Coefficient of variation and Sharpe ratio

CFA level I / Quantitative Methods: Basic Concepts / Statistical Concepts and Market Returns / Coefficient of variation and Sharpe ratio

The coefficient of variation (CV) is the ratio of the standard deviation of the data to its mean value. It gives us the value of risk per unit of return.

CV = s/X

where s is the sample standard deviation and X is the sample mean.

The higher is the coefficient of variation; the riskier is the security because we are taking more risk per unit of return.

The Sharpe ratio is the ratio of excess return over the risk-free rate to the standard deviation. It gives us the value of excess return per unit of risk.

S_h = (R-bar_p- R-bar_f)/s_p

where R-bar_p is the mean return to the portfolio, R-bar_f is the mean return to a risk-free asset, and s_p is the standard deviation of return on the portfolio.

The higher is the Sharpe ratio; the better is the portfolio because we will receive higher mean excess return per unit of risk.

Example 11: Calculating Coefficient of Variation and Sharpe

The mean return and the standard deviation of two portfolios are given below:

	Portfolio A	Portfolio B
Mean return	18.00 percent	14.00 percent
Standard deviation	30.00 percent	21.00 percent

Compute the coefficient of variation and Sharpe ratio of the portfolios. Assume the mean risk-free rate to be 5 percent.

Solution:

	Portfolio A	Portfolio B
Mean return	18.00 percent	14.00 percent
Standard deviation	30.00 percent	21.00 percent
Mean return - Mean riskfree rate	13.00 percent	9.00 percent
Sharpe ratio (R_p-R_f)/s_p	0.4333	0.4286
Coefficient of variation (s/X)	1.67	1.50

Please note that we get different results from Sharpe ratio and coefficient of variation. Portfolio A is better per Sharpe Ratio, and Portfolio B is better per coefficient of variation. Both measures should provide the similar results. But the results can be different because the Sharpe ratio compares the excess return over the risk-free rate to the risk whereas the coefficient of variation compares the total return to the risk.

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