SomeFormulas

SomeFormulas - n-1 out of the sum instead of Pk n-1 ± Our...

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CHAPTER 1 Some Useful Formulas 1. Variable Payments Assume that we receive a series of payments at the end of the year where each payment increases (or decreases) by a factor of k each year. Thus if the first payment is P , then the subsequent payments would be kP,k 2 P,k 3 P,. .. . Theorem 1 . Under the payment scheme described above, if we deposit the payments into an account earning interest i , then the total accumulation after the n th payment is A = Pk n - 1 ( j k ) n - 1 j k - 1 = Pj n - 1 ± k j ² n - 1 k j - 1 = P j n - k n j - k where j = 1 + i Proof Our payments accumulate to A = Pj n - 1 + Pkj n - 2 + · · · + Pj 0 k n - 1 = Pk n - 1 ³ j k ´ n - 1 + ³ j k ´ n - 2 + · · · + 1 ! = Pk n - 1 ( j k ) n - 1 j k - 1 = Pk n - 1 k ± ( j k ) n - 1 ² j - k = P j n - k n j - k proving the first and third equality in the theorem. (We used formula (7) on page 8 of the notes with x = j k in step 3.) The second equality follows by factoring Pj
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Unformatted text preview: n-1 out of the sum instead of Pk n-1 . ± Our solution to Example 1 on the 2009 Final is a good example of the use of this formula. 1 2 1. SOME USEFUL FORMULAS 2. Disappearing Deductibles On p. 29 Brown gives an example of a disappearing deductible. He does not, however, give a general formula. Here it is: Assume that up to A dollars, we have a deductible of D o . Our deductible disappears linearly for claims between A and B , vanishing at B . Then for a claim of X between A and B the deductible is D = B-X B-A D o . You can check that this is correct by noting that at X = A this formula yields D o while at X = B , it yields 0....
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This note was uploaded on 04/25/2011 for the course MA 170 taught by Professor Staff during the Spring '08 term at Purdue.

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SomeFormulas - n-1 out of the sum instead of Pk n-1 ± Our...

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