Geometric Average Return and Compound Growth

Why Compound Growth Needs Its Own Average

A sequence of investment returns changes wealth period by period. Each return applies to the wealth remaining after the previous period, so a gain or loss changes the base on which the next return is earned. The geometric average summarizes that sequence as the constant per-period return that would produce the same ending wealth.

This measure is useful when the question concerns the compound growth of an investment over several periods. It is different from simply finding the arithmetic mean of the listed returns: the arithmetic mean does not, in general, reproduce the sequence’s ending value.

Definition: For $n$ equal-length periods with periodic total returns $R_1,R_2,\ldots,R_n$, the geometric average return $G$ is the constant per-period return that gives the same ending wealth as the full sequence. Assuming each period’s wealth factor $1+R_t$ is positive, $$ 1+G=\left[\prod_{t=1}^{n}(1+R_t)\right]^{1/n}, \qquad G=\left[\prod_{t=1}^{n}(1+R_t)\right]^{1/n}-1. $$

Here $R_t$ is the return for period $t$, written as a decimal in calculations; for example, an 8% return is $0.08$. The formula assumes returns cover equal-length periods and are total returns, with any cash distributions treated consistently in the wealth calculation. It also assumes no outside contributions or withdrawals during the measurement interval. If the periods have different lengths, this formula gives a constant rate per listed period, not necessarily a rate per year.

For ordinary investment returns, $R_t$ cannot be less than $-1$. If any period has a return of exactly $-100\%$, ending wealth is zero; the sequence does not have a positive compound-equivalent growth factor. The geometric-average formula above is intended for positive wealth factors, so $R_t>-1$ for every period.

Deriving the Compound-Equivalent Rate

The formula follows directly from the way wealth changes. Suppose the initial investment is $W_0$. After the first period it is multiplied by $1+R_1$; after the second, by $1+R_2$; and so on. Thus, the realized sequence’s ending wealth is $W_0$ multiplied by all the period wealth factors.

W_n=W_0(1+R_1)(1+R_2)\cdots(1+R_n) =W_0\prod_{t=1}^{n}(1+R_t)

Now let $G$ be a constant return earned in every period. The ending wealth under that constant rate is $W_0(1+G)^n$. By definition, it must equal the ending wealth under the actual sequence. Equating the two expressions and dividing by the positive initial wealth gives:

W_0(1+G)^n=W_0\prod_{t=1}^{n}(1+R_t) \quad\Longrightarrow\quad (1+G)^n=\prod_{t=1}^{n}(1+R_t)

Taking the positive $n$th root gives $1+G$, and subtracting 1 gives $G$. This establishes the formula and explains its interpretation: the geometric average is not an arbitrary summary, but the constant rate that exactly matches compound growth.

Calculating the Geometric Average

A reliable calculation has two stages. First multiply the wealth factors, rather than averaging the returns. Then take the root corresponding to the number of periods and subtract 1. Keep full precision through the calculation and round only the final rate.

Convert returns to wealth factors.
For each return $R_t$, calculate $1+R_t$. A return of $-20\%$, for example, gives a factor of $0.80$.

Multiply the factors.
Their product is the ratio of ending wealth to starting wealth.

Take the $n$th root and subtract 1.
The result is the constant return per listed period with the same compound effect.

Worked Example: A Gain, a Loss, and a Recovery

A hypothetical fund earns returns of $10\%$, $-20\%$, and $25\%$ in three successive annual periods. Find its geometric average annual return. If the initial investment is $\$1{,}000$, also check the ending value using the calculated average. Use the compound-growth method, and avoid treating the arithmetic average as the compound-equivalent rate.

Step 1: Convert each return to a wealth factor. The factors are $1.10$, $0.80$, and $1.25$.

Step 2: Multiply the factors. $$ 1.10(0.80)(1.25)=0.88(1.25)=1.10 $$ The sequence therefore multiplies wealth by $1.10$. The actual ending value is $\$1{,}000(1.10)=\$1{,}100$.

Step 3: Take the cube root. There are three periods, so $$ G=(1.10)^{1/3}-1=1.032280\ldots-1=0.032280\ldots $$ Rounded to four decimal places as a percentage, the geometric average annual return is 3.2280%.

Check using the constant rate. The unrounded rate gives $$ \$1{,}000(1+G)^3=\$1{,}000(1.10)=\$1{,}100. $$ The ending value agrees with the actual sequence. The method avoids the mistake of reporting the arithmetic mean, $(10\%-20\%+25\%)/3=5\%$, as the rate that reproduces the ending value.

Interpreting the Result and Checking It

The geometric average answers a specific question: what single rate, applied in every period, would produce this sequence’s ending wealth? It does not say that the investment actually earned that rate in each period. The actual returns can vary substantially while having the same product of wealth factors.

A useful check is to compare the sequence’s wealth factor with $(1+G)^n$. They should be equal before any rounding. If the initial investment is known, multiply it by either factor to verify the ending wealth. A disagreement usually signals a missed negative sign, an incorrect number of periods, or an arithmetic average substituted for a geometric one.

Worked Example: Compound Growth From Four Returns

A hypothetical account begins with $\$25{,}000$ and earns four quarterly returns of $12\%$, $-15\%$, $30\%$, and $-5\%$. Find the geometric average return per quarter and the account’s ending value. Assume the given returns are total returns for the quarters, with no contributions or withdrawals. Use full precision until the final rate.

Step 1: Find the wealth factors. They are $1.12$, $0.85$, $1.30$, and $0.95$.

Step 2: Multiply them in sequence. $$ 1.12(0.85)=0.952,\qquad 0.952(1.30)=1.2376,\qquad 1.2376(0.95)=1.17572 $$ Thus, the account’s ending-to-starting wealth ratio is $1.17572$. The ending value is $$ \$25{,}000(1.17572)=\$29{,}393.00. $$

Step 3: Take the fourth root. $$ G=(1.17572)^{1/4}-1=1.041300\ldots-1=0.041300\ldots $$ The geometric average return is 4.130% per quarter, rounded to four significant figures.

Check. The unrounded quarterly rate satisfies $(1+G)^4=1.17572$, so applying it for four quarters to $\$25{,}000$ also gives $\$29{,}393.00$. This rate is per quarter because the four observations are quarterly; it is not an annual rate.

Solving for a Missing Period Return

The same relationship can be rearranged when the compound-equivalent rate is known but one period’s return is missing. The target compound factor is $(1+G)^n$. Divide it by the product of the known wealth factors to obtain the missing factor, then subtract 1. This approach is useful when a required ending value or average rate imposes a constraint on the final period.

Worked Example: Finding a Return Needed to Meet a Growth Target

A hypothetical investment has quarterly returns of $10\%$, $-5\%$, $20\%$, and an unknown fourth return. The four-quarter geometric average must be $5\%$. Find the required fourth return. Assume there are no contributions or withdrawals and all four returns refer to equal-length quarters. Use the target compound factor rather than averaging the known returns.

Step 1: Calculate the total factor required. A $5\%$ geometric average for four periods requires $$ (1.05)^4=1.21550625. $$

Step 2: Multiply the known factors. $$ (1.10)(0.95)(1.20)=1.045(1.20)=1.254. $$ Let $R_4$ be the missing return. Its factor must satisfy $$ 1.254(1+R_4)=1.21550625. $$

Step 3: Solve and convert the factor to a return. $$ 1+R_4=\frac{1.21550625}{1.254}=0.969303\ldots, \qquad R_4=0.969303\ldots-1=-0.030697\ldots $$ The required fourth-period return is $-3.0697\%$, rounded to four decimal places as a percentage.

Check. The four factors then multiply to $1.21550625$, whose fourth root is $1.05$. The result meets the specified geometric-average target. The method avoids incorrectly averaging only the three known returns or dividing the target growth by four.

Common Trap: Averaging Returns Instead of Growth Factors

Consider two successive returns of $50\%$ and $-50\%$. Their arithmetic mean is zero. But starting with $\$100$, the first return raises wealth to $\$150$, and the second reduces it to $\$75$. The ending wealth is only $75\%$ of the starting amount.

The correct compound-equivalent rate is found from the wealth factors $1.50$ and $0.50$: $$ G=(1.50 \times 0.50)^{1/2}-1 =\sqrt{0.75}-1 =-0.133974\ldots $$ So the geometric average is $-13.3974\%$, rounded to four decimal places as a percentage. The zero arithmetic mean does not reproduce the loss in wealth. This example highlights why the correct method depends on the question: for an equivalent compound-growth rate, multiply factors, take the root, and subtract 1.

Other frequent errors include taking the $n$th root of the product of returns $R_t$ instead of the product of $1+R_t$, and using the wrong value of $n$. The root counts the number of periods represented by the returns. A final check using $(1+G)^n$ can catch both errors.

Key takeaway: The geometric average is the per-period compound-equivalent return. Multiply the period wealth factors, take the root for the number of periods, and subtract 1. Verify that the resulting constant rate reproduces the same ending wealth.

Check Your Understanding

For each problem, use the compound-growth relationship and keep full precision until the final answer.

A hypothetical investment earns $5\%$ and then $10\%$ in two equal-length periods. Find its geometric average return per period. (Answer: 7.4709%)
An account starts at $\$8{,}000$, earns $25\%$ in one period, and then loses $20\%$ in the next. Find its ending value and geometric average return per period. (Answer: $\$8{,}000.00$; 0.0000%)
A three-period investment earns $21\%$, $-10\%$, and an unknown return. What third-period return gives the sequence a geometric average of $10\%$? (Answer: 22.2222%)
A hypothetical account earns $10\%$, $-10\%$, $10\%$, and $-10\%$ over four equal-length periods. Find its geometric average return per period. (Answer: $-0.5013\%$)
A portfolio begins with $\$12{,}000$ and earns $50\%$ in the first period and $-20\%$ in the second. Find its ending value and its geometric average return per period. (Answer: $\$14{,}400.00$; 9.5445%)

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