Assume that, in any month, the return on the stock market is either 6% or -2%, a
ID: 2750953 • Letter: A
Question
Assume that, in any month, the return on the stock market is either 6% or -2%, and these outcomes occur with equal probabilities. The riskless rate (cash) is 1% per month. The stock/cash mix of a manager who attempts to time the market is either 75/25 or 25/75 in a given month, depending on whether the manager predicts a good or bad market for that month. What is the Sharpe measure of the manager's portfolio if in fact he possesses no timing ability? Explain how, without timing ability, you could construct a portfolio having the same standard deviation but higher expected return than the manager's portfolio. What is the Sharpe measure of your portfolio in b? What is the Sharpe measure of a portfolio that maintains a constant mix of half cash and half stock?Explanation / Answer
Return from the stock = 0.5 * 6% + 0.5 * -2 = 2%
When stock/cash mix is 75/25
Return = 0,75* 2 + 0.25 * 1 = 1.75%
When stock/cash mix is 25/75
Return = 0.25 * 2 + 0.75 *1 = 1.25%
The standard deviation is [(6 - 2)^2 + (-2 -2 )^2/ 2]^1/2 = 4
Sharpe ratio = Return - risk free return / standard deviation
= 2 -1 / 4 = 0.25