# De-compound

*Compounding.*

To compound means:

- to calculate interest, return or growth on an accumulating total,
- including earlier amounts of interest, return or growth.

De-compounding is the reverse process.

It calculates a compound rate per period for a shorter period of time, based on a total rate of growth for a longer period.

**Example: de-compounding calculation**

Total growth for a two-year period = 69%.

What is the average compound rate of growth per year, for the two years?

Rate per year = (End amount / Starting amount)^{(1/n)} - 1

*Where:*

n = number of years

Let's say the starting amount = $100m

Total growth for two years = $100m x 0.69

= $69m

End amount = $100m + $69m

= $169m

The rate per year is:

= (169 / 100)^{(1/2)} - 1

= 1.69^{(1/2)} - 1

= **30%**

Check:

Growth for the first year:

= $100m x 0.30

= $30m

Balance at the end of the first year:

$100m + $30m

= $130m

Growth for the second year:

= $130m x 0.30

= $39m

Balance at the end of the second year:

$130m + $39m

= $169m

Total percentage growth for the two-year period:

169/100 - 1

= 1.69 - 1

= 0.69

= **69%**