Average Returns Can Be Calculated Using Or Arithmetic Average

Calculating average returns is a cornerstone of investment analysis, offering a simplified view of an investment's historical performance. The arithmetic average, also known as the mean, is one of the most common methods used for this purpose. While straightforward, understanding its nuances, strengths, and limitations is crucial for investors looking to make informed decisions.

Introduction to Average Returns

Average return represents the sum of returns over a period divided by the number of periods. This calculation provides a single number that summarizes how an investment has performed on average during that time frame. Investors use this metric to compare different investment options, assess historical performance, and project potential future returns.

The allure of average return lies in its simplicity. However, this simplicity can be misleading if not understood in context. The type of average used – arithmetic, geometric, or dollar-weighted – significantly impacts the results and the interpretation of those results. This article will focus on the arithmetic average, exploring its calculation, applications, strengths, weaknesses, and comparison to other averaging methods.

Calculating the Arithmetic Average Return

The arithmetic average return is calculated by summing the returns for each period in the dataset and dividing by the number of periods. The formula is as follows:

Arithmetic Average Return = (R1 + R2 + R3 + ... + Rn) / n

Where:

R1, R2, R3, ..., Rn are the returns for each period (e.g., year, quarter, month).
n is the number of periods.

Example:

Let's say an investment generated the following annual returns over five years:

Year 1: 10%
Year 2: 15%
Year 3: -5%
Year 4: 8%
Year 5: 12%

The arithmetic average return would be calculated as follows:

Arithmetic Average Return = (10% + 15% + (-5%) + 8% + 12%) / 5 = 40% / 5 = 8%

This means the investment yielded an average return of 8% per year over the five-year period.

Step-by-Step Breakdown:

Gather the Returns: Collect the returns for each period you want to analyze. Ensure the returns are expressed in the same format (e.g., percentages or decimals).
Sum the Returns: Add up all the individual returns. Be mindful of negative returns; treat them as negative numbers in the summation.
Count the Periods: Determine the number of periods included in your calculation. This is simply the number of data points you have.
Divide the Sum by the Number of Periods: Divide the total sum of returns by the number of periods. The result is the arithmetic average return.
Express as a Percentage: Multiply the result by 100 to express the average return as a percentage.

Applications of Arithmetic Average Return

The arithmetic average return is a versatile tool used in various aspects of investment analysis:

Performance Comparison: Investors use it to compare the performance of different investments or investment managers over the same time period. A higher arithmetic average return generally indicates better performance, although it’s crucial to consider risk.
Historical Analysis: Analyzing the historical arithmetic average return of an investment can provide insights into its past performance and potential future trends. This data helps investors understand the investment's behavior under different market conditions.
Benchmarking: Investors can compare the arithmetic average return of their portfolio to relevant benchmarks, such as the S&P 500 or a specific sector index. This comparison helps assess whether the portfolio is outperforming, underperforming, or matching the market.
Simple Projections: While not recommended as a primary forecasting tool, the arithmetic average return can be used for rudimentary estimations of potential future returns. This is particularly useful for illustrating scenarios or setting initial expectations.
Academic Research: The arithmetic average return is frequently used in academic studies to analyze investment performance, test financial theories, and model market behavior.

Strengths of the Arithmetic Average

Despite its limitations, the arithmetic average possesses several advantages:

Simplicity: The calculation is straightforward and easy to understand, even for those without a strong financial background. This makes it accessible to a wide range of investors.
Ease of Calculation: The formula requires only basic arithmetic operations, making it easy to calculate manually or with simple spreadsheet software.
Widely Used: Its widespread adoption makes it easy to compare results across different sources and analyses. The familiarity of the metric allows for quick understanding and interpretation.
Suitable for Short-Term Analysis: Over shorter time periods, the arithmetic average provides a reasonable estimate of average performance, especially when returns are relatively stable.
Unbiased Estimator: In statistical terms, the arithmetic average is an unbiased estimator of the expected return over a single period, assuming returns are independent and identically distributed.

Weaknesses and Limitations of the Arithmetic Average

The arithmetic average has significant limitations that investors must be aware of:

Ignores Compounding: The arithmetic average does not account for the effects of compounding, which can significantly impact long-term investment returns. It assumes that returns are earned on the initial investment amount each period, ignoring the fact that earnings can be reinvested to generate further returns.
Susceptible to Distortion by Extreme Values: The arithmetic average is highly sensitive to extreme values, or outliers. A single unusually high or low return can disproportionately influence the average, leading to a misleading representation of overall performance.
Overestimates Long-Term Returns: Due to its failure to account for compounding, the arithmetic average typically overestimates the actual long-term return an investor can expect to achieve. This can lead to unrealistic expectations and poor investment decisions.
Does Not Reflect Actual Wealth Growth: The arithmetic average doesn't accurately reflect how an investor's wealth actually grows over time. It doesn't consider the impact of volatility or the sequence of returns, which can significantly affect the final outcome.
Misleading for Volatile Investments: For investments with highly volatile returns, the arithmetic average can be particularly misleading. The swings in returns are smoothed out, obscuring the true risk and potential for loss.

Arithmetic Average vs. Geometric Average

The geometric average is another common method for calculating average returns. Unlike the arithmetic average, the geometric average does account for compounding. It represents the average return needed to achieve the same cumulative return over the period.

The formula for the geometric average return is:

Geometric Average Return = [(1 + R1) * (1 + R2) * (1 + R3) * ... * (1 + Rn)]^(1/n) - 1

Where:

R1, R2, R3, ..., Rn are the returns for each period (e.g., year, quarter, month).
n is the number of periods.

Key Differences and When to Use Each:

Compounding: The geometric average considers compounding, while the arithmetic average does not.
Long-Term vs. Short-Term: The geometric average is generally more appropriate for evaluating long-term investment performance, as it reflects the actual rate of wealth accumulation. The arithmetic average is more suitable for shorter periods or for estimating the expected return over a single period.
Volatility: In volatile markets, the geometric average provides a more conservative and realistic estimate of average returns. The arithmetic average can be overly optimistic in such environments.
Interpretation: The geometric average represents the constant rate of return needed to achieve the same final wealth as the actual investment. The arithmetic average represents the simple average of the periodic returns.

Example (Continuing from the previous example):

Using the same annual returns:

Year 1: 10%
Year 2: 15%
Year 3: -5%
Year 4: 8%
Year 5: 12%

The geometric average return would be calculated as follows:

Geometric Average Return = [(1 + 0.10) * (1 + 0.15) * (1 - 0.05) * (1 + 0.08) * (1 + 0.12)]^(1/5) - 1

Geometric Average Return = [1.10 * 1.15 * 0.95 * 1.08 * 1.12]^(0.2) - 1

Geometric Average Return = [1.3698]^(0.2) - 1

Geometric Average Return = 1.065 - 1 = 0.065 or 6.5%

In this example, the geometric average return (6.5%) is lower than the arithmetic average return (8%). This illustrates how the geometric average accounts for the impact of compounding and volatility, providing a more realistic view of long-term performance.

Dollar-Weighted Return (Internal Rate of Return)

Another method for calculating average returns is the dollar-weighted return, also known as the Internal Rate of Return (IRR). This method takes into account the timing and amount of cash flows into and out of an investment.

Key Differences and When to Use:

Cash Flows: The dollar-weighted return considers the impact of deposits and withdrawals on investment performance. The arithmetic and geometric averages do not.
Investor's Perspective: The dollar-weighted return reflects the actual return earned by the investor, taking into account their specific investment decisions.
Portfolio Performance: The dollar-weighted return is often used to evaluate the performance of a portfolio where the investor has made contributions or withdrawals over time.

When to Use:

The dollar-weighted return is most appropriate when evaluating the performance of a portfolio where the investor has control over the timing and amount of cash flows. It provides a more accurate representation of the investor's actual return than the arithmetic or geometric average in such cases.

Practical Considerations and Best Practices

Time Period: Choose a sufficiently long time period for analysis to capture a representative sample of market conditions. A longer time period will provide a more reliable estimate of average returns.
Data Consistency: Ensure that the data used for calculation is consistent and accurate. Inaccurate data will lead to misleading results.
Risk Assessment: Always consider risk alongside average returns. A higher average return may be accompanied by higher risk, which could make the investment unsuitable for some investors.
Diversification: Diversify your portfolio across different asset classes to reduce risk and improve overall returns.
Professional Advice: Seek advice from a qualified financial advisor before making investment decisions. A financial advisor can help you assess your risk tolerance, set realistic goals, and choose appropriate investments.
Understand the Limitations: Recognize the limitations of all averaging methods, including the arithmetic average. Use them as one tool among many when evaluating investment opportunities.
Consider Inflation: When evaluating long-term returns, consider the impact of inflation. Real returns (adjusted for inflation) provide a more accurate picture of the investment's purchasing power.
Sequence of Returns: Be aware of the impact of the sequence of returns. A series of negative returns early in the investment period can significantly reduce the final outcome, even if the average return is positive.
Use in Conjunction with Other Metrics: Don't rely solely on average returns. Consider other metrics such as standard deviation, Sharpe ratio, and drawdowns to get a more complete picture of investment performance.

Examples and Case Studies

Let's examine a few examples to illustrate the application and limitations of the arithmetic average:

Case Study 1: Comparing Two Stocks

Stock A: Annual returns of 8%, 10%, 12%, 14%, 16% (Arithmetic Average: 12%)
Stock B: Annual returns of 2%, 20%, 5%, 25%, 8% (Arithmetic Average: 12%)

Both stocks have the same arithmetic average return of 12%. However, Stock B is clearly more volatile. An investor focused solely on the arithmetic average might mistakenly believe the two stocks are equally attractive. Further analysis, considering risk metrics like standard deviation, is crucial.

Case Study 2: Impact of a Single Outlier

Investment: Annual returns of 5%, 7%, 6%, 8%, -20% (Arithmetic Average: 1.2%)

The negative return of -20% significantly drags down the arithmetic average. While this reflects the impact of the loss, it might not accurately represent the overall performance if the other returns are more typical.

Case Study 3: Long-Term Investment vs. Short-Term Trading

For a long-term investor, the geometric average would be more useful for understanding the compounded growth rate. For a short-term trader making decisions on a weekly or monthly basis, the arithmetic average might provide a reasonable estimate of expected return over that single period.

Common Misconceptions

Misconception: A high arithmetic average guarantees high future returns.
- Reality: Past performance is not indicative of future results. Market conditions can change, and there is no guarantee that an investment will continue to perform as it has in the past.
Misconception: The arithmetic average is the only way to calculate average returns.
- Reality: There are multiple methods for calculating average returns, each with its own strengths and weaknesses. The appropriate method depends on the specific context and the goals of the analysis.
Misconception: The arithmetic average accounts for compounding.
- Reality: The arithmetic average does not account for compounding. The geometric average is the appropriate method for considering the effects of compounding.
Misconception: A higher average return always means a better investment.
- Reality: A higher average return may be accompanied by higher risk. Investors must consider their risk tolerance and investment goals when evaluating investment opportunities.

Conclusion

The arithmetic average is a fundamental tool for calculating average returns. Its simplicity and ease of use make it a valuable starting point for investment analysis. However, investors must be aware of its limitations, particularly its failure to account for compounding and its susceptibility to distortion by extreme values. For long-term investment analysis, the geometric average is generally a more appropriate measure. By understanding the strengths and weaknesses of the arithmetic average and using it in conjunction with other metrics, investors can make more informed and effective decisions. Remember to always consider risk, seek professional advice, and diversify your portfolio to achieve your financial goals.