# Interest Calculator Compound Investment by npq16003

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```									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
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

Funded in part by the National Science Foundation and Indiana University 1995
Life Insurance and Investment Counseling
Sample Solution

Policy   Annual    Contract    New Money      11% of      Amount of    Actual Cost      Death        Actual
Year    Premiu     Fund          into       Previous      Premium      of "Term"     Benefit of    Cost of
m                    Contract     Balance         into         Part of    the "Term"      "Term"
Fund                      Contract     Insurance     Insurance    Insurance
Fund
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

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

Funded in part by the National Science Foundation and Indiana University 1995
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      Annual        Contract        Cash            Death
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

Funded in part by the National Science Foundation and Indiana University 1995
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 benefitthere 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.

Funded in part by the National Science Foundation and Indiana University 1995
Life Insurance and Investment Counseling
Background Unit 1Compound 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                   Interest Earned                             New Balance
(Years)
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                   Interest Earned                             New Balance
(Years)
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).

Funded in part by the National Science Foundation and Indiana University 1995
Life Insurance and Investment Counseling
Background Unit 2Life 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
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   Cost of Death     Cost of     Total    Interest on      Amount in         Death        Cash
Year       Benefit      Investmen   Premium    Investment      Contract Fund      Benefit      Value
t
1          \$500          \$100       \$600      \$0*.10=\$0      \$100+\$0=100      \$100000+\$100    \$100
= \$100100
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