ASM Study Manual for SOA MLC

1.1 Functions and moments......Page 25
1.2 Probability distributions......Page 26
1.2.2 Uniform distribution......Page 27
1.3 Variance......Page 28
1.4 Normal approximation......Page 29
1.5 Conditional probability and expectation......Page 31
 Exercises......Page 33
 Solutions......Page 37
2.1 Probability notation......Page 43
2.2 Actuarial notation......Page 46
2.3 Life tables......Page 48
2.4 Mortality trends......Page 49
 Exercises......Page 51
 Solutions......Page 56
3 Survival Distributions: Force of Mortality......Page 61
 Exercises......Page 67
 Solutions......Page 75
4.1 Mortality laws that may be used for human mortality......Page 85
4.1.1 Gompertz's law......Page 88
4.1.2 Makeham's law......Page 89
4.2.1 Exponential distribution, or constant force of mortality......Page 90
4.2.2 Uniform distribution......Page 91
4.2.3 Beta distribution......Page 92
 Exercises......Page 94
 Solutions......Page 97
5.1.1 General......Page 103
5.1.2 Special mortality laws......Page 105
5.2 Curtate......Page 109
 Exercises......Page 114
 Solutions......Page 122
6.1 Percentiles......Page 135
6.2 Recursive formulas for life expectancy......Page 136
 Exercises......Page 138
 Solutions......Page 142
7.1 Uniform distribution of deaths......Page 151
7.2 Constant force of mortality......Page 156
 Exercises......Page 158
 Solutions......Page 164
8 Survival Distributions: Select Mortality......Page 175
 Exercises......Page 179
 Solutions......Page 188
9 Supplementary Questions: Survival Distributions......Page 201
 Solutions......Page 203
10.1 Review of Financial Mathematics......Page 209
10.2 Moments of annual insurances......Page 210
10.3 Standard insurances and notation......Page 211
10.4 Illustrative Life Table......Page 213
10.5 Constant force and uniform mortality......Page 215
10.6 Normal approximation......Page 217
 Exercises......Page 218
 Solutions......Page 233
11.1 Definitions and general formulas......Page 249
11.2 Constant force of mortality......Page 250
 Exercises......Page 257
 Solutions......Page 267
12.1 Uniform survival function......Page 277
12.2 Other mortality functions......Page 279
12.2.1 Integrating tn e-dt (Gamma Integrands)......Page 280
12.3 Variance of endowment insurance......Page 281
12.4 Normal approximation......Page 282
 Exercises......Page 283
 Solutions......Page 291
13.1 Introduction......Page 301
13.2 Probabilities for continuous insurance variables......Page 302
13.3 Distribution functions of insurance present values......Page 306
13.4 Probabilities for discrete variables......Page 307
13.5 Percentiles......Page 309
 Exercises......Page 311
 Solutions......Page 316
14.1 Recursive formulas......Page 327
14.2 Varying insurance......Page 329
 Exercises......Page 336
 Solutions......Page 345
15.1 Uniform distribution of deaths......Page 357
15.2 Claims acceleration approach......Page 359
 Exercises......Page 360
 Solutions......Page 363
16 Supplementary Questions: Insurances......Page 367
 Solutions......Page 368
17.1 Annuities-due......Page 371
17.2 Annuities-immediate......Page 376
17.3 1/mthly annuities......Page 379
17.4 Actuarial Accumulated Value......Page 380
 Exercises......Page 382
 Solutions......Page 394
18.1 Whole life annuity......Page 405
18.2 Temporary and deferred life annuities......Page 408
18.3 n-year certain-and-life annuity......Page 411
 Exercises......Page 412
 Solutions......Page 418
19.1 Whole Life and Temporary Life Annuities......Page 427
19.2 Other Annuities......Page 429
19.3 Typical Exam Questions......Page 430
19.4 Combinations of Annuities and Insurances with No Variance......Page 432
 Exercises......Page 434
 Solutions......Page 444
20.1 Probabilities for continuous annuities......Page 459
20.2 Distribution functions of annuity present values......Page 462
20.3 Probabilities for discrete annuities......Page 463
20.4 Percentiles......Page 464
 Exercises......Page 467
 Solutions......Page 471
21.1.2 Arithmetically increasing annuities......Page 479
21.2 Recursive formulas......Page 481
 Exercises......Page 483
 Solutions......Page 488
22.1 Uniform distribution of deaths assumption......Page 495
22.2 Woolhouse's formula......Page 496
 Exercises......Page 499
 Solutions......Page 504
23 Supplementary Questions: Annuities......Page 511
 Solutions......Page 513
24.1 Future loss......Page 519
24.2 Net premium......Page 520
 Exercises......Page 522
 Solutions......Page 532
25.1 Premium formulas......Page 541
25.2 Expected value of future loss......Page 543
25.3 International Actuarial Premium Notation......Page 544
 Exercises......Page 545
 Solutions......Page 553
26 Premiums: Net Premiums Paid on a 1/mthly Basis......Page 563
 Exercises......Page 564
 Solutions......Page 568
27 Premiums: Net Premiums for Fully Continuous Insurances......Page 573
 Exercises......Page 576
 Solutions......Page 582
28.1 Gross future loss......Page 589
28.2 Gross premium......Page 590
 Exercises......Page 592
 Solutions......Page 599
29.1.1 Variance of net future loss by formula......Page 605
29.1.2 Variance of net future loss from first principles......Page 607
29.2 Variance of gross future loss......Page 608
 Exercises......Page 610
 Solutions......Page 616
30.1 Variance of net future loss......Page 625
30.2 Variance of gross future loss......Page 627
 Exercises......Page 628
 Solutions......Page 634
31.1.1 Fully continuous insurances......Page 643
31.1.3 Annuities......Page 647
31.1.4 Gross future loss......Page 650
31.2 Percentiles......Page 651
 Exercises......Page 652
 Solutions......Page 655
32.1 The portfolio percentile premium principle......Page 665
 Exercises......Page 667
 Solutions......Page 669
33 Supplementary Questions: Premiums......Page 673
 Solutions......Page 676
34 Reserves: Prospective Net Premium Reserve......Page 685
 Exercises......Page 690
 Solutions......Page 697
35.1 Gross premium reserve......Page 707
35.2 Expense reserve......Page 709
 Exercises......Page 711
 Solutions......Page 714
36.1 Retrospective Reserve Formula......Page 719
36.2 Relationships between premiums......Page 721
36.3 Premium Difference and Paid Up Insurance Formulas......Page 723
 Exercises......Page 725
 Solutions......Page 731
37.1 Annuity-ratio formula......Page 739
37.2 Insurance-ratio formula......Page 740
37.3 Premium-ratio formula......Page 741
 Exercises......Page 742
 Solutions......Page 751
38 Reserves: Variance of Loss......Page 763
 Exercises......Page 765
 Solutions......Page 772
39.1 Net premium reserve......Page 779
39.2 Insurances and annuities with payment of reserve upon death......Page 782
39.3 Gross premium reserve......Page 786
 Exercises......Page 788
 Solutions......Page 808
40 Reserves: Modified Reserves......Page 827
 Exercises......Page 828
 Solutions......Page 832
41.1 Reserves on semicontinuous insurance......Page 839
41.2 Reserves between premium dates......Page 840
41.3 Thiele's differential equation......Page 842
41.4 Policy alterations......Page 845
 Exercises......Page 849
 Solutions......Page 860
42 Supplementary Questions: Reserves......Page 875
 Solutions......Page 877
43.1 Introduction......Page 883
43.2 Definition of Markov chains......Page 886
43.3 Discrete Markov chains......Page 888
 Exercises......Page 890
 Solutions......Page 894
44 Markov Chains: Continuous—Probabilities......Page 899
44.1 Probabilities—direct calculation......Page 900
44.2 Kolmogorov's forward equations......Page 903
 Exercises......Page 907
 Solutions......Page 914
45.1 Premiums......Page 923
45.2 Reserves......Page 926
 Exercises......Page 929
 Solutions......Page 940
46.1 Probabilities......Page 951
46.2 Life tables......Page 953
46.3 Examples of Multiple Decrement Probabilities......Page 955
46.4 Discrete Insurances......Page 956
 Exercises......Page 957
 Solutions......Page 970
47.1 (j)x......Page 979
47.2 Probability framework for multiple decrement models......Page 981
47.3 Fractional ages......Page 983
 Exercises......Page 984
 Solutions......Page 993
48 Multiple Decrement Models: Associated Single Decrement Tables......Page 1003
 Exercises......Page 1008
 Solutions......Page 1012
49.2 Uniform in the multiple-decrement tables......Page 1021
49.3 Uniform in the associated single-decrement tables......Page 1025
 Exercises......Page 1028
 Solutions......Page 1032
50 Multiple Decrement Models: Discrete Decrements......Page 1039
 Exercises......Page 1042
 Solutions......Page 1048
51 Multiple Decrement Models: Continuous Insurances......Page 1053
 Exercises......Page 1056
 Solutions......Page 1067
52 Supplementary Questions: Multiple Decrements......Page 1081
 Solutions......Page 1082
53.1 Markov chain model......Page 1085
53.2 Independent lives......Page 1087
53.3 Joint distribution function model......Page 1089
 Exercises......Page 1091
 Solutions......Page 1097
54 Multiple Lives: Last Survivor Probabilities......Page 1103
 Exercises......Page 1108
 Solutions......Page 1114
55 Multiple Lives: Moments......Page 1121
 Exercises......Page 1127
 Solutions......Page 1130
56 Multiple Lives: Contingent Probabilities......Page 1137
 Exercises......Page 1143
 Solutions......Page 1150
57 Multiple Lives: Common Shock......Page 1159
 Exercises......Page 1160
 Solutions......Page 1163
58.1 Joint and last survivor insurances......Page 1167
58.2 Contingent insurances......Page 1172
 Exercises......Page 1174
 Solutions......Page 1191
59.1 Introduction......Page 1207
59.2 Three techniques for handling annuities......Page 1208
 Exercises......Page 1211
 Solutions......Page 1221
60 Supplementary Questions: Multiple Lives......Page 1231
 Solutions......Page 1233
61.1 Calculating the contribution for a defined contribution plan......Page 1241
61.2 Service table......Page 1244
61.3 Valuing pension plan benefits......Page 1245
61.4 Funding the benefits......Page 1249
 Exercises......Page 1253
 Solutions......Page 1264
62 Interest Rate Risk: Replicating Cash Flows......Page 1273
 Exercises......Page 1276
 Solutions......Page 1281
63 Interest Rate Risk: Diversifiable and Non-Diversifiable Risk......Page 1287
 Exercises......Page 1290
 Solutions......Page 1292
64.1 Introduction......Page 1295
64.2 Asset Shares......Page 1296
 Exercises......Page 1301
 Solutions......Page 1308
65.1 Profits by policy year......Page 1315
65.2 Profit measures......Page 1317
65.3 Determining the reserve using a profit test......Page 1320
65.4 Handling multiple-state models......Page 1321
 Exercises......Page 1324
 Solutions......Page 1331
66 Profit Tests: Participating Insurance......Page 1339
 Exercises......Page 1344
 Solutions......Page 1347
67.1 How universal life works......Page 1349
67.2 Profit tests......Page 1356
67.4 Comments on reserves......Page 1362
 Exercises......Page 1363
 Solutions......Page 1373
68 Profit Tests: Gain by Source......Page 1383
 Exercises......Page 1386
 Solutions......Page 1392
69.1 Chapter 1......Page 1395
69.2 Chapter 3......Page 1396
69.4 Chapter 13......Page 1397
70 Supplementary Questions: Entire Course......Page 1399
 Solutions......Page 1421
Practice Exams......Page 1449
1 Practice Exam 1......Page 1451
2 Practice Exam 2......Page 1461
3 Practice Exam 3......Page 1471
4 Practice Exam 4......Page 1483
5 Practice Exam 5......Page 1493
6 Practice Exam 6......Page 1503
7 Practice Exam 7......Page 1515
8 Practice Exam 8......Page 1525
9 Practice Exam 9......Page 1537
10 Practice Exam 10......Page 1547
11 Practice Exam 11......Page 1557
12 Practice Exam 12......Page 1567
13 Practice Exam 13......Page 1579
Appendices......Page 1589
Solutions for Practice Exam 1......Page 1591
Solutions for Practice Exam 2......Page 1604
Solutions for Practice Exam 3......Page 1614
Solutions for Practice Exam 4......Page 1626
Solutions for Practice Exam 5......Page 1640
Solutions for Practice Exam 6......Page 1652
Solutions for Practice Exam 7......Page 1664
Solutions for Practice Exam 8......Page 1678
Solutions for Practice Exam 9......Page 1690
Solutions for Practice Exam 10......Page 1702
Solutions for Practice Exam 11......Page 1714
Solutions for Practice Exam 12......Page 1725
Solutions for Practice Exam 13......Page 1736
B.1 Solutions to CAS Exam 3, Spring 2005......Page 1749
B.2 Solutions to CAS Exam 3, Fall 2005......Page 1753
B.3 Solutions to CAS Exam 3, Spring 2006......Page 1756
B.4 Solutions to CAS Exam 3, Fall 2006......Page 1760
B.5 Solutions to CAS Exam 3, Spring 2007......Page 1763
B.6 Solutions to CAS Exam 3, Fall 2007......Page 1767
B.7 Solutions to CAS Exam 3L, Spring 2008......Page 1770
B.8 Solutions to CAS Exam 3L, Fall 2008......Page 1773
B.9 Solutions to CAS Exam 3L, Spring 2009......Page 1776
B.10 Solutions to CAS Exam 3L, Fall 2009......Page 1779
B.11 Solutions to CAS Exam 3L, Spring 2010......Page 1782
B.12 Solutions to CAS Exam 3L, Fall 2010......Page 1785
B.13 Solutions to CAS Exam 3L, Spring 2011......Page 1788
B.14 Solutions to CAS Exam 3L, Fall 2011......Page 1790
B.15 Solutions to CAS Exam 3L, Spring 2012......Page 1793
B.16 Solutions to SOA Exam MLC, Spring 2012......Page 1796
B.17 Solutions to CAS Exam 3L, Fall 2012......Page 1804
B.18 Solutions to SOA Exam MLC, Fall 2012......Page 1808
B.19 Solutions to CAS Exam 3L, Spring 2013......Page 1815
B.20 Solutions to SOA Exam MLC, Spring 2013......Page 1819
B.21 Solutions to CAS Exam 3L, Fall 2013......Page 1827
B.22 Solutions to SOA Exam MLC, Fall 2013......Page 1831
B.23 Solutions to CAS Exam LC, Spring 2014......Page 1839
B.24.1 Multiple choice section......Page 1844
B.24.2 Written answer section......Page 1848
B.25 Solutions to CAS Exam LC, Fall 2014......Page 1855
B.26.1 Multiple choice section......Page 1859
B.26.2 Written answer section......Page 1864
B.27 Solutions to CAS Exam LC, Spring 2015......Page 1870
B.28.1 Multiple choice section......Page 1874
B.28.2 Written answer section......Page 1879
B.29 Solutions to CAS Exam LC, Fall 2015......Page 1885
B.30.1 Multiple choice section......Page 1890
B.30.2 Written answer section......Page 1895
B.31 Solutions to CAS Exam LC, Spring 2016......Page 1902
B.32.1 Multiple choice section......Page 1905
B.32.2 Written answer section......Page 1910
C Exam Question Index......Page 1917