
(1− tFeff,I,P,e
)+ etFeff,I,P,e
)βhE
(D.4)
βhE
=
(1−tseFCG
)2Cov
[rCGi,n
,rCGm,n
]+(1−tseFD
)(1−tseFCG
)(Cov
[rCGi,n
,rDm,n
]+Cov
[rDi,n
,rCGm,n
])+(1−tseFD
)2Cov
[rDi,n
,rDm,n
](1−tseF
CG
)2V ar
[rCGm,n
]+2(1−tseF
D
)(1−tseF
CG
)Cov
[rDm,n,rCGm,n
]+(1−tseF
D
)2V ar
[rDm,n
]
Based on equilibrium on the capital market we developed the Tax International Capital
Asset Pricing Model with homogeneous expectations. In order to quantify the e�ect of ex-
change gains taxation we regard the Tax-IntCAPM with homogeneous expectations where
exchange gains taxation is regarded as irrelevant. The Tax-IntCAPM under irrelevance
of exchange gains taxation is de�ned as follows:
E [rCGi,n](1− tFeff,CG
)+ E [rCGi,n] e+ E [rDi,n]
(1− tFeff,D
)+ E [rDi,n] e
−rI,n(1− tFeff,I
)− rI,ne
=(E[rCGm,n]
(1− tFeff,CG
)+ E [rCGm,n] e+ E [rDm,n]
(1− tFeff,D
)+ E [rDm,n] e
− rI,n(1− tFeff,I
)− rI,ne
)βhEexE .
(D.5)
βhEexE
=
(1−teFCG
)2Cov
[rCGi,n
,rCGm,n
]+(1−teFD
)(1−teFCG
)(Cov
[rCGi,n
,rDm,n
]+Cov
[rDi,n
,rCGm,n
])+(1−teFD
)2Cov
[rDi,n
,rDm,n
](1−teF
CG
)2V ar
[rCGm,n
]+2(1−teF
D
)(1−teF
CG
)Cov
[rDm,n,rCGm,n
]+(1−teF
D
)2V ar
[rDm,n
]
173
D Tax System International Capital Asset Pricing Model with Homogeneous
Expectations
We de�ne the terms.
teFCG ≡ tFeff,CG + e teFD ≡ tFeff,D + e. (D.6)
We implement the Tax-IntCAPM under homogeneous expectations and irrelevance of
exchange gains tax.
−0.0419 = (1−0.0401)20.0479+(1−0.0401)(1−0.0601)(−0.0473+(−0.0048))+(1−0.0601)2(−0.0046)(1−0.0401)20.0457+2(1−0.0401)(1−0.0601)0.0447+(1−0.0601)20.0478
E[rCGi,n] (1− 0.13) + E[rCGi,n](−0.0899)
+ E[rDi,n] (1− 0.15) + E[rDi,n](−0.0899)
−0.0252 (1− 0.13)− 0.0252(−0.0899)
= (−0.1606 (1− 0.13) + (−0.1606)(−0.0899)
+0.005 (1− 0.15) + 0.005(−0.0899)
− 0.0252 (1− 0.13)− 0.0252(−0.0899)) (−0.0419)
174
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