Bewertung von Wertpapieren in Mittelwert-Varianz-Modellen unter Ber

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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_at_web.de Vien
na University of Technology
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Contents

• 1 Initial question
• 2 Standard model (CAPM)
• 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
(nlt8), 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-s-plane,
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µ(r) rf s(r) Figure 2.1 The portfolio
frontier under A1 and A2.
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Capital market equilibrium Following Huang und
Litzenberger (1988) investors will undertake
risky investments if and only if Ermvp gt 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 gt 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
portfolio rf s(r) Figure 2.2 The capital
market line on the µ-s-plane ( Ermvp gt rf ) .
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µ(r) rf minimum variance
portfolio tangency portfolio
s(r) Figure 2.3 The portfolio frontier for
Ermvp lt rf .
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Huang and Litzenberger (1988) Suppose that rf
gt 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 lt 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-s-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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• 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 ? 0, ro , ro gt 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, no
f(Er, Vr, ro, c1, c2, ) , no ? (0, 1) , c1,
c2, constants. Idea no 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 ? 0, ro , W1,at (1
rf,at) Wo (1 no) W1 , no ? (0, 1) , Wo
initial wealth, W1, W1,at end of period
wealth before and after taxes, no wealth tax
rate on riskless assets, rf , rf,at risk-free
rates before and after taxes, ? rf,at (1
rf) (1 no) 1 . (1)
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Why wealth tax ? In the worst case, Ermvp 0 ,
all riskless rates must be negative, rf,at lt
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 !
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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 no , if rf 0
, no ? (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
• There exists a general capital market
  equilibrium.
• There is a value go? (-1, ro) with the
  following properties
• go max rf,at and go lt Ermvp .
• Asset pricing is independent of rf , rf ? 0,
  ro.

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• Proof
• ? (2) In equilibrium, the HLC must be
  fulfilled after taxes rf,at golt Ermvp .
• (2) ? (3) By contradiction (here only for Ermvp
  gt 0),
• (a) assume go f(rf) , rf ? 0, ro ,
• (b) in equilibrium must be
• go max rf,at a Ermvp , a ? (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
(1ro)(1no)1 ro,at , which is identical
with equation (1).
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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
corresponding zero 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)
ßjm (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, ßjm
ß-factor.
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µ(r) hyperbolic frontier after
taxes anticipated market portfolio arbitr
ary portfolio q on the upper branch go
s(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, no 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
ßjm (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 no as control variable Because of
ro,avt ro no , ? Erj ro no ßjm
(Erm ro no) , ? if share prices rise, no is
low and riskless assets are taxed
moderately, ? if share prices stagnate, no 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.