Taxation and International Capital Asset Pricing Theory · Taxation and International Capital Asset...

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Model under

Restrictions

Chapter 7

Tax International


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


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


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


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


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


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


(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


3.2 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


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


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


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


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

www.destatis.de

www.cbr.ru/eng/analytics/macro/


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


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


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


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


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


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


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.

26


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


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


(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).

29


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


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.

31


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


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


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


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


(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


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


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


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


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


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


(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


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


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


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


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


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


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

www.kpmg.com/Global/en/IssuesAndInsights/ArticlesPublications/TIES/Pages/Taxation-of-international-executives-2010.aspx

www.kpmg.com/Global/en/IssuesAndInsights/ArticlesPublications/TIES/Pages/Taxation-of-international-executives-2010.aspx


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


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


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


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


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


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


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


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


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


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


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


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


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


5.2 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


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


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

www.oanda.com/convert/fxhistory


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


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


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


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


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


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


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


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


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


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


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




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


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



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


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

www.de.global-rates.com/zinssatze/eonia/2008.aspx


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

www.deutsche-boerse.de

www.de.global-rates.com/zinssatze/eonia/2008.aspx


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


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

www.cbr.ru/eng/currency_base/monthly.aspx


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


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


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


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


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


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


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


derivative equal to zero.188

∂E[U((1 + rI) +

∑Ii=1 ωi (rCGi + rDi − rI)

)]∂ωi

= 0

E

∂U((1 + 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


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


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


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


), rDCGi

](5.65)

We decompose the covariance of the domestic investor by application of the linearity prop-

95


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


] (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


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


5.4.2 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


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


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


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


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


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


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,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


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


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


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.


)− eET FCG



(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



)− eET FCG



(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


]+ Cov


]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


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


)− 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


(AFW − ET FI

)+ eET FI

)︸︷︷︸worldmarket equity risk premium

Cov[rCG,CGt,aggr + rD,CGt,aggr , rCGi,n

]+ Cov


]Cov


]+ Cov


]︸︷︷︸

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


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


(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


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.


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


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


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


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.


)− eET FCG,dt

+rDi,n (1− TD,dt) + rDi,ne(AFW,dt − ET FD,dt


(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


5.5.2 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


)+ 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


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


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


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


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


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) +


)∂ωi

(rCGi + rDi − rI)

− λ1 + λ2 = 0

for iε 1..I.

(6.11)

121


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


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


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,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


can be derived (S. chapter 5.4.2 and appendix C.2.1).


)− eET FCG



(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


]+ Cov


])(6.21)

If we put the derived formulations of the left hand side 6.19 and right hand side 6.21

together, we have


)− eET FCG



(AFW − ET FI


=M0

PD0 A

Wπ

(Cov


]+ Cov


]).

(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


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


(AFW − ET FI


=M0

P0AWπ

(Cov


]+ Cov


]).

(6.23)

We divide equation 6.22 by equation 6.23, so we have the Tax-IntCAPM under short sale

and borrowing restrictions.


)− eET FCG



(AFW − ET FI


=(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


)Cov

[rCG,CGt,aggr + rD,CGt,aggr , rCG,n

]+ Cov


]Cov


]+ Cov


]

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


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


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


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,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


and separate the tax and exchange rates from the covariance terms.(1 + πF

)2AF


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


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


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


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


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


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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....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

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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


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


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

www.mscibarra.com/products/indices/equity/definitions

www.mscibarra.com/products/indices/equity/definitions


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.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


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


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.


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:


)− eET FCG



(AFW − ET FI

)+ eET FI

(C.12)

158


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


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:


)− eET FCG,dt

+rD,n (1− TD,dt) + rD,ne(AFW,dt − ET FD,dt


(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


Cov

[I∑i=1


)+ ωD0

(1 + rDI

), rDD

](C.20)


Cov

[I∑i=1


), 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


]

+Cov

[I∑i=1

rDi,nωDi

1− tDD1 + πD


] (C.23)

161


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


Cov

[I∑i=1


)+ 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


), 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


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


)+ ωF0

(1 + rFI

), rFD

](C.32)


Cov

[I∑i=1


), 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


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 + πF

,

rFD,n


)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


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,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.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


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


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).


)− eET FCG

+ E[rDi,n] (1− TD) + E[rDi,n]e(AFW − ET FD


(AFW − ET FI

)+ eET FI

=M0

PD0 A

Wπ

(Cov


]+ 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


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


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


(AFW − ET FI

)+ eET FI

=M0

P0AWπ

(Cov


]+ Cov


]).

(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


Model.


)− eET FCG



(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


(AFW − ET FI

)+ eET FI

)Cov


]+ Cov


]Cov


]+ Cov


]

(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


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


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


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)

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