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The **Sharpe ratio** is the average return earned in *excess* of the risk-free rate per unit of volatility or total risk.

Subtracting the risk-free rate from the mean return, the performance associated with risk-taking activities can be isolated.

One intuition of this calculation is that a portfolio engaging in “zero risk” investment, such as the purchase of **U.S. Treasury bills** (for which the expected return is the risk-free rate),** has a Sharpe ratio of exactly zero. **

### Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.

The Sharpe ratio has become the most widely used method for calculating risk-adjusted return

### Cons:

- it can be inaccurate when applied to portfolios or assets that do not have a normal distribution of expected returns. Many assets have a high degree of kurtosis (‘fat tails’) or negative skewness.
- Tends to fail when analyzing portfolios with significant non-linear risks, such as options or warrants.
- Uses the standard deviation of returns in the denominator as its proxy of total portfolio risk, which assumes that returns are normally distributed.

Modern Portfolio Theory states that adding assets to a diversified portfolio that have correlations of less than 1 with each other can decrease portfolio risk without sacrificing return. Such diversification will serve to increase the Sharpe ratio of a portfolio.

**Sharpe ratio = (Mean portfolio return − Risk-free rate) / Standard deviation of portfolio return:**

## Applications of the Sharpe Ratio

The Sharpe ratio is often used to compare the change in a portfolio’s overall risk-return characteristics when a new asset or asset class is added to it.

Example:

A portfolio manager is considering adding a hedge fund allocation to his existing 50/50 investment portfolio of stocks and bonds which has a **Sharpe ratio of 0.67**. If the **new** portfolio’s allocation is 40/40/20 stocks, bonds and a diversified hedge fund allocation (perhaps a fund of funds), the **Sharpe ratio increases to 0.87.**

Implications of the above example:

- Although the diversified hedge fund investment is risky as a standalone exposure, it actually improves the
**risk-return characteristic**of the combined portfolio. - Thus adding a diversification benefit
- Should be added to portfolio.
- If it decreased the Sharpe ratio, then it should not be added to the portfolio (even adding T-bills at this point would be better, since T-bills’ Sharpe ratio = 0)

The Sharpe ratio can also be “gamed” by hedge funds or portfolio managers seeking to boost their apparent risk-adjusted returns history. This can be done by:

- Lengthening the measurement interval: This will result in a lower estimate of volatility. For example, the annualized standard deviation of daily returns is generally higher than that of weekly returns, which is, in turn, higher than that of monthly returns.
- Compounding the monthly returns but calculating the standard deviation from the
*not*compounded monthly returns. - Writing out-of-the-money puts and calls on a portfolio: This strategy can potentially increase return by collecting the option premium without paying off for several years. Strategies that involve taking on default risk, liquidity risk, or other forms of catastrophe risk have the same ability to report an upwardly biased Sharpe ratio. An example is the Sharpe ratios of market-neutral hedge funds before and after the 1998 liquidity crisis.)
- Smoothing of returns: Using certain derivative structures, infrequent marking to market of illiquid assets, or using pricing models that understate monthly gains or losses can reduce reported volatility.
- Eliminating extreme returns: Because such returns increase the reported standard deviation of a hedge fund, a manager may choose to attempt to eliminate the best and the worst monthly returns each year to reduce the standard deviation.

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