Dynamic Mean Semi-variance Portfolio Selection

Transcription

1 Dynamic Mean Semi-variance Portfolio Selection Ali Lari-Lavassani and Xun Li The Mathematical and Computational Finance Laboratory Department of Mathematics and Statistics University of Calgary Calgary, Alberta, TN 1N4, Canada {lavassan, Abstract. In real investment situations, one desires to only minimize downside risk or portfolio loss without affecting the upside potentials. This can be accomplished by mean semi-variance optimization but not by mean variance. In the Black-Scholes setting, this paper proposes for the very practical yet intractable dynamic mean semi-variance portfolio optimization problem, an almost analytical solution. It proceeds by reducing the multi-dimensional portfolio selection problem to a one-dimensional optimization problem, which is then expressed in terms of the normal density, leading to a very simple and efficient numerical algorithm. A numerical comparison of the efficient frontier for the mean variance and semi-variance portfolio optimization problem is presented. 1 Introduction Multi-period and continuous-time mean variance portfolio selection have been extensively studied, see e.g. [3], [1], [9], [15], [11], [13], [7]. Dynamic programming was successfully used in the continuous case in [15] reducing the problem to the stochastic Ricatti differential equation within the framework of linear quadratic stochastic control. This reduction does not, however, apply to the continuous time mean semi-variance problem, which remains an open problem. In real investment situations, one desires to only minimize downside risk or portfolio loss without affecting the upside potentials. This can be accomplished by mean semi-variance optimization but not mean variance. The aim of this paper is to provide for this intractable problem of continuous time mean semivariance portfolio optimization, an analytical approach, reducing or compressing the multi-dimensional portfolio selection problem to a one dimensional optimization problem, and expressing it in terms of the normal density, leading to a very simple and efficient numerical implementation and algorithm. This research was partially supported by the National Science and Engineering Research Council of Canada, and the Network Centre of Excellence, Mathematics of Information Technology and Complex Systems. P.M.A. Sloot et al. Eds.: ICCS 003, LNCS 657, pp , 003. c Springer-Verlag Berlin Heidelberg 003

2 96 A. Lari-Lavassani and X. Li Mean Semi-variance Portfolio Model In this paper, we denote by M the transpose of a matrix or vector M =m ij, by M = i,j m ij its norm and by IRm the m-dimensional real space. We consider a Black-Scholes type financial market where m + 1 assets are traded continuously on a finite horizon [0,T]. One asset is a bond S 0 t, whose dynamics is governed by the ordinary differential equation ds 0 t =rs 0 tdt, t [0,T], S 0 0 > 0, 1 where r > 0 is the interest rate of the bond. The remaining m assets are stocks, and their prices are modeled by the system of stochastic differential equations ds i t =S i tb i dt + m j=1 σ ijdw j t, t [0,T], S i 0 > 0, where b i > r is the drift or appreciation rate, σ ij are the volatility coefficients, satisfying the non-degeneracy condition σσ > 0 and W t W 1 t,, W m t is an m-dimensional Brownian motion. A dynamic portfolio consists of m i=0 N its i t, where N i t is the number of shares invested in the ith asset S i. If an initial wealth X 0 > 0 is invested across the assets in this market, the total wealth at time t 0 denoted by Xt, can be shown, e.g. [5], to follow the dynamics { m dxt =Xtr1 i=1 π i+ m i=1 b iπ i dt + m X0 = X 0, m j=1 i=1 σ ijπ i dw j t, 3 where π i denotes the proportion of the wealth Xt invested in the i-th stock, that is π i Xt =N i ts i t, i =1,,m. Then Xt = m i=0 π ixt, where we let π 0 =1 m i=1 π i be the proportion of the investment in the bond, and call π := π 1,,π m IR m the portfolio selection. Note that even though π i is constant the actual portfolio N i t is dynamic and changes over time t. Wedo not constrain π i to take positive values, in other words, short-selling of stocks is allowed. Finally, transaction costs and consumptions are not considered here. The system 3 can be written in the vectorized form dxt =Xtr +b r1 πdt + σ πdwt, X0 = X 0, 4 where 1 is the m-dimensional column vector with each component equal to 1. To simplify notation further, we let µ = r +b r1 π and η = σ π. Then dxt =Xtµdt + ηdwt, X0 = X 0. 5 The first and second moments of Xt can be easily calculated, EXt =X 0 e µt, Var Xt =X 0 e µt e η t 1. 6 In finance, given a notion of risk, portfolio optimization consists of selecting the portfolio with a given return and minimum risk. In this work, risk is measured by the semi-variance of the terminal wealth, that is E[maxEXT XT, 0] > 0. 7

3 Dynamic Mean Semi-variance Portfolio Selection 97 We note that this captures risk on the undesirable downside, when EXT > XT, while leaving the upside EXT <XT unaffected. For a prescribed target expected terminal wealth EXT C, the investor s aim is to minimize the above downside risk. On the other hand, the investor expects a return above the risk free investment consisting of π i = 0 for i =1,,m, and whose associated wealth process X satisfies dxt =rxtdt, X0 = X 0, and has for solution, XT =X 0 e rt. This leads to the following natural assumption, C X 0 e rt. The wealth X and portfolio π are called admissible if they satisfy the linear stochastic differential equation 3. The mean semi-variance portfolio optimization problem can then be formulated as 1 min π IR subject to E[maxEXT XT, 0] m { EXT C, X,π is admissible. 8 Problem 8 is called feasible if there exists at least one admissible pair satisfying EXT C. Given C, the optimal strategy π of 8 is called an efficient strategy and the pair EXT,E[maxEXT XT, 0] is called an efficient point. The set of all efficient points, when the parameter C runs over [X 0 e rt, +, is called the efficient frontier. We emphasize again that the efficient frontier depends on the notion of risk under consideration. After some calculations in Section 3, we will reduce the m-dimensional mean semi-variance portfolio selection problem 8 to the following one-dimensional optimization problem, expressed in terms of the normal density function min ε IR X 0 e rt Γ ε, { X0 e subject to r+εθt C, ε>0, 9 where ε T Γ ε =e [3Φ εθt + e εt Φ 3ε ] T, 10 θ = σ 1 b r1 and Φx = 1 x π e y dy is the standard normal distribution function. The main result of this work is Theorem 1 Denote the market price of risk by θ = σ 1 b r1. The efficient strategy of the mean semi-variance portfolio selection problem 8 corresponding to the expected terminal wealth EXT C is given by 1 Clearly, E[maxEXT XT, 0] is convex in π. π = εθ 1 σσ 1 b r1, 11

4 98 A. Lari-Lavassani and X. Li where ε is the optimal solution of 9, that can be obtained numerically. Moreover, the efficient frontier is E[maxEXT XT, 0] =EXT [3Φ ε T ] +e εt Φ 3 ε T.1 Remark 1. The important fact about this result is that the only unknown in the expression of the optimal portfolio π, which could have an arbitrary large number of assets, is ε, which is the solution of the one-dimensional optimization problem 9. In other words, we have reduce the portfolio selection problem on m + 1 assets to a one-dimensional problem. Remark. We note that it is not possible to carry out the optimization problem 9 analytically since Φ is the standard normal distribution function. However, numerical techniques can be fruitfully used. For this, one can use standard polynomial approximations, such as the one with six-decimal-place accuracy see [4], and use a numerical optimization routine, such as Matlab Optimization Toolbox. Φx = { 1 Φ xa 1 k + a k + a 3 k 3 + a 4 k 4 + a 5 k 5,x 0, 1 Φ x, x < 0, 13 where Φ x = 1 π e x, k = 1 1+γx, γ = , a 1 = , a = , a 3 = , a 4 = , a 5 = Fig. 1.

5 Dynamic Mean Semi-variance Portfolio Selection 99 We numerically apply the above algorithm to an example, whose details can be found in Section 4, to run a comparative valuation of the efficient frontier for the mean semi-variance case studied here, versus the classical Markowitz style mean variance portfolio selection. The results are plotted in Figure 1. What is interesting to note in this figure, is that to reach the same level of terminal wealth, a level of risk almost 6 orders of magnitude higher must be taken with mean variance compared to semi-variance, since the latter leaves the up-side strategies open. 3 Proof of the Main Results We first recall a multi-dimensional version of Itô s lemma see, e.g., [14], [10] Lemma 1 Given an m-dimensional process x satisfying dxt = µt, xtdt + νt, xtdw t, and a real valued function ϕ, C [0,T] IR m, we have dϕt, xt = ϕ t t, xtdt + ϕ x t, xt dxt + 1 tr[νt, xt ϕ xx t, xtνt, xt]dt. 14 Using this lemma and after some calculations, see [8], one can find the density function of the wealth process Xt in 5 to be 1 ψxt = Xt η exp ln Xt ln X0 µ 1 η t πt η t. 15 We now reduce part of the semi-variance s expression 8 in continuous time. Using an idea introduced in [5], the starting point is to project the problem onto the family of ellipsoids, ε = η = σ π. 16 Lemma Given an admissible solution π of problem 8, the semi-variance can be expressed as where gε, T =3Φ E[maxEXT XT, 0] = X 0 gε, T e r+b r1 πt, 17 ε T + e εt Φ 3ε T.

6 100 A. Lari-Lavassani and X. Li Proof. First, the semi-variance can be written as the following integral expressions = = E[maxEXT XT, 0] 0 EXT 0 [maxext XT, 0] ψxt dxt EXT XT ψxt dxt EXT =EXT ψxt dxt EXT + EXT 0 0 XT ψxt dxt. EXT 0 XT ψxt dxt Using 15, one can calculate the above, in term of the standard normal distribution E[maxEXT XT, 0] =EXT Φ EXT X 0 e µt Φ +X0 e µ+ η T Φ ln EXT ln X 0 µ 1 η T η T ln EXT ln X 0 µ+ 1 η T ηt T ln EXT ln X 0 µ+ 3 η T η T Using 6, the above expression can be rearranged as E[maxEXT XT, 0] = X0 e [Φ µt η T Φ = X0 e [3Φ µt η T η T + e η T Φ + e η T Φ ] 3 η T.. ] 3 η T Now, substituting 16 in the above yields the desired result. Note that the function gε, T is one-dimensional. The exponential part in 17 has still m-variables. To further reduce dimension, we introduce the following intermediary optimization problem. Any fixed ε>0 defines an ellipsoid. Using Lemma and 6, problem 8, projected onto this ellipsoid, can be transformed into the following optimization problem min X0 gε, T e r+b r1 πt, π X 0 e r+b r1 πt C, subject to σ π = ε, π IR m. 18

7 Dynamic Mean Semi-variance Portfolio Selection 101 Proposition 1 The optimal solution of problem 18 is given by π = ε θ σσ 1 b r1, 19 where θ = σ 1 b r1 denotes the market price of risk. Proof. We introduce some transformations. Problem 18 is clearly equivalent to min X0 gε, T e r+b r1 πt, π b r1 π C, 0 subject to σ π = ε, π IR m, where C = 1 T ln C X 0 r. By hypothesis 7, E[maxEXT XT, 0] > 0, therefore, using Lemma, we have, gε, T > 0. Hence, we can further reduce problem 0 to min b r1 π, π b r1 π C, subject to σ π = ε, π IR m. 1 We will finally solve problem 1. Let us introduce the Lagrange multipliers µ 0 and λ 0 of 1 Lπ, µ, λ =1+µb r1 π µ C λ σ π ε. A simple square completion calculation, see [8], yields σσ Lπ, µ, λ = λ π 1+µ λ σσ 1 b r1 π 1+µ λ σσ 1 b r1 + 1+µ 4λ b r1 σσ b r1 µ C + λε. It can be verified, see the above reference, that this problem has for minimum, π = 1+µ λ σσ 1 b r1. Substituting the solution into σ π = ε, we conclude that 1+µ λ = ε θ, which, together with, implies 19. Solving 8 amounts to minimizing the solution 19 of 18 over all possible ε, we can more generally write problem 8 as where Γ ε =gε, T e εθt. min ε subject to X0 e rt Γ ε, { X0 e r+εθt C, ε>0, 3

8 10 A. Lari-Lavassani and X. Li It remains to verify that this problem is convex. Since the objective function X0 e rt Γ ε of 9 is transformed from E[maxEXT XT, 0] via the linear transformation 19, X0 e rt Γ ε is also convex in ε. Indeed, let Gπ =E[maxEXT XT, 0], Hε =X0 e rt Γ ε. Then Gπ =Hε for π, ε related by 19. For α 0, 1, ˆε, ε IR and π, π IR m, using the linearity of 19, we have Then, αˆπ +1 α π =[αˆε +1 α ε]σσ 1 b r1. Gαˆε+1 α ε =Hαˆπ+1 α π αhˆπ+1 αh π =αgˆε+1 αh ε. In addition, the set defined in 9 is convex. Therefore, the optimization problem 9 has a unique minimum solution. 4 Example We now consider an example to illustrate the results of the previous section, with interest rate r = % and m = 3 stocks. The time granularity for all parameters is yearly. The yearly drift, volatility and correlation matrix ρ of the 3 stocks are listed below, that is, drift b i volatility ν i stock 1 4% 0% stock 5% 5% stock 3 6% 30% ρ = ds i t =b i S i tdt + ν i S i tdz i t, t [0,T], i =1,, 3, 4 where zt := z 1 t,z t,z 3 t are correlated Brownian motions with dztdzt = ρdt. We need the volatility matrix σσ of the standard Brownian motion for the dynamics of these assets, as in. Let ν be the diagonal matrix with diagonal entries, ν 1,ν,ν 3. Comparing the volatility coefficients of and 4, yields the vector equality, νdzt = σdwt. Multiplying each side by the transpose, and using the fact that, dztdzt = ρdt and dw tdw t = Idt, where I is the 3-dimensional identity matrix, results in σσ = νρν. Therefore σσ 1 =νρν 1 = ν 1 ρ 1 ν = We then have and σσ 1 b r1 =0.676, ,

9 Dynamic Mean Semi-variance Portfolio Selection 103 θ = σ 1 b r1 = We then numerically solved problem 9 in this case, using Matlab Optimization Toolbox. We discretized time into daily time steps going 5 years forward, and the optimal ε was accordingly computed at the daily frequency. The results are plotted in Figure. Then substituting every optimal ε into 11 yields the optimal strategy. Finally, the corresponding efficient frontier was obtained using 1 and plotted as a surface, in Figure 1, for the interest rate r = %, time horizon up to T = 5 years, initial wealth X 0 =1, 000, 000 and terminal wealth C [X 0 e 0.0T,, 000, 000]. On the same Figure, we have also plotted the efficient frontier corresponding to the mean-variance optimal portfolio, see [8] for more detail. For a fixed T, the cross section of the surface is the efficient frontier. Note that a higher level of wealth C corresponds to higher semi-variance, which decreases as T increases, as a larger portion of the wealth can be invested in the bond. 7 x 10 5 Daily Optimal Strategies Obtained Numerically Optimal ε Time Horizon in Days Fig.. 5 Conclusion Portfolio optimization under mean semi-variance is more appropriate than its classical counter part mean variance, but is however a lot more complicated. In continuous-time, and for constant parameters, we proposed in this work an approach which compressed or reduced the multi-dimensional problem, as many as the assets, to a one-dimensional problem, for which one can use numerical routines quite efficiently. Extending this work to time dependent parameters adds considerable complexity to the problem. This is accomplished in [8]. Undertaking

10 104 A. Lari-Lavassani and X. Li numerical comparisons between various down side risk portfolio optimization approaches would be very valuable for practical applications. We intend to pursue this elsewhere. References 1. Duffie, D., Richardson, H.: Mean-variance hedging in continuous time. Annals of Applied Probability, Fishburn, P.: Mean-risk analysis with risk associated with below-target returns. American Economic Review, Hakansson, N.H.: Multi-period mean-variance analysis: Toward a general theory of portfolio choice. Journal of Finance, Hull, J.: Options, Futures, and Other Derivatives, 5th edn. Prentice Hall, New Jersey, Emmer, S., Klüppelberg, C., Korn, R.: Optimal portfolios with bounded capital at risk. Mathematical Finance, Lari-Lavassani, A., Li, X., Ware, A., Dmitrasinovic-Vidovic, G.: Dynamic portfolio selection under downside risks. Working paper, the Mathematical and Computational Finance Laboratory, University of Calgary, Lari-Lavassani, A., Li, X.: Dynamic mean-variance portfolio selection with borrowing constraint. Preprint, the Mathematical and Computational Finance Laboratory, University of Calgary, Lari-Lavassani, A., Li, X.: Continuous-time mean semi-variance portfolio selection. Preprint, the Mathematical and Computational Finance Laboratory, University of Calgary, Li, D., Ng, W.L.: Optimal dynamic portfolio selection: Multi-period mean-variance formulation. Mathematical Finance, Li, X.: Indefinite Stochastic LQ Control with Financial Applications. Dissertation, the Chinese University of Hong Kong, Li, X., Zhou, X.Y., Lim, A.E.B.: Dynamic mean-variance portfolio selection with no-shorting constraints. SIAM Journal on Control and Optimization, Markowitz, H.: Portfolio selection. Journal of Finance, Steinbach, M.C.: Markowitz Revisited: Mean-Variance Models in Financial Portfolio Analysis. SIAM Review, J. Yong and X.Y. Zhou. Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York, Zhou, X.Y., Li, D.: Continuous time mean-variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization,