Question

Person A opens an IRA at age 25, contributes $2000/yr for 10 years, but makes no additional contributions thereafter. Person B waits until age 35 to open an IRA and contributes $2000/yr for 30 years. There is no initial investment in either case, and the contributions are made continuously. Determine the return rate r for which the two IRA's have equal value at age 65.

Answer 1

For Person 1: IRA = (P((1 + r)^n1 - 1)/r)(1 + r)^n2; where P = 2000, n1 = 10, n2 = 30
For Person 2: IRA = P((1 + r)^n - 1)/r; where P = 2000, n = 30

(2000((1 + r)^10 - 1)/r)(1 + r)^30 = 2000((1 + r)^30 - 1)/r
log(2000((1 + r)^10 - 1)/r)(1 + r)^30 = log(2000((1 + r)^30 - 1)/r)
log 2000 + log((1 + r)^10 - 1) - log r + log(1 + r)^30 = log 2000 + log((1 + r)^30 - 1) - log r
log((1 + r)^30 - 1) - log((1 + r)^10 - 1) = log(1 + r)^30
log(((1 + r)^30 - 1) / ((1 + r)^10 - 1)) = log(1 + r)^30
log((1 + r)^20 + (1 + r)^10 + 1) = log(1 + r)^30
(1 + r)^20 + (1 + r)^10 + 1) = (1 + r)^30
(1 + r)^30 - (1 + r)^20 - (1 + r)^10 - 1 = 0
let (1 + r)^10 be x, then
x^3 - x^2 - x - 1 = 0
x = 1.839286755
(1 + r)^10 = 1.839286755
1 + r = 10th root of 1.839286755 = 1.063
r = 1.063 - 1 = 0.063

Therefore, the required rate is 6.3%.