Actuarial Education Services

The Practical Part FM Seminar
Text Box: Annuities Payable More Frequently Than Interest Is Compounded These annuities are sometimes referred to as mth-ly annuities. In this tip we illustrate how to use the “upper m” actuarial notation associated with these annuities. Example 1: A 10-year annuity has monthly payments of 1. Which of the following does NOT represent the present value of this annuity one month before the first payment? In the answer choices, i is an annual effective interest rate (aeir) and j is the corresponding monthly effective interest rate (meir). (A)  (B)  (C)  (D)  (E)   Remarks: If a problem such as this is given on an actuarial exam, it will likely be the case that the aeir is given, so that we could get numeric results. Then the quickest way to an answer would be to first determine the equivalent meir and use the expression in answer choice (A) along with the TVM calculator buttons to compute the present value. Actuarial symbols such as those in answer choices (B) and (D) are used in practice, especially by retirement actuaries. Most normal forms of payment for defined benefit pension plans are life annuities payable monthly, and so you will see the “upper 12” notation if you work in the retirement practice of a consulting firm. We can also combine the “upper m” notation with arithmetically increasing annuities. Consider the symbol:  where i is a semiannual effective interest rate. First, let’s describe the payment stream that is being evaluated. Since i is a seir, the “angle 4” implies this annuity lasts for 4 semiannual periods. The coefficient of the actuarial symbol is understood to be 1, which means the total payment for the first semiannual period will equal 1, and since it is an increasing annuity, the total payment for the second semiannual period will equal 2, the total payment for the third semiannual period will equal 3, and the total payment for the fourth semiannual period will equal 4. Since this is an “upper 2” annuity, each total semiannual payment will be split evenly into 2 payments. Therefore, the payment stream consists of quarterly payments for two years in the amounts 0.5, 0.5, 1, 1, 1.5, 1.5, 2, and 2. The symbol itself, being an annuity immediate present value symbol, represents the value of this payment stream one (payment) period, quarter, before the first payment. Let’s use an exam type problem to further illustrate this concept. Example 2: A 10-year annuity has monthly payments for the first year equal to 1 each, monthly payments for the second year equal to 2 each, and so on, with monthly payments for the tenth year equal to 10 each. Symbolically express the present value of this annuity one month before the first payment of 1, using an aeir of i. Remarks: The expression on the right hand side of this last equation is the closed rule formula for the present value of this type of increasing annuity immediate. Notice that removing the “upper 12” on both sides of the equation gives the closed rule formula for the present value of the basic increasing annuity immediate in which compounding period and payment period are equal. Many people would solve this problem as follows: first compute the equivalent meir j. Then replace the first year’s monthly payments of 1 each with the single payment at time 0 of “a-angle 12 at rate j”. Likewise, replace the second year’s monthly payments of 2 each with the single payment at time 1 of “2 a-angle 12 at rate j”. Continue, finally replacing the tenth year’s monthly payments of 10 each with the single payment at time 9 of “10 a-angle 12 at rate j”. After all this the result is that we now have one of those basic increasing annuities in which compounding period and payment period are equal. We would then have  A good exercise is to pick an arbitrary aeir and show that this expression gives the same value as the expression in the solution of Example 2. It’s nice to see different approaches to the same problem, and that’s what we show in our seminar, so that you can choose the best one(s) for you.

Phone: 850-668-6686

Fax: 850-668-6676

E-mail: joeboor@embarqmail.com

Solution:  Clearly the first year’s payments total 12, the second year’s payments total 24, and so on, until the tenth year’s payments total 120.  Since i is an aeir and we have a 10 year annuity, we use “angle 10 at rate i”, and since the payments are monthly, we use “upper 12” notation.  The value of this annuity one month before the first payment is then