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An arithmetic average adds period returns and divides by the number of periods; it does not show what happened to a fixed starting sum. Wealth compounds by multiplying each period's growth factor, so a large loss can more than undo an earlier gain even when the simple average is positive. The geometric average expresses the constant rate that reproduces the actual ending value.
Two averages answer different questions
The arithmetic mean describes the average of the individual period returns. The geometric mean describes the realised rate per period over the whole sequence, assuming no external cash flows. The difference matters when a planner explains historical account growth: applying the arithmetic mean as though it had been earned every period can imply an ending value the account never reached.
CFP Board includes quantitative investment concepts and return measures in its exam knowledge topics. CFA Institute also discusses the distinction between arithmetic and geometric return measures in its explanation of compounded growth. The underlying calculation is ordinary compounding, independent of which exam asks for it.
An illustrative account that loses value
Take an invented account starting at £100, with a 50% return in the first period and a 40% loss in the second. Assume equal-length periods, no deposits or withdrawals, and returns already include any income. These figures are illustrative, not historical performance.
| Start | Given | £100 |
| After first period | £100 × 1.50 | £150 |
| After second period | £150 × 0.60 | £90 |
The arithmetic mean is positive 5% per period: the 50% gain and 40% loss sum to 10 percentage points, then divide by two. Yet the account ends £10 below its starting value. The second-period loss is applied to £150, a larger base than the initial £100.
Check the geometric result against the account: £100 multiplied by the two growth factors is £90. A constant loss of about 5.13% in each of the two equal periods leads to the same ending value. Compounding positive 5% for both periods would instead produce £110.25, so it cannot describe this realised path.
How to avoid the exam trap
- When asked for the simple average period return, add the period returns and divide by their count.
- When asked for realised compound growth, multiply the growth factors first, then take the root for the number of equal periods.
- Check whether there were deposits or withdrawals. The clean wealth calculation above assumes none; external flows need separate treatment.
Keep each return tied to its own starting value. A 40% fall after a 50% rise removes £60 from £150; it is not a £40 reversal of the original £100. For a broader route through investment planning, see the CFP program page and Omni's study methodology.
Sources
- CFP® Exam: What You'll Be Tested On · cfp.net · Retrieved
- The Myth of Volatility Drag (Part 2) · rpc.cfainstitute.org · Retrieved
