ILLUSTRATIVE LIFE TABLE: BASIC FUNCTIONS AND NET SINGLE PREMIUMSBASED ON THE FIFTH PERCENTILESLi-Fei Huang

lhuang@mail.mcu.edu.tw

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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