This paper investigates the optimal control of asset allocation on a defined contribution pension plan. In our model, the plan member is allowed to invest in a risk-free asset (bank account), a risky asset (stock) and an inflation-linked bond. The dynamics of the wealth in our model take into account a certain proportion of the client’s salary paid as the contribution towards the pension fund. By applying the Hamilton-Jacobi-Bellman equation we find the explicit solutions for the CARA and CRRA utility functions. This helps us to calculate the investment strategies associated with the stock and inflation-linked bond. Finally, a numerical simulation is presented to illustrate the behaviour of the model.
It is very imperative to assess the performance of pension fund managers or investment managers in relation to the large amounts of funds that pension managers are handling, and the scale of competition this market is facing. From the investor’s point of view, the need is even more significant when one looks at the increasing volatility that the financial market is subjected to.
A number of authors have dealt with the problem of managing the financial resources of a pensioner after retirement, which arises from the fact that whole life annuities are felt by policyholders to be “poor value for money” and have investigated the other alternatives available to a retiree [
Most of the studies left a literature gap as their models focused on investors investing in only one risky asset (being the stock), and so our study aimed to fill this gap by introducing another risky asset (being an inflation-linked bond). Our study has some similarities with the work of [
The main objective of the present study is to find the optimal investment strategies for a defined contribution (DC) pension with stochastic salary. This work is concerned with maximizing the optimal investment strategy for DC Pension with a stochastic salary. The present research attempts to:
1) Study optimal control of assets allocation on a defined contribution pension plan.
2) Apply the Hamilton-Bellman equation to find the explicit solutions for the CARA and CRRA utility function.
3) Obtain the optimal investment strategies (cash, inflation-linked bond and the stock) for the three investments under the exponential utility function.
We assume that the instantaneous nominal interest rate r ( t ) is stochastic and follows an Ornstein-Uhlenbeck process (see [
d r ( t ) = b ( a − r ( t ) ) d t + σ r d W 1 ( t ) , r ( 0 ) = r 0 , (1)
where W 1 ( t ) is a standard Brownian motion and a , b and σ are strictly positive constants.
The riskless asset or bank account is given by the following SDE
d B A ( t ) B A ( t ) = r ( t ) d t , B A ( 0 ) = B A 0 . (2)
Based on the work of [
d S ( t ) S ( t ) = ( r ( t ) + κ η λ r σ S ) d t + σ S η ( t ) d W 1 ( t ) , S ( 0 ) = S 0 . (3)
where κ , η , λ r and σ S are positive constants see [
We define the inflation rate process as:
d P ( t ) P ( t ) = i d t + σ P d W 2 ( t ) , P ( 0 ) = P 0 . (4)
The constant i is the expected rate of inflation and σ P represents its volatility. W 2 ( t ) is another Brownian motion under the physical measure, which generates uncertainty in the price level and is independent of W 1 ( t ) . The correlation between the two Brownian motions is given by d W 1 d W 2 = 0 .
The inflation-linked bond is given by the SDE
d I ( t ) I ( t ) = ( r ( t ) + ψ σ r λ r κ η + σ P λ P ) d t + ψ σ r d W 1 ( t ) + σ P d W 2 ( t ) . (5)
The salary of the client is given by the following SDE
d L ( t ) L ( t ) = ( ϖ L ( t ) + γ ( t ) ) d t + ρ L , 1 ( t ) d W 1 ( t ) + ρ L , 2 ( t ) d W 2 ( t ) , L ( 0 ) = L 0 , (6)
where the instantaneous mean of the salary μ L ( t ) = ϖ L ( t ) + γ ( t ) has two components, where the former ϖ L ( t ) is caused by inflation and the latter γ ( t ) is caused by other factors like economy growth. On the other hand ρ L ,1 ( t ) and ρ L ,2 ( t ) are two volatility factors that determine the effect of stock and inflation on salary.
Each employee contributes a constant proportion of his salary at the rate c ( t ) , then the process of contribution Γ ( t ) is given by
Γ ( t ) = c ( t ) L ( t ) d t . (7)
The dynamics of the pension wealth are given by
d V ( t ) = ( 1 − π 1 − π 2 ) V ( t ) d B A ( t ) B A ( t ) + π 1 V ( t ) d S ( t ) S ( t ) + π 2 V ( t ) d I ( t ) I ( t ) + Γ ( t ) , V ( 0 ) = V 0 , (8)
where V ( t ) is the wealth of pension fund at time t ∈ [ 0,1 ] and π 1 and π 2 respectively represent the proportion of the pension fund invested in the Stock and Bond. The proportion of the pension fund invested in the riskless asset is given by 1 − π 1 − π 2 .
Substituting (2), (3), (5) and (7) above into (8) we get
d V ( t ) = V ( t ) ( ( r ( t ) + π 1 λ r σ S κ η + π 2 ( ψ σ r λ r κ η + σ P λ P ) ) d t + ( π 1 σ S η ( t ) + π 2 ψ σ r ) d W 1 + π 2 σ P d W 2 ) + c ( t ) L ( t ) d t , V ( 0 ) = V 0 . (9)
Now taking X = V P to be the real wealth process and then applying chain rule to it by Ito i.e.
d X = d ( V P ) = V P ( d V V − d P P − d V V d P P + ( d P P ) 2 ) . (10)
Substituting (4) and (9) into (10) above we get
d X ( t ) = X ( t ) ( ( r ( t ) + π 1 λ r σ S κ η − i + σ P 2 + π 2 ( ψ σ r λ r κ η + σ P λ P − σ P 2 ) ) d t + ( π 1 σ S η ( t ) + π 2 ψ σ r ) d W 1 + ( π 2 σ P − σ P ) d W 2 ) + c ( t ) d t . (11)
Let H : [ 0, T ] × ℝ ↦ ℝ denote a value function. Then H ( ⋅ ) (see [
H ( t , x ) = sup u ∈ A ( t , x ) J ( t , x , u ( t ) ) . (12)
The HJB equation associated with (11) is given by
sup { d H } = sup { ∂ H ∂ t d t + ∂ H ∂ X d t + 1 2 ∂ 2 H ∂ X 2 ( d X ) 2 } , (13)
which simplifies to
H ( t , x ) = sup { H t + x ( r ( t ) + π 1 λ r σ S κ η − i + σ P 2 + π 2 ( ψ σ r λ r κ η + σ P λ P − σ P 2 ) ) H x + c + 1 2 x 2 ( π 1 2 σ S 2 η + π 2 2 ψ 2 σ r 2 + 2 π 1 π 2 σ S σ r ψ η + π 2 2 σ P 2 + σ P 2 − 2 σ P 2 π 2 ) H x x } = 0. (14)
Differentiating (14) with respect to π 1 and π 2 we get the first order maximizing conditions for the optimal strategies π 1 * and π 2 * below:
x σ S λ r κ η H x + x 2 σ S 2 η π 1 * H x x + x 2 σ S σ r ψ η π 2 * H x x = 0 (15)
( x σ P λ P + x ψ σ r λ r κ η − x σ P 2 ) H x + ( x 2 ψ 2 σ r 2 π 2 * + x 2 σ S σ r ψ η π 1 * + x 2 σ P 2 π 2 * − x 2 σ P 2 ) H x x = 0. (16)
(15) and (16) simplify to
σ S λ r κ η H x + x σ S 2 η π 1 * H x x + x σ S σ r ψ η π 2 * H x x = 0 (17)
( σ P λ P + ψ σ r λ r κ η − σ P 2 ) H x + ( x ψ 2 σ r 2 π 2 * + x σ S σ r ψ η π 1 * + x σ P 2 π 2 * − x σ P 2 ) H x x = 0, (18)
for x ≠ 0 .
We then solve (17) and (18) for π 1 * and π 2 * to obtain
π 2 * = H x x H x x − λ P x σ P H x H x x + 1 (19)
and
π 1 * = − σ r ψ η x σ S η H x H x x + λ P σ r ψ η x σ P σ S η H x H x x − σ r ψ η σ S η − κ λ r x σ S H x H x x . (20)
Substituting (19) and (20) into (14) above we get
H ( t , x ) = H t + ( x r ( t ) − x i + x σ P λ P + c ) H x − 1 2 H x 2 H x x ( η λ r 2 κ 2 + σ P 2 + λ P 2 + 2 λ P σ P ) , (21)
a nonlinear second order partial differential equation difficult to solve in this form.
The section demonstrates the explicit solutions for the CARA utility functions via power transformation technique and variable change method. It also illustrates the explicit solution for the CRRA utility function.
Assuming that the plan member takes an exponential utility function:
U ( x ) = − 1 q e − q x ; ( q > 0 ) (22)
The absolute risk aversion, q, of a decision maker with utility described in (22) is constant and (22) is a CARA utility. We conjecture a solution to (21) with the following form
H ( t , x ) = − 1 q e − q ( a ( t ) x + g ( t ) ) (23)
with the boundary conditions given by a ( T ) = 1 and g ( T ) = 0 .
The Constant relative risk aversion of an investor is given by:
H ( t , x ) = b ( t ) ( x − a ( t ) ) γ γ (24)
where the degree of risk aversion parameter γ ∈ ( − ∞ , 0 ) ∪ ( 0 , 1 ) with initial conditions a ( N + T ) = 0 and b ( N + T ) = 0 .
The main results are obtained using the utility functions given in (23) and (24) above. They are presented in the following theorems:
Theorem 1. Suppose the utility function is given as in (22). Then the optimal investment strategies for the bond and stock are respectively given as:
π 2 * = − e − ς ( T − t ) x q + λ P e − ς ( T − t ) q + 1 (25)
and
π 1 * = σ r ψ η e − ς ( T − t ) q x σ S η − λ P σ r ψ η e − ς ( T − t ) q x σ P σ S η + k λ r e − ς ( T − t ) x σ S q (26)
Proof. See Appendix A.
Theorem 2. Suppose the utility function is given as in (24). Then the optimal investment strategies for the bond and stock are respectively given as:
π 2 * = x − 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) x ( γ − 1 ) − λ P x σ P x − 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) γ − 1 + 1 (27)
and
π 1 * = − σ r ψ η x σ S η x − 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) γ − 1 + λ P σ r ψ η x σ P σ S η x − 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) γ − 1 − σ r ψ η σ S η − κ λ r x σ S x − a ( t ) γ − 1 (28)
Proof. See Appendix B.
In this chapter, we illustrate the relationship between the optimal investment strategies and the parameters in our model. The graphs were done using MATLAB program. The parameter values are as follows unless otherwise stated:
T = 40 ; t = 0 ; i = 0.8 ; σ P = 0.005 ; σ r = 0.6 ; σ s = 0.6 ; λ P = 0.2 ; λ r = 0.4 ; ψ = 0.2 ; q = 20 ; η = 3 ; x = 1 ; κ = 500.
The figures show the gesture of optimal investment proportions from initial time to retirement. The parameters are given by the previous paragraph. Figures 1-3 give us the trends how the optimal investment strategies in the three assets change with time. As time runs, the optimal amount invested in cash and bond decrease as shown in
Figures 4-6 show the gesture of optimal investment proportions from initial time to retirement using the CRRA technique. The parameters are stated as follows:
κ = 1500 ; i = 0.03 ; x = 0.5 ; σ P = 0.04 ; λ P = 0.05 ; λ r = 0.03 ; a = 0.005 ; φ = − 0.3 ; r = 0.5 ; N = 0.025 ; c = 0.49 ; T = 40 ; γ = 0.0003 ; σ S = 0.03 ; σ r = 0.5 ; η = 0.0002.
The above figures show the gesture of optimal investment proportions from initial time to retirement. The parameters are given by the previous paragraph. Figures 4-6 above give us the trends how the optimal investment strategies in the three assets change with time. As time runs, the optimal amount invested in cash and bond increase as shown in
We studied optimal control of asset allocation on a defined contribution pension plan. We considered three different investment strategies which include cash, inflation-linked bond and the stock. We obtained the optimal investment strategies for the three investments under the exponential utility function. The dynamics of the wealth in our model considered a certain proportion of the client’s salary paid as the contribution towards the pension fund. By applying the Hamilton-Bellman equation we found the explicit solutions for the CARA and CRRA utility functions. This helped us to calculate the investment strategies associated with the stock and inflation-linked bond. Finally, a numerical simulation was presented to illustrate the behaviour of the model.
I would like to thank Dr. Othusitse Basimanebotlhe for his guidance through each stage of the write-up of this article. I extended gratitude to the members of JAMP for their valuable comments and support.
The authors declare no conflicts of interest.
Keganneng, O. and Basimanebotlhe, O. (2022) Optimal Control of Assets Allocation on a Defined Contribution Pension Plan. Open Access Library Journal, 9: e7970. https://doi.org/10.4236/oalib.1107970
Proof. Taking the partial derivatives of (23) we obtain:
H t = − q ( a t x + g t ) H
H x = − q a ( t ) H
H x x = q 2 a ( t ) 2 H
where a t and g t denote derivatives of a ( t ) and g ( t ) respectively.
Introducing these derivatives in (21) above we get:
( q a t + q a − q a i + q a σ P λ P ) x + q g t + q a c + 1 2 ( η λ r 2 κ 2 + ( σ P + λ P ) 2 ) = 0 (29)
splitting (29) into two equations we get:
( q a t + q a − q a i + q a σ P λ P ) x = 0 (30)
and
q g t + q a c + 1 2 ( η λ r 2 κ 2 + ( σ P + λ P ) 2 ) = 0 (31)
Simplifying (30) and (31) respectively we get:
( a t + a − a i + a σ P λ P ) = 0 (32)
and
g t + a c + 1 2 q ( n λ r 2 κ 2 + ( σ P + λ P ) 2 ) = 0 (33)
Solving (32) and (33) we get:
a ( t ) = e ς ( T − t ) (34)
where ς = 1 − i + σ P λ P .
And
g ( t ) = ρ t (35)
where ρ = a c + η λ r 2 κ 2 + ( σ P + λ P ) 2 2 q
H x H x x = − e − ς ( T − t ) q (36)
where ς = 1 − i + σ P λ P
Substituting (36) into (19) and (20) we obtain the optimal investment strategies for the inflation-linked bond and stock respectively as:
π 2 * = − e − ς ( T − t ) x q + λ P e − ς ( T − t ) q + 1 (37)
and
π 1 * = σ r ψ η e − ς ( T − t ) q x σ S η − λ P σ r ψ η e − ς ( T − t ) q x σ P σ S η + k λ r e − ς ( T − t ) x σ S q (38)
Proof. Taking the partial derivatives of (24) we obtain:
H t = b ′ ( t ) ( x − a ( t ) ) γ γ − b ( t ) ( x − a ( t ) ) γ − 1 a ′ ( t ) (39)
H x = b ( t ) ( x − a ( t ) ) γ − 1 (40)
H x x = b ( t ) ( γ − 1 ) ( x − a ( t ) ) γ − 2 (41)
substituting the above into (21) we get
b ′ ( t ) ( x − a ( t ) ) γ γ − b ( t ) ( x − a ( t ) ) γ − 1 a ′ ( t ) + x r ( t ) b ( t ) ( x − a ( t ) ) γ − 1 − x i b ( t ) ( x − a ( t ) ) γ − 1 + x σ P λ P b ( t ) ( x − a ( t ) ) γ − 1 + c b ( t ) ( x − a ( t ) ) γ − 1 − 1 2 b ( t ) ( x − a ( t ) ) γ γ − 1 ( η λ r 2 κ 2 + σ P 2 + λ P 2 + 2 λ P σ P ) = 0 (42)
splitting (42) into terms containing ( x − a ( t ) ) γ and ( x − a ( t ) ) γ − 1 respectively gives:
b ′ ( t ) ( x − a ( t ) ) γ γ + r ( t ) b ( t ) ( x − a ( t ) ) γ + σ P λ P b ( t ) ( x − a ( t ) ) γ − 1 2 b ( t ) ( x − a ( t ) ) γ γ − 1 ( η λ r 2 κ 2 + σ P 2 + λ P 2 + 2 λ P σ P ) = 0 (43)
and
− b ( t ) ( x − a ( t ) ) γ − 1 a ′ ( t ) + ( r ( t ) a ( t ) − x i ) b ( t ) ( x − a ( t ) ) γ − 1 + ( σ P λ P a ( t ) + c ) b ( t ) ( x − a ( t ) ) γ − 1 − x i b ( t ) ( x − a ( t ) ) γ − 1 + x σ P λ P b ( t ) ( x − a ( t ) ) γ − 1 + c b ( t ) ( x − a ( t ) ) γ − 1 = 0 (44)
simplifying (43) and (44) we get
b ′ ( t ) γ + r ( t ) b ( t ) + σ P λ P b ( t ) − 1 2 b ( t ) γ − 1 ( η λ r 2 κ 2 + σ P 2 + λ P 2 + 2 λ P σ P ) = 0 (45)
and
− a ′ ( t ) + r ( t ) a ( t ) + σ P λ P a ( t ) − 2 x i + x σ P λ P + 2 c = 0 (46)
from (45) we have:
b ′ ( t ) + ( γ ( r ( t ) + σ P λ P ) − 1 2 γ γ − 1 ( η λ r 2 κ 2 + σ P 2 + λ P 2 + 2 λ P σ P ) ) b ( t ) = 0 (47)
let
B = γ ( r ( t ) + σ P λ P ) − 1 2 γ γ − 1 ( η λ r 2 κ 2 + σ P 2 + λ P 2 + 2 λ P σ P ) (48)
then (47) becomes
b ′ ( t ) + B b ( t ) = 0 (49)
solving (49) using integrating factor (IF) by the below steps results in:
I F = p ( t ) = e ∫ B d t = e B t
b ′ ( t ) e B t + B e B t b ( t ) = 0 (50)
d [ b ( t ) e B t ] = 0
integrating both sides gives
b ( t ) = 0 + C e − B t
using initial conditions gives
1 = C e − B ( N + T )
and so
b ( t ) = e B ( N + T − t ) (51)
substituting (48) back into (51) gives
b ( t ) = e ( γ ( r ( t ) + σ P λ P ) − 1 2 γ γ − 1 ( η λ r 2 κ 2 + σ P 2 + λ P 2 + 2 λ P σ P ) ) ( N + T − t ) (52)
then (46) is solved as:
a ′ ( t ) − ( r ( t ) + σ P λ P ) a ( t ) = − 2 x i + x σ P λ P + 2 c (53)
let
A = r ( t ) + σ P λ P and K = − 2 x i + x σ P λ P + 2 c (54)
then
a ′ ( t ) − A a ( t ) = K (55)
multiplying the Equation (55) above by integrating factor p ( t ) = e ∫ − A d t = e − A t gives:
a ′ ( t ) e − A t − A a ( t ) e − A t = K e − A t (56)
d [ a ( t ) e − A t ] = K e − A t
integrating both sides gives
a ( t ) e − A t = K − A e − A t + C (57)
a ( t ) = − K A + C e A t (58)
using initial conditions a ( N + T ) = 0 yields:
0 = − K A + C e A ( N + T ) (59)
and so:
a ( t ) = − K A + K A e − A ( N + T − t ) (60)
Now substituting (54) into (60) above gives:
a ( t ) = 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) (61)
and therefore we have
H x H x x = x − a ( t ) γ − 1 (62)
substituting (62) into (19) and (20) above gives
π 2 * = x − a ( t ) x ( γ − 1 ) − λ P x σ P x − a ( t ) γ − 1 + 1 (63)
and
π 1 * = − σ r ψ η x σ S η x − a ( t ) γ − 1 + λ P σ r ψ η x σ P σ S η x − a ( t ) γ − 1 − σ r ψ η σ S η − κ λ r x σ S x − a ( t ) γ − 1 (64)
now substituting (61) into (63) and (64) gives optimal investment strategies for the bond and stock respectively as:
π 2 * = x − 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) x ( γ − 1 ) − λ P x σ P x − 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) γ − 1 + 1 (65)
and
π 1 * = − σ r ψ η x σ S η x − 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) γ − 1 + λ P σ r ψ η x σ P σ S η x − 2 x i − x σ P λ P − 2 c r ( t ) + σ P λ P + − 2 x i + x σ P λ P + 2 c r ( t ) + σ P λ P e ( − r ( t ) − σ P λ P ) ( N + T − t ) γ − 1 − σ r ψ η σ S η − κ λ r x σ S x − a ( t ) γ − 1 (66)