Why can an average return be zero while the ending balance is lower?

Define equal periods and no cash flows

Start with a return for each equally long period and state what each return includes. For this original exercise, use two one-year periods, one currency, no contributions or withdrawals, and no separate fees, taxes, or distributions. All percentages are invented.

The CFA Institute Research Foundation’s mean-return formulas distinguish a simple average from compound annual growth. The distinction is about calculation. It does not establish that either historical summary is a suitable forecast.

Average the fictional percentages

Assume Year 1 returns +20% and Year 2 returns −20%. The arithmetic mean is (20% − 20%) ÷ 2 = 0% per year. That correctly describes the simple average of these two stipulated annual returns.

Now follow an opening $1,000. After Year 1 it is $1,200. A 20% decline in Year 2 removes $240, leaving $960. The dollar rise and dollar fall differ because each percentage applies to a different opening balance.

Link the growth factors

The cumulative factor is 1.20 × 0.80 = 0.96, giving a cumulative return of −4%. To find the constant annual rate with the same two-year result, solve (1 + g)² = 0.96. Thus g = √0.96 − 1, or approximately −2.0204% per year.

The cumulative −4%, annual geometric −2.0204%, and arithmetic mean 0% are three different labels. Replacing one with another makes the report misleading even when each number was calculated correctly. State the period beside every percentage.

Compare a path with the same arithmetic mean

In a separate invented path, both annual returns are 0%. Its arithmetic mean is also 0%, but $1,000 remains $1,000 and its geometric mean is 0%. The shared arithmetic average did not determine the shared ending balance.

For another arithmetic check, +25% followed by −20% gives a product of 1.25 × 0.80 = 1.00. The ending balance returns to its start while the arithmetic mean is +2.5%. A positive simple average therefore need not imply positive cumulative growth.

A useful report card records the raw period returns, the simple average, the product of growth factors, the number and length of periods, and the annualization method. Another reader can then reproduce the same result without guessing whether the word average refers to percentages or linked growth.

Method, sources, and limits

The examples and square-root calculation were checked using decimal arithmetic. The CFA Institute source was opened on 26 October 2026 for the mathematical definitions; none of its historical return data is used here. These worksheets do not include cash-flow timing or estimate future returns. If money enters or leaves between periods, reconcile those flows before interpreting the balance change, and identify the return method actually used.