Dynamic Asset Allocation

Transcription

1 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Dynamic Asset Allocation Chapter 18: Transaction costs Claus Munk Aarhus University August 2012

2 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Transaction costs We follow Davis & Norman (1990, Math. Operations Research) and consider following model a single risky asset (a stock) and a riskfree asset, proportional transaction costs when trading the stock, constant investment opportunities are constant, investor has an infinite time horizon, CRRA utility of consumption Many relevant extensions Fixed costs? Fixed and proportional costs? Multiple assets? Finite time horizon? Trading costs for durable goods? Transaction costs + stochastic investment opportunities?

3 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Outline 1 The model 2 Solution w/o transaction costs 3 Solution w/ transaction costs 4 Some extensions

4 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Assumptions Risk-free bank account constant r; no costs. A single risky asset (the stock) with price dynamics dp t = P t [µ dt + σ dz t ]. Buying one unit costs (1 + a)p t, selling one unit provides (1 b)p t, where a, b 0. Investment strategy in the stock is represented by the pair of processes (L, U) Lt: cumulative amounts of stock purchased on [0, t] Ut: cumulative amounts of stock sold on [0, t] (amounts measured by the listed price, costs subtracted from the bank account) S 0t : bank account; S 1t : value of the stocks owned at time t (measured at the listed unit price at time t). The dynamics is ds 0t = (rs 0t c t ) dt (1 + a) dl t + (1 b) du t, S 00 = x, (1) ds 1t = µs 1t dt + σs 1t dz t + dl t du t, S 10 = y. (2) Here c t is the consumption rate at time t.

5 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Assumptions, cont d Solvency condition: after eliminating his position in the stock, you must have non-negative wealth. If S1t > 0, the requirement is S 0t + (1 b)s 1t 0, i.e., S 1t 1 S 1 b 0t. If S1t < 0, the requirement is S 0t + (1 + a)s 1t 0, i.e., S 1t 1 S 1+a 0t. The solvency region is therefore S = { (x, y) R 2 : x + (1 b)y 0, x + (1 + a)y 0 }. The set of admissible consumption and trading strategies is U(x, y) = {(c, L, U) : (S 0t, S 1t ) S for all t 0 (a.s.), c t 0} For preferences, assume infinite horizon and power utility with γ > 1 denoting the relative risk aversion. Let J(x, y) = sup (c,l,u) U(x,y) E x,y [ 0 ] e δt 1 1 γ c1 γ t dt.

6 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Take Merton to the limit... For the case without transaction costs (a = b = 0), we solved the similar problem for a finite time horizon in Chapter 6. Let λ = (µ r)/σ. If the constant A = δ + r(γ 1) γ + 1 γ 1 2 γ 2 λ 2 is positive, the limit as T of the solution is J(x, y) = 1 1 γ A γ (x + y) 1 γ, c = A[x + y], where x + y is the total wealth. We have πt = S 1t S 0t +S 1t and hence π = λ γσ, S 1t = π S 0t 1 π = λ γσ λ, corresponding to a straight line through the origin in the (S 0, S 1 )-space, the Merton line.

7 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Theorem 18.1 The value function J(x, y) has the following properties: 1 J is concave, i.e., for θ [0, 1] J (θx 1 + [1 θ]x 2, θy 1 + [1 θ]y 2 ) θj(x 1, y 1 ) + [1 θ]j(x 2, y 2 ). 2 J is homogeneous of degree 1 γ, i.e., for k > 0 J(kx, ky) = k 1 γ J(x, y). It follows that J(x, y) = k γ 1 J(kx, ky) for any k > 0. Consequently, and, similarly, So, for all k > 0, J x(x, y) J ( ) (x, y) = k γ 1 J(kx, ky) = k γ J x(kx, ky) x x J y(x, y) J y (x, y) = k γ J y(kx, ky). J y(kx, ky) Jy(x, y) = J x(kx, ky) J x(x, y). Hence, J y/j x is constant along any straight line through the origin.

8 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Heuristic solution Assume trading strategies are of the form L t = t 0 l s ds, U t = t 0 u s ds; l s, u s [0, K ] for some constant K. In particular, dl t = l t dt and du t = u t dt. HJB-eq.: { 1 δj(x, y) = sup c 0,l [0,K ],u [0,K ] 1 γ c1 γ + J x [rx c (1 + a)l + (1 b)u] + J y[µy + l u] Jyyσ2 y 2} { } 1 = sup c 0 1 γ c1 γ cj x + sup {(J y (1 + a)j x) l} l [0,K ] + sup {((1 b)j x J y) u} + r xj x + µyj y + 1 u [0,K ] 2 σ2 y 2 J yy. The first-order conditions imply { K, if J y (1 + a)j x, l = 0, otherwise, u = { 0, if J y > (1 b)j x, K, otherwise.

9 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Heuristic solution, cont d Trading strategy: J y (1 + a)j x : buy stocks (1 + a)j x > J y > (1 b)j x : do not trade stocks J y (1 b)j x : sell stocks. This divides the solvency region into three regions: a buying region, a no trade region, and a selling region. Boundaries: B: points with J y (x, y) = (1 + a)j x (x, y)... a straight line! (slope denoted 1/ω B ) S: points with J y (x, y) = (1 b)j x (x, y)... a straight line! (slope denoted 1/ω S ; ω B ω S ) Draw graph...

10 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Heuristic solution, cont d Do not trade when 1 ω B S 1t S 0t 1 ω S. The fraction of wealth invested in the stock is π t = S 1t /(S 0t + S 1t ), which will then satisfy 1 1 π t. 1 + ω B 1 + ω S Except for extreme cases, the Merton weight π = λ γσ falls between the boundaries.

11 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Heuristic solution, cont d Inside the no trade region, HJB-eq. simplifies to { } 1 δj = sup c 0 1 γ c1 γ cj x + r xj x + µyj y σ2 y 2 J yy = γ 1 1 γ J1 γ x + r xj x + µyj y σ2 y 2 J yy. Exploit homogeneity of the value function: ( ) x J y, 1 = HJB-eq. becomes ( ) 1 γ ( ) 1 J(x, y) J(x, y) = y 1 γ x J y y, 1 y 1 γ ψ ( ) x. y 1 2 σ2 ω 2 ψ (ω) + (r µ + γσ 2 )ωψ (ω) ( δ + (γ 1)µ 1 ) 2 σ2 γ(γ 1) ψ(ω)+ γ 1 γ ψ (ω) 1 1 γ = 0, ω [ω S, ω B ].

12 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Heuristic solution, cont d In the selling region, we must have J(x, y) constant along any line of slope 1/(1 b), so that J(x, y) = F(x + [1 b]y) for some function F. Then J x = F and J y = (1 b)f so that J y = (1 b)j x. Inserting J x and J y, we see that ψ (ω)(ω + 1 b) = (1 γ)ψ(ω), ψ(ω) = A 1 (ω + 1 b)1 γ 1 γ for a constant A. Hence, J(x, y) = y 1 γ ψ(x/y) = A 1 1 γ (x + [1 b]y)1 γ. Similarly, ψ(ω) = B 1 (ω a)1 γ 1 γ for some constant B in the buying region, i.e., J(x, y) = B 1 1 γ (x + [1 + a]y)1 γ.

13 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions The final solution To sum up, we have to find constants ω B, ω S, A, B and a function ψ so that 1 2 σ2 ω 2 ψ (ω) + (r µ + γσ 2 )ωψ (ω) + γ 1 γ ψ (ω) 1 1 γ ( δ + (γ 1)µ 1 ) 2 σ2 γ(γ 1) ψ(ω) = 0, ω [ω S, ω B ], ψ(ω) = A 1 1 γ (ω + 1 b)1 γ, ω ω S, ψ(ω) = B 1 1 γ (ω a)1 γ, ω ω B. Davis & Norman show that (under a technical condition) a solution to this problem will lead to the optimal strategies as described above. The optimal consumption rate will be c t = S 1t ( ψ (S 0t /S 1t ) ) 1/γ. Davis & Norman confirm that a solution to the problem exists.

14 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions The final solution, cont d At the boundaries, we have the so-called value-matching conditions ψ(ω S ) = A 1 1 γ (ω S + 1 b) 1 γ, ψ(ω B ) = B 1 1 γ (ω B a) 1 γ. The so-called smooth-pasting conditions ensure that the derivative of ψ at ω S is the same from the left and from the right, and equivalently at ω B. Therefore ψ (ω S ) = A(ω S + 1 b) γ, ψ (ω B ) = B(ω B a) γ. Numerical solution techniques are required!

15 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Finite time horizon Note: vertical axis shows stock-bond ratio Source: Gennotte and Jung, Management Science, 1994

16 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Proportional and fixed costs Source: Øksendal and Sulem, SIAM Journal of Control and Optimization, 2002

17 The model Solution w/o transaction costs Solution w/ transaction costs Some extensions Multiple assets Source: Muthuraman and Kumar, Mathematical Finance, 2006

18 Asset allocation over the life cycle: How much do taxes matter? Holger Kraft 1 Marcel Marekwica 2 Claus Munk 3 1 Goethe University Frankfurt, Germany 2 Copenhagen Business School, Denmark 3 Aarhus University, Denmark

19 Outline 1 Introduction 2 Model and problem 3 Optimal policies 4 Gains from tax-optimization 5 Comparative statics 6 The end

20 Contribution Derive and study optimal consumption and portfolio decisions in a life cycle model with labor income and taxation of realized capital gains Assess the importance of tax-timing, i.e., exploiting the realization-based feature of capital gains taxation vs. mark-to-market taxation Assess the importance of taking into account taxes on financial returns at all

21 Asset allocation with income Hakansson (Econ-70), Merton (JET-71): risk-free income Bodie-Merton-Samuelson (JEDC-92): spanned income; labor supply decisions Viceira (JF-01): unspanned income Cocco-Gomes-Maenhout (RFS-05): life cycle income profiles Munk-Sørensen (JFE-10): interest rates and income Yao-Zhang (RFS-05), Van Hemert (REE-10), Kraft-Munk (MS-11): housing decisions and income Note: no taxes!

22 Asset allocation with taxes Constantinides (Econ-83): wash sales; shorting-the-box Dammon-Spatt-Zhang (RFS-01): our model w/o income (well...) Gallmeyer-Kaniel-Tompaidis (JFE-06): multiple stocks Ehling et al. (wp-10): asymmetric taxation of gains and losses DeMiguel-Uppal (ManSci-05): exact share identification vs. average purchase price Dammon-Spatt-Zhang (JF-04), Zhou (JEDC-09), Gomes-Michaelidis-Polkovnichenko (RED-09): taxable vs. tax-deferred accounts; assume either inappropriate income or mark-to-market taxation of capital gains in taxable account Note: realization-based taxation not studied in life cycle setting with reasonable model of labor income! Note: no welfare analysis!

23 Our main conclusions For investors assuming mark-to-market taxation the welfare gains from switching to the fully optimized portfolio policy are less than 0.5% of wealth Tax-timing considerations have modest impact on optimal portfolios Expected utility little sensitive to small variations in portfolios [Brennan-Torous (EN-99), Rogers (FiSt-01)] An investor completely ignoring taxation of investment profits will gain less than 2% by switching to the fully optimal portfolio policy Exception: very old investors with strong bequest motives if capital gains are forgiven at death (as in the U.S.)

24 Assumptions Discrete-time model with one-year time intervals Single consumption good, constant inflation rate i = 3% Individual lives for at most T = 100 years with deterministic unconditional survival probabilities F t C t : nominal consumption in period t Time-additive expected CRRA utility (bequest considered later): [ T ( )] E β t C t F t U t=t 0 (1 + i) t, U(c) = c1 γ 1 γ Benchmark parameters γ = 4, β = 0.96

25 Securities Always a one-period risk-free asset, pre-tax return r = 4%, tax rate τ i = 35% One risky asset (stock index) with price P t capital gain g t+1 = P t+1 P t 1; mean µ = 6%, std-dev σ = 20% 1 + g t+1 lognormally distributed realized capital gains taxed at rate τ g = 20% constant dividend yield d = 2%, taxed at rate τ i q t: number of stocks hold from t to t + 1 tax basis Pt is an average historical purchases price: P t if P Pt t 1 P t (realize loss; buy new) = if Pt 1 < P t (unrealized gains) q t 1 P t 1 +max(q t q t 1,0)P t q t 1 +max(q t q t 1,0) Capital gains subject to taxation at time t is [ ] G t = χ {P t 1 P t }q t 1 + χ {P t 1 <P t } max (q t 1 q t, 0) (P t Pt 1)

26 Labor income Pre-tax labor income I t, taxed at rate τ i Stochastic income in working life, i.e., up to age J = 65 Income growth rate f t+1 = I t+1 I t 1 has std-dev σ L = 15% and mean µ(t + 1) calibrated to age-profile of high-school graduates in the U.S. Capital gains 1 + g t+1 and income growth 1 + f t+1 jointly lognormally distributed with correlation ρ = 0 Retirement income is a fraction λ = 68.2% of pre-retirement income (inflation-adjusted): I J+m = λ(1 + i) m I J

27 The optimization problem Objective: max {Ct,q t,b t } T t=t 0 E Disposable wealth: [ T t=t 0 β t F t U ( Ct (1+i) t )] W t = I t (1 τ i ) + q t 1 P t (1 + d (1 τ i )) + B t 1 (1 + (1 τ i ) r) Budget constraint: C t + τ g G t + q t P t + B t W t Conditions: q t 0, B t 0, C t > 0 State variables: W t, P t, P t 1, q t 1, and I t Reduce complexity by normalization. New state variables: after-tax income-to-wealth ratio i t = (1 τ i ) I t W t, entering relative equity exposure s t = q t 1P t W t, basis-price ratio p t = P t 1 P t Cons/wealth ratio c t = Ct qt Pt W t, exiting equity exposure α t = W t, and relative bond investment b t = Bt W t only depend on t, i t, s t, and pt

28 Numerical approach Solve problem numerically using backward induction on a grid for the state variables 31 grid points each for s, p, and i; 80 timepoints million optimization problems With a Matlab implementation on a single pc, it takes around one week to solve the optimization problem

29 Inputs Assume benchmark parameter values introduced earlier Assume individual is currently of age t 0 = 20 No initial equity holdings Initial wealth consists entirely of labor income over the first year, i 0 = 100%

30 Optimal consumption: state-dependence 0.4 Consumption policies, age=25, s=50% Consumption wealth ratio p*=0.85 p*=0.55 p*= Income to wealth ratio

31 Optimal consumption: state-dependence 0.7 Consumption policies, s=50%, p*= Consumption wealth ratio Age= Age=40 Age=60 Age=80 45 degree line Income to wealth ratio

32 Optimal investment: state-dependence 1 Investment policies, age=25, s=50% Equity exposure p*=0.85 p*=0.55 p*= Income to wealth ratio

33 Optimal investment: state-dependence 1 Investment policies, s=50%, p*= Equity exposure Age=25 Age=40 Age=60 Age= Income to wealth ratio

34 Life cycle profile 3 Consumption Income Wealth Life cycle profile 30 Consumption, Income Wealth level Age Note: averages based on 10,000 simulations

35 Life cycle profile: income/wealth Evolution of income to wealth ratio over the life cycle Mean 5th percentile 95th percentile Income to wealth Age Note: based on 10,000 simulations

36 Life cycle profile: basis-price ratio Evolution of initial basis price ratio over the life cycle Mean 5th percentile 95th percentile 1 Basis price ratio Age Note: based on 10,000 simulations

37 Life cycle profile: consumption-wealth ratio Evolution of consumption wealth ratio over the life cycle Mean 5th percentile 95th percentile Consumption wealth ratio Age Note: based on 10,000 simulations

38 Life cycle profile: equity exposure 1.1 Evolution of equity exposure over the life cycle Equity exposure Mean 5th percentile 95th percentile Age Note: based on 10,000 simulations

39 Suboptimal policies considered No tax-timing: Best consumption and portfolio policies assuming mark-to-market taxation of capital gains, i.e., G t = q t (P t P t 1 ) Optimal consumption policy + best portfolio policy assuming mark-to-market taxation No taxes on returns at all: Best consumption and portfolio policies assuming zero taxes on returns (capital gains, interest and dividend payments) Optimal consumption policy + best portfolio policy assuming zero taxes on returns

40 Measure of suboptimality Welfare gain: extra total wealth (current wealth + future income) the investor following a suboptimal policy needs to obtain the same utility if she continues with the suboptimal policy instead of shifting to the optimal policy

41 State-dependent welfare gains: no tax-timing 0.45 Welfare gain for investor ignoring tax timing, age=25, s=67% p*=0.85 p*=0.55 p*= Welfare gain Income to wealth ratio

42 State-dependent welfare gains: no taxes Welfare gain for investor ignoring taxes, age=25, s=67% p*=0.85 p*=0.55 p*= Welfare gain Income to wealth ratio

43 Welfare effects over the life cycle: no tax-timing Welfare gain investor ignoring tax timing Suboptimal portfolio strategy Suboptimal consumption portfolio strategy Welfare gain Age Note: based on 10,000 simulations

44 Welfare effects over the life cycle: no tax-timing cons/wealth equity exposure welfare gains Age optimal heuristic optimal heuristic cons+port port only % 78.9% 100% 100% 0.23% 0.17% % 20.7% 100% 100% 0.32% 0.24% % 12.2% 97% 98% 0.41% 0.30% % 10.2% 90% 91% 0.42% 0.29% % 10.7% 86% 86% 0.35% 0.21% % 14.9% 87% 87% 0.24% 0.10% % 28.6% 91% 90% 0.17% 0.04% % 62.7% 93% 94% 0.08% 0.00% Note: averages based on 10,000 simulations

45 Welfare effects over the life cycle: no taxes 6 Welfare gain for investor ignoring taxes Suboptimal portfolio strategy Suboptimal consumption portfolio strategy 5 4 Welfare gain Age Note: based on 10,000 simulations

46 Welfare effects over the life cycle: no taxes cons/wealth equity exposure welfare gains Age optimal heuristic optimal heuristic cons+port port only % 79.8% 100% 100% 1.50% 0.96% % 22.3% 100% 100% 1.91% 1.28% % 13.4% 93% 88% 1.97% 1.41% % 11.2% 82% 74% 1.57% 1.18% % 11.7% 77% 70% 1.00% 0.78% % 16.2% 81% 75% 0.53% 0.40% % 30.5% 89% 85% 0.24% 0.16% % 68.4% 95% 95% 0.06% 0.03% Note: averages based on 10,000 simulations

47 Robustness In the paper we demonstrate that the results are robust to relevant variations in key inputs: initial income/wealth ratio volatility of labor income tax rates risk aversion, Epstein-Zin preferences

48 Robustness wrt. bequest? So far, no utility of bequest in line with empirical study of Hurd (Econ-89) With bequest motive of strength k, maximize [ T ( ) ( )] E β t Ct Wt+1 B F t U + β (F (1 + i) t t F t+1 ) k U (1 + i) t+1 t=0 where Wt+1 B is the wealth bequeathed to descendants at death. In U.S.: unrealized gains are forgiven at death Other countries: unrealized capital gains taxed at death

49 Bequest motive; taxation of capital gains at death Welfare gain investor ignoring tax timing k=0 k=5 k= Welfare gain Age Note: based on 10,000 simulations

50 Bequest motive; forgiveness of capital gains at death Welfare gain investor ignoring tax timing k=0 k=5 k=10 1 Welfare gain Age Note: based on 10,000 simulations

51 Conclusion Optimal stock-bond allocation over the life cycle is only little affected by the realization-based feature of capital gains taxation or by return taxation at all The realization-based feature of capital gains taxation is of minor importance to individual investors Modification: some effect for very old investors with strong bequest motives if capital gains are forgiven at death Even if you ignore taxation of financial returns completely, you will only lose relatively little The limited computational resources are probably better spent on adding other relevant state and decision variables (e.g., time-varying returns, housing) than on including the complicated real-life rules for taxation of financial returns