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Austrian Working Group on Banking and Finance (AWG) 23. Workshop (12. - 13. 12. 2008)

Vienna University of Technology

Mean-Variance Asset Pricing after Variable Taxes

Christian Fahrbach christian.fahrbach@web.de

Vienna University of Technology

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Contents

1 Initial question

2 Standard model (CAPM)

3 Model extension

4 CAPM after variable taxes

5 Conclusion

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1 Initial Question

Facing stagnation and high volatility on stock markets, the following question arises:

Are risky assets still attractive (competitive) compared with deposit and current accounts, call money, bonds and other assets with low risk?

If not,

is it possible to set up a favourable tax system, to stimulate risky investments and to stabilize financial markets?

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2 The Standard Model (CAPM)(Sharpe 1964, Lintner 1965)

(A1) There is a finite number of risky assets,short selling is allowed unlimitedly.

(A2) There is a riskless asset, which can be lent and borrowed unlimitedly.

Given: Er: n-vector of expected returns (n<∞),

Vr: covariance matrix,

rf: risk-free rate (non-stochastic).

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What are the mean-variance optimal portfolios under A1 and A2?

Optimization with Lagrange function (Merton 1972)

Solution: two half lines on the µ-σ-plane,

. 1) ..., 1, (1,

, ) r (E )(V ) r (E H

, (r) H r μ(r)

T

f1T

f

f

1

1rr1r -

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µ(r)

rf

σ(r)

Figure 2.1: The portfolio frontier under A1 and A2.

(r) H rf

(r) H rf

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Capital market equilibrium

Following Huang und Litzenberger (1988) investors will undertake risky investments if and only if

Ermvp > rf ,

Ermvp = A / C ,

A = 1T (Vr)-1 Er ,

C = 1T (Vr)-1 1 ,

rmvp: rate of return on the (global) minimum variance portfolio.

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Case 1: If Ermvp > rf , all investors buy portfolios on the capital market line

(i.e., a linear combination of the market portfolio and the riskless asset).

Case 2: If Ermvp ≤ rf , all investors put all their money into the riskless asset.

In this case, a market portfolio and therefore a pricing formula for risky assets according to the CAPM does not exist !

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µ(r)

market portfolio

minimum variance portfoliorf

σ(r)

Figure 2.2: The capital market line on the µ-σ-plane( Ermvp > rf ) .

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µ(r)

rf

minimum variance portfolio

tangency portfolio σ(r)

Figure 2.3: The portfolio frontier for Ermvp < rf .

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Huang and Litzenberger (1988):

“Suppose that rf > A/C. Then no investor holds a strictly positive amount of the market portfolio. This is inconsistent with market clearing. Thus in equilibrium, it must be the case that rf < A/C and the risk premium of the market portfolio is strictly positive“.

Remark: Whether or not this condition is fulfilled on real markets is an empirical issue.

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Conclusion

● Equilibrium does not exist a priori.

● The location of the riskless rate compared with the hyperbolic portfolio frontier in the μ-σ-plane is decisive.

● The CAPM is not a general equilibrium model.

● Is it possible to deduce equilibrium solutions for asset pricing in case the Huang-Litzenberger condition (HLC) is not fulfilled?

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3 Model Extension

How to extend the model?

→ Keep the model as simple as possible,

→ make further assumptions which allow the deduction of general equilibrium solutions for asset pricing.

Assertion: It suffices to modify the assumptions about risk-free lending and its taxation !

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Further Assumptions

(A2*) There are several riskless assets

(deposit and current accounts, call money, etc.),

short-selling is not allowed

(i.e., restricted borrowing due to Black 1972).

Definition 3.1: All possible risk-free rates are defined on

rf Є [0, ro] , ro > 0 ,

ro : overnight rate.

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(A3) Riskless assets are flat taxed (endbesteuert).

(i.e., all investors face the same riskless rates after taxes).

(A4) Riskless assets are variably taxed.

Definition 3.2: Variable wealth tax on riskless assets,

o: = f(Er, Vr, ro, c1, c2, …) , o Є (0, 1) ,

c1, c2, … : constants.

Idea: o contains all relevant information to ensure equilibrium after taxes.

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How to define a wealth tax on riskless assets?

W1 = (1 + rf) Wo , rf Є [0, ro] ,

W1,at = (1 + rf,at) Wo = (1 – o) W1 , o Є (0, 1) ,

Wo : initial wealth,W1, W1,at : end of period wealth before and after taxes,o : wealth tax rate on riskless assets,rf , rf,at : risk-free rates before and after taxes,

↔ rf,at = (1 + rf) (1 – o) – 1 . (1)

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Why wealth tax ?

In the worst case, Ermvp = 0 ,

all riskless rates must be negative, rf,at < 0 rf,at ,

→ this can not be done with a yield tax according to current tax law but with a wealth tax, that is

→ only a wealth tax allows the deduction of general equilibrium solutions for asset pricing !

. r1

r ν

o

oo

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Characteristics of a wealth tax on riskless assets:

• riskless rates can become negative after taxes,

• interest-free riskless assets (cash, current accounts, call money etc.) are also taxed, that is

rf,at = – o , if rf = 0 , o Є (0, 1) .

→ the interest-free riskless rate is always negative after taxes.

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Tax allowance (Freibetrag):

Money (cash, current accounts, call money etc.), which is used for payment transactions remains untaxed

(as long as the deposited amount does not exceed two to three monthly salaries).

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4 CAPM after Variable Taxes

Equilibrium Theorem 4.1:

Under A1 – A4 the following assertions are equivalent:

(1) There exists a general capital market equilibrium.

(2) There is a value goЄ (-1, ro) with the following properties:

go = max rf,at and go < Ermvp .

(3) Asset pricing is independent of rf , rf Є [0, ro].

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Proof:

(1) ↔ (2) : In equilibrium, the HLC must be fulfilled after taxes: rf,at ≤ go< Ermvp .

(2) → (3) : By contradiction (here only for Ermvp > 0),

(a) assume go* = f(rf) , rf Є [0, ro] ,

(b) in equilibrium must be:

go* = max rf,at = a · Ermvp , a Є (0, 1) ,

↔ contradiction to (a), because rmvp is an exogenious market value

→ go ≠ f(rf) . “

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The hypothetical value go …

• guarantees general equilibrium,

• is independent of ro ,

• is not yet implemented in a real economy,

→ see proposition 4.1,

• is still unknown,

→ see proposition 4.2.

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Proposition 4.1: Under A1 - A4 and the tax rate

the following equation holds:

go = ro,avt = max rf,at ,

ro,avt :overnight rate after variable taxes .

Proof: Rearranging (2) gives go = (1+ro)(1–o)–1 = ro,at , which is identical with equation (1).

(2) , )r (-1, gfor r1

g r ν oo

o

ooo

-

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Proposition 4.2: Given an arbitrary portfolio “q“, which is efficient under A1 (without A2 or A2*), then

go ≤ Erz(q) ,

go ≤ Erq – Rra Var(rq) ,

Rra : aggregate relative risk aversion(see Huang and Litzenberger 1988),

rz(q): rate of return on the correspondingzero covariance portfolio,

provide under A1 – A4 necessary and sufficient conditions for equilibrium.

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Zero-Beta-CAPM after variable taxes (Black 1972):

Choose a portfolio “q“ on the upper branch of the hyperbolic frontier, then

Erj = Erz(m) + βjm (Erm – Erz(m))

for Erz(m) ≥ go ,

if go = Erz(q) or

go = Erq – Rra Var(rq) ,

where rj : rate of return on asset “j“,

rm : anticipated market portfolio,

βjm : β-factor.

, r1

g r ν

o

ooo

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µ(r) hyperbolic frontier after taxes

anticipated market portfolioarbitrary portfolio “q” on the upper branch

go σ(r)

Figure 4.1: Anticipated equilibrium after variable taxes

( go = max rf,at = ro,avt = Erz(q) ).

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Remarks:

• The original portfolio “q“ is not efficient before taxes.

• The anticipated market portfolio is a convex combination of efficient portfolios on the hyperbolic frontier.

• The overnight rate before taxes ro is still relevant to calculate the variable tax rate, o = f(ro), but not in the CAPM after variable taxes, Erj ≠ f(ro).

• Asset pricing after variable taxes depends exclusively on capital market parameters.

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Asset pricing in practice:

If all investors combine risky and riskless assets,

Erj =ro,avt + βjm (Erm – ro,avt) ,

for ro,avt = Erz(index)

or ro,avt = Erindex – Rra ∙ Var(rindex) ,

where rindex : rate of return on a share index,

Rra : aggregate relative risk aversion.

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Interpreting o as control variable:

Because of ro,avt ≈ ro – o ,

→ Erj ≈ ro – o + βjm (Erm – ro + o) ,

→ if share prices rise, o is low and riskless assets are taxed moderately,

→ if share prices stagnate, o is high and riskless assets are taxed stronger, to give risky assets a chance to recover !

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5 Conclusion

• The variable tax has to be evaluated on the basis of current capital market data.

• How to tax bonds, if riskless assets are variably taxed?

• Option pricing after variable taxes (?)

• A variable tax on riskless assets can compensate for stagnation on stock markets:

→ While taxing riskless assets stronger, there is more scope for the firms to consolidate their profits and to attract potential investors.