Question

Use the following table to complete questions 1-3. S&P 500 returns and the risk-free rate of return can take on the following possible values with the given probabilities.

State Pi S&P 500 Returns Risk-Free Rate

Recession 0.25 -10% 8%

Low Pullback 0.25 -5% 8%

No Growth 0.25 0% 8%

Low Growth 0.25 5% 8%


1. What is the expected value of S&P 500 returns? What is the expected value of risk-free returns?

2. What is the variance and standard deviation of both sets of returns?

3. What is the Sharpe ratio of each potential investment?

Answer 1

(1)

Expected value of S&P 500 returns = 0.25*(-10%) + 0.25*(-5%) + 0.25*(0%) + 0.25*(5%)

Expected value of S&P 500 returns = -2.50%

Expected value of Risk-free rate = 0.25*(8%) + 0.25*(8%) + 0.25*(8%) + 0.25*(8%)

Expected value of Risk-free rate = 8%

(Intuitively too, the expected value would be 8%, as it remains the same through all the states of economy)

(2)

Variance and Standard deviation of S&P 500 :

The Variance for S&P 500 is calculated as follows:

Variance = 0.003125

Also, Variance = Standard deviation²

Thus, Standard Deviation = Square-root of 0.003125

Standard Deviation = 5.59%

Variance and Standard deviation of Risk-free rate :

As the risk-free rate is same during all the states of economy, we need not calculate its variance and standard deviation as they would both be zero.

(3)

Sharpe Ratio is calculated as:

Sharpe Ratio = (Rp-Rf)/ Std dev. of portfolio

The Sharpe ratio for each potential investment under the 4 different states can be calculated as follows: