multi-period mean-variance portfolio selection with ... - Springer Link
J Syst Sci Complex (2011) 24: 140–155
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION WITH MARKOV REGIME SWITCHING AND UNCERTAIN TIME-HORIZON∗ Huiling WU · Zhongfei LI
DOI: 10.1007/s11424-011-9184-z Received: 1 July 2009 / Revised: 6 May 2010 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2011 Abstract This paper investigates a multi-period mean-variance portfolio selection with regime switching and uncertain exit time. The returns of assets all depend on the states of the stochastic market which are assumed to follow a discrete-time Markov chain. The authors derive the optimal strategy and the efficient frontier of the model in closed-form. Some results in the existing literature are obtained as special cases of our results. Key words Dynamic programming, Markov regime switching, mean-variance, portfolio selection, uncertain time-horizon.
1 Introduction Portfolio selection is to search a best allocation of wealth among different assets in markets. The mean-variance model pioneered by Markowitz[1] provides a fundamental basis for portfolio selection and has stimulated hundreds of extensions and applications. One of them is to extend the original static mean-variance model to discrete-time dynamic mean-variance models. This line is full of challenges and enormous difficulties and has not made a great breakthrough until the paper of Li and Ng[2] , which derived the analytical optimal portfolio policy by using an embedding technique. Since then, there has been a fast development in the dynamic mean-variance theory. For instance, Guo and Hu[3] investigated a multi-period mean-variance portfolio optimization problem with uncertain exit time. Leippold, Trojani, and Vanini[4] studied a multi-period mean-variance model for asset-liability management. Zhu, Li, and Wang[5] investigated multi-period mean-variance portfolio with risk control over bankruptcy. Some other generalizations can be found in [6]. The works mentioned above only consider one state of the market mode that reflects the state of the underlying economy, the mood of investors in the market, and other economic factors. In the real world, however, the market mode may have a finite number of states, such as Huiling WU School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China. Email: [email protected]. Zhongfei LI Corresponding author. Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, China. Email: [email protected]. ∗ This research is supported by the National Science Foundation for Distinguished Young Scholars under Grant No. 70825002, the National Natural Science Foundation of China under Grant No. 70518001, and the National Basic Research Program of China 973 Program, under Grant No. 2007CB814902. This paper was recommended for publication by Editor Shouyang WANG.
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
141
“bullish” and “bearish”, and could switch among them. This is so called regime switching. In a market with regime switching, the market parameters, such as the bank interest rate, stocks appreciation rates, and volatility rates, depend on the market mode and can be quite different in different states. Recently, there has been a growing interest in portfolio selection with regime switching. For example, Zhou and Yin[7] investigated a continuous-time mean-variance port[9] ¨ folio selection model with regime switching while Yin and Zhou[8] and C ¸ akmak and Ozekici [10] ¨ considered a discrete-time version. C ¸ elikyurt and Ozekici generalized the model of C ¸ akmak [9] ¨ and Ozekici by considering quadratic utility function and safety-first rule. Wei and Ye[11] studied a multi-period mean-variance portfolio optimization model with bankruptcy control in a stochastic market. Chen, Yang, and Yin[12] investigated an asset-liability management problem with regime switching by adopting the techniques of Zhou and Yin[7] . For more detailed discussions on this subject, the interested readers are referred to Costa and Araujo[13] . In the above mentioned mean-variance models with regime switching, the time horizon of the whole investment process is always assumed to be certain. But an investment horizon is practically never known with certainty at the beginning of initial investment decisions because the investor might be forced to abandon his/her original investment plan for some unexpected affairs. Therefore, recently some researchers pay their attention to mean-variance portfolio selection with uncertain time horizon. For instance, Martellini and Uroˇsevi´c[14] generalized Markowitz[1] to the situations involving an uncertain exit time and get some useful conclusions when the exit time is either independent of or dependent on the asset returns. Yi, Li, and Li[15] derived an analytical optimal strategy and efficient frontier for the mean-variance model of a multi-period asset-liability management problem under the assumption that the uncertain exit time follows a given distribution. For some early works on uncertain time horizon, one can refer to Yaari[16] , Hakansson[17], Merton[18] , and Richard[19] . As far as we know, there is no literature on dynamic mean-variance portfolio selection with both regime switching and uncertain exit time. This paper will focus on this study. We derive the optimal strategy and the efficient frontier in closed-form for a multi-period mean-variance portfolio selection problem with both regime switching and uncertain exit time. This paper proceeds as following. In Section 2, our problem and primary notations are described. An auxiliary problem is also constructed. Section 3 is devoted to the optimal strategy of the auxiliary problem. Sections 4 and 5 are concerned with the explicit expressions of the optimal strategy and the efficient frontier for the original problem, respectively. The same known results in the literature are given as special cases in Section 6. Section 7 concludes the paper.
2 Notations and Problem Formulation We assume that an investor joins the market at time 1 with an initial wealth x1 and plans to process his/her investment in planed T consecutive time periods. But he/she has to abandon his/her investment plan at an uncertain time τ for some reasons. We assume that it is known that Pt = P (τ = t), t = 1, 2, · · · , T ; PT +1 = P (τ ≥ T + 1). We let Sn denote the state of the market at the beginning of period n (i.e., the time interval [n, n+1]) and suppose that {Sn , n = 1, 2, · · · } is a time-homogeneous Markov chain with regime space S = {1, 2, 3, · · · , L} and transition matrix Q. To make readers better understand the economic meaning of market states and transition matrix, here we can give a realistic example. As we know, the movements of a market on the whole can be viewed as a composition of a primary movement and secondary movement. In the primary movement, the market trends
142
HUILING WU · ZHONGFEI LI
include broad upward and broad downward which might last several years. In the secondary market movement, the market trends include significant decline in a bull market and strong recovery in a bear market which might last several weeks or months. Let Sn = (Sn,1 , Sn,2 ), where Sn,1 ∈ {a, b} represents the primary market trend at time n (state a and b represent up-trend and down-trend, respectively) and Sn,2 ∈ {c, d} represents the secondary market trend at time n (state c and d represent decline in a bull market and recovery in a bear market, respectively). Let 1, 2, 3, 4 denote the state (a, c), (a, d), (b, c), (b, d), respectively, then the market states could switch among {1, 2, 3, 4}. If the market state is 1, it means that the primary market trend is broad upward but has significant decline, and so on. Market states will change over time, when market state at time n is i, it will switch to j (i, j ∈ {1, 2, 3, 4}) at time n + 1 with probability Q(i, j). One-step transition probability Q(i, j), an element of transition matrix Q, can be determined by historical data. There are m + 1 assets in the market whose returns depend on the states of the market. The riskless asset has deterministic and positive return rf (i) and the m risky assets have random returns Rn (i) = (Rn,1 (i), Rn,2 (i), · · · , Rn,m (i)) in period n at state Sn = i, where Rn,k (i) denotes the random return of the kth asset in period n at state Sn = i, and the superscript stands the transpose of a matrix or vector. We make the following assumptions throughout this paper. A1 For any i ∈ S, the random returns R1 (i), R2 (i), · · · , RT (i) are identically distributed with the same distribution function. In different periods, the random returns are independent, i.e., as long as n = m, Rn (i) is independent to Rm (j) for all i, j ∈ S. A2 The Markov chain S and the returns are mutually independent in the following sense: Pn (Sn+1 = j, Rn (Sn ) ∈ B) = Pn (Sn+1 = j) Pn (Rn (Sn ) ∈ B) for all B ∈ B(Rm ), j ∈ S and n = 1, 2, · · · , T ; Pn Sn+1 = j, rnf (Sn ) ∈ (a, b) = Pn (Sn+1 = j) Pn rnf (Sn ) ∈ (a, b) for all (a, b) ∈ R, j ∈ S and n = 1, 2, · · · , T , where Pn is the probability based on the information up to time n and B(Rm ) is the Borel σ-algebra on Rm . A3 0 < PT +1 ≤ 1. It’s a rational assumption because if PT +1 = 0, then the investor will definitely quit his/her investment before time T + 1 and it’s not necessary to consider the investment in T consecutive time periods. A4 Short selling is allowed for all risky assets in all periods. Unlimited borrowing and lending are permitted for the riskless asset with the interest rate equal to the rate of return of the riskless asset during that period. Transaction costs are not taken into account. A5 Capital additions or withdrawals are forbidden for all assets in all periods. Furthermore, we use the following notations in this paper. N1 Q(i, j): an element of transition matrix Q, is the one-step transition probability from i to j (i, j ∈ S). Qm (i, j) is the m-step transition probability, which can be obtained as an element of matrix Qm , the mth power of Q. rk (i) = E[Rn,k (i)]: the mean return of the kth risky asset at state i; σkl (i) = Cov (Rn,k (i), Rn,l (i)): The covariance between the kth and lth risky assets at state i; e Rn,k (i) = Rn,k (i) − rf (i): the excess return of the kth risky asset at period n at state i; e e e Rne (i) = Rn,1 (i), Rn,2 (i), · · · , Rn,m (i) ; e rke (i) = E Rn,k (i) = rk (i) − rf (i); e (i)) , which is assumed to be nonzero. re (i) = (r1e (i), r2e (i), · · · , rm
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
143
e e Cov Rn,k (i), Rn,l (i) = Cov (Rn,k (i), Rn,l (i)) = σkl (i), σ(i) = (σkl (i))k,l=1,2,··· ,m , which is assumed to be positive definite. Here we might remind that in the notations σ(i) and re (i), i doesn’t denote the ith row of a matrix or the ith entry of a vector but means that the matrix or vector depends on the market state i, which is different from the notations about matrix or vector in N3–N6 as below. N2 Ei (Z) = E(Z|S1 = i), Di (Z) = Ei (Z 2 ) − [Ei (Z)]2 . N3 Denote by Q(i) the ith row of Q. In a general way, we let a(i) be the ith component of a vector a except the notations in N1, and let diag(a) be the diagonal matrix whose diagonal elements are the components of a. Further let 1 = (1, 1, · · · , 1) . N4 For transition matrix Q and a L-dimension column vector a, we define the matrix n Qa = Qdiag(a), the matrix Qna = nth power of Qa , and the vector Qa = Qna 1. In particular, 0 1 we define Qa = 1, Qa = Qa = Qa 1 = Qa. N5 If vectors a, b, c have the same dimension, then a · c/b denotes a vector whose ith entry 2 is a(i)c(i)/b(i) and a2 a vector with (a2 )(i) = [a(i)] . N6 h, g and q are L-dimension column vectors whose ith component are, respectively, h(i) = re (i) E −1 [Rne (i)Rne (i) ] re (i), g(i) = rf (i) (1 − h(i)) , q(i) = rf2 (i) (1 − h(i)) . hm , hm and αm (m = 2, 3, · · · , T +1) are column vectors whose ith component are, respectively, T +1
hm (i) = k=m T +1
k−m
Pk Qg
(i)
k−m (i) k=m Pk Qq
T +1
k−m
k=m Pk Qg
hm (i) = T +1
h(i),
(i)
2
h(i), (i)
2 k−m T +1 (i) k=m Pk Qg h(i). αm (i) = k−m T +1 (i) k=m Pk Qq k=m
k−m
Pk Qq
In addition, we define αT +1 (i) = PT +1 h(i), αT +2 (i) = 0. n For the sake of convenience, we set k=m Ak = 0 if n < m for any {Ak }. Now, we define Wn as the wealth available for investment at time n (n = 1, 2, · · · , T + 1) and un = (un,1 , un,2 , · · · , un,m ) as the amounts invested in the risky assets (1, 2, · · · , m) at time n (n = 1, 2, · · · , T ). Then, the wealth dynamics is Wn+1 = rf (Sn )Wn + Rne (Sn ) un ,
n = 1, 2, · · · , T.
The investor wants to find an optimal investment strategy in the uncertain time-horizon to maximize his/her final wealth while minimize his/her risk. Given S1 = i ∈ S, we formulate the portfolio selection problem as: max Ei (W(T +1)∧τ ) − ωDi (W(T +1)∧τ ) P (ω) s.t. Wn+1 = rf (Sn )Wn + Rne (Sn ) un , n = 1, 2, · · · , T,
144
HUILING WU · ZHONGFEI LI
where ω > 0 is a given real number and represents the risk aversion factor. Since the objective function of problem P (ω) contains the term Di (W(T +1)∧τ ), the stochastic dynamics approach cannot be applied directly. We introduce the following auxiliary problem: ⎧ 2 ⎨ max Ei λW (T +1)∧τ − ωW(T +1)∧τ A(λ, ω) ⎩ s.t. W = r (S )W + Re (S ) u , n = 1, 2, · · · , T. n+1
f
n
n
n
n
n
Let d(u, ω) = 1 + 2ωEi (WT +1∧τ )u , and define Π A (λ, ω) = {u|u is an optimal policy of A(λ, ω)}, Π (ω) = {u|u is an optimal policy of P (ω)}. The relationship between ΠA (λ, ω) and Π (ω) is summarized in the theorem below. Theorem 1 For any u∗ ∈ P (ω), then u∗ ∈ ΠA (d(u∗ , ω), ω); Conversely, if u∗ ∈ ΠA (λ∗ , ω), then a necessary condition for u∗ ∈ Π (ω) is λ∗ = 1 + 2ωEi (WT +1∧τ )|u∗ . The proof is similar to that of Liand Ng[2] and hence omitted here. Theorem 1 implies that Π (ω) ⊆ ΠA (λ, ω). It is clear that the optimal portfolio policy of λ
P (ω) can be generated by solving the auxiliary problem A(λ, ω) which is separable in the sense of dynamic programming. Therefore, we can find a suitable λ∗ which can make u ∈ ΠA (λ∗ , ω) become the optimal strategy of problem P (ω). The second part of Theorem 1 gives the necessary condition that λ∗ should satisfy.
3 Solution to the Auxiliary Problem A(λ, ω) It is obvious that the auxiliary problem can be equivalently written as ⎧ T +1 ⎪ ⎪ 2 ⎨ max E Pn (λWn − ωWn ) i A(λ, ω) n=1 ⎪ ⎪ ⎩ s.t. W e n = 1, 2, · · · , T. n+1 = rf (Sn )Wn + Rn (Sn ) un , We first demonstrate the properties of E [Rne (i)Rne (i) ], h(i), g(i) and q(i) to show the existence of the optimal strategy of A(λ, ω). Lemma 1 E [Rne (i)Rne (i) ] is a positive definite matrix for i ∈ S and n = 1, 2, · · · , T . Proof This is straightforward because E [Rne (i)Rne (i) ] = σ(i) + re (i)re (i) and σ(i) is assumed to be positive definite. Lemma 2 0 < h(i) < 1, g(i) > 0 and q(i) > 0 for i ∈ S. Proof Since σ(i)−1 re (i)re (i) σ(i)−1 , E −1 [Rne (i)Rne (i) ] = σ(i)−1 − 1 + re (i) σ(i)−1 re (i) we have h(i) =
re (i) σ(i)−1 re (i) . 1 + re (i) σ(i)−1 re (i)
Hence, 0 < h(i) < 1 because re (i) = 0 and σ(i) is positive definite. This together with the definition of g(i) and q(i) immediately implies that g(i) > 0 and q(i) > 0. We introduce the following lemma to achieve some primary results of this paper.
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
Lemma 3 For any vector a in N 6, n n−1 Ei a(Sk ) = Qa (i),
145
n = 2, 3, · · ·.
k=2
Proof
When n = 2, Ei
2
a(Sk ) = Ei [a(S2 )] =
1
Q(i, j)a(j) = Qa (i).
j∈S
k=2
Suppose the conclusion holds for n. Then for n + 1, n+1 n+1 Ei a(Sk ) = Q(i, j)a(j)E a(Sk ) S2 = j k=2 j∈S k=3 n = Q(i, j)a(j)Ej a(Sk ) j∈S
=
k=2 n−1 Qa (i, j)Qa (j)
n−1
= (Qa Qa
)(i)
j∈S n
1)(i) = (Qna 1)(i) = Qa (i), = (Qa Qn−1 a that is, the conclusion is true for N + 1. By induction, the conclusion holds for all n ≥ 2. Now, we begin to derive the optimal policy of the auxiliary problem by using the dynamic programming approach. Define the value functions T +1 2 fn (i, xn ) = max E Pk (λWk − ωWk ) Sn = i, Wn = xn , n = 1, 2, · · · , T + 1. un ,··· ,uT k=n
Then we have the Bellman’s equation fn (i, xn ) = max E Pn (λxn − ωx2n ) + fn+1 (Sn+1 , Wn+1 ) Sn = i, Wn = xn un 2 e = max E Pn (λxn − ωxn ) + Q(i, j)fn+1 (j, rf (i)xn + Rn (i) un ) un
(1)
j∈S
for n = T, T − 1, · · · , 1, with the boundary condition fT +1 (i, xT +1 ) = PT +1 (λxT +1 − ωx2T +1 ).
(2)
Theorem 2 The value functions are given by fm (i, xm ) = −ωm (i)x2m + λm (i)xm + αm (i),
m = 1, 2, · · · , T + 1,
(3)
where ωm (i) = ωq(i)
T +1
k−(m+1)
Pk Qq
(i) + ωPm ,
(4)
(i) + λPm ,
(5)
k=m+1
λm (i) = λg(i)
T +1
k−(m+1)
Pk Qg
k=m+1
αm (i) =
T +1 λ2 λ2 k−(m+2) Q αm+1 (i) + Qαk (i). 4ω 4ω k=m+2
(6)
HUILING WU · ZHONGFEI LI
146
The corresponding optimal policy is given by ⎤ ⎡ k−(m+1) T +1 (i) λ k=m+1 Pk Qg e e (i)Rm (i) ]re (i) um (i, xm ) = ⎣ − rf (i)xm ⎦ E −1 [Rm k−(m+1) +1 2ω Tk=m+1 Pk Qq (i)
(7)
for m = 1, 2, · · · , T and i ∈ S. Proof Obviously, (3) holds true for m = T + 1. For m = T , Q(i, j)PT +1 fT (i, xT ) = max E PT (λxT − ωx2T ) + uT
j∈S
2 · λ (rf (i)xT + RTe (i) uT ) − ω (rf (i)xT + RTe (i) uT ) = max PT (λxT − ωx2T ) + PT +1 λrf (i)xT − ωrf2 (i)x2T uT
+λre (i) uT − 2ωrf (i)xT re (i) uT − ωuT E [RTe (i)RTe (i) ] uT ]} .
(8)
The optimization problem in (8) gives the first order condition λre (i) − 2ωrf (i)xT re (i) − 2ωE [RTe (i)RTe (i) ] uT = 0. Since E [RTe (i)(RTe (i)) ] is positive definite according to Lemma 1, the optimal solution of (8) exists and is λ − rf (i)xT E −1 [RTe (i)RTe (i) ]re (i). (9) uT (i, xT ) = 2ω Substituting (9) into (8) yields fT (i, xT ) = −ω [PT +1 q(i) + PT ] x2T + λ[PT +1 g(i) + PT ]xT + PT +1 h(i) = −ωT (i)x2T + λT (i)xT + αT (i),
λ2 4ω (10)
where ωT (i) = ωPT +1 q(i) + ωPT , λT (i) = λPT +1 g(i) + λPT , λ2 λ2 PT +1 h(i) = αT +1 (i). 4ω 4ω Hence, (3) and (7) hold true for m = T . Now, we assume that (3) and (7) are true for m = n+1. Then for m = n, αT (i) =
fn (i, xn ) 2 e Q(i, j)fn+1 (j, rf (i)xn + R (i) un ) = max E Pn (λxn − ωxn ) + un
j∈S
= max Pn (λxn − un
ωx2n )
−ω n+1 (i) rf2 (i)x2n + 2rf (i)xn re (i) un + un E [Rne (i)Rne (i) ] un
n+1 (i) [rf (i)xn + re (i) un ] + α n+1 (i) , +λ
(11)
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
147
where ω n+1 (i) =
Q(i, j)ωn+1 (j)
j∈S
=
Q(i, j) ωq(j)
j∈S
Pk
+ ωPn+1
T +1
k−(n+2)
Q(i, j)q(j)Qq
(j) + ωPn+1
j∈S
k=n+2
=ω
k−(n+2) Pk Qq (j)
k=n+2
T +1
=ω
T +1
Pk Qq · Qk−(n+2) 1 (i) + ωPn+1 q
k=n+2 T +1
=ω
k−(n+1)
Pk Qq
(i),
k=n+1
n+1 (i) = λ
Q(i, j)λn+1 (j)
j∈S
=
Q(i, j) λg(i)
j∈S T +1
=λ
T +1
k−(n+2) Pk Qg (j)
+ λPn+1
k=n+2 k−(n+1)
Pk Qg
(i),
k=n+1
α n+1 (i) =
j∈S
=
j∈S
=
Q(i, j)αn+1 (j) T +1 λ2 λ2 k−(n+3) Q Q(i, j) αn+2 (j) + Qαk (j) 4ω 4ω
k=n+3
λ2 λ2 Q(i, j)αn+2 (j) + 4ω 4ω j∈S
=
λ2 4ω
T +1
T +1
Q(i, j) Qk−(n+3) Qαk (j)
k=n+3 j∈S
Qk−(n+2) Qαk (i).
k=n+2
We can find ω n+1 (i) > 0 according to Lemma 2 and the assumption PT +1 > 0. Hence, the optimal solution of (11) exists and is n+1 (i) λ − rf (i)xn E −1 Rne (i) (Rne (i)) re (i) un (i, xn ) = 2 ωn+1 (i) ⎡ ⎤ k−(n+1) T +1 λ k=n+1 Pk Qg (i) =⎣ (12) − rf (i)xn ⎦ E −1 Rne (i) (Rne (i)) re (i). k−(n+1) T +1 2ω k=n+1 Pk Qq (i)
HUILING WU · ZHONGFEI LI
148
Substituting (12) into (11), we get fn (i, xn ) = − [ωPn +
ω n+1 (i)q(i)] x2n
n+1 (i) 2 λ h(i) + α n+1 (i) + λPn + λn+1 (i)g(i) xn + 4 ωn+1 (i)
= −ωn (i)x2n + λn (i)xn + αn (i),
(13)
where ωn (i) = ωPn + ω n+1 (i)q(i) = ωPn + ωq(i)
T +1
k−(n+1)
Pk Qq
(i),
k=n+1
n+1 (i)g(i) = λPn + λg(i) λn (i) = λPn + λ
T +1
k−(n+1)
Pk Qg
(i),
k=n+1 T +1 n+1 (i)]2 [λ λ2 λ2 h(i) + α n+1 (i) = αn (i) = αn+1 (i) + Qk−(n+2) Qαk (i). 4 ωn+1 (i) 4ω 4ω k=n+2
Equations (12) and (13) means that (3) and (7) hold true for m = n + 1. By induction, the conclusions of the theorem are true.
4 Solution to the Original Problem In order to obtain the solution of the original problem, we introduce the following lemmas. Lemma 4 Under the optimal strategy (7) of the auxiliary problem, T +1 n T +1
λ n−k n−2 Ei W(T +1)∧τ = Qk−2 hk · Qg (i) + P1 x1 + g(i)x1 Pn Pn Qg (i), 2ω n=2 n=2 k=2
T +1 T +1 n
λ2 n−k n−2 2 k−2 2 2 = Q (i) + P Ei W(T P h · Q x + q(i)x Pn Qq (i). n k 1 q 1 1 +1)∧τ 4ω 2 n=2 n=2 k=2
Proof
By (7), we have
Wn+1 =rf (Sn )Wn + Rne (Sn ) un = 1 − Rne (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ) rf (Sn )Wn T +1 k−(n+1) λ k=n+1 Pk Qg (Sn ) e Rn (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ), + T +1 k−(n+1) 2ω k=n+1 Pk Qq (Sn ) 2 2 = 1 − Rne (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ) rf2 (Sn )Wn2 Wn+1 T +1 k−(n+1) λ k=n+1 Pk Qg (Sn ) e Rn (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ) + k−(n+1) T +1 ω k=n+1 Pk Qq (Sn ) e −1 e · 1 − Rn (Sn ) E [Rn (Sn )Rne (Sn ) ] re (Sn ) rf (Sn )Wn ⎡ ⎤2 k−(n+1) T +1 2 P Q (S ) λ ⎣ k=n+1 k g n ⎦ + 4ω 2 T +1 P Qk−(n+1) (S ) k n q k=n+1 · re (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] Rne (Sn )Rne (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ).
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
149
Hence, E ( Wn+1 | S1 , S2 , · · · , Sn ) = g(Sn )E ( Wn | S1 , S2 , · · · , Sn ) +
λ hn+1 (Sn ), 2ω
2 2 S1 , S2 , · · · , Sn = q(Sn )E Wn2 S1 , S2 , · · · , Sn + λ hn+1 (Sn ). E Wn+1 4ω 2 Noting that E ( Wn | S1 , S2 , · · · , Sn ) = E ( Wn | S1 , S2 , · · · , Sn−1 ) , E Wn2 S1 , S2 , · · · , Sn = E Wn2 S1 , S2 , · · · , Sn−1 ,
the above equations are recursive equations. By repeatedly using them we can obtain n
E ( Wn+1 | S1 , S2 , · · · , Sn ) = x1
g(Sk ) +
k=1
E
2 S1 , S2 , · · · Wn+1
, Sn =
n
x21
k=1
n+1 n λ hk (Sk−1 ) g(Sl ), 2ω k=2
l=k
n+1 n λ2 q(Sk ) + hk (Sk−1 ) q(Sl ). 4ω 2 k=2
l=k
Under the assumption that S1 = i, we have Ei (Wn+1 ) = Ei [E ( Wn+1 | S1 = i, S2 , · · · , Sn )] n n+1 n λ Ei g(Sk ) + hk (Sk−1 ) g(Sl ) = g(i)x1 Ei 2ω k=2
n−1
= g(i)x1 Qg
n−1
= g(i)x1 Qg
k=2
(i) +
(i) +
λ 2ω
n+1
l=k
n+1−k
Qk−2 (i, j)hk (j)Qg
(j) (by Lemma 3)
k=2 i∈S
n+1
λ k−2 n+1−k Q hk · Q g (i), 2ω k=2
that is, n−2
Ei (Wn ) = g(i)x1 Qg
(i) +
n
λ k−2 n−k Q hk · Q g (i), 2ω
n = 2, 3, · · · , T + 1.
(14)
k=2
Similarly, we can get n
2 λ2 k−2 n−k 2 n−2 Ei Wn = q(i)x1 Qq (i) + Q (i), h · Q k q 4ω 2
n = 2, 3, · · · , T + 1.
k=2
Substituting (14) and (15) into T +1 T +1 Ei W(T +1)∧τ = Ei Pn Wn = Pn Ei (Wn ) n=1
and
Ei
2 W(T +1)∧τ
= Ei
T +1 n=1
n=1
Pn Wn2
=
T +1 n=1
Pn Ei Wn2 ,
(15)
HUILING WU · ZHONGFEI LI
150
respectively, we get the conclusion of the lemma. Lemma 5 0
0. Hence, the equality in (16) holds and it’s obvious n=2 Pn k=2 Qk−2 hk · Qq
T +1 n k−2 n−k To prove the inequality n=2 Pn k=2 Q (i) < 1, we discuss the following hk · Qq two cases. When T = 1, T +1
Pn
n=2
n 2
n−k 2−k Qk−2 hk · Qg (i) = P2 Qk−2 hk · Qg (i) k=2
T +1 h2 (i) =
k=2 T +1 k=2
k=2
0 = P2 Q0 h2 · Qg (i) = P2 h2 (i), k−2
Pk Qg
(i)
k−2 Pk Qq (i)
2−2
h(i) =
P2 Qg
(i)
h(i) 2−2 P2 Qq (i)
= h(i),
then by Lemma 2, we have T +1 n=2
n
n−k Qk−2 hk · Qg (i) = P2 h2 (i) = P2 h(i) ≤ h(i) < 1; Pn k=2
When T ≥ 2, the investment will process at least two periods. So we can claim that Di (W(T +1)∧τ ) > 0 since Di (W(T +1)∧τ ) measures the risk of the investment at the terminal time and the risk cannot be completely hedged when all the primal parameters, including the
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
151
return of the riskless asset, depend on the market states modulated by the Markov chain. By Lemma 4 we have Di (W(T +1)∧τ ) =
T +1 T +1 n
λ2 n−k n−2 k−2 2 2 Q (i) + P P h · Q x + q(i)x Pn Qq (i) n k 1 1 q 1 4ω 2 n=2 n=2 k=2 #2 T +1 n T +1
λ n−k n−2 k−2 − Q hk · Qg (i) + P1 x1 + g(i)x1 Pn Pn Qg (i) . 2ω n=2 n=2
(17)
k=2
In particular, when x1 = 0 we have 0 < Di (W(T +1)∧τ ) T +1 n
λ2 n−k k−2 = Q (i) − P h · Q n k q 4ω 2 n=2
k=2
i.e.,
T +1 n=2
n
n−k Qk−2 hk · Qg (i) Pn
#2 T +1 n
λ n−k k−2 Q hk · Q g (i) , Pn 2ω n=2 k=2
#2
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION WITH MARKOV REGIME SWITCHING AND UNCERTAIN TIME-HORIZON∗ Huiling WU · Zhongfei LI
DOI: 10.1007/s11424-011-9184-z Received: 1 July 2009 / Revised: 6 May 2010 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2011 Abstract This paper investigates a multi-period mean-variance portfolio selection with regime switching and uncertain exit time. The returns of assets all depend on the states of the stochastic market which are assumed to follow a discrete-time Markov chain. The authors derive the optimal strategy and the efficient frontier of the model in closed-form. Some results in the existing literature are obtained as special cases of our results. Key words Dynamic programming, Markov regime switching, mean-variance, portfolio selection, uncertain time-horizon.
1 Introduction Portfolio selection is to search a best allocation of wealth among different assets in markets. The mean-variance model pioneered by Markowitz[1] provides a fundamental basis for portfolio selection and has stimulated hundreds of extensions and applications. One of them is to extend the original static mean-variance model to discrete-time dynamic mean-variance models. This line is full of challenges and enormous difficulties and has not made a great breakthrough until the paper of Li and Ng[2] , which derived the analytical optimal portfolio policy by using an embedding technique. Since then, there has been a fast development in the dynamic mean-variance theory. For instance, Guo and Hu[3] investigated a multi-period mean-variance portfolio optimization problem with uncertain exit time. Leippold, Trojani, and Vanini[4] studied a multi-period mean-variance model for asset-liability management. Zhu, Li, and Wang[5] investigated multi-period mean-variance portfolio with risk control over bankruptcy. Some other generalizations can be found in [6]. The works mentioned above only consider one state of the market mode that reflects the state of the underlying economy, the mood of investors in the market, and other economic factors. In the real world, however, the market mode may have a finite number of states, such as Huiling WU School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China. Email: [email protected]. Zhongfei LI Corresponding author. Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, China. Email: [email protected]. ∗ This research is supported by the National Science Foundation for Distinguished Young Scholars under Grant No. 70825002, the National Natural Science Foundation of China under Grant No. 70518001, and the National Basic Research Program of China 973 Program, under Grant No. 2007CB814902. This paper was recommended for publication by Editor Shouyang WANG.
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
141
“bullish” and “bearish”, and could switch among them. This is so called regime switching. In a market with regime switching, the market parameters, such as the bank interest rate, stocks appreciation rates, and volatility rates, depend on the market mode and can be quite different in different states. Recently, there has been a growing interest in portfolio selection with regime switching. For example, Zhou and Yin[7] investigated a continuous-time mean-variance port[9] ¨ folio selection model with regime switching while Yin and Zhou[8] and C ¸ akmak and Ozekici [10] ¨ considered a discrete-time version. C ¸ elikyurt and Ozekici generalized the model of C ¸ akmak [9] ¨ and Ozekici by considering quadratic utility function and safety-first rule. Wei and Ye[11] studied a multi-period mean-variance portfolio optimization model with bankruptcy control in a stochastic market. Chen, Yang, and Yin[12] investigated an asset-liability management problem with regime switching by adopting the techniques of Zhou and Yin[7] . For more detailed discussions on this subject, the interested readers are referred to Costa and Araujo[13] . In the above mentioned mean-variance models with regime switching, the time horizon of the whole investment process is always assumed to be certain. But an investment horizon is practically never known with certainty at the beginning of initial investment decisions because the investor might be forced to abandon his/her original investment plan for some unexpected affairs. Therefore, recently some researchers pay their attention to mean-variance portfolio selection with uncertain time horizon. For instance, Martellini and Uroˇsevi´c[14] generalized Markowitz[1] to the situations involving an uncertain exit time and get some useful conclusions when the exit time is either independent of or dependent on the asset returns. Yi, Li, and Li[15] derived an analytical optimal strategy and efficient frontier for the mean-variance model of a multi-period asset-liability management problem under the assumption that the uncertain exit time follows a given distribution. For some early works on uncertain time horizon, one can refer to Yaari[16] , Hakansson[17], Merton[18] , and Richard[19] . As far as we know, there is no literature on dynamic mean-variance portfolio selection with both regime switching and uncertain exit time. This paper will focus on this study. We derive the optimal strategy and the efficient frontier in closed-form for a multi-period mean-variance portfolio selection problem with both regime switching and uncertain exit time. This paper proceeds as following. In Section 2, our problem and primary notations are described. An auxiliary problem is also constructed. Section 3 is devoted to the optimal strategy of the auxiliary problem. Sections 4 and 5 are concerned with the explicit expressions of the optimal strategy and the efficient frontier for the original problem, respectively. The same known results in the literature are given as special cases in Section 6. Section 7 concludes the paper.
2 Notations and Problem Formulation We assume that an investor joins the market at time 1 with an initial wealth x1 and plans to process his/her investment in planed T consecutive time periods. But he/she has to abandon his/her investment plan at an uncertain time τ for some reasons. We assume that it is known that Pt = P (τ = t), t = 1, 2, · · · , T ; PT +1 = P (τ ≥ T + 1). We let Sn denote the state of the market at the beginning of period n (i.e., the time interval [n, n+1]) and suppose that {Sn , n = 1, 2, · · · } is a time-homogeneous Markov chain with regime space S = {1, 2, 3, · · · , L} and transition matrix Q. To make readers better understand the economic meaning of market states and transition matrix, here we can give a realistic example. As we know, the movements of a market on the whole can be viewed as a composition of a primary movement and secondary movement. In the primary movement, the market trends
142
HUILING WU · ZHONGFEI LI
include broad upward and broad downward which might last several years. In the secondary market movement, the market trends include significant decline in a bull market and strong recovery in a bear market which might last several weeks or months. Let Sn = (Sn,1 , Sn,2 ), where Sn,1 ∈ {a, b} represents the primary market trend at time n (state a and b represent up-trend and down-trend, respectively) and Sn,2 ∈ {c, d} represents the secondary market trend at time n (state c and d represent decline in a bull market and recovery in a bear market, respectively). Let 1, 2, 3, 4 denote the state (a, c), (a, d), (b, c), (b, d), respectively, then the market states could switch among {1, 2, 3, 4}. If the market state is 1, it means that the primary market trend is broad upward but has significant decline, and so on. Market states will change over time, when market state at time n is i, it will switch to j (i, j ∈ {1, 2, 3, 4}) at time n + 1 with probability Q(i, j). One-step transition probability Q(i, j), an element of transition matrix Q, can be determined by historical data. There are m + 1 assets in the market whose returns depend on the states of the market. The riskless asset has deterministic and positive return rf (i) and the m risky assets have random returns Rn (i) = (Rn,1 (i), Rn,2 (i), · · · , Rn,m (i)) in period n at state Sn = i, where Rn,k (i) denotes the random return of the kth asset in period n at state Sn = i, and the superscript stands the transpose of a matrix or vector. We make the following assumptions throughout this paper. A1 For any i ∈ S, the random returns R1 (i), R2 (i), · · · , RT (i) are identically distributed with the same distribution function. In different periods, the random returns are independent, i.e., as long as n = m, Rn (i) is independent to Rm (j) for all i, j ∈ S. A2 The Markov chain S and the returns are mutually independent in the following sense: Pn (Sn+1 = j, Rn (Sn ) ∈ B) = Pn (Sn+1 = j) Pn (Rn (Sn ) ∈ B) for all B ∈ B(Rm ), j ∈ S and n = 1, 2, · · · , T ; Pn Sn+1 = j, rnf (Sn ) ∈ (a, b) = Pn (Sn+1 = j) Pn rnf (Sn ) ∈ (a, b) for all (a, b) ∈ R, j ∈ S and n = 1, 2, · · · , T , where Pn is the probability based on the information up to time n and B(Rm ) is the Borel σ-algebra on Rm . A3 0 < PT +1 ≤ 1. It’s a rational assumption because if PT +1 = 0, then the investor will definitely quit his/her investment before time T + 1 and it’s not necessary to consider the investment in T consecutive time periods. A4 Short selling is allowed for all risky assets in all periods. Unlimited borrowing and lending are permitted for the riskless asset with the interest rate equal to the rate of return of the riskless asset during that period. Transaction costs are not taken into account. A5 Capital additions or withdrawals are forbidden for all assets in all periods. Furthermore, we use the following notations in this paper. N1 Q(i, j): an element of transition matrix Q, is the one-step transition probability from i to j (i, j ∈ S). Qm (i, j) is the m-step transition probability, which can be obtained as an element of matrix Qm , the mth power of Q. rk (i) = E[Rn,k (i)]: the mean return of the kth risky asset at state i; σkl (i) = Cov (Rn,k (i), Rn,l (i)): The covariance between the kth and lth risky assets at state i; e Rn,k (i) = Rn,k (i) − rf (i): the excess return of the kth risky asset at period n at state i; e e e Rne (i) = Rn,1 (i), Rn,2 (i), · · · , Rn,m (i) ; e rke (i) = E Rn,k (i) = rk (i) − rf (i); e (i)) , which is assumed to be nonzero. re (i) = (r1e (i), r2e (i), · · · , rm
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
143
e e Cov Rn,k (i), Rn,l (i) = Cov (Rn,k (i), Rn,l (i)) = σkl (i), σ(i) = (σkl (i))k,l=1,2,··· ,m , which is assumed to be positive definite. Here we might remind that in the notations σ(i) and re (i), i doesn’t denote the ith row of a matrix or the ith entry of a vector but means that the matrix or vector depends on the market state i, which is different from the notations about matrix or vector in N3–N6 as below. N2 Ei (Z) = E(Z|S1 = i), Di (Z) = Ei (Z 2 ) − [Ei (Z)]2 . N3 Denote by Q(i) the ith row of Q. In a general way, we let a(i) be the ith component of a vector a except the notations in N1, and let diag(a) be the diagonal matrix whose diagonal elements are the components of a. Further let 1 = (1, 1, · · · , 1) . N4 For transition matrix Q and a L-dimension column vector a, we define the matrix n Qa = Qdiag(a), the matrix Qna = nth power of Qa , and the vector Qa = Qna 1. In particular, 0 1 we define Qa = 1, Qa = Qa = Qa 1 = Qa. N5 If vectors a, b, c have the same dimension, then a · c/b denotes a vector whose ith entry 2 is a(i)c(i)/b(i) and a2 a vector with (a2 )(i) = [a(i)] . N6 h, g and q are L-dimension column vectors whose ith component are, respectively, h(i) = re (i) E −1 [Rne (i)Rne (i) ] re (i), g(i) = rf (i) (1 − h(i)) , q(i) = rf2 (i) (1 − h(i)) . hm , hm and αm (m = 2, 3, · · · , T +1) are column vectors whose ith component are, respectively, T +1
hm (i) = k=m T +1
k−m
Pk Qg
(i)
k−m (i) k=m Pk Qq
T +1
k−m
k=m Pk Qg
hm (i) = T +1
h(i),
(i)
2
h(i), (i)
2 k−m T +1 (i) k=m Pk Qg h(i). αm (i) = k−m T +1 (i) k=m Pk Qq k=m
k−m
Pk Qq
In addition, we define αT +1 (i) = PT +1 h(i), αT +2 (i) = 0. n For the sake of convenience, we set k=m Ak = 0 if n < m for any {Ak }. Now, we define Wn as the wealth available for investment at time n (n = 1, 2, · · · , T + 1) and un = (un,1 , un,2 , · · · , un,m ) as the amounts invested in the risky assets (1, 2, · · · , m) at time n (n = 1, 2, · · · , T ). Then, the wealth dynamics is Wn+1 = rf (Sn )Wn + Rne (Sn ) un ,
n = 1, 2, · · · , T.
The investor wants to find an optimal investment strategy in the uncertain time-horizon to maximize his/her final wealth while minimize his/her risk. Given S1 = i ∈ S, we formulate the portfolio selection problem as: max Ei (W(T +1)∧τ ) − ωDi (W(T +1)∧τ ) P (ω) s.t. Wn+1 = rf (Sn )Wn + Rne (Sn ) un , n = 1, 2, · · · , T,
144
HUILING WU · ZHONGFEI LI
where ω > 0 is a given real number and represents the risk aversion factor. Since the objective function of problem P (ω) contains the term Di (W(T +1)∧τ ), the stochastic dynamics approach cannot be applied directly. We introduce the following auxiliary problem: ⎧ 2 ⎨ max Ei λW (T +1)∧τ − ωW(T +1)∧τ A(λ, ω) ⎩ s.t. W = r (S )W + Re (S ) u , n = 1, 2, · · · , T. n+1
f
n
n
n
n
n
Let d(u, ω) = 1 + 2ωEi (WT +1∧τ )u , and define Π A (λ, ω) = {u|u is an optimal policy of A(λ, ω)}, Π (ω) = {u|u is an optimal policy of P (ω)}. The relationship between ΠA (λ, ω) and Π (ω) is summarized in the theorem below. Theorem 1 For any u∗ ∈ P (ω), then u∗ ∈ ΠA (d(u∗ , ω), ω); Conversely, if u∗ ∈ ΠA (λ∗ , ω), then a necessary condition for u∗ ∈ Π (ω) is λ∗ = 1 + 2ωEi (WT +1∧τ )|u∗ . The proof is similar to that of Liand Ng[2] and hence omitted here. Theorem 1 implies that Π (ω) ⊆ ΠA (λ, ω). It is clear that the optimal portfolio policy of λ
P (ω) can be generated by solving the auxiliary problem A(λ, ω) which is separable in the sense of dynamic programming. Therefore, we can find a suitable λ∗ which can make u ∈ ΠA (λ∗ , ω) become the optimal strategy of problem P (ω). The second part of Theorem 1 gives the necessary condition that λ∗ should satisfy.
3 Solution to the Auxiliary Problem A(λ, ω) It is obvious that the auxiliary problem can be equivalently written as ⎧ T +1 ⎪ ⎪ 2 ⎨ max E Pn (λWn − ωWn ) i A(λ, ω) n=1 ⎪ ⎪ ⎩ s.t. W e n = 1, 2, · · · , T. n+1 = rf (Sn )Wn + Rn (Sn ) un , We first demonstrate the properties of E [Rne (i)Rne (i) ], h(i), g(i) and q(i) to show the existence of the optimal strategy of A(λ, ω). Lemma 1 E [Rne (i)Rne (i) ] is a positive definite matrix for i ∈ S and n = 1, 2, · · · , T . Proof This is straightforward because E [Rne (i)Rne (i) ] = σ(i) + re (i)re (i) and σ(i) is assumed to be positive definite. Lemma 2 0 < h(i) < 1, g(i) > 0 and q(i) > 0 for i ∈ S. Proof Since σ(i)−1 re (i)re (i) σ(i)−1 , E −1 [Rne (i)Rne (i) ] = σ(i)−1 − 1 + re (i) σ(i)−1 re (i) we have h(i) =
re (i) σ(i)−1 re (i) . 1 + re (i) σ(i)−1 re (i)
Hence, 0 < h(i) < 1 because re (i) = 0 and σ(i) is positive definite. This together with the definition of g(i) and q(i) immediately implies that g(i) > 0 and q(i) > 0. We introduce the following lemma to achieve some primary results of this paper.
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
Lemma 3 For any vector a in N 6, n n−1 Ei a(Sk ) = Qa (i),
145
n = 2, 3, · · ·.
k=2
Proof
When n = 2, Ei
2
a(Sk ) = Ei [a(S2 )] =
1
Q(i, j)a(j) = Qa (i).
j∈S
k=2
Suppose the conclusion holds for n. Then for n + 1, n+1 n+1 Ei a(Sk ) = Q(i, j)a(j)E a(Sk ) S2 = j k=2 j∈S k=3 n = Q(i, j)a(j)Ej a(Sk ) j∈S
=
k=2 n−1 Qa (i, j)Qa (j)
n−1
= (Qa Qa
)(i)
j∈S n
1)(i) = (Qna 1)(i) = Qa (i), = (Qa Qn−1 a that is, the conclusion is true for N + 1. By induction, the conclusion holds for all n ≥ 2. Now, we begin to derive the optimal policy of the auxiliary problem by using the dynamic programming approach. Define the value functions T +1 2 fn (i, xn ) = max E Pk (λWk − ωWk ) Sn = i, Wn = xn , n = 1, 2, · · · , T + 1. un ,··· ,uT k=n
Then we have the Bellman’s equation fn (i, xn ) = max E Pn (λxn − ωx2n ) + fn+1 (Sn+1 , Wn+1 ) Sn = i, Wn = xn un 2 e = max E Pn (λxn − ωxn ) + Q(i, j)fn+1 (j, rf (i)xn + Rn (i) un ) un
(1)
j∈S
for n = T, T − 1, · · · , 1, with the boundary condition fT +1 (i, xT +1 ) = PT +1 (λxT +1 − ωx2T +1 ).
(2)
Theorem 2 The value functions are given by fm (i, xm ) = −ωm (i)x2m + λm (i)xm + αm (i),
m = 1, 2, · · · , T + 1,
(3)
where ωm (i) = ωq(i)
T +1
k−(m+1)
Pk Qq
(i) + ωPm ,
(4)
(i) + λPm ,
(5)
k=m+1
λm (i) = λg(i)
T +1
k−(m+1)
Pk Qg
k=m+1
αm (i) =
T +1 λ2 λ2 k−(m+2) Q αm+1 (i) + Qαk (i). 4ω 4ω k=m+2
(6)
HUILING WU · ZHONGFEI LI
146
The corresponding optimal policy is given by ⎤ ⎡ k−(m+1) T +1 (i) λ k=m+1 Pk Qg e e (i)Rm (i) ]re (i) um (i, xm ) = ⎣ − rf (i)xm ⎦ E −1 [Rm k−(m+1) +1 2ω Tk=m+1 Pk Qq (i)
(7)
for m = 1, 2, · · · , T and i ∈ S. Proof Obviously, (3) holds true for m = T + 1. For m = T , Q(i, j)PT +1 fT (i, xT ) = max E PT (λxT − ωx2T ) + uT
j∈S
2 · λ (rf (i)xT + RTe (i) uT ) − ω (rf (i)xT + RTe (i) uT ) = max PT (λxT − ωx2T ) + PT +1 λrf (i)xT − ωrf2 (i)x2T uT
+λre (i) uT − 2ωrf (i)xT re (i) uT − ωuT E [RTe (i)RTe (i) ] uT ]} .
(8)
The optimization problem in (8) gives the first order condition λre (i) − 2ωrf (i)xT re (i) − 2ωE [RTe (i)RTe (i) ] uT = 0. Since E [RTe (i)(RTe (i)) ] is positive definite according to Lemma 1, the optimal solution of (8) exists and is λ − rf (i)xT E −1 [RTe (i)RTe (i) ]re (i). (9) uT (i, xT ) = 2ω Substituting (9) into (8) yields fT (i, xT ) = −ω [PT +1 q(i) + PT ] x2T + λ[PT +1 g(i) + PT ]xT + PT +1 h(i) = −ωT (i)x2T + λT (i)xT + αT (i),
λ2 4ω (10)
where ωT (i) = ωPT +1 q(i) + ωPT , λT (i) = λPT +1 g(i) + λPT , λ2 λ2 PT +1 h(i) = αT +1 (i). 4ω 4ω Hence, (3) and (7) hold true for m = T . Now, we assume that (3) and (7) are true for m = n+1. Then for m = n, αT (i) =
fn (i, xn ) 2 e Q(i, j)fn+1 (j, rf (i)xn + R (i) un ) = max E Pn (λxn − ωxn ) + un
j∈S
= max Pn (λxn − un
ωx2n )
−ω n+1 (i) rf2 (i)x2n + 2rf (i)xn re (i) un + un E [Rne (i)Rne (i) ] un
n+1 (i) [rf (i)xn + re (i) un ] + α n+1 (i) , +λ
(11)
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
147
where ω n+1 (i) =
Q(i, j)ωn+1 (j)
j∈S
=
Q(i, j) ωq(j)
j∈S
Pk
+ ωPn+1
T +1
k−(n+2)
Q(i, j)q(j)Qq
(j) + ωPn+1
j∈S
k=n+2
=ω
k−(n+2) Pk Qq (j)
k=n+2
T +1
=ω
T +1
Pk Qq · Qk−(n+2) 1 (i) + ωPn+1 q
k=n+2 T +1
=ω
k−(n+1)
Pk Qq
(i),
k=n+1
n+1 (i) = λ
Q(i, j)λn+1 (j)
j∈S
=
Q(i, j) λg(i)
j∈S T +1
=λ
T +1
k−(n+2) Pk Qg (j)
+ λPn+1
k=n+2 k−(n+1)
Pk Qg
(i),
k=n+1
α n+1 (i) =
j∈S
=
j∈S
=
Q(i, j)αn+1 (j) T +1 λ2 λ2 k−(n+3) Q Q(i, j) αn+2 (j) + Qαk (j) 4ω 4ω
k=n+3
λ2 λ2 Q(i, j)αn+2 (j) + 4ω 4ω j∈S
=
λ2 4ω
T +1
T +1
Q(i, j) Qk−(n+3) Qαk (j)
k=n+3 j∈S
Qk−(n+2) Qαk (i).
k=n+2
We can find ω n+1 (i) > 0 according to Lemma 2 and the assumption PT +1 > 0. Hence, the optimal solution of (11) exists and is n+1 (i) λ − rf (i)xn E −1 Rne (i) (Rne (i)) re (i) un (i, xn ) = 2 ωn+1 (i) ⎡ ⎤ k−(n+1) T +1 λ k=n+1 Pk Qg (i) =⎣ (12) − rf (i)xn ⎦ E −1 Rne (i) (Rne (i)) re (i). k−(n+1) T +1 2ω k=n+1 Pk Qq (i)
HUILING WU · ZHONGFEI LI
148
Substituting (12) into (11), we get fn (i, xn ) = − [ωPn +
ω n+1 (i)q(i)] x2n
n+1 (i) 2 λ h(i) + α n+1 (i) + λPn + λn+1 (i)g(i) xn + 4 ωn+1 (i)
= −ωn (i)x2n + λn (i)xn + αn (i),
(13)
where ωn (i) = ωPn + ω n+1 (i)q(i) = ωPn + ωq(i)
T +1
k−(n+1)
Pk Qq
(i),
k=n+1
n+1 (i)g(i) = λPn + λg(i) λn (i) = λPn + λ
T +1
k−(n+1)
Pk Qg
(i),
k=n+1 T +1 n+1 (i)]2 [λ λ2 λ2 h(i) + α n+1 (i) = αn (i) = αn+1 (i) + Qk−(n+2) Qαk (i). 4 ωn+1 (i) 4ω 4ω k=n+2
Equations (12) and (13) means that (3) and (7) hold true for m = n + 1. By induction, the conclusions of the theorem are true.
4 Solution to the Original Problem In order to obtain the solution of the original problem, we introduce the following lemmas. Lemma 4 Under the optimal strategy (7) of the auxiliary problem, T +1 n T +1
λ n−k n−2 Ei W(T +1)∧τ = Qk−2 hk · Qg (i) + P1 x1 + g(i)x1 Pn Pn Qg (i), 2ω n=2 n=2 k=2
T +1 T +1 n
λ2 n−k n−2 2 k−2 2 2 = Q (i) + P Ei W(T P h · Q x + q(i)x Pn Qq (i). n k 1 q 1 1 +1)∧τ 4ω 2 n=2 n=2 k=2
Proof
By (7), we have
Wn+1 =rf (Sn )Wn + Rne (Sn ) un = 1 − Rne (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ) rf (Sn )Wn T +1 k−(n+1) λ k=n+1 Pk Qg (Sn ) e Rn (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ), + T +1 k−(n+1) 2ω k=n+1 Pk Qq (Sn ) 2 2 = 1 − Rne (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ) rf2 (Sn )Wn2 Wn+1 T +1 k−(n+1) λ k=n+1 Pk Qg (Sn ) e Rn (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ) + k−(n+1) T +1 ω k=n+1 Pk Qq (Sn ) e −1 e · 1 − Rn (Sn ) E [Rn (Sn )Rne (Sn ) ] re (Sn ) rf (Sn )Wn ⎡ ⎤2 k−(n+1) T +1 2 P Q (S ) λ ⎣ k=n+1 k g n ⎦ + 4ω 2 T +1 P Qk−(n+1) (S ) k n q k=n+1 · re (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] Rne (Sn )Rne (Sn ) E −1 [Rne (Sn )Rne (Sn ) ] re (Sn ).
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
149
Hence, E ( Wn+1 | S1 , S2 , · · · , Sn ) = g(Sn )E ( Wn | S1 , S2 , · · · , Sn ) +
λ hn+1 (Sn ), 2ω
2 2 S1 , S2 , · · · , Sn = q(Sn )E Wn2 S1 , S2 , · · · , Sn + λ hn+1 (Sn ). E Wn+1 4ω 2 Noting that E ( Wn | S1 , S2 , · · · , Sn ) = E ( Wn | S1 , S2 , · · · , Sn−1 ) , E Wn2 S1 , S2 , · · · , Sn = E Wn2 S1 , S2 , · · · , Sn−1 ,
the above equations are recursive equations. By repeatedly using them we can obtain n
E ( Wn+1 | S1 , S2 , · · · , Sn ) = x1
g(Sk ) +
k=1
E
2 S1 , S2 , · · · Wn+1
, Sn =
n
x21
k=1
n+1 n λ hk (Sk−1 ) g(Sl ), 2ω k=2
l=k
n+1 n λ2 q(Sk ) + hk (Sk−1 ) q(Sl ). 4ω 2 k=2
l=k
Under the assumption that S1 = i, we have Ei (Wn+1 ) = Ei [E ( Wn+1 | S1 = i, S2 , · · · , Sn )] n n+1 n λ Ei g(Sk ) + hk (Sk−1 ) g(Sl ) = g(i)x1 Ei 2ω k=2
n−1
= g(i)x1 Qg
n−1
= g(i)x1 Qg
k=2
(i) +
(i) +
λ 2ω
n+1
l=k
n+1−k
Qk−2 (i, j)hk (j)Qg
(j) (by Lemma 3)
k=2 i∈S
n+1
λ k−2 n+1−k Q hk · Q g (i), 2ω k=2
that is, n−2
Ei (Wn ) = g(i)x1 Qg
(i) +
n
λ k−2 n−k Q hk · Q g (i), 2ω
n = 2, 3, · · · , T + 1.
(14)
k=2
Similarly, we can get n
2 λ2 k−2 n−k 2 n−2 Ei Wn = q(i)x1 Qq (i) + Q (i), h · Q k q 4ω 2
n = 2, 3, · · · , T + 1.
k=2
Substituting (14) and (15) into T +1 T +1 Ei W(T +1)∧τ = Ei Pn Wn = Pn Ei (Wn ) n=1
and
Ei
2 W(T +1)∧τ
= Ei
T +1 n=1
n=1
Pn Wn2
=
T +1 n=1
Pn Ei Wn2 ,
(15)
HUILING WU · ZHONGFEI LI
150
respectively, we get the conclusion of the lemma. Lemma 5 0
0. Hence, the equality in (16) holds and it’s obvious n=2 Pn k=2 Qk−2 hk · Qq
T +1 n k−2 n−k To prove the inequality n=2 Pn k=2 Q (i) < 1, we discuss the following hk · Qq two cases. When T = 1, T +1
Pn
n=2
n 2
n−k 2−k Qk−2 hk · Qg (i) = P2 Qk−2 hk · Qg (i) k=2
T +1 h2 (i) =
k=2 T +1 k=2
k=2
0 = P2 Q0 h2 · Qg (i) = P2 h2 (i), k−2
Pk Qg
(i)
k−2 Pk Qq (i)
2−2
h(i) =
P2 Qg
(i)
h(i) 2−2 P2 Qq (i)
= h(i),
then by Lemma 2, we have T +1 n=2
n
n−k Qk−2 hk · Qg (i) = P2 h2 (i) = P2 h(i) ≤ h(i) < 1; Pn k=2
When T ≥ 2, the investment will process at least two periods. So we can claim that Di (W(T +1)∧τ ) > 0 since Di (W(T +1)∧τ ) measures the risk of the investment at the terminal time and the risk cannot be completely hedged when all the primal parameters, including the
MULTI-PERIOD MEAN-VARIANCE PORTFOLIO SELECTION
151
return of the riskless asset, depend on the market states modulated by the Markov chain. By Lemma 4 we have Di (W(T +1)∧τ ) =
T +1 T +1 n
λ2 n−k n−2 k−2 2 2 Q (i) + P P h · Q x + q(i)x Pn Qq (i) n k 1 1 q 1 4ω 2 n=2 n=2 k=2 #2 T +1 n T +1
λ n−k n−2 k−2 − Q hk · Qg (i) + P1 x1 + g(i)x1 Pn Pn Qg (i) . 2ω n=2 n=2
(17)
k=2
In particular, when x1 = 0 we have 0 < Di (W(T +1)∧τ ) T +1 n
λ2 n−k k−2 = Q (i) − P h · Q n k q 4ω 2 n=2
k=2
i.e.,
T +1 n=2
n
n−k Qk−2 hk · Qg (i) Pn
#2 T +1 n
λ n−k k−2 Q hk · Q g (i) , Pn 2ω n=2 k=2
#2
