Rebeccah is 65 and planing to retire next month. She can select a pension of $1745 monthly guaranteed for the rest of her life, but not indexed for inflation, or take a lump sum of $312 000. Assume she can invest the lump sum at five percent annually and draw the same income as the pension. How long does she need to live in order for the monthly pension to be the better choice?
A)89
B)90
C)92
D)93

Question

Quizplus · Accepted Answer

To determine which option is better, we need to calculate the present value of the pension and compare it to the lump sum.

First, we calculate the present value of the pension using the formula for a level payment annuity:

PV = PMT * (1 - (1 + r)^-n) / r

Where PV is the present value, PMT is the monthly payment, r is the monthly interest rate, and n is the number of payments.

PV = 1745 * (1 - (1 + 0.05/12)^-(12*20)) / (0.05/12) = $235,650.47

Therefore, the lump sum option is better if its present value is greater than $235,650.47.

PV of lump sum = $312,000

Since the lump sum option has a greater present value, it is the better choice unless Rebeccah expects to live longer than the break-even point. To calculate the break-even point, we need to find the number of months it takes for the present value of the pension to exceed $312,000.

PV of pension after k months = 1745 * (1 - (1 + 0.05/12)^-(12*k)) / (0.05/12)

Solving for k:

PV of pension after k months = 312,000

1745 * (1 - (1 + 0.05/12)^-(12*k)) / (0.05/12) = 312,000

1 - (1 + 0.05/12)^-(12*k) = 178.680017

(1 + 0.05/12)^-(12*k) = -177.680017

12*k = log(-177.680017) / log(1 + 0.05/12) = 155.338797

k = 12.9449

Therefore, the break-even point is 12.9449 months or approximately 92 months (since Rebeccah cannot live a fraction of a month). If Rebeccah expects to live longer than 92 months, then the pension option is better. Otherwise, the lump sum option is better.

So, the best choice for Rebeccah depends on her life expectancy and personal financial goals.

Rebeccah Is 65 and Planing to Retire Next Month