Excess Returns: Meaning, Risk, and Formulas for Calculating

A detailed exploration of excess returns, including their meaning, associated risks, and the formulas used for calculation. Understand excess returns in the context of investment performance and benchmarking.

Excess returns are the returns generated by an investment that exceed the returns of a chosen benchmark or proxy. Typically, this proxy could be a market index, such as the S&P 500, or the risk-free rate, like the return on U.S. Treasury bills. The concept of excess returns is crucial for investors as it provides a measure of how well an investment performs relative to expectations set by the benchmark.

Defining Excess Returns§

Excess returns are calculated as the difference between the actual return of an investment and the return of the benchmark. Mathematically, it can be expressed as:

$$\text{Excess Return} = R_i - R_b$$

where $R_i$ is the return of the investment, and $R_b$ is the return of the benchmark.

Risk and Excess Returns§

Risk-Adjusted Excess Returns§

To accurately gauge performance, investors often turn to risk-adjusted measures of excess returns, such as the Sharpe Ratio or the Treynor Ratio. These metrics account for the investment’s risk to offer a clearer picture of its performance.

Sharpe Ratio§

The Sharpe Ratio adjusts excess returns for the risk (volatility) of the investment:

$$\text{Sharpe Ratio} = \frac{R_i - R_f}{\sigma_i}$$

where $R_f$ is the risk-free rate and $\sigma_i$ is the standard deviation of the investment’s returns.

Treynor Ratio§

The Treynor Ratio considers the investment’s systematic risk:

$$\text{Treynor Ratio} = \frac{R_i - R_f}{\beta_i}$$

where $\beta_i$ represents the investment’s sensitivity to market movements.

Systematic vs. Unsystematic Risk§

It is essential to understand the different types of risks when evaluating excess returns:

• Systematic Risk: Also known as market risk, this affects the entire market and cannot be diversified away.
• Unsystematic Risk: This is specific to a particular company or industry and can be mitigated through diversification.

Formulas for Calculating Excess Returns§

Simple Excess Returns Formula§

To calculate simple excess returns:

$$\text{Excess Return} = R_i - R_b$$

CAPM-based Excess Returns§

Another approach leverages the Capital Asset Pricing Model (CAPM):

$$R_i = R_f + \beta(R_m - R_f)$$

where $R_m$ is the return of the market portfolio.

The excess return, in this case, is:

$$\text{Excess Return}_{\text{CAPM}} = R_i - (R_f + \beta(R_m - R_f))$$

Applicability§

Excess returns are a fundamental concept in performance evaluation across various investment types, including stocks, bonds, and mutual funds. They provide insight into whether active management or security selection has generated value beyond passive strategies.

Examples§

• An investment fund generates a return of 8% over a year, while the market benchmark index returns 5%. The excess return is:

  $$\text{Excess Return} = 0.08 - 0.05 = 0.03 \text{ or } 3\%$$

• For a stock with a beta of 1.2, if the market return is 10% and the risk-free rate is 2%, using the CAPM approach, the excess return might be calculated as:

  $$R_i = 2\% + 1.2(10\% - 2\%) = 11.6\%$$

  Therefore, if the actual return was 15%, the CAPM excess return is:

  $$\text{Excess Return}_{\text{CAPM}} = 15\% - 11.6\% = 3.4\%$$

FAQs§

What is the difference between absolute and relative returns?

Absolute returns refer to the actual gains or losses made by an investment over a specific period, without comparison to any benchmark. In contrast, relative returns compare the investment’s performance against a benchmark to show performance in context.

Can excess returns be negative?

Yes. If an investment underperforms its benchmark, the excess returns will be negative, indicating underperformance relative to the benchmark.

Why are excess returns important?

Excess returns help investors and portfolio managers assess the effectiveness of their investment strategies. They are also instrumental in performance attribution analysis and determining the contribution of active management.

Related Terms§

• Alpha: A measure of an investment’s performance on a risk-adjusted basis.
• Beta: Represents an investment’s volatility in relation to the market.
• Benchmark: A standard against which the performance of a security, mutual fund, or investment manager can be measured.
• Risk-Free Rate: The theoretical return on an investment with zero risk, often represented by government Treasury bonds.
• Market Risk Premium: The additional return expected from holding a risky market portfolio instead of risk-free assets.

References§

• Fama, E. F., & French, K. R. (1992). The Cross-Section of Expected Stock Returns. Journal of Finance.
• Sharpe, W. F. (1966). Mutual Fund Performance. Journal of Business.

Summary§

Excess returns are a vital measure in evaluating the performance of investments relative to benchmarks. Understanding how to calculate and interpret excess returns helps investors make informed decisions and assess the value added by active management. The risk-adjusted metrics further refine this evaluation, providing deeper insights into the overall efficiency and performance of investment strategies.