Question

A) We have the following historical returns on a portfolio. Assume the monthly risk-free rate in the same time period was 3%. Estimate the Sharpe ratio of this portfolio.

month	return
1	10%
2	5%
3	-2%
4	3%
5	15%

B) Consider the same historical record above, what is the stock's geometric average monthly return in that time period?

Answer 1

Answer A.

Expected Return = [0.10 + 0.05 + (-0.02) + 0.03 + 0.15] / 5
Expected Return = 0.31 / 5
Expected Return = 0.062 or 6.20%

Variance = [(0.10-0.062)^2 + (0.05-0.062)^2 + (-0.02-0.062)^2 + (0.03-0.062)^2 + (0.15-0.062)^2] / 4
Variance = 0.01708 / 4
Variance = 0.00427

Standard Deviation = (0.00427)^(1/2)
Standard Deviation = 0.0653 or 6.53%

Sharpe Ratio = [Expected Return - Risk-free Rate] / Standard Deviation
Sharpe Ratio = (0.062 - 0.03) / 0.0653
Sharpe Ratio = 0.49

Answer B.

Geometric Return = [(1+0.10) * (1+0.05) * (1-0.02) * (1+0.03) * (1+0.15)]^(1/5) - 1
Geometric Return = 1.340736^(1/5) - 1
Geometric Return = 1.0604 - 1
Geometric Return = 0.0604 or 6.04%