Insurance: Mathematics and Economics 70 (2016) 237–244
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Insurance: Mathematics and Economics
journal homepage: www.elsevier.com/locate/ime
A stochastic Nash equilibrium portfolio game between two DC
pension funds
Guohui Guan, Zongxia Liang ∗
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
highlights
• Study stochastic Nash equilibrium portfolio game of two DC pension funds.
• Derive closed-forms of the Nash equilibrium portfolio strategies.
• Give numerical analysis to investigate evolutions of the Nash equilibrium strategies.
article
info
Article history:
Received July 2015
Received in revised form
June 2016
Accepted 25 June 2016
Available online 1 July 2016
JEL classification:
C73
C61
G11
abstract
In this paper, we study the stochastic Nash equilibrium portfolio game between two pension funds under
inflation risks. The financial market consists of cash, bond and two stocks. It is assumed that the price
index is derived through a generalized Fisher equation while the bond is related to the price index to
hedge the risk of inflation. Besides, these two pension managers can invest in their familiar stocks. The
goal of the pension managers is to maximize the utility of the weighted terminal wealth and relative
wealth. Dynamic programming method is employed to derive the Nash equilibrium strategies. In the end,
a numerical analysis is presented to reveal the economic behaviors of the two DC pension funds.
© 2016 Elsevier B.V. All rights reserved.
MSC:
91A15
91A30
91B51
91G10
Submission classifications:
IB13
IB81
IE11
Keywords:
Defined contribution pension plan
Stochastic portfolio game
Nash equilibrium
Inflation risk
Dynamic programming method
1. Introduction
Pension fund management has attracted more and more attention and becomes a popular subject in recent years. Pension fund
∗
Corresponding author.
E-mail addresses: [email protected] (G. Guan), [email protected]
(Z. Liang).
http://dx.doi.org/10.1016/j.insmatheco.2016.06.015
0167-6687/© 2016 Elsevier B.V. All rights reserved.
can be viewed as a saving vehicle before retirement to ensure retirement income and thus has great importance. Therefore, an efficient pension fund is very important for the individual and society,
and there are many works studying the management of pension
fund. Usually, there are mainly two kinds of pension fund, classified by how the contribution and benefit are set in the plan. The
first one is defined contribution (DC) pension plan, which involves
a fixed contribution rate before retirement. However, the benefit is
not fixed for DC pension plan but is determined by the investment
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G. Guan, Z. Liang / Insurance: Mathematics and Economics 70 (2016) 237–244
performance of the pension plan. The second one is the defined
benefit (DB) pension plan. In the DB pension plan, the benefit is
fixed while the contribution should be adjusted continuously to
keep balance. Thus, the sponsors will undertake the risks in the DB
pension plan. Recently, in contrast with the DB pension plan, the
DC pension plan has attracted more attention and develops more
and more fast.
In the DC pension plan, the portfolio of the plan before retirement is very important to ensure the benefit after retirement.
There are many works investigating the optimal portfolios before retirement to maximize the expectation of utility of terminal
wealth. Vigna and Haberman (2001) firstly studied the optimal DC
pension plan in a discrete model. In their further work (cf. Haberman and Vigna, 2002), they also paid attention to the risk measures for the DC plan. Boulier et al. (2001) established an efficient
continuous financial model for the DC plan with stochastic interest
rate. They successfully derived the optimal investment strategies
in a portfolio insurance problem for the pension manager. Later,
Deelstra et al. (2003) considered a more general financial model
and solved the optimization problem by introducing auxiliary processes and martingale method.
Since the accumulation phase of a DC pension plan is often
very long, about 20–40 years, the risks in the market will influence
the pension fund heavily, mainly the risks of inflation and interest
rate. Zhang et al. (2007) firstly analyzed the economic behavior of
a DC pension manager under inflation. After that, Han and Hung
(2012) obtained the optimal allocations of a pension plan with
CRRA utility preference under inflation and interest rate risks.
Yao et al. (2013) initially solved the mean–variance problem for
a pension manager with inflation risk. Dynamic programming
method was applied to obtain the efficient frontier in their work.
Wu et al. (2015) derived the closed form solutions for the pension
manager under inflation risk in a time-inconsistent mean–variance
framework. Moreover, the risks of mortality and contribution for a
pension manager were introduced in Yao et al. (2014).
Besides the risks of the market, some papers also consider the
risks of stock for a pension plan. The stock price in the previous
papers follows a geometric Brownian motion, which does not
characterize the features of stock well. Gao (2009) characterized
the stock price by a constant elasticity of variance (CEV) model.
The Legendre transform and dynamic programming method were
combined in his work to derive the optimal strategies both before
and after retirement. Guan and Liang (2014) established a financial
model for DC pension manager under interest rate and volatility
risks. The optimization goal in the paper was to maximize the CRRA
utility of terminal wealth over an annuity guarantee. Besides, in
more work (cf. Guan and Liang, 2015), a thorough research was
conducted for the DC pension plan under stochastic interest rate
and mean-reverting returns.
In most papers, a pension manager is only concerned with the
maximization of the expectation of the utility of terminal wealth.
However, in the real market, there exists competition between
different pension managers. The pension manager is concerned
about relative performance and considers the terminal wealth
and relative wealth at the same time. Therefore, if the pension
manager intends to behave better than the other manager to
attract more attention, it is more realistic to take into account
the other manager’s economic behavior. Some existing works
are concerned with the competition between different managers.
The goal of the manager is to maximize the expectation of the
utility of weighted terminal wealth and relative wealth. However,
there may not exist optimal investment strategies achieving the
managers’ goal at the same time, hence they often search the Nash
equilibrium strategies for different competitors. Browne (2000)
firstly solved the problem of portfolio games for two investors
and derived the equilibrium strategies for some specified games.
Meanwhile, Bensoussan and Frehse (2000) studied the regularity
condition for the existence of Nash strategies in a stochastic games
among N players by dynamic programming method. Basak and
Makarov (2014) investigated the case of competition between two
investors. In their work, the investor was cared about the relative
wealth when it was above a level. They beautifully employed the
martingale method to derive the Nash equilibrium strategies. Later,
they (cf. Basak and Makarov, 2013) explored the relation between
competition and asset specialization. The competition between
two insurance companies was studied in Bensoussan et al. (2014).
In their work, proportional reinsurance could be purchased and
one insurance company intended to maximize the utility of the
difference between the insurance company and the other one.
They also applied the dynamic programming method to derive the
Nash equilibrium strategies. Later, Meng et al. (2015) extended the
model to the case when the surplus process of the insurer was
characterized by a nonlinear (quadratic) risk control process.
In this paper, we consider the competition between two DC
pension managers. One pension manager will try to have a better
performance than the other manager to attract more attention.
In the market, since the time of a DC plan is often long, we take
into account the influence of inflation risk. The financial market
contains cash, bond and two stocks. The bond is related to the
price index and can help hedge the risk of inflation. However, since
the two managers are willing to invest in their familiar stocks,
the stocks the two managers invest in are not the same. The goal
of the pension manager is to maximize the utility of his terminal
wealth and the relative wealth w.r.t. the other pension manager. So
we need to solve two different optimization problems. However,
since there hardly exist optimal investment strategies for these
two problems at the same time, we search the Nash equilibrium
strategies by dynamic programming method, i.e., each manager is
assumed to know the equilibrium strategy of the other manager,
and no one will change his own strategy. In the end of the paper,
we present the numerical analysis to show the evolutions of the
Nash strategies and wealths.
The rest of this paper is organized as follows: The financial
market and the structure of the pension fund are presented in
Section 2. In Section 3, we study the competition between two
pension managers and derive the Nash equilibrium strategies.
Section 4 shows the Nash equilibrium strategies and evolution of
the wealths. Section 5 is a conclusion.
2. The financial market and the pension management
In this section, let (Ω , F , {Ft }t ∈[0,T ] , P) be a filtered complete
probability space. Ft represents the information of the market
available before time t. Besides, [0, T ] is a fixed time horizon
and the pension managers can adjust their investment strategies
continuously within [0, T ]. In what follows, we assume that all the
processes are well-defined and adapted to {Ft , t ∈ [0, T ]}.
2.1. The financial market
In this paper, we consider the inflation risk for the pension
funds, which can help hedge the risk of inflation in the long run
of a pension fund. The risks of inflation and the financial market
are presented in this section. In fact, there exist many treasury
inflation-protected securities in the market to hedge the risk of
inflation and we introduce a particular asset named inflationindexed zero coupon bond in our market. Thus, the financial
market in our work consists of cash, treasury inflation-protected
securities and two stocks. The price of the risk-free (i.e., cash) asset
S0 (t ) is the following:
dS0 (t )
S0 (t )
= rn (t )dt ,
S0 (0) = S0 ,
(2.1)
G. Guan, Z. Liang / Insurance: Mathematics and Economics 70 (2016) 237–244
where S0 > 0 is the initial price of cash and rn (t ) denotes the
nominal interest rate in the financial market.
In order to characterize the inflation risk, we study the
generalized Fisher equation here. The Fisher equation stems from
Fisher (1930) and can well describe the relationship between the
nominal interest rate rn (t ), the real interest rate rr (t ) and the price
index I (t ). The price index I (t ) here reflects a reduction in the
purchasing power per unit of money. The original Fisher equation
is a discrete time model and the continuous-time model presented
in Kwak and Lim (2014) is:
dI (t )
I (t )
= (rn (t ) − rr (t ))dt + σI dWI (t ),
where BI (t ) is a standard Brownian motion under a risk-neutral
measure m, and the risk of price index is characterized by BI (t ).
Similar to Guan and Liang (2014), we present the following
extended continuous-time Fisher equation given by (cf. Zhang
et al., 2007):

1



E[i(t , t + 1t )|Ft ],
rn (t ) − rr (t ) = lim
1t →0 1t

i(t , t + 1t ) = I (t + 1t ) − I (t ) ,

I (t )
(2.2)
where 
E is the expectation under risk neutral measure 
P and i(t , t +
1t ) is the inflation rate from time t to t + 1t.
Denote the market price of risk of WI (t ) by λI . Then, by Girsanov
theorem we can obtain the model of the stochastic price index I (t )
w.r.t. original measure P by
dI (t )
I (t )
= (rn (t ) − rr (t ))dt + σI [λI dt + dWI (t )], I (0) = I0 .
(2.3)
Besides, an inflation-indexed zero coupon bond is introduced to
hedge the risk of the inflation. An inflation-indexed zero coupon
bond P (t , T ) is a contract at time t with final payment of real
money $1 at maturity T . Different from the general zero-coupon
bond, P (t , T ) delivers I (T ) at maturity T . Therefore, based on the
pricing formula of derivatives, the price of P (t , T ) is P (t , T ) =
T



E exp(− t rn (s)ds)I (T )|Ft . Since the nominal interest rate in our
model is deterministic, a simple calculation can show that the
explicit form of P (t , T ) is
 
P (t , T ) = I (t ) exp −
T

rn (s)ds .
t
Moreover, P (t , T ) also satisfies the following backward stochastic
differential equation:

 dP (t , T ) = r (t )dt + σ [λ dt + dW (t )],
n
I
I
I
P (t , T )

P (T , T ) = I (T ).
(2.4)
Assume that there are two pension fund managers in the financial
market, indexed by i = 1, 2. Each pension manager can invest in
the cash and inflation-indexed zero coupon bond. However, the
pension manager i can also allocate money in the third assets: two
stocks. The price Si (t ) of the ith stock is as follows:



dSi (t )


= rn (t )dt + σSi1 λI dt + dWI (t )

Si (t )


+ σSi2 λSi dt + dWSi (t ) ,



Si (t ) = Si ,
239
correlation coefficient between WS1 (t ) and WS2 (t ) is ρ ∈ [−1, 1].
We can see that the stocks the two managers can invest are not the
same. This is quite natural: a manager is more willing to invest in
his familiar asset. Thus, S1 (t ) and S2 (t ) are not the same and they
are related to the correlation coefficient ρ ∈ [−1, 1].
2.2. The pension management
In this subsection, we consider the continuous contribution
in the pension fund. The defined contribution pension fund can
be viewed as a saving vehicle for retirement. Before retirement,
the contributor contributes a continuous wealth into the fund.
This increases the wealth of the pension fund. The pension fund
managers need to manage the wealth of contribution well in the
financial market. Since the risk of inflation exists in the financial
market, we assume that the contribution rate of the pension fund
i at time t increases with the price index, i.e., ci I (t ), where ci >
0, i = 1, 2.
Apart from the contribution rate, the pension managers also
participate in the financial market continuously. The pension
manager i can invest in S0 (t ), P (t , T ) and Si (t ). Assume that for the
pension manager i, i = 1, 2, the proportions of money invested in
the cash, inflation-indexed zero coupon bond and stock at time t
are denoted by u0i (t ), uPi (t ) and uSi (t ), respectively. Besides, there
are no transaction costs or taxes in the market, and short buying
is also allowed. Then, the wealth of the pension manager i with
investment behavior is as follows:

dS0 (t )


dX (t ) = ci I (t )dt + u0i (t )Xi (t )

 i
S0 ( t )
dSi (t )
dP (t , T )
+ uSi (t )Xi (t )
,
+ uPi (t )Xi (t )



P
(
t
,
T
)
Si (t )

Xi (0) = Xi .
(2.6)
Xi ≥ 0 represents the initial wealth of pension manager i. Substituting (2.1), (2.4), (2.5) into Eq. (2.6), we can obtain the following
compact form of the wealth for pension manager i:

dXi (t ) = ci I (t )dt + rn (t )Xi (t )dt




+ uPi (t )Xi (t )σI [λI dt + dWI (t )]
+ uSi (t )Xi (t )σSi1 [λI dt + dWI (t )]


+ uSi (t )Xi (t )σSi2 [λSi dt + dWSi (t )],


Xi (0) = Xi .
(2.7)
In Eq. (2.7), the relation 1 = u0i (t ) + uPi (t ) + uSi (t ) is applied. Denote ui (t ) = (uPi (t ), uSi (t ))T . ui (t ) represents investment strategies. We call ui (t ) an admissible strategy if it satisfies the following
conditions:
(i) uPi (t ) and uSi (t ) are progressively measurable w.r.t. (Ω , F ,
{Ft }t ∈[0,T ] , P).
T
(ii) E{ 0 [uPi (t )2 σI2 + uSi (t )2 σS2i1 + uSi (t )2 σS2i2 ]dt } < +∞.
(iii) Eq. (2.7) has a unique strong solution for the initial data
(t0 , I (0), Xi ) ∈ [0, T ] × (0, +∞) × (0, +∞).
Denote the set of all admissible investment strategies ui (t )
by Πi . We search the optimal investment strategies within the
admissible strategies.
3. The competition
3.1. The competition between pension managers
(2.5)
where σSi1 and σSi2 are positive constants and represent the
volatilities of the stocks. WSi (t ) is a standard Brownian motion
on (Ω , F , {Ft }t ∈[0,T ] , P) and independent of WI (t ). Moreover,
λSi represents the market price of risk of WSi (t ). Besides, the
In the financial market, the two pension managers intend to
maximize the utility of the weighted terminal wealth and related
wealth to attract more attention. Denote the real wealth of the
X (T )
pension fund i by Yi (T ) = Ii(T ) . Since competition exists, we
assume that the pension managers are concerned with their real
wealths as well as the ratio between their wealth and the wealth of
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G. Guan, Z. Liang / Insurance: Mathematics and Economics 70 (2016) 237–244
the other manager. The optimization goal of the pension manager
i is as follows:
max E[Ui Yi (T )(1−θi ) Ri (T )θi ]



(Xi (t ), ui (t )) satisfy Eq. (2.7)
subject to ui (t ) ∈ Πi ,
Xi (T ) ≥ 0,
Y (T )
(3.1)
Y (T )
where R1 (T ) = Y1 (T ) and R2 (T ) = Y2 (T ) represent the real relative
2
1
wealths of the manager 1 and 2, respectively. θi ∈ [0, 1] is the
weight of the manager i’s preference over the competition. If θi
is large, the manager i becomes more competitive and is more
concerned with the relative wealth towards the other pension
manager. When θi = 0, the optimization problem is the classical
optimization problem without competition.
We assume that the two managers have different preferences
over their wealth and relative wealth. The utility function Ui (·) of
the pension manager i is defined by
Ui (x) =
x1−γi
1 − γi
, γi > 0 and γ ̸= 1.
where

σI
0
,
Σi =
σSi1 σSi2


WI (t )
.
Wi (t ) =
WSi (t )

Dynamic programming method is applied to solve the competition
problem. For pension manager 1, set
V (t , y1 , y2 )


 
= max E U1 Y1 (T )Y2 (T )−θ1 |Y1 (t ) = y1 , Y2 (t ) = y2 .

u1 (·)
V (t , y1 , y2 ) represents the optimal expectation of utility of pension
manager 1 given the states of financial market at time t and the investment strategy 
u2 (t ) of pension manager 2 within [0, T ].
Using the standard stochastic dynamic programming method,
we can obtain the associated HJB equation for pension manager 1.
The following result presents the HJB equation.
The associated HJB equation for pension manager 1 is as
follows:
uT1 Σ1 Λ1 ]
sup Vt + Vy1 [c1 + rr (t )y1 + y1

Ui (x) is the standard CRRA utility function.

u1 (·)∈
1
1
3.2. The Nash equilibrium strategies
+ Vy1 y1 y21
uT1 Σ1 Σ1T
u1 + Vy2 [c2 + rr (t )y2 + y2
uT2 Σ2 Λ2 ]
As is stated above, the two pension managers need to achieve
their own optimization goal. However, the goal in their optimization problems involves the wealth of the other pension manager.
The two pension managers have different optimization goals and
the optimization problem of one manager is very closely related
to the other one’s strategy. Therefore, there do not exist optimal strategies satisfying these two pension managers’ goals at the
same time and we introduce the notion of Nash equilibrium strategies here. Nash equilibrium states that if the two pension managers adopt some strategies, one’s utility will not be improved if
the other one keeps his strategy. Then the two pension managers’
strategies constitute Nash equilibrium. The mathematical explanation of Nash equilibrium is as follows:
Definition 3.1. The pair (u∗1 (t ), u∗2 (t )), t ∈ [0, T ] is called Nash
equilibrium strategy if manager 1’s optimal strategy is u∗1 (t ) after
manager 2 adopts the strategy u∗2 (t ) and vice versa, i.e., u∗1 (t ) and
u∗2 (t ) solve the following problems, respectively:
2
1
uT2 Σ2 Σ2T
u2
+ Vy2 y2 y22
2



1 0
+ Vy1 y2 y1 y2
uT1 Σ1
Σ2T
u2 = 0.
0 ρ
Proposition 3.1. The optimal feedback strategy 
u∗1 (t ) of pension
manager 1 when pension manager 2 adopts strategy 
u2 (t ) is:

u∗1 (t ) =
1 Y1∗ (t ) + D1 (t )
γ1
Y1∗ (t )
(Σ1−1 )T Λ1
θ1 (1 − γ1 ) Y1∗ (t ) + D1 (t )
Y2 (t )
(Σ −1 )T
γ1
Y1∗ (t )
Y2 (t ) + D2 (t ) 1


1 0
×
Σ2T
u2 (t ).
0 ρ
−

max E[U2 Y2 (T )Y1∗ (T )−θ2 ],

(3.3)
The derivation for the HJB equation for manager 2 is similar. We
only need to exchange subscripts 1 and 2 in Eq. (3.3).
From Eq. (3.3), the optimal investment strategy of pension
manager 1 given the investment behavior of pension manager 2
can be obtained. We have the following proposition.
max E[U1 Y1 (T )Y2∗ (T )−θ1 ],


λI − σI
,
Λi =
λSi


(3.4)
where Y1∗ (T ) and Y2∗ (T ) are the real wealths of the pension
manager 1 and 2 w.r.t. strategy u∗1 (t ) and u∗2 (t ), respectively.
Similarly, the optimal feedback strategy 
u∗2 (t ) of pension manager 2 when pension manager 1 adopts strategy 
u1 (t ) is:
In order to derive the Nash equilibrium strategy, we need to
obtain the compact form of the differential of Yi (t ). Applying Ito’s
formula to Eqs. (2.3) and (2.7), the differential of Yi (t ) is as follows:

u∗2 (t ) =
dYi (t ) = ci dt + rr (t )Yi (t )dt + [uPi (t ) − 1]Yi (t )σI
× [(λI − σI )dt + dWI (t )]
+ uSi (t )Yi (t )σSi1 [(λI − σI )dt + dWI (t )]
+ uSi (t )Yi (t )σSi2 [λSi dt + dWSi (t )].
Let
γ2
Y2∗ (t )

i.
Rewrite the differ-
dYi (t ) = ci dt + rr (t )Yi (t )dt + Yi (t )
ui (t )T Σi Λi dt + dWi (t ) , (3.2)


(Σ2−1 )T Λ2
θ2 (1 − γ2 ) Y2∗ (t ) + D2 (t )
Y1 (t )
(Σ −1 )T
∗
γ2
Y2 (t )
Y1 (t ) + D1 (t ) 2


1 0
×
Σ1T
u1 (t )
0 ρ
T
s
where Di (t ) = ci t exp(− t rr (u)du)ds, i = 1, 2.
−
Proof. See Appendix.

ui (t ) = ui (t ) − (1, 0)T .
We call 
ui (·) a admissible strategy if ui (·) ∈
ential of Yi (t ) in a more compact form:
1 Y2∗ (t ) + D2 (t )
(3.5)
We can see from Proposition 3.1 that one’s optimal strategy
given the other one’s strategy is composed of two parts. The first
part is the case where θi = 0, i.e., one pension manager’s optimization goal is not related to the other one’s behavior. The optimization problem is a pure problem maximizing the expectation of
G. Guan, Z. Liang / Insurance: Mathematics and Economics 70 (2016) 237–244
241
terminal utility. Thus, in the case θi = 0, Proposition 3.1 also
presents the optimal strategy of a pension manager with CRRA utility preference under inflation risk. The second part in Eqs. (3.4) and
(3.5) is related to the other pension manager’s wealth and strategies. Thus, the second part shows the effect of competition on the
investment strategies.
By Proposition 3.1, we can derive the Nash equilibrium
strategies for the pension managers 1 and 2.
Proposition 3.2. The Nash equilibrium pair (
u∗1 (t ),
u∗2 (t )) for these
two pension managers is as follows:

u∗1 (t ) =
1 Y1∗ (t ) + D1 (t )
γ1
Y1∗ (t )
Ξ1−1 (Σ1−1 )T Λ1
θ1 (1 − γ1 ) Y1∗ (t ) + D1 (t ) −1 −1 T
Ξ1 (Σ1 ) ΦΛ2 ,
γ1 γ2
Y1∗ (t )
1 Y2∗ (t ) + D2 (t ) −1 −1 T

u∗2 (t ) =
Ξ2 (Σ2 ) Λ2
γ2
Y2∗ (t )
θ2 (1 − γ2 ) Y2∗ (t ) + D2 (t ) −1 −1 T
−
Ξ2 (Σ2 ) ΦΛ1 ,
γ1 γ2
Y2∗ (t )
−
(3.6)
Fig. 1. Nash equilibrium strategies for investor 1.
where
Ξi
Φ

θ1 θ2 (1 − γ1 )(1 − γ2 ) −1 T 1
(Σi )
=I−
0
γ1 γ2


1 0
=
.
0 ρ
0
ρ2

ΣiT ,
(3.7)
Proof. Eqs. (3.4) and (3.5) show the relation between the two
pension fund managers. Combining them, we can obtain the Nash
equilibrium strategies directly. Since we only obtain the Nash equilibrium strategies for 
u∗i (t ),
we present the following proposition for the original strategies. We
can see that the Nash equilibrium strategies 
u∗i (t ) is proportional
to
Yi∗ (t )+Di (t )
.
Y ∗ (t )
i
Especially when there is no contribution, i.e., ci = 0,
we have Di (t ) = 0. In this case, the original problem is a classical
self-financing optimization problem and the optimal strategies are
all deterministic.
Proposition 3.3. The Nash equilibrium pair (u∗1 (t ), u∗2 (t )) for these
two pension managers are:
u∗1 (t )
u∗2 (t )
=
u∗1 (t ) + (1, 0)T ,

= u∗2 (t ) + (1, 0)T ,
(3.8)
where 
u∗1 (t ) and 
u∗2 (t ) are calculated by Eq. (3.6).
4. Sensitivity analysis
In this section, we use the Monte Carlo Methods (MCM) to study
the evolutions of the Nash equilibrium strategies for manager 1
and manager 2. Unless otherwise stated, the parameters we adopt
are as follows: θ1 = 0.5, θ2 = 0.5, ρ = 0.5, γ1 = 2, γ2 =
2, rn = 0.1, rr = 0.045, λI = 0.2, λS1 = 0.2, λS2 = 0.2, σI =
0.1, σS11 = 0.06, σS12 = 0.2, σS21 = 0.05, σS22 = 0.08,
c1 = 1, c2 = 2, I0 = 1, T = 20.
4.1. Nash equilibrium strategies
Firstly, we present the Nash equilibrium strategies of manager
1 and manager 2. In Figs. 1–5, we simulate 10 000 tracks of
the optimal strategies and calculate the mean of the 10 000
tracks. Figs. 1–4 show the mean of optimal strategies while Fig. 5
illustrates the mean of optimal wealth with respect to time.
Fig. 2. Nash equilibrium strategies for investor 2.
Fig. 1 reveals the evolution of the Nash equilibrium strategies
for pension manager 1. The manager 1 firstly holds a short
position in the cash, then short less proportion of money in cash
rapidly during the accumulation phase. The mean proportion of
money in cash increases from −8 to about −1 at terminal time.
However, the manager always holds a long position in the bond
and stock. The proportions invested in the bond and stock all have
a declining tendency. The manager invests most in stock, about 6.
The proportion in stock decreases to be less than the proportion in
the bond at time 2. At terminal time, the manager invests about 1
in the stock and bond.
The Nash equilibrium strategies of pension manager 2 are
illustrated in Fig. 2. As is indicated by Fig. 2, the pension manager
2 invests heavily in the stock, which is 20 at initial time. The
evolution of proportion in cash is similar with the case of investor 1,
increasing from −18 to about −2. However, manager 2 maintains
the proportion in the bond at zero all the time. Pension fund
manager 1 is different from manager 2 in two aspects: the
contribution rate and the volatilities of the stock. Manager 1 is
faced with lower contribution rate and higher volatilities. Thus, in
contrast to manager 2, manager 1 will allocates fewer in the stock
and more in the bond to achieve the optimal utility.
We are also concerned about the influence of θ1 and θ2 on the
nash equilibrium strategies. The cases when θ1 = θ2 = 0 and θ1 =
θ2 = 1 are illustrated in Figs. 3 and 4. When θ1 = θ2 = 0, the nash
equilibrium strategies coincide with the case of no competition.
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G. Guan, Z. Liang / Insurance: Mathematics and Economics 70 (2016) 237–244
Fig. 5. Evolution of mean wealth.
Fig. 3. θ1 = θ2 = 0.
Fig. 6. Evolution of mean wealth.
Fig. 4. θ1 = θ2 = 1.
As is showed in Fig. 3, the monotonicity of proportions invested in
the three assets is the same as Fig. 1. Nevertheless, compared with
Fig. 1, manager 1 invests less in the risk assets, i.e., [0.8, 5] in stock
and [−0.8, −6] in cash. Fig. 4 illustrates the case when the pension
managers are extremely competitive. In this situation, proportion
in stock varies from 6 to 0.8 while proportion in bond varies from
6 to 2 at terminal time. Besides, manager 1 short largely in the
cash. Combining Figs. 1, 3 and 4, we can observe that when θ1 , θ2
increase, the pension managers become competitive. Therefore, in
order to behave better than the other, they will adopt more risky
investment strategies.
4.2. Evolution of optimal wealth
This subsection investigates the evolutions of the optimal
wealths of the two managers. The mean wealths of manager 1
and manager 2 are illustrated in Fig. 5. Particular paths of the
wealth of manager 1 and manager 2 are showed in Fig. 6. The mean
wealth of the pension manager 1 increases from 1 to about 60 at
terminal time. However, since the pension manager 2 has larger
contribution rate, the mean wealth of the manager 2 increases
faster, which is only 1 at initial time and about 120 at terminal time.
This shows that although the two managers intend to act better
than the other, the manager with more contribution can achieve
larger expectation of terminal wealth. Fig. 6 shows the paths of
the wealths of the manager 1 and manager 2. We can see that the
evolutions of the two managers’ wealths are similar. However, the
wealth of the manager 2 fluctuates more heavily since the manager
2 allocates more in the risky assets.
5. Conclusion
In this paper, we consider competition for two pension fund
managers. Besides the terminal real wealth, the pension manager
is also cared about the relative wealth, that is, the wealth divided
by the other manager’s wealth. Thus, the goal of the pension
manager is to maximize the expectation of the CRRA utility of
the product of the real terminal wealth and relative wealth. In
this paper, the inflation risk is also taken into account. In the
case of competition, dynamic programming method is applied to
derive the Nash equilibrium strategies. The results show that the
equilibrium strategies for one manager is related to the parameters
of the other manager.
Acknowledgments
The authors gratefully thank the anonymous referees for their
many valuable suggestions, which made our paper more interest-
G. Guan, Z. Liang / Insurance: Mathematics and Economics 70 (2016) 237–244
ing and accurate. The authors thank the members of the group of
Stochastic Analysis, Insurance Mathematics, Insurance Economics
and Mathematical Finance at the Department of Mathematical
Sciences, Tsinghua University for their feedback and useful conversations. The authors also gratefully acknowledge the support
of the National Natural Science Foundation of China (Grant No.
11471183).
Substituting 
u∗1 (t ) and the differentials of V into the HJB
equation (3.3), we have
(1 − θ1 )rr (t ) +
− θ1
f ′ (t )
(1 − γ1 )f (t )
y2
y2 + D2 (t )
1
θ1 [1 − γ1 ]
y2
−
Proof. We prove Proposition 3.1 based on the HJB equation (3.3).
We solve HJB equation (3.3) in the following steps. By the boundary
condition
+ θ1 [θ1 (1 − γ1 ) + 1]
V (T , y1 , y2 ) =
1 − γ1
+
,
we conjecture that the optimal utility function for investor 1 is as
follows:
V (t , y1 , y2 ) =
(A.1)
where Di (t ) = ci
T
  s

exp −
rr (u)du ds,
t
(1 − γ1 )[−c1 + rr (t )D1 (t )]
V
y1 + D1 (t )
θ1 (1 − γ1 )[−c2 + rr (t )D2 (t )]
−
V
y2 + D2 (t )
(y1 + D1 (t ))1−γ1 (y2 + D2 (t ))−θ1 (1−γ1 ) ′
+
f (t ),
1 − γ1
y1 + D1 (t )
Vy1 y1 = −γ1 (1 − γ1 )
,
Vy2 = −θ1 (1 − γ1 )
V
(y1 + D1 (t ))2
Vy1 y2 = −θ1 (1 − γ1 )
Vy 1
y1 Vy1 y1
−
y2 Vy1 y2
y1 Vy1 y1
V
(y2 + D2 (t ))
(y1 + D1 (t ))(y2 + D2 (t ))
y1
1
2γ1
θ12 [1 − γ1 ]2
y22
(y2 + D2 (t ))2

uT2 Σ2 Σ2T
u2

uT2 Σ2 Φ 2 Σ2T
u2 = 0.
(A.4)
Since we require that f (t ) is deterministic and not related with
y1 , y2 , we can observe from Eq. (A.4) that the condition that 
u2 is
y +D (t )
proportional to 2 y 2 is needed.
The explicit form of f (t ) can be derived by integrating w.r.t. t in
Eq. (A.4) with the boundary condition f (T ) = 1.
Combining Eqs. (3.4) and (3.5), we can show that the Nash
equilibrium strategies 
u∗1 (t ) and 
u∗2 (t ) are indeed proportional
y +D (t )
and 2 y 2 , respectively. Therefore, substituting the
to 1 y 1
1
2
Nash equilibrium 
u∗2 (t ) into Eq. (A.4), we can obtain an ordinary
differential equation satisfied by f (t ), from which we get the
expression of f (t ):
T

(1 − θ1 )rr (s) +
1
γ1
1
2γ1
ΛT1 Λ1 (s)
θ1 [1 − γ1 ]ΛT1 (s)ΦΣ2T (s)G(s)
V
y2 + D2 (t )
+ θ1 [θ1 (1 − γ1 ) + 1](G(s))T Σ2 (s)Σ2T (s)G(s)
2
 
1 2
2
T
2 T
θ1 [1 − γ1 ] (G(s)) Σ2 (s)Φ Σ2 (s)G(s) ds ,
+
2γ1
,
G(t ) =
,
2
.
1
γ2
−
Ξ2−1 (t )(Σ2−1 )T (t )Λ2 (t )
θ2 (1 − γ2 ) −1
Ξ2 (t )(Σ2−1 )T (t )ΦΛ1 (t ).
γ1 γ2
Thus, the closed form of V (t , y1 , y2 ) can be derived by (A.1). By
the similar technique, we can derive the Nash equilibrium utility
function for investor 2. (Σ1−1 )T Λ1
References
( Σ1 )
−1 T
1 y1 + D1 (t )
γ1
(y2 + D2 (t ))2
1

1
0
0
ρ

 (t ).
Σ2T u2
(A.2)
Substituting the differentials of V (t , y1 , y2 ) into the last equation,
we can obtain the optimal feedback control for pension manager 1
as follows:

u∗1 (t ) =
2
− θ1 (G(s))T Σ2 Λ2 (s) −
Using the first order condition of the HJB equation (3.3), the
relationship between 
u∗1 (t ) and V (t , y1 , y2 ) is

u∗1 (t ) = −
y22
where
V
2
ΛT1 ΦΣ2T
u2
t
,
Vy2 y2 = θ1 (1 − γ1 )[1 + θ1 (1 − γ1 )]
y2 + D2 (t )


f (t ) = exp (1 − γ1 )
Vt =
V
ΛT1 Λ1
1
y +D (t )
i = 1, 2.
t
Differentiating V (t , y1 , y2 ), we can obtain the following expressions:
Vy1 = (1 − γ1 )
1
2γ1
2
(y1 + D1 (t ))1−γ1 (y2 + D2 (t ))−θ1 (1−γ1 )
f (t ),
1 − γ1

γ1
+

uT2 Σ2 Λ2
Appendix. Proof of Proposition 3.1
1−γ −θ (1−γ1 )
y1 1 y2 1
243
(Σ1−1 )T Λ1
θ1 (1 − γ1 ) y1 + D1 (t )
y2
(Σ −1 )T
γ1
y1
y2 + D2 (t ) 1


1 0
×
Σ2T
u2 (t ).
0 ρ
−
Thus, Eq. (3.4) follows. Similarly, Eq. (3.5) also holds.
(A.3)
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