An approximate solution for the power utility optimization under predictable returns
AAn approximate solution for the power utilityoptimization under predictable returns
Dmytro Ivasiuk
Department of Statistics, European University Viadrina, Frankfurt(Oder), GermanyE-mail: [email protected]
Abstract
This work presents an approximate solution of the portfolio choiceproblem for the investor with a power utility function and the pre-dictable returns. Assuming that asset returns follow the vector au-toregressive process with the normally distributed error terms (whatis a popular choice in financial literature to model the return path)it comes up with the fact that portfolio gross returns appear to benormally distributed as a linear combination of normal variables. Asit was shown, the log-normal distribution seems to be a good proxyof the normal distribution in case if the standard deviation of the lastone is way much smaller than the mean. Thus, this fact is exploited toderive the optimal weights. Besides, the paper provides a simulationstudy comparing the derived result to the well-know numerical solu-tion obtained by using a Taylor series expansion of the value function.
Keywords: approximate solution, power utility, utility maximization, nu-merical solution.
The current paper discusses portfolio optimization as the utility of wealthmaximization. The classical problem stands for deriving the value functiongiven by the whole investment period with the use of the Bellman backwardmethod (see Pennacchi, 2008; Brandt, 2009). One can find it as the typicalsolution for the investment strategies for different types of utility functions(see Bodnar et al., 2015a,b). However, most of the time it is hard to succeed1 a r X i v : . [ q -f i n . P M ] N ov nd in order to derive an analytical solution it is required to have some sortof specific assumptions because of the difficulty to perform the calculations(see Bodnar et al., 2015b, 2018; Campbell and Viceira, 2002). On the otherhand, one can always use numerical or approximative results if no informationon returns is considered (see Brandt et al., 2005; Broadie and Shen, 2017).However, the quadratic utility has a closed-form multi-period solution thatdoes not require any assumptions (see Bodnar et al., 2015a).This work deals with the power utility of wealth function U ( W ) = W − γ − γ , γ >
0, which is classified as constant relative risk aversion (CRRA) utility.The interpretation of the CRRA invented by John W. Pratt stands for theperson who’s investing decision does not rely on his or her initial wealth (seeCampbell and Viceira, 2002; Pennacchi, 2008), which is, in fact, observablein findings of the paper.To derive the optimal weights it is considered that asset returns followa multivariate autoregressive model with normally distributed error termswhat has become a popular choice in recent literature (see Bodnar et al.,2015a,b; Barberis, 2000; Brandt and Santa-Clara, 2006). Consequently, itcomes up with the multivariate normal returns, thus the portfolio returnswill be normally distributed as a linear combination of normal variables.However, it gives no benefits in case of the power utility function. Instead,we would like to have the log-normal distribution for the portfolio grossreturns so one can easily calculate the expected value of a power functionof the log-normal variable. It is notable that the log-normal distribution isoften used to model stock returns in the recent literature (see C¸ etinkaya andThiele, 2016; Herzel and Nicolosi, 2019). Hence, the log-normal distributionis used as an approximation of the normal distribution. Precisely speaking,the idea is to model the returns using the proxy distribution which behavessimilar to the original one.The main result of the paper provides approximate single-period optimalportfolio weights for the power utility function given by the multivariatenormal distribution on returns. The approximation is based on a similarbehaviour of the normal N ( µ, σ ) and log-normal ln N (ln µ, σ /µ ) densitiesif σ/µ approaches zero (see Bodnar et al., 2018). For instance, let µ =1 and σ = 0 .
04 then both densities look pretty similar (Figure 1) and inpractice, simulated samples from the corresponding log-normal distributionwould rarely fail to reject the normality hypothesis testing. Besides, a similarsituation is observed at a stock market trading: the portfolio gross returnsfluctuate around value 1 within a small interval. It is also notable that thecorrect definition of the approximation requires a positive definite µ , which isitself a fair assumption since nobody expects to lose all the invested money.2 .8 0.9 1.0 1.1 1.2 N(1, 0.0016)lnN(0, 0.0016)
Figure 1: Normal and Log-normal densitiesAs it is known, the power utility optimization has no analytical solution,therefore for an empirical study, it was decided to compare the derived resultwith the well known numerical solution (see Brandt et al., 2005), since bothof them are an approximation. From the theoretical point of view, an invest-ment strategy is said to be better than the other one if the correspondingexpected utility of the final wealth provides a higher value. Thus it is donein the same way. Simulating the return path from a vector autoregressivemodel with normally distributed error terms, one can calculate the final util-ity of wealth for both strategies and examine the outcomes. Even though thenumerical solution provides a multi-period strategy unlike the derived resultis build under a single-period setup, nevertheless, the last one outperforms.Outcomes for different combinations of the risk aversion parameter γ and theinvestment horizon T have always shown the numerical solution receiving asmaller sample mean for the final utility of wealth, what means, that in caseof a normally distributed returns the proposed solution is a better strategycomparing to the numerical one. 3 Framework
Below one can find a framework used to present the portfolio selection prob-lem for a power utility function. Let r denote the one-period random returnvector on risky assets and let r f be the return vector on the risk-free assetand ω stands for portfolio weights allocated between risky assets. Therefore,if R f = 1 + r f is the gross return on the risk-free asset and R = r − r f isthe vector of excess return on the risky assets, then the investor’s wealth atthe end of investment period is W ( R f + ω (cid:48) R ), where W defines the initialwealth. So it is considered that all the money is distributed between riskyassets and one risk-free asset without any consumption. Also the mean vec-tor and the variance-covariance matrix of the excess returns are given withthe next way: ˜ µ = E [ r − r f ], Σ = V ar [ r − r f ].The main purpose of the paper is to derive the optimal portfolio weightsin order to maximise the expected utility of the final wealth: ω ∗ = arg max { ω } E [ U ( W )] . (1)Precisely speaking, this is a partial case of the classical multi-period maxi-mization problem (see, Brandt and Santa-Clara, 2006; Pennacchi, 2008; Bod-nar et al., 2015a). The utility U ( · ) is considered as a power function of wealth U ( W ) = W − γ − γ , γ > , γ (cid:54) = 1 , (2)which belongs to the constant relative risk aversion family.To solve the optimization problem, a multivariate autoregressive modelfor the behaviour of asset returns is considered, what has become a popularchoice in a recent literature (see, Barberis, 2000; Campbell and Viceira, 2002;Bodnar et al., 2015b). For instance, if the excess returns follow the VARprocess of order one R t = ϕ + Φ R t − + ε t , ε t ∼ N (0 , Σ ) , (3)then the portfolio gross returns R f + ω (cid:48) R t will follow the normal distribu-tion with the mean R f + ω (cid:48) ( ϕ + Φ R t − ) and the variance ω (cid:48) Σ ω as a linearcombination of normally distributed random variables. Precisely speaking,only the normality of error terms is important, since the idea behind is toapproximate the normal distribution with the log-normal because of the con-venience to calculate the expected value of a power function of a log-normalrandom variable: E (cid:2) X λ (cid:3) = exp (cid:18) αλ + 12 β λ (cid:19) , if X ∼ ln N ( α, β ) , λ ∈ R . (4)4 Approximate solution of the power utilityoptimization
The main finding of this paper presents the approximate close form solutionof the single-period optimal portfolio choice problem for the power utilityinvestor given by the assumption that the excess returns follow the multi-variate normal distribution. Indeed, considering the regressive process (3)one can conclude that the excess returns will follow the multivariate nor-mal distribution with the mean vector ϕ + Φ R t − and the covariance matrix Σ . Particularly any model with a multivariate normal distribution structuresatisfies the given result. Theorem 1.
Assume that the excess returns R follow the multivariate nor-mal distribution with the covariance matrix Σ and the mean vector ˜ µ . If γ ≥ µ (cid:48) Σ − ˜ µ , (5) then the approximate solution of the optimization problem (1) with the powerutility function (2) is given by ω ∗ approximate = Σ − ˜ µ R f (cid:16) γ − − (cid:112) ( γ − − γ −
1) ˜ µ (cid:48) Σ − ˜ µ (cid:17) γ − (cid:0) ˜ µ (cid:48) Σ − ˜ µ (cid:1) , (6) where R f corresponds to the gross returns on the risk-free asset. The proof of the theorem one can find it the
Appendix section. Notethat given by the assumption (5), optimal weights presented in Theorem 1 donot exist for all the values of the relative risk aversion parameter. However,having a minimum level of γ different from zero (by the definition) can beexplained by the fact, that considering the log-normal distribution we boundthe portfolio gross returns from bellow when the normal distribution, on theother hand, is unbounded. Thus, it might be interpreted as an investor re-jecting extremely risky trading situations. Besides, the next section providesempirical results for the lower bound of γ gathered as a histogram, whichgives the idea of its possible location. As an empirical study, the approximate weights derived in Theorem 1 werecompared to the well known numerical result introduced by Brandt et al.(2005). The last one is a universal solution obtainable with the use of a5aylor series expansion of the value function, which can be applied to anygiven utility. However, it is a time consuming iterative procedure, sincegood accuracy can be achieved after a large number of repetitions. It is alsonotable that for a simplicity a constant return of 0.01 on the risk-free assetis considered.Presented below, one can find the numerical solution as the fourth-orderexpansion suggested by Brandt et al. (2005): ω t ( i + 1) ≈ − E t ∂ U (cid:16) ˆ W T (cid:17) ∂ ˆ W T T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) R t +1 R (cid:48) t +1 W t − × (cid:40) E t ∂U (cid:16) ˆ W T (cid:17) ∂ ˆ W T T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) R t +1 + 12 E t ∂ U (cid:16) ˆ W T (cid:17) ∂ ˆ W T T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) ( ω (cid:48) t ( i ) R t +1 ) R t +1 W t + 16 E t ∂ U (cid:16) ˆ W T (cid:17) ∂ ˆ W T T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) ( ω (cid:48) t ( i ) R t +1 ) R t +1 W t (cid:41) , (7)where ˆ ω s defines already calculated weights ( s = t + 1 , . . . , T −
1) andˆ W T = W t R f T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) . (8)After calculating the derivatives of the power utility and substituting ˆ W T one has: ω t ( i + 1) ≈ γ (cid:40) E t (cid:34) T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) − γ R t +1 R (cid:48) t +1 (cid:35)(cid:41) − × (cid:40) R f E t (cid:34) T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) − γ R t +1 (cid:35) + γ ( γ + 1)2 R f E t (cid:34) T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) − γ ( ω (cid:48) t ( i ) R t +1 ) R t +1 (cid:35) − γ ( γ + 1)( γ + 2)6 R f E t (cid:34) T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) − γ ( ω (cid:48) t ( i ) R t +1 ) R t +1 (cid:35) (cid:41) , (9)6here ω t (0) ≈ γ (cid:40) E t (cid:34) T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) − γ R t +1 R (cid:48) t +1 (cid:35)(cid:41) − × R f E t (cid:34) T − (cid:89) s = t +1 (cid:0) ˆ ω (cid:48) s R s +1 + R f (cid:1) − γ R t +1 (cid:35) (10)and ω T − ( i + 1) ≈ γ { E T − [ R T R (cid:48) T ] } − × (cid:40) R f E T − [ R T ] + γ ( γ + 1)2 R f E T − (cid:104)(cid:0) ω (cid:48) T − ( i ) R T (cid:1) R T (cid:105) − γ ( γ + 1)( γ + 2)6 R f E T − (cid:104)(cid:0) ω (cid:48) T − ( i ) R T (cid:1) R T (cid:105) (cid:41) , (11)with ω T − (0) ≈ R f γ { E T − [ R T R (cid:48) T ] } − E T − [ R T ] . (12)Hereby, this is a complete backward scheme on how to obtain the numericalsolution of a multi-period investment strategy similar to the Bellman method.Like it was mentioned before, there are no wealth components in the weightsformula because of the constant relative risk aversion utility function. Theonly task left is to calculate the expectations where the law of large numbersis applied (see Barberis, 2000): if f ( x ) defines a probability density functionof a random variable x , then the expectation E [ g ( x )] can be approximatedby 1 N N (cid:88) i =1 g ( y i )where { y i } Ni =1 is a large sample from the distribution given by f ( x ). Forinstance, to do all the calculations N = 10 and i = 20 was taken.A model for a stock, a bond, and a state variable, suggested by Brandtand Santa-Clara (2006) was considered as an excess returns path simulation: (cid:20) R t z t +1 (cid:21) = ln (cid:0) r st +1 (cid:1) ln (cid:0) r bt +1 (cid:1) z t +1 = . . − . + . . . × z t + ε st +1 ε bt +1 ε zt +1 , (13)7 . − . − . − . − . − . . . . . . . . g = , T = Fn(x) N u m e r i c A pp r o x i m a t e − . − . − . − . . . . . . . . g = , T = Fn(x) N u m e r i c A pp r o x i m a t e − . − . − . . . . . . . . g = , T = Fn(x) N u m e r i c A pp r o x i m a t e − . − . − . − . − . . . . . . . . g = , T = Fn(x) N u m e r i c A pp r o x i m a t e Figure 2: Empirical distribution function of the power utility for the numer-ical and the derived strategies for several combinations of γ and T . − . − . − . . . . . . . . g = , T = Fn(x) N u m e r i c A pp r o x i m a t e − . − . − . − . − . . . . . . . . g = , T = Fn(x) N u m e r i c A pp r o x i m a t e − . − . − . − . − . − . . . . . . . . g = , T = Fn(x) N u m e r i c A pp r o x i m a t e − . − . − . . . . . . . . g = , T = Fn(x) N u m e r i c A pp r o x i m a t e Figure 3: Empirical distribution function of the power utility for the numer-ical and the derived strategies for several combinations of γ and T γ -0.18679 -0.06217 -0.06122 -0.02953 -0.02901 -0.01623 -0.01596 NumericalApproximate12 -0.14346 -0.13735 -0.03482 -0.03283 -0.01261 -0.01154 -0.00502 -0.00474
NumericalApproximate18 -0.10872 -0.09965 -0.01957 -0.01738 -0.00519 -0.00451 -0.00152 -0.00135
NumericalApproximate24 -0.08294 -0.07191 -0.01103 -0.00925 -0.00216 -0.00183 -0.00047 -0.00039
NumericalApproximateTable 1: The sample means of the power utility of the final wealth for thenumerical solution and the derived strategy for several combinations of γ and T with ε st +1 ε bt +1 ε zt +1 ∼ M V N , . . − . . . . − . . . . (14)According to Theorem 1, the mean vector is˜ µ = (cid:20) . . (cid:21) + (cid:20) . . (cid:21) × z t and the covariance matrix is given by Σ = (cid:20) . . . . (cid:21) , which are used in the comparison study.Table 1 presents sample means of the power utility of a final wealth forseveral combinations of γ and T for the numerical (Numerical) solution andthe portfolio weights from Theorem 1 (Approximate). Despite the fact thatthe expression (6) provides the one-period constraint, one can see that thecorresponding values are very close, which means that the derived resultprovides a good level of approximation which even outperforms the numericalone. Besides, Figure 2 and Figure 3 describe the behaviour fo the empiricalcumulative distribution function of both strategies for several combinationsof the risk aversion parameter γ and the investment range T .The last part of the simulation study refers to Theorem 1 where it is saidthat the solution exists only in case if the risk aversion parameter γ exceeds10 min D en s i t y Figure 4: Histogram of the lower bound of γ -values in Theorem 1some lower bound. Figure 4 presents the histogram of the minimum γ -values(5) estimated form 10 samples given by the model (13). One can observethat most of the values are located between one and two, however, there aresome points outside of the interval which do not exceed four, meanwhile,the literature provides the median value of the relative risk aversion beingapproximately 7 (see Pennacchi, 2008). Lemma 1.
Assume that the portfolio gross return R f + ω (cid:48) R at the end ofthe investment period is normally distributed, i.e. R f + ω (cid:48) R ∼ N ( R f + ω (cid:48) ˜ µ , ω (cid:48) Σ ω ) . Let R f + ω (cid:48) ˜ µ > and √ ω (cid:48) Σ ω R f + ω (cid:48) ˜ µ → .Then R f + ω (cid:48) R (approx.) ∼ ln N (cid:18) ln ( R f + ω (cid:48) ˜ µ ) , ω (cid:48) Σ ω ( R f + ω (cid:48) ˜ µ ) (cid:19) Proof of Lemma 1.
According to Bodnar et al. (2018), assuming that µ > σ/µ →
0, the difference between the distribution functions of N ( µ, σ )11nd ln N (cid:16) ln µ, σ µ (cid:17) approaches zero. Proof of Theorem 1.
For the power utility function, one has E [ U ( W )] = E (cid:34) W − γ ( R f + ω (cid:48) R ) − γ − γ (cid:35) = W − γ − γ E (cid:104) ( R f + ω (cid:48) R ) − γ (cid:105) . Assuming that R ∼ N ( ˜ µ , Σ ), given by Lemma 1 it holds: E [ U ( W )] ≈ W − γ − γ exp (cid:34) (1 − γ ) ln ( R f + ω (cid:48) ˜ µ ) + (1 − γ ) ω (cid:48) Σ ω ( R f + ω (cid:48) ˜ µ ) (cid:35) . (15)In order to maximize the expected utility (15) it is required to derive thefirst order conditions (FOCs) with respect to ω .The FOCs are: ∂∂ ω (cid:34) ln ( R f + ω (cid:48) ˜ µ ) + 1 − γ ω (cid:48) Σ ω ( R f + ω (cid:48) ˜ µ ) (cid:35) = 0 (16)The partial derivation leads to:˜ µω (cid:48) R f + ˜ µ + (1 − γ ) Σ ω ( R f + ω (cid:48) ˜ µ ) − ω (cid:48) Σ ω ( R f + ω (cid:48) ˜ µ ) ˜ µ ( R f + ω (cid:48) ˜ µ ) = , or ˜ µ + (1 − γ ) Σ ω ( R f + ω (cid:48) ˜ µ ) − ω (cid:48) Σ ω ˜ µ ( R f + ω (cid:48) ˜ µ ) = . (17)Let J := ˜ µ Σ − ˜ µ , X := R f + ω (cid:48) ˜ µ , and Y := ω (cid:48) Σ ω ( R f + ω (cid:48) ˜ µ ) . (18)Multiplying (17) by ˜ µ (cid:48) Σ − and ω (cid:48) it follows: X − R f + (1 − γ ) R f Y = 0 , (19) J + (1 − γ ) (cid:16) X − R f X − Y J (cid:17) = 0 . (20)Next, substituting Y in the second equation for Y in the first one transformsto: J + (1 − γ ) (cid:18) X − R f X + X − R f R f (1 − γ ) J (cid:19) = 0 (21)12nd, consequently, J R f X + (1 − γ ) R f ( X − R f ) + ( X − R f ) XJ = 0 , (22)or J X + (1 − γ ) R f X − (1 − γ ) R f = 0 , (23)The roots of the quadratic equation (23) with respect to X are given by X ± = R f γ − ± √D J , (24)where D = ( γ − − γ − J, and Y ± = R f − X ± R f (1 − γ ) . (25)Moreover, equation (24) shows that the quadratic equation has a solution ifand only if D ≥ γ ∈ (0 , ∪ [1 + 4 J, ∞ ).Since the FOCs only provide the critical points, it is necessary to discovera maximum of the objective function (15). In order to do that, it is sufficientto compare the argument of the exponent in (15) for both combinations of( X + , Y + ) and ( X − , Y − ):(1 − γ ) ln X + + (1 − γ ) Y + − (1 − γ ) ln X − − (1 − γ ) Y − (25) = (1 − γ ) (cid:20) ln X + X − + X − − X + R f (cid:21) (24) = (1 − γ ) (cid:34) ln γ − √D γ − − √D − √D J (cid:35) (26)Next is shown that (26) is positive for γ > J : ∂∂γ (cid:34) ln γ − √D γ − − √D − √D J (cid:35) = (cid:18) ∂∂γ D √D (cid:19) (cid:16) γ − − √D (cid:17) − (cid:18) − ∂∂γ D √D (cid:19) (cid:16) γ − √D (cid:17) γ − J − ∂∂γ D J √D = 2 ∂∂γ D √D ( γ − − √D γ − J − ∂∂γ D√D ( γ − J ( γ −
1) = − √D J ( γ − . (27)13eanwhile (cid:34) ln γ − √D γ − − √D − √D J (cid:35) γ =1+4 J = ln 4 J J = 0 . (28)Thus, the second multiplier in (26) is a monotonically decreasing and a neg-ative function of γ for γ > J and, thus, (26) is positive. Hence, fro all γ > J the maximum of (15) is attained at ( X − , Y − ).It is notable that R f + ω (cid:48) ˜ µ or X is required to be positive, but for0 < γ < X − becomes negative, consequently ( X + , Y + ) is an only candidatefor an extrema point. In order to investigate the type of the extrema wecalculate the second derivative of the objective function given by ω andanalyse a sign of the quadratic form: ∂ ∂ ω (cid:34) ln ( R f + ω (cid:48) ˜ µ ) + 1 − γ ω (cid:48) Σ ω ( R f + ω (cid:48) ˜ µ ) (cid:35) (17) = ∂∂ ω (cid:20) ˜ µ + (1 − γ ) Σ ω ( R f + ω (cid:48) ˜ µ ) − ω (cid:48) Σ ω ˜ µ ( R f + ω (cid:48) ˜ µ ) (cid:21) (25) = (1 − γ ) ∂∂ ω (cid:20) Σ ω R f + ω (cid:48) ˜ µ + ω (cid:48) ˜ µ R f (1 − γ ) ˜ µ (cid:21) = (1 − γ ) (cid:34) Σ ( R f + ω (cid:48) ˜ µ ) − Σ ω ˜ µ (cid:48) ( R f + ω (cid:48) ˜ µ ) + ˜ µ ˜ µ (cid:48) R f (1 − γ ) (cid:35) (17) = (1 − γ ) Σ X − (cid:16) Y − − γ (cid:17) X ˜ µ ˜ µ (cid:48) X + ˜ µ ˜ µ (cid:48) R f (1 − γ ) (25) = (1 − γ ) Σ X − (cid:16) − γ − XR f (1 − γ ) − − γ (cid:17) X ˜ µ ˜ µ (cid:48) X + ˜ µ ˜ µ (cid:48) R f (1 − γ ) = (1 − γ ) (cid:20) Σ X + 2 ˜ µ ˜ µ (cid:48) R f (1 − γ ) (cid:21) . (29)Thus, the quadratic form is positive definite for γ ∈ (0 ,
1) and in this case( X + , Y + ) provide a minima of the value function (15).As the result, given by (17), (18), (24) and (25) single period approximateoptimal portfolio wights are: ω ∗ approximate = Σ − ˜ µ R f (cid:16) γ − − (cid:112) ( γ − − γ −
1) ˜ µ (cid:48) Σ − ˜ µ (cid:17) γ − (cid:0) ˜ µ (cid:48) Σ − ˜ µ (cid:1) (30)14 eferenceseferences