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### Subject 8. The Sharpe Measure of Risk-Adjusted Performance

The Sharpe Measure is a more precise return-risk measure than standard deviation. It recognizes the existence of a risk-free return, a return for virtually zero standard deviation. It measures the reward to total volatility trade-off.

It is defined as: • (rp - rf) = the extra reward that investors receive for the added risk taken, called the excess return on portfolio p
• σp = the standard deviation of the portfolio returns

Note that the numerator of the Sharpe measure recognizes the existence of a risk-free return. Portfolios with large Sharpe ratios are preferred to those with smaller ratios because we assume that investors prefer return and dislike risk. The Sharpe ratio is also called the reward-to-variability ratio.

Example

The mean monthly return on T-bills (the risk-free rate) is 0.25%. The mean monthly return on the S&P 500 is 1.30% with a standard deviation of 7.30%. Calculate the Sharpe measure for the S&P 500 and interpret the results.

Sharpe measure = (1.30 - 0.25)/7.30 = 0.144
Interpretation: The S&P 500 earned 0.144% of excess return per unit of risk, where risk is measured by standard deviation.

#### Practice Question 1

The mean return in 2011 for a stock mutual fund of midsize companies was 5.3% and the mean return for a risk-free bond was 1.0 %. If the Sharpe measure for the mutual fund is 2, what is the standard deviation of the stock mutual fund?

A. 2.15
B. 3.15
C. 4.34

Sharpe measure = (portfolio mean return - risk-free mean return)/ standard deviation of the portfolio.

#### Practice Question 2

A natural resource fund has a mean total return of 22.5% and a standard deviation of 60.70%. Given that a risk-free asset has a mean return of 2.469%, what is the value of the Sharpe measure?

A. 0.33
B. 0.37
C. 9.11

#### Practice Question 3

The Sharpe ratio uses ______ as a measure of volatility.

A. standard deviation
B. variance
C. beta

It uses standard deviation, not beta, as a measure of volatility. This is one of its limitations.

#### Practice Question 4

Which statement is false regarding Sharpe ratio?

A. Sharpe ratio can be used to rank portfolios.
B. Sharpe ratio cannot be applied to assets with negative beta.
C. If the investor does not have any other assets than the portfolio, then the use of total risk is appropriate in the Sharpe ratio.

We can compare the Sharpe ratios of portfolios in order to rank them.

B is false. It does not even use beta.

#### Practice Question 5

Which statement about the Sharpe ratio is false?

A. The Sharpe ratio cannot be applied to risk-free assets.
B. A portfolio with a Sharpe ratio of 0.8 is 2 times better than a portfolio with a Sharpe ratio of 0.4.
C. The Sharpe ratio for one stock can be different among different investors.

A is true. The zero standard deviation of such assets cannot be used as the denominator.

B is false. The Sharpe ratio can be used to rank portfolios but does not give any information about the economic significance of differences.C is true. If it is used as an ex ante measure of expected return and risk, it will likely vary among different investors.

#### Practice Question 6

The mean return in 2011 for a stock mutual fund of large companies was 7.2% and the mean return for a risk-free bond was 1.0 %. If the Sharpe measure for the mutual fund is 0.50, what is the standard deviation or risk of the stock mutual fund?

A. 3.2
B. 12.4
C. 16.4