Errata for Actuarial Mathematics for Life Contingent Risks

Transcription

1 Errata for Actuarial Mathematics for Life Contingent Risks David C M Dickson, Mary R Hardy, Howard R Waters Note: These errata refer to the first printing of Actuarial Mathematics for Life Contingent Risks. Some corrections may have been incorporated in subsequent printings. Chapter 4 Page 84: Towards the bottom of the page the sentence that begins If we calculate q = should begin If we calculate 1 q Page 85: The second displayed equation should read ( ) v 1/ 1 q p 9 11 A () 130 Page 100: Equation (4.33) should read = j A 1 x:n i Page 101: The final sentence of Section 4.7 should read : Similarly, A [x]:n denotes the EPV of a benefit of 1 payable at the end of the year of death within n years, of a newly selected life age x, or at age x + n if (x) survives. Page 106: The answers to Exercise 4.1 are: (a) (b) (c)

2 Chapter 5 (d) Page 132: In equation (5.40) g (0) should be g (0). Page 133: The three references to second and higher order derivatives should be replaced with third and higher order derivatives. Chapter 6 Page 149: Solution 6.5. Remove the select brackets for age 65 in the calculation of ä [45]:20, which should read ä [45]:20 = ä [45] l 65 l [45] v 20 ä 65 = Page 175: The answer to Exercise 6.15(c) should be $ Chapter 7 Page 2: Solution 7. part (b), µ should be Page 227: In Exercise 7.14(c), replace the display equation with d t V dt = ( µ [60]+t + δ ) tv µ [60]+t for 0 < t < 20 Page 228: The solution to Exercise 7.2(e) should be $ Page 229: The solution to Exercise 7.7(g), t = 10, should read t = 10 : $ $ Page 229: The solution to Exercise 7.10(d) should be $263.37/9=$ Chapter 8 Page 254: All terms denoted V k on this page should be denoted k V. The corrected page follows. 2

3 We can achieve this by writing: and d dt t V (0) = ( t V (0) t h V (0) )/h + o(h)/h d dt t V (1) = ( t V (1) t h V (1) )/h + o(h)/h Putting these expressions into formulae (8.18) and (8.19), multiplying through by h, rearranging and ignoring terms which are o(h), gives the following two (approximate) equations: t hv (0) = t V (0) (1 δh) P h + hµ 01 x+t( t V (1) t V (0) ) + hµ 02 x+t(s t V (0) ) t hv (1) = t V (1) (1 δ h) + Bh + hµ 10 x+t( t V (0) t V (1) ) + hµ x+t(s t V (1) ). These equations, together with the starting values at time n the step size, h, and the premium rate, P, can be used to calculate successively: n hv (0), n hv (1), n 2hV (0), n 2hV (1),..., 10V (0), 10V (1),..., 0V (0) (1) For n = 20, h = 1/ and P = $5 500, we get 10V (0) = $18 084; 10V (1) = $ ; 0V (0) = $3 815 (2) For n = 20, h = 1/ and P = $6 000, we get: 10V (0) = $ V (1) = $ ; 0V (0) = $2 617 (ii) Let P be the premium calculated using the principle of equivalence. Then for this premium we have by definition 0 V (0) = 0. Using the results in part (i) and assuming 0V (0) is (approximately) a linear function of P, we have: P So that P $ Page 279: In Exercise 8.1 the labels on the transition intensities have been flipped. The intensities should read µ 01 x = 10 5 for all x and µ 02 x = A + Bc x 3

4 Page 280: In Exercise 8.2(b), add payable continuously at the end of the third bullet, describing the disability income benefit. Page 281: Exercise 8.4(a) should read Write down the Kolmogorov forward differential equation for t p 00 xy in the joint life model illustrated in Figure Page 285: In Exercise 8.17(c) the first equation should be t 1 p xy t p xy = 1 m m m (1 p xy) m 2t + 1 q x q y Page 286: In Exercise 8.18 (a), the first line of the question should read Show that, for independent lives (x) and (y) Page 288: The solution to Exercise 8.2(b)(i) should be $ Page 289: The solution to Exercise 8.15 (b) should be $76 846, and to (c) should be $ Chapter 9 Page 322: In Exercise 9.8 replace the first paragraph with the following: m 2 In a pension plan, a member who retires before age 65 has their pension reduced by an actuarial reduction factor. The factor is expressed as a rate oer month, k, say, and is then applied to reduce the member s pension to ((1 t k)b where B is the accrued benefit, based on service and final average salary at the date of early retirement. Page 322: In Exercise 9.10, replace the sentence His salary in the year to valuation is $ with His salary in the year following valuation is projected to be $ Page 323: In Exercise 9.11 replace the bullet point starting She could retire at with the following: She could retire at 60.5, with an actuarial reduction applied to her pension of 0.5% per month up to age 62. That is, if her accrued benefit at retirement, based on her salary in the 2 years prior to retirement is B, her reduced pension would be ( )B. Page 325: The answer to Exercise 9.8(a) is 0.43% and to Exercise 9.8(b) is 0.53%. Page 325: The answers to Exercise 9.11 are (a) 41.3% 47.6% 52.1% 4

5 (b) $ $ $ Chapter 10 (c) $ Page 343: In Solution 10.5(a) the expected present value of the loss should be E 50 ā 65 + P (ĪĀ) 1 50:15 P ā 50:15 Page 349 : Exercise 10.4(a) should read: The coefficient of variation for a random variable X is defined as the ratio of the standard deviation of X to the mean of X. Let X denote the aggregate loss on a portfolio, so that X = N j=1 X j. Assume that, for each j, X j > 0 and X j has finite mean and variance. Show that, if the portfolio risk is diversifiable, then the limiting value of the coefficient of variation of aggregate loss X, as N 0, is zero. Page 352: In the solution to Exercise 10.2, the forward rates in the table should be , and (the numbers given in the text are 1+f(k, k + 1)). Chapter 11 Page 373: The answers to Exercise 11.3 are: (a) ( , 60.16, , , ) (b) ( , 60.16, , , ) (c) $ (d) 7.8% (e) 3 years (f) No (The IRR is 42%). (g) Yes Appendix B Page 478: In formula (B.1) f (a) and f (b) should be replaced with f (a) and f (b). References to second and higher order derivatives should be changed to third and higher order derivatives. Page 478: In three cases the term left-hand side should be replaced by the term right-hand side. 5