# How much annual income would need to be generated over a 5 year period, to be equivalent to...

How much annual income would need to be generated over a 5 year period, to be equivalent to $4,000 over an 8 year period? Consider 6% nominal interest (compounded annually) for both cases. ## Annual Percentage Rate and Effective Annual Rate: The annual percentage rate is the nominal annual interest rate, that is often the quoted rate on financial products. The annual percentage rate is equal to the effective annual rate if interest compounds exactly once a year. ## Answer and Explanation: The annual income would be$5,896.73.

The two cash flows are equivalent if their present values are the same. We can use the following formula to compute the present value of an annuity with periodic payment {eq}M {/eq} for {eq}T{/eq} periods, given periodic return {eq}r{/eq}:

• {eq}\displaystyle \frac{M(1 - (1 + r)^{-T})}{r} {/eq}

Denote the annual income over the 5 year period by {eq}M {/eq}. Then we have:

• {eq}\displaystyle \frac{M(1 - (1 + 6\%)^{-5})}{6\%} = \frac{4000*(1 - (1 + 6\%)^{-8})}{6\%}\\ 4.212363786 * M = 24839.17524\\ M = 5,896.73 {/eq}