Question

Danny wants to accumulate a sum of money at age 65 so he can retire. In order to accomplish this goal, he can deposit 117 per month at the beginning of the month or 118 per month at the end of the month. Calculate the annual effective rate of interest earned by Danny.

Answer 1

PMT = Payment; N = Period in months ; R = Monthly rate

Future value of annuity due = Future value of Annuity

(PMT x ((1+R)^N-1)/R) x (1+R) = PMT x ((1+R)^N-1)/R

117 x ((1+R)^N-1)/R) x (1+R) = 118 x ((1+R)^N-1)/R

(1+R) = (118 x ((1+R)^N-1)/R) / (117 x ((1+R)^N-1)/R))

(1+R) = (118 x ((1+R)^N-1)/R) / (117 x ((1+R)^N-1)/R))

Above bold thing will get cancelled out.

1+R = 118/117

R = 118/117 -1

R = Monthly rate = 0.85470%;

Annual Nominal rate = 0.85470% x 12 = 10.256410%

Annual Effective Rate = (1+ 0.85470%)^12 -1 = 10.752542% = ~ 10.75%