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Text Box: Valuing Sums-Part 2: Calculating Increasing and Decreasing Annuities Using A Georithmic Sums Method Note: This tip uses a method similar to the one in the book, but we think presents it in a fresh way. The trick is of course to multiply the sum of the increasing/decreasing annuity terms by 1-v [or (1+i)-1 in accumulation function circumstances]. We’ve got the details below, plus a lead-in of pure arithmetic sums. In last week’s tip, we introduced geometric sums, arithmetic sums, and georithmic sums, and we gave a verbal formula for valuing geometric sums. In this week’s tip, we start with a verbal formula for valuing arithmetic sums. An arithmetic sum is a sum in which there is a common difference between consecutive terms. Said another way, the same amount is added to each term to get the next term in the sum. A verbal formula for the value of an arithmetic sum is ΣArithmetic= (Average of 1st and Last Term)×(Number of Terms) -Steve Example 1: Irene deposits 100 at the beginning of each year for 20 years into an account in which each deposit earns simple interest at a rate of 10% from the time of the deposit. Other than these deposits Irene makes no other deposits or withdrawals from the account until exactly 25 years after the first deposit was made, at which time she withdraws the full amount in the account. Determine the amount of Irene’s withdrawal. (A) 5,100 (B) 6,150 (C) 6,240 (D) 9,224 (E) 10,147 Solution: The amount of Irene’s withdrawal is the sum of the simple interest accumulated values of the payments using the focal point 25 years after the first deposit. The first deposit of 100 accumulates for 25 years and therefore has accumulated value equal to 100(1+0.1(25)), the second deposit of 100 accumulates for 24 years and therefore has accumulated value equal to 100(1+0.1(24)), and so on. Finally, the 20th and last deposit of 100 accumulates for 6 years and therefore has accumulated value equal to 100(1+0.1(6)). Therefore, the amount of Irene’s withdrawal is:  Using the verbal formula above, we have , and so the amount of Irene’s withdrawal is 2000 + 100(0.1)(310) = 5,100. Note from Steve: The process of valuing each payment at the focal point and then adding to get the total value of a stream of payments is what I call VEP. For example, the present value of an n-year annuity due with annual payments of 1 is , whereas the accumulated value of an n-year annuity immediate with annual payments of 1 is . I’ve seen this process called basic principles, or first principles, but I think VEP is more descriptive of what is actually done. Georithmic Sums: The next examples illustrate georithmic sums. Unfortunately, I don’t have a simple verbal formula for valuing georithmic sums. Instead, there is a simple process to follow to value such sums, as illustrated in the examples. Example 2: The sum  is what I call a georithmic sum, since ignoring the coefficients would result in a geometric sum (in this case with common ratio v), whereas considering only the coefficients produces an arithmetic sum. You should recognize this sum as the present value of a 5-year increasing annuity immediate. Letting  denote the value of the sum, we can get a formula for  as follows:  If we multiply both numerator and denominator by 1 + i, then we get the well-known formula for the present value of a 5-year increasing annuity immediate: . Let’s do another example to make the process clear. Example 3: Larry is scheduled to receive payments of 6, 5, 4, 3, 2, and 1 at times 0, 2, 4, 6, 8, and 10, respectively. Let  denote the value of these payments immediately prior to the first payment. Then using a periodic discount factor, v, we get . This is a georithmic sum with common ratio . Using the process above, we get  The last equality follows from last week’s tip; namely,  Using this process, you can quickly derive formulas for the annuity symbols , , , and , among others. Hopefully I’ve convinced you how useful it is to know how to value geometric, arithmetic, and georithmic sums, and hopefully after reading these tips, you can do so easily.