Equal percentages act on different amounts
Two invented years. No deposits or withdrawals; all income stays invested. Costs and taxes excluded.
Euro scale starts at zero. Lines connect annual observations only; they do not show what happened within each year.
In A, the €2,400 loss acts on €12,000, after a €2,000 gain. The balance ends 4% lower even though the two annual percentages average to zero.
Calculate recovery from what remains
No cash flows, costs or taxes. Both bars use the same euro scale.
The required €2,000 gain is a larger share of the smaller remaining balance. This describes the required gain, not whether recovery will happen or when.
Compare other losses and recovery requirements
| Loss | Amount left | Gain needed |
|---|---|---|
| 10% | €9,000 | 11.1% |
| 20% | €8,000 | 25% |
| 30% | €7,000 | 42.9% |
| 50% | €5,000 | 100% |
An investment can keep falling or fail entirely. Even recovery to the original euro amount leaves less purchasing power if prices have risen. A hoped-for recovery does not fund a bill due earlier.
Choose the average for the question
- Arithmetic mean: 0%
- The simple average of the opening +20% and −20% yearly rates.
- Geometric mean: about −2.02%
- The constant annual compound rate connecting €10,000 to €9,600. Neither actual year earned this rate.
Connect a cumulative return to an annual rate
About −2.02% compounded twice reproduces the opening example’s 4% cumulative loss.
Separately, with no flows, costs or taxes, €10,000 growing to €12,100 over two years gives a 21% cumulative return and a 10% annualised return: €10,000 × 1.10 × 1.10 = €12,100. Dividing 21% by two gives 10.5%, which does not reproduce the ending value through compounding.
An “average annual return” on a factsheet may mean an annualised compound figure. Read the method and period rather than assume which average it uses.
Look between the starting and ending values
At the year-1 points in the opening graph, A has €4,000 more available than B. A payment due then cannot use the shared year-2 endpoint. Taking money out changes the base for the next return.
Read historical dates without treating them as an entry signal
Annualised returns hide drawdowns and payment deadlines. Starting just before a fall differs from starting just after it; annualising an unusually strong short period can look impressive. Selective favourable dates warrant reading the full period, not choosing an entry date from hindsight.
Compare the same dates and currency, with income and comparable costs included. Past averages promise no future return or recovery timetable. Test lower and uneven returns against spending: funding year twenty does not necessarily fund year five. The next lesson adds external cash flows.