You own an annuity due contract that will pay you $3,000 per year for 12 years.You need money to pay back a loan in 5 years,and you are afraid if you get the annuity payments annually you will spend the money and not be able to pay back your loan.You decide to sell your annuity for a lump sum of cash to be paid to you five years from today.If the interest rate is 8%,what is the equivalent value of your 12-year annuity if paid in one lump sum five years from today?

Question

Quizplus · Accepted Answer

To find the equivalent value of the annuity, we need to find the present value of the annuity as of 5 years from today. We can use the formula for the present value of an annuity due to find the present value of the annuity for the first 5 years (which is not affected by the interest rate).

PV of annuity due = Payment x ((1 - (1 + r)^-n) / r) x (1 + r)

PV of annuity for the first 5 years = $3,000 x ((1 - (1 + 0.08)^-5) / 0.08) x 1.08 = $12,394.58

To find the present value of the annuity after the first 5 years, we can use the formula for the present value of a lump sum:

PV of lump sum = FV / (1 + r)^n

FV of annuity payments for the remaining 7 years = $3,000 x ((1 + 0.08)^7 - 1) / 0.08 = $29,491.17

PV of annuity after 5 years = $29,491.17 / (1 + 0.08)^5 = $20,481.57

Total present value of the annuity = PV of annuity for the first 5 years + PV of annuity after 5 years = $12,394.58 + $20,481.57 = $32,876.15

Therefore, the equivalent value of the annuity is $32,876.15, which is closest to choice C ($35,876).

You Own an Annuity Due Contract That Will Pay You