 # Time Value of Money

This is a question from QBank: Optimal Insurance is offering a deferred annuity that promises to pay 10 percent per annum with equal annual payments beginning at the end of 10 years and continuing for a total of 10 annual payments. For an initial investment of \$100,000, what will be the amount of the annual payments? Anyone know why the payments are made at the beginning of Year 10? Question seems ambiguous.

0…1…2…3…4…~ … 9…10…11…12… ~…19 In year 0, you pay \$100k. You earn 10% every year starting Year 1 and onwards. However, you don’t start receiving payments until end of year 9 in the diagram above (or beginning of year 10), and you receive 10 payments, the last of which will be end of year 19. So you need to compute the future value of \$100k at end of year 9 (or beginning of year 10). From this FV, you calculate what the payment is for (PV=\$FV sum, N=10, Begin Mode, I=10%).

you can also calc like this: calculate PV of level payment of 1 (N=10, PMT=1, FV=0, i=10), which gives you PV = -6.144. This is the “forward” PV (future value) at the beginning of year 9. Now calculate the PV of this amount: -6.144/(1+10%)^9 = 2.60589. So if you invest 2.60589, you can have the annuity payment of 1. If you invest 100k, you can get 100,000/2.60589 = 38,374.

Can someone verify my solution? At end of year 10 FV is 259,374. At end of year 10/beg of year 11 we begin receiving payments. PV = 259,374, n = 10, i = 10, BEGIN mode I get PMT of 23,579.

Hmm, I’m not getting that. At beginning of year 10 (end of year 9) I get FV is 235,794.77 This is calculated by PV = -100,000 I/Y = 10%, P/Y = 1, N = 9 (because you give money NOW, not in one year, so it’s equivalent to cash flow at the end of 9 years or beginning of 10 years) Payment is made to you at BEGINNING of year 10, which is an annuity due Mode = BGN, FV = 0, PV = 235,794.77 CPT PMT = 34,885.92 per annum. Exactly as how Dreary described. Isura: I believe you are receiving payments at te beginning of year 10, not year 11