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# Comparison of Risk asjusted Returns

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To be able to compare returns on your portfolio with a benchmark index, you need to equalize the differences in risk and return. For example, a 5 percent return on a portfolio of Treasury securities is not the same as a 5 percent return on a portfolio of small-cap growth stocks because of the greater risk in the latter portfolio. Investors would prefer the 5 percent return from the low-risk Treasury security portfolio.

Three different measures adjust returns for the risk associated with a portfolio:
* Sharpe index
* Treynor index
* Jensen index

These measures allow you to compare different returns on a riskadjusted basis.

## The Sharpe Index

The Sharpe index is a risk-adjusted measure of performance that standardizes the risk premium of a portfolio using the standard deviation of the portfolio return.

Sharpe index = (portfolio return - risk-free rate)/ standard deviation of portfolio returns

The risk-adjusted return of a portfolio is the risk premium for a portfolio (portfolio return - risk-free return) divided by the standard deviation of the portfolio’s returns. For example, suppose that a portfolio has an annual return of 7 percent and an annual standard deviation of 20 percent. The average Treasury bill rate during the year was 3.5 percent. What is the Sharpe index?

Sharpe index = (rp - rf )/σp
= (0.07 - 0.035)/0.2 = 0.175

The result is put in perspective when it is compared with another portfolio or the market. Assume that during the same year the market return is 6 percent, with a standard deviation of 18 percent and a risk-free rate of 3.5 percent. The Sharpe index for the market is 0.1389:

Sharpe index = (0.06 - 0.035)/ 0.18 = 0.1389

The performance of the portfolio is superior to the return of the market because of the higher result (0.175 compared with 0.1389). The higher return of the portfolio more than balances the higher standard deviation or risk (20 percent for the portfolio versus 18 percent for the market).

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