Time Value of Money for Annuities - please help!

CFA Level I
#1

This topic is really bothering me, I can’t seem to find help anywhere. I can’t understand why annuities are discounted for one fewer period than I believe they should be.

Example: Philanthropist sets up scholarship fund, the first scholarship is paid out 4 years from today and will be awarded annually after that, calculate money that you need to put in today to ensure that scholarship can be paid every year in perpetuity following first payment 4 years from today.

Here is how I approach the problem. First, find the value of perpetuity in the future. Second, discount it back to the present. I thought you had to discount it for 4 years, but apparently you need to discount it for 3 years and I can’t understand why. To me, it seems that something paid out “four years from today” means that 4 entire periods go by and you should therefore compound for all 4 of those periods. Taking this example to an extreme, if the problem said that the scholarship is paid out “one year from today,” by the logic given in the solution you woulnd’t discount the value at all. This makes no sense because something paid out “one year from today” means that an entire year goes by, no? Maybe I am not understanding the vocabulary behind annuities, could that be the problem? When annuities are paid “a year from today,” does it mean that they are paid right away and no time goes by?

I’m not posting conrete numbers because that doesn’t matter to me. I just want to understand the logic behind why you would discount the value of the annuity for 3 periods instead of 4 periods. And please don’t tell me to put calculator in BGN mode, I want to understand how annuities work.

Thanks for the help, really appreciate it

#2

If you draw a time line it may help.

If you use the formula for an ordinary perpetuity (V0 = P/r), then the payment 4 years from today is at the end of year 4, and the formula gives you the value at the beginning of year 4; i.e., at the end of year 3. Thus, you discount it 3 years to get the value today.

If you use the formula for a perpetuity due (V0 = P(1 + r)/r), then the payment 4 years from today is (still) at the end of year 4 (beginning of year 5), and the formula gives you the value at the beginning of year 5; i.e., at the end of year 4. In this case, you discount it 4 years to get the value today. But because the value is greater than the value of the ordinary annuity (by a factor of (1 + r)), you’ll get the same result.

#3

Thank you so much - i get it now. I didn’t understand that the formula for an ordinary perpetuity gives you the value at the end of year 3, not year 4.

Cheers S2000

#4

Glad to be of some help.

Best of luck on the exam.