Portfolios that Contain Risky Assets 17: Fortune s Formulas

Transcription

1 Portfolios that Contain Risky Assets 7: Fortune s Formulas C. David Levermore University of Maryland, College Park, MD Math 420: Mathematical Modeling April 2, 208 version c 208 Charles David Levermore

2 Portfolios that Contain Risky Assets Part II: Stochastic Models. Independent, Identically-Distributed Models 2. Growth Rate Mean and Variance Estimators 3. Law of Large Numbers (Kelly) Objectives 4. Kelly Objectives for Markowitz Portfolios 5. Central Limit Theorem Objectives 6. Optimization of Mean-Variance Objectives 7. Fortune s Formulas 8. Utility Function Objectives

3 Fortune s Formulas Introduction 2 Efficient Frontier 3 Parabolic Objectives 4 Quadratic Objectives 5 Reasonable Objectives 6 Comparisons Objectives 7 Seven Lessons Learned

4 Introduction We now consider some settings in which the optimization problem can be solved analytically. Specifically, we will derive explicit formulas for the solutions to the maximization problems for the family of parabolic objectives Γp χ (f) = µ rf + m T f 2 ft Vf χ f T Vf, (.a) the family of quadratic objectives Γq χ (f) = µ rf + m T f ( 2 µrf + m T f ) 2 2 ft Vf χ f T Vf, (.b) and the family of reasonable objectives Γ χ r (f) = log ( + µ rf + m T f ) 2 ft Vf χ f T Vf, (.c) considered over their natural domains of allocations f for unlimited leverage portfolios with one risk-free asset.

5 Introduction Recall that: µ rf is the return on the risk-free asset; m is the sample excess return mean vector, which is given in terms of the sample return mean vector m by m = m µ rf ; V is the sample return covariance matrix; χ is the nonnegative caution coefficient chosen by the investor. Recall too that m and V are computed from a return history {r(d)} D d= and a choice of positive weights {w(d)} D d= that sum to by m = D w(d) r(d), V = d= D w(d) ( r(d) m ) ( r(d) m ) T. d=

6 Introduction In the previous lecture we saw that the maximizer f for such a problem will correspond to a point (σ, µ ) on the efficient frontier. Moreover, we saw that (σ, µ ) is the point in the σµ-plane where the level curves of the objective are tangent to the efficient frontier. While this geometric picture gave insight into how optimal portfolio allocations arise, we have not yet computed them. The explicit formulas derived in this lecture for the maximizer f will confirm the general picture developed in the previous lecture. They will also give insight into the relative merits of the different families of objectives in (.). In particular, the maximizers when χ = 0 give different realizations of the Kelly Criterion so-called fortune s formulas. The maximizers when χ > 0 will be corresponding fractional Kelly strategies. We will derive and analyze these formulas after reviewing the efficient frontier for unlimited leverage portfolios with one risk-free asset.

7 Efficient Frontier Recall that for unlimited leverage portfolios without risk-free assets the frontier is the hyperbola in the right-half of the σµ-plane given by ( ) σ = σ 2 µ 2 µmv mv +, (2.2a) ν as where the so-called frontier parameters σ mv, µ mv, and ν as are given by σmv 2 = T V, µ mv = T V m T V, (2.2b) νas 2 = m T V m (T V m) 2 T V. This so-called frontier hyperbola has vertex (σ mv, µ mv ) and asymptotes µ = µ mv ± ν as σ for σ 0. The positive definiteness of V insures that σ mv > 0 and ν as > 0.

8 Efficient Frontier If we introduce one risk-free asset with risk-free return µ rf < µ mv then the efficient frontier becomes the tangent half-line given by µ = µ rf + ν tg σ for σ 0, (2.3a) where the slope is ν tg = m T V m ( ) = ν µmv µ 2 as + rf. (2.3b) ν as σ mv This slope is the so-called Sharpe ratio of the efficient frontier. Remark. The Sharpe ratio of any portfolio with return mean µ and volatility σ is defined as µ µ rf. σ Clearly ν tg is the Sharpe ratio of every portfolio on the efficient frontier (2.3a). Moreover, ν tg is the largest possible Sharpe ratio for any portfolio.

9 Efficient Frontier The efficient frontier (2.3a) is tangent to the frontier hyperbola (2.2a) at the point (σ tg, µ tg ) where ( ) σ tg = σ νas σ 2 mv + mv, µ µ mv µ tg = µ mv + ν as 2 σmv 2. rf µ mv µ rf The unique tangency portfolio associated with this point has allocation f tg = σ 2 mv µ mv µ rf V m. (2.4) Every portfolio on the efficient frontier (2.3a) can be viewed as holding a position in this tangency portfolio and a position in a risk-free asset.

10 Efficient Frontier We can select a particular portfolio on this efficient frontier by identifying an objective function to be maximized. In subsequent sections we derive and analyze explicit formulas for the maximizers for each family member of the parabolic, quadratic, and reasonable objectives given in (.).

11 Parabolic Objectives First we consider the maximization problem f = arg max { Γ χ p (f) : f R N}, (3.5a) where Γ χ p (f) is the family of parabolic objectives parametrized by χ 0 and given by Γ χ p (f) = µ rf + m T f 2 ft Vf χ f T Vf. (3.5b) If f 0 then the gradient of Γ χ p (f) is where σ = f T Vf > 0. f Γ χ p (f) = m Vf χ σ Vf,

12 Parabolic Objectives By setting this gradient equal to zero we see that if the maximizer f is nonzero then it satisfies where σ = f T Vf > 0. 0 = m σ + χ σ Vf, Upon solving this equation for f we obtain f = All that remains is to determine σ. σ σ + χ V m. (3.6)

13 Parabolic Objectives Because σ = f T Vf we have σ 2 = f T Vf = we conclude that σ satisfies σ 2 (σ + χ) 2 mt V m = (σ + χ) 2 = ν 2 tg. Because σ > 0 and χ 0 we see that and that σ is determined by σ 2 (σ + χ) 2 ν 2 tg, 0 χ < ν tg, (3.7) σ + χ = ν tg.

14 Parabolic Objectives Then the maximizer f given by (3.6) becomes f = ( χ ) V m. (3.8) ν tg Remark. Kelly investors take χ = 0, in which case (3.8) reduces to f = V m. (3.9) Formula (3.9) is often called fortune s formula in the belief that it is a good approximation to the Kelly strategy. In this view formula (3.8) gives an explicit fractional Kelly strategy for every χ (0, ν tg ). However, we will see that formula (3.9) gives an allocation that can be far from the Kelly strategy, and generally leads to overbetting.

15 Parabolic Objectives The foregoing analysis did not yield a maximzier when χ ν tg. To treat that case we will use the Cauchy inequality in the form m T f m T V m f T Vf. (3.0) When χ ν tg the positive definiteness of V, the fact χ ν tg, the Sharpe ratio formula (2.3b), and the above Cauchy inequality imply Therefore f = 0 when χ ν tg. Γ χ p (f) = µ rf + m T f 2 ft Vf χ f T Vf µ rf + m T f χ f T Vf µ rf + m T f ν tg f T Vf = µ rf + m T f m T V m f T Vf µ rf = Γp χ (0).

16 Parabolic Objectives Therefore the solution f of the maximization problem (3.5) is ( χ ) V m if χ < ν f = ν tg, tg 0 if χ ν tg. (3.) This solution lies on the efficient frontier (2.3a). It allocates f χ tg times the portfolio value in the tangent portfolio f tg given by (2.4) and f χ tg times the portfolio value in a risk-free asset, where ( f χ tg = χ ) µmv µ rf ν tg σmv 2. (3.2)

17 Quadratic Objectives Next we consider the maximization problem f = arg max { Γ χ q (f) : f R N}, (4.3a) where Γ χ q (f) is the family of quadratic objectives parametrized by χ 0 and given by Γq χ (f) = µ rf + m T f ( 2 µrf + m T f ) 2 2 ft Vf χ f T Vf. (4.3b) If f 0 then the gradient of Γ χ q (f) is where σ = f T Vf > 0. f Γ χ q (f) = ( µ rf ) m m m T f Vf χ σ Vf,

18 Quadratic Objectives By setting this gradient equal to zero we see that if the maximizer f is nonzero then it satisfies where σ = 0 = ( µ rf ) m m m T f σ + χ Vf, σ f T Vf > 0. After multiplying this relation by V and bringing the terms involving f to the left-hand side, we obtain σ + χ σ f + V m m T f = ( µ rf ) V m. (4.4)

19 Quadratic Objectives Now multiply this by σ m T and use the Sharpe ratio formula (2.3b), m T V m = ν 2 tg, to obtain which implies that ( σ + χ + ν 2 tg σ ) m T f = ( µ rf ) ν 2 tg σ, m T f = ( µ rf ) ν 2 tg σ σ + χ + ν 2 tg σ. When this expression is placed into (4.4) we can solve for f to find f = ( µ rf ) All that remains is to determine σ. σ σ + χ + ν 2 tg σ V m. (4.5)

20 Quadratic Objectives Because σ = f T Vf we have σ 2 = f T Vf = we conclude that σ satisfies = ( µ rf ) 2 σ 2 ( ( + ν 2 tg ) σ + χ ) 2 mt V m ( µ rf ) 2 σ 2 ( ( + ν 2 tg ) σ + χ ) 2 ν tg 2, ( ( + ν 2 tg ) σ + χ ) 2 = ( µrf ) 2 ν 2 tg.

21 Quadratic Objectives Because σ > 0 and χ 0 we see that and that σ is determined by 0 χ < ( µ rf ) ν tg, (4.6) ( + ν 2 tg) σ + χ = ( µ rf ) ν tg. Then the maximizer f given by (4.5) becomes f = ( µ rf χ ) ν tg + νtg 2 V m. (4.7)

22 Quadratic Objectives Remark. Kelly investors take χ = 0, in which case (4.7) reduces to f = µ rf + νtg 2 V m. (4.8) Formula (4.8) differs significantly from formula (3.9) whenever the Sharpe ratio ν tg is not small. Sharpe ratios are often near and sometimes can be as large as 3. So which of these should be called fortune s formula? Certainly not formula (3.9)! To see why, set f = V m into the quadratic objective (4.3b) with χ = 0 to obtain Γq 0 ( V m ) = µ rf + 2 ν tg 2 ( 2 µrf + νtg) 2 2, which can be negative when ν tg is near. So formula (3.9) can overbet!

23 Quadratic Objectives The foregoing analysis did not yield a maximzier when χ ( µ rf ) ν tg. In that case the positive definiteness of V, the fact χ ( µ rf ) ν tg, the Sharpe ratio formula (2.3b), and the Cauchy inequality (3.0) imply Γq χ (f) = µ rf + m T f ( 2 µrf + m T f ) 2 2 ft Vf χ f T Vf µ rf + m T f ( 2 µ 2 rf + 2µ rf m T f ) χ f T Vf = µ rf 2 µ 2 rf + ( µ rf ) m T f χ f T Vf µ rf 2 µ rf 2 + ( µ rf ) m T f ( µ rf ) ν tg f T Vf ( = µ rf 2 µ rf 2 + ( µ rf ) m T f m T V m ) f T Vf µ rf 2 µ 2 rf = Γ χ q (0). Therefore f = 0 when χ ( µ rf ) ν tg.

24 Quadratic Objectives Therefore the solution f of the maximization problem (4.3) is ( µ f = rf χ ) V m ν tg + ν 2 if χ < ( µ rf ) ν tg, tg 0 if χ ( µ rf ) ν tg. (4.9) This solution lies on the efficient frontier (2.3a). It allocates f χ tg times the portfolio value in the tangent portfolio f tg given by (2.4) and f χ tg times the portfolio value in a risk-free asset, where ( f χ tg = µ rf χ ) ν tg + νtg 2 µ mv µ rf σ 2 mv. (4.20)

25 Reasonable Objectives Next we consider the maximization problem f = arg max { Γ χ r (f) : f R N, + µ rf + m T f > 0 }, (5.2a) where Γ χ r (f) is the family of reasonable objectives parametrized by χ 0 and given by Γ χ r (f) = log ( + µ rf + m T f ) 2 ft Vf χ f T Vf. (5.2b) Because Γ χ r (f) as f approaches the boundary of the domain being considered in (5.2a), the maximizer f must lie in the interior of the domain. If f 0 then the gradient of Γ χ r (f) is f Γ χ r (f) = + µ m Vf χ σ Vf, where µ = µ rf + m T f and σ = f T Vf > 0.

26 Reasonable Objectives By setting this gradient equal to zero we see that if the maximizer f is nonzero then it satisfies where µ = µ rf + m T f and σ = Because σ = f T Vf we have σ 2 = f T Vf = f = + µ σ σ + χ V m, (5.22) = f T Vf > 0. ( + µ ) 2 σ 2 (σ + χ) 2 mt V m ( + µ ) 2 σ 2 (σ + χ) 2 ν 2 tg.

27 Reasonable Objectives From this we conclude that µ and σ satisfy (σ + χ) 2 = Because σ > 0 and χ 0 we see that ν 2 tg ( + µ ) 2. and that we can determine σ in terms of µ from σ + χ = ν tg. + µ 0 χ < ν tg + µ, (5.23) Then the maximizer f given by (5.22) becomes ( f = χ ) V m, (5.24) + µ ν tg

28 Reasonable Objectives Because µ = µ rf + m T f, by the Sharpe ratio formula (2.3b) we have ( µ = µ rf + m T f = µ rf + χ ) m T V m + µ ν tg ( = µ rf + χ ) νtg 2. + µ ν tg This can be reduced to the quadratic equation ( ) ( ) νtg 2 + µrf νtg + χ =, + µ + µ ν tg = ( + µrf + µ 2 ν tg ν tg which has the unique positive root ) χ ( + µrf ν tg χ ) 2. (5.25)

29 Reasonable Objectives Then condition (5.23) is satisfied if and only if 0 < ν tg + µ χ = 2 ( + µrf ν tg ) ( + χ + + µrf + 4 ν tg ) 2 χ. This inequality holds if and only if 0 < + 4 ( + µrf ν tg ) 2 χ ( + µrf 4 ν tg ) 2 + χ = + µ rf χ. ν tg This holds if and only if χ satisfies the bounds 0 χ < ν tg + µ rf. (5.26)

30 Reasonable Objectives By using (5.25) to eliminate µ from the maximizer f given by (5.24) we find f = ( ) ( ) + µrf 2 + χ + + µrf + χ V m. 2 4 ν tg This becomes f = ν tg ( χ ) + µ rf ν tg D ( ν tg χ, + µ rf ν tg ) V m, (5.27a) where D(χ, y) = 2 ( + χ y) + 2 ( χ y) 2 + 4y 2. (5.27b)

31 Reasonable Objectives Remark. Kelly investors take χ = 0, in which case (5.27) reduces to f = 2 ( + µ rf ) + V m. (5.28) 2 ( + µ rf ) 2 + 4νtg 2 This candidate for fortune s formula will be compared with the others later.

32 Reasonable Objectives The foregoing analysis did not yield a maximzier when ( + µ rf ) χ ν tg. The definiteness of V, the concavity of log(x), the fact ( + µ rf ) χ ν tg, the Sharpe ratio formula (2.3b), and Cauchy inequality (3.0) imply Γ χ r (f) = log ( + µ rf + m T f ) 2 ft Vf χ f T Vf log( + µ rf ) + mt f + µ rf χ f T Vf log( + µ rf ) + mt f ν tg + µ rf ( = log( + µ rf ) + m T f + µ rf log( + µ rf ) = Γr χ (0). Therefore f = 0 when ( + µ rf ) χ ν tg. + µ rf f T Vf m T V m f T Vf )

33 Reasonable Objectives Therefore the solution f of the maximization problem (5.2) is ( χ ) V m ( ) if χ < ν tg, + µ rf ν tg ν tg + µ rf f = D χ, + µ rf 0 if χ ν tg, + µ rf where D(χ, y) was defined by (5.27b). (5.29) This solution lies on the efficient frontier (2.3a). It allocates f χ tg times the portfolio value in the tangent portfolio f tg given by (2.4) and f χ tg times the portfolio value in a risk-free asset, where ( f χ tg = χ ) ( ) µ mv µ rf + µ rf ν tg ν tg σ 2. (5.30) mv D χ, + µ rf

34 Comparisons The maximizers for the parabolic, quadratic, and reasonable objectives are given by (3.), (4.9), and (5.29) respectively. They are ( f p χ ) V m if χ < ν = ν tg, tg (6.3a) 0 if χ ν tg, ( f q µ = rf χ ) V m ν tg + ν 2 if χ < ( µ rf ) ν tg, tg (6.3b) 0 if χ ( µ rf ) ν tg, ( χ ) V m ( ) if χ < ν tg, + µ rf ν tg ν tg + µ rf f r = D χ, + µ (6.3c) rf 0 if χ ν tg. + µ rf

35 Comparisons Fact. If µ rf [0, ) then f q is the most conservative of these allocations and f p is the most agressive. Proof. First observe that because µ rf [0, ) we have These inequalities imply that and that µ rf + µ rf. ( µ rf ) ν tg ν tg + µ rf ν tg, (6.32) µ rf χ ν tg Each of these inequalities is strict when µ rf (0, ). χ χ. (6.33) + µ rf ν tg ν tg

36 Comparisons Recall from (5.27b) that D(χ, y) = 2 ( + χ y) + 2 ( χ y) 2 + 4y 2. (6.34) For every y > 0 we have ) χ D(χ, y) = 2 ( y χ y > 0, ( χ y) 2 + 4y 2 whereby D(χ, y) is an strictly increasing function of χ. Hence, for every χ [0, y) we have < D(0, y) D(χ, y) < D(y, y) = + y 2. (6.35) Therefore when µ rf 0 we have ( ) ν tg νtg 2 < D χ, < + + µ rf ( + µ rf ) 2 + ν tg 2 if χ < ν tg. (6.36) + µ rf

37 Comparisons The inequalities (6.32) imply that f q given by (6.3b) has the smallest critical value of χ at which it becomes 0, and that f p given by (6.3a) has the largest critical value of χ at which it becomes 0. The inequalities (6.33) and (6.36) imply that the factor multiplying V m in the expression for f q given by (6.3b) is smaller than the factor multiplying V m in the expression for f r given by (6.3c), which is smaller than the factor multiplying V m in the expression for f p given by (6.3a). Hence, f q is more conservative than f, r which is more conservative than f p. Remark. The risk-free return µ rf is usually much smaller than the Sharpe ratio ν tg. This means that the main differences between the maximizers given by formulas (6.3) arise due to their dependence upon ν tg.

38 Comparisons In practice µ rf is often small enough that it can be neglected except in m. By setting µ rf = 0 in (6.3) we get ( f p χ ) V m if χ < ν = ν tg, tg (6.37a) 0 if χ ν tg, ( f q χ ) V m = ν tg + ν 2 if χ < ν tg, tg (6.37b) 0 if χ ν tg, ( f r χ ) V m if χ < ν = ν tg D(χ, ν tg ) tg, (6.37c) 0 if χ ν tg, where D(χ, y) is given by (6.34). These all are nonzero for χ < ν tg and all vanish for χ ν tg. We see from (6.35) that < D(χ, ν tg ) < + ν 2 tg.

39 Comparisons We now use formulas (6.37b) and (6.37c) to isolate the dependence of the maximizers f q and f r upon ν tg. Fact 2. For every χ [0, ν tg ) we have νtg 2 + νtg 2 D(χ, ν tg) + ν 2 tg <, (6.38) where the left-hand side is a strictly decreasing function of ν tg. Proof. By (6.35) we have + νtg 2 > D(χ, ν tg ) D(0, ν tg ) = νtg 2. The inequalities (6.38) follow. The task of proving the left-hand side of (6.38) is a strictly decreasing function of ν tg is left as an exercise.

40 Comparisons We now use Fact 2 to show that f q and f r are close when ν tg 2 3. Fact 3. If ν tg 2 3 then for every χ [0, ν tg) we have 2 3 D(χ, ν tg) + νtg 2 <. (6.39) Proof. By the monotonicity asserted in Fact 2 if ν tg 2 3 then νtg 2 + νtg = = 2 3. Then (6.39) follows from inequality (6.38) of Fact 2.

41 Comparisons Remark. A Kelly investor would set χ = 0, in which case (6.3) gives f p = V m, (6.40a) f q = µ rf + νtg 2 V m, f r = 2 ( + µ rf ) + V m. 2 ( + µ rf ) 2 + 4νtg 2 (6.40b) (6.40c) This is the case for which the difference between f q and f r is greatest. To get a feel for this difference, when µ rf = 0 and ν tg = 2 these become f q = 3 V m, f r = 2 V m, while when µ rf = 0 and ν tg = 6 these become f q = 7 V m, f r = 3 V m.

42 Seven Lessons Learned Here are some insights that we have gained.. The Sharpe ratio ν tg and the caution coefficient χ play a large role in determining the optimal allocation. In particular, when χ ν tg the optimal allocation is entirely in risk-free assets. 2. The risk-free return µ rf plays a role in determining the optimal allocation mainly through m. 3. For any choice of χ the maximizer for the quadratic objective is more conservative than the maximizer for the reasonable objective, which is more conservative than the maximizer for the parabolic objective. 4. The maximizer for a parabolic objective is agressive and will overbet when the Sharpe ratio ν tg is not small.

43 Seven Lessons Learned 5. The maximizers for quadratic and reasonable objectives are close when the Sharpe ratio ν tg is not large. As χ approaches ν tg, the maximizers for the quadratic and reasonable objectives will become closer. 6. We will have greater confidence in the computed Sharpe ratio ν tg when the tangency portfolio lies towards the nose of the efficient frontier. This translates into greater confidence in the maximizers for the quadratic and reasonable objectives. 7. Analyzing the maximizers for both the quadratic and reasonable objectives gave greater insights than analyzing each of them separately. Together they are fortune s formulas.