# Equivalent Uniform Annual Cost

## Table of Contents

## Present Worth to Annual Worth #

For a sum of money \(\small P\) invested at a compounding interest rate \(\small i\) for \(\small n\) years, the equivalent uniform annual cost (a.k.a “ordinary annuity, capital recovery cost”) \(\small A\) is given by:

$$A = P \cdot (A/P, i, n) = P \left[\frac{i(1 + i)^n}{(1 + i)^n - 1}\right] $$

Where

- \(\small P\) is the present value,
- \(\small i\) is the interest rate, and
- \(\small n\) is the number of periods

This can be rewritten to find P (“series present worth”):

$$P = A \cdot (P/A, i, n) = A \left[\frac{(1 + i)^n - 1}{i(1 + i)^n}\right]$$

## Future Worth to Annual Worth #

If the future worth is known instead, the formula is

$$A = F \cdot (A/F, i, n) = F \left[\frac{i}{(1 + i)^n - 1}\right]$$

Rearranging again, F can be found instead if A is known:

$$F = A \cdot (F/A , i, n) = A \left[\frac{(1 + i)^n - 1}{i}\right]$$

## Sources #

Note that these sources use a slightly rearranged form of the formula, but it is equivalent.