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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 ERCENTILES Li-Fei Huang lhuang@mail.mcu.edu.tw Department of Applied Statistics and Information Science Ming Chuan University, Taiwan

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O UTLINE 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

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I NTRODUCTION - SYMBOLS FOR NUMBER OF SURVIVORS newborns ( ) is the cohorts 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

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I NTRODUCTION - 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.

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I NTRODUCTION - 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.

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T HE 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%.

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T HE 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.

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T HE 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.

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T HE FIFTH PERCENTILE OF NUMBER OF SURVIVORS AT AGE 0 TO AGE 10 AgeexactApprox. 0103500.000N/A 10.9795780.0204223428.52434143414.259 20.9782630.0217373423.91934093409.228 30.9770660.0229343419.72934053404.661 40.9759670.0240333415.88634013400.481 50.9749500.0250503412.32633973396.617 60.9739980.0260023408.99233933393.005 70.9730950.0269053405.83333903389.586 80.9722290.0277713402.80033873386.309 90.9713870.0286133399.85333833383.128 100.9705590.0294413396.95633803380.005

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T HE FIFTH PERCENTILE OF NUMBER OF SURVIVORS AT AGE 76 TO AGE 85 AgeexactApprox. 760.5117150.4882851791.00317421741.856 770.4828140.5171821689.86316411640.732 780.4530360.5469641585.62615371536.681 790.4225160.5774841478.80714311430.235 800.3914360.6085641370.02713261322.029 810.3600040.6399961260.01312161212.800 820.3284540.6715461149.58911041103.383 830.2970490.7029511039.673 995 994.702 840.2660730.733927 931.257 888 887.751 850.2358250.764175 825.386 784 783.572

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T HE FIFTH PERCENTILE OF NUMBER OF SURVIVORS AT AGE 101 TO AGE 110 AgeexactApprox. 1010.0023700.9976308.2974 3.0640 1020.0013340.9986664.6691 0.6166 1030.0007100.9992902.4860-0.6070 1040.0003560.9996441.2450-1.0901 1050.0001670.9998330.5840-1.1730 1060.0000730.9999270.2540-1.0752 1070.0000290.9999710.1020-0.9238 1080.0000110.9999890.0380-0.7816 1090.0000040.9999960.0130-0.6721 1100.0000010.9999990.0040-0.5975

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L IFE 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.

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L IFE 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.

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NOTICE because the insurance companies have to pay more insurance if many insured dont survive. because the insurance companies can pay fewer annuities if many insured dont survive.

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T HE FIFTH PERCENTILE OF THE PRESENT - VALUE RANDOM VARIABLES AT AGE 0 TO AGE 10 AgeNew 0116.7100816.800950.0541470.049003 10.97550317.0708717.098190.0337240.032178 20.97406517.0602717.087030.0343240.032810 30.97276017.0467217.073140.0350910.033596 40.97156617.0304317.056700.0360140.034526 50.97046217.0115817.037860.0370800.035593 60.96943016.9903517.016750.0382820.036788 70.96845316.9668716.993510.0396110.038103 80.96751716.9412616.968230.0410610.039534 90.96660816.9136216.940990.0426250.041076 100.96571616.8840216.911860.0443010.042725

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T HE FIFTH PERCENTILE OF THE PRESENT - VALUE RANDOM VARIABLES AT AGE 46 TO AGE 55 AgeNew 460.90475313.8865113.954590.2139710.210118 470.90065713.7218113.791360.2232940.219357 480.89625513.5513513.622350.2329430.228923 490.89152113.3750813.447520.2429200.238820 500.88642613.1929813.266830.2532280.249047 510.88094113.0053513.080270.2638660.259607 520.87503212.8112612.887580.2748340.270499 530.86866712.6116912.689600.2861310.281721 540.86180712.4063612.485560.2977530.293270 550.85441412.1953512.275810.3096970.305143

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T HE FIFTH PERCENTILE OF THE PRESENT - VALUE RANDOM VARIABLES AT AGE 94 TO AGE 103 AgeNew 940.0346962.707712.945020.8467340.833301 950.0249282.519502.788850.8573870.842141 960.0172312.330082.640590.8681090.850533 970.0113742.136012.500200.8790940.858479 980.0070881.932392.367590.8906200.865985 990.0040911.712252.242650.9030800.873058 1000.0021061.466622.125230.9169840.879704 1010.0008751.189862.015170.9326490.885934 1020.00017611.912290.9433960.891757 103011.8163910.897185

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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

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A NOTHER 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.

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T HE FIFTH PERCENTILE OF THE PRESENT - VALUE FOR 100 INSURED AT AGE 20 OR AGE 40 Age 2 20 6.52850.014303 8.1769 6.7253 4016.13240.04863318.605816.4673

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CONCLUSION 1 T he insurance companies can preserve more money for - approximated insured who may not survive to prevent losing lots of money.

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CONCLUSION 2 T he insurance companies can sell both insurances and annuities to balance the income and the payment.

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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.

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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

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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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T HE END Thank you for your watching!

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