Mean return

What Is Mean Return?

Mean return, also known as the arithmetic mean return, is a fundamental metric in investment performance measurement that represents the simple average of a series of returns over a specified period. It is calculated by summing the returns for each period and dividing by the number of periods. This statistical measure falls under the broader category of quantitative finance and is widely used to provide a quick snapshot of an investment's past performance without accounting for the effects of compounding. While the mean return offers an intuitive understanding of average performance, its application and interpretation vary depending on the context, especially when evaluating performance over multiple periods or in comparison to other return measures.

History and Origin

The concept of using mean returns in finance gained significant prominence with the development of portfolio theory. Harry Markowitz, often considered the father of modern portfolio theory (MPT), introduced the idea of optimizing portfolios based on their expected return and standard deviation in his seminal 1952 paper, "Portfolio Selection," published in The Journal of Finance.⁵ Markowitz's work revolutionized investment management by demonstrating that investors could achieve a better risk-return tradeoff through diversification, focusing on the portfolio as a whole rather than individual assets. This framework inherently relied on the arithmetic mean to represent the expected returns of individual assets and portfolios for a single period. The mean return thus became a cornerstone of modern financial analysis, providing a concise measure for assessing average profitability.

Key Takeaways

• Mean return calculates the simple average of a set of periodic returns.
• It offers an easy-to-understand measure of an investment's average past performance.
• The mean return does not account for the effect of compounding, making it less suitable for long-term, multi-period performance evaluation.
• It is often used in portfolio optimization models, such as Modern Portfolio Theory, to estimate expected single-period returns.
• Regulatory bodies like the SEC often require specific return calculations and presentations in performance reporting.

Formula and Calculation

The mean return (arithmetic mean) is calculated by summing all the individual periodic returns and dividing by the total number of periods.

The formula for the mean return is as follows:

Where:

• (\bar{R}) = Mean Return
• (R_i) = Return in period (i)
• (n) = Total number of periods

For instance, if an investment had annual returns of 10%, -5%, and 15% over three years, the mean return would be calculated by summing these percentages and dividing by three. The individual (R_i) values are expressed as decimals (e.g., 0.10, -0.05, 0.15). This calculation provides a simple average, offering a clear initial look at the investment's typical performance without accounting for the sequence or reinvestment of returns over time.

Interpreting the Mean Return

The mean return provides a straightforward interpretation of the average periodic change in an investment's value. For example, a stock with a mean return of 8% over five years suggests that, on average, it gained 8% per year during that period. This metric is particularly useful for understanding the average return over discrete, non-compounding periods or for single-period expected return estimates in models like the Capital Asset Pricing Model (CAPM).

However, it is crucial to understand that the mean return does not reflect the actual cumulative wealth accumulation over multiple periods due to the impact of volatility and compounding. A high mean return might seem appealing, but if individual period returns are highly dispersed (meaning high volatility), the actual wealth generated might be significantly less than what the simple average suggests. Therefore, while mean return offers a quick summary, it should be considered alongside other measures of investment performance to gain a comprehensive understanding.

Hypothetical Example

Consider an investor, Sarah, who purchased a mutual fund. She wants to calculate its mean annual return over the past four years. The fund's annual returns were:

• Year 1: +20%
• Year 2: -10%
• Year 3: +30%
• Year 4: +5%

To calculate the mean return:

1. Convert the percentages to decimals: 0.20, -0.10, 0.30, 0.05.
2. Sum the returns: (0.20 + (-0.10) + 0.30 + 0.05 = 0.45).
3. Divide by the number of periods (4 years): (0.45 / 4 = 0.1125).

The mean annual return for Sarah's mutual fund is 0.1125, or 11.25%.

This 11.25% represents the arithmetic average of the annual returns. If Sarah had initially invested $10,000, this mean return does not directly tell her how much her investment is worth after four years due to the effect of compounding returns, where gains or losses in one period impact the base for the next. To accurately reflect the growth of her portfolio, she would need to consider each year's return sequentially or use a measure like the geometric mean return. This distinction is critical for evaluating true wealth accumulation versus simple average performance, and often informs decisions regarding asset allocation and rebalancing strategies.

Practical Applications

Mean return finds several practical applications in finance and investing, particularly where a simple average of periodic performance is needed or for single-period estimations.

• Performance Benchmarking: Investment managers often report mean returns to compare their fund's average performance against a benchmark index over specific periods.
• Portfolio Optimization: In models like Modern Portfolio Theory, mean returns are used as an input for the expected return of individual assets when constructing an optimal portfolio designed to balance risk and return.
• Regulatory Reporting: While more sophisticated measures are also required, basic return calculations, often starting with the arithmetic mean, form part of financial disclosures. The U.S. Securities and Exchange Commission (SEC) has provided guidance on performance reporting for investment advisers, emphasizing the clear presentation of performance metrics in marketing materials, including distinguishing between gross and net returns.⁴
• Financial Modeling: Mean returns are used in various financial modeling and forecasting scenarios to estimate average outcomes for discrete periods.
• Risk Management: As a component of risk management assessments, mean return, alongside measures of volatility, helps analysts understand the typical payoff associated with a particular investment. Investors often discuss how to calculate their personal returns, seeking straightforward methods to understand their investment growth.³

Limitations and Criticisms

While useful, the mean return has significant limitations, particularly when assessing multi-period investment performance. The primary criticism is its failure to account for the effect of compounding, which can lead to an overstatement of actual wealth growth over time. Because the arithmetic mean averages discrete returns without considering that the base investment changes after each period's gain or loss, it can paint a misleadingly optimistic picture, especially with volatile returns.

For instance, if an investment gains 50% in one year and loses 50% in the next, the mean return is 0% ((50% + (-50%)) / 2 = 0%). However, an initial $100 investment would become $150 after the first year ($100 * 1.50) and then $75 after the second year ($150 * 0.50), resulting in an actual loss of 25%. The mean return in this case, 0%, significantly overstates the true outcome. This phenomenon is often referred to as "volatility drag" or "variance drain."²

This limitation means that the mean return is generally inappropriate for calculating the average growth rate of an investment over a long time horizon or for forecasting future portfolio values when returns are reinvested. Academic research highlights that compounding at the arithmetic average historical return can lead to an upwardly biased forecast of future portfolio value.¹ For accurate representation of cumulative wealth, especially with fluctuating returns, the geometric mean return is typically preferred. The divergence between the two measures becomes more pronounced as return volatility increases.

Mean Return vs. Geometric Mean Return

The terms "mean return" and "geometric mean return" are often confused, but they serve different purposes in investment performance analysis.

The primary distinction lies in how each accounts for compounding. The mean return provides a statistical average of individual periods, while the geometric mean return calculates the constant annual rate at which an investment would have grown if it had compounded at the same rate each period. For investors focused on the actual growth of their capital over time, especially over several years, the geometric mean return is generally considered a more accurate and realistic measure.

FAQs

Q1: When should I use mean return instead of geometric mean return?

Mean return is most appropriate for single-period analysis or for estimating the expected return of an asset for a specific upcoming period, especially in models where returns are not compounded over multiple time steps. It can also be used when analyzing discrete, non-compounding events, such as the average daily price change of a stock.

Q2: Does mean return account for dividends or interest?

Yes, when calculating the mean return for an investment, each periodic return ((R_i)) should include both capital gains (or losses) and any income generated, such as dividends for stocks or interest for bonds. This ensures that the calculation reflects the total return for each period.

Q3: Why is the mean return often higher than the geometric mean return?

The mean return tends to be higher than the geometric mean return because it does not account for the impact of volatility and compounding. When returns fluctuate (both up and down), the simple average (mean return) will always be greater than or equal to the compound average (geometric mean return). The greater the fluctuation in returns, the larger the difference between the two. This effect is often called "volatility drag."

Q4: Can I use mean return to forecast future investment performance?

While mean return is sometimes used as a simple estimate for future returns, it has limitations, especially for long-term forecasts. For multi-period forecasts where returns are reinvested, the geometric mean, or a rate somewhere between the arithmetic and geometric mean, may provide a more accurate and less biased forecast of cumulative wealth, particularly over longer time horizons. Investors typically rely on historical data for such estimations, but past performance is not indicative of future results.