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.
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