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Text Box: Valuing Sums – Part 1: Calculating Level Annuities -Even the Hard Ones The following 3 types of sums often appear on Exam FM/2. In the next two tips we will focus on valuing these sums. Geometric Sums – These will show up when valuing level annuities using compound interest. Arithmetic Sums – These will show up when valuing level annuities using simple interest. Georithmic Sums (‘OK, you got me. I made this word up.’-Steve) – These will show up when valuing, using compound interest, annuities in which the payments form an arithmetic progression. In this week’s tip, we focus on valuing the geometric sums involved in level annuities-the a’s and ä’s. A geometric sum is a sum in which there is a common ratio between consecutive terms. Said another way, the same amount is multiplied to each term to get the next term in the sum. A verbal formula for the value of a geometric sum is  Example 1: The sum  is geometric with common ratio, v. The first term is 1 and the first omitted term is . You should recognize this sum as the value, at the time of the first payment, of n periodic payments of 1 each. That is, it’s the present value of an annuity due. Using the above verbal formula for geometric sums, we have . Since , the periodic effective discount rate, we get the well-known formula . Example 2: Similar to example 1, the sum  is a geometric sum with common ratio, v. This time the first term is v and the first omitted term is . So . You should recognize this sum as the value, one period before the first payment, of n periodic payments of 1 each. That is, it’s the present value of an annuity immediate. Now multiply both numerator and denominator by . We get  (numerator)  (denominator) We then get the well-known formula  Example 3: Let’s determine an expression for the present value of a 40-year annuity whereby payments of 1 are made at the end of each 5 year period. A term-by-term present value of the payments is . This is a geometric sum with common ratio,  , first term, , and first omitted term, . Therefore, using the above verbal formula for geometric sums, . Similar to what we did in Example 2, we can multiply both numerator and denominator by  to get the following ratio of actuarial symbols: . Remark: If the sum is an infinite sum (a series), then there are no omitted terms and we would just use 0 for the first omitted term. For example, the expression  is the present value of a perpetuity due. It is a geometric sum with common ratio, v, first term, 1, and no omitted terms. Then our verbal formula gives the following well-known formula for the present value of a perpetuity due: .