# You are valuing an investment that will pay you $26,000 per year for the first 9 years,$34,000...

## Question:

You are valuing an investment that will pay you $26,000 per year for the first 9 years,$34,000 per year for the next 11 years, and \$47,000 year the following 14 years (all payments are the end of each year). Another similar risk investment alternative is an account with a quoted annual interest rate of 9.00% with monthly compounding of interest. What is the value in today's dollars of the set of cash flows you have been offered?

## Compounding Frequency and Effective Annual Rate:

Compounding frequency is the number of times interest is calculated in a year. For example, wit monthly compounding interest is calculated 12 times a year. The higher the compounding frequency, all else the same, the higher the effective annual rate.

We first compute the effective annual rate:

• {eq}(1 + 9\%/12)^{12} - 1 = 9.38\% {/eq}

The investments provides cash flows that are an annuity. 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}

Applying the formula, the present value of the investment is:

• {eq}\displaystyle \frac{26,000(1 - (1 + 9.38\%)^{-9})}{9.38\%} + \frac{34,000*(1 - (1 + 9.38\%)^{-11})}{9.38\%(1 + 9.38\%)^{10}} + \frac{47,000*(1 - (1 + 9.38\%)^{-14})}{9.38\%(1 + 9.38\%)^{13}} = 357,904.32 {/eq}