Work out an annualised return and explain what it means
The idea
Annualising restates a return earned over one period as the equivalent rate per year, so that returns over different horizons can be compared. It compounds rather than multiplies: a return r over a period that occurs m times a year annualises to (1 + r)ᵐ - 1. Annualising a very short period magnifies both the return and any noise in it, which is why a strong week expressed as an annual rate is close to meaningless. The convention makes comparison possible; it does not make a short-period result any more informative than it was.
Formulas
Annualising a return
(1 + r)ᵐ − 1
- r
- the return over one period
- m
- how many of those periods there are in a year
It compounds rather than multiplies. Annualising a very short period magnifies the noise in it by the same factor as the return.
A practice question
Why are returns annualised?
- To make returns over different holding periods comparable — correct
- To increase the reported returnWhy not: It enlarges a positive return and worsens a negative one, so it is no flattering device. The purpose is a common scale.
- To remove the effect of compoundingWhy not: It relies on compounding rather than removing it — that is why a 3% quarter annualises to 12.55% and not 12%.
Different horizons cannot be ranked as they stand. Expressing each as a rate per year puts them on one scale — it does not change the underlying performance.
A harder one
A quarterly return of 3% is annualised. What is the annual equivalent?
- 12.00%Why not: 3% x 4 — simple multiplication, which ignores that each quarter compounds on the balance the last one left.
- 12.55% — correct
- 3.00%Why not: The quarterly figure quoted back, unconverted. Four quarters have to be compounded together.
(1.03)⁴ − 1 = 12.55%. Multiplying by four gives 12% and ignores that each quarter compounds on the last.
The hardest tier
Why is annualising a single strong week's return usually misleading?
- Weekly returns cannot be compoundedWhy not: They can, and the arithmetic here is perfectly valid. It is the inference from it that does not follow.
- Annualisation requires at least a year of data to be validWhy not: No minimum applies — the conversion is defined for any period. The problem is how little evidence one week contains, not whether the formula applies.
- It projects a short, noisy result across a year as though it would repeat — correct
Compounding a week's noise fifty-two times produces a large number that reflects randomness rather than expected performance. The arithmetic is correct and the implication is not.
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