Question

Nicki​ Johnson, a sophomore mechanical engineering​ student, receives a call from an insurance​ agent, who believes that Nicki is an older woman ready to retire from teaching. He talks to her about several annuities that she could buy that would guarantee her an annual fixed income. The annuities are as follows in the popup​ window: If Nicki could earn 9 percent on her money by placing it in a savings​ account, should she place it instead in any of the​ annuities? Which​ ones, if​ any? Why?

What rate of return could Nicki earn on her money if she place it in annuity A with $6,500 payment per year and 16 years duration?

ANNUITY	INITIAL PAYMENT INTO ANNUITY ​(AT t​ = 0)	AMOUNT OF MONEY RECEIVED PER YEAR	DURATION OF ANNUITY ​(YEARS)
A	​$50,000	​$6,500	16
B	​$70,000	​$7,500	22
C	​$70,000	​$8,000	20

Answer 1

In this problem, we need to find out our IRR Internal Rate of Return.

IRR is the rate which is earned on the investment during a period of time to make the PV of Inflows of the investment equal to present values of Outflows on the investment. In other words, it provides a rate of return which is earned on an investment given it's initial investment and annual cash inflows.

Formula to calculate IRR is

PV of Outflow = Inflow 1 /(1+IRR/100)¹ + ...+ Inflow N /(1+IRR/100)ⁿ

Where IRR stands for our rate and N stands for the No of Period for which cash-flow is received.

We are performing calculations in excel for simplicity. IRR formula is provided above.

For Annuity A

Outflow 50000, N (Term ) = 16 Years, Inflow 6500 Per year

PV of Outflow = Inflow 1 /(1+IRR/100)¹ + ...+ Inflow N /(1+IRR/100)¹⁶

50000 = 6500/(1+IRR/100)¹ + ...+ 6500 /(1+IRR/100)¹⁶

Solving for IRR we get,

IRR for Annuity A = 10.29%

For Annuity B

Outflow 70000, N (Term ) = 22 Years, Inflow 7500 Per year

PV of Outflow = Inflow 1 /(1+IRR/100)¹ + ...+ Inflow N /(1+IRR/100)²²

70000 = 7500/(1+IRR/100)¹ + ...+ 7500 /(1+IRR/100)²²

Solving for IRR we get,

For Annuity B, IRR is 9.15 %

For Annuity C

Outflow 70000, N (Term ) = 20 Years, Inflow 8000 Per year

PV of Outflow = Inflow 1 /(1+IRR/100)¹ + ...+ Inflow N /(1+IRR/100)²⁰

70000 = 8000/(1+IRR/100)¹ + ...+ 8000 /(1+IRR/100)²⁰

Solving for IRR, we get.

Now IRR for Annuity C is 9.60%

Answer

If Nicki want to purchase an annuity, she should purchase annuity A, as it is having the Most Highest IRR i.e. 10.29 %, it means that she will earn a rate of 10.29 % on her investments, while she is earning only 9% in her savings account.

Nicki could earn a rate of return of 10.29 %, if she goes with Annuity A.

P.S. Please provide your valuable feedback.