Megs pension plan is an annuity with a guaranteed return of

Meg\'s pension plan is an annuity with a guaranteed return of 3% per year (compounded quarterly). She would like to retire with a pension of $50.000 per quarter for 15 years. If she works 35 years before retiring, how much money must she and her employer deposit each quarter? $

Solution

P * (1 + 0.03/4) + P * (1 + 0.03/4)^2 + ... + P * (1 + 0.03/4)^(35 * 4) = K

((K - 50000) * (1 + 0.03/4) - 50000) * (1 + 0.03/4) - 50000) .... = 0

P * 1.0075 * (1 + 1.0075 + 1.0075^2 + ... + 1.0075^139) = K
P * 1.0075 * (1.0125 + ... + 1.0125^140) = 1.0075 * K
K * 1.0075 - K = P * 1.0075 * (1.0125^140 - 1)
K * 0.0075 = P * 1.0075 * (1.0075^140 - 1)
K = P * (10075 / 75) * (1.0075^140 - 1)
K = P * 134.3 * (1.0075^140 - 1)

K = 50000 + 50000 * 1.0075 + 50000 * 1.0075^2 + ... + 50000 * 1.0075^(69)
K = 50000 * (1 + 1.0075 + 1.0075^2 + ... + 1.0075^69)
K * 1.0075 = 50000 * (1.0075 + 1.0075^2 + ... + 1.0075^70)
K * 1.0075 - K = 50000 * (1.0075^70 - 1)
K * 0.0075 = 50000 * (1.0075^70 - 1)
K = 50000 * 70 * (1.0075^70 - 1)
K = 3500000 * (1.0075^70 - 1)

P * 134.3 * (1.0075^140 - 1) = 5 * 7 * 10^5 * (1.0075^70 - 1)
P * 134.3 * (1.0075^140 - 1) = 35 * 10^5 * (1.0075^80 - 1)
P = 35 * 10^5 * (1.0075^70 - 1) / (134.3 * (1.0075^140 - 1)
P = 9698.39869

13341.399 cents per quarter