Question

Calculating the following:

1. Suppose a 60-year old person wants to purchase an annuity from an insurance company that pay $25,000 per year until the end of that person’s life. The insurance company expects this person’s life.  The insurance company expects this person to live for 20 more years and would be willing to pay 4.5 percent on the annuity. How much should the insurance company ask the person to pay for the annuity?
2. What if the person is expected to live for 30 more years?  If the same 4.5 percent interest rate applies, how much should this person be charged for the annuity?
3. In each case, what is the difference in the purchase price of the annuity if the distribution payments are made at the beginning of the year?

Answer 1

Formula for present value of annuity is PV of annuity = A[1-(1/(1+r)^n / r]
Where r = rate of interest
A= annuity
n= no. of years
a.) here A = 25000$ , n = 20 years , r = 4.5%
PV(annuity) = 25000[1-(1/(1+4.5%)^20 / 4.5%]
=25000[1- (1/(1+0.045)^20 / 0.045]
=25000[1-(1/1.045)^20 /0.045]
=25000[1-0.4146 / 0.045]
=25000[0.585357/0.045]
=25000(13.00794)
=3,25,198$
Insurance co. should ask person to pay $325198
b.) here A = 25000$ , n = 30 years , r = 4.5%
PV(annuity) = 25000[1-(1/(1+4.5%)^30 / 4.5%]
=25000[1- (1/(1+0.045)^30 / 0.045]
=25000[1-(1/1.045)^30 /0.045]
=25000[1-0.267 / 0.045]
=25000[0.733/0.045]
=25000(16.28889)
=4,07,222$
Insurance co. should ask person to pay $407222
c.) If payments are made in begining of the year than the fprmula will be PV of annuity = A[1-(1/(1+r)^n / r] (1+r)
thus amount to be ask if person is expected to live for 20 years = 325198 * (1.045)
=339831$
Difference = 339831-325198 = $14633
mount to be ask if person is expected to live for 30 years = 407222 * (1.045)
=425547$
Difference = 425547-407222 = $18325