Equilibrium investment strategy for defined-contribution pension schemes with generalized mean–variance criterion and mortality risk

Insurance: Mathematics and Economics 64 (2015) 396–408

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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

Equilibrium investment strategy for defined-contribution pension schemes with generalized mean–variance criterion and mortality risk Huiling Wu a , Yan Zeng b,∗ a

China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, PR China

b

Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, PR China

highlights • • • • •

The multi-period equilibrium investment strategy for DC pension schemes is studied. A generalized mean–variance criterion is adopted in the objective function. Stochastic salary flow and stochastic mortality rate are considered in the model. Analytical expressions for equilibrium strategy and value function are derived. The effects of the mortality risk on our derived results are illustrated.

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Article history: Received January 2015 Received in revised form June 2015 Accepted 24 July 2015 Available online 31 July 2015 Keywords: Defined-contribution pension scheme Equilibrium investment strategy Mortality risk Generalized mean–variance criterion Time-inconsistency

abstract This paper studies a generalized multi-period mean–variance portfolio selection problem within the game theoretic framework for a defined-contribution pension scheme member. The member is assumed to have a stochastic salary flow and a stochastic mortality rate, and is allowed to invest in a financial market with one risk-free asset and one risky asset. The explicit expressions for the equilibrium investment strategy and equilibrium value function are obtained by backward induction. In addition, some sensitivity analysis and numerical illustrations are provided to show the effects of mortality risk on our results. © 2015 Elsevier B.V. All rights reserved.

1. Introduction According to the contract of contributions and benefits, enterprise pension schemes are mainly divided into defined-benefit (DB) schemes and defined-contribution (DC) schemes. In a DB scheme, financial risk and longevity risk are mainly faced by the sponsor. In this case, the benefits are predetermined while the contributions are initially set and then subsequently adjusted to keep the scheme in balance. However, in a DC scheme, the members mainly face financial and longevity risks, with the contributions fixed in advance and the benefits dependent on the investment performance of the fund during the accumulation phase. Nowadays, many countries are making efforts to transfer longevity risk

∗

Corresponding author. Tel.: +86 20 84110516; fax: +86 20 84114832. E-mail addresses: [email protected] (H. Wu), [email protected] (Y. Zeng). http://dx.doi.org/10.1016/j.insmatheco.2015.07.007 0167-6687/© 2015 Elsevier B.V. All rights reserved.

and are finding ways to guarantee the sustainability of a sufficient retirement income. According to the contract, DC plans have an advantage over DB plans in that they can ease the pressure on social security programs by transferring investment and longevity risks to the retirees. As a result, DC schemes are playing an increasingly important role in social security programs, with many countries preferring DC schemes over DB schemes. For a DC scheme member, the main income after retirement comes from the pension fund, and is therefore heavily dependent on the investment performance of the fund during the accumulation phase. Hence, the analysis of the asset allocation in the accumulation phase of a DC scheme is very important. Numerous studies have examined the optimal investment strategies in the accumulation phase of a DC scheme using different objective functions and financial market settings. For example, to maximize the expected utility from the terminal wealth under the CRRA or CARA utility functions, Cairns et al. (2006) investigate the asset allocation strategies to DC scheme members with asset, salary and interest rate risk, and propose a novel form of terminal utility function

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

by incorporating habit formation. Zhang et al. (2007) and Zhang and Ewald (2010) aim to maximize the power utility of the terminal value of a DC pension fund under inflation risk. Giacinto et al. (2011) propose and investigate a model of optimal allocation for a DC pension plan with a minimum guarantee in the continuoustime setting. Korn et al. (2011) study an optimal portfolio selection problem for a DC scheme member in a hidden Markov-modulated economy. Within the framework of prospect theory, Blake et al. (2013) consider the optimal dynamic investment strategies for DC pension plans when the plan members are loss averse. Recently, some scholars have also examined the optimal investment strategy for a DC pension plan with the mean–variance criterion proposed by Markowitz (1952). Højgaad and Vigna (2007) compare a mean–variance model with a target-based model, and show that the target-based model can be formulated as a mean– variance model. Nkeki (2013) studies a mean–variance DC pension management problem with time-dependent salary, and compares the optimal portfolios under quadratic utility function, power utility function and exponential utility function. He and Liang (2013) introduce the return of premium clauses into the portfolio model with the mean–variance criterion for a DC pension plan during the accumulation phase, and derive a time-consistent investment strategy within the game theoretic framework. Menoncin and Vigna (2013) consider a mean–variance investment problem for a DC pension plan with a stochastic interest rate in the accumulation phase. Guan and Liang (2015) generalize this problem to the case with stochastic interest rate, stochastic volatility and stochastic salary. Vigna (2014) compares the mean–variance efficient portfolios with the optimal portfolios maximizing the expected CARA and CRRA utilities, which are proved to be not mean–variance efficient. Yao et al. (2014) consider a multi-period mean–variance investment problem for the accumulation phase of a DC pension scheme. For more information about optimal portfolio selection for a DC pension scheme under the mean–variance criterion, interested readers are referred to Vigna (2009) and Nkeki (2012). However, in the above-mentioned literature, except for He and Liang (2013), the optimal investment strategy with the mean–variance criterion is time-inconsistent, which is only optimal at the initial time. That is, the optimal strategy at time m does not agree with that at time n, where n > m, because the mean– variance criterion does not have the iterated expected property. Therefore, the optimal strategy in the classical mean–variance model is usually called the pre-commitment strategy. In recent years, the pre-commitment strategy has been criticized for lacking rationality on the basis that investment psychology and taste change over time. For this reason, there has seen a recent upsurge of interest in studying the time-consistent strategy for the mean–variance problem, see Stroz (1956), Björk and Murgoci (2010), Basak and Chabakauri (2010), Wu (2013), Björk et al. (2014), Björk and Murgoci (2014), Bensoussan et al. (2014) and references therein. To the best of our knowledge, for the discrete-time multiperiod mean–variance investment problem in the accumulation phase of a DC pension scheme, only Yao et al. (2014) consider an optimal pre-commitment investment strategy that is time-inconsistent, and no studies have examined the corresponding time-consistent investment strategy. Therefore, this paper presents the first study of this strategy. Specifically, we consider a multi-period mean–variance investment problem for a DC scheme member in the accumulation phase, and attempt to derive the time-consistent equilibrium investment strategy within the game theoretic framework. We assume that the member has a stochastic salary flow and a stochastic mortality rate, and she can invest her wealth in a financial market consisting of one risk-free asset and one risky asset. Due to the mortality risk, the time horizon of the member is uncertain. In addition, we assume that the objective of the scheme member is to maximize the weighted sum

397

of a linear combination of the expectation and variance of the terminal wealth. In our paper, the objective function of the member varies over time, according to the weighted coefficients measured by the corresponding exit probabilities. Similar to Björk and Murgoci (2010, 2014) and Bensoussan et al. (2014), we take this problem as a non-cooperative game and derive the closed-form expressions for the equilibrium strategy and equilibrium value function by backward induction. We also present some special cases of our results and the relationship between the expectation and variance of the terminal wealth. Moreover, we provide some sensitivity analysis and numerical illustrations, which show that some properties of the optimal strategy in the case without mortality risk do not hold in the case with mortality risk. Compared with Yao et al. (2014) and other existing literature, this paper makes four main contributions: (1) a generalized mean–variance criterion is first introduced into the optimal investment model of DC pension schemes, which makes our optimization problem more general; (2) the mechanism of defining the exit probabilities is quite different from that in the existing literature, in that we assume that the exit probabilities depend on the starting times and the future times, which also makes our optimization problem time-inconsistent; (3) the time-consistent equilibrium investment strategy is first considered and derived explicitly for the multi-period mean–variance investment problem in the accumulation phase of a DC pension scheme; and (4) to the best of our knowledge, we are the first to consider the time-consistent equilibrium strategy for the multi-period mean–variance investment problem with uncertain time horizon. The remainder of this paper is organized as follows. The assumptions and problem formulation are described in Section 2. The equilibrium investment strategy and equilibrium value function are derived explicitly by backward induction in Section 3, and some special cases of our results and properties of our equilibrium strategy are also presented. The expectation and variance of the terminal wealth and the relationship between them are provided in Section 4. In Section 5, we provide some numerical illustrations to show the effects of mortality risk on our results and produce some interesting findings. Conclusions are given in Section 6. Proofs of the propositions and theorems are given in Appendices A–E. 2. Problem formulation Consider a financial market that consists of one risk-free asset and one risky asset. Over period [n, n + 1), the risk-free asset has f a deterministic and positive return rn and the risky asset has a random return Rn . The member enters the financial market at time 0 and plans to invest her wealth in the market within T consecutive time periods. However, due to the mortality risk, the member does not know exactly the time when she will exit the market. The dynamics of the member’s salary before her death are given as: Yn+1 = Qn Yn ,

n = 0, 1, . . . , T − 1,

(2.1)

where Yn is the member’s salary at time n and Qn is an exogenous nonnegative random variable representing the stochastic growth rate of the member’s salary over period [n, n + 1). At time n, contributions are paid as a nonnegative deterministic proportion cn of Yn . Denote ηn = E(Qn ), and suppose that Qn is independent of Qm , ∀m ̸= n. Let π := {πn , n = 0, 1, . . . , T − 1} be an investment strategy, where πn is the amount invested in the risky asset at time n. Then the wealth of the member under strategy π evolves over time according to: Wnπ+1 = Wnπ + cn Yn − πn rnf + Rn πn





= Wnπ rnf + πn Ren + cn rnf Yn ,

n = 0, 1, . . . , T − 1,

(2.2)

398

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

where Wnπ is the wealth at time n and Ren = Rn − rn is the excess return of the risky asset over period [n, n+1). Denote by rne = E(Ren ) the expected excess return, and suppose that Ren is independent of Rem , ∀m ̸= n. Moreover, {Ren , n = 0, 1, . . . , T − 1} and {Qn , n = 0, 1, . . . , T − 1} are assumed to be mutually independent. Eq. (2.2) results in the explicit formula of the wealth at a future time k + 1 when given Wnπ = wn and Yn = yn , i.e., f

Wkπ+1 = wn

k 

f rm +

m=n

+ yn

k  m=n

k 

m−1

cm

m=n



k 

πm Rem

Qi

i =n

f

rl

l=m+1 k 

f

rj ,

(2.3)

j =m

which is used in the following sections. Because the member might die before the terminal time T , we denote by K = K (x) the number of future time periods the member will survive when she is aged x, which is a random variable and takes integer values 0, 1, 2, . . .. Suppose that at time 0, the member is aged x. Then K (n + x) = k means that she is alive at time n but she dies during the time interval (n + k, n + k + 1] and exits the market at time n + k + 1 for any n = 0, 1, . . . , T − 1, k = 0, 1, . . . , T − n − 2. Let k| qn+x

= Pr(K (n + x) = k)

be the probability that the member exits the market at time n + k + 1 on the premise of living at time n, and

which is used in the recursions of the equilibrium value function. Problem (2.5) is time-inconsistent in the sense that Bellman optimality principle does not hold, with a consequence that the control law is optimal at time m but it is not optimal at least at some time n with m + 1 ≤ n ≤ T − 1. The time-inconsistency of problem (2.5) results from two factors: the mean–variance utility, which is known as one of the time-inconsistent utilities, and the inconsistent viewpoints on the exit probabilities when the member is at different starting times. This implies that even if we adopt the classic time-consistent utilities, such as the power utility and exponential utility, the mechanism of defining the exit probabilities also makes our problem time-inconsistent. Remark 2.1. The objective function (2.5) is also used in Yi et al. (2014), who consider a multi-period mean–variance portfolio selection problem with uncertain exit time and a mean-field formulation. A similar form of objective function (2.5) is also mentioned in Costa and Araujo (2008), who consider a generalized multi-period mean–variance portfolio selection problem with T the form of minu∈U t =1 α(t ) [E(V (t )) − ρ(t )Var(V (t ))] . Other similar forms of our objective function are considered in Zhu et al. (2004), Cui et al. (2014) and He et al. (2015). The difference between the above five papers and our paper is the decision-making mechanism. They focus on the pre-commitment investment strategies, which are time-inconsistent and are only optimal at time 0. We study the time-consistent equilibrium strategy for a DC pension plan with stochastic salary flow and stochastic mortality rate, which they do not consider.

T −n −2 T −n −1 p n +x

=1−



k| qn+x

(2.4)

k=0

j

be the probability of exiting the market at time T , where i (·) = 0 if j < i. By the definitions of k| qn+x and T −n−1 pn+x , the exit probabilities depend on the starting times and the future times, which implies that at different starting times, the member will have different exit probabilities for the future times. For example, k+1| qn+x ̸= k| qn+1+x , where k+1| qn+x and k| qn+1+x are the exit probabilities at time n + k + 2 for the member at time n and at time n + 1, respectively. The mechanism of defining the exit probabilities is quite different from that in the existing literature with an uncertain time horizon where the exit probabilities are only estimated at initial time 0. Moreover, since the scheme member at different starting times has different viewpoints about the future exit probabilities, our definition of the exit probability makes the decision-making problem time-inconsistent. A more detailed explanation of time-inconsistency is presented below (2.5). At any time n, under the condition that the member is alive, the objective of the member is to maximize the weighted sum of a linear combination of the expectation and variance of the wealth at the time of exiting the market:   T −1    π  π     q E W − ω Var W m−n−1| n+x wn ,yn wn ,yn m m max , (2.5) πn ,...,πT −1          m=n+1 + T −n−1 pn+x Ewn ,yn WTπ − ωVarwn ,yn WTπ where Ewn ,yn (·) = E (·|Wn = wn , Yn = yn ), w is the risk aversion coefficient and the weighted coefficients are the corresponding exit probabilities. Specifically, if the member is alive at time T − 1, she will certainly exit the market at time T with probability 1 regardless of whether she is dead or still alive at time T . We can also obtain 0 pT −1+x = 1 by (2.4). Therefore, the objective of the member at time T − 1 is π

π

max EwT −1 ,yT −1 WT T −1 − ωVarwT −1 ,yT −1 WT T −1



πT −1









,

As mentioned above, for the time-inconsistent problem, because the optimal control law at time m is not optimal at least at some time n with m + 1 ≤ n ≤ T − 1, the member cannot persuade her future incarnations to accept a strategy that is not optimal for themselves. Therefore, here we follow Björk and Murgoci (2010) and the relevant references therein and develop a time-consistent solution within the game theoretic framework. We define Jn (wn , yn ; π )

:=

T −1 

m−n−1| qn+x

Ewn ,yn Wmπ − ωVarwn ,yn Wmπ











m=n+1

     + T −n−1 pn+x Ewn ,yn WTπ − ωVarwn ,yn WTπ ,

(2.6)

and view the decision-making process as a non-cooperative game with one distinct decision-maker, referred to as decision-maker n, over each period [n − 1, n). At each time n, the member can only choose πn to maximize Jn (wn , yn ; π ) given that her future incarnations choose their own optimal strategies on the premise of living to that time. Under this circumstance, at each time the member is satisfied and has no impulse to deviate from the socalled subgame perfect Nash equilibrium strategy defined below. Definition 2.1. Let πˆ be a fixed control law. For any n 0, 1, . . . , T − 1, define

=

π¯ (n) := (πn , πˆ n+1 , . . . , πˆ T ), where πn is arbitrarily selected. Then πˆ is said to be a subgame perfect Nash equilibrium strategy (or equilibrium strategy) if for all n < T , it satisfies max Jn (wn , yn ; π¯ (n)) = Jn (wn , yn ; πˆ (n)), πn

where πˆ (n) := (πˆ n , πˆ n+1 , . . . , πˆ T ). In addition, if equilibrium strategy πˆ exists, the equilibrium value function is defined as Vn (wn , yn ) = Jn (wn , yn ; πˆ (n)).

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

399

3. Equilibrium strategy and equilibrium value function

Theorem 3.1. For problem (2.5), the equilibrium strategy is

To obtain an equilibrium strategy, we start with a recursive formula for equilibrium value function Vn . Denote qn+x = 0| qn+x and pn+x = Pr{K (x) ≥ n + 1|K (x) ≥ n} = 1 − qn+x . By Definition 2.1 and (2.6), the recursive formula of the equilibrium value function Vn (wn , yn ) is given in the following proposition.

πˆ n =

1 2ω

rne

 ,

ϖn

Vn (wn , yn )

 =

T −2 

m−n| qn+x

Vn (wn , yn )



(3.2)

   = Ewn ,yn hn+1,m Wnπˆ+n 1 , Yn+1 , hn,n (wn , yn ) = wn ,

(3.3)

n = 0, 1, . . . , T .

Proof. See Appendix A.

(3.4)



T −1 

m 

f

rj ,

ϕn+1,m−1 Qn m=n+2   + T −n−1 pn+x Var ϕn+1,T −1 Qn , m−n−1| qn+x Var

T −2

 ϖn =

(3.5)

j=l

i=n

l=n

µn =

ηi

m−n| qn+x

m=n m−n| qn+x

m=n

φn,m =

m 

+ T −n −1 p n +x

f rk

 k=n+1

( ) + T −n−1 pn+x

k=n+1

T −1



,

(3.7)

f rk 2

( )

T −2 

rl ,

T −2 

1

T −1  l=n

µl

l−1 





ξn =

+ T −n−1 pn+x



f rk

rnf wn + An yn

k=n+1

4ω

pn+x ξn+1 +

T −2 

m−n| qn+x

m=n f rk

−ω

T −2 



+ T −n −1 p n +x

k=n+1





T −1

m 

f rk

rne πˆ n

k=n+1 m 

m−n| qn+x

(3.10)

(rkf )2

k=n+1



T −1

+ T −n−1 pn+x T −2 

(3.14)



T −1 f rk



1

(3.9)

m=n

ξn = T −n−1 pn+x φn,T −1 +

φn,m−1 ,



m=n

pm+x E (Qm )2 ,

2ω

  ξT −1 = πˆ T −1 rTe−1 − ωVar πˆ T −1 ReT −1 ,

× ϕn,m ,

1

f

rk + ϕn,m−1 yn +

k=n+1

m=n

Bn =



m 

m−n| qn+x

(3.8)

m−n| qn+x



f rk 2

( )

Var Ren (πˆ n )2

 

k=n+1 m−n| qn+x φn,m ,

(3.13)

4ω which can be regarded as the trade off between the expected excess return and the loss (measured by the variance) of the risky investment at time T − 1. Then by (B.4) and (B.7), we notice that 4ω

f

k

n = 0, 1, . . . , T − 1,

can be taken as the weighted sum of the returns by investing the wealth wn and the expected contribution in the risk-free asset until the exiting time. The second part −ωBn (yn )2 can be regarded as the loss of the value function resulting from the random salary process. For the third part 41ω ξn , it first follows from (B.2) in Appendix B that

1

k=n+1

Var(Rek ) l=k+1

An = T −n−1 pn+x ϕn,T −1 +

(3.6) T −1

f rk 2

m m  ϖk (r e )2  k =n

f rk



k=n+1

T −2



m 



ξn ,

From (3.12), we find that at any time n, the member will invest more wealth in the risky asset as the risk aversion coefficient w becomes smaller or the Sharpe ratio of the risky asset over period [n, n+1) becomes larger. We also find that the equilibrium strategy πˆ n is independent of the current wealth wn , the current salary yn , the contribution ratio cn and the future Sharpe ratio of the risky asset, but is dependent on the return of the risk-free asset over future periods and the exit probabilities at time n + 1, n + 2, . . . , T . According to (3.13), we find that the equilibrium value function Vn (wn , yn ) is a linear function of the current wealth wn and a quadratic function of the current salary yn , and consists of three parts. The first part

m=n

cl

rnf wn

n = 0, 1, . . . , T − 1, m = n + 1, n + 2, . . . , T .

 To proceed, we define l−1 m  

1 4ω

k=n



n = 0, 1, . . . , T − 1, m = n + 1, . . . , T ,

f rk

k=n+1

m−1

hn,m (wn , yn ) = wn

where hn,m (wn , yn ) = Ewn ,yn Wmπˆ

+ T −n−1 pn+x

k=n+1

Proof. See Appendix B.

πT −1



and

VT −1 (wT −1 , yT −1 )

ϕn,m =

f rk

+ An yn − ωBn (yn )2 +

(3.1)

  π   π  = max EwT −1 ,yT −1 WT T −1 − ωVarwT −1 ,yT −1 WT T −1 ,



T −1

m 

m=n

  π    pn+x Ewn ,yn Vn+1 Wn+n1 , Yn+1     πn   πn        + qn+x Ewn ,yn Wn+1 − ωVarwn ,yn Wn+1     T − 1  = max   πn  , πn   −ω  m−n−1| qn+x Varwn ,yn hn+1,m Wn+1 , Yn+1        m=n+2     πn  − ω · T −n−1 pn+x Varwn ,yn hn+1,T Wn+1 , Yn+1

(3.12)

the equilibrium value function is

Proposition 3.1.

n = 0, 1, . . . , T − 2,

n = 0, 1, . . . , T − 1,

Var Ren

(3.11)

m=n

for n = 0, 1, . . . , T − 1, m = n, . . . , T and AT = BT = ξT = 0. Using these notations, we obtain the equilibrium strategy and value function as given in Theorem 3.1.

=

1 4ω

pn+x ξn+1 +

T −2 

  m−n| qn+x

m=n

 − ωVar πˆ n Ren

k=n+1

m  k=n+1



m



E πˆ

e n Rn

f

rk

 f rk

400

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

 



T −1



+ T −n−1 pn+x E πˆ

e n Rn

Proposition 3.2 reveals that under the equilibrium strategy, the member will increase the amount invested in the risky asset at time n as the exiting probabilities at time n + 1, n + 2, . . . , T − 1 become larger. Specifically, if the member will certainly die over period (n, n + 1], i.e., qn+x = 1, then by (3.12), the amount invested

f rk

k=n+1





T −1



− ωVar πˆ n Ren

f

rk

.

re

k=n+1

ξ can be taken as the random return of the The term πˆ n Ren in risky investment in excess of that of the risk-free investment at time n + 1, which is called the excess return of the risky investment in this paper for convenience. If the scheme member exits the m f market at time m + 1(m = n, . . . , T − 1), the term πˆ n Ren k=n+1 rk can be taken as the total wealth obtained at time m + 1 by investing the random wealth πˆ n Ren in the risk-free asset until the time the member exits the market. By the expressions of 41ω ξT −1 and 41ω ξn , 1 4ω n

we can take 41ω ξn as the accumulated trade off between the excess return and the loss by the risky investment from time n to time T . Remark 3.1. On the one hand, if the contribution ratios cn = 0, n = 0, 1, . . . , T − 1, then the equilibrium value function (3.13) is reduced to

 Vn (wn , yn ) =

T −2 

m−n| qn+x

m =n

+

1 4ω

m 



T −1 f rk

+ T −n−1 pn+x

k=n+1



f rk

rnf wn

in the risky asset becomes 21ω Var(nRe ) ; if the member will certainly n exit the market at time T , i.e., T −n−1 pn+x = 1, then the amount invested in the risky asset becomes πˆ nc . By Proposition 3.2, the amount invested in the risky asset with mortality risk is in the middle of the amounts for the two extreme scenarios mentioned above. 4. Expectation and variance of the terminal wealth In this section, in order to find some distinct properties of our equilibrium results, we seek the explicit expressions for the expectation and variance of the terminal wealth, and then analyze the relationship between them. Since we do not know exactly the time when the member exits the market, according to the form of the objective function (2.5), we define the expectation of the terminal wealth starting from time n, denoted by ETWn , as the weighted sum of the expected wealth at each time n + 1, n + 2, . . . , T , i.e.,

k=n+1

ξn ,

(3.15)

but the equilibrium strategy does not change. On the other hand, when the mortality risk is not considered, i.e., T −n−1 pn+x = 1 for n = 0, 1, . . . , T − 1, the equilibrium value function (3.13) is reduced to

ETWn :=

T −2 

Vn (wn , yn ) = wn









Wkπˆ+1 + T −n−1 pn+x Ewn ,yn WTπˆ ,

k=n

and the variance of the terminal wealth starting from time n is defined as VTWn :=

T −2 

k−n| qn+x Varwn ,yn



Wkπˆ+1



k=n

T −1



k−n| qn+x Ewn ,yn

f rk

+ yn ϕn,T −1 − ω(yn )

  + T −n−1 pn+x Varwn ,yn WTπˆ .

2

k=n T −1

×



l−1  

µl

E (Qm )2 +



m=n

l=n

T −1 1  (rke )2

4ω k=n Var(Rek )

By (2.3), (3.5) and (3.12), we have

.

(3.16)



Ewn ,yn Wkπˆ+1



= wn

πˆ = c n

rne

1

−1 2ω T

f

rk

Var(Ren )

,

+

k=n

T −1 1  (rke )2

4ω k=n Var(Rek )

,

(3.18)

In addition, we present some properties of the equilibrium strategy with mortality risk, which is summarized in the following proposition. Proposition 3.2. πˆ n is increasing along with n, n + 1, . . . , T − 2. Specifically, 1

rne

m=n

+ yn ϕn,k +

f rm

m=n

k 

f

rl + yn ϕn,k

l=m+1 k

k e 2  ) 1  ϖm (rm

2ω m=n Var(Rem ) l=m+1

,

ETWn = Wnf +

1 2ω

ξn ,



f

rl .

into the

(4.1)

where

which is the same as the equilibrium value function without mortality risk in Wu (2013).

πˆ nc ≤ πˆ n ≤

πˆ m rme



T −1

Vn (wn , yn ) = wn



k 

In view of (3.9) and (3.11), substituting Ewn ,yn Wkπˆ+1 expression of ETWn yields

which is the same as that in Wu (2013). Furthermore, when cn = 0, n = 0, 1 . . . , T − 1 and there is no mortality risk, Vn (wn , yn ) can be further simplified as f rk

= wn

(3.17)

k=n+1



f rm +

m=n k

The equilibrium strategy (3.12) is reduced to 1

k 

m−n| qn+x

for m =

Wnf

= wn

 T −2 

k−n| qn+x

k  m=n

k=n



T −1 f rm

+T −n−1 pn+x



f rm

+ yn An . (4.2)

m=n

f

By (3.11) and (4.2), Wn can be obtained by investing the current wealth wn and the expected contributions at each period in the risk-free asset until the time she exits the market, and 21ω ξn can be taken as the accumulated trade off between the excess return and the loss by the risky investment from time n to time T . In addition, because the equilibrium strategy does not depend on the salary yn , the risky derivation 21ω ξn is not related to the parameters of salary, such as cn and ηn . f

where πˆ nc is given by (3.17).

Proposition 4.1. If wn ≥ 0, then Wn  is decreasing with f T −1 f k−n| qn+x , k = n, n + 1, . . . , T − 2, and Wn ≤ wn m=n rm + yn ϕn,T −1 , where ϕn,T −1 is given by (3.5).

Proof. See Appendix C.

Proof. See Appendix D.

2ω Var(Ren )





H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

Proposition 4.1 shows that the higher the mortality risk, the less f the Wn . This is reasonable because the higher the mortality risk, the shorter the investment time horizon, and the member derives less investment return from the risk-free asset. Alternatively, by Proposition 3.2, a higher mortality risk results in more wealth being invested in the risky asset and then a less wealth being invested in the risk-free asset. However, for the second part ξn , it is not clear whether or not ξn is decreasing along with k−n| qn+x because ϖn is increasing along with k−n| qn+x for k = n, n + 1, . . . , T − 2. In other words, a larger k−n| qn+x leads to more wealth being invested in the risky asset and a shorter time horizon, and then the effect of k−n| qn+x on ξn is ambiguous. In next section, we will show the relationship between ξn and k−n| qn+x by numerical analysis. At the same time, according to (3.13) and (4.2), the equilibrium value function Vn (wn , yn ) can be rewritten as 1 ξn . (4.3) 4ω Since Vn (wn , yn ) = ETWn − ωVTWn , by (4.1) and (4.3), we have Vn (wn , yn ) = Wnf − ωBn (yn )2 +

1 (4.4) ξn + Bn (yn )2 . 4ω2 Therefore, at time n, the relationship between the expectation and variance of the terminal wealth is summarized in Proposition 4.2.

VTWn =

1

[ETWn − Vn (wn , yn )] =

ω

Proposition 4.2. The relationship between the expectation and variance of the terminal wealth for Problem (2.5) is



f

ETWn − Wn

VTWn =

2 + Bn (yn )2 ,

ξn

(4.5)

where Bn and ξn are given by (3.10) and (3.11), respectively. Proof. See Appendix E.



By (4.5), it is clear that the scheme member cannot eliminate all risks even if she puts all of her wealth in the risk-free asset. This phenomenon is reasonable. Since the scheme member has to face the random salary process and mortality risk, in our model she cannot fully hedge all risks by investing in the financial market. Remark 4.1. (i) When the contribution ratios are zero, i.e., cn = 0, n = 0, 1, . . . , T − 1, according to (4.1) and (4.4), the expectation and variance of the terminal wealth are respectively reduced to ETWn = wn

 T −2 

k−n| qn+x

k 

+T −n−1 pn+x

m=n

k =n



T −1 f rm



f rm

+

m=n

1 2ω

ξn , (4.6)

VTWn =

1 4ω2

ξn ,

(4.7)

and the relationship between ETWn and VTWn is reduced to VTWn



ETWn − wn

 T −2 

k−n| qn+x

k=n

=

k 

T −1

f

rm + T −n−1 pn+x

m=n



f

2

rm

m=n

ξn

. (4.8)

(ii) When the mortality risk is not considered, i.e., T −n−1 pn+x = 1 for n = 0, 1, . . . , T − 1, by (4.1) and (4.4), the expectation and variance of the terminal wealth are reduced to T −1

ETWn = wn



f rm + yn ϕn,T −1 +

m=n

VTWn =

T −1 e 2 1  (rm )

4ω2

Var(Rem ) m=n

T −1 e 2 1  (rm )

,

(4.9)

E [(Qm )2 ],

(4.10)

2ω m=n Var(Rem )

+ (yn )2

T −1  l =n

µl

l −1  m=n

401

and the relationship between ETWn and VTWn is simplified as



T −1

ETWn − wn

 m=n

VTWn =

T −1

 m=n

+ (yn )2

T −1 

µl

l =n

f

rm − yn ϕn,T −1

2

e 2 ) (rm Var(Rem )

l−1 

E[(Qm )2 ].

(4.11)

m=n

5. Numerical analysis In this section, we present some numerical analysis for our derived equilibrium strategy, equilibrium value function, and the expectation and variance of the terminal wealth. For convenience but without loss of generality, we assume that for all n = f 0, 1, . . . , T − 1, the risk-free return rn ≡ 1.03, the expectation and variance of Rn are 1.09 and 0.2, respectively, the contribution ratios cn ≡ 0.08, ηn = E(Qn ) ≡ 1.05, the risk aversion ω = 0.25, the initial wealth w0 = 1 and the initial income y0 = 1. In addition, qy , y = 25, 26, . . . , 70 based on the life table of males from England and Wales in 2011 are given in Table 1. 5.1. Numerical analysis of the equilibrium strategy In this subsection, we show the effects of the mortality risk on the equilibrium strategy by numerical analysis. We conduct the analysis using three numerical examples. First, we present a numerical example for members who enter the market at the same age but with different expected time horizons. Suppose that the members enter the market aged 25 and T = 32, 33, 34, 35, respectively. Denote by πˆ n and πˆ nc the equilibrium strategies with and without mortality risk, respectively, where the expression of πˆ nc is given in (3.17). Fig. 1 shows the difference between πˆ n and πˆ nc , and reveals that: (i) πˆ n ≥ πˆ nc is always true, which is consistent with the result in Proposition 3.2; (ii) πˆ n − πˆ nc is increasing as the time horizon becomes longer, which we think is reasonable because the longer time horizon leads to more uncertainty due to the mortality risk and then results in more wealth being invested in the risky asset; and (iii) πˆ n − πˆ nc is increasing for a while and then decreases until the end of the decision making process. A possible explanation for (iii) is as follows. In our numerical analysis, the risk-free return, the expected return and variance of the risky asset are assumed to be constant over time. Consequently, the equilibrium strategy πˆ n only depends on the mortality risk and the remaining investment time periods. Furthermore, either a higher mortality risk or a shorter remaining time horizon leads to more wealth being invested in the risky asset. In the beginning, when the scheme member has a relatively long investment time horizon, with our parameter settings, the effect of the mortality risk dominates that of the remaining time horizon. Consequently, at the beginning of the investment process, although both πˆ n and πˆ nc rise over time, πˆ n rises faster than πˆ nc because the former is affected by the mortality risk while the latter is not. This is a possible reason why the distance between πˆ n and πˆ nc shows an increasing trend at the beginning. However, the latter decreasing line attributes to a much weaker effect of the mortality risk when the remaining time is running short. Specially, at the final time period, πˆ T −1 is the same as πˆ Tc−1 and the distance between them is zero. Second, we provide a numerical example for the members with different starting ages but the same expected time horizon. Suppose that there are six members with starting ages of 25, 27, 29, 31, 33 and 35, respectively, and their expected time horizon is T = 25.

402

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

Table 1 Life table of the males from England and Wales in 2011a . Age y qy

25 0.00059

26 0.00056

27 0.00056

28 0.00068

29 0.00067

30 0.00071

31 0.0076

Age y qy

32 0.00075

33 0.0008

34 0.00093

35 0.00096

36 0.00112

37 0.00125

38 0.00129

Age y qy

39 0.00132

40 0.00147

41 0.00156

42 0.00183

43 0.00176

44 0.00204

45 0.00214

Age y qy

46 0.00234

47 0.00232

48 0.0027

49 0.00283

50 0.00303

51 0.00339

52 0.0039

Age y qy

53 0.00437

54 0.00451

55 0.00507

56 0.00563

57 0.00605

58 0.00695

59 0.00704

Age y qy

60 0.00801

61 0.00853

62 0.00943

63 0.01023

64 0.01163

65 0.01165

66 0.01393

Age y qy

67 0.01494

68 0.01619

69 0.0178

70 0.02077

– –

– –

– –

a

Data from: http://www.mortality.org/.

Fig. 3. Equilibrium strategy in Wu (2013) without mortality risk. Fig. 1. Distance πˆ n − πˆ with different expected time horizon. c n

Fig. 2. Distance between the equilibrium strategies with different starting ages.

We take the strategy for the youngest member aged 25 as the benchmark and analyze the distances between the other members’ strategies and the benchmark. Fig. 2 shows the corresponding distances, and demonstrates that the older members invest more wealth in the risky asset than the younger members even if they have the same expected time horizon. We believe that this result

arises because an older member has a higher mortality risk than a younger one. However, when the mortality risk is neglected, πˆ nc does not have the same properties as πˆ n . We explain this as follows. By (3.17), the equilibrium strategy without mortality risk is only affected by the uncertainty in the financial market, which implies that no matter how old the scheme members are, as long as they are in the same financial environment, they invest the same amount in the risky asset. Specifically, suppose that there are two members with starting ages of 25 and 35, and denote by πˆ nc,1 and πˆ nc,2 their equilibrium strategies, respectively. By (3.17), we have πˆ nc,2 − πˆ nc,1 = 0 for n = 0, 1, . . . , T − 1, which shows that the results in Fig. 2 do not hold for the case without mortality risk. Finally, we use a numerical example to compare πˆ n with πˆ nc . Our study indicates that, when the mortality risk is considered, a classical property of the equilibrium strategy for the case without mortality risk no longer holds. For example, Wu (2013) considers the equilibrium strategy for a multi-period mean–variance portfolio selection problem without mortality risk, and shows that if {R0 , R1 , . . . , RT −1 } is independently identically distributed and the f risk-free return rn is constant over time, then the amount invested in the risky asset is independent of the expected time horizon T and is only dependent on the remaining time to the terminal time, i.e., the strategies are the same as long as the remaining time to the terminal time is the same. Fig. 3 shows the property of the equilibc rium strategies in Wu (2013), where πˆ 14 corresponding to T = 20, c for example, is the same as πˆ 24 corresponding to T = 30. However, for the equilibrium strategy with mortality risk, the situation changes. For example, suppose that the member enters the market aged 25 and T = 20, 30, respectively. Table 2 shows that when

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

403

Table 2 Comparison of the equilibrium strategies with mortality risk. T = 20

n = 14 0.5179

n = 15 0.5333

n = 16 0.5492

n = 17 0.5656

n = 18 0.5826

n = 19 0.6000

T = 30

n = 24 0.5182

n = 25 0.5336

n = 26 0.5494

n = 27 0.5657

n = 28 0.5827

n = 29 0.6000

πˆ n πˆ n

T = 20, the amount invested in the risky asset at time 14 is 0.5179 whereas the amount at time 24 is 0.5182 when T = 30. In the last period, the amount invested in the risky asset is also 0.6000 because the exit probability and the remaining time remain the same. The mortality risk is clearly the reason for this change. 5.2. Numerical analysis of the equilibrium value function Because the equilibrium value function consists of the expectation and variance of the terminal wealth, i.e., Vn (wn , yn ) = ETWn − wVTWn , in this subsection we first present some numerical analysis for the expectation and variance of the terminal wealth, and then for the equilibrium value function. For convenience, but without loss of generality, we only show numerical examples for the related results at time 0, i.e., ETW0 , VTW0 and V0 (w0 , y0 ) := V0 . In the previous section, we found that it is difficult to show the effect of the mortality risk on ξ0 , i.e., the second part of ETW0 , by mathematical analysis. Then it is also difficult to determine the effect of the mortality risk on ETW0 . Here, we show the effect of the mortality risk on ETW0 and ξ0 by presenting a numerical example with the mortality rate given in Table 1. Suppose that there are eleven members whose ages are between 25 and 45 when they enter the market at time 0, and that they all have the same expected time horizon T = 15. The exit probability at time T , T −1 px , in Table 3 is derived according to Table 1. Table 3 shows f f that: (i) W0 is increasing along with T −1 px . In other words, W0 is decreasing along with n| qx , n = 0, 1, . . . , T − 2, which is consistent with the result in Proposition 4.1; and (ii) 21ω ξ0 is also increasing with T −1 px in the numerical example. This, together with (i) above, results in an increasing expectation of the terminal wealth ETW0 with respect to T −1 px . Table 3 also reveals that the older members have a lower expected terminal wealth than the younger ones even if they have the same expected time horizon. This is due to the fact that the older members have a higher mortality risk than the younger ones. For the variance of the terminal wealth given by (4.4), if the growth rate of the member’s salary, Qn , is deterministic, then we have VTW0 = 4ω1 2 ξ0 . Together with Table 3, we find that the variance of the terminal wealth VTW0 is increasing with T −1 px . Because Table 3 only shows the results of VTW0 without salary risk, we further present the analysis of VTWn , n = 0, 1, . . . , T − 1 for a member aged 25 at time 0. Furthermore, we want to find the effects of the stochastic salary on VTWn . In Fig. 4, the ‘‘-+-’’ line corresponds to the case with deterministic Qn and the ‘‘-o-’’ line corresponds to the case with Var(Qn ) ≡ 0.2. From Fig. 4, we find that: (i) when the salary risk is ignored, the member suffers less risk; (ii) the longer the remaining time, the more significant the effect of the salary

Fig. 4. Effect of salary risk on VTWn . Table 4 Equilibrium value functions with different starting ages. E (Qn )2





1.1033 1.5025

(1)

(2)

(3)

(4)

(5)

(6 )

V0

V0

V0

V0

V0

V0

3.9536 1.6742

3.9513 1.6747

3.9484 1.6753

3.9449 1.6760

3.9406 1.6767

3.9355 1.6775

risk on the variance of the terminal wealth; and (iii) the shorter the remaining time, the smaller the variance of the terminal wealth in the cases with and without salary risk. Next, we present a numerical example for the members with different starting ages but the same expected time horizon to show (1) (2) some properties of the equilibrium value function. Let V0 , V0 , (3)

(4)

(5)

(6)

V0 , V0 , V0 and V0 represent the equilibrium value functions at time 0 for the scheme members with the same expected time horizon T = 15 and aged 25, 27, 29, 31, 33 and 35, respectively. Table 4 shows the effects of the mortality risk and the salary risk on the equilibrium value function, and demonstrates that: if the risk of the salary’s growth rate (measured by the corresponding variance, where the expectation of the salary’s growth rate E(Qn ) is fixed in our numerical example) is relatively small, the younger member with lower mortality risk obtains a higher equilibrium value function. However, the situation changes when the variance of the salary’s growth rate becomes large because the equilibrium value function (3.13) is inversely proportional to the variance of the salary. If the salary risk is relatively high, staying longer in the market yields more salary risk. When the loss caused by a longer investment time horizon outweighs the gain, the older members with higher exit probabilities have a higher equilibrium value function. Finally, we provide a numerical example to show the effect of the expected time horizon on the equilibrium value function for a member aged 25 at time 0. Wu (2013) shows that the equilibrium value function is increasing with respect to the expected time horizon T , where the salary risk is not considered. However, the

Table 3 Expected terminal wealth starting from time 0. T −1 px (Age)

1.000

0.988(25)

0.987(27)

0.985(29)

0.982(31)

0.979(33)

f W0

3.7043 0.2700 4.2443

3.6884 0.2686 4.2256

3.6864 0.2684 4.2232

3.6837 0.2682 4.2201

3.6805 0.2679 4.2163

3.6764 0.2676 4.2116

0.976(35)

0.973(37)

0.970(39)

0.962(41)

0.955(43)

0.946(45)

3.6718 0.2672 4.2062

3.6666 0.2667 4.2000

3.6608 0.2663 4.1934

3.6534 0.2656 4.1846

3.6447 0.2649 4.1745

3.6336 0.2639 4.1614

ξ0

ETW0 T −1 px (Age) f

W0

ξ0

ETW0

404

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

Team (2014A030312003), National Science Foundation of Guangdong Province of China (No. 2013010011959), Guangdong Natural Science for Distinguished Young Scholar and the Program for Innovation Research in Central University of Finance and Economics. We are also indebted to the anonymous referees for the criticisms and valuable comments. Appendix A. Proof of Proposition 3.1 Proof. By (2.6), we first have Jn (wn , yn ; π ) T −1 

=

m−n−1| qn+x

Ewn ,yn Wmπ − ωVarwn ,yn Wmπ











m=n+1

     + T −n−1 pn+x Ewn ,yn WTπ − ωVarwn ,yn WTπ , n = 0, 1, . . . , T − 1 Fig. 5. Effect of expected time horizon T on V0 .

and π

Jn+1 Wn+n 1 , Yn+1 ; π



equilibrium value function with salary risk given by (3.13) may be decreasing along with the expected time horizon T . Therefore, the property of V0 in Wu (2013) might not hold in our paper. Fig. 5 indicates that a longer time horizon may not always have a higher equilibrium value function. When the salary risk is relatively high, the additional gain might not compensate the loss caused by longer investment time horizon. Therefore, the equilibrium value function declines as T increases. Under this situation, a wise member should also consider a suitable expected time horizon for her investment plan. 6. Conclusions In this paper, we study a multi-period investment problem for a DC pension scheme member during the accumulation phase. The member is assumed to have a stochastic salary flow and a stochastic mortality rate, which results in our optimization problem with an uncertain time horizon. Moreover, the member can invest in a financial market with one risk-free asset and one risky asset, and her objective is to maximize the weighted sum of a linear combination of the expectation and variance of the wealth at the time of exiting the market, where the weighted coefficients are the corresponding exit probabilities. We consider our investment problem as a non-corporate game and derive the explicit expressions of the equilibrium investment strategy and equilibrium value function by backward induction. We also provide some mathematical analysis of our results and present some properties of our equilibrium strategy. In addition, we show the expectation and variance of the terminal wealth and their relationship, and present some numerical illustrations to reveal the effect of mortality risk on our results. However, our paper also has some limitations: (1) we do not consider inflation, which may have a significant effect on the investment performance of DC pension schemes; (2) the returns of the risk-free asset and the risky asset are not assumed to depend on the regimes of the financial market; and (3) the risk aversion coefficient w is assumed to be constant, although it is always not a constant and may depend on the wealth and age of the DC pension plan members. In future works, we will relax these assumptions and consider more general models.

=

T −1 



m−n−2| qn+1+x



EW πn ,Yn+1 Wmπ − ωVarW πn ,Yn+1 Wmπ n+1 n+1

m=n+2



 + T −n−2 pn+1+x EW πn

,Y n+1 n+1







WTπ − ωVarW πn ,Yn+1 WTπ n+1





,

n = 0, 1, . . . , T − 2. Then we have π

Jn (wn , yn ; π ) = pn+x Ewn ,yn Jn+1 Wn+n 1 , Yn+1 ; π + Jn (wn ,yn ; π ) − pn+x E w n ,y n × Jn+1 Wnπ+n 1 , Yn+1 ; π







   = pn+x Ewn ,yn Jn+1 Wnπ+n 1 , Yn+1 ; π   πn    + qn+x Ewn ,yn Wn+1 − ωVarwn ,yn Wnπ+n 1 T −1    π + m−n−1| qn+x Ewn ,yn Wm m=n+2   − ωVarwn ,yn Wmπ      + T −n−1 pn+x Ewn ,yn WTπ − ωVarwn ,yn WTπ  T −1   − pn+x Ewn ,yn m−n−2| qn+1+x EW πn ,Yn+1 n+1 m=n+2       × Wmπ − ωVarW πn ,Yn+1 Wmπ n+1     − pn+x Ewn ,yn T −n−2 pn+1+x EW πn ,Yn+1 WTπ n+1    , − ωVarW πn ,Yn+1 WTπ n+1

n = 0, 1, . . . , T − 2, JT −1 (wT −1 , yT −1 ; πT −1 )

 π   π  = EwT −1 ,yT −1 WT T −1 − ωVarwT −1 ,yT −1 WT T −1 .

Rearranging the terms of Jn (wn , yn ; π )(n = 0, 1, . . . , T − 2) results in π

Jn (wn , yn ; π ) = pn+x Ewn ,yn Jn+1 Wn+n 1 , Yn+1 ; π







     + qn+x Ewn ,yn Wnπ+n 1 − ωVarwn ,yn Wnπ+n 1 +

T −1 

m−n−1| qn+x

Ewn ,yn Wmπ − ωVarwn ,yn Wmπ









m=n+2

Acknowledgments This research is supported by grants from National Natural Science Foundation of China (Nos. 71231008, 11301562, 71201173, 71471045), Guangdong Natural Science for Research





− pn+x

T −1 

m−n−2| qn+1+x Ewn ,yn

m=n+2

 × EW πn

,Y n+1 n+1

Wmπ − ωVarW πn ,Yn+1 Wmπ n+1











H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

 π

 π

+ T −n−1 pn+x Ewn ,yn WT − ωVarwn ,yn WT −pn+x · T −n−2 pn+1+x Ewn ,yn      × EW πn ,Yn+1 WTπ − ωVarW πn ,Yn+1 WTπ . 





−ω

n+1

T −1 

Varwn ,yn Wmπ





m−n−1| qn+x

hn,n (wn , yn ) = wn ,



− Ewn ,yn VarW πn ,Yn+1 Wm n+1    − ω · T −n−1 pn+x Varwn ,yn WTπ    . − Ewn ,yn VarW πn ,Yn+1 WTπ

π

Jn (wn , yn ; π ) = pn+x Ewn ,yn Jn+1 Wn+n 1 , Yn+1 ; π



     + qn+x Ewn ,yn Wnπ+n 1 − ωVarwn ,yn Wnπ+n 1 T −1    π 2 −ω m−n−1| qn+x Ewn ,yn Wm m=n+2

 − Ewn ,yn EW πn

,Y n+1 n+1

+ω

T −1 

Wmπ



 m−n−1| qn+x

2 

Ewn ,yn Wmπ





,



   = Ewn ,yn hn+1,m Wnπˆ+n 1 , Yn+1 ,

n = 0, 1, . . . , T .

πn

n+1





Vn (wn , yn ) = max Jn (wn , yn ; (πn , πˆ n+1 , . . . , πˆ T −1 ))

Further using the expression of variance Var(Z ) = E(Z )2 − [E(Z )]2 yields



WTπ



Since Vn (wn , yn ) = maxπn Jn (wn , yn ; (πn , πˆ n+1 , . . . , πˆ T −1 )) = Jn (wn , yn ; (πˆ n , πˆ n+1 , . . . , πˆ T −1 )), then we have

 π



,Yn+1



n = 0, 1, . . . , T − 1, m = n + 1, . . . , T ,

m=n+2





Denote





EW πn ,Yn+1 Wmπ n+1

n = 0, 1, . . . , T − 2.

hn,m (wn , yn ) = Ewn ,yn Wmπˆ

 π Jn (wn , yn ; π ) = pn+x Ewn ,yn Jn+1 Wn+n 1 , Yn+1 ; π  πn   πn   + qn+x Ewn ,yn Wn+1 − ωVarwn ,yn Wn+1 −ω



 − ω · T −n−1 pn+x Varwn ,yn EW πn

Since pn+x · m−n−2| qn+1+x = m−n−1| qn+x and pn+x · T −n−2 pn+1+x = T −n−1 pn+x , by the tower property of conditional expectations, we can rewrite Jn (wn , yn ; π ) as



m−n−1| qn+x Varwn ,yn

m=n+2

n+1

n+1

T −1 

405

  π    pn+x Ewn ,yn Jn+1 Wn+n 1 ,Yn+1 ; πˆ       π π  + qn+x Ew ,y W n − ωVarw ,y W n    n n n n n+1 n+1    T − 1     π ˆ = max − ω m−n−1| qn+x Varwn ,yn EW πn ,Yn+1 Wm πn   n+1     m=n+2          − ω · T −n−1 pn+x Varwn ,yn EW πn ,Yn+1 WTπˆ , n+1   π    pn+x Ewn ,yn Vn+1 Wn+n 1 , Yn+1      πn    πn         + qn+x Ewn ,yn Wn+1 − ωVarwn ,yn Wn+1  T −1  = max   πn  . πn   −ω  m−n−1| qn+x Varwn ,yn hn+1,m Wn+1 , Yn+1        m=n+2   πn    − ω · T −n−1 pn+x Varwn ,yn hn+1,T Wn+1 , Yn+1 Therefore, the recursive formula (3.1) of the equilibrium value function Vn (wn , yn ) is obtained. 

2

m=n+2

− Ewn ,yn



EW πn ,Yn+1 Wmπ n+1



 2

Appendix B. Proof of Theorem 3.1 Proof. When n = T − 1, we have

  2 − ω · T −n−1 pn+x Ewn ,yn WTπ   2  − Ewn ,yn EW πn ,Yn+1 WTπ n+1    2 + ω · T −n−1 pn+x Ewn ,yn WTπ    2 − Ewn ,yn EW πn ,Yn+1 WTπ ,

VT −1 (wT −1 , yT −1 )

  π π = max EwT −1 ,yT −1 (WT T −1 ) − ωVarwT −1 ,yT −1 (WT T −1 ) πT −1     f f E rT −1 wT −1 + πT −1 ReT −1 + cT −1 rT −1 yT −1   = max  πT − 1 − ωVar rTf −1 wT −1 + πT −1 ReT −1 + cT −1 rTf −1 yT −1

n+1

n = 0, 1, . . . , T − 2. In view of the tower property of conditional expectations, Jn (wn , yn ; π ) can be further simplified as

 π Jn (wn , yn ; π ) = pn+x Ewn ,yn Jn+1 Wn+n 1 , Yn+1 ; π      + qn+x Ewn ,yn Wnπ+n 1 − ωVarwn ,yn Wnπ+n 1 

+ω

T −1 

m−n−1| qn+x

m=n+2

−ω

T −1 







π

Ewn ,yn EW πn ,Yn+1 Wm n+1

m−n−1| qn+x Ewn ,yn

m=n+2









2 π

+ ω · T −n−1 pn+x Ewn ,yn EW πn ,Yn+1 WT n+1    π 2 π − ω · T −n−1 pn+x Ewn ,yn EW n ,Yn+1 WT n+1   πn  = pn+x Ewn ,yn Jn+1 Wn+1 , Yn+1 ; π      + qn+x Ewn ,yn Wnπ+n 1 − ωVarwn ,yn Wnπ+n 1 

πT −1

The optimal solution of maxπT −1 πT −1 rTe−1 − ω(πT −1 )2 Var ReT −1 is



πˆ T −1 = 2

  π 2

EW πn ,Yn+1 Wm n+1

= rTf −1 wT −1 + cT −1 rTf −1 yT −1    + max πT −1 rTe−1 − ω(πT −1 )2 Var ReT −1 .

1

rTe−1

2ω Var ReT −1





.



(B.1)

Substituting (B.1) back into VT −1 and (3.3), respectively, yields VT −1 (wT −1 , yT −1 )

= rTf −1 wT −1 + cT −1 rTf −1 yT −1  e 2 +

1

r T −1

4ω Var ReT −1



 1

= rTf −1 wT −1 + AT −1 yT −1 + ξT −1 , 4ω   hT −1,T (wT −1 , yT −1 ) = EwT −1 ,yT −1 WTπˆ = rTf −1 wT −1 + cT −1 rTf −1 yT −1

(B.2)

406

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

+



1

rTe−1

2

2ω Var ReT −1



T −2

= rTf −1 wT −1 + ϕT −1,T −1 yT −1 +

1 2ω

Jn wn , yn ; πn , πˆ n+1 , . . . , πˆ T −1



φT −1,T −1 .





=

−ω

m−n−1| qn+x Varwn ,yn

π

hn+1,m Wn+n 1 , Yn+1





1 2ω





m=n  T −2 

T −2 

2ω





T −1



f rk 2

( ) + T −n−1 pn+x

f rk 2

( )

k=n+1

m−n| qn+x

f rk

 pn+x An+1 ηn +

T −2 

 cn rnf

T −1 

cl

l −1 

+ pn+x T −2 

+

m−n| qn+x T −1 

cl

l−1 

j =l

m 

cl

l−1 

ηi

m=n+1  T −2  m=n

l =n

m−n| qn+x

m=n

cl

l −1 

ηi

m 

f

rj

j=l

i=n



T −1 f rk

+ T −n−1 pn+x



f rk

cn rnf

k=n+1

T −1

ηi

i=n



f

rj

j =l

m l −1  

cl

l =n

cn rnf

j =l

m 

cl

 f rk

f rj

k=n+1

= T −n−1 pn+x

 k=n+1



ηi

m 

T −1 l −1  

f

rj

j=l

i=n

+ T −n−1 pn+x

l=n+1

m−n| qn+x

m 

T −1

i=n

m−n| qn+x



+

f

rj

k=n+1

T −2 

T −2 



T −1 f rk

ηi

i =n

m 

f

rj .

(B.5)

j =l

Comparing (B.5) with (3.9), the coefficient of yn in (B.5) is

 pn+x An+1 ηn +

T −2  m=n

= An .

m−n| qn+x



T −1

m 

f rk



+ T −n−1 pn+x

k=n+1

f rk

− ω · T −n−1 pn+x Var π

 k=n+1

f rk

k=n+1

pn+x E (Qn )2 Bn+1 + µn = pn+x E (Qn )2



+ ϕn+1,T −1 Qn yn .

cn rnf

Moreover, referring to (3.10), the coefficient of (yn )2 in (B.4) can be written as







T −1

cn rnf

T −1

ηi

l=n+1 m 

f rk

T −1  

µl

l =n +1 e n Rn

(B.4)

k=n+1

i=n

m−n−1| qn+1+x



+ T −n−1 pn+x

k=n+1

l =n +1 T −2 

f rk

m=n+2



yn



T −1

m 

m−n| qn+x

= pn+x · T −n−2 pn+1+x

+

  ϕn+1,m−1 Qn yn

f rk

k=n+1

m=n

l=n+1

k=n+1



+ T −n−1 pn+x

k=n+1

= T −n−1 pn+x

m−n−1| qn+1+x     m=n+1    m T −1    f f  ×  + p r r T −n−2 n+1+x = pn+x   k k   k=n+2 k=n+2      × rnf +1 wn rnf + πn rne + cn rnf yn    1 ξn+1 + An+1 ηn yn − ωBn+1 E (Qn )2 (yn )2 + 4ω    + qn+x wn rnf + πn rne + cn rnf yn − ω(πn )2 Var Ren   T −1 m −1   f e −ω rk m−n−1| qn+x Var πn Rn



T −1

m 

m=n



m−n−1| qn+x Var

k=n+1

k=n+1

f

rk

T −2 

T −1 

rne πn

By (3.5) and (3.9), the coefficient of yn in (B.4) can be written as

T −1

m=n+2

f rk

    1 − ω pn+x E (Qn )2 Bn+1 + µn (yn )2 + pn+x ξn+1 . 4ω

+

Substituting (2.1) and (2.2) into the above formula and simplifying it results in





+ T −n−1 pn+x

m=n+1

φn+1,T −1 + ϕn+1,T −1 Yn+1 .

Jn wn , yn ; πn , πˆ n+1 , . . . , πˆ T −1



T −1 f rk

m 

m−n| qn+x

rnf wn

k=n+1

k=n+1

m=n



k=n+1

−ω

+



1

+ T −n−1 pn+x

m 

m−n| qn+x

 f rk

  × Var Ren (πn )2 + pn+x An+1 ηn

φn+1,m−1 + ϕn+1,m−1 Yn+1 





k=n+1

− ω · T −n−1 pn+x Varwn ,yn Wnπ+n 1



+

f rk

k=n+1

T −2 





+



m=n

   − ω · T −n−1 pn+x Varwn ,yn hn+1,T Wnπ+n 1 , Yn+1   T −2  m − n − 1|qn+1+x     m=n+1    m T −1    f f f πˆ n  = pn+x Ewn ,yn  rk + T −n−2 pn+1+x rk rn+1 Wn+1    k=n+2  k=n+2   1 2 ξn+1 + An+1 Yn+1 − ωBn+1 (Yn+1 ) +      4ω + qn+x Ewn ,yn Wnπ+n 1 − ωVarwn ,yn Wnπ+n 1  T −1 m −1   f πn −ω rk m−n−1| qn+x Varwn ,yn Wn+1 m=n+2

m−n| qn+x

 T −1

m 

m=n

−ω

m=n+2

+

T −2 



   = pn+x Ewn ,yn Vn+1 Wnπ+n 1 , Yn+1      + qn+x Ewn ,yn Wnπ+n 1 − ωVarwn ,yn Wnπ+n 1 T −1 





(B.3)

Since BT −1 = µT −1 = 0, (B.1)–(B.3) indicate that (3.12)–(3.14) hold for n = T − 1. Now we assume that (3.12)–(3.14) hold for T − 1, . . . , n + 1. Then for n, by (3.1) and (3.13), we have Jn wn , yn ; πn , πˆ n+1 , . . . , πˆ T −1

T −2

Since pn+x m=n+1 m−n−1| qn+1+x = m=n+1 m−n| qn+x and pn+x · = T −n−1 pn+x , referring to (3.6), we can rewrite T −n−2 pn+1+x  Jn wn , yn ; πn , πˆ n+1 , . . . , πˆ T −1 as



×

l −1  m=n+1

pm+x E (Qm )2 + µn





H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

=

T −1 

l −1 

µl

 pm+x E (Qm )2







m=n

l =n

= Bn .

m−1



= wn

πn

1

+

πˆ n

=

1

m 

m−n| qn+x

m=n

2ω 

T −1

f

k=n+1

T −2

m m−n| qn+x

m= n

T −1

rne

( ) + T −n−1 pn+x

f rk 2

( )



k=n+1

Var

 .

Vn (wn , yn ) =

T −2 

Ren

m−n| qn+x

m=n



+ T −n−1 pn+x

k=n+1

f rk

rnf wn

Now the last step is to prove that the constant term of Vn (wn , yn ) can be written as 41ω ξn . By (3.8) and (3.11), we have 1 4ω

pn+x ξn+1 +

× =

=

=

4ω



m−n| qn+x

m=n



f

rk + T −n−1 pn+x

k=n+1

1

f

rk

4ω

1 4ω

φn,m−1 .

 T −2

rne 1 Λn,2 , 2ω Λn,1 Var(Ren )

πˆ n =

(C.1)

where T −2 

Λn,1 =



T −2 

Λn,2 =

m 

m−n| qn+x

m=n

2ω

 ϖ( )  ( )

( ) −

k=n+1



m 

m−n| qn+x



T −1 f rk 2



f rk 2

( )

T −1

k=n+1



f

rk −

k=n+1

f

rk

m−n| qn+x



T −1

m 

f rk



+ T −n−1 pn+x

k=n+1

f rk

k=n+1

ϖn (rne )2 Var(Ren )

ϖ( )  ( )

m−n| qn+x

T −1



+

k=n+1

f

rk .

k=n+1 f k=n+1 rk

m

∂ πˆ n

∂m−n|qn+x    r | − r f |T −1 Λn,1 − (r f )2 |m − (r f )2 |T −1 Λn,2 =  2 Λn,1   r f |T −1 − r f |m  f T −1 = r | + r f |m Λn,2 − Λn,1 .  2 Λn,1   Next, we simplify the term r f |T −1 + r f |m Λn,2 − Λn,1 . By (C.1), rne

f m

f

k

Var(Rek ) l=k+1

rl

r f |T −1 + r f |m Λn,2 − Λn,1



ξn .

(B.7)



=

T −2 

l−n| qn+x



r f |T −1 + r f |m

r f |l − r f |T −1





l =n

For hn,m (wn , yn ), m = n + 1, . . . , T , by (3.5), (3.8), (3.12) and (3.14), we have

−

T −2 

l−n| qn+x

 f 2l  (r ) | − (r f )2 |T −1

l =n





πˆ

hn,m (wn , yn ) = Ewn ,yn hn+1,m (Wn+n 1 , Yn+1 )

 = E w n ,y n

πˆ Wn+n 1

 k=n+1

+

1 2ω



m−1

φn+1,m−1

f rk

+

m−1



ck

k=n+1

k−1  i=n+1

  + r f |m + r f |T −1 r f |T −1 − (r f )2 |T −1 

m−1

ηi



f rj

Yn+1

= r f |m · r f |T −1 +

T −2 

l−n| qn+x

r f |T −1 · r f |l



l =n

j =k

+ r f |m · r f |l − r f |m · r f |T −1 − (r f )2 |l   = T −n−1 pn+x r f |m · r f |T −1



−1   m f rk = wn rnf + cn rnf yn + πˆ n rne k=n+1

and

f 2 k=n+1 rk ,

we have

m m  ϖk (r e )2  k=n

(rkf )2 ,

k=n+1



T −1



+

m

Var(Ren )



e 2 T −1 k rk f r T −n−1 pn+x Var Rek l=k+1 l k=n

4ω m=n

2ω

= ( ) then differentiating πˆ n with respect to m−n| qn+x (m = n, n + 1, . . . , T − 2), we have

m

m=n

T −2 1 

1

f

rk

Proof. Substituting T −n−1 pn+x = 1 − m=n m−n| qn+x into the equilibrium strategy (3.12) and rearranging the terms yield

(r ) |

T −1

1

f

2ω Var(Ren ) k=n+1

rk + ϕn,m−1 yn +

m=n

T −1 −1  ϖk (rke )2 T f r Var(Rek ) l=k+1 l k=n+1



T −2 

m−1 1 ϖn (rne )2 

Appendix C. Proof of Proposition 3.2

f 2 m

T −2



yn

j =k

Now it is clear that (3.12)–(3.14) also hold for n. By mathematical induction, we complete the proof of Theorem 3.1. 

k=n+1

m e 2 k rk f pn+x r m−n−1| qn+1+x Var Rek l=k+1 l m=n+1 k=n+1

1

ηi

i=n

f rj

Furthermore, for convenience, denote r f |m =

pn+x · T −n−2 pn+1+x

4ω

4ω

+





T −1

m

ϖ ( ) ( )

4ω

+

T −2

e 2 n rn Var Ren

1

+

1

ck

k=n

φn+1,m−1



m−1



1 2ω

k=n+1

1

pn+x ξn+1 + An yn − ωBn (yn )2 + 4ω   T −1 T − 2 m   f 1 f rk rk + T −n−1 pn+x + m−n| qn+x 4ω m=n k=n+1 k=n+1 ϖn (rne )2 × . Var(Ren )



+

ηn yn +



T −1 f rk

k−1 



k=n

k=n+1

m 



m −1

f rk

f

rj

j=k

φn+1,m−1 +



= wn

(B.6)

Substituting (B.6) into (B.4) and arranging the terms results in





m−1

rk

k=n+1

f rk 2



f



rk + T −n−1 pn+x

2ω

m−1

ηi

i=n+1

k=n

obviously exists and is



k−1 

ck

k=n+1

   The optimal solution of max Jn wn , yn ; πn , πˆ n+1 , . . . , πˆ T −1

T −2

m−1

+

407



+

T −2  l =n

l−n| qn+x





r f |T −1 · r f |l − (r f )2 |l + r f |m · r f |l ≥ 0.



408

H. Wu, Y. Zeng / Insurance: Mathematics and Economics 64 (2015) 396–408

Therefore, πˆ n is increasing along with m−n| qn+x for m = n, n + 1, . . . , T − 2. For the inequalities in Proposition 3.2, since T −2

 m=n

1≥

T −1

f

m 

m−n| qn+x

m =n



r k + T −n −1 p n +x

k=n+1

T −2



m 

m−n| qn+x

f

rk

k=n+1

f rk 2

( ) + T −n −1 p n +x

k=n+1

T −1

 k=n+1

≥ f rk 2

( )

1 T −1



, f rk

k=n+1

together with equilibrium strategy πˆ n given by (3.12), we find that the inequalities in Proposition 3.2 hold.  Appendix D. Proof of Proposition 4.1 Proof. We only need to prove that T −2 

k−n| qn+x

k 

T −1

m=n

k=n



f rm +T − n − 1 p n + x

f rm

m =n

is decreasing along with k−n| qn+x (k = n, n + 1, . . . , T − 2), and Proposition 4.1 can be proved in a similar way. Because T −2 

k−n| qn+x

T −1 f rm + T −n−1 pn+x

m=n

k=n

=

k 

T −2 



f rm

m=n

 k−n| qn+x

k  m=n

k =n



T −1 f rm

−



f rm

T −1

+

m=n



f rm ,

m=n

it is obviously decreasing along with k−n| qn+x (k = n, n + 1, . . . , T − f

2). Specially, when k−n| qn+x = 0(k = n, n + 1, . . . , T − 2), Wn arrives at its maximum value, i.e., the inequality holds.  Appendix E. Proof of Proposition 4.2 Proof. According to (4.1), we have 1 2ω

f

=

ETWn − Wn

ξn

.

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