Question on TVM problem

Good day, colleagues!

I learn on Kaplan Shweser 2011 CFA Level 1. Now I`m on TVM concept.

Please explain me test question on page 132 #13:

An investor will receive an annuity of $4,000 a year for ten years. The first payment is to be received five years from today. At a 9% discount rate, this annuity`s worth today is closest to: A. $16,684 B. $18,186 C. $25,671.


Two steps: (1) Find the PV of the 10-year annuity: N=10, I/Y=9, PMT=-4000, FV=0,CPT-PV=25,670.63. This is the present value as of the end of Year 4, (2) Discount PV of the annuity back FOUR years: N=4, PMT=0, FV=-25,670.63, I/Y=9, CPT-PV=18,185.72.

I can not understand, why we need to discount PV of the annuity back FOUR years???

(1) if you discout the 10 years annuity using ordinary annuity, you would have to deposit the money in the beginning of year 4 to get the money in year 5 (End of year payment). In other words, to recieve it in 5 years, you need to deposit the money in year 4. Year 4 is thus the year you need (x) amount.

(2) Another way to do it, is that since the first payment comes in year 5… you can find PV using annuity due or beginning payment for 10 years. This gives your a PV of 27980.99. Changing back to End payment, then discounting back 5 years gives you the 18,186.

TVM is simple when you think in terms of timing and when you get the money. If you master timing, you master TVM!

Hope that helps

Which is simply discounting back four years. This is confusing and probably shouldn’t be done.

When you’re first learning TVM it is almost always helpful to draw a timeline. It’s generally easier to stick with only ordinary annuities, so the first payment occurs at t=1 and t=0 is right now.

In this case, the first payment occurs at t=5. Since you want the first payment to be at t=1, you should discount 4 years.

Always remember what time your present value refers to. You want the present value now, and the present value you calculate is for t=4 since no payment is payment initially.

I agree. Better not to change the calculator settings and try doing it with the help of timeline.

Let’s see it in another way.

I have 18,185.72 at time 0

I invest it for four years and at the end of year 4, I have 25670.63 at 9%

I invest that money again for 10 years at end of t4 of beggining of t5 and at the end of year 5 I get the first PMT of 4000. I keep on getting that amount for 10 years at 9%.

This is what I wanted! The question simply asks for a reversal of it. That is why you discount the amount to 4 years to get the PV at time 0.

For any such problems, drawing a timeline would be of great help to simplify the understanding of cash flow…

Here, total time horizon is 14 years…i.e. first 4 years from now, when we won’t have any cash flow and 10 years thereafter when we will get $4000 each year.

First of all draw the timeline…

t = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Payment = 0 0 0 0 0 ($4000 each year till year 14…)

Now, calculate the PV of cash inflow till year ending ‘4’ by using PVIFA formula or calculator…

ie. ((4000*(1+9%)^(14-4))-1)/(9%*(1+9%)^(14-4)) = $ 25670.63

PV of cash flow at year 0 shall be= 25670.63/(1+9%)^4= $ 18185.72


Kailas Kale