What is Performance Evaluation?

Achieving a specific return on investments doesn’t provide a complete picture of the portfolio performance. Is an annual return of 10% or 50% good? Does a return of -10% or -20% necessarily mean poor performance?

Benchmark Comparison

One method to evaluate performance is by comparing it to a passive investment in a benchmark. Typically, a benchmark is an index that serves as a reference point for assessing the performance of an investment or portfolio.

For example, if the investment strategy focuses on large-cap U.S. companies, it would be good to compare its return and risk with those of the S&P 500.

If the S&P 500 realized a return of 10% with a standard deviation of 10%, and the portfolio returned 12% with a standard deviation of 10%, then the strategy outperformed the benchmark. It realized a 2% higher return at identical risk. In financial terms, this return above the benchmark is known as alpha.

Similarly, in a falling market, a loss of 10% can be considered a good result if the benchmark recorded a loss of 15% (with identical risk parameters).

Performance should be considered in terms of both return and risk. One of the most used indicators for evaluating portfolio performance is the Sharpe ratio.

Sharpe Ratio

The Sharpe Ratio measures the risk-adjusted return of an investment or portfolio. It shows how much excess return (above the risk-free rate) is generated per unit of risk, where risk is represented by the standard deviation of the investment’s returns.

The Sharpe ratio formula is:

\[
\text{Sharpe Ratio} = \frac{R_p – R_f}{\sigma_p}
\]

Where:

  • Rp: Return of the portfolio
  • isk-free rate
  • σp: Standard deviation of the portfolio

A higher Sharpe Ratio indicates a more favorable risk-adjusted performance, meaning the investment provides better returns for the level of risk taken.

Treynor Ratio

In previous chapters, it was mentioned that standard deviation accounts for both systematic and unsystematic risk. However, a well-diversified portfolio can largely or entirely eliminate unsystematic risk. The Treynor ratio measures the risk premium relative to systematic risk (beta).

The Treynor ratio formula is:

\[
\text{Treynor Ratio} = \frac{R_p – R_f}{\beta_p}
\]

Where:

  • Return of the portfolio
  • isk-free rate
  • βp: Portfolio’s beta, which measures the sensitivity of the portfolio’s returns to the overall market (systematic risk)

The Treynor Ratio shows how much excess return (above the risk-free rate) is generated per unit of systematic risk. A higher Treynor Ratio indicates better risk-adjusted performance, specifically in relation to the investment’s exposure to overall market movements.

M-Squared (Modigliani-Modigliani Measure)

The M-Squared measure, developed by Franco Modigliani and Leah Modigliani, is a risk-adjusted performance metric that allows for the comparison of portfolio performance with a benchmark. It adjusts the portfolio’s return to match the risk level of the benchmark, making it easier to compare performance.

Formula:

\[
M^2 = (R_p – R_f) \frac{\sigma_m}{\sigma_p} + R_f
\]

Where:

  • M^2: The M-Squared measure
  • ortfolio return
  • isk-free rate
  • σm: Standard deviation of the market
  • σp: Standard deviation of the portfolio

The M-Squared measure adjusts the Sharpe Ratio to express performance in percentage terms, allowing for direct comparison with the market. A higher M-Squared value indicates better risk-adjusted returns.

Jensen’s Alpha

Jensen’s Alpha, developed by Michael Jensen, is a performance measure that calculates the excess return a portfolio generates over its expected return, as predicted by the Capital Asset Pricing Model (CAPM). This metric helps investors understand how much a portfolio has outperformed or underperformed its expected return, given its level of systematic risk (beta).

The formula for Jensen’s Alpha is:

\[
\alpha = R_p – \left[ R_f + \beta_p (R_m – R_f) \right] \]

Where:

  • α: Jensen’s Alpha
  • Rp: Portfolio return
  • Rf: Risk-free rate
  • βp: Portfolio’s beta (systematic risk)
  • Rm: Market return

A positive Jensen’s Alpha indicates that the portfolio has outperformed its expected return based on its level of risk, while a negative alpha suggests underperformance. This metric helps investors evaluate the effectiveness of active management strategies.

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