How to Calculate the Average Return on an Investment

A key metric for assessing investment performance is the average return, which provides a concise summary of an investment’s profitability over a specified period.

Fundamentals of Investment Returns

An investment return represents the gain or loss generated from an investment over a certain timeframe. This can be expressed as a dollar amount or, more commonly, as a percentage of the initial investment.

While a single period’s return offers a snapshot, it often does not provide a complete picture of an investment’s long-term behavior. Investments frequently experience fluctuations, with periods of gains and losses. To gain a more comprehensive understanding of performance across multiple periods, averaging returns becomes necessary.

Averaging helps to smooth out these fluctuations, offering a generalized view of an investment’s typical performance. Different methods exist for calculating average returns, each suited for distinct analytical purposes. The choice of method depends on what aspect of performance an investor aims to evaluate.

Calculating Arithmetic Average Return

The arithmetic average return is a straightforward calculation representing the simple average of a series of returns over multiple periods. It is determined by summing all the periodic returns and then dividing the total by the number of periods. This method provides a general sense of an investment’s performance in any given period.

The formula for the arithmetic average return (AAR) is:

AAR = (R1 + R2 + … + Rn) / n

Here, R1, R2, up to Rn represent the returns for each individual period, and ‘n’ is the total number of periods.

Consider an example with a stock that generated annual returns over three years: Year 1: 20%, Year 2: -10%, Year 3: 15%. To calculate the arithmetic average return, sum these percentages: 20% + (-10%) + 15% = 25%. Then, divide by the number of periods, which is three: 25% / 3 = 8.33%.

This method is useful for understanding the average return over individual, independent periods. It is often applied when forecasting the expected return for a single future period. However, a limitation of the arithmetic average is that it does not account for the effect of compounding, where returns from one period influence the base for subsequent periods. It typically overstates the true growth rate of an investment over time, especially when returns are volatile.

Calculating Geometric Average Return

The geometric average return, also known as the compound annual growth rate (CAGR), provides a more accurate measure of an investment’s performance over multiple periods by considering the effect of compounding. This method accounts for the fact that gains or losses in one period affect the capital available for the next period. It is particularly relevant for long-term investments where returns are reinvested.

The formula for the geometric average return (GAR) is:

GAR = [(1 + R1) (1 + R2) … (1 + Rn)]^(1/n) – 1

In this formula, R1, R2, up to Rn are the decimal returns for each period, and ‘n’ is the number of periods. The “1 +” before each return converts the percentage return into a growth factor, and the “minus 1” at the end converts the final growth factor back into a percentage return.

Let’s use the same example as before: Year 1: 20% (0.20), Year 2: -10% (-0.10), Year 3: 15% (0.15).
First, add 1 to each return:
Year 1: 1 + 0.20 = 1.20
Year 2: 1 + (-0.10) = 0.90
Year 3: 1 + 0.15 = 1.15
Next, multiply these values together: 1.20 0.90 1.15 = 1.242.
Then, raise this product to the power of 1 divided by the number of periods (which is 3): (1.242)^(1/3) ≈ 1.0749.
Finally, subtract 1: 1.0749 – 1 = 0.0749, or 7.49%.

Comparing this to the arithmetic average of 8.33%, the geometric average of 7.49% is lower. This difference highlights how the geometric average provides a more realistic representation of the actual compounded growth an investor would have experienced. When investment returns fluctuate significantly, the geometric mean will be considerably different from the arithmetic mean, offering a more accurate reflection of the true return over time. It is considered a more accurate measure of long-term performance because it incorporates the impact of volatility and compounding.

Selecting the Appropriate Average Return

The choice between arithmetic and geometric average returns depends on the specific question an investor is trying to answer. Each method serves a different purpose in analyzing investment performance. Understanding these distinctions helps in accurately interpreting historical data and making future projections.

The arithmetic average return is typically used for understanding the average return over individual, discrete periods. It is also suitable for making simple forecasts of a single future period’s performance, assuming returns are independent. This average can be helpful for quick assessments, particularly for short-term investment analysis.

Conversely, the geometric average return is generally preferred for understanding the true compounded annual growth rate of an investment over multiple periods. It provides a more accurate measure of how an initial investment would have grown over time, assuming reinvestment of returns. This makes it the standard for comparing the historical performance of different investments or portfolios, especially over longer horizons where compounding plays a significant role. The geometric mean reflects the actual wealth accumulation from an investment.