Actuarial Education Services

The Practical Part FM Seminar
Text Box: A Shortcut For Valuing Annuities In Which Payments Increase or Decrease Exponentially A common problem on Exam FM/2 involves finding the present value of an annuity in which the payments form a geometric progression. By valuing each payment (VEP) it is easy to see that the present value of such an annuity can be given by a geometric sum. (We’ll do examples shortly.) We can then use the formula for valuing a geometric sum, given in a previous tip, to find a single term expression for the present value of the annuity. However, the next step would be to actually calculate the value of the resulting single term expression, and this step can be quite tedious. In this week’s tip we show how to calculate, using only basic actuarial symbols, the present value of annuities in which the payments form a geometric progression. We will be able to do so by tweaking the interest rate used for discounting the payments, and using the fact that an expression of the form 1 + X + X 2 + ... + X n -1 can be thought of as either the VEP expression for the present value of an annuity due, or the VEP expression for the accumulated value of an ordinary annuity. Let’s illustrate the tip with a couple of examples. Example 1: A 20-year annuity with annual payments has first payment equal to 100 and each subsequent payment is 5.06% more than its preceding payment. Find the present value of the annuity immediately before the first payment using an annual effective interest rate (aeir) of 3%.  (B)  (C)  (D)  (E)  Solution: Note that the first payment is 100, the second payment is 100(1.0506), and so forth, until the final payment is 100(1.0506)19. The VEP expression for the present value of this annuity, immediately before the first payment, is  Since the ratio of 1.0506 to 1.03 is greater than 1, we can think of this ratio as an accumulation factor (1 + i) for some “new” aeir i. Then we get  where  So, the correct answer is ‘E’. Example 2: A 20-year annuity with annual payments has first payment equal to 100 and each subsequent payment is 4% less than its preceding payment. Find the present value of the annuity one year before the first payment using an annual effective interest rate (aeir) of 8%. Solution: Note that the first payment is 100, the second payment is 100(0.96), and so forth, until the final payment is 100(0.96)19. The VEP expression for the present value of this annuity, one year before the first payment, is  Since the ratio of 0.96 to 1.08 is less than 1, we can think of this ratio as a discount factor v for some “new” aeir i. Then we get  where We get PV = 754.31. Remarks: 1. We could have solved Example 2 just as we did Example 1, thinking of the ratio of 0.96 to 1.08 as a new accumulation factor (1 + i). However, since the ratio of 0.96 to 1.08 is clearly less than 1, the resulting new aeir, i, would be negative. We invite you to check that using this negative i then  2. Likewise, we could have solved Example 1 just as we did Example 2, thinking of the ratio of 1.0506 to 1.03 as a new discount factor v. However, since the ratio of 1.0506 to 1.03 is clearly greater than 1, the resulting new aeir, i, would be negative. We invite you to check that using this negative i then  3. Some of you may know a general formula to value these types of annuities. We will cover the general formula in our seminar for those participants that prefer memorization. We know that people have different learning styles, and so in our seminar we present the material from several different a-angles. J A Note from Joe: The tip above presents two clear, concise, examples of the geometric-type annuities. These annuities are often best valued using a different interest rate (the ‘annual effective interest rate’) than the one that is initially specified in the problem. Another great example of this is the ‘present value of the dividends’ formula for pricing a stock, with the added (and often realistic) complication that the dividends increase over time. We of course cover that formula in our seminar as well. I just mention it because it is likely the most common and well-known practical use of this aspect of the study material. We hope this is helpful. If you have anything you’d like to add or clarify, please email us at joeboor@emarqmail.com.

Phone: 850-668-6686

Fax: 850-668-6676

E-mail: joeboor@embarqmail.com