Compounding Periods Confusion

CFA Level I Quant
#1

Hey guys, solving a retirement problem and I do not agree with the “N” they are using:

“Billy will save $1,000 per year for the next 15 years (t = 1 to t = 15)”

So I just want to see the FV of this annuity, but the formula they use they make N = 15. However drawing this on a timeline shows you that the first $1,000 is compounded 14 times to reach t = 15 and not 15 times…

However, when I manually solve it (I compound the first cash flow 14 times etc up until the last cash flow) and I got the same answer they did, I never compounded 15 times. So on the exam, how do I know what to do?

P.S: Formula they used is: FVannuity = 1000*((1+8%)15-1)/0.08))

#2

An immediate annuity has payments of 1 payable at times 1, 2, 3, … n. You can accumulate each payment with interest or you can use the formula. The FV formula takes advantage of the fact that the payments are part of a geometric series. Since your series is 1+1.08+1.082+… + 1.0814, then the sum of the series is simply (1- 1.0815)/(1-1.08) = (1.0815 -1)/0.08.

#3

Thank you for the explanation!