Continuous-time mean–variance asset–liability management with endogenous liabilities

Insurance: Mathematics and Economics 52 (2013) 6–17

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Continuous-time mean–variance asset–liability management with endogenous liabilities✩ Haixiang Yao a , Yongzeng Lai b,∗ , Yong Li c a

School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, China

b

Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada, N2L 3C5

c

UQ Business School, University of Queensland, Brisbane, Australia

article

info

Article history: Received July 2012 Received in revised form September 2012 Accepted 5 October 2012 Keywords: Endogenous liabilities Mean–variance Asset–liability management Efficient frontier Hamilton–Jacobi–Bellman equation

abstract This paper investigates a continuous-time mean–variance asset–liability management problem with endogenous liabilities in a more general market where all the assets can be risky. Different from exogenous liabilities that cannot be controlled, the endogenous liabilities can be controlled by various financial instruments and investors’ decisions. For example, a company can raise fund by issuing different kinds of bonds. Types and quantities of the bonds are controlled by the company itself. Investors optimize allocation not only for their assets, but also for their liabilities under our model. This makes the analysis of the problem more challenging than in the setting based on exogenous liabilities. In this paper, we first prove the existence and uniqueness of the solution to the associated Riccati-type equation by using the Khatri–Rao product technique and the relevant stochastic control theory; we then derive closed form expressions of the efficient strategy and the mean–variance efficient frontier by using the Lagrange multiplier method and the Hamilton–Jacobi–Bellman equation approach, and we next discuss two degenerated cases; finally, we present some numerical examples to illustrate the results obtained in this paper. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The seminal paper of Markowitz (1952) on mean–variance (M–V) portfolio selection lays down the foundation for modern financial portfolio theory. The most important contribution of Markowitz’s M–V model is the introduction of quantitative analysis approaches to portfolio selection and risk management. However, the classical Markowitz’s model considers only the case of single-period. It is natural to extend the Markowitz’s M–V model to cases of multi-period and continuous-time. Adopting the martingale method, Isabelle and Roland (1998) construct the efficient strategy and the efficient frontier for continuoustime M–V model under the assumption that a zero-coupon bond of maturity T exists. Li and Ng (2000) derive the analytical

✩ This research is supported by grants from Humanity and Social Science Foundation of Ministry of Education of China (No. 10YJC790339), Natural Science Foundation of Guangdong Province (No. S2011010005503), Guangdong Colleges and Universities Subject Construction Special Foundation (Scientific and Technological Innovation), Philosophy and Social Science Foundation of Guangdong Province (No. 09O-19), and an internal grant of Wilfrid Laurier University. ∗ Corresponding author. Tel.: +1 519 884 0710; fax: +1 519 884 9738. E-mail addresses: [email protected] (H. Yao), [email protected] (Y. Lai), [email protected] (Y. Li).

0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.10.001

solution to the multi-period M–V model by using an embedding technique. Zhou and Li (2000) adopt the embedding technique and the stochastic linear quadratic (LQ) control theory to derive the expressions of the efficient strategy and the efficient frontier in closed forms. From then on, many scholars devote their efforts to extend the dynamic M–V model to cases with a variety of more realistic conditions. Bielecki et al. (2005) consider a continuoustime M–V portfolio selection problem with bankruptcy control. Fu et al. (2010) consider a continuous time M–V portfolio selection problem with different borrowing–lending rates. Dai et al. (2010) investigate continuous time M–V model with proportional transaction costs. Chiu and Wong (2011) investigate a continuoustime M–V portfolio selection problem in a market where the prices of assets are cointegrated. Under multi-period M–V criterion, Costa and Oliveira (2012) consider a portfolio selection problem with Markovian jumps and multiplicative noises. Wu and Li (2012) study a multi-period M–V portfolio optimization with Markov regime-switching and a stochastic cash flow. Nowadays, the asset–liability management (ALM) problem is of both theoretical interest and practical importance, and has received considerable attention for the last decade in the actuarial and the financial literature. ALM under the M–V criterion was first studied by Sharpe and Tint (1990) in a static setting. For more discussion on ALM, readers are referred to Gerber and Shiu (2004), Decamps et al. (2006) and Consiglio et al. (2008). Most of these

H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

authors focus primarily on the optimal dividend pay-out and ruin problems of ALM. In recent years, along with the break through of solving of the dynamic M–V model, many scholars adopt the dynamic M–V model to study the ALM problem. The paper by Leippold et al. (2004) is the first one to consider the multi-period M–V ALM problem. The efficient strategy and frontier are derived by using the embedding technique and the geometric approach in that paper. Yi et al. (2008) and Chen and Yang (2011) extend the work of Leippold et al. (2004) to the cases with uncertain exit time and Markov regime switching market, respectively. Based on the continuous-time M–V framework, Chiu and Li (2006) and Xie et al. (2008) investigate the ALM problem with uncontrollable liability. Chiu and Li (2006) describe the liability as geometric Brownian motion, while Xie et al. (2008) describe the liability as Brownian motion with drift. Chen et al. (2008) and Xie (2009) extend the work of Chiu and Li (2006) and Xie et al. (2008) to the case with regime-switching, respectively. It is assumed in the literature mentioned above that there is only one liability for selection, and the liability is exogenous and uncontrollable. In this case, it needs only to consider the portfolio optimization for assets but keep the liability fixed. However, such an assumption is too restrictive for many financial institutions. In many cases, there exist more than one liabilities for them to select, and the liabilities can be controlled by various financial instruments and their decisions. Indeed, these institutions can optimize allocation not only for their assets, but also for their liabilities. For example, a company can raise fund by issuing different kinds of bonds, which differ in terms of durations, returns and risks. Thus, the company needs to decide the quantity of each bond under the condition that total liability value is fixed. Another example is that banks and insurers devise many different financial and insurance products, which can also be regarded as different liabilities for them to select. Therefore, in their ALM, endogenous liability management and exogenous liability management share the same practical significance. To the best of our knowledge, in dynamic setting, only Leippold et al. (2011) consider multi-period M–V ALM problem with endogenous liabilities and obtain the geometrical structure for optimal policies and M–V efficient frontier. There appears no research work in the literature on continuous-time version of the ALM problem with endogenous liabilities. In addition, to our knowledge, all the studies in the literature about continuous-time M–V portfolio selection consider only the markets with a riskfree asset. However, in the real world and on a relative long time horizon, a risk-free asset hardly exists due to the stochastic nature of real interest rates and the inflation risk. Under the continuoustime setting, this paper studies the ALM problem with endogenous liabilities under the M–V model in a more general market where all the assets can be risky. Under our model, we consider not only the optimal allocation for multiple risky assets, but also the optimal allocation for multiple risky liabilities simultaneously. The main difficulty of the problem under our setting is that increasing the control (decision) variables about liability allocation makes the optimization problem more difficult to be solved. Thus, the stochastic control method in Chiu and Li (2006) and Xie et al. (2008) cannot be used directly to solve the new problem under our model. Fortunately, by using the Lagrange multiplier method and the Hamilton–Jacobi–Bellman (HJB) equation approach, we can convert the M–V ALM problem to solve Riccati-type equations (see Eqs. (14)–(16)). However, due to the fact that in this case Eq. (14) is matrix-valued and highly nonlinear differential Riccati-type equation, its explicit solution cannot be derived. Furthermore, although Eq. (14) has been studied extensively in the literature (e.g., Chen et al., 1998; Chen and Zhou, 2000 and Rami et al., 2001), the existing results in the literature cannot guarantee the existence of a solution to this Riccati-type equation since neither the state nor the control weighting matrices nor

7

the terminal matrix is positive definite. This is another essential difficulty in our model. To overcome this difficulty, we adopt the Khatri–Rao matrix product technique and the relevant stochastic control theory to prove the existence and uniqueness of the solution to Eq. (14) under certain conditions. The remainder of our paper is organized as follows. In Section 2, the market setting for our problem is described and the continuous-time M–V ALM problem with endogenous liabilities is formulated. In Section 3, this original problem is converted to an unconstrained continuous-time stochastic control problem by introducing a Lagrange multiplier, and it is shown that the solution to the corresponding HJB equation can be constructed via Riccatitype equation and two other equations. In Section 4, the existence and uniqueness of the solution to the Riccati-type equation (14) is proved. In Section 5, closed form expressions of the efficient strategy and the efficient frontier are derived. In Section 6, two degenerated cases are discussed, and some results in the existing literature are obtained as degenerated cases under our model. In Section 7, some numerical examples are presented to illustrate the results obtained in this paper. Conclusion is given in Section 8. 2. Model description Given a filtered probability space (Ω , P, F , {Ft }0≤t ≤T ), let W (t ) = (W1 (t ), W2 (t ), . . . , Wm (t ))′ be an m-dimensional standard Brownian motion defined on (Ω , P, F ) over [0, T ], and Ft = σ {W (s); 0 ≤ s ≤ t } be augmented by all the P-null sets in F , where F = FT . Let A′ be the transpose of matrix or vector A. Suppose that an investor can allocate his/her wealth among nA + 1 assets that can be all risky. The prices of these assets satisfy the following stochastic differential equations

    

 dPi (t ) = Pi (t ) ai (t )dt + Pi (0) = pi ,

m 

 σij (t )dWj (t ) ,

j =1

(1)

i = 0, 1, . . . , nA ,

where (a0 (t ), a1 (t ), . . . , anA (t )) and (σij (t ))m×m are the appreciation rate vector and the volatility matrix of these assets, respectively, which are assumed to be deterministic functions of time t. Besides, the investor can also allocate his/her liabilities among nL + 1 (≥2) financial instruments. For convenience, throughout this paper, we regard these nL + 1 financial instruments as nL + 1 different kinds of liabilities. Suppose that the prices of these liabilities satisfy the following stochastic differential equations

    

 dQi (t ) = Qi (t ) bi (t )dt + Qi (0) = qi ,

m  j =1

 υij (t )dWj (t ) ,

(2)

i = 0, 1, . . . , nL ,

where (b0 (t ), b1 (t ), . . . , bnL (t )) and (vij (t ))m×m are the appreciation rate vector and the volatility matrix of the liabilities, respectively, which are also assumed to be deterministic functions of time t. Remark 1. It needs to point out that our model include the case with a risk free asset. Without loss of generality, let the 0th asset be risk-free. We need only to set a0 (t ) = r (t ), σ0j (t ) = 0, j = 1, 2, . . . , m, where r (t ) denotes the risk-free interest rate. Remark 2. W (t ) = (W1 (t ), W2 (t ), . . . , Wm (t ))′ is used to describe all the random factors which influence the prices of assets and liabilities. Therefore, it should have m ≥ max{nA + 1, nL + 1}. However, it is unnecessary to assume that m = nA + nL + 2, which means that we may allow the market to be incomplete. Besides, m should be greater than or equal to nA + nL such that Assumption 3 is valid, i.e., the matrix Y (t ) defined in (7) is row full rank.

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H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

Let x(t ) and l(t ) be the total value of assets and the total value of liabilities at time t held by the investor, respectively. Let ui (t ) denote the amount invested in the ith asset, i = , nA . Then the amount invested in the 0th asset is 1, 2, . . . n x(t ) − i=A 1 ui (t ) . Assume that the market is self-financing and trading is continuous without transaction costs. Then, by (1), the total value x(t ) of assets follows the dynamics

 dx(t ) =

x(t )a0 (t ) +

nA 

 ui (t )(ai (t ) − a0 (t )) dt

+ x( t )

m 

σ0j (t )dWj (t )

j=1

+

nA m  

   ui (t ) σij (t ) − σ0j (t ) dWj (t ) .

j=1 i=1

(3)

and x(0) = x0 . Then the dynamics for the total value of assets can be simplified as

   dx(t ) = x(t )a0 (t ) + u′ (t )α(t ) dt      m m   + x(t ) σ0j (t )dWj (t ) + u′ (t )δj (t )dWj (t ) ,   j =1 j =1   x(0) = x0 .

(5)

where j = 1, . . . , m.

Denote L2F (t , T ; Rn ) the set of all Rn -valued and measurable stochastic processes f (s) adapted to {Fs }s≥t on [t , T ] such that  

T t

|f (s)|2 ds < +∞.

Definition 1. A strategy π = {π (t ); t ∈ [0, T ]} is called admissible if π (·) ∈ L2F (0, T ; Rn ), and the pair (Z (·), π (·)) satisfies (8). In this case, (Z (·), π (·)) is called an admissible pair. Denote U[0, T ] the set of all such admissible pairs over [0, T ]. We define the surplus of an investor at time t as S (t ) = x(t ) − l(t ) = c ′ Z (t ). The continuous-time M–V ALM problem refers to the problem of finding the optimal admissible strategy such that the variance of the terminal surplus is minimized for a given expected terminal surplus d, i.e.



min

(Z (·),π (·))∈U[0,T ]

Var[S (T )] = E[S 2 (T )] − d2 ,

s.t. E[S (T )] = d.

(6)

For convenience, we introduce the following notations

     σ0j (t ) δj (t )  L ( t ) = , Y ( t ) = , j j  υ0j ( t) ηj (t )           x(t ) 1 u(t )  Z (t ) = , c= , π (t ) = ,  v(t )    l(t )  −1  ′    a0 (t ) 0 α (t ) 0   , A(t ) = , B(t ) = ′  0 b ( t ) 0 β (t )   0     ′   δj (t ) 0 σ0j (t ) 0 Cj (t ) = , Dj (t ) = , 0 υ0j (t ) 0 ηj′ (t )    L(t ) = (L1 (t ), L2 (t ), . . . , Lm (t )) ,     Y (t ) = (Y1 (t ), Y2 (t ), . . . , Ym (t )) ,        2a0 (t ) a0 (t ) + b0 (t ) F (t ) =  ,   2b0 (t )  a0′(t ) + b0′ (t )    G(t ) = α (t ) β (t ) , α ′ (t ) β ′ (t )

Assumption 3. ∀t ∈ [0, T ], Y (t )Y ′ (t ) ≥ ε In , for some ε > 0, where In is the n × n identity matrix, n = nA + nL .

(4)

where vj (t ) denote the amount allocated in the jth liability at time t, and

 ′ v(t ) = (v    1 (t ), . . . , vnL (t )) , ′  β(t ) = b1 (t ) − b0 (t ), . . . , bnL (t ) − b0 (t ) ,  ′  η (t ) = υ1j (t ) − υ0j (t ), . . . , υnLj (t ) − υ0j (t ) ,   j j = 1, 2, . . . , m.

Assumption 2. All functions A(t ), B(t ), Cj (t ) and Dj (t ) are essentially bounded and measurable on [0, T ], for j = 1, . . . , m.

E

In exactly the same way, we can obtain the dynamics for the total value of liabilities

 dl(t ) =[l(t )b0 (t ) + v ′ (t )β(t )]dt    m m    ′ v (t )ηj (t )dWj (t ) , υ0j (t )dWj (t ) + + l(t )   j =1 j=1  l(0) = l0 ,

(8)

Assumption 1. For any t ∈ [0, T ], rank(L(t )) = rank(B(t )) = 2, namely, B(t ) and L(t ) are row full rank.

Denote

 u(t ) = (u1 (t ), . . . , unA (t ))′ ,      α(t ) = a1 (t ) − a0 (t ), . . . , an (t ) − a0 (t ) ′ , A  ′  δj (t ) = σ1j (t ) − σ0j (t ), . . . , σnAj (t ) − σ0j (t ) ,   j = 1, 2, . . . , m,

 dZ (t ) = [A(t )Z (t ) + B(t )π (t )]dt   m   + [Cj (t )Z (t ) + Dj (t )π (t )]dWj (t ),   j =1  Z (0) = z0 = (x0 , l0 )′ .

Suppose that M and N are symmetric matrices with the same order. Throughout this paper, we denote M > N (M ≥ N ) if and only if (iff) M − N is positive definite (semidefinite). In particular, M > 0 (M ≥ 0) iff M is positive definite (semidefinite). Similar to most existing studies, throughout this paper, we make the following assumptions.

i=1



From (4) and (5), Z (t ), the 2-dimensional vector formed by the total value of assets and the total value of liabilities satisfies the following dynamics

(9)

The solution of the optimization problem (9), π = {π (t ); t ∈ [0, T ]}, is called the efficient strategy for d ≥ dσmin , where dσmin is the expected terminal surplus corresponding to global minimum variance of terminal surplus over all feasible strategies. A point (Var∗ [S (T )], d) in the variance–mean coordinate plane is called an efficient point, and the collection of all the efficient points is called the efficient frontier. 3. Solution of the problem

(7)

The equality constraint E[S (T )] = d in (9) can be dealt with by Lagrange methods. We can transform the optimization problem (9) to the following unconstrained stochastic control problem by introducing a Lagrange multiplier λ as follows: min

(Z (·),π (·))∈U[0,T ]

E[S 2 (T )] − d2 + 2λ (E[S (T )] − d) .

Let us focus on the optimization problem (10) first. Since E[S 2 (T )] − d2 + 2λ (E[S (T )] − d)

= E[S 2 (T )] + 2λE[S (T )] − d2 − 2λd = E[Z ′ (T )cc ′ Z (T ) + 2λc ′ Z (T )] − d2 − 2λd,

(10)

H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

  and −d2 − 2λd is fixed, the optimization problem (10) is equivalent to



1

ϕ( ˙ t ) − g ′ (t )B(t ) 4

min

(Z (·),π (·))∈U[0,T ]

E[Z ′ (T )cc ′ z (T ) + 2λc ′ Z (T )].

(11)

For any time t ∈ [0, T ] with state variable Z (t ) = z, we consider a truncated optimization problem (11). Now the dynamics of Z (s), s ∈ [t , T ], becomes

 dZ (s) = [A(s)Z (s) + B(s)π (s)]ds   m   + [Cj (s)Z (s) + Dj (s)π (s)]dWj (s),   j =1  Z (t ) = z .

(8′ )

m 

9

 −1 D′j (t )Ω (t )Dj (t )

B′ (t )g (t ) = 0,

j =1

ϕ(T ) = 0,

(16)

where Ω (t ) is a 2 × 2 matrix-valued function, g (t ) is a 2 × 1 ˙ (t ) = vector-valued function, ϕ(t ) is a scalar function, and, Ω dϕ(t ) dg (t ) dΩ (t ) ˙ , g ( t ) = , ϕ( ˙ t ) = . Eq. (14) is called a stochastic dt dt dt Riccati equation in the literature (for example, m Yong and Zhou, 1999). The positive-definiteness condition j=1 D′j (t )Ω (t )Dj (t ) > 0 guarantees the existence of a solution to the HJB equation (13). With the above preparation, we have the following theorem.

The set of all the admissible pairs for this truncated problem is now denoted by U[t , T ], i.e.,

Theorem 1. If there exists a solution to Eq. (14), then the value function of the optimization problem (12) is given by

U[t , T ] = (Z (·), π (·))|π (·) ∈ L2F (t , T ; Rn ), and

V (t , z ) = z ′ Ω (t )z + λg ′ (t )z + λ2 ϕ(t ),



 (Z (·), π (·)) satisfies (8′ ) .

and the corresponding optimal strategy is

Define the corresponding value function V (t , z ) as V (t , z ) :=

min

(Z (·),π(·))∈U[t ,T ]

J (π (·); t , z ),

 (12)

π (t ) = − ∗

m 

 −1 Dj (t )Ω (t )Dj (t ) ′

j =1

where J (π (·); t , z ) := E[Z ′ (T )cc ′ Z (T ) + 2λc ′ Z (T )|Z (t ) = z ]. Obviously, if we set t = 0, then V (0, z0 ) is the optimal value of the optimization problem (11). By the principle of dynamic programming (see pp. 132–133 of Fleming and Soner, 2006 for more details), V (t , z ) satisfies the following HJB equation

     inf Vt + (A(t )z + B(t )π (t ))′ Vz   π (t )    m  ′  1   Cj (t )z + Dj (t )π (t ) Vzz + 2 j =1           × Cj (t )z + Dj (t )π (t ) = 0,     V (T , z ) = z ′ cc ′ z + 2λc ′ z , ∂ V (t ,z )

(17)

∂ V (t ,z )

(13)

∂ 2 V (t ,z )

where Vt = ∂ t , Vz = ∂ z , Vzz = ∂ z 2 . In order to derive the expression of the value function V (t , z ), we construct functions Ω (t ), g (t ) and ϕ(t ) on [0, T ] satisfying the following differential equations and conditions m    ˙ (t ) + A(t )Ω (t ) + Ω (t )A(t ) + Ω Cj (t )Ω (t )Cj (t )     j =1      m      − Ω (t )B(t ) + Cj (t )Ω (t )Dj (t )    j =1     −1   m  ′ × Dj (t )Ω (t )Dj (t )   j =1      m    ′ ′   × B (t )Ω (t ) + Dj (t )Ω (t )Cj (t ) = 0,    j =1   m       D′j (t )Ω (t )Dj (t ) > 0, t ∈ [0, T ], Ω (T ) = cc ′ ,

 ×

B (t ) (Ω (t )z + λg (t )) + ′

m 

 Dj (t )Ω (t )Cj (t )z ′

,

(18)

j =1

where Ω (t ), g (t ) and ϕ(t ) are the solutions to Eqs. (14)–(16), respectively. Proof. If the solution to Eq. (14) exists, then (15) also admits a solution on [0, T ] since it is a linear ordinary differential equation with bounded coefficients. Hence, the solution to Eq. (16) also exists when both solutions of Eqs. (14) and (15) exist. Since the dynamics of Z (t ) given by (8) satisfies the linear growth and Lipschitz conditions, according to classical verification theorem (see Theorem 3.1 in p. 157 of Fleming and Soner, 2006), we need only to prove that V (t , z ) defined by (17) is the solution to the HJB equation (13). It is obvious that V (t , z ) satisfies the boundary condition of the HJB equation (13). In the following, we prove that V (t , z ) also satisfies the equation in (13). Notice that A(t ), Cj (t ) and Dj (t ) are symmetric matrices, the equation in (13) is equivalent to Vt + z ′ A(t )Vz +



m 1 ′ z Cj (t )Vzz Cj (t )z 2 j =1



+ inf π (t ) B (t )Vz + ′

′

π (t )

m 

 Dj (t )Vzz Cj (t )z ′

j =1



m

 1 + π ′ (t ) D′j (t )Ω (t )Dj (t )π (t ) 2 j =1

(14)

= 0.

(19)

It follows that we need to verify that V (t , z ) satisfies Eq. (19). By (17), we have

j =1

    m     g˙ (t ) + A(t ) − Ω (t )B(t ) + Cj (t )Ω (t )Dj (t )     j =1  −1   m   ′ ′  × Dj (t )Ω (t )Dj (t ) B (t ) g (t ) = 0,     j = 1  g (T ) = 2c ,

˙ (t )z + λ˙g ′ (t )z + λ2 ϕ( Vt = z ′ Ω ˙ t ), Vz = 2Ω (t )z + λg (t ), (15)

Vzz = 2Ω (t ).

(20)

According to the positive-definiteness condition in (14), we have m  j =1

D′j (t )Vzz Dj (t ) = 2

m  j =1

D′j (t )Ω (t )Dj (t ) > 0.

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H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

As a result, the first order condition which is also a sufficient condition about π (t ) in the bracket of (19) gives the optimal strategy

π(t ) = −

 m 

 −1 Dj (t )Vzz Dj (t ) ′

Theorem 2. If there exists a solution to Eq. (14), then the value function of the optimization problem (24) is given by

j =1

 B (t )Vz + ′

×

m 

 Dj (t )Vzz Cj (t )z

.

′

(21)

Substituting (21) into (19), we have

−

m 1

2 j =1



1

Vz B(t ) + z ′

2

Cj (t )Vzz Dj (t )

 m 

j=1



Theorem 3. If there exists a solution Ω (t ) to Eq. (14), then it is unique and Ω (t ) ≥ 0.

 −1 Dj (t )Vzz Dj (t ) ′

j =1



m

B′ (t )Vz +

×



D′j (t )Vzz Cj (t )z

= 0.

(22)

j =1

z′

˙ + AΩ + Ω A + Ω

m 

−

ΩB +

m 

Cj Ω D j

 m 

j =1

 ×

BΩ+ ′

D′j Ω Dj

 Dj Ω Cj ′

z





m 

+ λ g˙ + A − Ω B +

˜ (t ) by the arbitrariness of z ∈ R2 . Hence, Thus, we have Ω (t ) = Ω the solution to Eq. (14) is unique. 

 Cj Ω Dj

The proof of the existence of the solution to Eq. (14) is much more difficult than that of the uniqueness of the solution to (14) as can be seen below. Some authors (e.g., Yong and Zhou, 1999) have studied a similar Riccati-type equation arising from a stochastic LQ control problem

j =1

 ×

−1  ′ m  D′j Ω Dj B′  g  z j=1





m  1 D′j Ω Dj + λ2 ϕ˙ − g ′ B 4 j =1



 −1

B g  = 0, ′

(23)

where the dependence on t of functions in (23) is omitted for simplicity. Besides, we rewrite 2z ′ AΩ z as z ′ (AΩ + Ω A)z in (23) due to the fact that AΩ is not always a symmetric matrix, whereas AΩ + Ω A is. According to the differential equations about Ω (t ), g (t ) and ϕ(t ) in (14)–(16), Eq. (23) is obviously true. Therefore, V (t , z ) = z ′ Ω (t )z + λg (t )z + λ2 ϕ(t ) is the solution to the HJB equation (13). As a consequence, it is the value function of the optimization problem (12). The corresponding optimal strategy given in (18) can be obtained by substituting (20) into (21).  4. The existence and uniqueness of the solution to Eq. (14)

−1

′ Due to the presence of the term in j=1 Dj (t )Ω (t )Dj (t ) Eq. (14), it is a matrix-valued, highly nonlinear differential equation. One cannot derive an explicit solution in general. In the following, we will prove the existence and uniqueness of the solution to Eq. (14) first. To this end, we first consider the following relevant optimal control problem:

m

V1 (t , z ) :=

min

(π(·), Z (·))∈U[t ,T ]

J1 (π (·); t , z ) ≥ 0.

˜ (t )z . V1 (t , z ) = z ′ Ω (t )z = z ′ Ω

j =1



min

(π (·), Z (·))∈U[t ,T ]

Hence Ω (t ) ≥ 0 by the arbitrariness of z ∈ R2 , t ∈ [0, T ]. We now prove the uniqueness of the solution to Eq. (14). ˜ (t ) to Eq. (14); then by Suppose that there is another solution Ω Theorem 2 again, we have

−1

j =1

m 

for any (π (·), Z (·)) ∈ U[t , T ]. Thus, z ′ Ω (t )z = V1 (t , z ) =

Cj Ω Cj

j=1



Proof. We first prove that Ω (t ) ≥ 0. For any t ∈ [0, T ] and z ∈ R2 , by Theorem 2, we have V1 (t , z ) = z ′ Ω (t )z. On the other hand, since cc ′ ≥ 0, so, J1 (π (·); t , z ) = E[Z ′ (T )cc ′ Z (T )|Z (t ) = z ] ≥ 0

Substituting (20) into (22) and simplifying the equation, we obtain



(25)

Theorem 2 can also be obtained by the classical stochastic LQ theory (see Yong and Zhou, 1999 for more details). Using Theorem 2, we can prove that the solution to Eq. (14) is unique.

z ′ Cj (t )Vzz Cj (t )z

m 

′

V1 (t , z ) = z ′ Ω (t )z , where Ω (t ) is the solution to Eq. (14).

j =1

Vt + z ′ A(t )Vz +

where J1 (π (·); t , z ) := E[Z ′ (T )cc ′ Z (T )|Z (t ) = z ]. The optimal control problem (24) is a special case of problem (12) with λ = 0. Thus, by Theorem 1, the expression of V1 (t , z ) can be obtained by the following theorem.

J1 (π (·); t , z ),

(24)

 P˙ (t ) +  Aˆ ′ (t )P (t ) + P (t )Aˆ (t ) + Cˆ ′ (t )P (t )Cˆ (t ) + Q (t )      − P (t )Bˆ (t ) + Cˆ ′ (t )P (t )Dˆ (t )     −1 × R(t ) + Dˆ ′ (t )P (t )Dˆ (t )        × Bˆ ′ (t )P (t ) + Dˆ ′ (t )P (t )Cˆ (t ) = 0,    ˆ ′ (t )P (t )Dˆ (t ) > 0, R(t ) + D P (T ) = H .

(26)

Wonham (1968a,b) proves that there exists a unique solution to Eq. (26) under the conditions H ≥ 0, Q (t ) ≥ 0, R(t ) > 0. Chen and Zhou (2000) further proves that there exists a solution to Eq. (26) when H ≥ 0, Q (t ) ≥ 0, R(t ) ≥ 0, and with at least one of them is strictly positive definite at any time. However, under our model, H = c ′ c ≥ 0, Q (t ) ≡ 0, R(t ) ≡ 0, which does not satisfy the sufficient condition as stated in Wonham (1968a,b) and Chen and Zhou (2000). Though Chen et al. (1998) and Rami et al. (2001) give the necessary and sufficient condition for the existence of solution to Eq. (26) for cases Cˆ (t ) = 0 and Cˆ (t ) ̸= 0, respectively, the condition involves the unknown matrix function P (t ). It is very difficult to verify whether these conditions are satisfied or not. To our knowledge, the results in the literature cannot obtain the existence of a solution to Eq. (14). To study the existence of solutions to Eq. (14), we introduce some concepts and results on matrices. Consider matrices M = (mij )s×t and N = (nij )s×t . Let M and N be partitioned with Mij and

H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

Nij as the (i, j)th block submatrices, respectively,  where M ij is of order si × tj and Nij is of order pi × qj , i si = s, tj = t , i pi = j  p, j qj = q. We give the definitions of Kronecker product and Khatri–Rao product of matrices (see Liu and Trenkler, 2008). Definition 2. The Kronecker product between M and N is defined as



m11 N

··· .. .

m1t N

ms1 N

···

mst N

 . M ⊗ N = (mij N ) =  ..





D′j (t )Ω (t )Cj (t ) = Ω (t ) ∗ Y (t )L′ (t ) ,



j=1

A(t )Ω (t ) + Ω (t )A(t ) = Ω (t ) ∗ F (t ),

Ω (t )B(t ) = Ω (t ) ∗ G(t ),

Definition 3. The Khatri–Rao product between partitioned matrices M and N is defined as M ∗ N = (Mij ⊗ Nij ) = 



.. . Ms1 ⊗ Ns1

··· .. . ···

M1t ⊗ N1t



Proof. Since Ω (t ) is a 2 × 2 matrix, w(t ), θ (t ) and γ (t ) are all scalars. By the definition of the Khatri–Rao product and direct computation, we have

 w(t )δj (t )δj′ (t ) θ (t )δj (t )ηj′ (t ) Dj (t )Ω (t )Dj (t ) = θ (t )ηj (t )δj′ (t ) γ (t )ηj (t )ηj′ (t )   = Ω (t ) ∗ Yj (t )Yj′ (t ) . 

Therefore,

 .. , . Mst ⊗ Nst

m 

where Mij⊗ Nij is of order si pi × tj qj and M ∗ N is of order  si pi × tj qj . When Mij and Nij are all scalars, the Khatri–Rao product is also called the Hadamard product (see Liu and Trenkler, 2008). For the general case, we have (M +N )∗C = M ∗C +N ∗C , but M ∗N ̸= N ∗M. However, if all Mij or all Nij are scalars, we also have M ∗ N = N ∗ M (see Liu, 1999). Let M + be the Moore–Penrose pseudoinverse of M satisfying MM + M = M ,

M + MM + = M + , +

′+

−1

M is unique for all matrices M, and when the inverse M of M exists, then M + = M −1 .Let M be a symmetric square matrix and M11 ′ M12

M12 M22

m 

  Ω (t ) ∗ Yj (t )Yj′ (t )

j =1

j =1

 = Ω (t ) ∗

m 

 Yj (t )Yj (t ) ′

j =1

  = Ω (t ) ∗ Y (t )Y ′ (t ) . Other identities can be verified similarly.



Dj (t )cc Dj (t ) = cc

m

Proposition 2. ∀t ∈ [0, T ].

′

j =1



′

′



  ∗ Y (t )Y ′ (t ) > 0 for

, where M11 and M22 are also

symmetric square matrices. The following lemmas are useful.

m

j=1

D′j (t )cc ′ Dj (t ) =

  M M12 = cc ′ = −11 −11 , ∗ Y (t )Y ′ (t ) . Set M = M11 ′ M 22 12 i.e., M11 = M22 = 1, M12 = −1. Then, M ≥ 0, M let N = Y (t )Y ′ (t ) = 11 > 0 and M 22 > 0. Furthermore,    

+

be partitioned as M =

D′j (t )Ω (t )Dj (t ) =

Proof. First, similar to Proposition 1, we have

(MM ) , (M M ) M . + ′+

B′ (t )Ω (t ) = Ω (t ) ∗ G′ (t ),

where L(t ), Y (t ), F (t ) and G(t )  are definedand partitioned as in (7), w(t ) θ(t ) Ω (t ) is partitioned as Ω (t ) = θ(t ) γ (t ) , and other matrices are

′

M11 ⊗ N11



appropriately partitioned.

..  , . 

where mij N is of order p × q and M ⊗ N is of order sp × tq.



11

m

cc

 ′

m

j=1 m j=1



m

δj (t )δj′ (t )

j=1 m

ηj (t )δj′ (t )







j=1

δj (t )ηj′ (t )

ηj (t )ηj′ (t )

. Then N > 0 by Assumption 3.

Lemma 1. Let M ≥ 0 be partitioned as above such that M11 > 0 and M22 > 0, and N is partitioned properly and N > 0. Then M ∗ N > 0.

Hence, it follows from Lemma 1 that cc ′ ∗ Y (t )Y ′ (t ) > 0.

Lemma 2. Let M ≥ P ≥ 0, N ≥ Q ≥ 0, and M , P , N and Q be partitioned matrices appropriately. Then M ∗ N ≥ P ∗ Q ≥ 0.

ℜ(t , Ω (t )) := A′ (t )Ω (t ) + Ω (t )A(t ) +



 −

Ω (t )B(t ) +

 ×

D′j (t )Ω (t )Dj (t ) = Ω (t ) ∗ Y (t )Y ′ (t ) ,





j =1 m 

j =1

m 

 Cj (t )Ω (t )Dj (t )

m 

 −1 Dj (t )Ω (t )Dj (t ) ′

j =1

 ×

B (t )Ω (t ) + ′

m 

 Dj (t )Ω (t )Cj (t ) , ′

j=1

then Eq. (14) can be rewritten as the following equivalent integral equation

Ω (t ) = cc ′ +

T



ℜ(s, Ω (s))ds,

and

t m 

D′j (t )Ω (t )Dj (t ) > 0,

(27) t ∈ [0, T ].

j=1

 Cj (t )Ω (t )Cj (t ) = Ω (t ) ∗ L(t )L′ (t ) , 

j =1 m 

Cj (t )Ω (t )Cj (t )

j=1

+ Lemma 4. M ≥ 0 is equivalent to M22 ≥ 0, M22 M22 M21 = M21 and + ′ M11 − M12 M22 M12 > 0.

m 

m  j =1

(C ′ ∗ D′ )(M ∗ N )+ (C ∗ D) ≤ (C ′ M + C ) ∗ (D′ N + D).

Proposition 1. For any t ∈ [0, T ], we have



Now let

Lemma 3. Suppose that M , N , C and D are partitioned properly and M ≥ 0, N ≥ 0, C = MM + C , and D = NN + D. Then

For Lemmas 1 and 2, see Liu (1999), for Lemma 3 see Liu (2002), and for Lemma 4 see Albert (1969). Before stating and proving the sufficient condition for the existence of solution to Eq. (14), we give following propositions first.



Cj (t )Ω (t )Dj (t ) = Ω (t ) ∗ L(t )Y ′ (t ) ,





From Proposition 1, we obtain

  ℜ(t , Ω (t )) = Ω (t ) ∗ F (t ) + Ω (t ) ∗ L(t )L′ (t )   −1 − (Ω (t ) ∗ H (t )) Ω (t ) ∗ Y (t )Y ′ (t )   × Ω (t ) ∗ H ′ (t ) ,

(28)

12

H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

where H (t ) = G(t ) + L(t )Y ′ (t ). Denote

such that

K (t ) = F (t ) + L(t )L′ (t ) − H (t ) Y (t )Y ′ (t )



−1

H ′ (t ).

(29)

E[Zπ′ 0 (T )cc ′ Zπ0 (T )|Zπ0 (t ) = z ] ≤ θ z ′ z . Therefore

Proposition 3. If Ω (t ) ≥ 0, then

⃗n ; t , z ) = E[Zπ′ (T )cc ′ Zπ0 (T )|Zπ0 (t ) = z ] z ′ Ω (t )z ≤ J1 (0 0

ℜ(t , Ω (t )) ≥ Ω (t ) ∗ K (t ).

≤ θ z′z.

Proof. Set C = M = Ω (t ), D = H (t ), N = Y (t )Y (t ) > 0. Then N −1 exists and N −1 = N + . By the assumption Ω (t ) ≥ 0, we have M = Ω (t ) ≥ 0. It is easy to verify that C = M = MM + M = MM + C and D = NN − D = NN + D. This means that M , N , C and D satisfy all ′

′

  −1 = the conditions in Lemma 3. Notice that Ω (t ) ∗ Y (t )Y ′ (t )   + Ω (t ) ∗ Y (t )Y ′ (t ) and Y (t )Y ′ (t ) > 0; therefore, by Lemma 3, we have

  ℜ(t , Ω (t )) ≥ Ω (t ) ∗ F (t ) + Ω (t ) ∗ L(t )L′ (t )    + − (Ω (t )Ω + (t )Ω (t )) ∗ H (t ) Y (t )Y ′ (t ) H ′ (t )  = Ω (t ) ∗ F (t ) + L(t )L′ (t ) − H (t )  −1 ′  × Y (t )Y ′ (t ) H (t ) = Ω (t ) ∗ K (t ). This completes the proof.



Under the condition K (t ) ≥ 0, we can prove the existence of a solution to Eq. (14). Theorem 4. If K (t ) ≥ 0 for any t ∈ [0, T ], then there exists a global solution to Eq. (14) on [0, T ], where K (t ) is defined in (29). Proof. By the boundary condition of Eq. (14), we have Ω (T ) = cc ′ . According to Proposition 2, we have m 

D′j (T )Ω (T )Dj (T ) =

j=1

m 

D′j (T )cc ′ Dj (T ) > 0.

j =1

This means that the positive-definiteness condition of Eq. (14) is satisfied on the boundary t = T . By the classical ordinary differential equation theory, Eq. (14) has a local solution Ω (t ) on some interval (t0 , T ], where t0 is a real number which is less than but sufficiently close to T . We assume that this local solution Ω (t ) can be extended to the maximal interval (tM , T ] within [0, T ] with tM ≤ t0 . In the following, we show that Ω (t ) is also the solution to Eq. (14) on [tM , T ] and tM = 0. By Theorem 2, Ω (t ) ≥ 0 for ∀t ∈ (tM , T ]. In the following, we prove that Ω (t ) is uniformly bounded on (tM , T ]. To this end, we need only to show that there exists a positive scalar θ > 0 independent of tM such that

Ω (t ) ≤ θ I2 ,

∀t ∈ (tM , T ],

By the arbitrariness of z, we have Ω (t ) ≤ θ I2 . This means that Ω (t ) is uniformly bounded on (tM , T ]. Now define Ω (tM ) as Ω (tM ) = limt →t + Ω (t ). Due to the uniM

form boundedness of Ω (t ) on tM , T ], this limit is well defined. In (m the following, we prove that j=1 D′j (tM )Ω (tM )Dj (tM ) > 0. By Theorem 2 we know that Ω (t ) ≥ 0 for ∀t ∈ (tM , T ]. Since by assumption K (t ) ≥ 0, thus by Lemma 2, we have Ω (t ) ∗ K (t ) ≥ 0. Therefore, it follows from Proposition 3 that ℜ(t , Ω (t )) ≥ Ω (t ) ∗ K (t ) ≥ 0. Thus, by the equivalent integral equation (27) of Eq. (14), we have

Ω (t ) = cc ′ +

(π(·), Z (·))∈U[t ,T ]



→ tM+ in the above equation, we obtain Ω (tM ) = T cc + t ℜ(s, Ω (s))ds ≥ cc ′ . As a consequence, Ω (tM ) ∗ M       Y (tM )Y ′ (tM ) ≥ cc ′ ∗ Y (tM )Y ′ (tM ) by Lemma 2. Therefore, by ′

Propositions 1 and 2, it follows that m 

D′j (tM )Ω (tM )Dj (tM ) = Ω (tM ) ∗ Y (tM )Y ′ (tM )



    ≥ cc ′ ∗ Y (tM )Y ′ (tM ) > 0. This means that the local solution Ω (t ) satisfies the positivedefiniteness condition of Eq. (14) at t = tM . Therefore, Ω (t ) is the solution to Eq. (14) on [tM , T ]. Namely, the maximal interval (tM , T ] of the local solution Ω (t ) within [0, T ] can be extended to [tM , T ]. In the following, we claim that tM = 0. Otherwise, the local solution Ω (t ) can still be extended to the left of [tM , T ]. This contradicts to the fact that [tM , T ] is the largest continuation interval of local solution Ω (t ) on [0, T ]. In summary, we have proved that the local solution Ω (t ) is also the global solution to Eq. (14) on [0, T ].  It is worth pointing out that K (t ) defined in (29) depends only on the market parameters F (t ), G(t ), L(t ) and Y (t ), but is independent of the unknown function Ω (t ). Thus, it is easy to check whether the condition K (t ) ≥ 0 in Theorem 4 is satisfied or not. To understand the implication of this condition better, we provide an equivalent condition in the following. Proposition 4. K (t ) ≥ 0 is equivalent to L(t )L′ (t ) Y (t )L′ (t )

L(t )Y ′ (t ) Y (t )Y ′ (t )



 +

F (t ) G′ (t )

G(t ) 0





 m  dZ (s) = A(s)Z (s)ds +  C (s)Z (s)dW (s),



j



K (t ) = F (t ) + L(t )L′ (t ) − H (t ) Y (t )Y ′ (t ) is equivalent to

j =1

j



j =1

⃗n in (8′ ), where 0⃗n is the zero vector of by Theorem 2. Set π (·) ≡ 0 order n. Then the stochastic system (8′ ) becomes

 Z ( t ) = z .

t ∈ (tM , T ].

≥ 0.

Proof. According to Assumption 3, Y (t )Y ′ (t ) > 0, so (Y (t )Y ′ (t ))−1 exists and (Y (t )Y ′ (t ))+ = (Y (t )Y ′ (t ))−1 . Thus by Lemma 4,

J1 (π (·); t , z )

= E[Z (T )cc ′ Z (T )|Z (t ) = z ] ′

ℜ(s, Ω (s))ds ≥ cc ′ ,

Take t



min

T

 t

where I2 is the identity matrix of order 2. Take any t ∈ [0, T ] and z ∈ R2 , we have V1 (t , z ) = z ′ Ω (t )z =

(31)

(30)

Let Zπ0 (s) be the solution to Eq. (30). Since (30) is a homogeneous linear stochastic differential equation, there exists a scalar θ > 0

L(t )Y ′ (t ), so

F (t ) + L(t )L′ (t ) H ′ (t )

H (t ) Y (t )Y ′ (t )



F (t ) + L(t )L′ (t ) H ′ (t )

H (t ) Y (t )Y ′ (t )





 =





H ′ (t ) ≥ 0

≥ 0. Since H (t ) = G(t ) +

L(t )L′ (t ) Y (t )L′ (t )

+ Therefore the proposition is true.

−1

F (t ) G′ (t )

L(t )Y ′ (t ) Y (t )Y ′ (t )



G(t ) . 0



H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

Though we cannot derive the analytical solution to Eq. (14) in general, we can obtain its numerical solution by using numerical methods. There are quite a few efficient methods to obtain numerical solutions to the first order ordinary differential equations, such as Eq. (14). Examples of these numerical methods are Euler’s method, Linear Multistep methods, Runge–Kutta methods, Taylor Series methods and Hybrid methods (see for example, Butcher, 2003 and Hairer and Wanner, 2010 for more details). After obtaining the solution Ω (t ) to Eq. (14), we can derive solutions to Eqs. (15) and (16) in terms of Ω (t ). For that purpose, let

 Ψ (t ) := A(t ) − Ω (t )B(t ) +

m 



Proof. Since Ω (t ) is the solution to Eq. (14),

′ j=1 Dj (t )Ω (t )  − 1 ′ Dj (t ) > 0. This implies that exists and j=1 Dj (t )Ω (t )Dj (t ) is also positive definite. By Assumption 1, B(t ) is row full rank, so   −1 m  ′ B(t ) Dj (t )Ω (t )Dj (t ) B′ (t ) > 0.

m

j =1

⃗2 for any t ∈ [0, T ) since c ̸= 0⃗2 . From (34) we know that g (t ) ̸= 0 Thus, Θ (t ) =

Cj′ (t )Ω (t )Dj (t )

13

m

1 4

g (t )B(t )

 m 

 −1 Dj (t )Ω (t )Dj (t )

B′ (t )g ′ (t ) > 0.

′

j =1

j =1

 ×

m 

−1 Dj (t )Ω (t )Dj (t )

B (t ).

′

′

(32)

j=1

Then Eq. (15) becomes g˙ (t ) + Ψ (t )g (t ) = 0,

g (T ) = 2c .

(33)

g (t ) = e

t

Ψ (s)ds

g (T ) = 2e

T t

Ψ (s)ds

c,

(34)

+∞

t

Θ (s)ds < 0 for any

2d − g ′ (0)z0

, (40) 2ϕ(0) by the first-order condition. Substituting it into (18), we obtain the efficient strategy for the original M–V model (9) as follows

Mk

where the matrix function eM is defined as eM = k=0 k! for any square matrix M. After obtaining Ω (t ) and g (t ), we obtain ϕ(t ) by integrating both sides of Eq. (16):

T

Proposition 5 tells us that ϕ(0) < 0. So, the expression of the objective function in (39) is an open-down parabola as a function of λ. Hence, the optimal solution to the optimization problem (39) exists and is given by

λ∗ =

By linear differential equation theory, we have T

Therefore, by (35), we have ϕ(t ) = − t ∈ [0, T ). 

 π (t ) = − ∗

m 

 −1 Dj (t )Ω (t )Dj (t ) ′

j =1



ϕ(t ) = −

T



Θ (s)ds,

(35)

t

where

Θ (t ) :=

1 4

 g (t )B(t )

m 

 −1 D′j (t )Ω (t )Dj (t )

  2d − g ′ (0)z0 g (t ) × B′ (t ) Ω (t )z + 2ϕ(0)  m  + D′j (t )Ω (t )Cj (t )z .

(41)

j =1

B′ (t )g ′ (t ).

(36)

j =1

Substituting (40) into (39), we find the optimal value of the M–V model (9), namely, the minimum variance as follows Var∗ [S (T )]

5. Efficient strategy and frontier If we set t = 0 in (17), then the optimal value function of the optimization problem (11) is V (0, z0 ) = z0′ Ω (0)z0 + λg ′ (0)z0 + λ2 ϕ(0).

(37)

From the discussion at the beginning of Section 3, we have that the optimal value of the optimization problem (10) is given by

Φ (z0 , λ) := V (0, z0 ) − d2 − 2λd

= z0′ Ω (0)z0 + λg ′ (0)z0 + λ2 ϕ(0) − d2 − 2λd.

(38)

According to the Lagrange dual theory (see Luenberger, 1968 for more details) the optimal value of the optimization problem (9) can be obtained by the following optimization problem Var∗ [S (T )] = max Φ (z0 , λ) λ



= max λ2 ϕ(0) + λ g ′ (0)z0 − 2d λ  + z0′ Ω (0)z0 − d2 . 

4

(39)

In order to show that there exists an optimal solution to the optimization problem (39), we need the following proposition.

ϕ(0) ̸= −1, ϕ(0) = −1.

By the definition of variance, we must have Var∗ [S (T )] ≥ 0 for any d. Thus, the case of ϕ(0) = −1 is impossible. Therefore, the minimum variance should be Var∗ [S (T )] = −

1 + ϕ(0)

 d−

g ′ (0)z0

2

ϕ(0) 2(1 + ϕ(0))   g (0)g ′ (0) z0 . + z0′ Ω (0) − 4(1 + ϕ(0))

Set d = dσmin := variance is Var∗min [S (T )] = z0′



Proposition 5. For any t ∈ [0, T ), we have ϕ(t ) < 0.

  2 g ′ (0)z0 1 + ϕ(0)    d− −   ϕ(0) 2(1 + ϕ(0))    g (0)g ′ (0) = + z0′ Ω (0) − z0 ,  4(1 + ϕ(0))      1  ′  −g (0)z0 d + Ω (0) + g (0)g ′ (0) z0 ,



g ′ (0)z0 , 2(1+ϕ(0))

Ω (0) −

(42)

then by (42), the global minimum g (0)g ′ (0)



z0 . 4(1 + ϕ(0)) Obviously, the investors will not choose the expected terminal surplus d less than dσmin . In summary, we obtain the following result. Theorem 5. Under the continuous-time M–V model (9) with endogenous liabilities, for given expected terminal surplus d(≥dσmin ), the corresponding efficient strategy and efficient frontier are given by (41) and (42), respectively.

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H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

6. Two degenerate cases

Solving Eqs. (47) and (48) above, we have

In the previous sections, we assume that nL + 1 ≥ 2. This means that there are at least 2 liabilities for selection. In the following, we discuss two degenerate cases: nL + 1 = 1 and nL + 1 = 0. Degenerate case 1: There is only nL + 1 = 1 liability for selection, which is the case with exogenous liability. In this case, some parameters can be simplified as

 m = 0, π (t ) = u(t ), Y (t ) = δ (t ),  Y (t ) = δ(t ) := (δ (t ), δ (t ), . . j. , δ (t ))j , 1 2  ′   ′m  δj (t ) α ( t )  B(t ) = , , Dj (t ) = 01×nA

(43)

01×nA

and the dynamics of the liability becomes

    

 dl(t ) = l(t ) b0 (t )dt +

m 

 υ0j (t )dWj (t ) ,

j =1

l(0) = l0 .

(44)

(50)

T ′ θ (t ) = e t (a0 (s)+b0 (s)+σ0 (s)υ0 (s)−J (s))ds .



Thus by (49), γ (t ) can be expressed by T ′ γ (t ) = e t (2b0 (s)+υ0 (s)υ0 (s))ds  T θ 2 (s)  s (2b0 (r )+υ0 (r )υ ′ (r ))dr 0 N (s) − et ds. w(s) t



(51)

On the other hand, g (t ) and ϕ(t ) are also given by (34) and (35), respectively. However, an abstract matrix series representation is used in (34), which is not convenient for computation. In the following, we  can  give more specific expressions for g (t ) and ϕ(t ). ψ(t ) φ(t ) , then by (15) and (43), we have

   H (t )ψ(t ) ˙ t) ψ(  = 0,  θ (t ) ˙ t ) + b0 (t )φ(t ) − G(t ) ψ(t ) φ( w(t )     ψ(T ) 2 = , φ(T ) −2 

w(t ) θ(t )

θ(t ) γ (t ) can be obtained. By (14) and (43),

we have





Let g (t ) =

This is the case considered by Chiu and Li (2006). Therefore, in some sense, we extend the model of Chiu and Li (2006). In addition, we consider a more general market where all the assets can be risky, whereas they suppose that there is a risk-free asset in the market. In this case, Ω (t ), g (t ) and ϕ(t ) are still determined by Eqs. (14)–(16). The  closed-form  expressions for each component function of Ω (t ) =

T ′ w(t ) = e t (2a0 (s)+σ0 (s)σ0 (s)−M (s))ds ,

 w( ˙ t ) θ˙ (t ) ˙ t ) γ˙ (t ) θ(   2a0 (t )w(t ) (a0 (t ) + b0 (t )) θ (t ) + 2b0 (t )γ (t ) (a0 (t ) + b0 (t )) θ (t )      ′ σ0 (t )σ0 (t )  w(t )  σ0 (t )υ0′ (t ) θ (t )  + σ0 (t )υ0′ (t ) θ (t ) υ0 (t )υ0′ (t ) γ (t )   M (t )w(t ) J (t )θ (t ) θ 2 (t )  = 0, − J (t )θ (t ) N (t ) w(t )

(52)

where



    −1 δ(t )δ ′ (t ) α(t ),  ′     −1 G(t ) = α (t ) + υ0 (t )δ ′ (t ) δ(t )δ ′ (t ) α(t ). H (t ) = a0 (t ) − α ′ (t ) + σ0 (t )δ ′ (t )

Therefore, ψ(t ) and φ(t ) can be obtained by solving the following ordinary differential equations

˙ t ) + H (t )ψ(t ) = 0, ψ( (45)

ψ(T ) = 2,

(53)

θ (t ) ˙ t ) + b0 (t )φ(t ) − G(t ) ψ(t ) = 0, φ( w(t )

φ(T ) = −2.

On the other hand, it follows from (16) and (43) that

ϕ( ˙ t) −

  −1 1 ψ 2 (t ) ′ α (t ) δ(t )δ ′ (t ) α(t ) = 0, 4 w(t )

ϕ(T ) = 0. (55)

Solving Eqs. (53)–(55), we obtain

where

T

  −1  M (t ) = α ′ (t ) + σ0 (t )δ ′ (t ) δ(t )δ ′ (t )    × α(t ) + δ(t )σ0′ (t ) ,        J (t ) = α ′ (t ) + υ0 (t )δ ′ (t ) δ(t )δ ′ (t ) −1     × α(t ) + δ(t )σ0′ (t ) , (46)   −1   N (t ) = α ′ (t ) + υ0 (t )δ ′ (t ) δ(t )δ ′ (t )     × α(t ) + δ(t )υ0′ (t ) ,    σ0 (t ) = (σ01 (t ), . . . , σ0m (t )) , υ0 (t ) = (υ01 (t ), . . . , υ0m (t )) .   1 −1 Notice that Ω (T ) = c ′ c = , then by (45), w(t ), −1 1 θ(t ), γ (t ) are determined by the following ordinary differential equations and terminal conditions

w( ˙ t ) + 2a0 (t ) + σ0 (t )σ0′ (t ) − M (t ) w(t ) = 0, 

(54)



w(T ) = 1,   ˙θ(t ) + a0 (t ) + b0 (t ) + σ0 (t )υ0′ (t ) − J (t ) θ (t ) = 0,

(47)

θ(T ) = −1,   θ 2 (t ) = 0, γ˙ (t ) + 2b0 (t ) + υ0 (t )υ0′ (t ) γ (t ) − N (t ) w(t ) γ (T ) = 1 .

(48)

ψ(t ) = 2e t H (s)ds ,    T φ(t ) = − 2e t b0 (s)ds +

G(s) t

ϕ(t ) = −

T

 t

s θ (s) ψ(s)e t b0 (z )dz ds , w(s)



 −1 ψ 2 ( s) ′  α (s) δ(s)δ ′ (s) α(s)ds. 4w(s)

(57)

(58)

Notice that Y (t ) = δ(t ) in this case, so from Assumption 2, we obtain that Y (t )Y ′ (t ) = δ(t )δ ′ (t ) > 0. Therefore, by (50) we have w(t ) > 0. This implies that m 

D′j (t )Ω (t )Dj (t ) = δ(t )δ ′ (t ) w(t ) > 0,





j =1

which guarantees that Theorems 1 and 5 also hold in this case. Degenerate case 2: nL + 1 = 0, namely, there is no liability. This is the case considered by Zhou and Li (2000). In addition, all the assets in our model can be risky, whereas there is a risk-free asset in their model. In this case, we have

 Z (t ) = x(t ),   L (t ) = σ (t ),

c = 1, π (t ) = u(t ), Yj (t ) = δj (t ),

Y (t ) = δ(t ),   B(t ) = α ′ (t ),

A(t ) = a0 (t ), Dj (t ) = δj′ (t ).

j

(49)

(56) T

0j

Cs (t ) = σ0j (t ),

(59)

H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

15

In this case, Ω (t ), g (t ) and ϕ(t ) are also determined by Eqs. (14)– (16), but they are all scalar functions now. Eqs. (14)–(16) in this case can be further simplified as follows

  ˙ (t ) + 2a0 (t ) + σ0 (t )σ0′ (t ) − M (t ) Ω (t ) = 0, Ω

(60)

Ω (T ) = 1, g˙ (t ) + H (t )g (t ) = 0,

ϕ( ˙ t) −

g (t ) 2

4Ω (t )

g (T ) = 2,

(61)

 −1 α ′ (t ) δ(t )δ ′ (t ) α(t ) = 0,

ϕ(T ) = 0.

(62)

Comparing (60) with (47), (61) with (53), we find that Ω (t ) and g (t ) in this case are exactly the same as w(t ) and ψ(t ) in Degenerate case 1, respectively. Namely, Ω (t ) = w(t ), g (t ) = ψ(t ). By (55) and (62), this further shows that ϕ(t ) is the same as that in Degenerate case 1. Remark 3. Since ϕ(0) in this case is the same as that in Degenerate case 1, the shape of the efficient frontier in the variance–mean coordinate plane is the same as that in Degenerate case 1. This means that exogenous liability only shifts the efficient frontier parallelly, but does not change the shape of the efficient frontier. Similar to Degenerate case 1, in this case we also have m 

D′j (t )Ω (t )Dj (t ) = δ(t )δ ′ (t ) w(t ) > 0,





j =1

which implies that Theorems 1 and 5 also hold in this case. Moreover, the efficient strategy and the efficient frontier can be further simplified as follows

  −1 π ∗ (t ) = δ(t )δ ′ (t )     2d − g (0)x0 × δ(t )σ0′ (t )x + x + g (t ) α(t ) , (63) 2ϕ(0)w(t ) and Var∗ [S (T )] = −

1 + ϕ(0)

g (0)x0



ϕ(0)  + w(0) −

d−

2

2(1 + ϕ(0))

g (0) 2

4(1 + ϕ(0))



x20 ,

(64)

respectively. 7. Numerical examples In this section, we provide some numerical examples to illustrate our results. Assume an investor with the initial total value of assets x0 = 3 and the initial total value of liabilities l0 = 2. Then the initial surplus is S0 = x0 − l0 = 1. The investor enters the market at time 0, and exits the market at time T = 5. In addition, suppose that the investor can invest two assets and allocate two liabilities, and the related market parameters are A(t ) = B(t ) =

0.0161 0

0 , 0.0153

0.0068 0

0 , −0.0064

 



−0.0528 0.0014   0.0923 0.0402 L(t )L′ (t ) = , 0.0402 0.1080   0.0978 −0.0040 ′ Y (t )Y (t ) = . −0.0040 0.1468 ′

L(t )Y ′ (t ) = Y (t )L′ (t )



=

Plugging the above data into the expression of K (t ) in (29), we obtain

K (t ) =

0.1001 0.0634



0.0634 0.0858



> 0,

i.e., K (t ) is positive definite for t > 0. This means that K (t ) satisfies the condition in Theorem 4. Hence, by the previous analysis, there exists a unique solution to each of the Eqs. (14)–(16). Since Eq. (14) cannot be solved analytically, we can only solve Eqs. (14)–(16) numerically. Eqs. (14)–(16) are ordinary differential equations, there are many numerical methods to deal with them, such as Runge–Kutta methods, Taylor Series methods and Hybrid methods (see Butcher, 2003 and Hairer and Wanner, 2010). For this numerical example, we use the MATLAB built-in ODE solver, ‘‘ode45’’, to obtain numerical solutions. This ODE solver is based on the classical Runge–Kutta method. To solve Eqs. (14)–(16), we divide the time interval [0, T ] into 45 subintervals evenly, and compute the corresponding function values of Ω (t ), g (t ) and ϕ(t ) at the corresponding points. Substituting these results into (42), we obtain the efficient frontier Var∗ [S (5)] = 317.2716 (d − 1.1809)2 + 3.1509. To distinguish from the degenerate cases, we call the above case as the General case. In the following, we discuss two degenerate cases. Degenerate case 1: There is only one liability (assume that it is the first liability in the General case). This is the case with exogenous liability. According to the analysis in Section 6, we can solve Ω (t ), g (t ) and ϕ(t ) explicitly. Substituting the data into the expressions of Ω (t ), g (t ) and ϕ(t ), we obtain the efficient frontier as follows Var∗ [S (5)] = 493.2647 (d − 1.1534)2 + 4.2013. Degenerate case 2: There is no liability. By the analysis in Section 6, we can also solve Ω (t ), g (t ) and ϕ(t ) explicitly in this case. Substituting the data into (64), we obtain the efficient frontier Var∗ [S (5)] = 493.2647 (d − 1.1035)2 + 0.4572.





Fig. 1. Efficient frontiers of terminal surplus for different cases.

−0.0119 , −0.0812 

The efficient frontiers for the above three cases are displayed in Fig. 1. From Fig. 1, we found that (i) compared with the efficient frontier of Degenerate case 1, the efficient frontier of the General case (namely, the case with endogenous liabilities) shifts to the upper left, namely, the investor can reduce the variance (risk) of terminal surplus for given expected terminal surplus, or increase the expected terminal surplus for given variance of terminal surplus in the General case; (ii) compared with the efficient frontiers of cases with liabilities (including General case

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H. Yao et al. / Insurance: Mathematics and Economics 52 (2013) 6–17

8. Conclusion

Fig. 2. Impact of exit time on the efficient frontier.

In this paper we investigate a continuous-time M–V ALM problem with endogenous liabilities in a more general market where all the assets can be risky, whereas studies in the literature on continuous-time M–V ALM consider only the case with exogenous liability and a risk-free asset. We optimize allocation not only for the assets, but also for the liabilities in this paper. By using the Lagrange dual theory and the HJB method, we solve the problem analytically, and derive closed form expressions of the efficient strategy and the M–V efficient frontier. By applying the Khatri–Rao matrix product technique and the relevant stochastic control theory, we prove the existence and uniqueness of the solution to Riccati-type Eq. (14) induced by the HJB equation. In addition, two degenerate cases of our model are discussed. Finally, some numerical examples are provided to illustrate the results obtained in this paper. In particular, the impacts of endogenous liabilities, exit time and the initial total value level of liabilities on the efficient frontier are demonstrated by numerical examples. The method introduced in this paper has generality and adaptability. It can be further used to study the continuoustime M–V ALM problems with endogenous liabilities under various realistic cases, such as regime-switching market environment, short-selling constraint, uncertain exit time, and so on. Acknowledgment The authors are grateful to the anonymous referee for giving them very useful suggestions and comments. References

Fig. 3. Impact of the initial total value level of liabilities on the efficient frontier.

and Degenerate case 1), the efficient frontier of Degenerate case 2 shifts to the lower left. In order to study the impact of exit time T on the efficient frontier of terminal surplus, we plot the efficient frontiers for cases with five different exit times T = 1, 2, 3, 4, and 5 in Fig. 2. Fig. 2 demonstrates that (i) when the exit time T increases from 1 to 5, the corresponding efficient frontier move to the upper right; (ii) the longer the exit time is, the bigger the global minimum variance and its corresponding expected terminal surplus are. In the following, we investigate the impact of the initial total value level of liabilities l0 on the efficient frontier. We let x0 and l0 change and fix S0 = 1. We select six different initial total value levels of liabilities l0 = 0, 1, 2, 3, 4, 5. Then, according to the relation x0 = S0 + l0 , we can obtain the corresponding initial total value levels of assets as x0 = 1, 2, 3, 4, 5, 6. The corresponding efficient frontiers are presented in Fig. 3. From Fig. 3 we find that the impact of the initial value of the total liability level l0 on the efficient frontier is similar to that of the exit time T . From the above analysis of numerical examples, we have the following conclusions. (i) Consideration of multiple liabilities for selection and allocation can reduce the risk for given expected return, or enhance the expected return for given risk. (ii) Liabilities (including exogenous and endogenous liabilities) substantially increase the global minimum variance of terminal surplus and its corresponding expected terminal surplus. (iii) The longer the time horizon of investment is, or the more the liabilities is, the higher the investment return the investor can obtain, meanwhile the more risk he or she needs to take.

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