Geometric growth
Why gains and losses compound rather than cancel.
Library note. Check the assumptions and further reading before applying a formula.
What Is This?
A return changes the amount you have for the next step. That makes a sequence of returns a multiplication problem. Logarithms turn those multiplications into additions. Here, r is the return in each period and g is the average log return. Convert g back with exp(g) minus one to get the constant per-period return that would produce the same final wealth.
Try an example
Start with £100. A 50% gain leaves £150; a 50% loss then leaves £75. The arithmetic average return is zero, but the money has fallen by a quarter. The equivalent steady return is about minus 13.4% per period.
Where it needs care
This expression requires each return to exceed minus 100%. A total loss makes log wealth undefined. A historical growth rate does not promise the same future rate.
Historical Context
Logarithms have long been used to simplify products. John Kelly used expected log wealth as the objective for repeated betting in his 1956 paper.
Real-World Applications
- Compare investment paths after fees.
- Describe compound changes in populations or repeated growth processes.
