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Unformatted text preview: ACTSC 232  Spring 2010  Assignment 3 SOLUTIONS 1. Suppose that Makeham’s law applies with A = 0 . 0001, B = 0 . 00035 and c = 1 . 075. Assume also that the effective rate of interest is 6% per year, and ω = 120. (a) Use Excel (your spreadsheet from Assignment 2 can be a start) to find the exact values of ¨ a x , ¨ a (4) x , ¨ a (12) x , and ¨ a (52) x for x = 50 and 100. You do not have to submit your spreadsheet electronically. ¨ a 50 = 8.463819 ¨ a 100 = 1.154875 (b) Using the two approximations (UDD and simplified), estimate the values of the quarterly, monthly, and weekly annuities for x = 50 and 100. Annuity Exact UDD Simplified ¨ a (4) 50 8.081375 8.081826 8.088819 ¨ a (12) 50 7.997601 7.998078 8.005486 ¨ a (52) 50 7.965497 7.965977 7.973434 ¨ a (4) 100 0.638511 0.770943 0.779875 ¨ a (12) 100 0.546358 0.687080 0.696542 ¨ a (52) 100 0.513260 0.654966 0.664491 (c) Compare your estimated values to the actual ones. Comment on the differences. Within each starting age, we can see that the values decrease when the payment frequency increases. This makes sense because later payments have slightly lower PVs and also in the year of death, less money is received upfront when the payments are more spread out. Both approximations preserve these inequal ities....
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 Summer '08
 MATTHEWTILL
 Actuarial Science, Annuity, Actuarial present value, Mortality drag, px µx+t dt

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