Question

Betty received 500,000 dollars from a life insurance policy to be distributed to her as an annuity in 10 annual level payments with the first payment made immediately.

Right after she receives her third payment, she is offered a perpetuity paying X dollars every month in lieu of the future annual payments. The first payment will be made in exactly one month.

Determine the value of X assuming an annual effective interest rate of 3%.

Answer 1

Since 1 payment of the annuity is to be made at present the remaining will be made over a period of 9 years. The equation would stand as follows

500000 = A + PVA_(0.03,9)

where PVA_(0.03,9) is the Present Value of annuity of 9 years @3% p.a.

500000 = A + A*[(1+r)ⁿ - 1] / [ r*(1+r)ⁿ]

500000 = A + A*[(1+0.03)⁹ - 1] / [ 0.03*(1+0.03)⁹]

500000 = A+ A*[1.3048-1] / 0.03914

500000 = A +A*0.3048/0.03914

500000 = A + A*7.7874

8.7874A = 500000

A = $ 56,899.65

The third payment is received at the end of year 2 whereas there are 7 payments of $ 56,899.65 each pending. The present value of these payments at the end of year 2 is calculated below:

PV = 56899.65 * [(1+0.03)⁷ - 1] / [0.03*(1+0.03)⁷]

= 56899.65 * [1.229874 - 1] / 0.036896

= 56899.65 * 0.229874 / 0.036896

= $354,503.20

Present value of perpetuity = Perpetuity amount / rate of interest

$354,503.20 = Perpetuity / 0.03

Perpetuity = $354,503.20 * 0.03

= $ 10,635.10

So the value of the perpetuity (X) to be paid each year is $ 10,635.10