Life Insurance and Investment Counseling
Oliver Bailey Bruce Urban Mike Lutz
Class: Pre-calculus, calculus, or others if background work on compound interest is done.
Materials: Graphing calculator or computer software capable of iteration.
Setting, Problem, Background Information, and Report Format: See student data sheet.
Teacher Notes: The main concept in this project is compound interest. It can be used at any level where compound interest is discussed. The lower the level of the class the more background information will be needed. The more advanced classes such as calculus or pre-calculus could be given the project without doing much additional background work, although experience has shown that all students, regardless of math background, seem to struggle with the idea of compound interest.
Included with this project are two "Background Units" that could be done: 1) prior to assigning the project, 2) during the class days while students are working on the assignment, or 3) not at all. You are the judge as to what your students need. If both Background Units are done prior to the project, the project should be relatively simple for advanced math students (which may defeat the idea of a modeling project). If no prior work is done, probably very few students would have any idea how to approach the problem. One approach you may wish to try with advanced classes is to assign the problem at the start of the unit and have it worth decreasing value each day as the class progresses through the background units and related textbook materials .
The background units included with this project are "sketchy" and "brief" by design. They are not intended to replace your textbook or your teaching. They are included so that you can quickly see what concepts the students need to understand in order to successfully complete the project.
Extensions: A teaching unit on compound interest and life insurance is an excellent place to do such things as:
1. iterations on calculators
2. spreadsheets on computers
3. probability
4. career work and outside speakers on actuarial science
5. career work and outside speakers on investment counseling
6. study of population growth
7. data analysis with curve fitting
Life Insurance and Investment Counseling
Sample Solution
Policy Year |
Annual Premium |
Contract Fund |
New Money into Contract Fund |
11% of Previous Balance |
Amount of Premium into Contract Fund |
Actual Cost of "Term" Part of Insurance |
Death Benefit of the "Term" Insurance |
Actual Cost of "Term" Insurance |
1 |
$678 |
$386 |
$386 |
$0.00 |
$386.00 |
$292.00 |
$49661 |
$292.00 |
2 |
$678 |
$806 |
$420 |
$42.46 |
$377.54 |
$300.46 |
$49321 |
$300.46 |
3 |
$678 |
$1265 |
$459 |
$88.66 |
$370.34 |
$307.66 |
$48979 |
$307.66 |
4 |
$678 |
$1766 |
$501 |
$139.15 |
$361.85 |
$316.15 |
$48636 |
$316.15 |
5 |
$678 |
$2313 |
$547 |
$194.26 |
$352.74 |
$325.26 |
$48294 |
$325.26 |
6 |
$678 |
$2912 |
$599 |
$254.43 |
$344.57 |
$333.43 |
$47954 |
$333.43 |
7 |
$678 |
$3568 |
$656 |
$320.32 |
$335.68 |
$342.32 |
$47617 |
$342.32 |
8 |
$678 |
$4284 |
$716 |
$392.48 |
$323.52 |
$354.48 |
$47283 |
$354.48 |
9 |
$678 |
$5067 |
$783 |
$471.24 |
$311.76 |
$366.24 |
$46954 |
$366.24 |
10 |
$678 |
$5924 |
$857 |
$557.37 |
$299.63 |
$378.37 |
$46631 |
$378.37 |
11 |
$678 |
$6861 |
$937 |
$651.64 |
$285.36 |
$392.64 |
$46317 |
$392.64 |
12 |
$678 |
$7886 |
$1025 |
$754.71 |
$270.29 |
$407.71 |
$46014 |
$407.71 |
13 |
$678 |
$9005 |
$1119 |
$867.46 |
$251.54 |
$426.46 |
$45726 |
$426.46 |
14 |
$678 |
$10230 |
$1225 |
$990.55 |
$234.45 |
$443.55 |
$45457 |
$443.55 |
15 |
$678 |
$11566 |
$1336 |
$1125.30 |
$210.70 |
$467.30 |
$45213 |
$467.30 |
16 |
$678 |
$13026 |
$1460 |
$1272.30 |
$187.74 |
$490.26 |
$44998 |
$490.26 |
17 |
$678 |
$14622 |
$1596 |
$1432.90 |
$163.14 |
$514.86 |
$44816 |
$514.86 |
18 |
$678 |
$16363 |
$1741 |
$1608.40 |
$132.58 |
$545.42 |
$44671 |
$545.42 |
19 |
$678 |
$18265 |
$1902 |
$1799.90 |
$102.07 |
$575.93 |
$44568 |
$575.93 |
20 |
$678 |
$20343 |
$2078 |
$2009.20 |
$68.85 |
$609.15 |
$44513 |
$609.15 |
Explanation of the Solution: Each year Sue pays $678 for insurance (column 2). Of this $678 part goes for the actual insurance ("term" part) and part goes for an investment. The third column (contract fund) is the amount the insurance company says that Sue would have in her investment account. The new money in the account (column 4, which is calculated by taking the current year's balance minus the previous year's balance) would come from two sources. It would either be interest paid on the account (which the company claims is 11%) or from new money Sue puts in the account as part of her insurance premium. Therefore, if 11% of the previous year=s balance is subtracted from the new money in the account (column 4 minus column 5), the remaining amount would have to be the part of the money that Sue paid that went into her investment account (column 6). If this amount (column 6) is subtracted from $678, the result is the amount Sue would be paying for the insurance part (column 7).
So, would Sue earn 11% on her investment? It could be regarded that way. But, if there is an 11% return on the investment, then the term insurance is decreasing in value while the cost is increasing (columns 8 and 9). This is logical. The probability of a person dying increases as age increases (actuarial science), so it would make sense that either the cost of insurance should increase or the death benefit decrease or both. Sue probably should compare the term insurance part of this policy with other term policies in order to see if she could do better. Terry's claim that the insurance is free after eight years does not seem to be true, but how could it?
Life Insurance and Investment Counseling
Student Data Sheets
Setting: In the early 1980's a new life insurance company, Fancy Life Insurance Company (FLICO) appeared. Their marketing scheme was to replace all of the whole-life policies in the country with FLICO term policies. Term policies are purchases of insurance only. They only pay when the insured person dies. At no time is there any "cash value" for the policy. On the other hand, whole-life policies have a cash value in addition to the life insurance. Term insurance is usually less expensive than comparable whole-life policies.
FLICO agents claimed that whole-life policies were really two policies disguised as one. They claimed that whole-life policies were part term insurance policies and part investment policies. Additionally, they claimed that insurance companies really gave a poor return on the investment part of the whole-life policies. FLICO's marketing scheme was to get people to cash in their whole-life policies sold by other companies, purchase FLICO term policies, and invest the money they saved ("the difference") in a program that would achieve a better investment return.
The marketing scheme was effective. In the competitive U.S. economy traditional insurance companies such as Dearborn Life Insurance Company (DLICO) responded. One of their agents, Terry Ticom, approached one of his customers, Sue Sultan, who was considering making the switch to FLICO. Terry told Sue that he had a policy that, if she wanted to invest some extra money, would get her an eleven percent (11%) return on her investment. The policy would be a $50,000 policy. Following is a table of information on the policy:
Policy Year |
Annual Premium |
Contract Fund |
Cash Value |
Death Benefit |
1 |
$678 |
$386 |
$4 |
$50,047 |
2 |
$678 |
$806 |
$399 |
$50,127 |
3 |
$678 |
$1,265 |
$831 |
$50,244 |
4 |
$678 |
$1,766 |
$1,306 |
$50,402 |
5 |
$678 |
$2,313 |
$1,827 |
$50,607 |
6 |
$678 |
$2,912 |
$2,502 |
$50,866 |
7 |
$678 |
$3,568 |
$3,260 |
$51,185 |
8 |
$678 |
$4,284 |
$4,079 |
$51,567 |
9 |
$678 |
$5,067 |
$4,965 |
$52,021 |
10 |
$678 |
$5,924 |
$5,924 |
$52,555 |
11 |
$678 |
$6,861 |
$6,861 |
$53,178 |
12 |
$678 |
$7,886 |
$7,886 |
$53,900 |
13 |
$678 |
$9,005 |
$9,005 |
$54,731 |
14 |
$678 |
$10,230 |
$10,230 |
$55,687 |
15 |
$678 |
$11,566 |
$11,566 |
$56,779 |
16 |
$678 |
$13,026 |
$13,026 |
$58,024 |
17 |
$678 |
$14,622 |
$14,622 |
$59,438 |
18 |
$678 |
$16,363 |
$16,363 |
$61,034 |
19 |
$678 |
$18,265 |
$18,265 |
$62,833 |
20 |
$678 |
$20,343 |
$20,343 |
$64,856 |
Age 65 |
$678 |
$37,460 |
$37,460 |
$83,210 |
Terry explained to Sue that he doubted whether she could find an investment anywhere that guaranteed her an 11% return as this policy did. Additionally, he explained that from the eighth year on the entire $678 premium (and even more) would go into her contract fund so that she was actually getting the life insurance free in addition to the 11% investment return.
Background Material and Definition of terms:
1. There are two separate quantities involved with insurance. One is the death benefit. It is paid when the insured person dies. The second quantity is any other kind of benefit payment. It is considered an investment (similar to a savings account). Term policies only have a death benefitCthere is no investment part of the policy. It is important in this problem to separate the two.
2. The annual premium is the amount the customer pays each year for the policy. For a policy other than a term policy, part of the premium goes toward the life insurance and part is an investment. The issue in this problem is to actually calculate how much of the premium goes toward the insurance and how much goes as an investment.
3. The contract fund is the amount the customer has in his/her investment account. (Theoretically, it should equal the cash value.)
4. The cash value is the amount that the policy holder would receive if he/she chose to terminate the policy. It represents the investment portion of the policy, so it is available to the policy holder.
5. The death benefit is the total amount that would be paid to the beneficiary upon the death of the insured (as long as the policy is still in effect). Logically, it would seem to equal the life insurance benefit plus the investment.
Problem: Analyze the data and answer these questions:
1. Would Sue be getting an 11% return on the money she invested in the investment portion of the policy as Terry claimed? If not, what is the rate?
2. Is the life insurance free after the eighth year as Terry claimed? In other words, is the "term"-insurance portion of the policy free?
3. What is the cost of the "term"-insurance portion each of the twenty years?
4. Why doesn't the amount in the contract fund always equal the cash value?
5. Why isn't the death benefit equal to $50,000 plus the cash value?
6. EXTRA CHALLENGE: How old was Sue when the table was developed for her?
Report Format: Type a report analyzing the problem. A table similar to the one provided with several additional columns breaking down the data further would probably be a useful aid in your explanation (which means you may wish to use a computer spreadsheet program). Be sure to explain in detail what the information in your table means.
Life Insurance and Investment Counseling
Background Unit 1CCompound Interest
Compound interest is the paying of interest on interest. It can probably be best understood by the study of a few examples:
Example 1: $1000 is placed in a savings account paying 6% annual percentage rate (APR) compounded annually. Analyze what happens in the account for the first four years.
Solution 1:
$1000 Invested at 6% Interest Compounded Annually
Elapsed Time (Years) |
Interest Earned |
New Balance |
0 |
0 |
$1000.00 |
1 |
$1000.00 * .06 = $60.00 |
$1000.00 + $60.00 = $1060.00 |
2 |
$1060.00 * .06 = $63.60 |
$1060.00 + $63.60 = $1123.60 |
3 |
$1123.60 * .06 = $67.42 |
$1123.60 + $67.42 = $1191.02 |
4 |
$1191.02 * .06 = $71.46 |
$1191.02 + $71.46 = $1262.48 |
Notice that each time interest is calculated it is calculated on the previous balance instead of just $1000. This makes the interest more each time than the $60 that it would be otherwise.
Example 2: $1000 is placed in a savings account paying 6% APR compounded semiannually. Analyze what happens in the account for the first four years.
Solution 2:
$1000 Invested at 6% Interest Compounded Semiannually
Elapsed Time (Years) |
Interest Earned |
New Balance |
0.0 |
0 |
$1000.00 |
0.5 |
$1000.00 * .03 = $30.00 |
$1000.00 + $30.00 = $1030.00 |
1.0 |
$1030.00 * .03 = $30.90 |
$1030.00 + $30.90 = $1060.90 |
1.5 |
$1060.90 * .03 = $31.83 |
$1060.90 + $31.83 = $1092.73 |
2.0 |
$1092.73 * .03 = $32.78 |
$1092.73 + $32.78 = $1125.51 |
2.5 |
$1125.51 * .03 = $33.77 |
$1125.51 + $33.77 = $1159.28 |
3.0 |
$1159.28 * .03 = $34.78 |
$1159.28 + $34.78 = $1194.06 |
3.5 |
$1194.06 * .03 = $35.82 |
$1194.06 + $35.82 = $1229.88 |
4.0 |
$1229.88 * .03 = $36.90 |
$1229.88 + $36.90 = $1266.78 |
Notice that the interest is now calculated twice per year instead of once. Also, notice that if the interest rate is 6% for the entire year it is 6/2 (or 3%) for each 6-month period. Additionally, note that there are now 8 payment periods (twice per year for four years) instead of the previous 4 payment periods (once per year for four years).
Life Insurance and Investment Counseling
Background Unit 2CLife Insurance & Compound Interest
Providing financial security for the future can be done in many different ways. One way is investing money that will yield a return, such as a savings account demonstrated in "Background Unit 1." Another way is purchasing life insurance. Life insurance pays a death benefit to the beneficiary listed in the policy upon the death of the insured. One problem with this type of investment is that it yields no financial return to the insured.
Historically, life insurance companies have wanted to provide both of these types of financial security. (Of course, what they really want is the income from these kinds of investments.) As both types of investments have often been sold in one policy the "lines have been blurry" as to what is being spent on the death benefit, what is being invested, and what the true rate of return is.
Let's look at a simple example to see how this might work: Mary is 30 years old and has two children, ages 3 and 7. Since her salary represents 2/3 of her family's income, she is concerned about how her family would live if she were to die suddenly. She contacts an insurance agent. The agent tells her that for $500 per year she can purchase $100,000 worth of life insurance. This means that upon Mary's death her husband (whom she listed as beneficiary in the policy) would receive $100,000. There would be no other financial benefit for the family.
Mary's agent tells her that she should also plan for her and her husband's retirement by making regular investments. He suggests that an excellent way of doing this would be to simply include this as part of the insurance policy. He said that he could write her a policy where she could pay the $500 per year for the death benefit and pay an additional $100 per year that he said would return her 10% APR. It would look like this:
Policy Year |
Cost of Death Benefit |
Cost of Investment |
Total Premium |
Interest on Investment |
Amount in Contract Fund |
Death Benefit |
Cash Value |
1 |
$500 |
$100 |
$600 |
$0*.10=$0 |
$100+$0=100 |
$100000+$100= $100100 |
$100 |
2 |
$500 |
$100 |
$600 |
$100*.10=$10 |
$100+100+10 |
$100210 |
$210 |
3 |
$500 |
$100 |
$600 |
$210*.10=$21 |
$210+100+21 |
$100331 |
$331 |
4 |
$500 |
$100 |
$600 |
$33.10 |
$464.100 |
$100464.10 |
$464.10 |
5 |
$500 |
$100 |
$600 |
$46.41 |
$610.51 |
$100610.51 |
$610.51 |
6 |
$500 |
$100 |
$600 |
$61.05 |
$771.56 |
$100771.56 |
$771.56 |
7 |
$500 |
$100 |
$600 |
$77.16 |
$948.72 |
$100948.72 |
$948.72 |
8 |
$500 |
$100 |
$600 |
$94.87 |
$1143.59 |
$101143.59 |
$1143.59 |
9 |
$500 |
$100 |
$600 |
$114.36 |
$1357.95 |
$101357.95 |
$1357.95 |
10 |
$500 |
$100 |
$600 |
$135.80 |
$1593.75 |
$101593.75 |
$1593.75 |
The annual premium is what Mary would pay each year for the policy. The contract fund is the amount that Mary has invested (similar to a savings account). This is the amount that is available to her "similar to" a savings account. (She does not need to die in order to get this money, but there are usually some restrictions.) The death benefit is what Mary's husband would receive in case of her death. Notice that he would receive the $100,000 that was the insurance policy plus the additional money that they had added in investments.
Funded in part by the National Science Foundation and Indiana University 1995