Journal of Mathematical Finance
Vol.3 No.1(2013), Article ID:28399,8 pages DOI:10.4236/jmf.2013.31012
Optimal Portfolio Strategy with Discounted Stochastic Cash Inflows
Department of Mathematics, Faculty of Physical Sciences, University of Benin, Benin City, Nigeria
Email: nkekicharles2003@yahoo.com
Received March 19, 2012; revised June 10, 2012; accepted June 25, 2012
Keywords: Optimal Portfolio; Stochastic Cash Inflows; InflationLinked Bond; Variational Form; Intertemporal Hedging Terms
ABSTRACT
This paper examines optimal portfolios with discounted stochastic cash inflows (SCI). The cash inflows are invested into a market that is characterized by inflationlinked bond, a stock and a cash account. It was assumed that inflationlinked bond, stock and the cash inflows are stochastic and follow a standard geometric Brownian motion. The variational form of Merton portfolio strategy was obtained by assuming that the investor chooses constant relative risk averse (CRRA) utility function. The intertemporal hedging terms that offset any shock to the SCI were obtained. A closed form solution to our resulting nonlinear partial differential equation was obtained.
1. Introduction
This paper consider the optimal portfolio strategies with valuing expected discounted cash inflows. It was assumed that the underlying assets and cash inflow process follow a standard geometric Brownian motion. The investment of the investor’s SCI into a cash account, an inflationlinked bond and a stock was considered. The SCI and stock price were correlated with inflation and stock market risk.
In a related literature, [1] studied an optimal investment problem in a continuoustime framework where the interest rate follows the CoxIngersollRoss dynamics. They obtained a closed form solution for the optimal investment strategy under a complete market framework. They assumed that the investor chooses CRRA utility function. [2] considered the process of finding the optimal portfolio, optimal consumption and the efficient frontier for a small agent in an economy. They also considered financial market that composed of two sources of uncertainties: an mdimensional Brownian motion and a continuous time Markov chain. They found theoretically, that the regimes of an economy have a significant impacts on the portfolio and consumption decision as well as on the efficient frontier of a small investor. [3] studied the present value and overcome the difficulty of independence by reversing the order of the cash flow. He found that similar recursive formulas for the present value is also applicable to the future value of the expected returns. [4] considered the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. They proposed a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. [5] studied how portfolio composition changes with individual wealth. [6] considered the present value of expected future contribution into the pension fund. [7] investigated the stochastic dynamics of depository financial institution assets, liabilities and capital under the influence of macroeconomic factors. They adopted dynamic programming techniques in their optimization process. They provided an analysis of the economic aspects of the depository financial institution modeling. [8] studied a portfolio problem of a fund manager who wants to maximize the expected utility of his terminal wealth in a complete financial market. He found that the optimal portfolio is formed by three components: a speculative, an hedging, and a preferencefree hedging component. He obtained a close form solution to the asset allocation problem. [9] considered the optimal portfolio selection problem with portfolio constraints. They derived the general utility function using the martingale approach. They found that using CRRA utility function, optimal policies can be obtained explicitly when there are minimum capital requirements. ([10,11]) studied the optimal portfolio management in the accumulation phase of a defined contribution pension scheme. They obtained portfolio values with hedging strategies for a pension plan member. In this paper, we consider the optimal portfolio and investment strategies involving cash inflow valuation over time. We obtain explicitly, analytical solution to our resulting HJB equation. The riskfree rate is assume to be deterministic.
The remaining parts of the paper is structured as follows: In Section 2, we described the structure of the financial market; In Section 3, we consider the dynamics of the discounted present value of expected SCI process; In Section 4, we consider the wealth process of the investor; In Section 5, we consider the valuation of the discounted present value of expected SCI process of the investor as well as the sensitivity of the present value of the SCI; In Section 6, we consider the optimization program and optimal portfolio and optimal solution of the investor’s wealth using CRRA utility function; Section 7 concludes the paper.
2. The Model
Let be a probability space. Let
, where
. The Brownian motions
is a 2dimensional process, defined on a given filtered probability space
, , where is the real world probability measure, the time period, the terminal time period, is the Brownian motion with respect to source of uncertainty arising from inflation and is the Brownian motion with respect to source of uncertainty arising from the stock market. It is assumed that the market is arbitragefree, complete and continuously open between time period and.
Financial Model
The dynamics for the cash account with the price at time is given by
(1)
where is the short term interest rate. The stock price at time is given by the dynamics:
(2)
where, is the expected growth rate of stock price,
and is the volatility of stock. The price of the inflationlinked bond is given by the dynamics:
(3)
where, is the volatility of inflationlinked bond, is the market price of inflation risk, is the inflation index at time and has tha dynamics:
where is the expected rate of inflation, which is the difference between nominal interest rate, real interest rate and is the volatility of inflation index. Since the market is complete, we have that
(4)
(5)
Therefore, the market price of market risk is given by
(6)
where, is the market price of stock market risk. The exponential process
is assumed to be a martingale. We now define the stateprice density function by
3. Dynamics of Stochastic Cash Inflows
The dynamics of the stochastic cash inflows with price process, is given by
(7)
where, is the volatility of the cash inflows and is the expected growth rate of the cash inflows. is the volatility arising from inflation and is the volatility arising from the stock market. Figure 1 presents the simulated diffusion paths of (7).
Figure 1 was obtained by setting,
, , , , ,
and.
Applying It lemma to (7), we obtain
(8)
Figure 1. Simulated diffusion paths of the stochastic cash inflows.
4. The Wealth Process
Let be the wealth process and
be the portfolio value at timewhere is the portfolio value in inflationlinked bond and is the portfolio value in stock at time t. Then, is the portfolio value in cash account at time. Therefore, the dynamics of the wealth process is given by
(9)
Substituting (1), (2) and (3) into (9), we obtain
(10)
5. The Discounted Value of SCI
In this section, we determine the value of expected discounted SCI.
Definition 1:
The discounted value of the expected future SCI is defined as
(11)
where, is the conditional expectation with respect to the Brownian filtration and
is the stochastic discount factor which adjusts for nominal interest rate and market price of risks for stock and inflationlinked bond.
Proposition 1:
Suppose is the discounted value of the expected future SCI, then
(12)
Proof. By definition 1, we have that
(13)
Applying change of variable on (13), we have
(14)
Applying parallelogram law and martingale principles on (14), we have
Therefore,
(15)
Integrating, we have
(16)
Figure 2 was obtained by setting, k = 0.099, r = 0.04, , , , , , and.
At, we obtain the present discounted value of future SCI to be
(17)
If and we allow i.e.,
For a deterministic case, we have the present discounted value future deterministic cash inflows to be
Figure 2. The flow of the discounted value of SCI.
(18)
Hence,
This shows that as, converges to
, provided.
For the deterministic case, it shows that as,
converges to, provided.
Figure 3 was obtained by setting, k = 0.099, r = 0.04, , , , , , and.
Figure 2 represents the flow of the discounted value of stochastic cash inflow in the investment at time and Figure 3 represents the present value of discounted future SCI at time
We now consider the sensitivity analysis of Proposition 2 establishes this fact.
Proposition 2:
Let, then
Proof. The results follow by taking the partial derivatives of with respect to, , , and, respectively.
Figure 3. The present discounted value of future SCI.
Table 1 shows the sensitivity analysis of the discounted value of the SCI.
Proposition 3:
Suppose that Proposition 1 holds, then
(19)
Proof. Finding the differential of both sides of (16) and then substitute (7), we have
Let be a value process at time. We defined
Table 1. Simulation of the sensitivity analysis.
as
(20)
where, satisfy (10) and satisfy (19).
Proposition 4:
Let satisfy (20), satisfy (10) and satisfy (19), then
(21)
Proof. Finding the differential of both sides of (20) and then substitute in (10) and (19), the result follows.
6. Optimization of the Value of Wealth Process
We define the general value function
where is the path of. Define to be the set of all admissible portfolio strategy that are Fprogressively measurable, that satisfy the integrability conditions and let be a concave function in such that satisfies the HJB equation
Therefore, by applying Itô lemma on (21), we obtain the following HJB equation:
(22)
subject to:
where,
(23)
Let be the solution of the HJB equation (22). Since the utility function is concave and the value function is smooth, i.e., , then (22) is welldefined. Hence, we have the following:
(24)
from (24), we have
(25)
Substituting (25) into (22), we obtain the following HJB equation:
(26)
This is the resulting HJB equation of our problem.
Proposition 5:
The solution of the HJB equation (26) is of the form
with
(27)
Proof. Finding the partial derivatives of
with respect to, , , ,
and and then substitute into (26), we obtain the following:
(28)
From (28), we obtain
(29)
Using (29), we obtain
(30)
Figure 4 was obtained by setting, , , , , , , , , and.
Figure 4 shows the expected value of utility of wealth at time for different values of. Here, measures the level of risk the investor is willing to take. Observe that the smaller the value of, the higher the expected value of wealth, and vice versa. Therefore, when, the expected value of utility of wealth, , when, , when, and when,. Hence, the lower the rate at which investors are risk averse, the more the wealth that will accrue to them, and vice versa, which is an intuitive result.
Proposition 6:
Suppose that is the solution of the HJB Equation (26), then the optimal portfolio in inflationlinked bond, stock and cash account are given by
(31)
Figure 4. The expected value of utility of wealth for different values of γ.
(32)
Proof. From (25), we have that
But,
Therefore,
It implies that
(33)
(34)
(35)
From (33), the first term represents the classical portfolio strategy while the second term represents the intertemporal hedging strategy that offset any shock to the SCI at time t. From (34),
represents the classical portfolio strategy at time and
represents the intertemporal hedging term that offset shock resulting from the SCI at time. From (35), observe that these hedging terms can be transfer to cash account at time or it can be reinvest in stock and in inflationlinked bond at time.
Figure 5 was obtained by setting, , , , , , , , , , and. The optimal portfolio value in inflationlinked bond at time is obtained as 0.16 (or 16%).
Figure 6 was obtained by setting, , , , , , , , , , and. The optimal portfolio value in stock at time is obtained as 1.001562 (or 100.1562%).

In Figures 57, we set, , , , , , , , , , and. Figure 5 shows the portfolio value in inflationlinked bond. We found that the optimal portfolio value in inflationlinked bond at time is 0.16 (or 16%). Figure 6 shows the portfolio value in stock. We found that the optimal portfolio value in stock at time is 1.001562 (or 100.1562%). Figure 7 shows the portfolio value in cash
Figure 5. Portfolio value in inflationlinked bond.
Figure 6. Portfolio value in stock.
Figure 7. Portfolio value in cash account.
account. We found that the optimal portfolio value in cash account at time t = 10 is −0.161562 (or −16.1562%).
7. Conclusion
The optimal portfolio strategy X with discounted stochastic cash inflows was considered. It was assumed that the cash inflow, stock and inflationlinked bond are stochastic and follow a standard geometric Brownian motion. The sensitivity analysis of the present value of the discounted cash inflows was carried out in this paper and the results are presented in Table 1. Analytical solution to the resulting HJB equation was obtained. It was found that the smaller the value of (which measure the level of risk the investor is willing to take), the higher the expected value of wealth, and vice versa. The optimal portfolio values in stock, inflationlinked bond and cash account were obtained. The resulting optimal portfolio values in stock and inflationlinked bond were found to involve intertemporal hedging terms that offset any shock to the SCI.
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