Positive average returns, but less money

A positive average return can sit beside a balance smaller than the amount you started with. Before treating the average as evidence of growth, ask how it was calculated.

The simple average of a list of returns describes that list. Following the money through those returns requires another step: apply each percentage to the balance then in the account.

Follow $100 through two periods

Consider an invented investment that gains 50 percent in one period and loses 40 percent in the next. The periods are equal in length. Start with $100, leave all returns invested, and make no deposits or withdrawals. Fees and taxes are excluded. These are supplied numbers for an explanation, not a forecast or a record of an actual investment.

The arithmetic average is positive:

(50% + −40%) ÷ 2 = 5% per period.

Now follow the balance:

• Start: $100.
• After the 50 percent gain: $150.
• After the 40 percent loss: $90.

The second percentage applies to $150. Forty percent of that balance is $60, so the loss exceeds the first period’s $50 gain. Across both periods, the original $100 has fallen by $10, a total return of −10 percent.

The 5 percent arithmetic average is correctly calculated. Using it as the growth rate of this starting capital would answer the wrong question.

Find the steady rate with the same ending

What constant rate, repeated for both periods, would turn $100 into $90?

The two actual growth factors are 1.50 and 0.60. Their product is 0.90. Taking its square root gives about 0.948683. Subtract one to express that factor as a return:

√(1.50 × 0.60) − 1 ≈ −5.13% per period.

This is the geometric average return. Repeating the unrounded rate twice takes $100 to $90. It is an equivalent steady rate, not a claim that either actual period returned −5.13 percent. The actual path still included a rise to $150.

Will Morrison’s explanation of arithmetic and geometric means, published by CFA Institute, follows this distinction between adding returns and multiplying growth factors.

“Per period” matters here. This example has not specified years or months. The figure is an annual compound rate only if each period is one year. Nor should you take the square root of the product of the raw percentages: the calculation uses one plus each return in decimal form.

What “volatility drag” describes

Compare the uneven pair with a separate invented pair: 5 percent followed by 5 percent. Both pairs have a 5 percent arithmetic average. The steady pair turns $100 into $105 and then $110.25. The uneven pair ends at $90.

That comparison helps explain the phrase “volatility drag”: an uneven return series can have a compound growth rate below its arithmetic average. For positive growth factors, the geometric average cannot exceed the arithmetic average; they match when the period returns are identical.

The gap is not an extra fee deducted from the account. It also does not establish that every more volatile investment will finish with less money than every steadier investment. That would require information about their returns, not just their variability. Will Morrison’s discussion of the terminology, also published by CFA Institute, cautions against treating the gap as a separate force pulling money away.

Reversing the order does not fix this example

With no deposits or withdrawals, reversing these same returns leaves the ending balance unchanged:

$100 × 0.60 × 1.50 = $90.

The intermediate balance is now $60 rather than $150. Those paths could feel very different, but rearranging this fixed pair does not change its product. The arithmetic and geometric averages are unchanged too.

Cash movements create a different question. Keep the same returns, but withdraw $10 immediately after the first period and make no other cash movements:

• Gain first: $100 becomes $150; withdraw $10; the remaining $140 loses 40 percent and ends at $84.
• Loss first: $100 becomes $60; withdraw $10; the remaining $50 gains 50 percent and ends at $75.

Each version has paid out the same $10, yet the remaining balances differ. The withdrawal changed how much money experienced the second return. These balances are not stand-alone investment-return percentages, and the difference should not be confused with the no-withdrawal calculation. Evaluating a person’s results with cash flows requires their amounts and dates.

Two measures for one completed record

A time-weighted return can be positive while a money-weighted return on the same account is negative. In David Allison's explanation, published by CFA Institute, the time-weighted measure removes the effect of external cash movements; the money-weighted measure reflects their timing and size.

Use a separate invented, completed two-year record in US dollars. Each period is exactly one year. The values and period returns already include every account fee; taxes are excluded. There are no withdrawals, distributions or contributions beyond those listed:

• Start: invest $100.
• End of the first year: a 20 percent return raises the value to $120. Immediately after that valuation, add $880, bringing the account to $1,000.
• End of the second year: a negative 10 percent return leaves a closing value of $900.

Total contributions are $980. The account ends $80 below that amount. The $880 addition was money supplied, not investment earnings.

The time-weighted calculation links the two period growth factors without counting that contribution as a gain: 1.20 × 0.90 − 1 = 8 percent over the complete two years. Its equivalent annual rate is √1.08 − 1, about 3.92 percent.

Here, calculate the annual money-weighted return using the internal rate of return. Find the annual rate r that would grow the initial $100 for two years and the later $880 for one year to the same $900 closing valuation:

100 × (1 + r)² + 880 × (1 + r) = 900.

Using a positive value for 1 + r gives r ≈ −7.46 percent a year. The calculation uses the closing valuation; it does not assert that $900 was withdrawn. Dividing the $80 loss by the $980 contributed would ignore when the later money entered.

The measures do not disagree about the transactions or the ending value. Only $100 earned the first year's 20 percent gain, adding $20. After the contribution, $1,000 incurred the second year's 10 percent loss, removing $100. The loss was applied to much more money than the earlier gain.

Repair a report saying, “The investment returned 8 percent, so my money grew 8 percent.” For this record, an accurate summary is: “The time-weighted return was 8 percent over two years, about 3.92 percent annually. The annual money-weighted return was about −7.46 percent. Contributions totaled $980; the closing value was $900.” Keep the method, period and external cash movements beside each figure before comparing it with another report.

Ask what the average is meant to show

When a return figure matters, ask which average it uses, what period it covers, and whether deposits, withdrawals, fees and taxes are included. For a fixed starting sum left invested, check the ending balance by multiplying the period growth factors. A positive simple average alone does not establish that the starting capital grew.

For a different kind of reading about money and ownership, read the opening of Infinite Upside: Building Wealth With Room to Live, Vijay Sheth’s business memoir. It follows a founder’s decisions and their continuing demands. If the story fits what you want to read, explore the editions.