Abstract
The aim of this paper is to maximize an investor’s terminal wealth which exhibits constant relative risk aversion (CRRA). Considering the fluctuating nature of the stock market price, it is imperative for investors to study and develop an effective investment plan that considers the volatility of the stock market price and the fluctuation in interest rate. To achieve this, the optimal investment plan for an investor with logarithm utility under constant elasticity of variance (CEV) model in the presence of stochastic interest rate is considered. Also, a portfolio with one risk free asset and two risky assets is considered where the risk free interest rate follows the Ornstein-Uhlenbeck (O-U) process and the two risky assets follow the CEV process. Using the Legendre transformation and dual theory with asymptotic expansion technique, closed form solutions of the optimal investment plans are obtained. Furthermore, the impacts of some sensitive parameters on the optimal investment plans are analyzed numerically. We observed that the optimal investment plan for the three assets give a fluctuation effect, showing that the investor’s behaviour in his investment pattern changes at different time intervals due to some information available in the financial market such as the fluctuations in the risk free interest rate occasioned by the O-U process, appreciation rates of the risky assets prices and the volatility of the stock market price due to changes in the elasticity parameters. Also, the optimal investment plans for the risky assets are directly proportional to the elasticity parameters and inversely proportional to the risk free interest rate and does not depend on the risk averse coefficient.
Highlights
The aim of this paper is to maximize an investor’s terminal wealth which exhibits constant relative risk aversion (CRRA)
In [6], the rein- siderations fluctuations in interest rate as well as the volatility surance problem and optimal investment under the constant elasticity of variance (CEV) model of the stock market prices. We do this by studying the optiwas studied. [1], studied the optimal investment problem with mal investment plans of an investor exhibiting the CRRA and taxes, dividend and transaction cost using different utility func- whose risky assets and risk free interest rate are modelled by tions under the CEV process. [7, 8] solved the optimal invest- the CEV process and the O-U process respectively
Furtherment problem for a defined contribution (DC) pension plan with more, the Legendre transformation and asymptotic technique return of premiums clauses under different assumptions and as- are used to determine asymptotic solutions of the optimal insumed that the stock market price follows the CEV process; vestment plan
Summary
Considering an investor with logarithm utility, substitute Solving equation (36) for h, we obtain the solution into (30) and (31) for the optimal investment plan using change of variable and asymptotic method. 4. Investor’s Optimal Investment Plan under Logarithm Utility g (t, r, s1, s2, z) =. Since our interest here is to determine the optimal investment plan for the investor with CRRA utility, we choose the logarithm utility function similar to the one in [20, 21]. To prove the Proposition above, we attempt to solve equations (37) and (38) by stating and proofing the following lemmas The solution of equation (37) is given as e Differentiating g with respect to r, s1, s2, z and substitute into equation (30) and (31), we have φ∗1. From Lemma 4, we observed that our result is similar to the one in [3] but the difference between our result and theirs is that their interest rate is a constant while ours is stochastic
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