Estimation, Evaluation, and Selection of Actuarial Models
Estimation, Evaluation, and Selection of Actuarial Models
Estimation, Evaluation, and Selection of Actuarial Models
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99<br />
Exercise 6 Using the raw data, the results are:<br />
value(x) r s S KM (x) H NA (x) S NA (x)<br />
27 20 1 0.95 0.0500 0.9512<br />
82 19 1 0.90 0.1026 0.9025<br />
115 18 1 0.85 0.1582 0.8537<br />
126 17 1 0.80 0.2170 0.8049<br />
155 16 1 0.75 0.2795 0.7562<br />
161 15 1 0.70 0.3462 0.7074<br />
243 14 1 0.65 0.4176 0.6586<br />
294 13 1 0.60 0.4945 0.6099<br />
340 12 1 0.55 0.5779 0.5611<br />
384 11 1 0.50 0.6688 0.5123<br />
457 10 1 0.45 0.7688 0.4636<br />
680 9 1 0.40 0.8799 0.4148<br />
855 8 1 0.35 1.0049 0.3661<br />
877 7 1 0.30 1.1477 0.3174<br />
974 6 1 0.25 1.3144 0.2686<br />
1,193 5 1 0.20 1.5144 0.2199<br />
1,340 4 1 0.15 1.7644 0.1713<br />
1,884 3 1 0.10 2.0977 0.1227<br />
2,558 2 1 0.05 2.5977 0.0744<br />
15,743 1 1 0.00 3.5977 0.0274<br />
When the deductible <strong>and</strong> limit are imposed, the results are as follows:<br />
value(x) r s S KM (x) H NA (x) S NA (x)<br />
115 18 1 0.9444 0.0556 0.9459<br />
126 17 1 0.8889 0.1144 0.8919<br />
155 16 1 0.8333 0.1769 0.8379<br />
161 15 1 0.7778 0.2435 0.7839<br />
243 14 1 0.7222 0.3150 0.7298<br />
294 13 1 0.6667 0.3919 0.6758<br />
340 12 1 0.6111 0.4752 0.6218<br />
384 11 1 0.5556 0.5661 0.5677<br />
457 10 1 0.5000 0.6661 0.5137<br />
680 9 1 0.4444 0.7773 0.4596<br />
855 8 1 0.3889 0.9023 0.4056<br />
877 7 1 0.3333 1.0451 0.3517<br />
974 6 1 0.2778 1.2118 0.2977<br />
Because 1000 is a censoring point <strong>and</strong> not an observed loss value, there is no change in the<br />
survival function at 1000.<br />
Exercise 7 Suppose the lapse was at time 1. Then the estimate <strong>of</strong> S(4) is (3/4)(2/3)(1/2) =<br />
0.25. If it is at time 2, the estimate is (4/5)(2/3)(1/2)=0.27. If it is at time 3, the estimate is<br />
(4/5)(3/4)(1/2) = 0.3. If it is at time 4, the estimate is (4/5)(3/4)(2/3) = 0.4. If it is at time 5,<br />
the estimate is (4/5)(3/4)(2/3)(1/2) = 0.20. Therefore, the answer is time 5.
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