Standard deviation of excess return

What Is Standard Deviation of Excess Return?

The standard deviation of excess return quantifies the volatility or dispersion of a portfolio's or investment's returns above a specified benchmark or risk-free rate. It is a key metric within portfolio theory and investment performance measurement, indicating the consistency and predictability of outperformance. Unlike the standard deviation of total returns, which measures overall volatility, the standard deviation of excess return focuses specifically on the variability of the additional return generated beyond a baseline. This measure helps investors and portfolio management professionals assess the risk associated with achieving returns superior to a passive investment or a market index.

History and Origin

The concept of evaluating investment performance relative to a benchmark or risk-free rate, and subsequently measuring the volatility of that difference, gained prominence with the advent of Modern Portfolio Theory (MPT) and related performance attribution models. Pioneering work by economists such as William F. Sharpe in the 1960s laid the groundwork for modern risk-adjusted return measures. Sharpe's development of the Sharpe ratio, for instance, directly incorporates the standard deviation of excess return by dividing the average excess return by its standard deviation to evaluate the reward per unit of risk taken. This metric gained widespread recognition, with William F. Sharpe detailing its evolution and application in various academic works, including a comprehensive overview in 1994.³

Key Takeaways

The standard deviation of excess return measures the variability of returns generated above a benchmark or risk-free rate.
It is a crucial indicator for assessing the risk of active investment strategies and the consistency of alpha generation.
A lower standard deviation of excess return suggests a more consistent outperformance relative to the chosen benchmark.
This metric is distinct from the overall standard deviation of returns, which reflects total portfolio volatility.
It is a component in several key performance measurement ratios, providing insight into the quality of active returns.

Formula and Calculation

The standard deviation of excess return is calculated by first determining the series of excess returns and then applying the standard deviation formula to that series.

Let:

(R_t) = Portfolio or investment return at time t
(B_t) = Benchmark or risk-free return at time t
(ER_t = R_t - B_t) = Excess return at time t
(\overline{ER}) = Average excess return over the period
(n) = Number of periods

The formula for the standard deviation of excess return ((\sigma_{ER})) is:

$\sigma_{ER} = \sqrt{\frac{\sum_{t=1}^{n} (ER_t - \overline{ER})^2}{n-1}}$

This calculation quantifies the dispersion of an investment's returns relative to its benchmark or the risk-free rate, providing a measure of the risk involved in achieving those excess returns.

Interpreting the Standard Deviation of Excess Return

Interpreting the standard deviation of excess return involves understanding what its magnitude implies about an investment's performance against a chosen baseline. A higher value indicates greater variability in the amount by which an investment outperforms (or underperforms) its benchmark. Conversely, a lower value suggests a more stable and predictable level of outperformance. For an active fund manager, a consistently low standard deviation of excess return, coupled with a positive average excess return, implies skillful active management rather than erratic swings that could be attributed to luck. Investors use this metric to gauge the reliability of a manager's ability to generate alpha and to evaluate if the additional risk taken to achieve those excess returns is justified.

Hypothetical Example

Consider two hypothetical active funds, Fund A and Fund B, both aiming to outperform a broad market index. Over five years, their annual excess returns (Fund Return - Index Return) are as follows:

Fund A Excess Returns: 3%, -1%, 5%, 2%, 1%
Fund B Excess Returns: 10%, -8%, 15%, -5%, 3%

Step 1: Calculate the average excess return for each fund.

Average Excess Return (Fund A) = (3 - 1 + 5 + 2 + 1) / 5 = 10 / 5 = 2%
Average Excess Return (Fund B) = (10 - 8 + 15 - 5 + 3) / 5 = 15 / 5 = 3%

Step 2: Calculate the standard deviation of excess return for each fund.
For Fund A:

(3-2)^2 = 1
(-1-2)^2 = 9
(5-2)^2 = 9
(2-2)^2 = 0
(1-2)^2 = 1
Sum of squared differences = 1 + 9 + 9 + 0 + 1 = 20
Variance = 20 / (5-1) = 20 / 4 = 5
Standard Deviation of Excess Return (Fund A) = (\sqrt{5}) (\approx) 2.24%

For Fund B:

(10-3)^2 = 49
(-8-3)^2 = 121
(15-3)^2 = 144
(-5-3)^2 = 64
(3-3)^2 = 0
Sum of squared differences = 49 + 121 + 144 + 64 + 0 = 378
Variance = 378 / (5-1) = 378 / 4 = 94.5
Standard Deviation of Excess Return (Fund B) = (\sqrt{94.5}) (\approx) 9.72%

In this example, Fund B has a higher average excess return (3% vs. 2%), but also a significantly higher standard deviation of excess return (9.72% vs. 2.24%). This indicates that while Fund B achieved higher overall outperformance, its year-to-year excess returns were much more volatile and less predictable than Fund A's. An investor might prefer Fund A for its more consistent outperformance, even if the average return on investment is slightly lower, due to the lower associated risk.

Practical Applications

The standard deviation of excess return is a vital tool across various facets of financial analysis and asset allocation:

Manager Selection and Evaluation: Institutional investors and wealth managers use this metric to evaluate the consistency of fund managers' ability to generate returns above a benchmark. A manager who consistently delivers positive excess returns with low volatility is often preferred over one whose excess returns fluctuate wildly, even if the average is high.
Performance Reporting: Many investment management firms adhere to industry standards, such as the Global Investment Performance Standards (GIPS) set by the CFA Institute, which promote fair representation and full disclosure of investment performance.² While GIPS does not specifically mandate the reporting of the standard deviation of excess return, it emphasizes transparent risk measurement alongside return figures, making this a relevant internal metric for firms.
Risk Management: It helps in understanding the specific risk associated with an active strategy, isolating it from general market risk. For example, the U.S. Securities and Exchange Commission (SEC) emphasizes clear disclosure on performance benchmarks to help investors understand how fund performance is measured against relevant indices.¹
Portfolio Construction: Understanding the standard deviation of excess return for individual assets or sub-portfolios can aid in building diversified portfolios where the consistency of active bets contributes positively to overall diversification rather than introducing undue variability.
Deriving Risk-Adjusted Ratios: As demonstrated by the Sharpe Ratio, the standard deviation of excess return is a critical input for many popular risk-adjusted return measures. The 3-Month Treasury Bill rate is commonly used as a proxy for the risk-free rate in such calculations, providing a baseline for excess return determination.

Limitations and Criticisms

While a widely used and valuable metric, the standard deviation of excess return has certain limitations. One key criticism, similar to that of standard deviation of total returns, is its assumption of a normal distribution of returns, which may not always hold true, particularly for alternative investments or during periods of market stress. It treats both positive and negative deviations from the average excess return equally, implying that large positive outperformance is as "risky" as large negative underperformance. Some argue that investors are primarily concerned with downside risk, which this measure does not explicitly distinguish. Furthermore, historical standard deviation of excess return does not guarantee future performance or variability, as market conditions and manager effectiveness can change over time. Its utility is also heavily dependent on the appropriateness of the chosen benchmark; an ill-suited benchmark can render the excess return and its variability meaningless.

Standard Deviation of Excess Return vs. Tracking Error

The standard deviation of excess return and tracking error are closely related terms, often used interchangeably, but with a subtle distinction in common usage.

Standard Deviation of Excess Return: This term broadly refers to the volatility of any return stream above a chosen baseline, whether that baseline is a risk-free rate, a custom benchmark, or even another portfolio. It focuses on the variability of the difference between an investment's return and any reference return.
Tracking Error: Specifically, tracking error (also known as active risk) measures the standard deviation of the difference between a portfolio's returns and the returns of its designated benchmark index. It is predominantly used in the context of active management to quantify how closely a portfolio tracks its benchmark or, conversely, how much an active manager deviates from it. A high tracking error suggests a highly active management style or significant divergence from the benchmark, while a low tracking error indicates the portfolio closely mirrors its index, as is common with passive or index funds.

In essence, tracking error is a specific application of the standard deviation of excess return, where the excess is explicitly measured against a benchmark index for the purpose of evaluating active management risk. While all tracking errors are standard deviations of excess return, not all standard deviations of excess return are necessarily referred to as tracking error (e.g., when comparing to a risk-free rate for a Sharpe Ratio calculation).

FAQs

What is the primary purpose of the standard deviation of excess return?

The primary purpose is to measure the consistency or variability of an investment's performance when compared to a baseline, such as a risk-free rate or a market index. It helps assess the risk associated with generating returns beyond a passive or benchmark investment.

How does it differ from the standard deviation of total return?

The standard deviation of total return measures the overall volatility of an investment's absolute returns. The standard deviation of excess return, conversely, specifically measures the volatility of the difference between the investment's return and a benchmark's return. It isolates the risk of outperformance or underperformance.

Can it be used to compare different investments?

Yes, it can be used to compare investments, especially when evaluating active managers or strategies against a common benchmark. A lower standard deviation of excess return, combined with a positive average excess return, can indicate a more efficient or consistent active strategy.

Is a high or low standard deviation of excess return preferable?

Generally, a lower standard deviation of excess return is preferable for an active manager, as it implies more consistent outperformance (or less volatile underperformance). However, a high standard deviation of excess return can be acceptable if it is accompanied by significantly higher average excess returns, indicating a strategy that takes more active market risk for potentially greater rewards.

What is the role of the risk-free rate in calculating excess return?

The risk-free rate (often represented by the yield on short-term government securities) serves as a baseline for determining excess return. By subtracting the risk-free rate from an investment's total return, you isolate the return that compensates for taking on investment risk, rather than simply for the passage of time.