### Basic Functions And Net Single Premiums Based On The Fifth

```ILLUSTRATIVE LIFE TABLE: BASIC
FUNCTIONS AND NET SINGLE
BASED ON THE FIFTH PERCENTILES
Li-Fei Huang
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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

 0 newborns
ℒ( x ) is the cohort’s number of survivors to age x
which follows a binomial distribution
 s(x) is the probability that a newborn can survive
to age x
 If only extremely rare newborns survive to age x ,
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 L0.05 ( x)

INTRODUCTION-SYMBOLS FOR LIFE
ANNUITY
x is the expected present-value of a whole life
a
annuity-due of 1 payable at the beginning of each
year while (x ) survives.
 Let a
139  1
x can be derived recursively by the equation:
 All a


ax   v
k 0


k

p x  1   v p x k 1 p x 1  1  p x v v
k
k
k 1
k 1
k 1

k 1
p x 1  1  p x v v k k p x 1  1  p x vax 1
k 0
The single premium a
0.05 ( x) that the insurance
companies should charge to prevent losing lots of
money will be computed.
(1)
INTRODUCTION-SYMBOLS FOR LIFE
INSURANCE

Ax is the expected present-value of a whole life
insurance of 1 payable at the end of year of death
issued to (x )
 Let A139  1

All Axcan be derived recursively by the equation:
Ax  vqx  vpx Ax1
(2)
 The single premium A
that the insurance
0.05 ( x)
companies should charge to prevent losing lots of
money will be computed.
THE ILLUSTRATIVE LIFE TABLE
The illustrative life table in the appendix of the
book “Actuarial Mathematics” was based on the
0.04 x
Makeham law 1000  x  0.7  0.0510  for ages
13-110, and the adjustment  x   0  0.978155
 The interest rate is 6%.

THE EXACT FIFTH PERCENTILE OF THE
NUMBER OF SURVIVORS
The exact fifth percentile of the number of
survivors satisfies the following equation:
L0.05 ( x )
 0 
 0 



s
(
x
)
(
1

s
(
x
))
 0.05


 0 

 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.

THE APPROXIMATED FIFTH PERCENTILE OF
THE NUMBER OF SURVIVORS

The approximated fifth percentile of the number
of survivors is calculated by
L0.05 ( x)   0  0.5  1.645  0 s( x)(1  s( x))

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.
THE FIFTH PERCENTILE OF NUMBER OF
SURVIVORS AT AGE 0 TO AGE 10
Age
x
px
qx
x
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
THE FIFTH PERCENTILE OF NUMBER OF
SURVIVORS AT AGE 76 TO AGE 85
px
qx
x
Age x
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
THE FIFTH PERCENTILE OF NUMBER OF
SURVIVORS AT AGE 101 TO AGE 110
Age
x
x
exact
Approx.
8.297
4
3.0640
0.998666
4.669
1
0.6166
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
px
qx
101
0.002370
0.997630
102
0.001334
103
LIFE ANNUITY: THE FIFTH PERCENTILE
Those approximated L0.05 ( x) in tables provide
the new survival function.
0.05 ( x) can be found
 Let a
0.05 (103)  1 , then all a
recursively by Eq. (1) using the new survival
function.

LIFE INSURANCE: THE FIFTH PERCENTILE
Those approximated L0.05 ( x) in tables
provide the new survival function.
 Let A
, then all A0.05 ( x) can be found
0.05 (103)  1
recursively by Eq. (2) using the new survival
function.

NOTICE
A0.05 ( x)  Ax
because the insurance
companies have to pay more insurance if many
insured don’t survive.
 a
0.05 ( x)  a
x because the insurance
companies can pay fewer annuities if many
insured don’t survive.

THE FIFTH PERCENTILE OF THE PRESENTVALUE RANDOM VARIABLES AT AGE 0 TO AGE
10
Age
x
New
s(x)
0.05 ( x)
a
x
a
A0.05 ( x)
Ax
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
THE FIFTH PERCENTILE OF THE PRESENTVALUE RANDOM VARIABLES AT AGE 46 TO AGE
55
Age
x
New
s(x) a0.05 ( x)
x
a
A0.05 ( x)
Ax
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
THE FIFTH PERCENTILE OF THE PRESENTVALUE RANDOM VARIABLES AT AGE 94 TO AGE
103
Age
x
New
s(x)
0.05 ( x)
a
x
a
A0.05 ( x)
Ax
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
THE FIFTH PERCENTILE OF THE
PRESENT-VALUE FOR MORE THAN 1
INSURED
There are 100 (x ) . 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 S0.05 ( x)
such that

 S  E (S ) S ( x)  100A 
x 
P
 0.05
 1  (1.645)  0.05
2
2 
 Var(S )
100
(
A

A
x
x) 

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 100A0.05 ( x) in this paper.
THE FIFTH PERCENTILE OF THE PRESENTVALUE FOR 100 INSURED AT AGE 20 OR AGE 40
Age
x
100Ax
2
Ax
S 0.05 ( x)
100A0.05 ( x)
20
6.5285
0.014303
8.1769
6.7253
40
16.1324
0.048633
18.6058
16.4673
CONCLUSION 1

T he insurance companies can preserve more
money for  x - approximated L0.05 ( x) insured
who may not survive to prevent losing lots of
money.
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 A0.05 ( x) for
each insured of a large group of customers.
 The new single premium A0.05 ( x) is just a little bit
higher than the actuarial present-value Ax so it
should be more acceptable than the usual normal
approximated fifth percentile.

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

THE END

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```