SomeFormulas

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## This note was uploaded on 04/25/2011 for the course MA 170 taught by Professor Staff during the Spring '08 term at Purdue.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online