Question

Selyn Cohen is 63 years old and recently retired. He wishes to provide retirement income for himself and is considering an annuity contract with the Philo Life Insurance Company. Such a contract pays him an equal-dollar amount each year that he lives. For this cash-flow stream, he must put up a specific amount of money at the beginning. According to actuary tables, his life expectancy is 15 years, and that is the duration on which the insurance company bases its calculations regardless of how long he actually lives. a. If Philo Life uses a compound annual interest rate of 5 percent in its calculations, what must Cohen pay at the outset for an annuity to provide him with $10,000 per year? (Assume that the expected annual payments are at the end of each of the 15 years.) b. What would be the purchase price if the compound annual interest rate is 10 percent? c. Cohen had $30,000 to put into an annuity. How much would he receive each year if the insurance company uses a 5 percent compound annual interest rate in its calculations? A 10 percent compound annual interest rate?

Answer 1

We need to find the PV of an ordinary Annuity for 15 years
that pays $10,000 each year end.
Formula for present value of an anuuity = PV= A [ {(1+k)n-1}/k(1+k)n]
PV = Present value of Annuity fund
A = Yearly installment =$10,000
k=interest rate=5% pa
n=periods=15 years
PV =10000*[1.05^15-1]/5%(1.05)^15=103,796.58
So Cohen needs to pay $103,796.58 at the outset for the
required annuity.

Ans b.
If the interest rate is 10% , we can find the PV of the
Annuity by the same formula.
Formula for present value of an anuuity = PV= A [ {(1+k)n-1}/k(1+k)n]
PV = Present value of Annuity fund
A = Yearly installment =$10,000
k=interest rate=10% pa
n=periods=15 years
PV =10,000*[1.10^15-1]/(10%*1.10^15)=$76,060.80
So Cohen needs to pay $76,060.80 at the outset for the
required annuity.

Ans c.
Given PV of Annuity =$30,000
Given interest rate =5% pa
Formula for present value of an anuuity = PV= A [ {(1+k)n-1}/k(1+k)n]
PV = Present value of Annuity fund=30,000
A = Yearly installment =??
k=interest rate=5% pa
n=periods=15 years
30,000=A*[1.05^15-1]/(5%*1.05^15)
A=2890.27
So Cohen will receive $2,890.57 per year for 15 years

Ans d.
Here interest rate is 10% pa
Formula for present value of an anuuity = PV= A [ {(1+k)n-1}/k(1+k)n]
PV = Present value of Annuity fund=30,000
A = Yearly installment =??
k=interest rate=10% pa
n=periods=15 years
30,000=A*[1.10^15-1]/(10%*1.10^15)
A=3,944.21
So Cohen will receive $3,944.21 per year for 15 years