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

1. 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

2. Contents 1 Initial question 2 Standard model (CAPM) • Model extension 4 CAPM after variable taxes 5 Conclusion

3. 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?

4. 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).

5. What arethe mean-variance optimal portfolios under A1 and A2? Optimization with Lagrange function (Merton 1972) Solution: two half lineson the µ-σ-plane,

6. µ(r) rf σ(r) Figure 2.1:The portfolio frontier under A1 and A2.

7. 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)-1Er , C = 1T (Vr)-11 , rmvp: rate of return on the (global) minimum variance portfolio.

8. 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 !

9. µ(r) market portfolio minimum variance portfolio rf σ(r) Figure 2.2: The capital market line on the µ-σ-plane ( Ermvp > rf ) .

10. µ(r) rf minimum variance portfolio tangency portfolio σ(r) Figure 2.3: The portfolio frontier for Ermvp < rf .

11. 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.

12. 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?

13. 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 !

14. Further Assumptions (A2*) There are severalriskless 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 rfЄ [0, ro] , ro > 0 , ro : overnight rate.

15. (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, …),noЄ (0, 1) , c1, c2, … : constants. Idea: nocontains all relevant information to ensure equilibrium after taxes.

16. How to define a wealth tax on riskless assets? W1 = (1 + rf) Wo , rfЄ [0, ro] , W1,at = (1 + rf,at) Wo = (1 – no) W1 , noЄ (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)

17. 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 !

18. 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 , noЄ (0, 1) . → the interest-free riskless rate is always negative after taxes.

19. 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).

20. 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 goЄ(-1, ro) with the following properties: • go= max rf,at and go < Ermvp . • Asset pricing is independent of rf , rfЄ [0, ro].

21. Proof: • ↔ (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) , rfЄ [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) . “

22. 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.

23. 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–no)–1 = ro,at , which is identical with equation (1).

24. 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.

25. 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.

26. µ(r) hyperbolic frontier after taxes anticipated market portfolio arbitrary portfolio “q” on the upper branch goσ(r) Figure 4.1: Anticipated equilibrium after variable taxes ( go = max rf,at = ro,avt = Erz(q) ).

27. 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. • Theovernight 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.

28. 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.

29. 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 !

30. 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.