Sharpe Ratio

Risk-adjusted performance measurement

The Sharpe ratio measures excess return per unit of risk, enabling comparison across investments with different risk profiles.

1 Abstract

The Sharpe ratio, developed by William Sharpe in 1966, measures the excess return per unit of risk. It remains the most widely used metric for risk-adjusted performance, allowing investors to compare investments with different volatility levels on a common scale (Sharpe, 1964).

2 Definition

The Sharpe ratio is defined as:

\[ SR = \frac{R_p - R_f}{\sigma_p} \]

Where:

• \(R_p\) = portfolio return
• \(R_f\) = risk-free rate
• \(\sigma_p\) = portfolio standard deviation (volatility)

The numerator \((R_p - R_f)\) is the excess return — the return above the risk-free rate. The denominator \(\sigma_p\) is the risk measured by volatility.

3 Interpretation

Sharpe Ratio	Interpretation
< 0	Negative excess return
0 - 1	Acceptable, risk may not be adequately compensated
1 - 2	Good risk-adjusted return
2 - 3	Very good
> 3	Excellent (rare, verify data)

4 Annualization

For daily returns, the annualized Sharpe ratio is:

\[ SR_{ann} = \sqrt{252} \cdot \frac{\bar{r}_d - r_f/252}{\sigma_d} \]

Where \(\bar{r}_d\) is the mean daily return and \(\sigma_d\) is the daily standard deviation.

5 Compute (Python)

We calculate the Sharpe ratio for several ETFs representing different asset classes.

	Ticker	Ann. Return (%)	Ann. Volatility (%)	Sharpe Ratio
0	SPY	13.42	17.07	0.55
1	QQQ	14.22	22.60	0.45
2	TLT	-7.70	16.02	-0.73
3	GLD	17.99	15.60	0.90
4	VNQ	5.32	18.83	0.07

6 Sharpe Ratio Comparison

7 Risk-Return Scatter

8 Rolling Sharpe Ratio

The Sharpe ratio varies over time. A rolling window shows how risk-adjusted performance evolves.

9 Limitations

• Assumes normal distribution: Penalizes upside volatility equally to downside
• Sensitive to time period: Different periods yield different results
• Risk-free rate choice: Results vary with the benchmark rate used
• Not suitable for: Strategies with non-normal returns (options, hedge funds)

For strategies with asymmetric returns, consider the Sortino ratio (uses downside deviation) or Calmar ratio (uses max drawdown).

10 Conclusion

The Sharpe ratio provides a standardized measure of risk-adjusted return, enabling comparison across assets with different volatility profiles. While widely used, it should be considered alongside other metrics, particularly for strategies with non-normal return distributions.

References

Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425–442.