**Document ID**: 01_19_99_1

**Date Received**: 1999-01-19 **Date Revised**: 1999-04-07 **Date
Accepted**: 1999-04-12

**Curriculum Topic Benchmarks**: M1.41., M1.4.8, M3.4.1, M8.4.12,
M1.3.11, M8.3.11

**Grade Level**: \[9-12\] High School

**Subject Keywords**: social security, algebra, problem-solving process,
computation, functions

**Rating**: moderate

**Grandpa's Social Security**

**By**: Alan L Jones, State University of New York at Binghamton, 3717
Wildwood Drive, Endwell NY 13760 e-mail: AlanLJones@aol.com

**From**: The PUMAS Collection http://pumas.jpl.nasa.gov

©1999, California Institute of Technology. ALL RIGHTS RESERVED. Based on
U.S. Gov\'t sponsored research.

This simple example shows how algebra can be useful in the real world:
**Should Grandpa start receiving his Social Security Benefits at age 62
or should he wait until age 65?**

**Background:** Most retired Americans are eligible to start receiving
Social Security benefits at age 62. However, if a retiree waits until
age 65 to receive benefits, the monthly check is larger.

**Exercise:** Determine how long a retiree has to live until the total
benefits are the same whether he retires at age 62 or age 65.

**Presentation:** Set up a situation for the students. Tell each one
that his or her grandfather has come to the student and asked if he
should take his retirement benefits when he reaches 62 or should he wait
until he reaches 65. He explains that he doesn\'t need the money to live
on at present but wants to have the maximum amount of money when he dies
to pass on to his grandchildren (meaning you!). He wants to know how
long he has to live to make it beneficial to wait until 65 to start
taking his Social Security payments.

Explain to each student roughly how Social Security works. The Federal
Government keeps track of how much you earn each year of your life. They
also collect Social Security payments from you and your employer each
year. When you approach retirement, you can have the Social Security
Administration tell you how much you will earn each month based on your
work history. The base amount is computed assuming you will retire and
begin receiving benefits at age 65. If you decide to retire at age 62,
this amount is reduced by 20%.

**Brute Force Approach:** The student might create a table of how much
Grandpa will have at the end of each year based on his year of starting
the benefits. Let\'s assume his payments will be \$1000 per month at age
65. We assume the students will then know that the payments will be
\$800 a month if he starts receiving them at age 62. Then the total
payments for the year will be 800x12 = \$9600 in the one case and
1000x12 = \$12,000 in the other case.

End of Start Total Start Total

Year at 62 at 65

/year /year

===== ====== ====== ====== ======

62 9600 9600 0 0

63 9600 19200 0 0

64 9600 28800 0 0

65 9600 38400 12000 12000

66 9600 48000 12000 24000

67 9600 57600 12000 36000

68 9600 67200 12000 48000

69 9600 76800 12000 60000

70 9600 86400 12000 72000

71 9600 96000 12000 84000

72 9600 105600 12000 96000

73 9600 115200 12000 108000

74 9600 124800 12000 120000

75 9600 134400 12000 132000

76 9600 144000 12000 144000

77 9600 153600 12000 156000

78 9600 163200 12000 168000

So we see that if Grandpa lives to the end of his 76th year, or in other
words, to his 77th birthday, then he will have received the same amount
of money regardless whether he started receiving at age 62 or age 65. If
he lives longer than 77, he does better by delaying the start of his
Social Security until age 65.

**Have Students Survey Folks on Social Security**: Instead of taking the
figure of \$1000 in the above, have each student talk to someone on
Social Security (a grandparent, an aunt, uncle, etc.) and find out what
their monthly Social Security payment is and have each student use this
figure or, perhaps, take all the amounts and use the mean.

**Plot the Two Cases:** Have the students plot the above two cases on a
plotting calculator or on paper and have them observe where the two
lines cross. This is the \"break-even\" point where Grandpa will do
equally well regardless of when he starts taking his benefit.

**Use a Symbol for the Yearly Amount**: You could now ask the students
to redo this table but make the amount Grandpa receives each year equal
to the symbol Y. For example:

End of Start Total Start Total

Year at 62 at 65

/year /year

===== ====== ====== ====== ======

62 0.8Y 0.8Y 0 0

63 0.8Y 1.6Y 0 0

64 0.8Y 2.4Y 0 0

65 0.8Y 3.2Y Y Y

66 0.8Y 4.0Y Y 2Y

\....

Let them determine when the break-even age is reached. Many students
will realize right away that it doesn\'t matter what the amount is, the
time to reach the break-even point is the same.

**Doing the Problem with Algebra:** Make an equation of the situation.
Let X be Grandpa\'s age and let Y be the amount of money received each
year. Then, if Grandpa takes his benefit at age 62, the amount he will
have received at age X for any age past 62 is: 0.8Y(X - 62)

The 0.8 reflects that he only gets 80% of the benefit he would have
received at age 65 if he had waited. If he had waited, the amount
received at any age X is: Y(X - 65)

Since we wanted to find at what age these benefits will be the same, set
these two expression equal to each other: 0.8Y(X - 62) = Y(X - 65)

If the students have had algebra, they should recognize that the Y can
be divided into both sides of the equation resulting in: 0.8(X - 62) =
X - 65

So, as we said above, the amount received per month does not matter. And
solving for X yields: X = 77

So, just as above, if Grandpa lives to 77 he gets the same amount.

Wow! This was a lot easier than constructing that table!

What if Grandpa wants to receive his benefits at age 62 and 6 months?
The amount that a benefit is reduced is not a flat 20% as stated above.
What the Social Security Administration actually does is reduce the
amount by 5/9 of one percent for each month before 65 that a person
begins receiving a benefit. Ask the class to compute what the break-even
rate is for some other starting age such as 62 years and 6 months.

**A Complication:** All of the above assumes that the payments do not
change each year. However, they do change. Each year the government
computes the cost of living and adjusts the Social Security payments
accordingly. Ask the student how the above would change if there was a
three percent cost of living adjustment each year.

**A Further Complication:** What if Granddad is very clever and figures
he can earn 10% per year by taking this payment and investing it. So,
with a cost of living adjustment of 3% and an assumed 10% return on
investments, now what age does he have to live to for a break-even
situation?

**What is Grandpa\'s life expectancy?** Another avenue of exploration
would be to ask the students what Grandpa\'s life expectancy is knowing
that he has lived to age 62. Help them find tables that give the average
life expectancy based on how long a person has lived. They should
appreciate the fact that a person\'s life expectancy at birth is quite
different from their life expectancy at other ages.

**And What About Grandma?** The students will find in looking up
actuarial tables that women have a longer life expectancy than men.
(They probably know this already; they probably don\'t know the
numbers.) What should Grandma do about her social security? Grandma has
the option of taking any social security she has earned or of taking
one-half of the amount the Grandpa gets. (If the amount she is entitled
to is more than Grandpa\'s, then their roles are reversed and Grandpa
has the option of taking his or one-half of the amount that Grandma
gets.) If Grandpa dies first, she continues to receive the same amount.