ACTSC232-A1 - ACTSC 232 Introduction to Actuarial...

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Unformatted text preview: ACTSC 232 Introduction to Actuarial Mathematics Assignment 1, Winter 2012 Assignment 1 consists of two parts. In Part I, the students are expected to construct one Excel spreadsheet to answer the given questions, and submit the file electronically to the Dropbox on course web site (learn.uwaterloo.ca). In Part II, the students are required to work out all the given questions and submit a hard copy of their solutions. Deadline: • Part I: by 11:59pm, Jan. 24th; submit ONE Excel file that contains all your answers in a single spreadsheet; the file should be submitted to the Dropbox on the course website. You are only allowed to submit your solutions for once. Therefore you’d better check your own solution carefully before you submit them. • Part II: in class, Jan. 19th (Thursday); submit your solutions to the instructor in class; assignment solutions submitted after the class are not acceptable. Note: Please place the present page on the top of your solutions to Part II as the cover and combine them together. Put your name and UW ID number very clearly in the corresponding blanks below. If you fail to do so, 5% of your grade will be taken off for Part II of this assignment. Last Name: First Name: UW ID No: Part I (10 points) Consider a generalized De Moivre’s survival model with a force of mortality function µx = α , 0<x<ω ω−x where α > 0, and ω > 0 is the limiting age. Assuming α = 1 , and ω = 120, construct the following 2 items in one Excel spreadsheet: (a) a column of x p0 for integer x from age 0 to age ω − 1; (b) a column of px for integer x from age 0 to age ω − 1; (c) a column of qx for integer x from age 0 to age ω − 1; (d) a column of ex for integer x from age 0 to age ω ; (Hint: you may use the recursive formula ex = px + px ex+1 ) (e) a column of the probability mass function Pr(K10 = x) for x = 0, 1, 2, . . . , ω − 10 − 1; (f) obtain the curtate expectation e10 and the mode of K10 from the p.m.f. you derived in (e). Part II (50 points) 1. Exercise 1.7 of the textbook, p.16 2. Exercise 2.2 of the textbook, p.36 3. Exercise 2.11 of the textbook, p.38 4. Show that d t px = t p x ( µ x − µ x+ t ) dx 5. For a population which contains equal numbers of males and females at birth: (i) For females, µf (t) = 0.08, x ≥ 0 x (ii) For males, µm (t) = 0.10, x ≥ 0 x For this population, calculate (a) q50 1 (b) ˚0 e (c) ˚20 and ˚20:10 e e 6. You are given that S0 (x) = 1 − (a) V ar(T30 ) ￿ ￿ x2 120 for 0 ≤ x ≤ 120, Calculate (b) ˚30:3 , the 3-year temporary complete life expectancy of a life (30) e (c) e30:3 , the 3-year temporary curtate life expectancy of a life (30) 7. Why do insurers generally require evidence of health from a person applying for life insurance but not for an annuity? Your answer should be roughly one paragraph in length. 2 ...
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