Insurance: Mathematics and Economics 54 (2014) 84–92

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Insurance: Mathematics and Economics

journal homepage: www.elsevier.com/locate/ime

Asset allocation for a DC pension fund with stochastic income andmortality risk: A multi-period mean–variance framework✩

Haixiang Yao a, Yongzeng Lai b,∗, Qinghua Ma a, Minjie Jian c

a School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, Chinab Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada, N2L 3C5c Department of Applied Mathematics, College of Science, South China Agricultural University, Guangzhou 510642, China

h i g h l i g h t s

• A multi-period mean–variance asset allocation for DC pension funds is studied.• Stochastic income and mortality risk are considered in the model.• Lagrange multiplier method and dynamic programming approach are used.• Explicit expressions of the efficient strategy and efficient frontier are derived.• Some special cases are discussed and some numerical analyses are presented.

a r t i c l e i n f o

Article history:Received June 2013Received in revised formAugust 2013Accepted 29 October 2013

Keywords:Asset allocationDefined contribution pension fundMulti-period mean–varianceStochastic incomeMortality risk

a b s t r a c t

This paper investigates an asset allocation problem for defined contribution pension fundswith stochasticincome and mortality risk under a multi-period mean–variance framework. Different from most studiesin the literature where the expected utility is maximized or the risk measured by the quadratic meandeviation is minimized, we consider synthetically both to enhance the return and to control the riskby the mean–variance criterion. First, we obtain the analytical expressions for the efficient investmentstrategy and the efficient frontier by adopting the Lagrange dual theory, the state variable transformationtechnique and the stochastic optimal control method. Then, we discuss some special cases under ourmodel. Finally, a numerical example is presented to illustrate the results obtained in this paper.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

According to the fund procurement and operation pattern, pen-sion funds can mainly be divided into two types: one is definedbenefit (DB) pension funds, the other is defined contribution (DC)pension funds. A DB pension fund is a pension fundwhere the ben-efits are set in advance by the sponsor. In such a fund, contribu-tions are set and constantly adjusted so as to ensure that the fund

✩ This research is supported by grants from the National Natural Science Founda-tion of China (No. 71271061, 61104138), Natural Science Foundation of GuangdongProvince (No. S2011010005503), Scientific and Technological Innovation Founda-tion ofGuangdongColleges andUniversities (No. 2012KJCX0050), Science andTech-nology Planning Project of Guangdong Province (No. 2012B040305009), NationalSocial Science Foundation of China (No. 11CGL051), the Business Intelligence KeyTeam of Guangdong University of Foreign Studies (No. TD1202), and National Sta-tistical Science Research Projects (No. 2013LY101).∗ Corresponding author. Tel.: +1 519 884 0710x2107; fax: +1 519 884 9738.

E-mail addresses: yaohaixiang@gdufs.edu.cn (H. Yao), ylai@wlu.ca (Y. Lai),mqh@mail.gdufs.edu.cn (Q. Ma), iamjianminjie@outlook.com (M. Jian).

0167-6687/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.insmatheco.2013.10.016

remains in balance, and the risk is borne by the organizer of thefund. A DC pension fund is a pension fund where contributions areset and benefits therefore depend solely on the accumulation scaleand the return of the investment, thus the financial risk is borne bythe members.

Historically, DB pension funds have been the more popularand preferred by workers since the management is easy andthe risk is borne by the sponsors of the fund. However, due tothe demographic evolution and the development of the capitalmarket, especially due to the population aging problem and thelongevity risk, DC has becomemore andmore popular in the globalpension market in recent years, and more and more countrieshave shifted completely or partially from a DB pension scheme toa DC pension scheme. As a result, the study of DC pension fundinvestment management has become a research hot topic in theactuarial and the financial literature over the past decade. Due tospace limitations, only some important research results are brieflyintroduced below.

By minimizing the risk measured by the quadratic meandeviation (also called the target-based criterion), Vigna and

H. Yao et al. / Insurance: Mathematics and Economics 54 (2014) 84–92 85

Haberman (2001) and Haberman and Vigna (2002) investigatemulti-period DC pension fund investment managements duringaccumulation phase, and obtain the optimal investment strategiesby using the dynamic programming method. Gerrard et al. (2004)extend the work of Vigna and Haberman (2001) to continuous-time setting and post-retirement case. Based on continuous-timeexpected utility maximization: Deelstra et al. (2003) and Giacintoet al. (2011) investigate optimal asset allocation for DC pensionfunds with stochastic interest rates and a minimum guaranteeprotection; Gao (2009) investigate the optimal portfolios for DCpension funds under a constant elasticity of variance (CEV) byapplying the Legendre transform and dual theory; Han and Hung(2012) consider a portfolio selection problem for DC pensionfund under inflation by using the stochastic control approach.Using quadratic risk minimization criterion, Gerrard et al. (2012)formulate and solve a stochastic control and optimal stoppingproblem of finding the optimal time of annuitization for a DCpension scheme in the de-cumulation phase. Based on target-driven framework and prospect theory in behavioral finance, Blakeet al. (2013) study the optimal asset allocation problem for DCpension funds.

As the contribution period in a pension fund is very long,generally from 20 to 40 years due to promotion prospect andchanges in economic environment, it is crucial to allow a stochasticterm structure for the income (salary). As the contribution is oftena fixed percentage of salary, the contribution is also stochastic.Therefore, it is crucial to take into account the income risk for aDC pension fundmanagement. Recently, by using continuous-timeutility maximization model, Cairns et al. (2006), Zhang and Ewald(2010) and Ma (2011) investigate the optimal asset allocationwith stochastic income for DC pension funds in different marketsettings, where the income process is described by a geometricBrownianmotion. Under the DC pension framework, Emms (2012)study the lifetime investment and consumption problem withstochastic income based on CARA utility maximization model.

A pension fund member may die before his retirement. Insuch a case, his pension plan has to be terminated due tomortality risks, such as traffic accident, fire hazard and seriousillnesses, etc. Although there are many studies in the literatureson ordinary portfolio selection and life insurance problems withuncertain time horizon, see for example, Martellini and Urosevic(2006), Pliska and Ye (2007) and Christophette et al. (2008),there are very limited studies aim at pension fund investmentmanagements. By maximizing the expected utility, Charupat andMilevsky (2002) consider the optimal asset allocation in a variableannuity contract that may have some relation to DC pension fund;Hainaut and Devolder (2007) consider the optimal dividend policyand the asset allocation of a DB pension fund under the mortalityrisk. By minimizing the risk which is a quadratic target-basedcost function, Hainaut and Deelstra (2011) investigate optimalcontribution rate of a DB pension fund with a stochastic mortalitywhich is modeled by a jump process. But all these studies focuson annuity contract and DB pension fund management problems.Research on DC pension fund portfolio selection problems withmortality risks is very limited.

Furthermore, researchers of the abovementioned studies eithermaximize the expected utility or minimize the risk which ismeasured by the quadratic mean deviation, but they do notapply the mean-risk bi-objective criteria to study the pensionfund investment problems. It is well known that the mean-riskframework has become one of the most basic frameworks inthe modern portfolio selection theory since Markowitz (1952)published his seminal work on the mean–variance portfolioselection. The mean–variance criterion is a mean-risk bi-objectivecriterion that strikes a balance between enhancing the return andcontrolling the risk, where the risk is measured by the variance of

the portfolio’s return. Along with the break through of solving ofthe dynamic mean–variance model (see Zhou and Li (2000) andLi and Ng (2000) for continuous-time case and multi-period case,respectively), the dynamic mean–variance model also become oneof the most important frameworks to study asset allocation andasset–liability management problems, see, for example, Fu et al.(2010), Costa and Oliveira (2012), Chen et al. (2008), Wu and Li(2011), Chiu and Wong (2013) and Yao et al. (2013a,b). But, toour knowledge, only very few studies in the literature apply thedynamicmean–variancemodels to study pension fund investmentproblems. By using the continuous-time mean–variance model,Delong et al. (2008) and Josa-Fombellida and Rincón-Zapatero(2008) study the optimal investment and contribution strategiesfor DB pension funds; Vigna (2012) investigates the portfolioselection problem for DC pension funds. But they only focus on thecontinuous-time case without touching the multi-period case.

To the best of our knowledge, no researchwork in the literatureinvestigates the multi-period version of DC pension fund assetallocation problems by using the mean–variance framework.This paper attempts to explore this topic. In this paper, weconsider a DC pension fund investment management problemby using the multi-period mean–variance model. In addition, weincorporate stochastic income and mortality risk into our model.Mathematically, the inclusion of stochastic income and mortalityrisk increases the difficulty in solving the Bellman equation whichcomes from the dynamic programming approach. Specifically,the inclusion of the stochastic income adds a new state variableto the model, namely, there are two state variables for ourproblem: the wealth and the wage income. Furthermore, afterincorporating the mortality risk, our asset allocation problem fora DC pension fund becomes more complicated. Therefore, addingthe stochastic income and the mortality risk into the multi-periodmean–variance model drastically increase the computationalcomplexity in obtaining the closed form solutions. These solutionscannot be obtained easily by using the embedding techniqueused in Li and Ng (2000) since the calculation procedure of theembedding technique is quite troublesome. Different from mostof the researches in the literature on multi-period mean–variancemodels which adopt the embedding technique, we use the statevariable transformation technique and the Lagrange dual theoryto solve the model synthetically. Compared to the embeddingtechnique, our approach is relatively simple in procedure settingsand calculations.

The remainder of the paper is organized as follows. The marketsetting is described in Section 2. The multi-period mean–varianceasset allocationmodel forDCpension fundswith stochastic incomeand mortality risk are set up in the same section. In Section 3,the original model is transformed into a standard multi-periodstochastic control problem by using the Lagrange multiplier, thenthe corresponding analytical solution is derived by using the statevariable transformation technique and the dynamic programmingapproach. Closed form expressions for the efficient investmentstrategy and the efficient frontier are obtained in Section 4 byusing the Lagrange dual theory. Some special cases are discussedin Section 5. In Section 6, some numerical analyses are presentedto illustrate our results. The paper is concluded in Section 7.

2. Model formulation

Consider a financial market consisting of n + 1 assets thatmay include a risk free asset or be all risky. Denote by ek =

(e0k, e1k, e

2k, . . . , e

nk)

′ the random returns of these n + 1 assets overperiod k (i.e., the time interval from time k to time k + 1), k =

0, 1, . . . , T − 1, where ′ represents the transpose of a matrixor a vector. This paper considers a multi-period asset allocationproblem for a DC pension fund. Suppose that a representative

86 H. Yao et al. / Insurance: Mathematics and Economics 54 (2014) 84–92

pension fund member enters a pension plan at time 0, and plansto retire at time T . Before his retirement, he needs to contributecertain amount of money at the beginning of every period in apredefined way. Upon his retirement, he can convert the wealthof his pension fund into annuity so that he can receive a scheduledpension at every period after his retirement. If the member diesbefore his retirement, his heir can withdraw all the money of thepension fund. Denote by x0 and y0 (>0) the initial fund paid andthe initial income, respectively. Let yk be thewage income receivedat time k. It is assumed in this paper that the wage income isstochastic and satisfies the following dynamics

yk+1 = qkyk, k = 0, 1, . . . , T − 1, (1)

where qk is an exogenous random variable representing thestochastic growth rate of the wage income over period k. It is alsoassumed that the wage income cannot be negative. So we supposethat qk > 0 almost surely for all k = 0, 1, . . . , T − 1. Supposeckyk is the amount that the member contributes at time k, whereck is a deterministic variable depending on k only. Let xk and zkdenote the wealth of the pension fund just before and just after hiscontribution at time k, respectively. Then we have zk = xk + ckyk.

Remark 1. In our model, ck can depend on k. Compared with thecase where ck is a constant c irrelevant to k, our model is moreadaptable and flexible.Wepoint out that itmeets the requirementsof the actual needs in most cases. For example, generally speaking,the pension fund member contributes every month, but one day isa period for the pension fund investment. Obviously, in this case,the pension fundmember needs not to contribute for every period.If over period k, he needs not to contribute, we can set ck = 0;otherwise, we can set ck > 0.

Remark 2. In order to make our model to be more general, we donot assume that ck ≥ 0. For example, when ck < 0, ckyk can beinterpreted as the consumption of the member or the distributionof the pension fund over period k, k = 1, 2, . . . , T . Therefore,our model can also be used to study the DC pension fund assetallocation problem in the de-cumulation phase.

Suppose that a pension fund can be invested in n + 1 assets inthe market. Denote by ui

k the amount invested in the ith asset overperiod k, i = 1, 2, . . . , n and k = 1, 2, . . . , T . Then incorporationof the contribution ckyk at the beginning of period k, the amountinvested in the 0th asset over period k is (xk + ckyk) −

ni=1 u

ik,

k = 1, 2, . . . , T . Therefore, xk follows the dynamics

xk+1 = e0k

(xk + ckyk) −

ni=1

uik

+

ni=1

eikuik

= (xk + ckyk) e0k + P ′

kuk, (2)

where Pk = (e1k − e0k, e2k − e0k, . . . , e

nk − e0k)

′ and uk =

(u1k, u

2k, . . . , u

nk)

′.From (1) and (2), it follows that

xk+1 + ck+1yk+1 = (xk + ckyk) e0k + ck+1qkyk + P ′

kuk. (3)

Notice that zk = xk + ckyk, then by (3), the dynamics of zk satisfies

zk+1 = zke0k + ck+1qkyk + P ′

kuk. (4)

Though the pension fund member plans to retires at time T .However, in reality, hemay die before T and in this case his pensionfund plan has to be terminated before the retired time T due tomortality risks, such as traffic accident and serious illness. Supposethe pension fundmember is alive at time t = 0 and denote by τ hislifetime (the time of death), where τ is a positive random variable.If he dies during the (k − 1)th time period, the actual terminated

time T τ of the pension fund plan is k; if he dies after T − 1, theactual terminated time T τ

= T . I.e.,

T τ=

k, k − 1 < τ ≤ k and 1 ≤ k ≤ T − 1,T , τ > T − 1. (5)

We now discuss how to find the probability (mass) function for T τ .Let S(t) (t > 0) be the survival probability of the pension fundmember, i.e.,

S(t) = Pr( τ ≥ t| τ > 0), (6)where, Pr(·) is the probability measure. It is well known that(see Charupat and Milevsky (2002) and Pliska and Ye (2007)) thesurvival probability S(t) can be expressed as the following form

S(t) = e− t0 (s)ds, (7)

where (s) is the instantaneous hazard rate (mortality intensity).By (5) and (6), the probability mass function of T τ is given by

pk := Pr(T τ= k) =

S(k − 1) − S(k), k = 1, . . . , T − 1,S(T − 1), k = T .

(8)

According to (7) and (8), we have

pk =

e−

k−10 (s)ds

− e− k0 (s)ds, k = 1, . . . , T − 1,

e− T−10 (s)ds > 0, k = T .

(9)

Let ℘k be the σ -field representing the information availabletill time k. We call the investment strategy u = {uk; k =

0, 1, . . . , T − 1} admissible if uk is measurable with respect to ℘k.Let Θk denote the collection of all admissible strategies startingat time k. Let E[·] and Var[·] denote the expectation operator andvariance operator, respectively. Throughout this paper, we makethe following assumptions similar to most literatures.

Assumption 1. E[eke′

k] > 0, i.e., E[eke′

k] is a positive definitematrix for k = 0, 1, . . . , T − 1.

Assumption 2. The random series Υk = (P ′

k, qk)′ for k =

0, 1, . . . , T − 1 are statistically independent.

Assumption 3. τ is statistically independent of Υk for k =

0, 1, . . . , T − 1.

Assumption 4. E[Pk] = 0, where 0 is the n dimension zero vectorfor k = 0, 1, . . . , T − 1.

Remark 3. Assumption 1 is used in Li and Ng (2000) and itmeans that the financial assets in the market are not redundant.Assumption 2 shows that the returns of financial assets andwage income are statistically independent among different timeperiods. Similar assumptions are also used in some literatures. Forexample, Li and Ng (2000) assume that the returns of the financialassets are statistically independent among different time periods.Assumptions 3 and 4 are used in Wu and Li (2011) and Yao et al.(2013b).

The optimal asset allocation problem for a DC pension fund underthe multi-period mean–variance framework refers to the problemof finding the optimal admissible investment strategy such that thevariance of the terminal wealth is minimized for a given expectedterminal wealth level d, i.e.,

minu∈Θ0

Var[xT τ ] := E[x2T τ ] − d2

,

s.t. E[xT τ ] = d, (1)–(2).(10)

The solution u∗= {u∗

k; k = 0, 1, . . . , T − 1} of Problem (10)is called an efficient investment strategy. The point (Var[xT τ ], d)corresponding to an efficient investment strategy on the variance-mean space is called an efficient point. The set of all the efficientpoints forms the efficient frontier in the variance-mean space.

H. Yao et al. / Insurance: Mathematics and Economics 54 (2014) 84–92 87

3. Solution scheme

For convenience, we define p0 = 0. By the law of totalprobability, under Assumption 3 we have

E[xT τ ] =

Ts=0

E[xT τ |T τ= s] Pr(T τ

= s) = E

T

s=0

psxs

,

E[x2T τ ] =

Ts=0

E[x2T τ |T τ= s] Pr(T τ

= s) = E

T

s=0

psx2s

.

(11)

Thus, Model (10) is equivalent tominu∈Θ0

E

T

s=0

psx2s

− d2

,

s.t. E

T

s=0

psxs

= d, (1)–(2).

(12)

The equality constraint ET

s=0 psxs

= d in Model (12) canbe eliminated by the Lagrange method. We first fix a Lagrangemultiplier 2µ and turn to solve the following optimization problemmin

u∈Θ0

E

T

s=0

psx2s

− d2 + 2µ

E

T

s=0

psxs

− d

,

s.t. (1)–(2).

(13)

To this end, we first focus on solving Problem (13). Since

E

T

s=0

psx2s

− d2 + 2µ

E

T

s=0

psxs

− d

= E

T

s=0

psx2s + 2µpsxs

− d2 − 2µd, (14)

and−d2 − 2µd

is fixed, Problem (13) is equivalent to the

following optimization problem in the sense that they share thesame optimal solution

minΘ0

E

T

s=0

psx2s + 2µpsxs

, s.t. (1)–(2). (15)

For convenience, we make a state variable transformation forProblem (15). Problem (15) can be transformed intomin

u∈Θ0E

T

s=0

ps (zs − csys)2 + 2µps (zs − csys)

,

s.t. (1) and (4)(16)

since xs = zs − cys. In the following, we adopt the dynamicprogramming approach to solve Problem (16).

Let fk(zk, yk) be the optimal value function of Problem (16)starting from time kwith initial states zk and yk, that is

fk(zk, yk) = minu∈Θk

E

T

s=k

ps (zs − csys)2

+ 2µps (zs − csys))

(zk, yk)

s.t. (1) and (4).

(17)

Then by the dynamic programming principle, we obtain theBellman equation for Problem (16) as follows

fk(zk, yk) = minuk

Epk (zk − ckyk)2

+ 2µpk (zk − ckyk) + fk+1(zk+1, yk+1)]= pkz2k + pkc2k y

2k − 2pkckzkyk + 2pkµzk − 2pkckµyk

+ minuk

Efk+1

zke0k + ck+1qkyk + P ′

kuk, qkyk

,

fT (zT , yT ) = pT (zT − cTyT )2 + 2µpT (zT − cTyT )= pT z2T − 2pT cT zTyT + pT c2T y

2T + 2µpT zT − 2µpT cTyT .

(18)

Setting t = 0, we find that the optimal values to Problem (13) and(16) are f0(z0, y0) and (f0(z0, y0) − d2 − 2µd), respectively, wherez0 = x0 + cy0.

To find the explicit expression of fk(zk, yk), we construct theseries wk, hk, αk, λk, γk and gk satisfying the recurrence relationsand boundary conditions as follows.

wk = pk + wk+1Ak, wT = pT , (19)hk = pk + hk+1Jk, hT = pT , (20)

αk = αk+1 −h2k+1

wk+1Dk, αT = 0, (21)

λk = λk+1Ck − pkck + wk+1ck+1Ck, λT = −cTpT , (22)γk = γk+1E[q2k] + pkc2k + (wk+1ck+1 + 2λk+1) ck+1Bk

−λ2k+1

wk+1

E[q2k] − Bk

,

γT = c2T pT ,

(23)

gk = gk+1E[qk] − pkck + hk+1

×

ck+1Mk −

λk+1

wk+1(E[qk] − Mk)

,

gT = −cTpT ,

(24)

where

Ak = E[(e0k)2] − E[e0kP

′

k]E−1

[PkP ′

k]E[e0kPk],Bk = E[q2k] − E[qkP ′

k]E−1

[PkP ′

k]E[qkPk],Ck = E[e0kqk] − E[e0kP

′

k]E−1 PkP ′

k

E[qkPk],

Dk = E[P ′

k]E−1

[PkP ′

k]E[Pk],Jk = E[e0k] − [e0kP

′

k]E−1

[PkP ′

k]E[Pk],Mk = E[qk] − E[qkP ′

k]E−1

[PkP ′

k]E[Pk],

(25)

and E−1[PkP ′

k] is the inverse matrix of E[PkP ′

k], i.e., E−1[PkP ′

k] =

(E[PkP ′

k])−1.

We now further derive the computational formulas for theserieswk, hk,αk,λk, γk and gk. For convenience, define

k−1i=k (·) = 0

andk−1

i=k (·) = 1. We give a useful lemma first.

Lemma 1. Suppose that the series {lk} satisfies the recursion formulalk = lk+1tk + sk, k = 0, 1, . . . , T − 1 and lT is given. Then

lk = lTT−1i=k

ti +T−1i=k

sii−1j=k

tj. (26)

This lemma can be proved easily by repeated backwarditerations.

By Lemma 1, (19) and (20), we havewk = pT

T−1i=k

Ai +

T−1i=k

pii−1j=k

Aj =

Ti=k

pii−1j=k

Aj,

hk = pTT−1i=k

Ji +T−1i=k

pii−1j=k

Jj =

Ti=k

pii−1j=k

Jj,(27)

88 H. Yao et al. / Insurance: Mathematics and Economics 54 (2014) 84–92

for k = 0, 1, . . . , T . After obtaining the expressions for wk and hk,by Lemma 1, (21) and (22), we can further express λk and αk interms of wk and hk as follows

λk = −cTpTT−1i=k

Ci +

T−1i=k

(wi+1ci+1Ci − cipi)i−1j=k

Cj

=

T−1i=k

wi+1ci+1

ij=k

Cj −

Ti=k

cipii−1j=k

Cj,

αk = −

T−1i=k

h2i+1

wi+1Di, k = 0, 1, . . . , T .

(28)

For k = 0, 1, . . . , T − 1, letηk = c2k pk + (wk+1ck+1 + 2λk+1) ck+1Bk

−λ2k+1

wk+1

E[q2k] − Bk

,

ξk = −ckpk + hk+1

ck+1Mk −

λk+1

wk+1(E[qk] − Mk)

.

(29)

Then, (23) and (24) can be simplified as followsγk = γk+1E[q2k] + ηk, γT = c2T pT ,gk = gk+1E[qk] + ξk, gT = −cTpT .

(30)

By Lemma 1 and (30), for k = 0, 1, . . . , T , we obtain the explicitexpressions for γk and gk

γk = c2T pTT−1i=k

E[q2i ] +

T−1i=k

ηi

i−1j=k

E[q2j ],

gk = −cTpTT−1i=k

E[qi] +

T−1i=k

ξi

i−1j=k

E[qj].(31)

It is well known that Assumption 1 implies that

Ak = E[(e0k)2] − [e0kP

′

k]E−1

[PkP ′

k]E[e0kPk] > 0,

see Li and Ng (2000). By (27), we have the following proposition.

Proposition 1. For k = 0, 1, . . . , T , we have wk > 0.

We are now ready to state and to prove the following theorem.

Theorem 1. Let z = zk and y = yk for simplicity. Then thesolution to Bellman equation (18), namely the optimal value functionof Problem (16), is given by

fk(z, y) = wkz2 + 2λkzy + γky2 + 2hkµz + 2gkµy + αkµ2, (32)

where wk, hk, αk, λk, γk and gk are defined by (19)–(24) for k =

0, 1, . . . , T .

Proof. We prove this theorem by mathematical induction on k.For k = T , by the boundary conditions of (19)–(24), we have

wT z2 + 2λT zy + γTy2 + 2hTµz + 2gTµy + αTµ2

= pT z2 − 2pT cT zy + pT c2T y2+ 2pTµz − 2pT cTµy.

On the other hand, according to the boundary condition of Bellmanequation (18), it follows that

fT (z, y) = pT z2 − 2pT cT zy + pT c2T y2+ 2pTµz − 2pT cTµy.

Therefore, for k = T , (32) holds.Suppose that for k + 1, (32) holds, i.e.,

fk+1(z, y) = wk+1z2 + γk+1y2 + 2λk+1zy + 2hk+1µz+ 2gk+1µy + αk+1µ

2.

Then for k, by Bellman equation (18), we obtain

fk(z, y) = pkz2 + pkc2k y2− 2pkckzy + 2pkµz − 2pkckµy

+ minuk

Efk+1

ze0k + ck+1qky + P ′

kuk, qky

= pkz2 + pkc2k y2− 2pkckzy + 2pkµz − 2pkckµy

+ minuk

Ewk+1

ze0k + ck+1qky + P ′

kuk2

+ γk+1q2ky2

+ 2λk+1ze0k + ck+1qky + P ′

kukqky + αk+1µ

2

+2hk+1µze0k + ck+1qky + P ′

kuk+ 2gk+1µqky

= pkz2 + pkc2k y

2− 2pkckzy + 2pkµz − 2pkckµy

+ wk+1z2E[(e0k)2] + wk+1c2k+1y

2E[q2k]

+ 2wk+1ck+1yzE[e0kqk] + γk+1y2E[q2k]

+ 2λk+1yzE[qke0k] + 2λk+1ck+1y2E[q2k]

+ 2hk+1µzE[e0k] + 2hk+1µck+1yE[qk]

+ 2gk+1µE[qk]y + αk+1µ2+ min

uk

wk+1u′

k

× EPkP ′

k

uk + 2

wk+1zE[e0kP

′

k] + (wk+1ck+1

+ λk+1) yE[qkP ′

k] + hk+1µE[P ′

k]uk. (33)

It is known from Proposition 1 that wk+1 > 0. On the other hand,it is known from Li and Ng (2000) that E

PkP ′

k

is positive definite

under Assumption 1. Therefore, the first order necessary condition(which is also sufficient) about uk gives the optimal strategy

u∗

k = −E−1 PkP ′

k

zE[e0kPk] + y

ck+1 +

λk+1

wk+1

× E[qkPk] + µ

hk+1

wk+1E[Pk]

. (34)

Substituting (34) into (33) yields

fk(z, y) = pkz2 + pkc2k y2− 2pkckzy + 2pkµz − 2pkckµy

+ wk+1z2E[(e0k)2] + wk+1c2k+1y

2E[q2k] + 2wk+1ck+1yzE[e0kqk]

+ γk+1y2E[q2k] + 2λk+1yzE[qke0k] + 2λk+1ck+1y2E[q2k]

+ 2hk+1µzE[e0k] + 2hk+1µck+1yE[qk] + 2gk+1µE[qk]y

+ αk+1µ2−wk+1zE[e0kP

′

k] + (wk+1ck+1 + λk+1) yE[qkP ′

k]

+ hk+1µE[P ′

k]

E−1 PkP ′

k

zE[e0kPk] + y

ck+1 +

λk+1

wk+1

× E[qkPk] + µ

hk+1

wk+1E[Pk]

.

Simplifying the above formula and by (25), we have

fk(z, y) = (pk + wk+1Ak) z2 + 2 [−pkck + (wk+1ck+1

+ λk+1) Ck] zy + 2 (pk + hk+1Jk) µz +

αk+1 −

h2k+1

wk+1Dk

µ2

+

pkc2k + γk+1E[q2k] +

wk+1c2k+1 + 2λk+1ck+1

Bk

−λ2k+1

wk+1

E[q2k] − Bk

y2

+ 2

− pkck + gk+1E[qk] + hk+1

ck+1Mk

−λk+1

wk+1(E[qk] − Mk)

µy.

H. Yao et al. / Insurance: Mathematics and Economics 54 (2014) 84–92 89

Then, by (19)–(24), it follows that

fk(z, y) = wkz2 + 2λkczy + γkc2y2 + 2hkµz + 2gkµcy + αkµ2.

This means that (31) holds for k. By the Principle of MathematicalInduction (PMI), (32) holds true for k = 0, 1, . . . , T , and thetheorem is proved. �

By the proof of Theorem 1, we have the following result.

Theorem 2. The optimal strategy for Problem (16) is given by (34).

4. Efficient investment strategy and efficient frontier

By the previous analysis in Section 3, we know that the optimalvalue of Problem (13) is

H(z0, y0, µ) = f0(z0, y0) − d2 − 2µd, (35)

where z0 = x0 + cy0. By Theorem 1 and (35), it follows that

H(z0, y0) = w0z20 + 2λ0z0y0 + γ0y20+ 2h0µz0 + 2g0µy0 + α0µ

2− d2 − 2µd

= α0µ2+ 2µ (h0z0 + g0y0 − d) + w0z20

+ 2λ0z0y0 + γ0y20 − d2. (36)

According to the Lagrange dual theory (see Luenberger (1968)),the optimal value of Problem (10) (which is equivalent to Problem(12)) can be obtained by maximizing H(z0, y0, µ) over µ, i.e.,

Var∗[xT τ ] = maxµ

H(z0, y0, µ). (37)

In order to show the existence of the solution to for Problem (37),we give the following proposition first.

Proposition 2. αk < 0 for k = 0, 1, . . . , T − 1.

Proof. It is known from Li and Ng (2000) that under Assumption 1,E[PkP ′

k] is positive definite. Then so is E−1[PkP ′

k]. By Assumption 4,E[Pk] = 0, then Dk = E[P ′

k]E−1

[PkP ′

k]E[Pk] > 0. By (9), pT > 0,and by Proposition 1, wk+1 > 0 for k = 0, 1, . . . , T − 1. Therefore,by (27) and (28), we have

αk = −

T−1i=k

h2i+1

wi+1Di ≤

h2T

wTDT−1 = −pTDT−1 < 0,

for k = 0, 1, . . . , T−1. This completes the proof of the proposition.�

Proposition 2 shows thatα0 < 0. Therefore, by (36), the optimalsolution of optimization problem (37) exists. By the first-ordercondition, we obtain the optimal solution as follows

µ∗= −

h0z0 + g0y0 − dα0

. (38)

Substituting (38) into (34) and noticing that z = zk, y = yk andzk = xk+ckyk, we obtain the optimal strategy of themean–variancemodel (10), namely the efficient investment strategy as follows

u∗

k = −E−1 PkP ′

k

(xk + ckyk) E[e0kPk]

+ yk

ck+1 +

λk+1

wk+1

E[qkPk]

−(h0 (x0 + c0y0) + g0y0 − d) hk+1

α0wk+1E[Pk]

. (39)

Again substituting (38) into (37) and noticing that z0 = x0 + c0y0,the optimal value of the mean–variance model (10), namely, theminimum variance is obtained as follows

Var∗[xT τ ] = −1 + α0

α0

d −

h0 (x0 + c0y0) + g0y01 + α0

2

+ w0 (x0 + c0y0)2 + 2λ0 (x0 + c0y0) y0 + γ0y20

−1

1 + α0(h0 (x0 + c0y0) + g0y0)2. (40)

Setting d = dσmin :=h0(x0+c0y0)+g0y0

1+α0, we can obtain the global

minimum variance

Var∗min[xT τ ] := w0 (x0 + c0y0)2 + 2λ0 (x0 + c0y0) y0

+ γ0y20 −1

1 + α0(h0 (x0 + c0y0) + g0y0)2. (41)

We summarize the above results in the following theorem.

Theorem 3. For a given expected terminal wealth E[xT τ ] = d(d ≥ dσmin ), the efficient investment strategy and the efficient frontierof the multi-period mean–variance DC pension funds investmentmanagement problem (10) with stochastic income and mortality riskare given by (39) and (40), respectively.

5. Some special cases

The model in the previous section is referred to as the Generalcase. In this section, we discuss some special cases of our model.

Special case 1: The case of no pension contribution. In this case,we need only to set ck = 0 for k = 0, 1, . . . , T . Our modeldegenerates to an ordinary multi-period mean–variance portfolioselection model. It is known from (28) that λk = 0 for k =

0, 1, . . . , T . Then by (29), ηk = ξk = 0 for k = 0, 1, . . . , T − 1.Hence, by (31), it follows that γk = gk = 0 for k = 0, 1, . . . , T .Therefore, in this case, by (39) and (40), the efficient investmentstrategy and the efficient frontier can be simplified as

u∗

k = −E−1 PkP ′

k

xkE[e0kPk] −

(h0x0 − d) hk+1

α0wk+1E[Pk]

, (42)

and

Var∗[xT τ ] = −1 + α0

α0

d −

h0x01 + α0

2

+

w0 −

h20

1 + α0

x20, (43)

respectively, where wk, hk and αk are also given by (27) and (28).Special case 2: The terminal time is deterministic. We need only

to set the mortality intensity (s) = 0 on [0, T ]. Then by (9), itfollows that

pi = 0, i = 1, 2, . . . , T − 1; pT = 1.

In this case, wk, hk, αk, λk, γk and gk can be simplified as

wk =

T−1i=k

Ai, hk =

T−1i=k

Ji, αk = −

T−1i=k

T−1j=i+1

J2jAj

Di,

λk = −cTT−1i=k

Ci +

T−1i=k

wi+1ci+1

ij=k

Cj,

γk = c2TT−1i=k

E[q2i ] +

T−1i=k

ηi

i−1j=k

E[q2j ],

gk = −cTT−1i=k

E[qi] +

T−1i=k

ξi

i−1j=k

E[qj],

(44)

90 H. Yao et al. / Insurance: Mathematics and Economics 54 (2014) 84–92

whereηk = (wk+1ck+1 + 2λk+1) ck+1Bk −

λ2k+1

wk+1

E[q2k] − Bk

,

ξk = hk+1

ck+1Mk −

λk+1

wk+1(E[qk] − Mk)

.

(45)

The efficient investment strategy and the efficient frontier are alsogiven by (39) and (40), respectively.

Special case 3: The case with a risk-free asset in the market.Suppose that the 0th asset is the risk-free asset. Then e0k is a non-stochastic constant, k = 0, 1, . . . , T − 1. By (25), we have

Ak = (e0k)2 (1 − Dk) , Ck = e0k (E[qk] − Dk) ,

Jk = e0k (1 − Dk) .(46)

According to (27), (28) and (46), the expressions of wk, hk and λkcan be rewritten as

wk =

Ti=k

pii−1j=k

(e0j )2 1 − Dj

,

hk =

Ti=k

pii−1j=k

e0j1 − Dj

,

λk =

T−1i=k

wi+1ci+1

ij=k

e0jE[qj] − Dj

−

Ti=k

cipii−1j=k

e0jE[qj] − Dj

.

(47)

In this case, αk, γk and gk are again given by (28) and (31). Inaddition, the efficient investment strategy and the efficient frontierare still given by (39) and (40), respectively.

6. Numerical illustration

In this section,we provide somenumerical analyses to illustratethe results obtained in this paper.

Consider a pension fundmember who enters a DC pension fundscheme at time 0 with an initial fund paid x0 = 3 and plans toretire at time T = 6. Suppose that his initial income is y0 = 1 andhe needs to contribute 20% of his income at the beginning of everyperiod, i.e., ck = 0.2 for k = 0, 1, . . . , T . Due to the mortality risk,the actual terminating time of his pension fund scheme is T τ . Theprobability distribution of T τ is defined by (5)–(9). For simplicity,we assume that themortality intensity (s) is independent of times, and always equal to 0.1, i.e., (s) = 0.1 for s ∈ [0, T ]. Then by(9), The probability distribution of T τ is given by

p1 = 0.0952, p2 = 0.0861, p3 = 0.0779,p4 = 0.0705, p5 = 0.0638, p6 = 0.6065.

Suppose that the pension fund can be invested in four risky assets(indexed as 0, 1, 2, 3). For convenience, suppose that the marketparameters are independent of time k and are listed as follows

E[PkP ′

k] =

0.2365 0.0719 0.11840.0719 0.3449 0.13780.1184 0.1378 0.3262

,

E[e0kqk] = 1.1689, E[(e0k)2] = 1.2468,

E[e0k] = 1.0430, E[qk] = 1.0284, E[q2k] = 1.4257,E[qkPk] = (−0.0083, 0.0220, 0.0657)′,E[Pk] = (−0.0255, 0.0015, 0.0004)′,E[e0kPk] = (−0.0827, −0.0924, −0.0446)′.

Substituting the above data into (27), (28) and (31) gives

α0 = −0.0127, γ0 = 2.7557, g0 = 0.8813,w = (2.5164, 2.0926, 1.6610, 1.3097,

1.0243, 0.7932, 0.6065),λ = (1.9533, 1.2562, 0.7611, 0.4054,

0.1560, −0.0128, −0.1213),h = (1.1807, 1.1405, 1.0097, 0.8922,

0.7866, 0.6917, 0.6065),

(48)

where w = (w0, w1, . . . , w6), λ = (λ0, λ1, . . . , λ6), h =

(h0, h1, . . . , h6). Then, substituting (48) into (39), we obtain theefficient investment strategy as

u∗

k =

0.31760.2323

−0.0766

(xk + 0.2yk) + yk

0.2 +

λk+1

wk+1

0.16540.0075

−0.2646

+(4.6595 − d) hk+1

0.0127wk+1

0.1342−0.0150−0.0437

,

where the values of wk+1, λk+1 and hk+1 (k = 0, 1, . . . , 5) aregiven by (48). Plugging (48) into (40), we obtain the efficientfrontier as

Var∗[xT τ ] = 77.9389 (d − 4.7192)2 + 19.0355.

In the following, we discuss some special cases. The relatedparameters are the same as above unless stated otherwise.

Special case 1: The case without pension contribution. In thiscase, ck = 0 for k = 0, 1, . . . , T . Then by (42) and (43), we obtainthe efficient investment strategy

u∗

k =

0.31760.2323

−0.0766

xk +

(3.5421 − d) hk+1

0.0127wk+1

0.1342−0.0150−0.0437

,

and the efficient frontier

Var∗[xT τ ] = 77.9389 (d − 3.5875)2 + 9.9403,

where the values of wk+1 and hk+1 (k = 0, 1, . . . , 5) are alsogiven by (48). From Fig. 1, comparedwith the General case, we findthat the efficient frontier shifts to the lower left but the shape ofthe efficient frontier is invariant. It is not difficult to understandthis. Because in this case there is no pension contribution, and sothe wealth accumulation for the pension fund is reduced. So, for agiven risk (measured by variance) level of its terminal wealth, thepension fund has lower expected terminal wealth level. Therefore,the efficient frontier moves down.

Special case 2: The terminal time of the pension fund isdeterministic. In this case (s) = 0 for all s ∈ [0, T ]. Then by(9) pi = 0, i = 0, 1, . . . , 5, p6 = 1. Substituting the data into (44),we have

α0 = −0.0159, γ0 = 4.1160, g0 = 1.1921,w = (3.0239, 2.5147, 2.0911, 1.7389, 1.4461, 1.2025, 1),λ = (2.7619, 1.8650, 1.1807, 0.6645, 0.2805, 0, −0.2),h = (1.2310, 1.1891, 1.1486, 1.1095, 1.0717, 1.0352, 1).

(49)

Then by (49), (39) and (40), we obtain the efficient investmentstrategy

u∗

k =

0.31760.2323

−0.0766

(xk + 0.2yk) + yk

0.2 +

λk+1

wk+1

0.16540.0075

−0.2646

+(5.1312 − d) hk+1

0.0159wk+1

0.1342−0.0150−0.0437

,

H. Yao et al. / Insurance: Mathematics and Economics 54 (2014) 84–92 91

Fig. 1. Efficient frontiers for the General case and Special cases 1–3.

and the efficient frontier

Var∗[xT τ ] = 61.9407 (d − 5.2141)2 + 26.0029,

where the values of wk+1, λk+1 and hk+1 (k = 0, 1, . . . , 5) aregiven by (49). Fig. 1 shows that the efficient frontier moves tothe upper right compared with the General case. The economicimplications are stated as follows. Since there is no mortalityrisk in this case, the pension scheme would not be forced toterminate before the retirement of the member. Then, comparedwith the General case, it takes longer time for the pension fundto accumulate its wealth. Therefore, for a given risk level of theterminal wealth, the pension fund has higher expected terminalwealth level. This explains the upward movement of the efficientfrontier in this case.

Special case 3: There is a risk-free asset in the market. Supposethat the 0th asset is risk-free. Then, e0k is a constant. Suppose thate0k = 0.015, k = 0, 1, . . . , 5, and the market parameters for theother three assets are the same as above. In this case, the efficientfrontier is

Var∗[xT τ ] = 82.8673 (d − 4.5540)2 + 1.1508,

and the efficient investment strategy is

u∗

k =

0.0952−0.0412−0.0610

(xk + 0.2yk) − yk

0.2 +

λk+1

wk+1

0.10190.23940.3028

+(4.4997 − d) hk+1

0.0119wk+1

0.0924−0.0400−0.0592

,

where the values of wk+1, λk+1 and hk+1 (k = 0, 1, . . . , 5) aregiven as follows

w = (1.3142, 1.2419, 1.0837, 0.9427,0.8172, 0.7056, 0.6065),

λ = (1.0139, 0.7147, 0.4802, 0.2839,0.1210, −0.0128, −0.1213),

h = (1.1380, 1.1077, 0.9855, 0.8754,0.7762, 0.6869, 0.6065).

(50)

We point out that although there is a risk-free asset in the marketin this case, the global minimum variance Varmin = 1.1508 isstrictly larger than zero. The reason is as follows: not only therisky asset can cause risk, but also the two factors, stochasticincome and mortality risk, can generate the integral risk duringthe accumulation process of the pension fund. So, even if all thewealth is invested in the risk-free asset in this case, the global

minimum variance are also strictly larger than 0. But comparedwith the globalminimumvariance 19.0355 in the General case, theglobalminimumvariance 1.1508 in this case has been dramaticallyreduced.

Fig. 1 indicates that when the expected terminal wealth islarger than that corresponding to the intersection point of the twoefficient frontiers in the General case and Special case 3, for thesame expected terminal wealth level, the variance of the Generalcase is smaller than that of Special case 3. It is also not difficult tounderstand this. That is because there is onemore risky asset in theGeneral case, namely there is one more risk source in the Generalcase. Obviously, inclusion of one more risk source is beneficial tohedge the integral risk of the pension fund. This explains the factthatwhen the expected terminalwealth is big enough, the varianceof the terminal wealth is smaller than that of Special case 3.

7. Conclusion

Starting from the actual requirements of asset allocation forDC pension funds, we consider synthetically the two objectives—to enhance the yield and to control the risk for a problem ofDC pension fund investment management in this paper. Morespecifically, we establish the multi-period mean–variance assetallocationmodel for a DC pension fundwith stochastic income andmortality risk in this paper. By adopting the dynamic programmingmethod and the Lagrange dual theory, we obtain the closed-form expressions for the efficient investment strategy and theefficient frontier. We also discuss some special cases of our model.Finally, we present a numerical example to illustrate the resultsobtained in this paper. We find some interesting results vianumerical analyses, and also give the corresponding economicinterpretations. Our model can be extended in several ways. Forexample, we can extend our model to the cases withmore realisticconditions, such as stochastic market environment, parameteruncertainty, and minimum guarantee protection. Using the multi-periodmean–variancemodel to studyDB pension fund investmentmanagement problems is another research topic in the near future.

Acknowledgments

The authors are grateful to the anonymous referee(s) for givingthem very useful suggestions and comments.

References

Blake, D., Wright, D., Zhang, Y., 2013. Target-driven investing: optimal investmentstrategies in defined contribution pension plans under loss aversion. J. Econom.Dynam. Control 37, 195–209.

Cairns, A.J.G., Blake, D., Dowd, K., 2006. Stochastic lifestyling: optimal dynamic assetallocation for defined contribution pension plans. J. Econom. Dynam. Control30, 843–877.

Charupat, N., Milevsky, M.A., 2002. Optimal asset allocation in life annuities: a note.Insurance Math. Econom. 30, 199–209.

Chen, P., Yang, H.L., Yin, G., 2008. Markowitz’s mean–variance asset–liabilitymanagement with regime switching: a continuous-time model. InsuranceMath. Econom. 43, 456–465.

Chiu, M.C., Wong, H.Y., 2013. Mean–variance principle of managing cointegratedrisky assets and random liabilities. Oper. Res. Lett. 41, 98–106.

Christophette, B.S., Nicole, E.K., Monique, J., Lionel, M., 2008. Optimal investmentecisions when time-horizon is uncertain. J. Math. Econom. 44, 100–1113.

Costa, O.L.V., Oliveira, A.D., 2012. Optimal mean–variance control for discrete- timelinear systemswithMarkovian jumps andmultiplicative noises. Automatica 48,304–315.

Deelstra, G., Grasselli, M., Koehl, P.F., 2003. Optimal investment strategies in thepresence of a minimum guarantee. Insurance Math. Econom. 33, 189–207.

Delong, Ł, Gerrard, R., Haberman, S., 2008. Mean–variance optimization problemsfor an accumulation phase in a defined benefit plan. Insurance Math. Econom.42, 107–118.

Emms, P., 2012. Lifetime investment and consumptionusing a defined-contributionpension scheme. J. Econom. Dynam. Control 36, 1303–1321.

Fu, C.P., Ali, L.L., Li, X., 2010. Dynamic mean–variance portfolio selection withborrowing constraint. European J. Oper. Res. 200, 313–319.

Gao, J.W., 2009. Optimal portfolios for DC pension plans under a CEV model.Insurance Math. Econom. 44, 479–490.

92 H. Yao et al. / Insurance: Mathematics and Economics 54 (2014) 84–92

Gerrard, R., Haberman, S., Vigna, E., 2004. Optimal investment choices post-retirement in a defined contribution pension scheme. InsuranceMath. Econom.35, 321–342.

Gerrard, R., Højgaard, B., Vigna, E., 2012. Choosing the optimal annuitization timepost retirement. Quant. Finance 12 (7), 1143–1159.

Giacinto, D.M., Federico, S., Gozzi, F., 2011. Pension funds with a minimumguarantee: a stochastic control approach. Finance Stoch. 15, 297–342.

Haberman, S., Vigna, E., 2002. Optimal investment strategies and risk measures indefined contribution pension schemes. Insurance Math. Econom. 31, 35–69.

Hainaut, D., Deelstra, G., 2011. Optimal funding of defined benefit pension plans. J.Pension Econ. Finance 10 (1), 31–52.

Hainaut, D., Devolder, P., 2007. Management of a pension fund under mortality andfinancial risks. Insurance Math. Econom. 41, 134–155.

Han, N.W., Hung, M.W., 2012. Optimal asset allocation for DC pension plans underinflation. Insurance Math. Econom. 51, 172–181.

Josa-Fombellida, R., Rincón-Zapatero, J.P., 2008. Mean–variance portfolio andcontribution selection in stochastic pension funding. European J. Oper. Res. 187,120–137.

Li, D., Ng, W.L., 2000. Optimal dynamic portfolio selection: multiperiodmean–variance formulation. Math. Finance 10, 387–406.

Luenberger, D.G., 1968. Optimization by Vector Space Methods. Wiley, New York.Ma, Q.P., 2011. On optimal pension management in a stochastic framework with

exponential utility. Insurance Math. Econom. 49, 61–69.

Markowitz, H., 1952. Portfolio selection. J. Finance 7 (1), 77–91.Martellini, L., Urosevic, B., 2006. Static mean–variance analysis with uncertain time

horizon. Manag. Sci. 52, 955–964.Pliska, S.R., Ye, J.C., 2007. Optimal life insurance purchase and consump-

tion/investment under uncertain lifetime. J. Bank. Finance 31, 1307–1319.Vigna, E., 2012. On efficiency of mean–variance based portfolio selection in defined

contribution pension schemes. Quant. Finance 1–22. iFirst.Vigna, E., Haberman, S., 2001. Optimal investment strategy for defined contribution

pension schemes. Insurance Math. Econom. 28, 233–262.Wu, H.L., Li, Z.F., 2011. Multi-period mean–variance-portfolio selection with

Markov regime switching and uncertain time-horizon. J. Syst. Sci. Complexity24, 140–155.

Yao, H.X., Lai, Y.Z., Li, Y., 2013a. Continuous-time mean–variance asset–liabilitymanagement with endogenous liabilities. Insurance Math. Econom. 52,6–17.

Yao, H.X., Zeng, Y., Chen, S.M., 2013b. Multi-period mean–variance asset–liabilitymanagement with uncontrolled cash flow and uncertain time-horizon. Econ.Model. 30, 492–500.

Zhang, A., Ewald, C.O., 2010. Optimal investment for a pension fund under inflationrisk. Math. Methods Oper. Res. 71, 353–369.

Zhou, X.Y., Li, D., 2000. Continuous-Time mean–variance portfolio selection: astochastic LQ framework. Appl. Math. Optim. 42, 19–33.