J. Nig. Soc. Phys. Sci. 2 (2020) 170–179

Journal of the
Nigerian Society

of Physical
Sciences

Original Research

On the Investment Strategy, Effect of Inflation and Impact of
Hedging on Pension Wealth during Accumulation and

Distribution Phases.

B. O. Osua,∗, K. N. C. Njokub, B. I. Oruha

aDepartment of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria.
bDepartment of Mathematics, Imo State University, Owerri, Nigeria.

Abstract

This paper studies the various results obtained in literature on the investment strategy, the effect of inflation and impact of hedging on the Pension
Wealth generation. The explicit solution of the constant relative risk aversion (CRRA) and constant absolute risk aversion (CARA) utility functions
are obtained, both in the accumulation and distribution phase, using Legendre transform, dual theory, and change of variable techniques. It is
established herein that the elastic parameter (β) is not equal to one (β , 1), based on the assumption of our model. Theorems are constructed and
proved on the various wealth investment strategies. Observations and significant results are made and obtained, respectively in the comparison
of our various utility functions and some previous results in literature. Sensitivity analysis and Simulations on the various utility functions and
optimal strategies during the accumulation and distribution phase are presented; when the existence of a elastic parameter that is not equal to
one, when there is existence of modifying factors, when there is need for diversification of investment, when there is no significant effect of
the choice of risk aversion strategy on investment returns during inflation period, when there is hedging ability of Inflation-indexed Bond and
Inflation-linked Stock and when there is insignificant effect of the orthogonal relationship between stock and time and nonpayment of pension
benefits on the satisfaction of the investors.

Keywords: Hedging, constant relative risk aversion (CRRA), constant absolute risk aversion (CARA), Defined Contribution (DC), Constant
elasticity of variance (CEV), optimal strategy.

Article History :
Received: 16 March 2020
Received in revised form: 29 April 2020
Accepted for publication: 29 April 2020
Published: 01 August 2020

c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: T. Latunde

1. Introduction

One of the major challenges that old people are faced with
is economic insecurity. By economic security we mean; having
access to information, good health condition, good education,
and many more, and security as it concerns work. Particularly,
with the work related economic security, it is only but natural
that at retirement age, they would become economically vul-

∗Corresponding author tel. no: +2348032628251
Email address: osu.bright@mouau.edu.ng (B. O. Osu )

nerable. This is true because at old age, they are mentally, eco-
nomically and physically inactive. Thus, in a bid to providing
this required economic security, the National Pension Scheme
was established [1].

Pension scheme, an economic security scheme that takes
care of people at retirement was introduced in 1951, by colo-
nial British administration, through an instrument called “Pen-
sion Ordinance” (The Nation, May, 2015). This Ordinance had
retroactive effect from 1946, and was only applicable to those
British officials that was posted to Nigeria (the Nation, May,
2015) and as such, attracted some legislation, such as; Pension

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Osu et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 170–179 171

law for Military, and many more. Roughly speaking, Pension
can be said to be a specific pull fund into which a sum of money
is paid during an employee’s active years, and that is used in
turn to service his/her periodic payments after retirement. Ac-
cording to the Encyclopedia Britannica, Pension is simply “a
series of periodic” money payments made to a person(s) who
retires from employment due to age barrier, challenges due to
disability, or the completion of an agreed period of service.
These payments continue as long as the beneficiary is alive,
and even sometimes, extends to a next-of-kin. Pension con-
sists of lump sum payment made to an employee upon his/her
disengagement from active service. Payments are mostly made
installmentally, every month. Pension scheme could be contrib-
utory (i.e., defined contribution plan), where a fixed amount of
money is invested through the pension wealth, or noncontribu-
tory (i.e., defined benefit plan), where a fixed amount of money
is paid to a person, regularly.

2. Literature Review

Recent publications in economic Journals and other rep-
utable Mathematics and Science Journals have brought to light,
variety of methods of optimizing investment strategies and re-
turns. For instance, some researchers have made various con-
tributions in this direction, particularly, in DC Pension Plan. In
[2] a work on stochastic life styling optimal dynamic asset al-
location for defined contribution pension plans was done. In
their work, various properties and characteristics of the opti-
mal asset allocation strategy, both with and without the pres-
ence of non-hedge-able salary risk were discussed. The signif-
icance of alternative optimal strategy by pension providers was
established. In [3], a defined contribution (DC) pension plan
investment problem during the accumulation phase under the
multi-period mean-variance criterion was investigated. In [4]
the optimal investment strategies for a DC pension fund under
the Hull-White interest rate model. Under this model, the pen-
sion fund manager can invest capital in the bank account, stock
index, and real estates was analysed. More so, [5] studied opti-
mal pension management in a stochastic framework, they came
out with a significant result.

In order to deal with optimal investment strategy, the need
for maximization of the expected utility of the terminal wealth
became necessary. Example, the Constant Relative Risk Aver-
sion (CRRA) utility function, and (or) the Constant Absolute
Risk Aversion (CARA) utility function were used to maximize
the terminal wealth. In [5, 6] and [7, 8], CRRA was used to
maximize terminal wealth. However, [9] used the CRRA and
the CARA to maximize terminal wealth, and this triggered our
research.

In [4] an optimal investment strategy for a DC pension fund
with a stochastic salary, under the affine interest rate model was
constructed. In this work, they introduced the notion of ” Rel-
ative Pension Wealth”, that is, where a pension plan member
only considers his/her post-retirement benefit, the ratio of the
pension wealth to his terminal salary. [9] considered a mini-
mum guarantee while obtaining their optimal policy, under in-
flationary market. This minimum guarantee served as an amor-

tization fund to make up for lost fund due to inflation. In 2009,
[10] obtained an optimal investment strategy for annuity con-
tracts, under the constant elasticity of variance (CEV) model.
In his work made an assertion without a proof that the elastic
parameter is not equal to 1. The work did not consider in its
pension wealth model the fact that the next-of-kin of the dead
contributors should be paid some benefits. Neither did it con-
sider categories of contributors, since in every employment pe-
riod, a certain group of people of the same cadre is employed,
therefore, in real life situation, the right thing should be mul-
tiple and categorized contributors. It did not also indicate that
contributors will not willingly withdraw from the scheme. Se-
quel to the above, [11, 12] came up with a modification. How-
ever, these modifications are not void of mistakes, omission and
limitations, maybe due to environmental and financial factors.

Hence, we further modify of this works, by considering dif-
ferent categories of contributors, with some other additional as-
sumptions made. Our task in this work is to compare the various
results obtained in literature, and categorically state the optimal
Policy in the various trading period.

3. Preliminaries

We start with a complete and frictionless financial market
that is continuously open over the fixed time interval [ 0, T ], for
T > 0 , representing the retirement time of any plan member.

We assume that the market is composed of the risk-free as-
set (cash), and risky asset (stock). Let (Ω, F, P) be a complete
probability space, where Ω is a real space and P is a probability
measure, {ws (s) , wt (t)} are two standard non-orthogonal Brow-
nian motions, {Fs (s) , Ft (t)} are right continuous filtrations
whose information are generated by the two standard Brown-
ian motions {ws (s) , wt (t)} , whose sources of uncertainties are
respectively to the stock market and time evolution.

3.1. Materials and Methods

Assume we represent u = us as the strategy and we define
the utility attained by the contributor from a given state y at time
t as

∅u (t, r, y) = Eu
[
U (Y (t)) : r (t) , y (t) = y

]
(1)

where t is the time, r is the short interest rate andy is the wealth.
Our interest here is to find the optimal value function

∅ (t, r, y) = S upu∅u (t, r, y) (2)

and the optimal strategy u∗ = u∗s such that

∅u∗ (t, r, y) = ∅ (t, r, y) (3)

3.2. Legendre Transformation

The Legendre transform and dual theory help to transform
the nonlinear partial differential equation that is formed due to
1, to a linear partial differential equation.

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Osu et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 170–179 172

Theorem 1 [13] : Let f : Rn → R be a convex function for
z > 0, then the Legendre transform is defined as;

L (z) = max
y
{ f (y) = zy} , (4)

where L (z) is the Legendre dual of f (y).
Since f (y) is convex, from theorem 1 we define the Legen-

dre transform

∅̂ (t, r, z) = sup {∅ (t, r, y) − zy} (5)
0 < y < ∞, 0 < t < T,

where ∅̂ is the dual of ∅ and z > 0 is the dual variable of
x. The value of ywhere this optimum is attained is denoted by
h (t, r, z) , so that

h (t, r, z) = in f
{
y : ∅ (t, r, y) ≥ zy + ∅̂ (t, r, z)

}
(6)

0¡t¡T,

The function h and φ̂ are closely related and can be referred to
as the dual of φ . These functions are related as follows

∅̂ (t, r, z) = ∅ (t, r, y) − zh (7)

where,

h (t, r, z) = y,∅y = z, h = −∅̂z. (8)

At terminal time T , we denote
Û = S up {U(y) − zy : 0 < y < T}
and
∅ (z) = S up

{
U (y) ≥ zy + U (z)

}
.

As a result

∅ (z) = U−1 (z) , (9)

where φ is the inverse of the marginal utility Uand note that
∅ (T, r, y) = U (y).
At terminal time T , we can define

h (T, r, y) = infy>0
{
y : U (y) ≥ zy + φ̂ (t, r, z)

}
and
φ̂ (t, r, y) = S upy>0 {U (y) − zy}
so that h (T, r, z) = U−1 (z).

4. The Model

This section introduces the financial market and proposes
the optimization problems in the Pension distribution phase.
(i.e., Pension payment/disbursement period).

4.1. The Financial Market
Here, we consider a financial market that consists of a risk-

free asset (i.e., cash in the bank) and a risky asset (stock).
Let the risk-free asset Ct , say, at any positive time, t, evolve

thus;

dCt = rCtdt (10)

where r represents constant rate of interest. Next, we denote
the price of the risky asset (stock) at any positive time r, by S t,
as in [11, 14, 15] thus;

dS t = µS tdt + kS
β+1
t dWt (11)

where µ(µ > r) represents the instantaneous rate of return on
stock, β (β≤ 0) is the elastic constant parameter, k is a constant,
kS βt represents the instantaneous volatility.

Let {Wt; t ≥ 0} denote a standard Brownian motion, defined
on a probability space, (Ω, F, P) where F = {Ft} is an aug-
mented filtration generated by the Brownian motion.

5. Model Assumption

Consistent with the Nigerian Pension Reform Act of 2004
[1], we make the following assumptions:

1. The Pension Scheme accumulates wealth.
2. There are different categories of contributions.
3. The contributors will not willingly withdraw from the

scheme.
4. Payments are made to the retirees.
5. An accumulated amount is paid to the Next-of-kin of the

dead contributors, at the instance of death by any contrib-
utor(s).

6. A certain amount is retained from the payment made to
the families of dead contributors, by the Pension man-
agers (i.e., management fee).

5.1. Model Formation (i.e., the Optimization Program)
The fund accruing from the contributors can be invested in

both Bank and stock. Particularly, the fund to be invested by
the fund manager is the surplus, which is the fund that is avail-
able after each period of routine disbursements. That is, let the
contribution process be:

dy = (1 + θi) Ci+1dt (12)

and the payment process

d j = bi+1dt + (ai+1 −η) dWs (13)

Then the surplus

dP = dy − d j = (1 + θi) Ci+1dt − (14)[
bi+1dt + (ai+1 −η) dWs

]
= (Ci+1 + θiCi+1 − bi+1) dt −

(ai+1 −η) dWs

Therefore, our task here is to construct an optimal investment
strategy for the assets for the remaining periods after retirement,
to enable us maximize the expected utility at each retirement
period.

Without loss of generality, the pension wealth is denoted by
at any time 0 < t < T < T + N and it evolves stochastically,
thus (See [12]):

dY (t) = usY (t)
dS t
dS t

+ (15)

(1 − us) Y (t)
dCt
dCt

+

(Ci+1 + θiCi+1) dt − (ai+1 −η) dWs

i = 0, 1, . . . , n − 1 and θi = 0,θ1 = 1,θ2 = 2, . . . ,θi (an integer)
= staff loading, where; ai+1 > 0 represents various amount that

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Osu et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 170–179 173

is paid to the Next-of-Kin of the dead contributors, bi+1 > 0
represents various amount paid to retired contributors, ci+1 > 0
represents various amount contributed, η is the service charge
deducted from the ai+1.

However, relevant to the provisions of the Nigerian Pension
Reform Act of 2004, on the eligibility condition for signing up
on the Pension Scheme, by both government and private sec-
tors, we have;

dY (t) = usY (t)
dS t
dS t

+ (16)

(1 − us) Y (t)
dCt
dCt

+

(Ci+1 + θiCi+1 − bi+1) dt − (ai+1 −η) dWs

i = 4, 5, . . . , n − 1, and θ4 = 4,θ5 = 5,θ6 = 6, . . . ,θn = n (a
positive integer)= staff loading.
Assuming, bi+1 = Ci+1 + rdCi+1; r < rd , and rd represents dis-
counted interest, then

dY (t) = usY (t)
dS t
dS t

+ (1 − us) Y (t)
dCt
dCt

+ (17)
(θi − rd ) Ci+1dt − (ai+1 −η) dWs,

i = 4, 5, . . . , n − 1, and θ4 = 4,θ5 = 5,θ6 = 6, . . . ,θn = n,θi > 0
(integer)= staff loading.
Taking into 12, 13 and 18, one obtains the wealth process

dY (t) usY (t)
[
µdt + kS βdWt

]
+ (1 − us) Y (t) rdt + (θi − rd ) Ci+1dt − (ai+1 −η) dWs
us Y (t) µdt + usY (t) kS

βdWt + Y (t) rdt + us Y (t) rdt
+ (θi − rd ) Ci+1dt − (ai+1 −η) dWs
(us Y (t) µ + Y (t) r − usY (t) r + (θi − rd ) Ci+1) dt
+us Y (t) kS

βdWt − (ai+1 −η) dWs (18)

i = 4, 5, . . . , n − 1, and θ4 = 4,θ5 = 5,θ6 = 6, . . . ,θn = n,θi > 0
(integer)= staff loading.
Based on the wealth process in 19, the Pension manager seeks
a strategy, u∗t , which maximizes the utility function, such that
u∗t = max E (U (Y (T ))) ,∀u(t). Where u(•) is an increasing
concave utility function, which satisfies the Inada conditions;
U′(+∞) = 0,, and U′(0) + ∞, (cf. [10]).

5.2. Equations for Utility Functions and Optimal Policies
For CRRA Utility Function in the Pension Distribution Phase

under Noninflationary Market

h (t, r, z) = z
1

p−1 e
d

r
1−p −r+

1
q(2−p) −

u2s
k2 (1−p)

+
u2s (2−p) j

2k2 (1−p)
e[T−t]

−ert
[
2k (θi − rd ) Ci+1 + αγus (ai+1 −η)

]
[T − t] (19)

and

u∗s =
αγ (ai+1 −η)

2X (t) k
− uz ( p − 1) z

2p−3
p−1

×e
d

r
1−p −r+

1
q(2−p) −

u2s
k2 (1−p)

+
u2s (2−p) j

2k2 (1−p)
e[T−t]

(20)

where as in [16], c1 =
1
2 (ai+1 −η)

2
+ 38 (αγ)

2 (ai+1 −η)
2, as

required. For CARA Utility Function in the Distribution Phase
under Noninflationary Market

g∗ (t.s.z) = q−1er(T−t) ln z + q−1er(T−t)
(∫

w (T ) dT−

∫
w (t) dt + e−

(αγ)2 k(ai−1−η)
2 q

∫
s(t)dt

)
−q−1er(T−t)

u2L
2k2

[∫
e−

(αγ)2 k(ai−1−η)
2 q

∫
s(t)dtdt

−

∫
e−

(αγ)2 k(ai−1−η)
2 q

∫
s(T )dT dT

]
+e−rt

[∫
ert

2k (θi − rd ) Ci+1 + αγuL (ai+1 −η)
2k

dt

−erT
2k (θi − rd ) Ci+1 + αγuL (ai+1 −η)

2k
dT

]
(21)

and

u∗s =
αγ (ai+1 −η)

2X (t) kS β
+ uLe

(T−t) αγ

2qX (t) S β−1
d

dS

×

(
e−(αγ)

2 k(an+1−η)q
∫

S (t)dt
∫

e−(αγ)
2 k(an+1−η)q

∫
S (T )dT dT

)
(22)

where q > 0, µ = uL + r (See [17]). For CRRA Utility Function
in the Accumulation Phase under Inflationary Market

u∗S =
σIp

σss
−


−λ1σ

s
s −λ2σ

I
sθI + σ

I
sσ

I
p + σ

I
s

(
θI −

σIp
2

)
σss


×

1
p − 1

e
{
a±(a2−2k2 H)1/2 k−12 t

}

e
{
a±(a2−2k2 H)1/2 k−12 T

}
 e

{
a±(a2−2k2 H)1/2 k−12 T

}

e
{
a±(a2−2k2 H)1/2 k−12 t

}

−
ρ∗z

1
p−1

k
(
1 − e−ρ∗(T−t)

)  . (23)
u∗B =

σIp

σI
−
σIpσ

I
s

σssσI

+


[(
σIs

)2
θI −σ

I
sσIσ

I
p −σ

I
sλ1σ

s
s

]
(σss)

2

−

(
σIs

)2
λ2θI +

(σIs)
2
σIp

2 +
σssσ

I
sσ

I
p

2

(σss)
2


1

p − 1

+
e

{
a±(a2−2k2 H)

1
2 k−12 t

}

e

{
a±(a2−2k2 H)

1
2 k−12 T

}
 e

{
a±(a2−2k2 H)

1
2 k−12 T

}

e

{
a±(a2−2k2 H)

1
2 k−12 t

}

−
ρ∗z

1
p−1

k
(
1 − e−ρ∗(T−t)

) 
+

θI
p − 1

e

{
a±(a2−2k2 H)

1
2 k−12 t

}

e

{
a±(a2−2k2 H)

1
2 k−12 T

}
 e

{
a±(a2−2k2 H)

1
2 k−12 T

}

e

{
a±(a2−2k2 H)

1
2 k−12 t

}

−
ρ∗z

1
p−1

σI k
(
1 − e−ρ∗(T−t)

)  ,
ρ∗ =

1
2
ρ5 + ρ1, σ

I
p = σ

s
p (24)

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Osu et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 170–179 174

where ρ1, ρ2, . . .ρ5 are as in [16].

H =
ρ5

2 (1 − p)
+

ρ1
1 − p

−
1
2
ρ5 −

1
2
ρ1 +

2
(
2θ2I

1
2

(
σIp

)2
− θIσ

I
p + ρ2 + ρ4

)
1 − p

−

(2 − p)
(
2θ2I

1
2

(
σIp

)2
− θIσ

I
p + ρ2 + ρ4

)
(1 − p)2

N (t) =
2b [t − T ]

k1
,

d1 =
4b
2k1

,

d2 = 0,
(25)

6. Sensitivity Analysis

For CRRA Utility Function in the Pension Distribution Phase
under Noninflationary Market; Setting αγ = 0 (that is, saying
that stock and time have orthogonal relationship), and ai+1−η =
0 (that is, no money is paid to the Next-of-kin of the dead con-
tributors) in equations ?? and 20 yield, respectively;

h (t, s, z) = z
1

p−1

e
d

r
1−p −r+

1
q(2−p) −

u2s
k2 (1−p)

+
u2s (2−p) j

2k2 (1−p)
e[T−t]

−ert [2k (θi − rd ) Ci+1] [T − t] , (26)

u∗s = −uz (p − 1) z
2p−3
p−1

e
−d

r
1−p −r+

1
q(2−p) −

u2s
k2 (1−p)

+
u2s (2−p) j

2k2 (1−p)
e[T−t]

(27)

and

h (t, s, z) = z
1

p−1

e
d

r
1−p −r+

1
q(2−p) −

u2s
k2 (1−p)

+
u2s (2−p) j

2k2 (1−p)
e[T−t]

−e−rt [2k (θi − rd ) Ci+1] [T − t] , (28)

u∗s = −us ( p − 1) z
2p−3
p−1

e
−d

r
1−p −r+

1
q(2−p) −

u2s
k2 (1−p)

+
u2s (2−p) j

2k2 (1−p)
e[T−t]

(29)

For CARA Utility Function in the Pension Distribution Phase
under Noninflationary Market, setting αγ = 0 (that is, saying
that stock and time have orthogonal relationship), and ai+1 −
η = 0 (that is, no money is paid to the Next-of-kin of the dead
contributors) in equations 21 and 22 yield, respectively;

h (t.s.z) = q−1
[
er(T−t)

(
ln z +

∫
w (T ) dT−∫

w (t) dt +
u2L
2k2

[∫
(dt − dT )

] +

Table 1. Parameters and their respective values .
Name of Parameters Symbol Used Values
Constant rate of interest r 0.02
Expected stock returns µ 0.10
Instantaneous stock returns us = uL 0.07
Stock volatility k 0.55
Risk aversion q = p 0.50
Rate of contribution rd 0.075
Management fee ηi 0.025

e−rt
[∫

ert
2k (θi − rd ) Ci+1

2k
dt

−erT
2k (θi − rd ) Ci+1

2k
dT

]
(30)

u∗L = uLe
r(T−t) (31)

and

h (t.s.z) = −q−1
[
er(T−t)

(
ln z +

∫
w (T ) dT

−

∫
w (t) dt +

u2L
2k2

[∫
(dt − dT )

]
+e−rt

[∫
ert

2k (θi − rd ) Ci+1
2k

dt

−erT
2k (θi − rd ) Ci+1

2k
dT

]
, (32)

u∗S = uLe
−iωt
−

αγ

2Y (t) S β−2
q−1 F

′

(S ) , (33)

where= F
′

(S ) = 1q
d

dS

(∫
dT

)
. For CARA Utility Function

in the Pension Accumulation Phase under Inflationary Market;
Setting σIp = σ

I
s = θI = 0 , makes the optimal strategies in

equations 23 and 24 would be of the form of the [4]. What this
means is that in the absence of inflation vis-a-vis risk premium
due to inflation, we would have a model of the form of [18].

7. Numerical Simulation

A numerical example of the proposed model was given to
demonstrate the dynamic behavior of a DC pension fund and
optimal investment strategy. Nigeria-National Pension Fund
Administration (NNPFA) real data was used to illustrate the ef-
ficiency of the model. The parameters used are summarized in
the table below, for T = 30. t = 0, 5, 15, 20, 25, 30, with i =
4, 5, 6, . . . , n − 1 and θ4 = 4,θ5 = 5,θ6 = 6,θ7 = 7, . . . ,θn = n.
Assume for perfect correlation αγ = 1, ai+1 = 274, 464.68 ×
30 = 8233940.4 as at 30th June (for April, 2016), 2019, X (t) =
0.07 × 8233940.4 + 82333940.4 = 576375.83 + 8233940.4 =
8810316.2 , β = 0, S (t) = 7657564.6, X (t0) = 8233940.4 ,
which is initial Pension wealth for 30 contributors, from Jan-
uary 2015 to April 2016 of IMSU Academic Staff of 2014 set
(TRUST FUND PFA Returns of June, 2019).

For CRRA Utility Function in the Pension Distribution Phase
under Noninflationary Market, below are various three-dimensional

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Osu et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 170–179 175

Table 2. The numerical values for the optimal policy and the Dual variable for the time period from 0 to 29th year, for the CRRA model.
U 1.57 1.52 1.51 1.50 1.49 1.39 0.38 0.38 0.34 0.32 0.46 0.42 0.41 0.34 0.32
Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
U 1.46 0.42 0.41 0.34 0.32 1.41 0.40 1.39 0.39 0.38 0.38 0.34 0.32 0.46 0.42
Z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
T 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Table 3. Values for optimal policy, dual variable and continuous time for CRRA model, when there is orthogonal relationship between time and stock.
U 0.47 0.042 0.041 0.040 0.039 0.038 0.038 0.34 0.32 0.46 0.42 0.41 0.34 0.32 0.46
Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
U 0.42 0.41 0.34 0.32 0.41 1.40 0.39 0.39 0.38 0.38 0.34 0.32 0.46 0.42 0.42
Z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Figure 1. The graph of dual variable and instantaneous time against
optimal policy for CRRA Mode.

tables (that is, Tables 1, 2 and 3 and their plots respectively in
Figures 1, 2 and 3).

Table 4 above are various three-dimensional used in the plot
of Figure 3.

For CARA Utility Function in the Distribution Phase un-
der Noninflationary Market, Table 6 below are various three-
dimensional used in the plot of Figure 6.

Table 7 above are various three-dimensional used in the plot
of Figure 7.

Table 8 above are various three-dimensional used in the plot
of Figure 8.

8. Discussion

For CRRA Utility Function in the Distribution Phase under
Noninflationary Market; Result shows that if we set αγ = 0 and
ai+1−η = 0 , then in equations 24 and 27, the orthogonal relation
between stock and time has the same effect on contributors’
satisfaction.

Figure 2. The graph of dual variable and instantaneous time against
optimal policy for CRRA Model, when there is orthogonal relationship
between time and stock.

In equations 26 and 28, we have the same negative effect
(i.e., a decline in stock investment). Considering the dual re-
lationship between pension wealth, it follows that the optimal
strategy will appreciate frictionless, which is unrealistic. There
exists a modifying factor which depends on time, only, just like
in [13] and this modifying factor controls investment decision
of the PPM. From proposition 4.4.1 in [14], result shows that
the elastic parameter, β is not equal to 1 (the elastic parameter
β , 1 ).

For CARA Utility Function in the Distribution Phase under
Noninflationary Market, results show that if we set αγ = 0 and
ai+1 −η = 0 then in equation 30 and 32, the orthogonal relation
between stock and time has the same effect on contributor’s sat-
isfaction. However, both have significant effect. In equations 31
and 33, investments made in stock slightly reduce, vis-a-vis the
stock returns. Though, more money is invested in cash account
than stock when there is orthogonal relationship between stock
and time. Here, there are minimal stock returns at each trading

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Table 4. Values for optimal policy, dual variable and continuous time for CRRA model, when no payment was made to the next of kin of the dead contributors.
U 0.047 0.042 0.041 0.040 0.039 0.039 0.038 0.38 0.34 0.32 0.46 0.42 0.41 0.34 0.32
Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
U 0.46 0.42 0.41 0.34 0.32 1.41 0.40 0.39 0.39 0.38 0.38 0.34 0.46 0.32 0.46
Z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Table 5. Values for optimal policy, dual variable and continuous time for CRRA model, for fixed values ranging from 1-3.
T Z U T Z U t z U
0 1 0.01126233265 0 2 0.1801973224 0 3 0.9122489445
1 1 0.01163316951 1 2 0.1861307122 1 3 0.9422867306
5 1 0.01324271030 5 2 0.2118833648 5 3 1.072659535
10 1 0.01557131915 10 2 0.2491411064 10 3 1.261276851
15 1 0.01830939246 15 2 0.2929502793 15 3 1.483060789
20 1 0.02152893078 20 2 0.3444628925 20 3 1.743843393
25 1 0.02531459532 25 2 0.4050335251 25 3 2.050482221
30 1 0.02976593415 30 2 0.4762549464 30 3 2.411040666
35 1 0.03500000000 35 2 0.5600000000 35 3 2.835000000

Table 6. Values for optimal policy, dual variable and continuous time for CARA model.
U 1.57 1.52 1.51 1.50 1.49 1.39 0.38 0.38 0.34 0.32 0.46 0.42 0.41 0.34 0.32
Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
U 1.46 0.42 0.41 0.34 0.32 1.41 0.40 1.39 0.39 0.38 0.38 0.34 0.32 0.46 0.42
Z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Table 7. Values for optimal policy, dual variable and continuous time for CARA model, when there is orthogonal relationship between time and stock.
U 0.47 0.042 0.041 0.040 0.039 0.038 0.038 0.34 0.32 0.46 0.42 0.41 0.34 0.32 0.46
Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
U 0.42 0.41 0.34 0.32 0.41 1.40 0.39 0.39 0.38 0.38 0.34 0.32 0.46 0.42 0.42
Z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Table 8. Values for optimal policy, dual variable and continuous time for CARA model, when there no payment was made to the next of kin of the dead contributors.
U 0.047 0.042 0.041 0.040 0.039 0.039 0.038 0.38 0.34 0.32 0.46 0.42 0.41 0.34 0.32
Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
U 0.46 0.42 0.41 0.34 0.32 1.41 0.40 0.39 0.39 0.38 0.38 0.34 0.46 0.32 0.46
Z 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
t 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

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Osu et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 170–179 177

Figure 3. The graph of dual variable and instantaneous time against
optimal policy for CRRA Model, when there is no payment made to
the next of kin of the dead contributors.

Figure 4. Two dimensional view of CRRA Model, for fixed values of
the dual variable.

period. This suggests that diversification of investments should
be advised in order to enhance investment returns.

There exists a modifying factor, which depends on both
time and the coefficient of the utility function, just like in [10],
and this modifying factor controls investment decision of the
PPM. Particularly, it reflects the investor’s decision to hedge
the volatility risk.

For CRRA Utility Function in the Accumulation Phase un-
der Inflationary Market, the effect of inflation on pension wealth
investment; the CRRA utility function has little or no effect on

Figure 5. The three dimensional view of the CRRA model, using Maple
of Table 5.

Figure 6. The graph of dual variable and instantaneous time against
optimal policy for CARA Model.

the investment strategy. Recall from [4], the coefficients d1, d2
degenerates to 4b2k1 and zero, in the absence of the coefficient of
the CRRA (i.e., as p → 0 ), however, in this work, even in the
presence of the coefficient of CRRA, the coefficients d1, d2 are
already degenerate. For the effect of hedging, using Inflation-
indexed bond and Inflation-linked stock, the Inflation-indexed
Bond and Inflation-linked Stock served as a hedging mecha-
nism against inflation. In the absence of the coefficient of the
CRRA (i.e., as → 0 ), the coefficients d1, d2 , still retains its
value (i.e., will never degenerate further than this). There exists
also a modifying factor, which depends on time, only, just like
in [12], for CRRA utility function, and this modifying factor
controls investment decision of the PPM. There exists a mod-
ifying factor which depends on both time and the coefficient

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Osu et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 170–179 178

Figure 7. The graph of dual variable and instantaneous time against
optimal policy for CARA Model, when there is orthogonal relationship
between time and stock.

Figure 8. The graph of dual variable and instantaneous time against
optimal policy for CARA Model, when no payment was made to the
next of kin of the dead contributors.

of the CARA utility function as in [10], for the CARA utility
function. This modifying factor controls investment decision of
the PPM. Particularly, it reflects the investor’s decision to hedge
the volatility risk. There exists an elastic parameter that takes
values other than unity. More so, the Inflation-linked Bond and
Inflation-indexed Stock served as hedging mechanism in the op-
timization of Pension wealth. Lastly, comparing the utilities in
the noninflationary market, during decummulation phase, we
conclude that the policy, u∗s (t) due to CRRA utility function
is the optimal policy, while in that of the inflationary market,
during accumulation phase, the policies, u∗s (t), and u

∗
B(t) due

to CRRA utility function is not the best fit, based on the as-
sumptions of our models .Based on the result above, there ex-
ists some optimal policies that can permit energetic retirees to
invest in risky assets.

9. Conclusion

For the noninflationary market, we studied and modified the
optimal investment strategy for annuity contract under the con-
stant elasticity of variance by [10], and proved that the elas-
tic parameter takes values other than unity. We constructed an
optimal investment strategy in the pension benefit distribution
phase, for both CRRA and CARA utility functions, and the as-
sociated utility functions (see equations ??, 20, 21, and 21).
We tested for sensitivity of some parameters in both the invest-
ment strategies and the utility functions (see sensitivity analy-
sis). We simulated and compared the expected utilities of the
accumulation phase (see Figures 1 through 8 above using Ta-
bles 1 through 8 as appropriate). Based on these comparisons,
we conclude that the investment strategy with the CRRA util-
ity function provides the optimal strategy, though, the invest-
ment returns is still low, and this suggests that diversification
of investment should be advised to enhance returns. More so,
there exists an investment strategy that permits investments into
risky assets, even after retirement. We conclude that both for
CRRA and CARA utility, the orthogonal relationship between
stock and time have the same effect on contributor’s satisfac-
tion. There exists a modifying factor which depends on both
time and the coefficient of CRRA utility function, and also de-
pends on time, only just like in [10], and this modifying factor
controls investment decision of the Pension Plan Member. Par-
ticularly, it reflects the investor’s decision to hedge the volatility
risk.

For inflationary market, we constructed an optimal invest-
ment strategy in the pension accumulation phase, for the CRRA
utility function and the associated expected utility functions
(see equations 23 and 24). We also tested for sensitivity of pa-
rameters in both the investment strategies and the utility func-
tions. We conclude that the CRRA utility approach has little
or no effect on the investment strategy, and this depicts the ef-
fect of inflation on optimal investment strategy. More so, the
Inflation-linked Bond and the Inflation-Indexed Stock serve as
hedging mechanism in the optimization of pension wealth.

As a result of uncertainties of the markets, the substitute in-
vestment and the encouragement of higher investment yield, for
the noninflationary markets, we recommend diversified invest-
ment of pension wealth into assets like government Bond and
even corporate (private) Bond. For the inflationary market (in
order to cushion the effect of inflation on pension wealth gener-
ation in a DC pension scheme) we recommend the investigation
of the effect of extra stochastic contributions.

Acknowledgments

We thank the referees for the positive enlightening com-
ments and suggestions, which have greatly helped us in making
improvements to this paper.

References

[1] Nigerian Pension Reform Act of (2004).

178



Osu et al. / J. Nig. Soc. Phys. Sci. 2 (2020) 170–179 179

[2] A. J. G. Cairns, D. Blake & K. Dowd “Stochastic lifestyling: optimal
dynamic asset allocation for defined contribution pension plans,” Journal
of Economic Dynamics & Control 30 (2006) 843.

[3] L. Wang & Z. Chen “Nash Equilibrium Strategy for a DC Pension Plan
with State-Dependent Risk Aversion: A Multi period Mean-Variance
Framework”, Hindawi Discrete Dynamics in Nature and Society Article
ID 7581231, (2018) 17 https://doi.org/10.1155/2018/7581231.

[4] Z. Chubing & R. Ximing “Optimal investment strategies for DC pension
with stochastic salary under affine interest rate model”, Hindawi Publish-
ing Corporation (2013) 15 http://dx.doi.org/10.1155/2013/297875.

[5] P. Battocchio & F. Menoncin “Optimal pension management in a stochas-
tic Framework”, Insurance 34 (2004) 79.

[6] J. F. Boulier, S. Huang & G. Taillard “Optimal management under
stochastic interest rates: the case of a protected defined contribution pen-
sion fund”, Insurance 28 (2001) 173.

[7] G. Deelstra, M. Grasselli & P. F. Koehl “Optimal investment strategies in
the presence of a minimum guarantee”, Insurance 33 (2003) 189.

[8] J. Gao “Stochastic optimal control of DC pension funds”, Insurance 42
(2008) 1159.

[9] J. Xiao, Z. Hong & C. Qin “The constant elasticity of variance (CEV)
model and the Legendre transform-dual solution for annuity contracts”,
Insurance 40 (2007) 302.

[10] J. Gao “Optimal portfolios for DC pension plans under a CEV model”,
Insurance: Mathematics and Economics 44 (2009) 479.

[11] O. Basimanebotlhe & X. Xue “Stochastic Optimal Investment under Infla-
tionary Market with Minimum Guarantee for DC Pension Plans”, Journal

of Mathematics Research 7 (2013) 1.
[12] J. Gao “Optimal investment strategy for annuity contracts under the con-

stant elasticity of variance (CEV) model”, Insurance 45 (2009) 9.
[13] K. N. C. Njoku & B. O. Osu “On the Modified Optimal Investment

Strategy for Annuity Contracts under the Constant Elasticity of Variance
(CEV) Model”, Earthline Journal of Mathematical Sciences 1 (2019) 63.

[14] B. O. Osu, K. N. C. Njoku & B. I. Oruh “On the Effect of Inflation and
Impact of Hedging on Pension Wealth Generation Strategies under the
Geometric Brownian Motion Model”, Earthline Journal of Mathematical
Sciences 1 (2009) 119.

[15] M. Jonsson & R. Sircar “Optimal investment problems and Volatility ho-
mogenization approximations”, in Modern Methodsin Scientific Comput-
ing and Applications 75 (2002) 255.

[16] E. E. Akpanibah, B.O. Osu, K. N .C. Njoku & E. O. Akak “Optimization
of Wealth Investment Strategies for DC Pension Fund with Stochastic
Salary and Extra Contributions”, International Journal of Partial Differ-
ential Equations and Applications 5 (2017) 33.

[17] K. N. C. Njoku, B. O. Osu, E. E. Akpanibah & R. N. Ujumadu “Effect of
Extra Contribution On Stochastic Optimal Investment Strategies for DC
Pension with Stochastic Salary under the Affine Interest Rate Model”,
Journal of Mathematical Finance 7 (2017) 821.

[18] P. K. Mwanakatwe, L Song & E Hagenimana “Management Strategies for
a Defined Contribution Pension Fund under the Hull-White Interest Rate
Model”, Advances in Intelligent Systems Research (AMMS) 153 (2017)
239.

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