Provide answers to the questions below

(i) In the context of the graduation of mortality data discuss the concepts of "smoothness" and "fidelity to data" and their relationship. [6] (ii) Comment on the graduation in the following table, mentioning briefly the limitations of any tests you apply. Exposed Actual Graduated mortality Expected Age to risk deaths rate 4x deaths 65 60,000 1,370 0.0225 1,350 66 50,000 1,200 0.0250 1,250 67 43,000 1,200 0.0276 1, 187 68 37,000 1,090 0.0304 1,125 69 30,000 1,010 0.0334 1,002 70 26,000 950 0.0368 957 71 23,000 980 0.0406 934 72 21,000 950 0.0449 943 73 20,000 1,000 0.0497 994 74 18,000 960 0.0551 992 328,000 10,710 10,734 The chi square value of the graduation is 7.29. You may assume that there are 7 degrees of freedom. [20] [Total 26](i) As there is no corresponding mortality table for female lives, in order to calculate survival probabilities for female policyholders, an actuary decides to use the AM92 table, but with an 'age rating' of 4 years applied, ie a female aged x is considered to experience mortality equivalent to a male aged x-4. (a) Explain the rationale underlying this approach. (b) Calculate the probability that a female policyholder aged 62 survives for at least the next 10 years. (ii) A male policyholder aged 65 is known to be in poor health, and it has been determined that his mortality is 200% of AM92 Ultimate, ie he is subject to q, values equal to twice those of the AM92 Ultimate table. Calculate the probability that this policyholder will die before age 67. A select life table is to be constructed with a select period of two years added to the ELT15 (Males) table, which is to be treated as the ultimate table. Select rates are to be derived by applying the same ratios select : ultimate seen in the AM92 table, ie: 9[x] (x1 qy and q(x]+1 = =*+1 qx41 qx q x+1 where the dash notation q' refers to AM92 mortality. Calculate the value of /[60] . The table below is part of a mortality table used by a life insurance company to calculate probabilities for a special type of life insurance policy. X [x] (x]+1 1[x1+2 /[x]+3 /x+4 51 1,537 1,517 1,502 1,492 1,483 52 1,532 1,512 1,497 1,487 1,477 53 1,525 1,505 1,490 1,480 1,470 54 1,517 1,499 1,484 1,474 1,462 55 1,512 1,492 1,477 1,467 1,453 (i) Calculate the probability that a policyholder who was accepted for insurance exactly 2 years ago and is now aged exactly 55 will die between age 56 and age 57. (ii) Calculate the corresponding probability for an individual of the same age who has been a policyholder for many years