Continuous-time portfolio selection with liability: Mean variance model and stochastic LQ approach

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1 Insurance: Mathematics and Economics 42 (28) Continuous-time portfolio selection with liability: Mean variance model and stochastic LQ approach Shuxiang Xie a, Zhongfei Li b,c,, Shouyang Wang d a Department of Probability and Statistics, School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 51275, PR China b Department of Risk Management and Insurance, Lingnan (University) College, Sun Yat-sen University, Guangzhou 51275, PR China c Institute for Quantitative Finance and Insurance, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 d Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 18, PR China Received October 26; received in revised form August 27; accepted 28 October 27 Abstract In this paper we formulate a continuous-time mean variance portfolio selection model with multiple risky assets and one liability in an incomplete market. he risky assets prices are governed by geometric Brownian motions while the liability evolves according to a Brownian motion with drift. he correlations between the risky assets and the liability are considered. he objective is to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. We derive explicitly the optimal dynamic strategy and the mean variance efficient frontier in closed forms by using the general stochastic linear-quadratic (LQ) control technique. Several special cases are discussed and a numerical example is also given. c 27 Elsevier B.V. All rights reserved. Keywords: Portfolio selection; Asset liability management; Continuous-time; Mean variance model; Stochastic linear-quadratic control 1. Introduction Since the pioneer work of Markowitz (1952) the mean variance (M V) portfolio selection model has inspired literally hundreds of extensions and applications. For example, Merton (1971) studied a continuous-time model with consumption; Koo (1998) took labor income into account; Li and Ng (2) studied a dynamic multi-period M V problem; Zhou and Li (2) investigated a continuous-time M V portfolio problem in a stochastic LQ framework; Zhu et al. (24) proposed a dynamic multi-period M V portfolio selection model with his research is supported in part by grants of the National Science Foundation of China (No , 75181), Program for New Century Excellent alents in University (NCE-4-798) of China, Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 2267), and National Basic Research Program of China (973 Program, No. 27CB81492). Corresponding author at: Department of Risk Management and Insurance, Lingnan (University) College, Sun Yat-sen University, Guangzhou 51275, PR China. el.: ; fax: addresses: xieshuxiang@gmail.com (S. Xie), lnslzf@mail.sysu.edu.cn (Z. Li), sywang@amss.ac.cn (S. Wang) /$ - see front matter c 27 Elsevier B.V. All rights reserved. doi:1.116/j.insmatheco bankruptcy prohibition, while Bielecki et al. (25) sought the same problem in a continuous-time setting. For more detailed discussions on this subject, one can refer to Li and Wang (21) and Wang and Xia (22). he above mentioned authors did not incorporate liability into their models. However, in most of the real-world situations, liability is an important factor which almost all investors should cope with. It is clear that the introduction of liability in a portfolio selection model will make it more practical. he existing literature on liability focused primarily on market models of firm s asset liability management (ALM), optimal dividend pay-out and ruin problems; see, for example, Sharpe and int (199), Keel and Muller (1995), Norberg (1999), Gerber and Shiu (24), and Decamps et al. (26). However, the research on dynamic M V portfolio selection with liability is limited, and it evokes recent concern. Leippold et al. (24) studied a multi-period asset liability management problem under the M V criteria, and derived explicit expressions for the optimal strategy and the M V efficient frontier by using a geometric approach and the embedding technique of Li and Ng (2). Chiu and Li (26) 转载

2 944 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) investigated a continuous-time asset liability management problem under the M V criteria and in the setting in which both the risky assets prices and the liability are assumed to be governed by the same Brownian motions, and provided analytical formulae for the optimal policy and the M V efficient frontier by employing the stochastic control theory. In this paper, we investigate the problem of continuoustime portfolio selection with liability in an incomplete financial market. In the market, m risky assets and one risk-free asset are traded continuously, the risky assets dynamic prices are driven by an n-dimensional Brownian motion with n m, the liability is dynamically exogenous and its driving factors include but do not equal to the ones of the risky assets prices. here exist correlations between the liability and the risky assets prices. As a special case, when the sum of these correlation coefficients squares is equal to one, both the risky assets prices and the liability are driven by the same Brownian motions as considered by Chiu and Li (26). Furthermore, we assume that the prices of the risky assets evolve according to geometric Brownian motions while the liability is governed by a Brownian motion with drift. he representation of a liability as a Brownian motion with drift can be found, for example, in Norberg (1999) and Decamps et al. (26). Additionally, in Henderson and Hobson (24), the stock price was described as a geometric Brownian motion and the auxiliary non-traded stock s price was submitted to a Brownian motion with drift. he non-traded stock discussed in that paper, just like the liability in this paper, is dynamically exogenous and is related to the price evolvement of the traded risky asset. It is noteworthy that Zhou and Li (2) employed the stochastic linear quadratic (LQ) technique to tackle a continuous-time pure investment problem under the complete market assumption, i.e., n = m. he diffusion terms in the wealth differential equation derived by Zhou and Li (2) are independent and identically distributed and their diffusion coefficients are homogeneously linear with respect to the control variables. So the optimal portfolio strategy and the M V efficient frontier for pure investment case can be derived by solving an auxiliary typical stochastic LQ control problem. We emphasize, however, that the introduction of a liability is by no means routine and does give rise to difficulties which are not encountered in the pure investment case of Zhou and Li (2). Firstly, due to the correlations between the risky assets and the liability, the wealth differential equation derived from our model cannot agree with the representation of the state equation in the general framework of stochastic LQ control, while the latter requires the independent property of the diffusion terms. Enlightened from Koo (1998), we overcome this difficulty by rewriting the diffusion term of the liability. Secondly, since the diffusion coefficients in the wealth differential equation derived from our model are nonhomogeneous with respect to the control variables, the stochastic LQ control framework stated in the pure investment case of Zhou and Li (2) is no longer valid for our model. By applying the more general stochastic LQ control technique in Yong and Zhou (1999) to our model, we introduce a stochastic LQ auxiliary control problem and derive its optimal feedback control. Eventually the optimal portfolio strategy and the efficient frontier for the original M V portfolio optimization problem with a liability are obtained in closed forms. he paper proceeds as follows. In Section 2 we model the continuous-time M V portfolio optimization problem with a liability. In Section 3 we introduce an auxiliary problem which is transformed into a stochastic LQ control problem. In Section 4 we derive the optimal feedback control of the auxiliary problem. Section 5 is devoted to derive explicit expressions of the optimal strategy and the M V efficient frontier for the problem formulated in Section 2. In Section 6 we discuss several special cases of our results. In Section 7 we demonstrate a numerical example to show the effect of the liability and the incompleteness in the market. Section 8 concludes the paper. 2. he model Let (Ω, F, P, F) be a complete filtered probability space. Assume that the filtration F = (F t ) t [, is generated by an (n + 1)-dimensional standard Brownian motion {(Wt, W t 1,..., W t n) : t [, } for a positive integer n, where the positive number is a fixed and finite time horizon, F = {, Ω}, F = F, and the superscript represents the transpose of a vector or a matrix. We denote by C([, ; R n k ) the class of R n k -valued continuous bounded deterministic functions on [,, and by L 2 F ([, ; Rm ) the class of all R m -valued, progressively measurable and square integral random variables on [, under P with norm ξ L 2 F := ( ) 1 E ξ t 2 2 dt <, ξ t L 2 F ([, ; Rm ). Consider an investor equipped with an initial endowment w > and an initial liability l (l R) at time t =. Denote by x the net initial wealth of the investor, i.e., x = w l. he investor is allowed to adjust his/her portfolio during the time interval [,, and short-selling is allowable. We consider a financial market in which (m + 1) assets are traded continuously within the time horizon [,. We label these assets by i =, 1,..., m, where m n and the th asset represents the risk-free asset. he risk-free asset s price A t is subject to the following (deterministic) ordinary differential equation (ODE): da t = r t A t dt, A = 1, (1) where r t C([, ; R + ) is the interest rate of the risk-free asset. he remaining m assets are risky and their price processes A 1 t, A2 t,..., Am t satisfy the following stochastic differential equations (SDEs): da i t = µ i t Ai t dt + σ i t Ai t dw t, A i = a i R, i = 1, 2,..., m, (2)

3 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) where σt i := (σt i1, σt i2,..., σt in ) C([, ; R 1 n ) is the volatility or diffusion of the ith risky asset, and W t := (Wt 1, W t 2,..., W t n). We denote by µ t := (µ 1 t, µ2 t,..., µm t ) C([, ; R m 1 ) the appreciation rate vector of the risky assets. Define the variance matrix ( ) σ t := σt 1, σ t 2,..., σ t m i j (σ t ) C([, ; R m n ). he investor s accumulative liability at time t is denoted by L t, and L t satisfies the following SDE: dl t = u t dt + v t db t, L = l, (3) where {B t : t [, } is a one-dimensional standard Brownian motion defined on the underlying probability space (Ω, F, P, F). In general, the dynamic evolution of the liability is not always independent of the risky assets prices. We denote by ρt j the correlation coefficient between B t and Wt j for j = 1, 2,..., n and let ρ t := (ρ 1 t, ρ2 t,..., ρn t ) C([, ; R n 1 ) be the correlation coefficient vector. It follows from the formula (2.6) given in Koo (1998) that the diffusion term of the liability, B t, can be expressed as the linear combination of Wt 1, W t 2,..., W t n and Wt : B t = ρ t W t + 1 ρ tρ t Wt. (4) Noting that B t and Wt, W t 1,..., W t n are standard Brownian motions, we have ρ t ρ t 1 for all t [,. When ρ t ρ t < 1, the risk from the liability cannot be eliminated by trading financial assets. By (4), we have dl t = u t dt + v t ρ t dw t + v t 1 ρ tρ t dwt, L = l. (5) Expression (5) includes three special cases. Firstly, if B t does not correlate with Wt j for j = 1, 2,..., n, that is, ρt j = for j = 1, 2,..., n, then B t is equal to Wt. Secondly, if n > m and ρ t ρ t = 1, then B t can be expressed as a linear combination of Wt 1, W t 2,..., W t n. his is the setup under which Chiu and Li (26) studied. In this case, the risky assets prices and the liability are driven by the same source of randomness. hirdly, if n = m and ρ t ρ t = 1, then the risk risen from the liability can be hedged completely by the m risky assets. hroughout this paper we assume that for i = 1, 2,..., m, j = 1, 2,..., n, r t, µ i t, σ t i j, u t, v t, ρt j are deterministic functions of t, µ i t > r t, and there exists ε > such that σ t σ t εi m for any t [,, where I m is the m m identity matrix. We assume that the trading of shares takes place continuously in a self-financing fashion (i.e., there is no consumption nor income) and that there are no transaction costs. We denote by X t the net wealth of the investor at time t [,, and by θt i the shares of asset i (i =, 1,..., m) held by the investor at time t. Let πt i := θt i Ai t be the dollar amount invested in asset i (i =, 1,..., m) at time t, and let π t := (πt 1, π t 2,..., π t m ). he process π := {π t : t [, } is also called a strategy. Clearly, πt, the amount invested in the risk-free asset, satisfies the equation πt = X t m i=1 πt i. Suppose that the trading of shares takes place at time t and the shares are fixed between the time interval (t, t +h. Without the liability, the increment of the wealth of the investor satisfies X t+h X t = m i= θt i (Ai t+h Ai t ). (6) aking the liability into consideration, the investor must pay a dollar amount (L t+h L t ) during (t, t + h, which reduces the wealth accordingly. In this case, (6) should be replaced by X t+h X t = m θt i (Ai t+h Ai t ) (L t+h L t ). (7) i= Letting h +, we get the continuous-time analogue of (7) as the following: dx t = m θt i dai t dl t, X = x >. (8) i= aking (1), (2) and (5) into account, we may write (8) as { dxt = (r t X t + b t π t u t )dt + (π t σ t + δ t )dw t + δ t dw X = x, where b t := (µ t r t 1) C([, ; R 1 m ), 1 := (1, 1,..., 1) }{{}, m δ t := v t ρ t C([, ; R n 1 ), δ t := v t 1 ρ tρ t. (1) he admissible strategy set with initial wealth x is defined as { } A(x) := π π t L 2 F ([, ; Rm ), (X t, π t ) satisfies (9). he objective of the investor is to determine an optimal strategy π A(x) such that the expected terminal wealth, E X, is maximized, and the variance of the terminal wealth, Var X, is minimized. his is the M V model and can be expressed as the bi-objective optimization problem: P min ( E X, Var X ). π A(x) It is known from Li and Ng (2) that P is equivalent to the following single objective optimization problem: P(λ) min ( E X + λvar X ), π A(x) where the parameter λ [, ) represents the weight imposed by the investor on the objective Var X. Define Π P(λ) := {π π is an optimal strategy of P(λ)}. 3. Auxiliary problem Since the objective function of the problem P(λ) contains the term Var X, the stochastic LQ control technique cannot t, (9)

4 946 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) be directly applied to P(λ). Similar to Zhou and Li (2), we introduce an auxiliary problem as follows: A(λ, ω) min E(λX 2 ωx ), π A(x) where ω R is given beforehand. Define Π A(λ,ω) := {π π is an optimal strategy of A(λ, ω)}. Following Zhou and Li (2), the relationship between problems P(λ) and A(λ, ω) can be shown as below. heorem 1 (Zhou and Li (2)). For any λ >, one has Π P(λ) ω R Π A(λ,ω). Moreover, if π Π P(λ), then π Π A(λ,ω) with ω = 1 + 2λE X, where X t is the wealth process corresponding to the strategy π. Let γ := 2λ ω and Y t := X t γ. hen (9) becomes the nonhomogeneously linear SDE dy t = (r t Y t + b t π t + r t γ u t )dt + (π t σ t + δ t )dw t + δt dw t, Y = y := x γ, (11) and the objective function of the auxiliary problem A(λ, ω) becomes E(λY 2 ω2 4λ ). Hence, the auxiliary problem A(λ, ω) is equivalent to minimizing ( ) λ J(π; γ ) := E 2 Y 2. (12) Furthermore, the admissible strategy set A(x) can be written as { } Λ(y) := π π t L 2 F ([, ; Rm ), (Y t, π t )satisfies (11). hus the auxiliary problem A(λ, ω) is equivalent to the following stochastic LQ control problem: A(γ ) min J(π; γ ). π Λ(y) he optimal cost functional of the problem A(γ ) is defined as J := J(π ; γ ) = inf J(π; γ ), π Λ(y) where π is the optimal solution of the problem A(γ ). 4. Solution to the auxiliary problem In this section we solve the auxiliary stochastic LQ control problem A(γ ), and hence give the optimal strategy to the auxiliary problem A(λ, ω). It should be noted that the problem A(γ ) is different from that of Zhou and Li (2). In particular, the diffusion coefficient vector π t σ t + δ t in the state equation (11) is non-homogeneous with respect to the control vector π t, whereas the one in Zhou and Li (2) is homogeneous. Enlighten from the general stochastic LQ optimal control theory in Yong and Zhou (1999), we obtain the following result. A solution of the stochastic LQ control problem A(γ ) will involve, in an essential way, the following Riccati equation: ϕ t + (2r t ς t )ϕ t =, ϕ = λ, (13) ϕ t σ t σ t >, a.e. t [,, along with the following adjoint ODE { φ t + (r t ς t )φ t + (r t γ + κ t )ϕ t =, φ =, where ϕ t := dϕ t dt, φ t := dφ t dt, κ t := b t h t u t C([, ; R), h t := (σ t σ t ) 1 σ t δ t C([, ; R m 1 ), ς t := b t τ t C([, ; R + ), (14) τ t := (σ t σ t ) 1 b t C([, ; Rm 1 ), (15) and the factor τ t is called the risk premium. heorem 2. If (14) and (15) have solutions ϕ t C([,, R + ) and φ t C([,, R), respectively, then the stochastic LQ problem A(γ ) has an optimal feedback control π (Y t ) = τ t (Y t + ϑ t ) h t, t [,. (16) Furthermore, the optimal cost functional is J = 1 2 [ (τ t ϑ t + h t ) (σ t σ t )(τ tϑ t + h t )ϕ t + 2(r t γ u t )φ t + v 2 t ϕ tdt ϕ y 2 + φ y, (17) where ϑ t := φ t ϕ t. Proof. First, we prove that the control given by (16) is an admissible control. Substituting (16) into (11), one has dy t = [(r t ς t )Y t ς t ϑ t + κ t + r t γ dt + ( τ t σ ty t τ t σ tϑ t h t σ t + δ t ) dwt + δt dw t, Y = y. (18) Noting that ϕ t C([,, R + ) and φ t C([,, R), we have ϑ t C([,, R). hus, the appreciation coefficient and the diffusion coefficients in (18) are Lipschitz continuous. By the heorem 2.9 given in Karatzas and Shreve (1991), we conclude that (18) admits a unique strong solution Yt such that E sup Yt 2 K (1 + y 2 ), t [, where K > is a constant associated with the terminal time. hus, we have shown that π (Y t ) Λ(y). Next, we prove that π (Y t ) is an optimal feedback control of state variable Y t. For any π Λ(y), let Y t be the state variable associated with the control vector π t. By applying Itô s formula to 1 2 ϕ ty 2 t and φ t Y t,

5 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) we get 1 2 d(ϕ tyt 2 ) = 1 [ ς t ϕ t Yt 2 + 2(b t π t + r t γ u t )ϕ t Y t dt [ π t 2 σ tσ t π t + 2π t σ tδ t + δ t δ t + (δt ) ϕ t dt + ( π t σ t + δ t ) ϕt Y t dw t + δt ϕ ty t dwt, (19) d(φ t Y t ) = [ς t φ t Y t (r t γ + κ t )ϕ t Y t + (b t π t + r t γ u t )φ t dt + ( π t σ t + δ t ) φt dw t + δt φ tdwt. (2) Integrating both (19) and (2) from to, taking expectations, adding them together, and noting (12), we get J(π; γ ) 1 2 ϕ y 2 φ y = 1 2 E {ς t ϕ t Y 2 t + 2(b t π t κ t u t )ϕ t Y t + 2ς t φ t Y t + 2(b t π t + r t γ u t )φ t + [π t σ tσ t π t + 2π t σ tδ t + δ t δ t + (δ t )2 ϕ t }dt = 1 2 E {[τ t )τ tϕ t Y 2 t + 2τ t )π tϕ t Y t + 2τ t (σ tσ t )(τ tϑ t + h t )ϕ t Y t + 2π t (σ tσ t )(τ tϑ t + h t )ϕ t + π t (σ tσ t )π tϕ t + ψ t (σ tσ t )(τ tϑ t + h t )ϕ t + [ ψ t (σ tσ t )(τ tϑ t + h t )ϕ t + 2(r t γ u t )φ t + vt 2 ϕ t}dt = 1 2 E (τ t Y t + π t + τ t ϑ t + h t ) (σ t σ t )ϕ t (τ t Y t + π t + τ t ϑ t + h t )dt [ (τ t ϑ t + h t ) (σ t σ t )(τ tϑ t + h t )ϕ t + 2(r t γ u t )φ t + v 2 t ϕ tdt. Noting that 1 2 E (τ t Y t + π t + τ t ϑ t + h t ) (σ t σ t )ϕ t (τ t Y t + π t + τ t ϑ t + h t )dt, we have J(π; γ ) 1 2 ϕ y 2 φ y 1 2 [ (τ t ϑ t + h t ) (σ t σ t ) (τ t ϑ t + h t )ϕ t + 2(r t γ u t )φ t + v 2 t ϕ tdt, and the equality holds if and only if τ t Y t + π t + τ t ϑ t + h t. his means that the feedback control given by (16) is an optimal control and the optimal cost functional is given by (17). he proof is completed. Noting that the third constraint in (13) is satisfied automatically due to the assumption σ t σ t εi m. We solve (13) and (14) and get ϑ t = t t κ s e s rzdz ds + γ (1 e t r z dz ). (21) Due to the equivalence of problems A(γ ) and A(λ, ω), heorem 2 also gives the optimal feedback control of the auxiliary problem A(λ, ω): π (X t ) = τ t (X t + ϑ t γ ) h t, t [,, (22) where τ t, h t are given by (15) and ϑ t is given by (21). 5. Solution to the original problem In this section we turn to derive the expressions for the optimal strategy and the efficient frontier of our original problem P(λ). Denote by Xt the wealth process under the optimal feedback control π of the auxiliary problem A(λ, ω). Substituting (22) into (9) yields dxt = [ (r t ς t )X t ς t ϑ t + ς t γ + κ t dt + [ τ t σ t(xt + ϑ t γ ) h t σ t + δ t dwt + δt dw t, (23) X = x. Applying Itô s formula to Xt 2 yields dxt 2 = [(2r t ς t )Xt 2 + 2κ t Xt + ς t (ϑ t γ ) 2 dt + [(δt )2 + δ t δ t δ t σ t (σ tσ t ) 1 σ t δ t dt +2 [ τ t σ t(xt + ϑ t γ ) h t σ t + δ t X t dw t + 2δt X t dw t, X 2 = x 2. (24) aking expectation on both sides of (23) and (24) results in, respectively, { de X t = [ (r t ς t )E X t ς t ϑ t + ς t γ + κ t dt, E X = x, (25) and de Xt 2 = [(2r t ς t )E Xt 2 + 2κ t E Xt + ς t (ϑ t γ ) 2 dt, +[(δt )2 + δ t δ t δ t σ t (σ tσ t (26) ) 1 σ t δ t dt E X 2 = x 2. he solution of ODE (25) is E Xt t t = xe (r z ς z )dz t + (ς s γ + κ s ς s ϑ s )e s (r z ς z )dz ds, t [,. (27) his leads to E X = α + βγ, (28) where α := κ t e t r z dz ςzdz dt + xe (r z ς z )dz, β := 1 e ς zdz. Similarly, by solving (26) we have E X 2 = x 2 e (2r z ς z )dz + [2κ t (E X t ) + ς t (ϑ t γ ) 2 e + t (2r z ς z )dz dt [ (δ t )2 + δ t δ t δ t σ t ) 1 σ t δ t e t (2r z ς z )dz dt.

6 948 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) By substituting (21) and (27) into the above equation, we finally get E X 2 = η + βγ 2, (29) where η := 2xe (r z ς z )dz ( + e ς zdz κ t e t r z dz dt κ t e t r z dz 2 dt) ςzdz + x 2 e (2r z ς z )dz [ + (δt )2 + δ t δ t δ t σ t (σ tσ t ) 1 σ t δ t e t (2r z ς z )dz dt. By heorem 1, any optimal solution of the problem P(λ) (as long as it exists) can be obtained via the solution π of the auxiliary problem A(λ, ω ) with ω = 1 + 2λ E X. According to (28), E X = α + βγ with γ = ω 2λ. he above two equations yield ω = 1 + 2λα 1 β. Hence, the optimal feedback control (i.e., the optimal strategy) of the problem P(λ) is given by (22) with γ = γ = 1 2λ(1 β) + α 1 β. Correspondingly, the variance of the terminal wealth is Var X = E X 2 (E X )2 = β(1 β)γ 2 2αβγ + η α 2. (3) By substituting γ = β 1 (E X α) (due to (28)) and the expressions of all the parameters α, β, η into (3), we finally obtain the efficient frontier in a closed form: Var X = e ( E X xe + [ ς zdz 1 e ς zdz rzdz + κ t e t r z dz dt ) } {{ } D 1 (δt )2 + δ t δ t δ t σ t (σ tσ t ) 1 σ t δ t e t, }{{} D 2 (31) 2 (2r z ς z )dz dt where δ t and δ t are given by (1), and κ t and ς t are defined as (15). Here, we note that e ς zdz 1 e ς zdz >, e rzdz > 1, e t (2r z ς z )dz >. We also claim that D 2. In fact, let H t := I n σ t (σ tσ t ) 1 σ t, and denote by ζt 1, ζ t 2,..., ζ t n the eigenvalues of the matrix H t. Since H t = H t, there exists an orthogonal matrix Q R n n such that H t = Q diag{ζ 1 t, ζ 2 t,..., ζ n t } Q. Because H t = H 2 t and Q Q = I n, we have Q diag{ζt 1, ζ t 2,..., ζ t n } Q = Q diag{ζt 1, ζ t 2,..., ζ t n } Q Q diag{ζt 1, ζ t 2,..., ζ t n } Q = Q diag{(ζt 1 )2, (ζt 2 )2,..., (ζt n )2 } Q. his leads to ζt j = or 1 for j = 1, 2,..., n. hus, H t is a non-negative definite matrix, i.e., H t. Further, we have δ t H tδ t = δ t δ t δ t σ t ) 1 σ t δ t. his implies that D 2 (δ t )2 e t (2r z ς z )dz dt. From the above analysis, we have the following result. heorem 3. he efficient frontier of the mean variance portfolio selection problem P(λ), if it ever exists, is given by (31) for E X D 1. he efficient frontier (31) reveals explicitly the tradeoff between the mean (i.e., return) and the variance (i.e., risk) of the terminal wealth. Once an expected return level is given, the risk the investor has to take is given by (31); and vice versa. In particular, if the investor s expected terminal wealth is D 1 and the optimal strategy is adopted, then the variance of the investor s terminal wealth will reach the globally minimal value D 2 ; and if the investor chooses a risk value of D 2 and adopts the optimal strategy, then his/her expected terminal wealth will be D 1. he relation (31) also indicates that the M V efficient frontier is the upper part of a parabola with a rightward hatch and a vertex (D 2, D 1 ) in the M V plane (with vertical axis being the mean and horizontal axis being the variance). he liability or the incompleteness of the market has effect on the M V efficient frontier and the optimal strategy, which will be discussed in detail in the following sections. 6. Special cases o compare our results with the ones in the existing literature, we discuss several special cases in this section. Special case 1. If the financial market is complete, i.e., n = m, then we have σ t ) 1 σ t = [σ t (σ t ) 1 [(σ t ) 1 σ t = I m, and thus δ t δ t = δ t σ t ) 1 σ t δ t. In this case, D 2 in (31) reduces to D 2 = = (δ t )2 e t (2r z ς z )dz dt vt 2 (1 ρ t ρ t (2r t)e z ς z )dz dt. (32)

7 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) able 1 he parameters Para. n m x r µ σ = (σ 1, σ 2 ) u v ρ = (ρ 1, ρ 2 ) Val (.15,.25).7.11 (.65,.1) hus, if the investor wishes to obtain the same expected return level in an incomplete market as in a complete market, then he/she should take more risk, and the amount of the additional risk is [ δ t δ t δ t σ t ) 1 σ t δ t e t (2r z ς z )dz dt. Special case 2. If the liability is independent of all the driving factors of the risky assets prices but is indeed random, then ρt j = for j = 1, 2,..., n and v t for t [,. his leads to h t = and κ t = u t for all t [,, and hence (31) becomes Var X = e ς zdz 1 e ς zdz 2 u t e t r z dt) dz + ( [E X xe r zdz v 2 t e t (2r z ς z )dz dt. (33) On the right-hand side of (33), the first item is a perfect square and the second item is strictly positive. Hence, one cannot find a risk-free portfolio in the efficient frontier. his is due to the fact that the risk resulting from the liability cannot be hedged by the risky assets existing in the market. Special case 3. If there exists no liability, then, for j = 1, 2,..., n, we have u t = v t = ρt j = for all t [,. his leads to h t = and κ t = for all t [,. hus, the efficient frontier (31) becomes Var X = e ς zdz 1 e ς zdz (E X xe r zdz ) 2. (34) his result is the same as the expression of M V efficient frontier derived by Zhou and Li (2). Furthermore, since the second item D 2 on the right-hand side of (31) is a non-negative constant, (31) and (34) imply that the presence of a liability results in an increase of risk the investor has to bear to achieve the same expected return level. Special case 4. If n = m and ρ t ρ t = 1 for all t [,, then the efficient frontier (31) involves a perfect square and hence becomes a straight line in the mean standard deviation plane: E X = xe rzdz + + e ς zdz κ t e t r z dz dt 1 e ς zdz σ X, (35) where σ X represents the standard deviation of X. In this case, there exists a risk-free portfolio, which has the variance Var X = and the expected return E X = xe r zdz + κ t e t r z dz dt. his is due to the fact that the risk risen from the liability can be hedged perfectly by the existing m risky assets. Similar to the standard situation without a liability, the line (35) is called the capital market line, and its slope k := the price of risk. 7. Numerical example e ςz dz 1 e ςz dz is called In this section we give a numerical example to illustrate our results. All the parameters are constants and hence we omit their subscript t. Example. Consider an investor whose initial fund is x = $1 million. he investor plans an investment on one bond and one stock in an incomplete market during the time horizon = 1 (year). he involved Brownian motions are (Wt, W t 1, W t 2). Other parameters are given in able 1. hus, we have ς = (µ r) 2 (σ σ ) 1, κ = v(µ r)(σ σ ) 1 σρ u, ( α = x + κ ) e (r ς) κ r r e ς, β = 1 e ς, ( κ ) 2 η = r + x e (2r ς) 2 κ ( κ ) r r + x e (r ς) + κ2 [ [ + v 2 1 ρ σ (σ σ ) 1 e (2r ς) σρ 2r ς 1, 2r ς and (31) reduces to e ς r 2 e ς Var X (E = 1 e ς X xer κ r er + κ r [ [ + v 2 1 ρ σ (σ σ ) 1 e (2r ς) σρ 2r ς 1. (36) 2r ς he effect of n and L on the M V efficient frontier We discuss the following four cases. Case 1. he market is incomplete and there is a liability. All the parameters are given in able 1. Case 2. he market is incomplete and there exists no liability. he parameters u = v =, σ = (.15,.25), ρ = (, ) and the other parameters remain the same as given in able 1. ) 2

8 95 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) Fig. 1. he effect of parameters n,, x, u, v, ρ on M V efficient frontier. One risky asset and one liability are considered. (a) is about the effect of incomplete market and the liability, where the common parameters are x = 1, = 1, r =.6, µ =.12. (a) C1 is the curve of Case 1 with n = 2, u =.7, v =.11, σ = (.15,.25), ρ = (.65,.1) ; (a) C2 is the curve of Case 2 with n = 2, u = v =, σ = (.15,.25), ρ = (, ) ; (a) C3 is the curve of Case 3 with n = 1, u =.7, v =.11, σ = (.15,.), ρ = (.65,.) ; (a) C4 is the curve of Case 4 with n = 1, u = v =, σ = (.15,.), ρ = (, ). (b) is about the effect of on M V efficient frontier, where the common parameters are x = 1, u =.7, v =.11, σ = (.15,.25), ρ = (.65,.1). (c) is about the effect of x on M V efficient frontier, where the common parameters are = 1, u =.7, v =.11, σ = (.15,.25), ρ = (.65,.1). (d) is about the effect of u on M V efficient frontier, where the common parameters are x = 1, = 1, v =.11, σ = (.15,.25), ρ = (.65,.1). (e) is about the effect of v on M V efficient frontier, where the common parameters are x = 1, = 1, u =.7, σ = (.15,.25), ρ = (.65,.1). (f) is about the effect of ρ on M V efficient frontier, where the common parameters are x = 1, = 1, u =.7, σ = (.15,.25), v =.11. Case 3. he market is complete and there is a liability. he parameters σ = (.15, ) and ρ = (.65,.) and the other parameters remain the same as given in able 1. Case 4. he market is complete and there exists no liability. he parameters σ = (.15, ), u = v = and ρ = (, ) and the others remain the same as given in able 1. All these data are the same as given in Zhou and Li (2). Substituting the data into (36), the efficient frontiers under the above four cases are given respectively by Var X 1 = (E X ) , (37) Var X 1 = (E X )2, (38) Var X 1 = (E X ) , (39) Var X 1 = (E X )2. (4) Fig. 1(a)C1 C4 display these curves. We can observe that the frontiers 1(a)C2 and C4 touch the vertical axis when v =. In particular, suppose that the investor wishes to obtain an expected return of 2% in one year, namely, E X1 = 1.2. hus, in Case 1 the investor has to bear the risk Var X and the standard deviation Var X1 is as high as %; in Case 2, Var X , and Var X % which is lower than the one in Case 1 by 3.589%; in Case 3, Var X , and Var X % which is lower than the one in Case 1 by %; in Case 4, we get the same results as given by Zhou and Li (2): Var X1.1112, and Var X % which is lower than the one in Case 2 by % and the one in Case 3 by %. hese indicate that to achieve the same level of expected return, the investor has to take more risk if the market is incomplete or there exists a liability.

9 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) Fig. 2. he effect of parameters x and on optimal investment amount of money in the stock. he common parameters are r =.6, µ =.12, σ = (.15,.25), u =.7, v =.11, ρ = (.65,.1). he top left panel (a) shows the one based on formula (41) with x = 1, = 1, t [, 1 and X t [1, 5. he top right panel (b) displays the one based on formula (45) with x = 4, = 1, t [, 1 and X t [4, 8. he bottom panel (c) shows the one based on formula (46) with x = 1, = 2, t [, 2 and X t [1, 5. he effect of, x, u, v, ρ on the M V efficient frontier By varying one parameter and fixing the others each time, we can demonstrate the effect of, x, u, v or ρ on the M V efficient frontier. Fig. 1(b) shows that longer maturities give higher expected returns under the same risk level. Fig. 1(c) (d) shows that the efficient frontier moves upwards and away from the horizontal (variance) axis when x (the initial endowment) increases or when u (the appreciation rate of the liability) decreases. Fig. 1(e) (f) show that the efficient frontier moves towards to the vertical (mean) axis when v (v being the volatility of the liability) decreases or when ρ ρ (the sum of correlation coefficients squares) increases. he effect of n and L on the optimal strategy We discuss the effect of n and L on the optimal strategy under the previous four cases. Suppose again that E X1 = 1.2. For Case 1, γ = γ = β 1 (E X1 α) , and the optimal strategy (22) is π (X t ).75882X t e.6t , t [, 1. (41) his function of time and wealth is depicted in Fig. 2(a) with time varying between and 1 year and the wealth between 1 and 5 million dollars. In particular, at the initial time t =, the dollar amount invested in the stock is π (1) $ million. his means that the investor should sell short the bond (i.e., borrow money) for an amount $ million and invest it together with his/her initial endowment $1 million in the stock at the initial time. For Case 2, γ and π (X t ).75882X t e.6t, t [, 1. (42) hus, the initial investment in the stock is π (1) $ million. It is lower than the one in Case 1 by $ million. For Case 3, γ and π (X t ) X t e.6t , t [, 1. (43) hus, the initial investment in the stock is π (1) $ million. It is higher than the one in Case 1 by $ million. For Case 4, γ and π (X t ) X t e.6t, t [, 1. (44) hus π (1) $ million. his result agrees with the one given by Zhou and Li (2). It is higher than the one in Case 2 by $ million and lower than the one in Case 3 by $ million. It is clearly that the initial borrowed amount of money will decrease greatly in the absence of a liability but increase a little

10 952 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) Fig. 3. he effect of the liability on optimal portfolio selection. he common parameters are r =.6, x = 1, = 2, µ =.12 and σ = (.15,.25). he time varies between and 2 years and the wealth between 1 and 5 million dollars. Each time one and only one of the parameters u, v and ρ is changed. he top left panel (a) shows the difference of dollar amount between four times the appreciation rate of the liability u =.28 and the case of base parameter u =.7; he top right panel (b) displays the difference of dollar amount between four times the volatility rate of the liability v =.44 and the case of base parameter v =.11; he bottom panel (c) shows the difference of dollar amount between one fourth of the sum of correlation coefficients square ρ = (.325,.5) and the case of base parameter ρ = (.65,.1). in a complete market. his shows that the introduction of a liability results in an increase of the initial investment in the risky asset, while the incompleteness of the market causes a decrease of the initial investment in the risky asset. he effect of x,, u, v, ρ on the optimal strategy First we suppose that the investor is equipped with much more initial endowment, for instance, x = $4 million, while the other parameters remain the same as given in able 1. hen the optimal strategy is π (X t ).75882X t e.6t , t [, 1. (45) his function is plotted in Fig. 2(b) with time varying between and 1 year and the wealth between 4 and 8 million dollars. Under the optimal strategy (45), in order to obtain an expected return of 2% in one year, the investor should invest π (4) $ in the stock at the initial time t =, that is, the investor should sell short the stock for an amount $ million and invest it together with his/her initial endowment $4 million in the bond at the initial time. hus, we can easily observe that the initial invest amount in the risky asset decreases with respect to one s initial endowment. Next we change the maturity time to = 2 (years) and keep the other parameters the same as in able 1. hus, the optimal strategy becomes π (X t ).75777X t e.6t , t [, 2. (46) his function of time and wealth is depicted in Fig. 2(c) with time varying between and 2 years and the wealth between 1 and 5 million dollars. Under the optimal strategy (46), to obtain an expected return of 2% in 2 years, the initial amount invested in the stock should be reduced to π (1) $ million. Finally we consider the effect of the liability on the optimal portfolio selection. o this end, we change one and only one of the parameters u, v, ρ of the liability each time. Fig. 3 plots the difference between the benchmark function (46) and a new function that is computed for the revised parameter. hese surfaces show that the initial investment amount in the risky asset increases with respect to the appreciation rate u, the volatility rate v, or the sum of correlation coefficients square ρ ρ. Details are stated as below. Fig. 3(a) shows the difference of portfolios when the appreciation rate of the liability is increased from u =.7 to u =.28. In the case of u =.28, the investor s optimal

11 S. Xie et al. / Insurance: Mathematics and Economics 42 (28) portfolio at the initial time t = is π (1) $ Comparing with the initial investment $ million when u =.7, one should increase $ million investment in the stock at the initial time t = so as to obtain the same mean return level. Fig. 3(b) shows the difference of portfolios when the volatility rate of the liability is increased from v =.11 to v =.44. In the case of v =.44, we have π (1) $ million, thus the investor should increase $ million investment in the stock at the initial time t = so as to obtain the same mean return level as the case v =.11. he final Fig. 3(c) shows the difference of portfolios when the correlation coefficient of the stock and the liability is decreased from ρ = (.65,.1) to ρ = (.325,.5). In the case of ρ = (.325,.5), the initial portfolio is π (1) $ million. Comparing with the initial investment $ million when ρ = (.65,.1), the investor can obtain the same mean return level by decreasing $ million investment in the stock at the initial time t =. 8. Concluding remarks By introducing a liability into a continuous-time mean variance portfolio selection problem, this paper extends the Markowitz s mean variance framework. In our model, the risky assets prices and the liability are driven by different Brownian motions, and their relationship is considered. By introducing an auxiliary problem and by making use of the general stochastic linear-quadratic technique, we have derived closed-form expressions for the optimal strategy and the M V efficient frontier of our model. Our results show that the M V efficient frontier is still a parabola in the mean variance plane, but not always a straight line in the mean standard-deviation plane. he results of Zhou and Li (2) can be obtained as a special case of our results when there is no liability and the market is complete. here are two interesting problems we have not addressed in this study. he first one is the problem with random interest/appreciation rates and volatility rates. he second one is the pricing of liabilities in an incomplete market. We shall leave them for a future research. Acknowledgements he authors thank the referees very much for their very valuable comments and suggestions. he authors are also grateful to Prof. Jiagang Ren and some other participants for their helpful discussions on this work during the Second International Conference on Risk Management and the hird International Conference on Financial Systems Engineering held in Guangzhou, China, December 9 12, 25, and the International Workshop on Mathematical Finance and Insurance held in Lijiang, China, May 27 June 3, 26. References Bielecki,.R., Jin, H.Q., Pliska, S.R., Zhou, X.Y., 25. Continuoustime mean-variance portfolio selection with bankruptcy prohibition. Mathematical Finance 15 (2), Chiu, M.C., Li, D., 26. Asset and liability management under a continuoustime mean variance optimization framework. Insurance: Mathematics and Economics 39 (3), Decamps, M., Schepper, A.D., Goovaerts, M., 26. A path integral approach to asset liability management. 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