Optimal Allocation under a Stochastic Interest Rate and the Costs from Suboptimal Allocation

Ditlev Muhlig Melgaard & Kristian Vandel Nørgaard

Student thesis: Master thesis


The purpose of this thesis is to analyse investor’s dynamic asset allocation strategies, when introducing a stochastic interest rate, and a non-constant market price of risk. This is followed by an evaluation of the costs of applying a suboptimal portfolio allocation strategy. The starting point is the classical static portfolio result of the mean-variance analysis developed by Markowitz (1952). This is followed by a demonstration of the intertemporal portfolio problem under constant investment opportunities from Merton (1969). The problem is solved by dynamic programming using the Hamiliton-Jacobian-Bellman equation to obtain portfolio result, the socalled myopic portfolio. The Merton’s portfolio problem serves as foundation for the extensions in this thesis. In the first extension, a dynamic portfolio choice model includes the interest rate as state variable, where interest rate is modelled by a one-factor Vasicek model, which introduces the set of stochastic investment opportunities. The result of this model is a closed-form solution and is considered for a CRRA-investor, which shows that investors should hold the myopic portfolio and a hedging portfolio, which contains assets that are correlated with the state variable, the interest rate. From the analysis, the model results show that the stock allocation is time-invariant and decreasing in risk aversion. The bond allocation is increasing in investment horizon and risk aversion, since bonds through a hedging term is used to minimize exposure from the interest rate risk. In the second extension, a dynamic asset allocation model is developed again with a stochastic interest rate as a state variable, where the market price of risk is an affine function of the state variable. A closed-form solution is obtained for the optimal portfolio choice and applied for a CRRA-investor. The result shows that the stock allocation is still decreasing in risk aversion, but it also starts to vary over time. This is because the stochastic interest rate enters directly into the portfolio weight of stocks. The bond allocation under the second extension is different from the former. With the market price of risk as an affine function of the stochastic interest rate, the bond allocation fluctuates due to the changes in the interest rate. As the risk aversion increases, the fluctuations will be smaller, and the two models will converge to the same bond allocation, which is increasing in the investment horizon. To evaluate the portfolio choice models, two loss functions are considered. The loss function evaluates a suboptimal investment strategy under the optimal assumptions in terms of welfare losses for the investor. It is shown that applying the result from constant investment opportunities under the assumptions of the first extension leads to marginal welfare losses. However, using the investment strategy of the first extension under the assumptions of second extension shows a significant increase in welfare losses for the investor.

EducationsMSc in Advanced Economics and Finance, (Graduate Programme) Final Thesis
Publication date2016
Number of pages172
SupervisorsClaus Munk