Transcript Bewertung von Wertpapieren in Mittelwert-Varianz

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
[email protected]
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
2
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,
r f:
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,
μ(r)  rf 
H  (r) ,
H  (Er  rf 1) (Vr ) (Er  rf 1) ,
T
-1
1T  (1, 1, ..., 1) .
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µ(r)
rf  H  (r)
rf
rf  H  (r)
σ(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 > 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 portfolio
rf
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-σplane is decisive.
●
The CAPM is not a general equilibrium model.
●
Is it possible to deduce equilibrium solutions
for asset pricing in case the HuangLitzenberger 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 :
ro > 0 ,
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, …) ,
c1, c2, … :
no Є (0, 1) ,
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 Є [0, ro] ,
W1,at = (1 + rf,at) Wo = (1 – no) W1 ,
no Є (0, 1) ,
Wo :
W1, W1,at :
no :
rf , rf,at :
initial wealth,
end of period wealth before and after taxes,
wealth tax rate on riskless assets,
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 < 0  rf,at ,
→ this can not be done with a yield tax according to
current tax law but with a wealth tax, that is
ro
νo 
.
1  ro
→ 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 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
ro - go
νo 
1  ro
for go  (-1, ro) ,
(2)
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–no)–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) + βjm (Erm – Erz(m))
for
Erz(m) ≥ go ,
if
ro  go
νo 
,
1  ro
where
go = Erz(q) or
go = Erq – Rra Var(rq) ,
rj :
rate of return on asset “j“,
rm : anticipated market portfolio,
βjm : β-factor.
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µ(r)
hyperbolic frontier after taxes
anticipated market portfolio
arbitrary portfolio “q” on the upper branch
σ(r)
go
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 + βjm (Erm – ro,avt) ,
for
or
ro,avt = Erz(index)
ro,avt = Erindex – Rra ∙ Var(rindex) ,
where
rindex :
Rra :
rate of return on a share index,
aggregate relative risk aversion.
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Interpreting no as control variable:
Because of
ro,avt ≈ ro – no ,
→
Erj ≈ ro – no + βjm (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.
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