Portfolio Selection with Random Liability and Affine Interest Rate in the Mean-Variance Framework

Hao Chang; Chunfeng Wang; Zhenming Fang

doi:10.21078/JSSI-2017-229-21

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Portfolio Selection with Random Liability and Affine Interest Rate in the Mean-Variance Framework

Hao Chang , Chunfeng Wang and Zhenming Fang

Published/Copyright: August 1, 2017

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From the journal Journal of Systems Science and Information Volume 5 Issue 3

Abstract

This paper studies a dynamic mean-variance portfolio selection problem with random liability in the affine interest rate environment, where the financial market consists of three assets: one risk-free asset, one risky asset and one zero-coupon bond. Assume that short rate is driven by affine interest rate model and liability process is described by the drifted Brownian motion, in addition, stock price dynamics is affected by interest rate dynamics. The investors expect to look for an optimal strategy to minimize the variance of the terminal surplus for a given expected terminal surplus. The efficient strategy and the efficient frontier are explicitly obtained by applying dynamic programming principle and Lagrange duality theorem. A numerical example is given to illustrate our results and some economic implications are analyzed.

Keywords: affine interest rate; random liability; mean-variance criterion; the efficient strategy; the efficient frontier; Lagrange duality theorem

1 Introduction

Mean-variance criterion and utility criterion are two fundamental methods dealing with the portfolio selection problems in the modern financial theory. It is all well-known that the pioneering work on the mean-variance portfolio selection problem was done by Markowitz[1]. But the optimal asset allocation problem that Markowitz considered was a simple and single-period discrete-time one and was studied in a static mean-variance framework. Therefore, multi-period and continuous-time dynamic portfolio choice problems have attracted great attentions of many authors since Markowitz. Bajeux-Besnainou and Portai[2] used a martingale approach to investigate a continuous-time dynamic mean-variance problem. Li and Ng[3] developed a linear quadratic (LQ) control approach and for the first time successfully solved the mean-variance portfolio selection problem in a multi-period environment. Zhou and Li[4] applied the LQ approach to a continuous-time mean-variance portfolio selection problem and obtained the closed-form solutions to the efficient strategy and the efficient frontier. Fu, et al.[5] used dynamic programming (DP) principle and Lagrange duality theorem to solve a continuous-time mean-variance problem with borrowing constraints and successfully derived the closed-form solutions to the optimal policy and the efficient frontier. Lim and Zhou[6] and Ferland and Watier[7] applied the backward stochastic differential equation (BSDE) theory to the mean-variance portfolio selection problems with random parameters and with extended CIR interest rates respectively. Those research results laid down solid foundations for further investigating the mean-variance portfolio selection problems. In summary, there are four main techniques in dealing with mean-variance portfolio selection problems: (i) The martingale approach; (ii) The LQ approach, (iii) DP principle and Lagrange duality theorem, (iv) BSDE theory.

In recent years, the asset and liability management (ALM) problems have been paid more and more attentions to. As a matter of fact, liability factor is often confronted with in the activity of investment. Some investors realized that it was very clear that the optimal strategies with liability environments are more practical. Therefore, it is very necessary for us to introduce liability factors into dynamic portfolio selection problems and investigate the optimal investment strategy. We can summarize research results on the ALM problems as follows. (i) Multi-period ALM problems. For example, Yi, et al.[8] studied an ALM problem with uncertain investment horizon, Chen and Yang[9] investigated an ALM problem with regime switching, Yao, et al.[10] considered an ALM problem with uncontrolled cash flow and uncertain horizon, Leippold, et al.[11] focused on an ALM problem with endogenous liability, and so on. (ii) Continuous-time ALM problems. For instance, Xie, et al.[12] studied an ALM problem by assuming liability process to be driven by the drifted Brownian motion, Zeng and Li[13] assumed liability process to be governed by Brownian motion with jump and investigated the optimal policy and the efficient frontier, Chen, et al.[14] devoted themselves to the ALM problem with regime switching, Yao, et al.[15] discussed the ALM problem with endogenous liabilities. These research results greatly enriched and expanded the ALM theories and laid down solid foundations for solving more sophisticated and realistic ALM problems. The concern was that the research results mentioned above were achieved under the assumption of constant interest rate.

It is all well-known that interest rate isn’t always fixed. Ever since the oil crisis in the last century, the interest rate has appeared more volatile in the many western countries. Therefore, it is very reasonable to incorporate stochastic interest rate model into the dynamic portfolio selection problems. In addition, it is very obvious that the optimal investment strategy under stochastic interest rate model will be more practical. Previous works on dynamic portfolio selection problems with stochastic interest rates included Korn and Kraft[16], Deelstra, et al.[17], Gao[18], Chang, et al.[19], Li and Wu[20], Liu[21], Chang and Rong[22], Guan and Liang[23,24]. Those models were all studied in the utility maximization framework. As far as we know, it is very difficult to derive the explicit expressions of the optimal policies and the efficient frontier in investigating dynamic mean-variance portfolio selection problems with stochastic interest rates. In the existing literatures, only few authors, for example, Lim and Zhou[6] and Ferland and Watier[7], successfully solved those problems, but the approach they used is just the BSDE theory. But if we introduce liability process into dynamic mean-variance portfolio selection problems with stochastic interest rates, it will give rise to many difficulties in constructing the BSDE such that we have no ways to obtain the closed-form solutions to the efficient strategy and the efficient frontier.

As a matter of fact, stochastic optimal control theory is also an effective approach in dealing with dynamic mean-variance portfolio selection problems with stochastic interest rates. In this paper, we introduce liability process into a dynamic mean-variance portfolio selection problem and assume interest rate to be driven by affine interest rate model, which includes the Vasicek model and the CIR model as special cases. Liability process is supposed to be governed by the drifted Brownian motion. The surplus is expected to be allocated in the three financial assets, which consists of one risk-free asset, one risky asset and one zero-coupon bond. The objective of the investors is to minimize the variance of the terminal surplus for a given the expectation of terminal surplus. In this paper, dynamic programming principle is used to derive the optimal investment strategies. After successfully solving the equation (22), we obtain the closed-form solutions to the optimal policies. By applying Lagrange duality theorem, we obtain the explicit expression of the efficient frontier. As a result, a numerical example is provided to illustrate these results. There have three main contributions in this paper: (i) We introduce liability factor into a dynamic portfolio selection problem with affine interest rate; (ii) We successfully solve the mean-variance model with stochastic interest rate; (iii) We use dynamic programming principle and Lagrange duality theorem altogether to successfully obtain the closed-form solutions to the efficient strategies and the efficient frontier.

The paper proceeds as follows. Section 2 describes the problem framework. In Section 3, we use Lagrange multiplier technique and dynamic programming principle to obtain the closed-form solution to the optimal policies. In Section 4, we obtain the explicit expression of the efficient frontier by applying Lagrange duality theorem. Section 5 gives a numerical example to demonstrate our results and Section 6 concludes the paper.

2 The Model

In this section, we present the problem formulation, which is composed of four parts: The financial market, the liability process, the wealth process and the optimization criterion. Assume that the financial market consists of three financial assets: One risk-free asset, one risky asset and one zero-coupon bond. In addition, an investor can buy or sell continuously without incurring any restriction as short-selling constraint or borrowing constraint or any transaction costs.

Assume that (W_r(t), W_S(t)) is a two-dimension standard and adapted independent Brownian motion defined on complete probability space (Ω, ℱ, ℙ, {ℱ_t}_{t∈[0, T]}) , where ℙ is the real world probability and the filtration {ℱ_t}_{t∈[0, T]} is the information structure generated by Brownian motion (W_r(t), W_S(t)). [0, T] represents the finite and fixed investment horizon.

The first asset is a risk-free asset (i.e., cash or bank account), whose price at time t is denoted by S₀(t), t ∈ [0, T], then S₀(t) evolves according to

dS0(t)=r(t)S0(t)dt,S0(0)=1,(1)

where short rate dynamics r(t) is described by the following stochastic differential equation (SDE):

dr(t)=(a−br(t))dt−k1r(t)+k2dWr(t),r(0)=r0>0,(2)

where a, b, k₁ and k₂ are positive constants.

Remark 1

SDE (2) have been investigated by Deelstra, et al.[17], Gao[18] and Chang, et al.[19]. Their papers show that term structure of interest rate is affine under SDE (2). In addition, the Vasicek[25] (resp. CIR model[26]) dynamics is one of special case of these dynamics, when k₁ (resp. k₂) is equal to zero.

The second asset is the stock. Its price, denoted by S₁(t), t ∈ [0, T], is given by (referring to Deelstra, et al.[17], Gao[18] and Chang, et al.[19]):

dS1(t)S1(t)=r(t)dt+σ1(dWS(t)+λ1dt)+σ2k1r(t)+k2(dWr(t)+λ2k1r(t)+k2dt),(3)

with initial condition S₁(0) = 1 and λ₁, λ₂ (resp. σ₁, σ₂) being constants (resp. positive constants), see Deelstra, et al.[17], Gao[18] and Chang, et al.[19].

The third asset is a zero-coupon bond with maturity T, whose price is denoted by B(t, T). The dynamics of B(t, T) is described by the following SDE (referring to [17–19]):

dB(t,T)B(t,T)=r(t)dt+σB(t,r(t))(dWr(t)+λ2k1r(t)+k2dt),B(T,T)=1,(4)

where

σB(t,r(t))=h(t)k1r(t)+k2,h(t)=2(em(T−t)−1)m−(b−k1λ2)+em(T−t)(m+b−k1λ2),m=(b−k1λ2)2+2k1.(5)

Suppose that an investor has the exogenous liability L(t) at time t, t∈ [0, T], then L(t) can be governed by the drifted Brownian motion:

dL(t)=udt+vdWS(t),L(O)=l0>0,(6)

where u and v are positive constants.

Assume that an investor is equipped with the initial wealth w₀ > 0 and has initial liability l₀ > 0 at t = 0, then the initial surplus of the investor is given by x₀ = w₀ − l₀ > 0. The surplus process at time t is denoted by X(t), and the capital amount invested the stock and the zero-coupon bond is denoted by π₁(t) and π₂(t) respectively. Under the investment strategy π(t) = (π₁(t), π₂(t)), the surplus process X(t) satisfies the following dynamics:

dX(t)=(X(t)r(t)+π1(t)σ1λ1+π1(t)σ2λ2(k1r(t)+k2)+π2(t)σBλ2k1r(t)+k2−u)dt+(π1(t)σ1−v)dWS(t)+(π1(t)σ2k1r(t)+k2+π2(t)σB)dWr(t),(7)

with the initial condition X(0) = x₀ > 0.

Definition 1

(Admissible strategy) An investment strategy (π₁(t), π₂(t)) is said to be admissible if the following conditions are satisfied:

(π₁(t), π₂(t)) is ℱ_t-progressively measurable, and
∫0Tπ12(t)dt<∞,∫0Tπ22(t)dt<∞;
E(∫0T((π1(t)σ1−v)2+(π1(t)σ2k1r(t)+k2+π2(t)σB)2)dt)<∞;
The SDE (6) has an unique solution under the investment strategy π(t) = (π₁(t), π₂(t)).

We denote the set of all admissible strategies π(t) = (π₁(t), π₂(t)) by Γ, The investors’ objective is to find an optimal portfolio (π₁(t), π₂(t)) such that the expected terminal surplus satisfies E(X(T)) = C, for some constant C ∈ ℝ, while the risk measured by the variance of the terminal surplus

VarX(T)=E(X(T)−E(X(T)))2=E(X(T)−C)2(8)

is minimized. The problem of finding such a portfolio (π₁(t), π₂(t)) is referred to as the mean-variance portfolio choice problem.

Therefore the mean-variance model can be formulated as a linearly constrained stochastic optimization problem:

minπ(t)∈Γ VarX(T)=E(X(T)−C)2s.t. E(X(T))=C.(9)

Finally, an optimal investment strategy of the above problem is called an efficient portfolio corresponding to some constant C ∈ ℝ, and the corresponding (C, VarX (T)) is called an efficient point, whereas the set of all the efficient points is called the efficient frontier.

3 The Optimal Portfolios

To find the optimal investment strategy for the problem (8) corresponding to the constraint E(X(T)) = C, we introduce a Lagrange multiplier 2 λ ∈ ℝ and arrive at the new objective function:

L^(π1(t),π2(t),λ)=E((X(T)−C)2+2λ(X(T)−C))=E(X(T)−(C−λ))2−λ2.(10)

Letting η = C − λ, we obtain the following stochastic control problem:

minL¯(π1(t),π2(t),η)=E(X(T)−η)2−(C−η)2.(11)

The link between the problems (9) and (11) is provided by Lagrange duality theorem (see Fu, et al.[5]), namely, we have

min VarX(T)=maxλ∈Rminπ(t)∈ΓL^(π1(t),π2(t),λ)=maxη∈Rminπ(t)∈ΓL¯(π1(t),π2(t),η).(12)

For fixed constants η and C, the problem (11) is clearly equivalent to

minπ(t)∈ΓE(X(T)−η)2.(13)

We define the value function H(t, r, x) as

H(t,r,x)=minπ(t)∈ΓE((X(T)−η)2|r(t)=r,X(t)=x)(14)

with the boundary condition H(T, r, x) = (x − η)².

For any function H(t, r, x)∈ ℂ^1,2,2([0, T]× ℝ × ℝ), we define a variational operator:

Aπ(t)H(t,r,x)=Ht+(rx+π1(t)σ1λ1+π1(t)σ2λ2σr2+π2(t)σBλ2σr−u)Hx+12((π1(t)σ1−v)2+(π1(t)σ2σr+π2(t)σB)2)Hxx+12σr2Hrr+(a−br)Hr−(π1(t)σ2σr2+π2(t)σBσr)Hrx,(15)

where σr=k1r(t)+k2, and H_t, H_x, H_xx, H_r, H_rr, H_rxrepresent first-order and second-order partial derivatives with respect to the variables t, r, x, respectively.

According to the conclusion obtained by Fleming and Soner[27], we can derive the following HJB equation for the problem (14):

minπ(t)∈Γ{Aπ(t)H(t,r,x)}=0,(16)

with terminal condition H(T, r, x) = (x − η)².

Differentiating (16) with respect to π₁(t) and π₂(t) respectively, we derive

π1∗(t)=−λ1σ1⋅HxHxx+vσ1,(17)

π2∗(t)=σrσB⋅σ2λ1−σ1λ2σ1⋅HxHxx+σrσB⋅HrxHxx−σ2σ1⋅σrσBv.(18)

Substituting (17) and (18) back into (16), we have

Ht+(rx+λ1v−u)Hx+(a−br)Hr+12σr2Hrr−12(λ12+λ22σr2)Hx2Hxx+λ2σr2HxHrxHxx−12σr2Hrx2Hxx=0,(19)

with terminal condition: H(T, r, x) = (x − η)².

We try to conjecture H(t, r, x) with the following structure

H(t,r,x)=f(t,r)(x−g(t,r))2,f(T,r)=1,g(T,r)=η.(20)

The partial derivatives with respect to H(t, r, x) are following:

Ht=ft(x−g)2−2f(x−g)gt,Hx=2f(x−g),Hxx=2f,Hr=fr(x−g)2−2f(x−g)gr,Hrx=2fr(x−g)−2fgr,Hrr=frr(x−g)2−4(x−g)frgr+2fgr2−2f(x−g)grr.(21)

Putting the above partial derivatives into (19) and considering

(rx+λ1v−u)Hx=2rf(x−g)2+2(rg+λ1v−u)f(x−g).

After some simplification, we obtain

(x−g)2(ft+(2r−λ12−λ22σr2)f+(a−br+2λ2σr2)fr+12σr2frr−σr2fr2f)−2f(x−g)2(gt−rg+(a−br+λ2σr2)gr+12σr2grr+u−λ1v)=0.(22)

In order to eliminate the dependence on x, we decompose (22) into the following two equations:

ft+(2r−λ12−λ22σr2)f+(a−br+2λ2σr2)fr+12σr2frr−σr2fr2f=0,f(T,r)=1;(23)

gt−rg+(a−br+λ2σr2)gr+12σr2grr+u−λ1v=0,g(T,r)=η.(24)

Lemma 1

Assume that a solution to (23) is of the structure f(t, r) = e^{D₁(t)+D₂(t)r}, with terminal conditions: D₁(T) = 0 and D₂(T) = 0, then we have the following conclusions:

Under the condition of
λ2∈R,b2<4k1,λ2≠bk1,b2=4k1,λ2<ξ1orλ2>ξ2,b2>4k1,
then D₂(t) and D₁(t) are given by (a8) and (a10) in the Appendix respectively.
Under the condition of
λ2=bk1,b2=4k1,λ2=ξ1orλ2=ξ2,b2>4k1,
then D₂(t) and D₁(t) are given by (a13) and (a14) in the Appendix respectively.
Under the condition ofξ₁ < λ₂ < ξ₂and b² > 4k₁, then D₂(t) and D₁(t) are given by (a16) and (a17) in the Appendix respectively.
Hereξ₁andξ₂are given by (a6) in the Appendix.

Proof

See the Appendix (A1).

In order to solve (24), we have the following lemma.

Lemma 2

Suppose that

g(t,r)=(u−λ1v)∫tTg~(s,r)ds+ηg~(t,r)

is a solution to (24), theng~(t,r)satisfies:

g~t−rg~+(a−br+λ2σr2)g~r+12σr2g~rr=0,g~(T,r)=1.(25)

Proof

See the Appendix (A2).

Lemma 3

Assume that a solution to (25) is of the formg~(t,r) = e^{D₃(t)+D₄(t)r}, with terminal conditions: D₃(T) = 0 and D₄(T) = 0, then D₄(t) and D₃(t) are given by (a26) and (a28) in the Appendix respectively.

Proof

See the Appendix (A3).

Further, we have

HxHxx=x−g,HrxHxx=D2(t)(x−g)−gr.

As a result, considering X(t) = x and r(t) = r, we draw the following conclusion.

Theorem 1

For given constantsηand C, the optimal investment strategies of the problems (9) and (11) are given by

π1∗(t)=−λ1σ1⋅(X(t)−g(t,r))+vσ1,(26)

π2∗(t)=σrσB⋅σ2λ1−σ1λ2σ1⋅(X(t)−g(t,r))+σrσB⋅(D2(t)(X(t)−g(t,r))−∂g(t,r)∂r)−σ2σ1⋅σrσBv,(27)

where

g(t,r)=(u−λ1v)∫tTeD3(s)+D4(s)rds+ηeD3(t)+D4(t)r,(28)

∂g(t,r)∂r=(u−λ1v)∫tTD4(s)eD3(s)+D4(s)rds+ηD4(t)eD3(t)+D4(t)r,(29)

and D₁(t), D₂(t), D₃(t) and D₄(t) are given byLemma 1andLemma 3, respectively.

4 The Efficient Frontier

In this section, we will use Lagrange duality theorem to derive the explicit expression of the efficient frontier and provide some special cases.

According to (14), the minimized value of (11) is written as f(0, r₀)(x₀ − g(0, r₀))². So the minimized value of (9) is given by

L¯min(π1∗(t),π2∗(t),η)=f(0,r0)(x0−g(O,r0))2−(C−η)2=f(0,r0)(x0−(u−λ1v)∫0Tg~(s,r(s))ds−ηg~(O,r0))2−(C−η)2=(f(0,r0)g~2(0,r0)−1)η2−2η(ψf(0,r0)g~(0,r0)−C)+ψ2f(0,r0)−C2,(30)

where

ψ=x0−(u−λ1v)∫0Tg~(s,r(s))ds.(31)

Noting that

f(0,r0)g~2(0,r0)−1=eD1(0)+2D3(0)+(D2(0)+2D4(0))r0−1,

we have the following conclusion.

Lemma 4

D₂(t), D₄(t) andD₁(t) + 2D₃(t) increases with respect to the variablet. Moreover, we haveD₂(t) < 0, D₄(t) < 0 andD₁(t)+2D₃(t) < 0, for ∀ t∈[0, T].

Proof

See the Appendix (A4).

According to Lemma 4, we have

f(0,r0)g~2(0,r0)−1=eD1(0)+2D3(0)+(D2(0)+2D4(0))r0−1<0.

Therefore, L¯min(π1∗(t),π2∗(t),η) can be maximized when η is given by

η∗=ψf(0,r0)g~(0,r0)−C(f(0,r0)g~2(0,r0)−1)=ψeD1(0)+D3(0)+(D2(0)+D4(0))r0−CeD1(0)+2D3(0)+(D2(0)+2D4(0))r0−1.(32)

In addition, the maximized value of L¯min(π1∗(t),π2∗(t),η) is

L¯maxmin(π1∗(t),π2∗(t),η∗)=f(0,r0)g~2(0,r0)1−f(0,r0)g~2(0,r0)(C−ψg~−1(0,r0))2(33)

In conclusion, we summarize the above results in the following proposition.

Theorem 2

For a given constant C ∈ ℝ, the efficient strategies for the original mean-variance problem (8) corresponding to E(X(T)) = C are given by

π1∗(t)=−λ1σ1⋅(X(t)−g(t,r))+vσ1,(34)

π2∗(t)=σrσB⋅σ2λ1−σ1λ2σ1⋅(X(t)−g(t,r))+σrσB⋅(D2(t)(X(t)−g(t,r))−∂g(t,r)∂r)−σ2σ1⋅σrσBv,(35)

with the efficient frontier given by

VarX(T)=1e−D1(0)−2D3(0)−(D2(0)+2D4(0))r0−1(E(X(T))−ψe−D3(0)−D4(0)r0)2,(36)

where

ψ=x0−(u−λ1v)∫0Tg~(s,r)ds,η∗=ψeD1(0)+D3(0)+(D2(0)+D4(0))r0−CeD1(0)+2D3(0)+(D2(0)+2D4(0))r0−1,(37)

g(t,r)=(u−λ1v)∫tTeD3(s)+D4(s)rds+η∗eD3(t)+D4(t)r,(38)

∂g(t,r)∂r=(u−λ1v)∫tTD4(s)eD3(s)+D4(s)rds+η∗D4(t)eD3(t)+D4(t)r.(39)

Here, D₁(t), D₂(t), D₃(t) andD₄(t) are given byLemma 1andLemma 3, respectively.

Remark 2

From Theorem 3, some conclusions are drawn as follows. (i) π1∗(t) is correlated with the parameters a, b, k₁, k₂, λ₁, σ₁, λ₂, u, v, but it is not correlated with the parameter σ₂. Although the parameter σ₂ has effect on the dynamics of stock price, π1∗(t) doesn’t depend on σ₂. It is greatly surprised us. (ii) π2∗(t) depends on all the parameters a, b, k₁, k₂, λ₁, σ₁, λ₂, u, v and σ₂. Although the dynamics of the zero-coupon bond and interest rate are driven by the same Brownian motion, π2∗(t) is still affected by the parameters of stock price. It also surprised us. (iii) The efficient frontier (36) depends on the parameters a, b, k₁, k₂, λ₁, λ₂, u, v, but doesn’t depend on σ₁ and σ₂. We can find from the dynamic equation of stock price that σ₁ and σ₂ represent the volatility resulted from the two different Brownian motions respectively, whereas the investors’ risks measured by VarX (T) don’t depend on the volatility of the stock. This implies that it is very necessary for us to analyze the sensitivity of model parameters on the efficient policies and the efficient frontier.

Remark 3

From (34)–(36), we can draw the following some conclusions. (i) The efficient policies and the efficient frontier are not deterministic functions as well, but are dynamic functions. They depend on the state of short rate r(t) and are affected by all the parameters of the liability and interest rate. (ii) The efficient frontier in the mean-standard deviation diagram is still a straight line, no matter at which state interest rate is. To be specific, let σ[X(T)] be the standard deviation of the terminal surplus, then (36) gives

E(X(T))=ψe−D3(0)−D4(0)r0+σ[X(T)]e−D1(0)−2D3(0)−(D2(0)+2D4(0))r0−1,(40)

which is also called the capital market line in the portfolio theory.

Remark 4

When σ[X(T)] = 0, then we have E(X(T)) = ψ e^{−D₃(0)−D₄(0)r₀}. Therefore, when C runs over [ψ e^{−D₃(0)−D₄(0)r₀}, + ∞), the efficient frontier is composed of all the points (C, VarX (T)).

In order to compare the conclusion from Theorem 3 with those in the existing literatures, we give the following three special cases.

Special case 1 If there is no liability, i.e., u = v = 0, then we get

ψ=x0,η∗=x0eD1(0)+D3(0)+(D2(0)+D4(0))r0−CeD1(0)+2D3(0)+(D2(0)+2D4(0))r0−1,g(t,r(t))=η∗eD3(t)+D4(t)r(t).(41)

Therefore the efficient strategies for the original mean-variance problem (8) are given by

π1∗(t)=−λ1σ1⋅(X(t)−η∗eD3(t)+D4(t)r(t)),(42)

π2∗(t)=σrσB . σ2λ1−σ1λ2σ1⋅(X(t)−η∗eD3(t)+D4(t)r(t))+σrσB⋅(D2(t)(X(t)−η∗eD3(t)+D4(t)r(t))−∂g(t,r)∂r),(43)

with the efficient frontier given by

VarX(T)=1e−D1(0)−2D3(0)−(D2(0)+2D4(0))r0−1(E(X(T))−x0e−D3(0)−D4(0)r0)2,(44)

where D₁(t), D₂(t), D₃(t) and D₄(t) are still given by Lemma 1 and Lemma 3, respectively.

Remark 5

When u = v = 0, our model is reduced to that studied by Ferland and Watier[7], who used backward stochastic differential equation (BSDE) theory to investigate a dynamic mean-variance portfolio choice problem in the CIR framework. Therefore, our work extends the conclusion obtained by Ferland and Watier[7].

Special case 2 If interest rate is a constant, i.e., a = b = k₁ = k₂ = 0, then the zero-coupon bond is reduced to one risk-free asset. Therefore, the zero-coupon bond should not be considered in the investment process such that π2∗(t) = 0. In addition, we obtain

D1(t)=−λ12(T−t),D2(t)=2(T−t),D3(t)=0,D4(t)=−(T−t).(45)

And it leads to that

ψ=x0−(u−λ1v)∫0Te−(T−s)r(s)ds,η∗=ψe(−λ12+r0)T−Ce−λ12T−1,g(t,r)=(u−λ1v)∫tTe−(T−s)r(s)ds+η∗e−(T−t)r(t).(46)

Finally, the efficient policies and the efficient frontier for the original mean-variance problem (8) are respectively given by

π1∗(t)=−λ1σ1⋅(X(t)−(u−λ1v)∫tTe−(T−s)r(s)ds−η∗e−(T−t)r(t))+vσ1,(47)

VarX(T)=1eλ12T−1(E(X(T))−(x0−(u−λ1v)∫0Te−(T−s)r(s)ds)e(T−t)r0)2.(48)

Remark 6

It can be easily seen that (47) and (48) coincide with the conclusions derived by Xie, et al.[12] when liability dynamics is completely positive correlation with stock price dynamics. It shows that our work extends the model of Xie, et al.[12] to stochastic affine interest rate environment, and successfully solves the dynamic mean-variance portfolio choice problem with stochastic interest rate.

Special case 3 If interest rate is reduced to be constant and liability is not considered, i.e., u = v = 0 and a = b = k₁ = k₂ = 0, then we have π2∗(t) = 0, and

ψ=x0,η∗=x0e(−λ12+r0)T−Ce−λ12T−1.(49)

As a result, the efficient strategies and the efficient frontier for the original mean-variance problem (8) are respectively given by

π1∗(t)=−λ1σ1⋅(X(t)+η∗e−(T−t)r(t)),(50)

VarX(T)=1eλ12T−1(E(X(T))−x0e(T−t)r0)2.(51)

Remark 7

Notice that (50) and (51) are consistent with the conclusions obtained by Zhou and Li[4]. It implies that our work is the most important extension of the model studied by Zhou and Li[4].

5 Numerical Illustrations

In this section, we provide a numerical example to illustrate the impact of the parameters of interest rate and liability on the efficient strategy and the efficient frontier. In the following numerical illustrations, we focus on the case of Δ₁ > 0 and k1λ22 > 2, i.e., λ₂∈ ℝ, b² < 4k₁ and k1λ22 > 2. Under these conditions, D₁(t), D₂(t), D₃(t) and D₄(t) are given by (a10), (a8), (a28) and (a26) in the Appendix respectively. The analysis in the other cases is similar to that in the case of λ₂∈ ℝ, b² < 4k₁ and k1λ22 > 2. Therefore, we assume that the main parameters are given by a = 0.018712, b = 0.2339, k₁ = 0.51, k₂ = 0.5, r(0) = 0.05, σ₁ = 0.2, λ₁ = 0.2, σ₂ = 0.02, λ₂ = 2, u = 0.07, v = 0.22, t = 0, T = 1, X(0) = 100.

Note that the values of a and b above mentioned come from the work of Deelstra, et al.[17]. In the following analysis we denote the optimal amount invested in the cash by π0∗(t) , denote the optimal amount invested in the stock by π1∗(t) , and denote the optimal amount in the zero-coupon bond by π2∗(t).

5.1 Sensitivity Analysis on the Efficient Strategy

From Figures 1 ∼ 6, we can draw the following some instructive conclusions.

Figure 1

The effect of b on the efficient strategy

Figure 2

The effect of k₁ on the efficient strategy

Figure 3

The effect of λ₁ on the efficient strategy

Figure 4

The effect of λ₂ on the efficient strategy

Figure 5

The effect of u on the efficient strategy

Figure 6

The effect of v on the efficient strategy

(a1) π0∗(t) decreases with respect to (w.r.t) the parameter b, while π1∗(t) and π2∗(t) increase w.r.t the parameter b. In fact, we can see that the expected value of interest rate becomes smaller as the value of b becomes larger. A smaller value of the expected interest rate leads to a less amount invested in the cash. Accordingly, the amount invested in the stock and zero-coupon bond will increase. Therefore, this conclusion is consistent with the economic implication of b.

(a2) π0∗(t) and π2∗(t) are decreasing functions of the parameter k₁, while π1∗(t) is increasing in k₁. As matter of fact, the larger the value of k₁, the larger interest rate risk, which will lead to the less amount invested in the cash and zero-coupon bond. On the contrary, the amount in the stock will be increasing. This conclusion conforms to our intuition.

(a3) The trend between π0∗(t) and the parameter λ₁ is same, while the trend between π1∗(t) and λ₁ is opposite, π2∗(t) almost remain unchanged in λ₁. It displays that the amount in the cash is increasing as the value of λ₁ is increasing, while the amount invested in the stock is decreasing. Meantime, λ₁ almost has no effect on the amount in the zero-coupon bond.

(a4) π0∗(t) increases w.r.t the parameter λ₂, π1∗(t) almost remains unchanged in λ₂, while π2∗(t) is a decreasing function in λ₂. It tells us that we should invest more money in the cash when the value of λ₂ becomes bigger, while invest less money in the zero-coupon bond. What surprised us is that λ₂ almost has no impact on the amount in the stock. We should keep this conclusion in mind in the practice of investment.

(a5) π0∗(t) is decreasing in u, while π1∗(t) and π2∗(t) are increasing in u. In reality, the bigger the value of u, the bigger the expected liability. In order to hedge the risk resulted from random liability, investors need increasing the amount in the risky assets. It is obvious that the amount invested in the risk-free asset is reduced.

(a6) π0∗(t) and π2∗(t) have decreasing trends in the parameter v, while π1∗(t) has an increasing trend in v. A larger value of v will cause a bigger volatility risk. It implies that investors need increasing the amount in the stock, while decreasing the amount in the cash and zero-coupon bond.

5.2 Sensitivity Analysis on the Efficient Frontier

Figures 7 ∼ 12 illustrate the relationships between the standard deviation σ[X(T)] and the expected return E(X(T)) = C. According to our observation on Figure 4 and Figure 5, we obtain the following some instructive conclusions:

Figure 7

The effect of b on the standard deviation

Figure 8

The effect of k₁ on the standard deviation

Figure 9

The effect of λ₁ on the standard deviation

Figure 10

The effect of λ₂ on the standard deviation

Figure 11

The effect of u on the standard deviation

Figure 12

The effect of v on the standard deviation

(b1) σ[X(T)] increases w.r.t the parameter b for a given expected return. From the points of economic implication of b, a larger value of b will lead to a greater amount in the stock and zero-coupon bond and a smaller amount in the cash. Therefore, investment yields will greatly increase. The principle of capital market line tells us that investors should undertake more risks.

(b2) σ[X(T)] first decreases then increases w.r.t. the parameter k₁ for a given expected return. This conclusion tells us that we need to pay close attention to the effect of k₁ on the risk level.

(b3) σ[X(T)] is a decreasing function in the parameter λ₁. According to (a3) in the previous subsection, the total wealth of investors is decreasing when the value of λ₁ is increasing. The less total wealth, the less the risks taken on. Therefore, σ[X(T)] measuring the risks is decreasing. This conclusion conforms to the principle of capital market line.

(b4) σ[X(T)] has a decreasing trend as the value of λ₂ rises under the same expected wealth level. From (a4) in the previous subsection, we find that the risk resulted from stocks remains unchanged and the risk from zero-coupon bond decreases all the time when the value of λ₂ increases. Therefore, we draw the conclusion that the comprehensive risks from financial market will decrease.

(b5) σ[X(T)] has the same trend as the parameter u for a fixed level of expected return. From the economic implications of u, which represents the expected level of liability. It means that the larger the value of u, the larger the value of liability. In order to hedge the risk caused by the liability, investors need to hold more shares of stocks and zero-coupon bonds. At the same time, it requires investors to take on more risks.

(b6) σ[X(T)] decreases w.r.t. the parameter v. Although a larger value of v leads to the more amount in the stock and the less amount in the cash and zero-coupon bond, the total risks investors need to take on is decreasing. We should keep this conclusion in mind.

6 Conclusions

Liability factor is often faced with in the investment activities. This paper takes liability factor into consideration and focuses on a dynamic mean-variance portfolio selection problem with random liability in an affine interest rate environment. Applying dynamic programming principle and Lagrange duality theorem, we obtain the explicit expressions of the efficient policies and the efficient frontier. Finally, a numerical example is provided to analyze the dynamic behavior of the efficient policies and the efficient frontier. Our model extends the works of Ferland and Watier[7], Xie, et al.[12], Korn and Kraft[16], and Gao[18]. Some interesting and instructive main conclusions are found as follows: (i) Although the parameter σ₂ has impacted on the dynamics of stock price, π1∗(t) doesn’t depend on σ₂; (ii) π2∗(t) not only depends on the parameters of zero-coupon bond but also depends on the parameters of stock price and liability process; (iii) The efficient frontier in the mean-standard deviation diagram is still a straight line, no matter at which state interest rate is.

As far as we know, there is no work on the dynamic mean-variance portfolio choice problem with stochastic affine interest rate model. In addition, we have investigated affine interest rate model and considered random liability factor and successfully obtained the closed-form solutions to the efficient policies and the efficient frontier. However, our model has also some limits: (i) The liability process is the simplest model and without jump, (ii) We only consider stochastic affine interest rate model and doesn’t consider stochastic volatility model, for example, the Heston model, (iii) Mean-variance framework is standard but is not the best criterion, and we can studied the problem under the other risk criterion, for example, the VaR (value at risk) constraint and down-side risk minimization criterion. In future research, we will devote ourselves to studying these problems.

Supported by National Natural Science Foundation of China (71671122), China Postdoctoral Science Foundation Funded Project (2014M560185, 2016T90203), Humanities and Social Science Research Fund of Ministry of Education of China (11YJC790006, 16YJA790004) and Tianjin Natural Science Foundation of China (15JCQNJC04000)

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Appendix

(A1) The proof of Lemma 1

Introducing f(t, r) = e^{D₁(t)+D₂(t)r} into (23) yields

eD1(t)+D2(t)r{D˙1(t)+(a+2λ2k2)D2(t)−12k2D22(t)−λ12−k2λ22+r(D˙2(t)+2−k1λ22+(2k1λ2−b)D2(t)−12k1D22(t))}=0.(a1)

Comparing the coefficients on the both sides of (a1), we get

D˙2(t)+2−k1λ22+(2k1λ2−b)D2(t)−12k1D22(t)=0,D2(T)=0;(a2)

D˙1(t)+(a+2λ2k2)D2(t)−12k2D22(t)−λ12−k2λ22=0,D1(T)=0.(a3)

The discriminant Δ₁ for the quadratic equation

12k1D22(t)−(2k1λ2−b)D2(t)−(2−k1λ22)=0(a4)

is given by Δ₁ = (2k₁λ₂ − b)² + 2k₁(2 − k1λ22 ). It is very obvious that (a4) has different roots depending on whether Δ₁ > 0, Δ₁ = 0 or Δ₁ < 0. These cases will be respectively discussed as follows.

If Δ₁ > 0, i.e.,
(2k1λ2−b)2+2k1(2−k1λ22)>0⇒λ2∈R,b2<4k1,λ2≠bk1,b2=4k1,λ2<ξ1 or λ2>ξ2,b2>4k1,(a5)
where
ξ1=bk1−2b2−8k12k1,ξ2=bk1+2b2−8k12k1,(a6)
then (a4) has two different real roots, which are respectively given by
m1,2=2k1λ2−bk1±Δ1k1.(a7)
Therefore (a2) can be rewritten as
D˙2(t)=12k1(D2(t)−m1)(D2(t)−m2)⇒1m1−m2∫tT(1D2(t)−m1−1D2(t)−m2)dD2(t)=12k1(T−t)⇒D2(t)=m1m2(1−exp⁡{12k1(m1−m2)(T−t)})m1−m2exp⁡{12k1(m1−m2)(T−t)}.(a8)
Using (a2)× k₂ − (a3) × k₁, we obtain
D˙1(t)=k2k1D˙2(t)−ak1+bk2k1D2(t)+k1λ12+2k2k1.(a9)
Solving the equation (a9), we derive
D1(t)=k2k1D2(t)−k1λ12+2k2k1(T−t)+ak1+bk2k1(m2(T−t)−2k1ln⁡m1−m2m1−m2exp⁡{12k1(m1−m2)(T−t)}).(a10)
If Δ₁ = 0, i.e.,
(2k1λ2−b)2+2k1(2−k1λ22)=0⇒λ2=bk1,b2=4k1,λ2=ξ1orλ2=ξ2,b2>4k1,(a11)
then (a4) has the unique real root, which is given by
m3=2k1λ2−bk1.(a12)
D˙2(t)=12k1(D2(t)−m3)2⇒1(D2(t)−m3)2D˙2(t)=12k1⇒D2(t)=k1m32(T−t)k1m3(T−t)−2.(a13)
According to (a9), we have
D1(t)=k2k1D2(t)−k1λ12+2k2k1(T−t)+ak1+bk2k1(m3(T−t)−2k1ln⁡22−k1m3(T−t)).(a14)
If Δ₁ < 0, i.e.,
(2k1λ2−b)2+2k1(2−k1λ22)<0⇒ξ1<λ2<ξ2,b2>4k1,(a15)
then (a4) has no real roots. But (a2) can be solved as follows.
D˙2(t)=12k1((D2(t)−m3)2+−Δ1k12)⇒D˙2(t)−Δ1k12((k1−Δ1(D2(t)−m3))2+1)=12k1⇒D2(t)=m3+−Δ1k1tan⁡(arctan⁡−k1m3−Δ1−−Δ12(T−t)).(a16)
According to (a9), we have
D1(t)=k2k1D2(t)−k1λ12+2k2k1(T−t)+ak1+bk2k1(m3(T−t)−2k1lncos⁡(arctan⁡−k1m3−Δ1)+2k1lncos⁡(arctan⁡−k1m3−Δ1−−Δ12(T−t))).(a17)
The proof is completed.

(A2) The proof of Lemma 2

The following two cases are discussed.

If u − λ₁v = 0, it is obviously that (25) holds.
If u − λ₁v ≠ 0, defining a variational operator for any function g(t, r) as
∇g(t,r)=−rg+(a−br+λ2σr2)gr+12σr2grr,
we rewrite (24) as
∂g(t,r)∂t+∇g(t,r)+u−λ1v=0,g(T,r)=η.(a18)

Noting that g(t,r)=(u−λ1v)∫tTg~(s,r)ds+ηg~(t,r), we derive

∂g(t,r)∂t=−(u−λ1v)g~(t,r)+η∂g~(t,r)∂t=(u−λ1v)(∫tT∂g~(s,r)∂sds−g~(T,r))+η∂g~(t,r)∂t,(a19)

∇g(t,r)=(u−λ1v)∫tT∇g~(s,r)ds+η⋅∇g~(t,r).(a20)

Putting (a19) and (a20) into (a18) yields

(u−λ1v)(∫tT(∂g~(s,r)∂s+∇g~(s,r))ds−g~(T,r)+1)+η(∂g~(t,r)∂t+∇g~(t,r))=0.(a21)

Thus, we get

∂g~(t,r)∂t+∇g~(t,r)=0,g~(T,r)=1.(a22)

In a result, we complete the proof.

(A3) The proof of Lemma 3

Inserting g~(t,r) = e^{D₃(t)+D₄(t)r} into (25), we obtain

eD3(t)+D4(t)r{D˙3(t)+(λ2k2+a)D4(t)+12k2D42(t)+r(D˙4(t)−1+(λ2k1−b)D4(t)+12k1D42(t))}=0.

After separating the variables, we obtain the following two equations:

D˙4(t)−1+(λ2k1−b)D4(t)+12k1D42(t)=0,D4(T)=0;(a23)

D˙3(t)+(λ2k2+a)D4(t)+12k2D42(t)=0,D3(T)=0.(a24)

Note that the discriminant Δ₂ for the quadratic equation:

−12k1D42(t)−(λ2k1−b)D4(t)+1=0(a25)

satisfies the condition Δ₂ = (λ₂k₁ − b)² + 2k₁ > 0.

Therefore, applying the same technique as the equation (a2), we derive

D4(t)=m4m5(1−exp⁡{−12k1(m3−m4)(T−t)})m4−m5exp⁡{−12k1(m3−m4)(T−t)},(a26)

where m₄ and m₅ are two different roots. In addition, we have

m4,5=λ2k1−b−k1±(λ2k1−b)2+2k1−k1.

Using the approach (a23) × k₁−(a24)× k₂, we obtain

D˙3(t)=k2k1D˙4(t)−ak1+bk2k1D4(t)−k2k1.(a27)

Solving (a27), we get

D3(t)=k2k1D4(t)+k2k1(T−t)+ak1+bk2k1(m5(T−t)+2k1ln⁡m4−m5m4−m5exp⁡{−12k1(m4−m5)(T−t)}).(a28)

Therefore, the conclusion is proved.

(A4) The proof of Lemma 4

Considering the different expressions under different conditions, we will discuss the following cases:

If (2k₁λ₂ − b)² + 2k₁(2 − k1λ22 ) > 0, D₂(t) is given by (a8). Differentiating D₂(t) with respect to (w.r.t.) t, we have
D˙2(t)=(k1λ22−2)(m1−m2)2exp⁡{12k1(m1−m2)(T−t)}(m1−m2exp⁡{12k1(m1−m2)(T−t)})2.(a29)
Notice that under the condition of k1λ22 > 2, we have D˙2(t) > 0. Therefore, D₂(t) increases with respect to t, which leads to D₂(t) < D₂(T) = 0.
If (2k₁λ₂ − b)² + 2k₁(2 − k1λ22 ) = 0, D₂(t) is given by (a13). Differentiating D₂(t) w.r.t. t, we get
D˙2(t)=2k1m32(k1m3(T−t)−2)2>0.(a30)
Therefore, D₂(t) is an increasing function w.r.t. t.
If (2k₁λ₂ − b)² + 2k₁(2 − k1λ22 ) < 0, D₂(t) is given by (a16). Differentiating D₂(t) w.r.t. t, we have
D˙2(t)=(−Δ1)22k1sec⁡2(arctan⁡−k1m3−Δ1−−Δ12(T−t))>0.(a31)
Therefore, D₂(t) is also an increasing function w.r.t. t.
According to (a26), we have
D˙4(t)=(m3−m4)2exp⁡{−12k1(m3−m4)(T−t)}(m3−m4exp⁡{−12k1(m3−m4)(T−t)})2>0.(a32)
Noting (a9) and (a27), we obtain
D˙1(t)+2D˙3(t)=k2k1(D˙2(t)+2D˙4(t))−ak1+bk2k1(D2(t)+2D4(t))+λ12k1.(a33)
Further, we find that D˙4(t) > 0, so we have D₄(t) < D₄(T) = 0.
Taking D˙2(t)>0,D˙4(t) > 0, D₂(t) < 0 and D₄(t) < 0 into consideration, it is very obvious to get D˙1(t)+2D˙3(t)>0, and we have D₁(t) + 2D₃(t) < D₁(T) + 2D₃(T) = 0.
Therefore, Lemma 4 holds.

Received: 2016-5-23

Accepted: 2016-12-19

Published Online: 2017-8-1

Articles in the same Issue

https://doi.org/10.21078/JSSI-2017-229-21

Keywords for this article

affine interest rate; random liability; mean-variance criterion; the efficient strategy; the efficient frontier; Lagrange duality theorem