QQ 102, 2 min read
Returns and compounding
After this lesson you can
- Compute simple and log returns
- Compound a sequence of returns
- Work out the gain needed to recover a loss
A return is the change in value as a fraction of where you started.
simple return = (end price - start price) / start price
100 to 110 is a return of 0.10, or 10%.
Returns multiply
Two periods of +10% do not make 20%. They compound:
1.10 x 1.10 = 1.21 a 21% gain
The same is true of losses, and it is not symmetric. Lose 50% and you have half. To get back you need to double, which is +100%.
gain needed to recover = loss / (1 - loss)
0.50 / (1 - 0.50) = 1.00
A 20% loss needs 25%. A 90% loss needs 900%.
Log returns
log return = ln(end price / start price)
Log returns add up across time. The log return over a year is the sum of the daily log returns, which makes the arithmetic easier. For small moves a log return and a simple return are nearly the same number. For large moves they differ: a simple return of +100% is a log return of about 0.69.
Average return overstates growth
Gain 50% then lose 50% and the average return is zero. You have 75% of what you started with.
1.50 x 0.50 = 0.75
The average of simple returns overstates what compounding delivers whenever returns vary. The more they vary, the bigger the gap.
Key points
- Returns multiply, they do not add
- A loss of L needs a gain of L / (1 - L) to recover
- Average return overstates growth when returns vary
Check your understanding
An account falls 25%. What gain gets it back to where it started?
A strategy gains 50% one year and loses 50% the next. Where does it end?
Further reading
- Hull, J. C. Options, Futures, and Other Derivatives. Pearson. The sections on continuous compounding.
- Campbell, J. Y., Lo, A. W. and MacKinlay, A. C. (1997). The Econometrics of Financial Markets. Princeton University Press. Chapter 1 defines returns carefully.