# T03F11 - Tutorial 3—ACTSC 232 Fall 2011 The ﬁrst two...

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Unformatted text preview: Tutorial 3—ACTSC 232, Fall, 2011 The ﬁrst two questions use this select table of mortality l[x] [ x] [40] 100000 [41] 99802 [42] 99597 [43] 99365 [44] 99120 l[x]+1 l[x]+2 l[x]+3 99899 99724 99520 99689 99502 99283 99471 99268 99030 99225 99007 98747 98964 98726 98429 lx+4 99288 99033 98752 98435 98067 x+4 44 45 46 47 48 1. (a) Calculate the probability that a life age 42 dies between ages 45 and 46 given that the life was selected at age 42. (b) Calculate the probability that a life age 42 dies between ages 45 and 46 given that the life was selected at age 41. (c) Calculate the probability that a life age 42 dies between ages 45 and 46 given that the life was selected at age 40. 2. The Index of selection is deﬁned as q[x]+k I (x, k ) = 1 − qx+k Calculate the index of selection at all durations for a life selected at age 44. (That is I (44, k ) for all k .) Comment brieﬂy on the result. ¯ 3. (a) Write A 1 x :n ¯ + n| Ax in single Actuarial notation. (b) Consistent with the actuarial notation seen in class, ¯ n| A 1 x:m would represent the expected present value of \$1 paid immediately on death of x over what interval? ¯ ¯ =c·A 1 and give the value of c. (c) Show that n| A 1 x :m x+n:m 4. Describe in words the insurances with the present values given (without consulting your notes!). You should specify how much beneﬁt is paid under every possible contingency. T is the future lifetime r.v. for a life age x. 0, T ≤5 T 20, 000 v , T ≤ 15, 5 < T ≤ 15 (a) Y1 = (b) Y2 = 10, 000 v T , 10, 000 v 15 , T > 15, 10, 000 v 15 , T > 15 5. For Y1 and Y2 above, express the expected present value of the beneﬁt (i.e. E[Y ]) using ‘A’ type actuarial functions. ¯ 6. You are given that, at an eﬀective rate of interest of 6% per year, Ax = 0.166117, ¯ ¯ Ax+5 = 0.20718, Ax+15 = 0.314208. You are also given that lx = 93132, lx+5 = 91641, lx+15 = 86409. Calculate the expected values of Y1 and Y2 . 1 ...
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