Homeowners Insurance

Oliver Bailey Mike Lutz John Ruebusch

Class: Calculus.

Materials: Graphing calculator or computer software such as Derive.

Goal: To enable students to see how ability with compound interest can assist them in making important financial decisions.

Time Required: Two class periods, plus time outside of class.

Background: Students should have a working knowledge of compound interest.

Setting: Sue Sultan needs to purchase homeowners insurance. Her regular insurance company, Dearborn Insurance Company, will sell her the \$100,000 worth of insurance that she needs for \$300 per year. Another company, Fancy Insurance Company, has a unique insurance policy. They will insure her home for \$100,000 for a single payment of \$3,000. Additionally, they tell her that if she ever wishes to cancel her insurance policy, they will reimburse her the entire \$3,000. Sue thinks that this is too good to be true so she asks for time to analyze the situation.

Problem: Your job is to analyze the two policies and advise Sue as to what she should do. Be sure to clearly state any assumptions that you make in coming to your conclusion and justify why (how) you made them. Write up your report in easy-to-understand language, because, although Sue is intellectually and physically gifted having PhD=s in four different areas, math is her lone human shortcoming. She does realize that her expense for the Fancy policy would be the revenue that she would lose by not having the \$3,000 to invest. Also, she realizes that her costs for the Dearborn policy are both the annual premium and the lost investment revenue.

Homeowners Insurance

(Sample Solution)

Some of the assumptions that students would probably need to make and, therefore, state:

1. Investment Rate of Return. Students probably need to make some assumption as to what a reasonable rate of return is on the investment. They should state what rate they are using, explain how they arrived at that figure, and have it be the one that was used in their calculations. A creative group may actually research past history of rates and use a variable rate throughout their solution. The examples that follow later in this document use an 8% rate.

2. Interest Calculation. Students need to decide whether they will use simple interest, compound interest, or disregard interest completely. It seems as if an analysis using compound interest would be preferred and earn a better score. Also, are they using interest calculations in pricing both policies or just one of them?

3. Changing Insurance Companies? Did they consider the possibility of switching companies at some particular time. (NOTE: Switching policies complicates the problem more than one might think if one is considering compounding, also. The cumulative amount that is used for compounding purposes would come from the original column which would mean that a reconstruction of data from that point would be necessary.)

4. Change in Premium. The examples that are provided below assume that neither company is going to change its rates.

5. Would the Client Invest the Money? An assumption would be made that the funds that are being calculated as earning investment interest would actually be invested.

Analysis using cost per year and compounding the investment interest on each policy:

The cost per year for the Fancy Insurance Company policy would be:

Year 1 .08 times \$3000 is \$240

Year 2 .08 times (\$3000 + 240) is \$259.20.

Year 3 .08 times (\$3000 + 240 + 259.20) is \$279.94.

Year 4 .08 times (\$3000 + 240 + 259.20 + 279.94) is \$302.33.

The data for the first twenty-five years are given in the table below.

The cost per year for the Dearborn Insurance Company policy would be:

Year 1 \$300 premium plus .08 times \$300 for lost investment revenue is \$324.00

Year 2 \$300 premium plus .08 times (300+324) in lost revenue is \$349.92.

Year 3 \$300 premium plus .08 times (300+324+349.92) in lost revenue is \$377.91.

Year 4 \$300 premium plus .08 times (300+324+349.92 + 377.91) totals \$408.15.

The data for the first twenty-five years are given in the table below.

 Policy year Fancy Ins Dearborn Ins Policy year Fancy Ins Dearborn Ins 1 \$240.00 \$324.00 14 \$652.71 \$881.16 2 \$259.20 \$349.92 15 \$704.93 \$951.65 3 \$279.94 \$377.91 16 \$761.32 \$1027.78 4 \$302.33 \$408.15 17 \$822.23 \$1110.01 5 \$326.52 \$440.80 18 \$888.00 \$1198.81 6 \$352.64 \$476.06 19 \$959.04 \$1294.71 7 \$380.85 \$514.15 20 \$1035.80 \$1398.29 8 \$411.32 \$555.28 21 \$1118.60 \$1510.15 9 \$444.22 \$599.70 22 \$1208.10 \$1630.96 10 \$479.76 \$647.68 23 \$1304.80 \$1761.44 11 \$518.14 \$699.49 24 \$1409.20 \$1902.36 12 \$559.59 \$755.45 25 \$1521.90 \$2054.54 13 \$604.36 \$815.89

Analysis using cumulative cost and compounding investment interest on each policy:

The total cost for the Fancy Insurance Company policy after one year is \$3000 times .08 which is \$240. The total cost for the policy after two years is \$240 plus \$3240 times .08 which is \$499.20. The total cost for the first three years is \$499.20 plus .08 times \$3499.20 which is \$799.14. The total cost for the first four years is \$779.14 plus .08 times 3799.14 which is \$1081.47. A function that models this data would be C(x) = \$3000 (1.08)n minus \$3000. The data for the first twenty-five years are given in the table below.

The total cost for the Dearborn policy after one year would be the \$300 premium plus .08 times \$300 which is \$324. The total cost for the Dearborn policy after two years would be the \$600 in premium plus the \$24 in lost investment the first year plus .08 times \$624 for the lost investment the second year which is a total of \$673.92. The total cost for the Dearborn policy after three years is the \$900 in premium plus the \$24 in lost investment the first year plus \$49.92 in lost investment the second year plus .08 times \$973.92 which is \$1051.83. The total cost for the Dearborn policy after four years is the \$1200 in premium plus the \$24 in lost investment the first year plus the \$49.92 in lost investment the second year plus the \$77.91 in lost revenue the third year plus .08 times \$1351.83 which is \$1459.98. This sequence can be defined recursively by: S0 = \$ 324 and Sn + 1 = Sn + 300 + .08 * (Sn + 300).

The data for the first twenty-five years are given in the table below.

 Policy year Fancy Ins Dearborn Ins Policy year Fancy Ins Dearborn Ins 1 \$240.00 \$324.00 14 \$5811.58 \$7845.63 2 \$499.20 \$673.92 15 \$6516.51 \$8797.28 3 \$779.14 \$1051.83 16 \$7277.83 \$9825.07 4 \$1081.47 \$1459.98 17 \$8100.05 \$10935.08 5 \$1407.98 \$1900.78 18 \$8988.06 \$12133.88 6 \$1760.62 \$2376.84 19 \$9947.10 \$13428.59 7 \$2141.47 \$2890.99 20 \$10982.87 \$14826.87 8 \$2552.79 \$3446.27 21 \$12101.50 \$16337.03 9 \$2997.01 \$4045.97 22 \$13309.62 \$17967.99 10 \$3476.77 \$4693.65 23 \$14614.39 \$19729.43 11 \$3994.92 \$5393.14 24 \$16023.54 \$21631.78 12 \$4554.51 \$6148.59 25 \$17545.42 \$23686.32 13 \$5158.87 \$6964.48

Analysis using cost per year and compounding 11.11% APR on each policy:

The analysis is the same as the first example except the interest rate has changed.

 Policy year Fancy Ins Dearborn Ins Policy year Fancy Ins Dearborn Ins 1 \$333.30 \$333.33 14 \$1311.10 \$1311.19 2 \$370.33 \$370.36 15 \$1456.70 \$1456.86 3 \$411.47 \$411.51 16 \$1618.60 \$1618.72 4 \$457.19 \$457.23 17 \$1798.40 \$1798.56 5 \$507.98 \$508.03 18 \$1998.20 \$1998.38 6 \$564.42 \$564.47 19 \$2220.20 \$2220.40 7 \$627.13 \$627.18 20 \$2466.90 \$2467.09 8 \$696.80 \$696.86 21 \$2740.90 \$2741.18 9 \$774.21 \$774.28 22 \$3045.40 \$3045.72 10 \$860.23 \$860.31 23 \$3383.80 \$3384.10 11 \$955.80 \$955.89 24 \$3759.70 \$3760.08 12 \$1062.00 \$1062.08 25 \$4177.40 \$4177.82 13 \$1180.00 \$1180.08

In the table above the total cost for the twenty-five years is \$38,778.22 for the Fancy policy and \$38,781.71 for the Dearborn policy. 11 1/9% seems to be the Abreak even@ point for the two policies. Using this analysis, if the investment rate is less than 11 1/9% one should purchase the Fancy policy. If the rate is greater than 11 1/9% one should purchase the Dearborn policy. If the rate is 18%, the total cost for twenty-five years is \$185,005.88 for the Fancy policy and \$121,281.63 for the Dearborn policy.

Funded in part by the National Science Foundation and Indiana University 1995