FIN 532 Investment Theory
Problem Set 4
Fall 2024
1 The tangible benefits of rebalancing
The point of this exercise is to quantify the benefits of rebalancing. Suppose that you are investing in two stock market indices, AAA and ZZZ. Both risky assets have returns that are normally distributed, with the same mean return (10%) and the same standard deviation (20%). Further, the correlation between the two risky returns is zero. Also, assume the risk-free rate is zero.
1. What is the optimal (i.e. mean-variance efficient) allocation among the two securities?
2. Now, we will examine the performance of two strategies over an investment horizon of 100 years. We will do so across many simulations (say 10,000 though you can choose any number you want). For each simulation,
(a) Simulate a vector of returns for the two risky assets, AAA and ZZZ. Use your software’s build in random number generator.For example, in Matlab, you can simulate a normal variable with mean μ and standard deviation σ via x = μ + σ × randn, where randn simulates a standard N (0, 1) variable.
(b) Compute the returns to your initial wealth of the buy and hold strategy—that is, the strategy in which you allocated optimally at t = 0 and then did not rebalance.
(c) Compute the returns to your initial wealth of the rebalanced strategy—that is, the strategy in which you rebalanced at the end of every year to the ‘optimal’ portfolio weights.
(d) Compute the realized Sharpe ratio for each of the two strategies above. In addi-tion, compute the average allocation to the first risky asset AAA in the buy and hold strategy (in the rebalancing strategy it should be constant by construction)
3. Plot a histogram of the realized Sharpe ratio across simulations of the two strategies. What is the average Sharpe ratio of each strategy? Also plot the mean portfolio weight to asset 1 of the buy and hold strategy. Discuss your findings. For the purposes of this exercise, it will be best to use a programming language like MATLAB, Octave, or R. It is possible to do it in Excel, though it will be rather painful. To get you started in Matlab, we have posted some sample code for simulating returns in GettingStartedhw4p1.m.