Question

After winning some money at a casino, Tony is considering purchasing an annuity that promises to pay him $300 at the end of each month for 12 months, then $350 at the end of each month for 24 months, and then $375 at the end of each month for 36 months. If the first payment is due at the end of the first month and interest is 7.5% compounded annually over the life of the annuity, find Tony’s purchase price.

Answer 1

The timeline of cash flows is :

Months 1 to 12 - $300

Months 13 to 36 - $350

Months 37 to 72 - $375

The present value of the 1st series of cash flows is calculated using the PV function in Excel with these inputs :

rate = 7.5% / 12 (Converting annual rate into monthly rate)

nper = 12 (12 monthly payments)

pmt = -300 (monthly payment. this is entered as a negative figure because in an annuity, the initial amount is a cash outflow and the monthly payments are cash inflows)

PV is calculated to be $3,457.92

The value of the 2nd series of cash flows at the end of 12 months from now is calculated using the PV function in Excel with these inputs :

rate = 7.5% / 12 (Converting annual rate into monthly rate)

nper = 24 (24 monthly payments)

pmt = -350 (monthly payment. this is entered as a negative figure because in an annuity, the initial amount is a cash outflow and the monthly payments are cash inflows)

The value of the 2nd series of cash flows at the end of 12 months from now is $7,777.85

The present value of this needs to be calculated by discounting it back to the present at 7.5% rate for 1 year

present value = $7,777.85 / 1.075 = $7,235.21

The value of the 3rd series of cash flows at the end of 36 months from now is calculated using the PV function in Excel with these inputs :

rate = 7.5% / 12 (Converting annual rate into monthly rate)

nper = 36 (36 monthly payments)

pmt = -375 (monthly payment. this is entered as a negative figure because in an annuity, the initial amount is a cash outflow and the monthly payments are cash inflows)

The value of the 2nd series of cash flows at the end of 12 months from now is $12,055.47

The present value of this needs to be calculated by discounting it back to the present at 7.5% rate for 3 years

present value = $7,777.85 / 1.075³ = $9,704.18

Purchase price is the sum of the present values of the 3 series of cash flows

Purchase price = $3,457.92 + $7,235.21 + $9,704.18 = $20,397.31