# A 13-year annuity pays $2,800 per month, and payments are made at the end of each month. The... ## Question: A 13-year annuity pays$2,800 per month, and payments are made at the end of each month. The interest rate is 12 percent compounded monthly for the first seven years, and 10 percent compounded monthly thereafter.

Required:

What is the present value of the annuity?

## Present value

Present value is a measure used in time value of money which shows the time preference of money which means that the money in future is less preferred over the money in present so future money should be compensated with some interest component.

The present value is to be calculated in to parts

First part ( from year 1 to 7 )i.e. 7 years

PV = {eq}\frac{E(1-(1+r)^-p)}{r} {/eq}

Where

E = monthly payment = $2800 r = monthly interest rate = 12%/12 = 1% p = months = 7 * 12 = 84 Therefore$ = {eq}\frac{2800(1-1.01^{-84})}{.01 } {/eq} = $158616 Second part ( from year7 to 13) i.e. for 6 years PV = {eq}\frac{E(1-(1+ R)^-p)}{R} * (1+r)^{-m} {/eq} Where E = monthly premium = 2800 R = monthly interest rate = 10%/12 = 0.833% p = months = 6 * 12 = 72 months r = monthly interest rate in lag period =1% m = lag period = 84 months Therefore PV = {eq}\frac{2800(1-1.00833^{-72})}{.00833} * 1.01^{-84} {/eq}=$65529

Now total PV of the annuity = $158616 +$65529 =\$224145