Profit Measures in Life Insurance

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Profit Measures in Life Insurance

Shelly Matushevski

Honors Project Spring 2011 The University of Akron

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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

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members of the deceased. The idea of life insurance as we now know it came from England in the 17th 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.

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II. Loss Function

For an individual, Table 1 shows an illustrative life table where the interest rate is .06.

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

d

x+k

/l

x

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 1 Illustrative Life Table: Basic Functions and Single Benefit Premiums at i=.06

Age lx dx 1000qx äx 1000Ax qx Ax 0 10,000,000 250,497 20.42 16.801 49 0.02042 0.049 5 9,749,503 43,915 0.98 17.0379 35.59 0.00098 0.03559 10 9,705,588 41,857 0.85 16.9119 42.72 0.00085 0.04272 15 9,663,731 45,929 0.91 16.7384 52.55 0.00091 0.05255 TABLE 2

Discrete Whole Life Age P(K=k) b PVE PVR PVC v^(k+1) P(K=k)*v^(k+1) (v^k)*(lk/l0) LF(K) 22 0.001097 1 0.94340 1.00000 0.03500 0.94340 0.00103 1.00000 0.96840 23 0.001134 1 0.89000 1.94340 0.03736 0.89000 0.00101 0.94236 0.90792 24 0.001174 1 0.83962 2.83339 0.03958 0.83962 0.00099 0.88801 0.85087 25 0.001219 1 0.79209 3.67301 0.04168 0.79209 0.00097 0.83676 0.79705 26 0.001268 1 0.74726 4.46511 0.04366 0.74726 0.00095 0.78843 0.74627 27 0.001321 1 0.70496 5.21236 0.04553 0.70496 0.00093 0.74286 0.69837

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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

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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

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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

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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

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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,

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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

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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%

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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

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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 IRR MIRR ROI 0.9966% 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.

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APPENDIX

TABLE 1 Illustrative Life Table: Basic Functions and Single Benefit Premiums at i=.06

Age lx dx 1000qx äx 1000Ax qx Ax 0 10,000,000 250,497 20.42 16.801 49 0.02042 0.049 5 9,749,503 43,915 0.98 17.0379 35.59 0.00098 0.03559 10 9,705,588 41,857 0.85 16.9119 42.72 0.00085 0.04272 15 9,663,731 45,929 0.91 16.7384 52.55 0.00091 0.05255 20 9,617,802 9,906 1.03 16.5133 65.28 0.00103 0.06528 21 9,607,896 10,201 1.06 16.4611 68.24 0.00106 0.06824 22 9,597,695 10,526 1.1 16.4061 71.35 0.0011 0.07135 23 9,587,169 10,881 1.13 16.3484 74.62 0.00113 0.07462 24 9,576,288 11,271 1.18 16.2878 78.05 0.00118 0.07805 25 9,565,017 11,698 1.22 16.2242 81.65 0.00122 0.08165 26 9,553,319 12,166 1.27 16.1574 85.43 0.00127 0.08543 27 9,541,153 12,678 1.33 16.0873 89.4 0.00133 0.0894 28 9,528,475 13,240 1.39 16.0139 93.56 0.00139 0.09356 29 9,515,235 13,854 1.46 15.9368 97.92 0.00146 0.09792 30 9,501,381 14,527 1.53 15.8561 102.48 0.00153 0.10248 31 9,486,854 15,263 1.61 15.7716 107.27 0.00161 0.10727 32 9,471,591 16,069 1.7 15.6831 112.28 0.0017 0.11228 33 9,455,522 16,951 1.79 15.5906 117.51 0.00179 0.11751 34 9,438,571 17,914 1.9 15.4938 122.99 0.0019 0.12299 35 9,420,657 18,969 2.01 15.3926 128.72 0.00201 0.12872 36 9,401,688 20,122 2.14 15.287 134.7 0.00214 0.1347 37 9,381,566 21,382 2.28 15.1767 140.94 0.00228 0.14094 38 9,360,184 22,757 2.43 15.0616 147.46 0.00243 0.14746 39 9,337,427 24,261 2.6 14.9416 154.25 0.0026 0.15425 40 9,313,166 25,902 2.78 14.8166 161.32 0.00278 0.16132 41 9,287,264 27,693 2.98 14.6864 168.69 0.00298 0.16869 42 9,259,571 29,646 3.2 14.551 176.36 0.0032 0.17636 43 9,229,925 31,776 3.44 14.4102 184.33 0.00344 0.18433 44 9,198,149 34,098 3.71 14.2639 192.61 0.00371 0.19261 45 9,164,051 36,625 4 14.1121 201.2 0.004 0.2012 46 9,127,426 39,377 4.31 13.9546 210.12 0.00431 0.21012 47 9,088,049 42,370 4.66 13.7914 219.36 0.00466 0.21936 48 9,045,679 45,622 5.04 13.6224 228.92 0.00504 0.22892 49 9,000,057 49,156 5.46 13.4475 238.82 0.00546 0.23882 50 8,950,901 52,988 5.92 13.2668 249.05 0.00592 0.24905 51 8,897,913 57,143 6.42 13.0803 259.61 0.00642 0.25961 52 8,840,770 61,642 6.97 12.8879 270.5 0.00697 0.2705 53 8,779,128 66,507 7.58 12.6896 281.72 0.00758 0.28172 54 8,712,621 71,760 8.24 12.4856 293.27 0.00824 0.29327 55 8,640,861 77,426 8.96 12.2758 305.14 0.00896 0.30514

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Table 1 Cont’d 56 8,563,435 83,527 9.75 12.0604 317.33 0.00975 0.31733 57 8,479,908 90,082 10.62 11.8395 329.84 0.01062 0.32984 58 8,389,826 97,113 11.58 11.6133 342.65 0.01158 0.34265 59 8,292,713 104,639 12.62 11.3818 355.75 0.01262 0.35575 60 8,188,074 112,671 13.76 11.1454 369.13 0.01376 0.36913 61 8,075,403 121,224 15.01 10.9041 382.79 0.01501 0.38279 62 7,954,179 130,300 16.38 10.6584 396.7 0.01638 0.3967 63 7,823,879 139,900 17.88 10.4084 410.85 0.01788 0.41085 64 7,683,979 150,015 19.52 10.1544 425.22 0.01952 0.42522 65 7,533,964 160,626 21.32 9.8969 439.8 0.02132 0.4398 66 7,373,338 171,703 23.29 9.6362 454.56 0.02329 0.45456 67 7,201,635 183,203 25.44 9.3726 469.47 0.02544 0.46947 68 7,018,432 195,065 27.79 9.1066 484.53 0.02779 0.48453 69 6,823,367 207,212 30.37 8.8387 499.7 0.03037 0.4997 70 6,616,155 219,546 33.18 8.5693 514.95 0.03318 0.51495 71 6,396,609 231,946 36.26 8.2988 530.26 0.03626 0.53026 72 6,164,663 244,269 39.62 8.0278 545.6 0.03962 0.5456 73 5,920,394 256,343 43.3 7.7568 560.93 0.0433 0.56093 74 5,664,051 267,970 47.31 7.4864 576.24 0.04731 0.57624 75 5,396,081 278,929 51.69 7.217 591.49 0.05169 0.59149 76 5,117,152 288,970 56.47 6.9493 606.65 0.05647 0.60665 77 4,828,182 297,822 61.68 6.6836 621.68 0.06168 0.62168 78 4,530,360 305,197 67.37 6.4207 636.56 0.06737 0.63656 79 4,225,163 310,798 73.56 6.161 651.26 0.07356 0.65126 80 3,914,365 314,327 80.3 5.905 665.75 0.0803 0.66575 81 3,600,038 315,496 87.64 5.6533 680 0.08764 0.68 82 3,284,542 314,046 95.61 5.4063 693.98 0.09561 0.69398 83 2,970,496 309,762 104.28 5.1645 707.67 0.10428 0.70767 84 2,660,734 302,488 113.69 4.9282 721.04 0.11369 0.72104 85 2,358,246 292,156 123.89 4.698 734.07 0.12389 0.73407 86 2,066,090 278,791 134.94 4.4742 746.74 0.13494 0.74674 87 1,787,299 262,541 146.89 4.2571 759.03 0.14689 0.75903 88 1,524,758 243,675 159.81 4.047 770.92 0.15981 0.77092 89 1,281,083 222,592 173.75 3.8442 782.41 0.17375 0.78241 90 1,058,491 199,815 188.77 3.6488 793.46 0.18877 0.79346 91 858,676 175,969 204.93 3.4611 804.09 0.20493 0.80409 92 682,707 151,748 222.27 3.2812 814.27 0.22227 0.81427 93 530,959 127,887 240.86 3.1091 824.01 0.24086 0.82401 94 403,072 105,091 260.73 2.945 833.3 0.26073 0.8333 95 297,981 84,004 281.91 2.7888 842.14 0.28191 0.84214 96 213,977 65,145 304.45 2.6406 850.53 0.30445 0.85053 97 148,832 48,867 328.34 2.5002 858.48 0.32834 0.85848 98 99,965 35,348 353.6 2.3676 865.99 0.3536 0.86599 99 64,617 24,568 380.2 2.2426 873.06 0.3802 0.87306 100 40,049 16,344 408.12 2.1252 879.7 0.40812 0.8797

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Table 1 Cont’d 101 23,705 10,366 437.28 2.0152 885.93 0.43728 0.88593 102 13,339 6,238 467.61 1.9123 891.76 0.46761 0.89176 103 7,101 3,543 498.99 1.8164 897.19 0.49899 0.89719 104 3,558 1,890 531.28 1.7273 902.23 0.53128 0.90223 105 1,668 941 564.29 1.6447 906.9 0.56429 0.9069 106 727 435 597.83 1.5685 911.22 0.59783 0.91122 107 292 184 631.64 1.4984 915.19 0.63164 0.91519 108 108 72 665.45 1.4341 918.82 0.66545 0.91882 109 36 25 698.97 1.3755 922.14 0.69897 0.92214 110 11 11 731.87 1.3223 925.15 0.73187 0.92515

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i= 0.06

TABLE 2

Discrete Whole Life Age P(K=k) b PVE PVR PVC v^(k+1) LF(K)

0 5 10 15 20 21 22 0.001097 1 0.94340 1.00000 0.03500 0.94340 0.96840 23 0.001134 1 0.89000 1.94340 0.03736 0.89000 0.90792 24 0.001174 1 0.83962 2.83339 0.03958 0.83962 0.85087 25 0.001219 1 0.79209 3.67301 0.04168 0.79209 0.79705 26 0.001268 1 0.74726 4.46511 0.04366 0.74726 0.74627 27 0.001321 1 0.70496 5.21236 0.04553 0.70496 0.69837 28 0.001379 1 0.66506 5.91732 0.04729 0.66506 0.65318 29 0.001443 1 0.62741 6.58238 0.04896 0.62741 0.61054 30 0.001514 1 0.59190 7.20979 0.05052 0.59190 0.57033 31 0.00159 1 0.55839 7.80169 0.05200 0.55839 0.53238 32 0.001674 1 0.52679 8.36009 0.05340 0.52679 0.49659 33 0.001766 1 0.49697 8.88687 0.05472 0.49697 0.46282 34 0.001866 1 0.46884 9.38384 0.05596 0.46884 0.43096 35 0.001976 1 0.44230 9.85268 0.05713 0.44230 0.40091 36 0.002097 1 0.41727 10.29498 0.05824 0.41727 0.37255 37 0.002228 1 0.39365 10.71225 0.05928 0.39365 0.34580 38 0.002371 1 0.37136 11.10590 0.06026 0.37136 0.32057 39 0.002528 1 0.35034 11.47726 0.06119 0.35034 0.29676 40 0.002699 1 0.33051 11.82760 0.06207 0.33051 0.27431 41 0.002885 1 0.31180 12.15812 0.06290 0.31180 0.25312 42 0.003089 1 0.29416 12.46992 0.06367 0.29416 0.23313 43 0.003311 1 0.27751 12.76408 0.06441 0.27751 0.21427 44 0.003553 1 0.26180 13.04158 0.06510 0.26180 0.19649 45 0.003816 1 0.24698 13.30338 0.06576 0.24698 0.17970 46 0.004103 1 0.23300 13.55036 0.06638 0.23300 0.16387 47 0.004415 1 0.21981 13.78336 0.06696 0.21981 0.14893 48 0.004753 1 0.20737 14.00317 0.06751 0.20737 0.13484 49 0.005122 1 0.19563 14.21053 0.06803 0.19563 0.12155 50 0.005521 1 0.18456 14.40616 0.06852 0.18456 0.10901 51 0.005954 1 0.17411 14.59072 0.06898 0.17411 0.09718 52 0.006423 1 0.16425 14.76483 0.06941 0.16425 0.08602 53 0.006929 1 0.15496 14.92909 0.06982 0.15496 0.07549 54 0.007477 1 0.14619 15.08404 0.07021 0.14619 0.06556 55 0.008067 1 0.13791 15.23023 0.07058 0.13791 0.05618

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Table 2 Cont’d 56 0.008703 1 0.13011 15.36814 0.07092 0.13011 0.04734 57 0.009386 1 0.12274 15.49825 0.07125 0.12274 0.03900 58 0.010118 1 0.11579 15.62099 0.07155 0.11579 0.03114 59 0.010903 1 0.10924 15.73678 0.07184 0.10924 0.02371 60 0.011739 1 0.10306 15.84602 0.07212 0.10306 0.01671 61 0.012631 1 0.09722 15.94907 0.07237 0.09722 0.01010 62 0.013576 1 0.09172 16.04630 0.07262 0.09172 0.00387 63 0.014576 1 0.08653 16.13802 0.07285 0.08653 -0.00201 64 0.01563 1 0.08163 16.22454 0.07306 0.08163 -0.00755 65 0.016736 1 0.07701 16.30617 0.07327 0.07701 -0.01279 66 0.01789 1 0.07265 16.38318 0.07346 0.07265 -0.01772 67 0.019088 1 0.06854 16.45583 0.07364 0.06854 -0.02238 68 0.020324 1 0.06466 16.52437 0.07381 0.06466 -0.02677 69 0.02159 1 0.06100 16.58903 0.07397 0.06100 -0.03092 70 0.022875 1 0.05755 16.65003 0.07413 0.05755 -0.03483 71 0.024167 1 0.05429 16.70757 0.07427 0.05429 -0.03852 72 0.025451 1 0.05122 16.76186 0.07440 0.05122 -0.04200 73 0.026709 1 0.04832 16.81308 0.07453 0.04832 -0.04528 74 0.02792 1 0.04558 16.86139 0.07465 0.04558 -0.04838 75 0.029062 1 0.04300 16.90697 0.07477 0.04300 -0.05130 76 0.030108 1 0.04057 16.94998 0.07487 0.04057 -0.05406 77 0.031031 1 0.03827 16.99054 0.07498 0.03827 -0.05666 78 0.031799 1 0.03610 17.02881 0.07507 0.03610 -0.05911 79 0.032383 1 0.03406 17.06492 0.07516 0.03406 -0.06143 80 0.03275 1 0.03213 17.09898 0.07525 0.03213 -0.06361 81 0.032872 1 0.03031 17.13111 0.07533 0.03031 -0.06567 82 0.032721 1 0.02860 17.16143 0.07540 0.02860 -0.06761 83 0.032275 1 0.02698 17.19003 0.07548 0.02698 -0.06945 84 0.031517 1 0.02545 17.21701 0.07554 0.02545 -0.07118 85 0.03044 1 0.02401 17.24246 0.07561 0.02401 -0.07281 86 0.029048 1 0.02265 17.26647 0.07567 0.02265 -0.07435 87 0.027355 1 0.02137 17.28912 0.07572 0.02137 -0.07580 88 0.025389 1 0.02016 17.31049 0.07578 0.02016 -0.07717 89 0.023192 1 0.01902 17.33065 0.07583 0.01902 -0.07846 90 0.020819 1 0.01794 17.34967 0.07587 0.01794 -0.07968 91 0.018335 1 0.01693 17.36762 0.07592 0.01693 -0.08083 92 0.015811 1 0.01597 17.38454 0.07596 0.01597 -0.08191 93 0.013325 1 0.01507 17.40051 0.07600 0.01507 -0.08294 94 0.01095 1 0.01421 17.41558 0.07604 0.01421 -0.08390 95 0.008753 1 0.01341 17.42979 0.07607 0.01341 -0.08482 96 0.006788 1 0.01265 17.44320 0.07611 0.01265 -0.08567 97 0.005092 1 0.01193 17.45585 0.07614 0.01193 -0.08649 98 0.003683 1 0.01126 17.46778 0.07617 0.01126 -0.08725 99 0.00256 1 0.01062 17.47904 0.07620 0.01062 -0.08797 100 0.001703 1 0.01002 17.48966 0.07622 0.01002 -0.08865

(20)

Table 2 Cont’d 101 0.00108 1 0.00945 17.49968 0.07625 0.00945 -0.08930 102 0.00065 1 0.00892 17.50913 0.07627 0.00892 -0.08990 103 0.000369 1 0.00841 17.51805 0.07630 0.00841 -0.09047 104 0.000197 1 0.00794 17.52646 0.07632 0.00794 -0.09101 105 9.8E-05 1 0.00749 17.53440 0.07634 0.00749 -0.09152 106 4.53E-05 1 0.00706 17.54188 0.07635 0.00706 -0.09200 107 1.92E-05 1 0.00666 17.54895 0.07637 0.00666 -0.09245 108 7.5E-06 1 0.00629 17.55561 0.07639 0.00629 -0.09288 109 2.6E-06 1 0.00593 17.56190 0.07640 0.00593 -0.09328 110 1.15E-06 1 0.00559 17.56783 0.07642 0.00559 -0.09366

Costs First Year Costs Renewal

Fixed Benefit Premium Fixed Benefit Premium

(21)

i= 0.06

TABLE 3

Discrete 30 Year Age P(K=k) b PVE PVR PVC LF(K) Term 0 1 5 1 10 1 15 1 20 1 21 1 22 0.001097 1 0.943396 1 0.055 0.948396 23 0.001134 1 0.889996 1.943396 0.059245283 0.852072 24 0.001174 1 0.839619 2.833393 0.063250267 0.7612 25 0.001219 1 0.792094 3.673012 0.067028554 0.675472 26 0.001268 1 0.747258 4.465106 0.070592975 0.594596 27 0.001321 1 0.704961 5.212364 0.073955637 0.518298 28 0.001379 1 0.665057 5.917324 0.077127959 0.446319 29 0.001443 1 0.627412 6.582381 0.080120716 0.378414 30 0.001514 1 0.591898 7.209794 0.082944072 0.314353 31 0.00159 1 0.558395 7.801692 0.085607615 0.253918 32 0.001674 1 0.526788 8.360087 0.088120392 0.196904 33 0.001766 1 0.496969 8.886875 0.090490936 0.143117 34 0.001866 1 0.468839 9.383844 0.092727298 0.092374 35 0.001976 1 0.442301 9.852683 0.094837073 0.044504 36 0.002097 1 0.417265 10.29498 0.096827428 -0.000657 37 0.002228 1 0.393646 10.71225 0.09870512 -0.043261 38 0.002371 1 0.371364 11.1059 0.100476529 -0.083454 39 0.002528 1 0.350344 11.47726 0.102147669 -0.121372 40 0.002699 1 0.330513 11.8276 0.103724216 -0.157143 41 0.002885 1 0.311805 12.15812 0.105211524 -0.19089 42 0.003089 1 0.294155 12.46992 0.106614645 -0.222726 43 0.003311 1 0.277505 12.76408 0.107938345 -0.25276 44 0.003553 1 0.261797 13.04158 0.109187118 -0.281095 45 0.003816 1 0.246979 13.30338 0.110365205 -0.307825 46 0.004103 1 0.232999 13.55036 0.111476609 -0.333043 47 0.004415 1 0.21981 13.78336 0.112525103 -0.356833 48 0.004753 1 0.207368 14.00317 0.113514248 -0.379276 49 0.005122 1 0.19563 14.21053 0.114447404 -0.400449 50 0.005521 1 0.184557 14.40616 0.115327739 -0.420424 51 0.005954 1 0.17411 14.59072 0.116158245 -0.439268

(22)

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Profit Measures in Life Insurance