Profit Measures in Life Insurance
Shelly Matushevski
Honors Project Spring 2011
The University of Akron
2
Table of Contents
I. Introduction ......................................................................................................................................... 3
II. Loss Function ....................................................................................................................................... 5
III. Equivalence Principle .......................................................................................................................... 7
IV. Profit Measures .................................................................................................................................. 8
a) Profit Margin ............................................................................................................................... 8
b) Internal Rate of Return ................................................................................................................ 9
c) Modified Internal Rate of Return................................................................................................ 10
d) Return on Investment ................................................................................................................ 12
e) Summary of Whole Life Profit Measures .................................................................................... 12
V. Term Life Insurance ........................................................................................................................... 13
VI. Conclusion........................................................................................................................................ 14
APPENDIX .............................................................................................................................................. 15
Works Cited........................................................................................................................................... 22
3
I. Introduction
Driving a car, skydiving, cooking dinner, reading a book-- each of these events has a certain risk
associated with them. Because of this risk, insurance was created to help manage the effects of a loss.
Without insurance, risk would put large burdens on individuals. For example, individuals would have to
maintain large emergency funds, the risk of a lawsuit may discourage innovation, and could cause the
individual to have excessive worry and fear (Rejda, 2011). Although there are a few ways to handle risk,
one of the most common methods for the average person is to buy insurance.
Insurance can be defined as “the pooling of fortuitous losses by transfer of such risks to insurers,
who agree to indemnify insureds for such losses, to provide other pecuniary benefits on their
occurrence, or to render services connected with the risk” (Rejda, 2011). Pooling losses together help to
spread the risk over the entire group, and risk reduction results because of the large number of
individuals in the group. Statistical theory says: as the number of exposures gets larger, predictions will
become more accurate, there is less deviation between the actual losses and the expected losses, and
the credibility of the prediction increases.
The two separate types of insurance that most people will purchase at some point in their life
can be classified into two separate groups: property and casualty (such as auto and home insurance) and
life insurance. The major difference between these two groups is that with property and casualty
insurance, it is not known whether a loss will occur, as opposed to life insurance where it is not a matter
of if a person will die but when.
There have been many different types of life insurance dating as far back as the Roman Empire,
although there have only been companies selling policies since the 1800s (Ajmera, 2009). The main
purpose of the original life insurance in Rome was to cover burial expenses and to assist the living family
4
members of the deceased. The idea of life insurance as we now know it came from England in the 17 th
century. The first life insurance company founded in the United States was in South Carolina, and called
The Philadelphia Presbyterian Synod. It was for the benefit of the ministers that worked there (Ajmera,
2009).
Today, life insurance has evolved quite a bit since its origin. While insurance agents and policy
makers are important, some of the most important people behind the scenes are actuaries. Actuaries
help a life insurance company by developing “health and long-term-care insurance policies by predicting
the likelihood of occurrence of heart disease, diabetes, stroke, cancer, and other chronic ailments
among a particular group of people who have something in common, such as living in a certain area or
having a family history of illness” (Statistics, 2011). This is beneficial to the company as well as the
consumer because it helps to keep premiums more accurate and fair. One of the most important tools
to a life insurance actuary is a life table, or mortality table, which will be a majority of the discussion in
this paper and can be found in the appendix. It will be used to perform many different calculations
including profit margin, internal rate of return, modified internal rate of return, and return on
investment, using Microsoft Excel.
Calculations such as those that will be discussed in this paper are extremely important to the
insurance industry because they help the insurance company accurately and fairly price their policies.
Pricing is a large part of what actuaries help with in insurance because the company wants to find the
best balance between their profits and costs, while still being able to have low enough prices that
consumers will want to buy their product. Some of the calculations in this paper will help show the types
of things that are considered when pricing a policy.
5
II. Loss Function
For an individual, Table 1 shows an illustrative life table where the interest rate is .06.
TABLE 1
Illustrative Life Table: Basic Functions and Single Benefit Premiums at i=.06
Age
lx
0
5
10
15
10,000,000
9,749,503
9,705,588
9,663,731
dx
1000qx
250,497
43,915
41,857
45,929
äx
20.42
0.98
0.85
0.91
1000Ax
16.801
17.0379
16.9119
16.7384
qx
49
35.59
42.72
52.55
Ax
0.02042
0.00098
0.00085
0.00091
0.049
0.03559
0.04272
0.05255
This table is the same table that is used for the third actuarial exam, MLC -- life contingencies (SOA,
2008) and the full table can be found in the appendix. These numbers will be used in all calculations
throughout this paper. The table spans age zero to age one hundred and ten and the calculations will be
using a status age x = 22. The curtate future lifetime variable K is the number of whole years an
individual survives. The first calculation performed is the column P(K=k). The formula for this is
dx+k /lx
where dx+k is the number of decrements in a given year, and lx is the initial number in the group. The
next column is the insurance benefit, which will just be one to keep the calculations simple. All
calculations for whole life insurance can be found in Table 2, which can be seen below
TABLE 2
Discrete Whole Life
Age
P(K=k)
22
23
24
25
26
27
0.001097
0.001134
0.001174
0.001219
0.001268
0.001321
b
PVE
1
1
1
1
1
1
0.94340
0.89000
0.83962
0.79209
0.74726
0.70496
PVR
1.00000
1.94340
2.83339
3.67301
4.46511
5.21236
PVC
0.03500
0.03736
0.03958
0.04168
0.04366
0.04553
v^(k+1) P(K=k)*v^(k+1) (v^k)*(lk/l0)
0.94340
0.89000
0.83962
0.79209
0.74726
0.70496
0.00103
0.00101
0.00099
0.00097
0.00095
0.00093
1.00000
0.94236
0.88801
0.83676
0.78843
0.74286
LF(K)
0.96840
0.90792
0.85087
0.79705
0.74627
0.69837
6
and the full table can be found in the appendix.
For an individual age x, the curtate future lifetime random variable K defines the number of
whole years lived. The loss function as a function of K is defined as
where PVE(K) is the present value of expenditures, PVR(K) is the present value of revenues, and PVC(K)
is the present value of costs. The loss function shows either the profit or loss depending on a company’s
costs, expenditures, and revenues. Ideally, the loss function should be less than zero, meaning that the
revenue being brought in is greater than expenditures and costs. For interest rate i we define the
discount value v = (1+i)-1. The present value benefit b payments at future time K+1 is
For discrete whole life insurance if benefit b = 1, the expected payment value is
The insurance is funded by annuity payments at the start of each surviving year. The present value for
unit premiums is
For a discrete whole life annuity the expected payment value is
7
The costs are defined as fixed costs, proportion of benefits, and proportion of premiums. Here, b is the
unit benefit, fR is the fixed cost renewal, rB,R is the proportion of benefits renewal, rP,R is the proportion
of premiums renewal, fI is the fixed cost initial, rB,I is the proportion of benefits initial, rP,I is the
proportion of premiums initial, and G is the loaded premium. The present value of costs is
for K = 1, 2, …
III. Equivalence Principle
The equivalence principle requires that parameters in the model are defined so that the
expectation of the loss function should be equal to zero giving
The equivalence principle allows us to solve our present value of cost equation for the loaded premium
G. An insurance premium is composed of two parts: the pure premium and the loaded premium. The
pure premium is the actual amount of the discounted expected loss and the loading is the amount of the
insurer’s costs and profits (Seog, 2010). To solve for the loaded premium in our present value of cost
equation we must be given values for the rest of the variables. For this example, I have chosen values for
the benefit, fixed cost renewal, proportion of benefits renewal, proportion of premiums renewal, fixed
cost initial, proportion of benefits initial, and proportion of premiums renewal. These values can be
found in Table 2.
To find the amount of the premium without costs we find the unit benefit premium
Px = Ax / äx
8
which gives Px = .004349. Multiplying the benefit of b = $100,000 by Px gives a premium (without costs)
of π = $434.90. A loaded premium G is found by including the costs in the loss function. A very useful
add-in that Microsoft Excel offers is one called Solver. Solver will be used to solve for the loaded
premium G by setting the loss function equal to zero. We will set our target cell equal to zero, which will
be the E(LF(K)), by changing the cell containing G, subject to the constraint that the value of G must be
greater than or equal to zero. This methods gives G = .008732, and when multiplied by the benefit of
$100,000, a loaded premium equal to $873.18. To find the value of the loading, we will take our loaded
premium G and subtract out the premium with no costs. The loading is equal to $438.28, meaning that
this is the amount that is equal to the insurer’s costs and profits.
IV. Profit Measures
a) Profit Margin (PM)
We apply various profit measures utilized in finance to the situation of discrete whole
life insurance. First, the profit margin is defined as
which is the negative expected value of the loss function divided by the expected present value
of revenues. The equation for the expected value of the loss function is given above and the
expected present value of revenue is
9
Two different fixed values of a general loaded premium G will be used to calculate the expected
value of the loss function including costs. Changing the value of G to .01 now produces a value
of -.0192 for the expected value of the loss function. For comparison, G was also changed to .05,
which gives a value of -0.62463 for the expected value of the loss function. The negative values
for the expectation of the loss function show that at this value of G, the company is making a
profit. The value of G = .01 means that the loaded premium is $1000.00, and a value of G = .05 is
a loaded premium of $5000.00, meaning that company has lower costs then when G = .01.
Comparing the profit margin of the two values of G shows a much higher profit margin for G =
.05 with PM = 0.03801 which would be expected since the loaded premium was so much higher.
The PM for G = .01 is 0.00117.
b) Internal Rate of Return (IRR)
Another good indicator of profit is the internal rate of return. IRR is defined as the
interest rate that causes the present value of the loss function to be equal to zero. First, we will
define
The loss function including costs will be computed for each year of life with a loaded premium
G= .01, and then the positive and negative loss functions will be separated to compute
)
10
In the preceding equations, A will be represented by the values of K for which the loss function
is negative. The positive values of the loss function will be represented by AC, or the
complement of A. To find RA, the expectation that the loss function will be less than zero, we will
compute
for all values of K where the loss function is negative. The same process will be followed for RAC,
the complement of RA, but using the values of K that are positive. For this example, the IRR was
calculated to be about 4.64%.
Two big advantages of using IRR include it being easy to use and understand as well as
being closely related to the net present value, and often resulting in the same decision for
investments. While IRR is a good profit measure, it does have short comings. The IRR may result
in multiple answers and usually cannot deal with nonconventional cash flows. It may also lead to
incorrect decisions in comparisons of mutually exclusive investments (Ross, Westerfield, &
Jordan, 2007). IRR is unable to be used when cash flows switch from negative to positive or vice
versa. When this problem arises it is better, and more appropriate to use the MIRR, or modified
internal rate of return (IRR, 2008).
c) Modified Internal Rate of Return (MIRR)
Modified internal rate of return assumes that the positive cash flows from a project are
reinvested at the IRR. The MIRR assumes that the positive cash flows are reinvested at the firm’s
cost of capital. This helps the MIRR to more accurately reflect the cost and profitability of a
project (MIRR, 2009). Assuming that the cash flows are reinvested, to calculate the MIRR all cash
flows are compounded to the end of the policy’s life, and then calculate the IRR (Ross,
11
Westerfield, & Jordan, 2007). The profits (over A, as defined above from the IRR process) will be
reinvested at rate α for m years so
will be the future value of RA, which was previously defined as the expectation of the loss
function where it is less than zero. The MIRR is defined as the rate where
Then solving for MIRR gives
This gives
Or if we set eα = 1+j, where j is the reinvestment interest rate, then we get
12
For this example m is equal to 88 and the reinvestment interest rate will be 8% with the loaded
premium G = .01. We will want to choose a higher interest rate than six percent because
otherwise, we would not want to reinvest. Using these numbers we get an MIRR of about
8.644%.
d) Return on Investment (ROI)
Another good measure of performance is the return on investment. The ROI is used to
measure the efficiency of an investment. To calculate the ROI we take the benefit, or return, of
an investment and divide it by the cost of the investment. This is shown by
Once calculated, if the ROI is not positive, or there are other investments with higher ROIs, then
the investment should not be undertaken (ROI, 2009). Again using a loaded premium of G = .01
and the interest rate at 6%, the ROI is calculated to be 0.0011714.
e) Summary of Whole Life Profit Measures
For our example with loaded premium G = .01 we found
PM
IRR
MIRR
ROI
0.117%
4.64%
8.644%
.11714%
13
which shows that overall the company will be making a profit on this policy with a favorable
profit measure and positive ROI. The value of MIRR is about double that of IRR.
V. Term Life Insurance
The above calculations and discussions involved only whole life insurance where benefit b is
paid at the end of the year of death. Another popular type of life insurance is term life insurance. Term
life insurance provides coverage with a fixed rate of payments for a limited period of time. It is the
simplest and least expensive type of policy to buy (Types of Life Insurance Explained, 2007). For this
example, we will take a status age x=22 and have them purchase a discrete 30 year term life insurance
policy.
As above, the present value of expenditures, present value of revenues, and present value of
cost is calculated. The formula for the present value of cost will be kept the same and the values for
each of fixed cost renewal, proportion of benefits renewal, proportion of premiums renewal, fixed cost
initial, proportion of benefits initial, and proportion of premiums initial will be kept the same.
One of the major benefits as stated above of term life insurance is lower premiums. When we
solve for the premium without costs we get Px = 0.00195, which is about half of the amount of
premiums for whole life. Then using the equivalence principle with costs included to solve for G, the
loaded premium, we get 0.0390545. This gives the loaded premium equal to $3,905.58 and the loading
equal to $3,710.53.
Also as we did with the whole life insurance, we can calculate the same profit measures. With a
loaded premium of G = .05, the calculated IRR is 1.9934% which is less than the IRR of whole life, but this
14
was expected. The MIRR was found to be 6.6273%, again less than the value of the MIRR of whole life.
The profit margin PM is 0.009966 and the ROI is 0.010067.
The following table summarizes the profit measures for term life insurance.
PM
0.9966%
IRR
MIRR
ROI
1.99%
6.63%
1.0067%
Comparing the whole life summary table and the term life summary table we can see that the IRR and
MIRR of the term life insurance is much lower than that of whole life. Conversely, the profit margin and
the ROI are higher for term than for whole life.
VI. Conclusion
Overall, the calculations performed here were extremely simplistic compared to some
calculations that are made in pricing a policy. Many other factors such as health, geographic area, age,
preexisting conditions, as well as other things could be taken into account to price a policy. Another
important thing to note is the type of policy also plays a large role in the price, shown here through the
calculations of whole life insurance versus term life insurance. Other types of life insurance such as
variable, universal, universal variable, joint, endowment, along with many others will each have their
own pricing and benefits. It is up to the consumer to decide which type is affordable and fits their
lifestyle. All in all, actuaries are an integral part of appropriately pricing and analyzing life insurance
calculations and policies.
15
APPENDIX
TABLE 1
Illustrative Life Table: Basic Functions and Single Benefit Premiums at i=.06
Age
lx
0
5
10
15
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
10,000,000
9,749,503
9,705,588
9,663,731
9,617,802
9,607,896
9,597,695
9,587,169
9,576,288
9,565,017
9,553,319
9,541,153
9,528,475
9,515,235
9,501,381
9,486,854
9,471,591
9,455,522
9,438,571
9,420,657
9,401,688
9,381,566
9,360,184
9,337,427
9,313,166
9,287,264
9,259,571
9,229,925
9,198,149
9,164,051
9,127,426
9,088,049
9,045,679
9,000,057
8,950,901
8,897,913
8,840,770
8,779,128
8,712,621
8,640,861
dx
250,497
43,915
41,857
45,929
9,906
10,201
10,526
10,881
11,271
11,698
12,166
12,678
13,240
13,854
14,527
15,263
16,069
16,951
17,914
18,969
20,122
21,382
22,757
24,261
25,902
27,693
29,646
31,776
34,098
36,625
39,377
42,370
45,622
49,156
52,988
57,143
61,642
66,507
71,760
77,426
1000qx
20.42
0.98
0.85
0.91
1.03
1.06
1.1
1.13
1.18
1.22
1.27
1.33
1.39
1.46
1.53
1.61
1.7
1.79
1.9
2.01
2.14
2.28
2.43
2.6
2.78
2.98
3.2
3.44
3.71
4
4.31
4.66
5.04
5.46
5.92
6.42
6.97
7.58
8.24
8.96
äx
16.801
17.0379
16.9119
16.7384
16.5133
16.4611
16.4061
16.3484
16.2878
16.2242
16.1574
16.0873
16.0139
15.9368
15.8561
15.7716
15.6831
15.5906
15.4938
15.3926
15.287
15.1767
15.0616
14.9416
14.8166
14.6864
14.551
14.4102
14.2639
14.1121
13.9546
13.7914
13.6224
13.4475
13.2668
13.0803
12.8879
12.6896
12.4856
12.2758
1000Ax
49
35.59
42.72
52.55
65.28
68.24
71.35
74.62
78.05
81.65
85.43
89.4
93.56
97.92
102.48
107.27
112.28
117.51
122.99
128.72
134.7
140.94
147.46
154.25
161.32
168.69
176.36
184.33
192.61
201.2
210.12
219.36
228.92
238.82
249.05
259.61
270.5
281.72
293.27
305.14
qx
0.02042
0.00098
0.00085
0.00091
0.00103
0.00106
0.0011
0.00113
0.00118
0.00122
0.00127
0.00133
0.00139
0.00146
0.00153
0.00161
0.0017
0.00179
0.0019
0.00201
0.00214
0.00228
0.00243
0.0026
0.00278
0.00298
0.0032
0.00344
0.00371
0.004
0.00431
0.00466
0.00504
0.00546
0.00592
0.00642
0.00697
0.00758
0.00824
0.00896
Ax
0.049
0.03559
0.04272
0.05255
0.06528
0.06824
0.07135
0.07462
0.07805
0.08165
0.08543
0.0894
0.09356
0.09792
0.10248
0.10727
0.11228
0.11751
0.12299
0.12872
0.1347
0.14094
0.14746
0.15425
0.16132
0.16869
0.17636
0.18433
0.19261
0.2012
0.21012
0.21936
0.22892
0.23882
0.24905
0.25961
0.2705
0.28172
0.29327
0.30514
16
Table 1 Cont’d
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
8,563,435
8,479,908
8,389,826
8,292,713
8,188,074
8,075,403
7,954,179
7,823,879
7,683,979
7,533,964
7,373,338
7,201,635
7,018,432
6,823,367
6,616,155
6,396,609
6,164,663
5,920,394
5,664,051
5,396,081
5,117,152
4,828,182
4,530,360
4,225,163
3,914,365
3,600,038
3,284,542
2,970,496
2,660,734
2,358,246
2,066,090
1,787,299
1,524,758
1,281,083
1,058,491
858,676
682,707
530,959
403,072
297,981
213,977
148,832
99,965
64,617
40,049
83,527
90,082
97,113
104,639
112,671
121,224
130,300
139,900
150,015
160,626
171,703
183,203
195,065
207,212
219,546
231,946
244,269
256,343
267,970
278,929
288,970
297,822
305,197
310,798
314,327
315,496
314,046
309,762
302,488
292,156
278,791
262,541
243,675
222,592
199,815
175,969
151,748
127,887
105,091
84,004
65,145
48,867
35,348
24,568
16,344
9.75
10.62
11.58
12.62
13.76
15.01
16.38
17.88
19.52
21.32
23.29
25.44
27.79
30.37
33.18
36.26
39.62
43.3
47.31
51.69
56.47
61.68
67.37
73.56
80.3
87.64
95.61
104.28
113.69
123.89
134.94
146.89
159.81
173.75
188.77
204.93
222.27
240.86
260.73
281.91
304.45
328.34
353.6
380.2
408.12
12.0604
11.8395
11.6133
11.3818
11.1454
10.9041
10.6584
10.4084
10.1544
9.8969
9.6362
9.3726
9.1066
8.8387
8.5693
8.2988
8.0278
7.7568
7.4864
7.217
6.9493
6.6836
6.4207
6.161
5.905
5.6533
5.4063
5.1645
4.9282
4.698
4.4742
4.2571
4.047
3.8442
3.6488
3.4611
3.2812
3.1091
2.945
2.7888
2.6406
2.5002
2.3676
2.2426
2.1252
317.33
329.84
342.65
355.75
369.13
382.79
396.7
410.85
425.22
439.8
454.56
469.47
484.53
499.7
514.95
530.26
545.6
560.93
576.24
591.49
606.65
621.68
636.56
651.26
665.75
680
693.98
707.67
721.04
734.07
746.74
759.03
770.92
782.41
793.46
804.09
814.27
824.01
833.3
842.14
850.53
858.48
865.99
873.06
879.7
0.00975
0.01062
0.01158
0.01262
0.01376
0.01501
0.01638
0.01788
0.01952
0.02132
0.02329
0.02544
0.02779
0.03037
0.03318
0.03626
0.03962
0.0433
0.04731
0.05169
0.05647
0.06168
0.06737
0.07356
0.0803
0.08764
0.09561
0.10428
0.11369
0.12389
0.13494
0.14689
0.15981
0.17375
0.18877
0.20493
0.22227
0.24086
0.26073
0.28191
0.30445
0.32834
0.3536
0.3802
0.40812
0.31733
0.32984
0.34265
0.35575
0.36913
0.38279
0.3967
0.41085
0.42522
0.4398
0.45456
0.46947
0.48453
0.4997
0.51495
0.53026
0.5456
0.56093
0.57624
0.59149
0.60665
0.62168
0.63656
0.65126
0.66575
0.68
0.69398
0.70767
0.72104
0.73407
0.74674
0.75903
0.77092
0.78241
0.79346
0.80409
0.81427
0.82401
0.8333
0.84214
0.85053
0.85848
0.86599
0.87306
0.8797
17
Table 1 Cont’d
101
102
103
104
105
106
107
108
109
110
23,705
13,339
7,101
3,558
1,668
727
292
108
36
11
10,366
6,238
3,543
1,890
941
435
184
72
25
11
437.28
467.61
498.99
531.28
564.29
597.83
631.64
665.45
698.97
731.87
2.0152
1.9123
1.8164
1.7273
1.6447
1.5685
1.4984
1.4341
1.3755
1.3223
885.93
891.76
897.19
902.23
906.9
911.22
915.19
918.82
922.14
925.15
0.43728
0.46761
0.49899
0.53128
0.56429
0.59783
0.63164
0.66545
0.69897
0.73187
0.88593
0.89176
0.89719
0.90223
0.9069
0.91122
0.91519
0.91882
0.92214
0.92515
18
i= 0.06
TABLE 2
Discrete Whole Life
Age
P(K=k)
0
5
10
15
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
0.001097
0.001134
0.001174
0.001219
0.001268
0.001321
0.001379
0.001443
0.001514
0.00159
0.001674
0.001766
0.001866
0.001976
0.002097
0.002228
0.002371
0.002528
0.002699
0.002885
0.003089
0.003311
0.003553
0.003816
0.004103
0.004415
0.004753
0.005122
0.005521
0.005954
0.006423
0.006929
0.007477
0.008067
b
PVE
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.94340
0.89000
0.83962
0.79209
0.74726
0.70496
0.66506
0.62741
0.59190
0.55839
0.52679
0.49697
0.46884
0.44230
0.41727
0.39365
0.37136
0.35034
0.33051
0.31180
0.29416
0.27751
0.26180
0.24698
0.23300
0.21981
0.20737
0.19563
0.18456
0.17411
0.16425
0.15496
0.14619
0.13791
PVR
PVC
1.00000
1.94340
2.83339
3.67301
4.46511
5.21236
5.91732
6.58238
7.20979
7.80169
8.36009
8.88687
9.38384
9.85268
10.29498
10.71225
11.10590
11.47726
11.82760
12.15812
12.46992
12.76408
13.04158
13.30338
13.55036
13.78336
14.00317
14.21053
14.40616
14.59072
14.76483
14.92909
15.08404
15.23023
0.03500
0.03736
0.03958
0.04168
0.04366
0.04553
0.04729
0.04896
0.05052
0.05200
0.05340
0.05472
0.05596
0.05713
0.05824
0.05928
0.06026
0.06119
0.06207
0.06290
0.06367
0.06441
0.06510
0.06576
0.06638
0.06696
0.06751
0.06803
0.06852
0.06898
0.06941
0.06982
0.07021
0.07058
v^(k+1)
0.94340
0.89000
0.83962
0.79209
0.74726
0.70496
0.66506
0.62741
0.59190
0.55839
0.52679
0.49697
0.46884
0.44230
0.41727
0.39365
0.37136
0.35034
0.33051
0.31180
0.29416
0.27751
0.26180
0.24698
0.23300
0.21981
0.20737
0.19563
0.18456
0.17411
0.16425
0.15496
0.14619
0.13791
LF(K)
0.96840
0.90792
0.85087
0.79705
0.74627
0.69837
0.65318
0.61054
0.57033
0.53238
0.49659
0.46282
0.43096
0.40091
0.37255
0.34580
0.32057
0.29676
0.27431
0.25312
0.23313
0.21427
0.19649
0.17970
0.16387
0.14893
0.13484
0.12155
0.10901
0.09718
0.08602
0.07549
0.06556
0.05618
19
Table 2 Cont’d
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
0.008703
0.009386
0.010118
0.010903
0.011739
0.012631
0.013576
0.014576
0.01563
0.016736
0.01789
0.019088
0.020324
0.02159
0.022875
0.024167
0.025451
0.026709
0.02792
0.029062
0.030108
0.031031
0.031799
0.032383
0.03275
0.032872
0.032721
0.032275
0.031517
0.03044
0.029048
0.027355
0.025389
0.023192
0.020819
0.018335
0.015811
0.013325
0.01095
0.008753
0.006788
0.005092
0.003683
0.00256
0.001703
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.13011
0.12274
0.11579
0.10924
0.10306
0.09722
0.09172
0.08653
0.08163
0.07701
0.07265
0.06854
0.06466
0.06100
0.05755
0.05429
0.05122
0.04832
0.04558
0.04300
0.04057
0.03827
0.03610
0.03406
0.03213
0.03031
0.02860
0.02698
0.02545
0.02401
0.02265
0.02137
0.02016
0.01902
0.01794
0.01693
0.01597
0.01507
0.01421
0.01341
0.01265
0.01193
0.01126
0.01062
0.01002
15.36814
15.49825
15.62099
15.73678
15.84602
15.94907
16.04630
16.13802
16.22454
16.30617
16.38318
16.45583
16.52437
16.58903
16.65003
16.70757
16.76186
16.81308
16.86139
16.90697
16.94998
16.99054
17.02881
17.06492
17.09898
17.13111
17.16143
17.19003
17.21701
17.24246
17.26647
17.28912
17.31049
17.33065
17.34967
17.36762
17.38454
17.40051
17.41558
17.42979
17.44320
17.45585
17.46778
17.47904
17.48966
0.07092
0.07125
0.07155
0.07184
0.07212
0.07237
0.07262
0.07285
0.07306
0.07327
0.07346
0.07364
0.07381
0.07397
0.07413
0.07427
0.07440
0.07453
0.07465
0.07477
0.07487
0.07498
0.07507
0.07516
0.07525
0.07533
0.07540
0.07548
0.07554
0.07561
0.07567
0.07572
0.07578
0.07583
0.07587
0.07592
0.07596
0.07600
0.07604
0.07607
0.07611
0.07614
0.07617
0.07620
0.07622
0.13011
0.12274
0.11579
0.10924
0.10306
0.09722
0.09172
0.08653
0.08163
0.07701
0.07265
0.06854
0.06466
0.06100
0.05755
0.05429
0.05122
0.04832
0.04558
0.04300
0.04057
0.03827
0.03610
0.03406
0.03213
0.03031
0.02860
0.02698
0.02545
0.02401
0.02265
0.02137
0.02016
0.01902
0.01794
0.01693
0.01597
0.01507
0.01421
0.01341
0.01265
0.01193
0.01126
0.01062
0.01002
0.04734
0.03900
0.03114
0.02371
0.01671
0.01010
0.00387
-0.00201
-0.00755
-0.01279
-0.01772
-0.02238
-0.02677
-0.03092
-0.03483
-0.03852
-0.04200
-0.04528
-0.04838
-0.05130
-0.05406
-0.05666
-0.05911
-0.06143
-0.06361
-0.06567
-0.06761
-0.06945
-0.07118
-0.07281
-0.07435
-0.07580
-0.07717
-0.07846
-0.07968
-0.08083
-0.08191
-0.08294
-0.08390
-0.08482
-0.08567
-0.08649
-0.08725
-0.08797
-0.08865
20
Table 2 Cont’d
101
102
103
104
105
106
107
108
109
110
0.00108
0.00065
0.000369
0.000197
9.8E-05
4.53E-05
1.92E-05
7.5E-06
2.6E-06
1.15E-06
Costs First Year
Fixed
Benefit Premium
0.01
0.02
0.5
1
1
1
1
1
1
1
1
1
1
0.00945
0.00892
0.00841
0.00794
0.00749
0.00706
0.00666
0.00629
0.00593
0.00559
17.49968
17.50913
17.51805
17.52646
17.53440
17.54188
17.54895
17.55561
17.56190
17.56783
0.07625
0.07627
0.07630
0.07632
0.07634
0.07635
0.07637
0.07639
0.07640
0.07642
0.00945
0.00892
0.00841
0.00794
0.00749
0.00706
0.00666
0.00629
0.00593
0.00559
Costs Renewal
Fixed
Benefit Premium
0.001
0.001
0.05
-0.08930
-0.08990
-0.09047
-0.09101
-0.09152
-0.09200
-0.09245
-0.09288
-0.09328
-0.09366
21
i= 0.06
TABLE 3
Discrete 30 Year
Term
Age
P(K=k)
0
5
10
15
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
0.001097
0.001134
0.001174
0.001219
0.001268
0.001321
0.001379
0.001443
0.001514
0.00159
0.001674
0.001766
0.001866
0.001976
0.002097
0.002228
0.002371
0.002528
0.002699
0.002885
0.003089
0.003311
0.003553
0.003816
0.004103
0.004415
0.004753
0.005122
0.005521
0.005954
b
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
PVE
PVR
0.943396
0.889996
0.839619
0.792094
0.747258
0.704961
0.665057
0.627412
0.591898
0.558395
0.526788
0.496969
0.468839
0.442301
0.417265
0.393646
0.371364
0.350344
0.330513
0.311805
0.294155
0.277505
0.261797
0.246979
0.232999
0.21981
0.207368
0.19563
0.184557
0.17411
1
1.943396
2.833393
3.673012
4.465106
5.212364
5.917324
6.582381
7.209794
7.801692
8.360087
8.886875
9.383844
9.852683
10.29498
10.71225
11.1059
11.47726
11.8276
12.15812
12.46992
12.76408
13.04158
13.30338
13.55036
13.78336
14.00317
14.21053
14.40616
14.59072
PVC
0.055
0.059245283
0.063250267
0.067028554
0.070592975
0.073955637
0.077127959
0.080120716
0.082944072
0.085607615
0.088120392
0.090490936
0.092727298
0.094837073
0.096827428
0.09870512
0.100476529
0.102147669
0.103724216
0.105211524
0.106614645
0.107938345
0.109187118
0.110365205
0.111476609
0.112525103
0.113514248
0.114447404
0.115327739
0.116158245
LF(K)
0.948396
0.852072
0.7612
0.675472
0.594596
0.518298
0.446319
0.378414
0.314353
0.253918
0.196904
0.143117
0.092374
0.044504
-0.000657
-0.043261
-0.083454
-0.121372
-0.157143
-0.19089
-0.222726
-0.25276
-0.281095
-0.307825
-0.333043
-0.356833
-0.379276
-0.400449
-0.420424
-0.439268
22
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