A personal account earmarked as a retirement supplement cont

A personal account earmarked as a retirement supplement contains $358,000. Suppose $300,000 is used to establish an annuity that earns 6%, compounded quarterly, and pays $6500 at the end of each quarter. How long will it be until the value of the annuity is $0? (Round your answer UP to the nearest quarter.) How many quarters?

Solution

The formula for periodic payments and annuity is A = PV [ r / { 1- ( 1 + r )^-n } ] where A is the periodic payment, PV is the present Value, r is the interest rate per period and n is the number of periods. Here, A = 6500, PV = 300000 and r = (6/4)* 1/100 = 0.015. Therefore, we have 6500 = 300000* 0.015 / [ 1 - ( 1 + 0.015)^-n ] or, 1 - ( 1 + 0.015)^-n = 300000* 0.015 / 6500 = 0.69231 . Then,  ( 1.015)^-n = 1 - 0.69231 = 0.30769. Therefore, -n log ( 1.015) = log 0.30769   or, n = (-) log 0.30769 / log ( 1.015) = (-) (- 0.511886618)/ 0.006466 = 79.16 = 79 on rounding off to the nearest quarter. The answer is 79 quarters.