Portfolio Performance Metrics

A guide to key statistical measures used for evaluating investment portfolio performance.

TL; DR

• Rate of Return: Measures the percentage change in value of an investment.

• Sharpe Ratio: Assesses performance adjusted for risk compared to a risk-free asset.

• Sortino Ratio: Similar to Sharpe but focuses only on downside risk.

• Alpha: Indicates the performance relative to a benchmark index.

• Beta: Measures the volatility of an investment in relation to the market.

• Treynor Ratio: Evaluates returns per unit of market risk.

• Jensen's Alpha: Uses CAPM to determine the expected return and compares it to the actual return.

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Understanding Portfolio Performance Metrics

When evaluating the performance of an investment portfolio, several statistical measures are commonly used to assess risk and return. These metrics provide investors with insights into how well their investments are doing relative to benchmarks or risk factors. Below, we introduce some of the key performance metrics.

Rate of Return

The rate of return (RoR) is the most basic measure of investment performance. It represents the percentage change in the value of an investment over a specified period.

$$RoR = \frac{Current\ Value - Initial\ Value}{Initial\ Value} \times 100$$

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The rate of return can be calculated for any investment, from stocks and bonds to real estate and mutual funds. It can be expressed on an annual basis (annualized return) or for any other period. It's important to compare the RoR of an investment with the appropriate benchmark to understand its performance in context.

Sharpe Ratio

The Sharpe ratio measures the performance of an investment compared to a risk-free asset, after adjusting for its risk.

$$Sharpe\ Ratio = \frac{R_p - R_f}{\sigma_p}$$

• $R_p$ is the return of the portfolio
• $R_f$ is the risk-free rate
• $\sigma_p$ is the standard deviation of the portfolio's excess return

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The Sharpe ratio is useful for comparing the risk-adjusted returns of different investments. A higher Sharpe ratio indicates a more favorable risk-return profile. It's important to use a consistent risk-free rate when comparing different investments.

Sortino Ratio

The Sortino ratio is similar to the Sharpe ratio but only considers downside risk.

$$Sortino\ Ratio = \frac{R_p - R_f}{\sigma_d}$$

• $\sigma_d$ is the standard deviation of the portfolio's negative asset returns

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The Sortino ratio is particularly useful for investors who are concerned about downside risk. Unlike the Sharpe ratio, it does not penalize the portfolio for upward price movements, focusing only on the variability of negative returns.

Alpha

Alpha measures the performance of an investment relative to a benchmark index.

$$Alpha = R_p - (R_f + \beta \times (R_m - R_f))$$

• $\beta$ is the portfolio's volatility relative to the market
• $R_m$ is the return of the market

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Alpha is often used to assess the value that a portfolio manager adds in terms of performance. A positive alpha indicates that the portfolio has outperformed the market after adjusting for risk, while a negative alpha suggests underperformance.

Beta

Beta measures the volatility of an investment in relation to the market.

$$Beta = \frac{Cov(R_p, R_m)}{Var(R_m)}$$

• $Cov(R_p,\ R_m)$ is the covariance between the portfolio returns and market returns
• $Var(R_m)$ is the variance of the market returns

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Beta is a key component in the Capital Asset Pricing Model (CAPM) and is used to determine the expected return of a portfolio based on its market risk. A beta greater than 1 indicates higher volatility than the market, while a beta less than 1 indicates lower volatility.

Treynor Ratio

The Treynor ratio measures returns earned in excess of that which could have been earned on a riskless investment per each unit of market risk.

$$Treynor\ Ratio = \frac{R_p - R_f}{Beta}$$

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The Treynor ratio is similar to the Sharpe ratio but uses beta as the denominator instead of standard deviation. This makes it a measure of risk-adjusted return based on systematic risk rather than total risk.

Jensen's Alpha

Jensen's Alpha is a version of the alpha metric that uses the Capital Asset Pricing Model (CAPM) to determine the expected return.

$$Jensen's\ Alpha = R_p - (R_f + Beta \times (R_m - R_f))$$

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Jensen's Alpha takes into account the expected return based on the market risk (as per CAPM) and compares it to the actual return of the portfolio. It's a measure of the manager's ability to generate excess returns.

Conclusion

These metrics are essential tools for investors to evaluate and compare the performance of their portfolios. Each metric provides a different perspective on risk and return, and when used together, they can offer a comprehensive view of a portfolio's performance. It's important to understand the context and limitations of each metric to make informed investment decisions.