An Introduction to Portfolio Performance Measures

How Do We Evaluate Portfolio Performance?

At the core of performance evaluation is the idea that investors prefer high returns but low risk.

• We therefore need measures which simultaneously reward for higher returns, but penalize for higher risk.

Measuring a portfolios return is easy. However, how do we measure a portfolio’s risk?

• Measuring risk is a little trickier, and will lead to multiple performance measures.

Classic Measures

In this presentation we’ll cover some classic and widely-used measures. They are:

• Sharpe Ratio
• Treynor Ratio
• Jensen’s Alpha
• Information Ratio

Sharpe Ratio (after William Sharpe)

This is probably the most famous performance measure. When evaluating past performance, it is defined as:

`SR = \frac{\bar{r_p} - r_f}{\sigma_p}` where `r_p`, `\sigma_p`, and `r_f` are the portfolio’s return and standard deviation, and the risk free return over the sample period.

• Usually the Sharpe Ratio is stated in annual terms (to do so multiply it by the square root of the number of periods in a year).

Notes on Using the Sharpe Ratio

• A higher Sharpe Ratio is better – reflecting higher returns and a lower standard deviation.

• Since the measure of risk is the standard deviation, the Sharpe Ratio should be used with portfolios of many, and not few, securities.

• Unfortunately, the Sharpe Ratio does not adjust for the use of leverage.

Sharpe Ratio App

The following interactive app will calculate the Sharpe Ratio for each select sector SPDR ETF over any year you choose, and create a scatter plot of the ETF’s return, standard deviation, and Sharpe Ratio. The last slide provides the sector description for each ticker.

• The axes are the return and standard deviation

• Higher Sharpe Ratios are signified by large blue points, and small Sharpe Ratios by smaller red points.

• Note the lowest Sharpe Ratios are not necessarily the ones with the lowest returns.

Treynor Ratio

For individual assets, the asset’s beta coefficient (see the CAPM presentation) is a better measure of risk than the asset’s standard deviation. This leads to the Treynor Ratio for historical data:

`TR = \frac{\bar{r_p} - r_f}{\beta_p}`where `\beta_p` is the portfolio’s beta coefficient, and the other terms are as defined earlier.

• A higher Treynor Ratio is better.

• The Treynor Ratio does account for leverage.

Jensen’s Alpha

Jensen’s Alpha is the expected return on the portfolio adjusted for the return earned for taking market risk. In other words, it is the return on a portfolio is excess of what the CAPM expects it to be:

`\alpha = \bar{r_p} - \beta_p(r_m - r_f)` where `\alpha` is the Greek letter alpha, and `r_m` is the return on the market portfolio.

• The larger the alpha the better.

• The CAPM predicts a zero alpha.

Information Ratio

A problem with Jensen’s Alpha is that a portfolio manager must depart from the market portfolio to generate alpha, and this introduces non-market, or residual risk, into the portfolio. The information ratio (`IR`) scales `\alpha` by this residual risk:

`IR = \frac{\alpha_p}{\sigma(\epsilon_p)}` where `\sigma(\epsilon_p)` is the standard deviation of the error term in a CAPM style regression equation.

• The larger the `IR` the better.

Which Measure Should I Use?

For well-diversified portfolios (with no firm-specific risk), the Sharpe Ratio is the preferred measure. If the portfolio contains leverage, or firm-specific risks that can be diversified away, then the Treynor Ratio, Jensen’s Alpha, and the Information Ratio become preferable.

• For this reason alpha-based measures are popular measures of hedge fund performance.

Agreement Among Measures

How might the rankings by each performance measure differ for a given set of portfolios?

• The following app will allow you to rank or select sector SPDR ETFs by each measure, over any year you choose.

• As you’ll see, there is a fair amount of agreement.

This means portfolio performance rankings are not too sensitive to the particular measure used.

Other Measures

There are other performance measures based on downside risk, and which take into account higher moments of the return distribution. Downside risk focuses on the probability that your returns will be below some threshold, and ‘higher moments’ refers to skewness and kurtosis (and above). Among such measures are the:

• Sortino Ratio
• Omega Ratio
• V2 Ratio

Tickers

• XLE: Energy
• XLV: Insurance
• XLU: Utilities
• XLK: Technology
• XLY: Consumer Discretionary
• XLB: Materials
• XLI: Industrials
• XLF: Financials
• XLP: Consumer Staples

Credits and Collaboration

Click the following links to see the code, line-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.

Learn more about how to contribute here.