Question

Ron is 25 years old and is retiring at the age of 65. When he retires, he will need a monthly income of $4,123 for 20 years. If Ron contributes 10% of his monthly income to a 401(k) paying 5.5% compounded monthly, will he reach his goal for retirement given that his monthly income is 3,142.23? If he does not make his goal then state by what amount he will need to supplement his income. Round all answers to the nearest cent.
a.
Ron will meet his monthly goal of exactly $4,123 for retirement.
b.
Ron will meet his monthly goal of $4,123 for retirement with an excess of $125.34.
c.
Ron will not make his monthly goal of $4,123 and will need $359.74 to supplement his monthly income when he retires.
d.
Ron will not make his monthly goal of $4,123 and will need $450.61 to supplement his monthly income when he retires.

Answer 1

Compute Ron’s savings at retirement and the income it can support.

1) Monthly contribution = 10% of 3,142.23 = 314.223 (dollars).
Monthly interest rate r = 0.055/12 = 0.0045833333. Number of contributions n = 40·12 = 480.

Future value of contributions at age 65:
FV = 314.223 * [ (1 + r)^480 − 1 ] / r ≈ $547,039.38.

2) To fund $4,123 per month for 20 years (N = 240) at the same monthly rate, the required amount at retirement is the present value
PV_required = 4,123 * [ 1 − (1 + r)^−240 ] / r ≈ $599,385.46.

Since FV < PV_required, he will not meet the goal. The monthly payment his savings can actually support is
Payment = FV / [ (1 − (1 + r)^−240) / r ] ≈ $3,763.26 (approx).

Shortfall = 4,123 − 3,763.26 ≈ $359.74 per month.

Answer: c. Ron will not make his monthly goal of $4,123 and will need $359.74 to supplement his monthly income when he retires.