i llustrative l ife t able : b asic f unctions a nd n et s ingle p remiums b ased o n t he f ifth p...
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ILLUSTRATIVE LIFE TABLE: BASIC FUNCTIONS AND NET SINGLE PREMIUMSBASED ON THE FIFTH PERCENTILESLi-Fei Huang
Department of Applied Statistics and Information Science
Ming Chuan University, Taiwan
OUTLINE
Introduction The fifth percentile of the number of
survivors The fifth percentile of the present-value
random variables The fifth percentile of the present-value for
more than 1 insured Conclusions References
INTRODUCTION-SYMBOLS FOR NUMBER OF SURVIVORS
newborns ℒ( ) is the cohort’s number of survivors to
age which follows a binomial distribution is the probability that a newborn can
survive to age If only extremely rare newborns survive to
age , the insurance companies have to pay more insurance earlier and lose lots of money.
The fifth percentile of the number of survivors is denoted by
0
x x
)(xs x
)(05.0 xL
x
INTRODUCTION-SYMBOLS FOR LIFE ANNUITY
is the expected present-value of a whole life annuity-due of 1 payable at the beginning of each year while survives.
Let All can be derived recursively by the
equation:
The single premium that the insurance companies should charge to prevent losing lots of money will be computed.
xa
)(x 1139 a
xa
)1(1111 1
1 1 0111
111
0
xxk k k
xkk
xxkk
xxkxk
kxk
kx avppvvppvvpppvpva
)(05.0 xa
INTRODUCTION-SYMBOLS FOR LIFE INSURANCE
is the expected present-value of a whole life insurance of 1 payable at the end of year of death issued to
Let All can be derived recursively by the
equation:
The single premium that the insurance companies should charge to prevent losing lots of money will be computed.
xA
)(x 1139 A
xA
)2(1 xxxx AvpvqA
)(05.0 xA
THE ILLUSTRATIVE LIFE TABLE
The illustrative life table in the appendix of the book “Actuarial Mathematics” was based on the Makeham law for ages 13-110, and the adjustment
The interest rate is 6%.
xx04.01005.07.01000
978155.00 x
THE EXACT FIFTH PERCENTILE OF THE NUMBER OF SURVIVORS
The exact fifth percentile of the number of survivors satisfies the following equation:
Each term of the equation is the product of some integers and some probabilities, and the product may become too large or too small to calculate if the multiplication is not in proper order.
To simplify the SAS program of finding the exact fifth percentile, the number of newborns is set to be 3,500 instead of 100,000.
05.0))(1()( 0
05.0 )(
0
0
xsxsxL
THE APPROXIMATED FIFTH PERCENTILE OF THE NUMBER OF SURVIVORS
The approximated fifth percentile of the number of survivors is calculated by
The approximated fifth percentiles are pretty close to the exact fifth percentiles in tables. For larger number of newborns, the approximated fifth percentile should also work well.
))(1)((645.15.0)( 0005.0 xsxsxL
THE FIFTH PERCENTILE OF NUMBER OF SURVIVORS AT AGE 0 TO AGE 10
Age exact Approx.
0 1 0 3500.000
N/A N/A
1 0.979578
0.020422
3428.524
3414 3414.259
2 0.978263
0.021737
3423.919
3409 3409.228
3 0.977066
0.022934
3419.729
3405 3404.661
4 0.975967
0.024033
3415.886
3401 3400.481
5 0.974950
0.025050
3412.326
3397 3396.617
6 0.973998
0.026002
3408.992
3393 3393.005
7 0.973095
0.026905
3405.833
3390 3389.586
8 0.972229
0.027771
3402.800
3387 3386.309
9 0.971387
0.028613
3399.853
3383 3383.128
10 0.970559
0.029441
3396.956
3380 3380.005
x xp xq x
THE FIFTH PERCENTILE OF NUMBER OF SURVIVORS AT AGE 76 TO AGE 85
Age exact Approx.
76 0.511715
0.488285
1791.003
1742 1741.856
77 0.482814
0.517182
1689.863
1641 1640.732
78 0.453036
0.546964
1585.626
1537 1536.681
79 0.422516
0.577484
1478.807
1431 1430.235
80 0.391436
0.608564
1370.027
1326 1322.029
81 0.360004
0.639996
1260.013
1216 1212.800
82 0.328454
0.671546
1149.589
1104 1103.383
83 0.297049
0.702951
1039.673
995 994.702
84 0.266073
0.733927
931.257
888 887.751
85 0.235825
0.764175
825.386
784 783.572
x xp xq x
THE FIFTH PERCENTILE OF NUMBER OF SURVIVORS AT AGE 101 TO AGE 110
Age exact Approx.
101 0.002370
0.997630
8.297 4 3.0640
102 0.001334
0.998666
4.669 1 0.6166
103 0.000710
0.999290
2.486 0 -0.6070
104 0.000356
0.999644
1.245 0 -1.0901
105 0.000167
0.999833
0.584 0 -1.1730
106 0.000073
0.999927
0.254 0 -1.0752
107 0.000029
0.999971
0.102 0 -0.9238
108 0.000011
0.999989
0.038 0 -0.7816
109 0.000004
0.999996
0.013 0 -0.6721
110 0.000001
0.999999
0.004 0 -0.5975
x xp xq x
LIFE ANNUITY: THE FIFTH PERCENTILE
Those approximated in tables provide the new survival function.
Let , then all can be found recursively by Eq. (1) using the new survival function.
)(05.0 xL
1)103(05.0 a )(05.0 xa
LIFE INSURANCE: THE FIFTH PERCENTILE
Those approximated in tables provide the new survival function.
Let , then all can be found recursively by Eq. (2) using the new survival function.
)(05.0 xL
1)103(05.0 A )(05.0 xA
NOTICE
because the insurance companies have to pay more insurance if many insured don’t survive.
because the insurance companies can pay fewer annuities if many insured don’t survive.
xAxA )(05.0
xaxa )(05.0
THE FIFTH PERCENTILE OF THE PRESENT-VALUE RANDOM VARIABLES AT AGE 0 TO AGE 10
Age New
0 1 16.71008
16.80095
0.054147
0.049003
1 0.975503
17.07087
17.09819
0.033724
0.032178
2 0.974065
17.06027
17.08703
0.034324
0.032810
3 0.972760
17.04672
17.07314
0.035091
0.033596
4 0.971566
17.03043
17.05670
0.036014
0.034526
5 0.970462
17.01158
17.03786
0.037080
0.035593
6 0.969430
16.99035
17.01675
0.038282
0.036788
7 0.968453
16.96687
16.99351
0.039611
0.038103
8 0.967517
16.94126
16.96823
0.041061
0.039534
9 0.966608
16.91362
16.94099
0.042625
0.041076
10 0.965716
16.88402
16.91186
0.044301
0.042725
x )(xs )(05.0 xa xa )(05.0 xA xA
THE FIFTH PERCENTILE OF THE PRESENT-VALUE RANDOM VARIABLES AT AGE 46 TO AGE 55
Age New
46 0.904753
13.88651
13.95459
0.213971
0.210118
47 0.900657
13.72181
13.79136
0.223294
0.219357
48 0.896255
13.55135
13.62235
0.232943
0.228923
49 0.891521
13.37508
13.44752
0.242920
0.238820
50 0.886426
13.19298
13.26683
0.253228
0.249047
51 0.880941
13.00535
13.08027
0.263866
0.259607
52 0.875032
12.81126
12.88758
0.274834
0.270499
53 0.868667
12.61169
12.68960
0.286131
0.281721
54 0.861807
12.40636
12.48556
0.297753
0.293270
55 0.854414
12.19535
12.27581
0.309697
0.305143
x )(xs )(05.0 xa xa )(05.0 xA xA
THE FIFTH PERCENTILE OF THE PRESENT-VALUE RANDOM VARIABLES AT AGE 94 TO AGE 103
Age New
94 0.034696
2.70771 2.94502 0.846734
0.833301
95 0.024928
2.51950 2.78885 0.857387
0.842141
96 0.017231
2.33008 2.64059 0.868109
0.850533
97 0.011374
2.13601 2.50020 0.879094
0.858479
98 0.007088
1.93239 2.36759 0.890620
0.865985
99 0.004091
1.71225 2.24265 0.903080
0.873058
100 0.002106
1.46662 2.12523 0.916984
0.879704
101 0.000875
1.18986 2.01517 0.932649
0.885934
102 0.000176
1 1.91229 0.943396
0.891757
103 0 1 1.81639 1 0.897185
x )(xs )(05.0 xa xa )(05.0 xA xA
THE FIFTH PERCENTILE OF THE PRESENT-VALUE FOR MORE THAN 1 INSURED
There are 100 . Each purchases a whole life insurance of 1 payable at the end of year of death. The interest rate is 6%.
Based on the usual normal approximation, the fifth percentile of the present-value is such that
)(x
)(05.0 xS
05.0)645.1(1
)(100
100)(
)(
)(22
05.0
xx
x
AA
AxS
SVar
SESP
ANOTHER CHOICE OF THE FIFTH PERCENTILE OF THE PRESENT-VALUE
Another choice of the fifth percentile of the present-value for more than 1 insured is suggested to be in this paper. )(100 05.0 xA
THE FIFTH PERCENTILE OF THE PRESENT-VALUE FOR 100 INSURED AT AGE 20 OR AGE 40
Age 2
20 6.5285 0.014303 8.1769 6.7253
40 16.1324 0.048633 18.6058 16.4673
x xA100 xA )(05.0 xS )(100 05.0 xA
CONCLUSION 1
T he insurance companies can preserve more money for - approximated insured who may not survive to prevent losing lots of money.
x )(05.0 xL
CONCLUSION 2
T he insurance companies can sell both insurances and annuities to balance the income and the payment.
CONCLUSION 3
T he insurance companies can charge for each insured of a large group of customers.
The new single premium is just a little bit higher than the actuarial present-value so it should be more acceptable than the usual normal approximated fifth percentile.
)(05.0 xA
)(05.0 xA
xA
REFERENCES 1
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986). Actuarial Mathematics. SOA.
Actuarial models of life insurance with stochastic interest rate. Wei, Xiang and Hu, Ping. Proceedings of SPIE - The International Society for Optical Engineering, v 7490, 2009, PIAGENG 2009 - Intelligent Information, Control, and Communication Technology for Agricultural Engineering
REFERENCES 2
Two approximations of the present value distribution of a disability annuity. Jaap Spreeuw. Journal of Computational and Applied Mathematics Volume 186, Issue 1, 1 February 2006, Pages 217-231
Modeling old-age mortality risk for the populations of Australia and New Zealand: An extreme value approach. Li, J.S.H. ,Ng, A.C.Y. and Chan, W.S. Mathematics and Computers in Simulation, v 81, n 7, p 1325-1333, March 2011
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