The formula
cumulative = ((1+i₁) × (1+i₂) × … × (1+iₙ) − 1) × 100
Where it comes from
Inflation rates cannot be added, they have to be multiplied, because each year's rise applies to prices that already include every previous rise. Adding them is always an underestimate, and the error grows with both the number of years and the size of the rates.
How to work it out by hand
- Turn each year's rate into a factor by adding 1 to the decimal
- Multiply all the factors together
- Subtract 1 and multiply by 100 for the cumulative percentage
- For the annual average, take the nth root of the product instead of dividing
What is worth knowing
The gap between adding and multiplying looks trivial over four moderate years — 18% against 19.14% — and becomes enormous over long stretches or high rates. Ten years at 10% is not 100% but 159%. This is also why the average inflation quoted for a period is a geometric mean and not an arithmetic one: the arithmetic average of those four years is 4.5%, but the rate that actually reproduces the outcome is 4.48%. Close here, far apart when the individual years vary a lot, which is exactly when people reach for the average.