Profit Measures in Life Insurance

Transcription

1 Profit Measures in Life Insurance Shelly Matushevski Honors Project Spring 2011 The University of Akron

2 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 d) Return on Investment e) Summary of Whole Life Profit Measures V. Term Life Insurance VI. Conclusion APPENDIX Works Cited... 22

3 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 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 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 dx 1000qx äx 1000Ax qx Ax 0 10,000, , ,749,503 43, ,705,588 41, ,663,731 45, 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 d x+k is the number of decrements in a given year, and l x 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) b PVE PVR PVC v^(k+1) P(K=k)*v^(k+1) (v^k)*(lk/l0) LF(K)

6 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 7 The costs are defined as fixed costs, proportion of benefits, and proportion of premiums. Here, b is the unit benefit, f R is the fixed cost renewal, r B,R is the proportion of benefits renewal, r P,R is the proportion of premiums renewal, f I is the fixed cost initial, r B,I is the proportion of benefits initial, r P,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 P x = A x / ä x

8 8 which gives P x = Multiplying the benefit of b = $100,000 by P x gives a premium (without costs) of π = $ 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 = , and when multiplied by the benefit of $100,000, a loaded premium equal to $ 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 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 for the expected value of the loss function. For comparison, G was also changed to.05, which gives a value of 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 $ , and a value of G =.05 is a loaded premium of $ , 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 = which would be expected since the loaded premium was so much higher. The PM for G =.01 is 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 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 A C, or the complement of A. To find R A, 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 R A C, the complement of R A, 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 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 R A, 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 e α = 1+j, where j is the reinvestment interest rate, then we get

12 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 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 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 P x = , 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 This gives the loaded premium equal to $3, and the loading equal to $3, 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 % which is less than the IRR of whole life, but this

14 14 was expected. The MIRR was found to be %, again less than the value of the MIRR of whole life. The profit margin PM is and the ROI is The following table summarizes the profit measures for term life insurance. PM IRR MIRR ROI % 1.99% 6.63% % 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 15 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, , ,749,503 43, ,705,588 41, ,663,731 45, ,617,802 9, ,607,896 10, ,597,695 10, ,587,169 10, ,576,288 11, ,565,017 11, ,553,319 12, ,541,153 12, ,528,475 13, ,515,235 13, ,501,381 14, ,486,854 15, ,471,591 16, ,455,522 16, ,438,571 17, ,420,657 18, ,401,688 20, ,381,566 21, ,360,184 22, ,337,427 24, ,313,166 25, ,287,264 27, ,259,571 29, ,229,925 31, ,198,149 34, ,164,051 36, ,127,426 39, ,088,049 42, ,045,679 45, ,000,057 49, ,950,901 52, ,897,913 57, ,840,770 61, ,779,128 66, ,712,621 71, ,640,861 77,

16 16 Table 1 Cont d 56 8,563,435 83, ,479,908 90, ,389,826 97, ,292, , ,188, , ,075, , ,954, , ,823, , ,683, , ,533, , ,373, , ,201, , ,018, , ,823, , ,616, , ,396, , ,164, , ,920, , ,664, , ,396, , ,117, , ,828, , ,530, , ,225, , ,914, , ,600, , ,284, , ,970, , ,660, , ,358, , ,066, , ,787, , ,524, , ,281, , ,058, , , , , , , , , , ,981 84, ,977 65, ,832 48, ,965 35, ,617 24, ,049 16,

17 17 Table 1 Cont d ,705 10, ,339 6, ,101 3, ,558 1, ,

18 18 i= 0.06 TABLE 2 Discrete Whole Life Age P(K=k) b PVE PVR PVC v^(k+1) LF(K)

19 19 Table 2 Cont d

20 20 Table 2 Cont d E E E E E E Costs First Year Costs Renewal Fixed Benefit Premium Fixed Benefit Premium

21 21 i= 0.06 TABLE 3 Discrete 30 Year Age P(K=k) b PVE PVR PVC LF(K) Term

22 22 Works Cited Ajmera, R. (December, ). History of Life Insurance. Retrieved February 17, 2011, from IRR. (n.d.). Retrieved 3 20, 2011, from moneyterms.co.uk: IRR- Internal Rate of Return. (2011, March 9). Retrieved March 10, 2011, from Think and Done- Financial articles, tools, and more: Law of Large Numbers. (n.d.). Retrieved February 17, 2011, from All Business- Business Glossary: Modified Internal Rate of Return-MIRR. (n.d.). Retrieved March 10, 2011, from Investopedia: Rejda, G. (2011). Principles of Risk Management and Insurance. Boston: Prentice Hall. Return on Investment- ROI. (n.d.). Retrieved March 31, 2011, from Investopedia: Ross, S., Westerfield, R., & Jordan, B. (2007). Fundamentals of Corporate Finance. McGraw Hill. Seog, S. H. (2010). The Economics of Risk and Insurance. Massachusetts: Wiley Blackwell. SOA. (2008). MLC Tables. Retrieved February 17, 2011, from Society of Actuaries: Statistics, B. o. (2011). Actuaries. Retrieved February 17, 2011, from Occupational Outlook Handbook: Types of Life Insurance Explained. (n.d.). Retrieved March 31, 2011, from Insurance Finder: