YOUR RETIREMENT PROGRAM
 AN INVESTIGATION INTO YOUR FINANCIAL FUTURE
 																					  BY
 														  JAMES S. KRALIK
 													WESTVILLE HIGH SCHOOL
 														 WESTVILLE, ILLINOIS

The Basics

Exponentiation

Recall that the expression "bn" (read "b to the nth power") represents an operation called exponentiation, or powering.  The variable b is called the base, n is called the exponent, and the expression bn is called a power.  So, 
53 = 5 x 5 x 5,
(-1/2)4 = (-1/2)(-1/2)(-1/2)(-1/2),
(1.05)2 = (1.05)(1.05),
and, in general, bn is b multiplied by itself n times.

Compound Interest & Return on Investment

We can apply the concept of exponentiation to the subject of interest rates and return on investment.  Suppose a person deposits $1000.00 (the principal) in a bank that pays interest at an annual rate of 4%.  Assuming that no money is withdrawn during the course of a year, at the end of the year the amount of money in the account will be:
1000 + .04(1000) = 1000(1+.04)
                = 1000(1.04)
                = 1040
Notice that to find the amount of money in the account at the end of one year, you did not have to add the interest to the principal.  You simply could have multiplied the principal by 1.04.

Now suppose you leave all of your money in for a second year.  At the end of the second year you will have:
1040(1.04) = [1000(1.04)](1.04)
          = 1000[(1.04)(1.04)]
          = 1000(1.04)2
          = 1081.60
Again, notice that we did not have to calculate the interest separately and add it to the principal.  We can simply calculate the amount of money in the account after n years by multiplying the principal by 1.04 n times.   

This leads us to a simple form of the compound ("interest on interest") interest formula.  Let P be the amount of money invested at an annual interest rate of r, compounded annually.  Let A be the total amount after t years.  Then:
																				A=P(1+r)t

The Individual Retirement Account

The Individual Retirement Account is one of the best investments available today.  Your annual contributions ($2000.00 per year maximum for an individual) are tax deductible in most cases, and your investment earnings are allowed to accumulate tax free until you begin to withdraw your money when you retire.  This is a tremendously powerful program that is worth investigating.

Assumptions

When trying to see into the future and making projections about the value of an investment twenty or thirty years from now, the assumptions that you make about your investment at the beginning will determine the projected value of your account when you retire.  Your assumptions generally fall into one of three categories: 1) the amount of your annual contribution;  2) the rate of return, or interest rate, of your investment; and 3) the conditions, or terms, of your investment, such as how often the interest on your investment is compounded.

For example, if you assume that you will put away $10,000.00 each year for twenty years and that your rate of return is 15% per year, you will have over ONE MILLION DOLLARS at the end of twenty years.

f[x_]:=10000*1.15^x
z=Sum[f[x], {x,1,n}];
t=Table[{30+n, z},{n,1,20}]//Round

Sounds real good!  But, ask yourself, "Self, how realistic is this projection?"  Will you really be able to sock away ten thousand dollars each and every year?  Is a 15% rate of return a realistic assumption in this day and age?  During the early eighties, a lot of investments were sold because salesmen were able to show some very attractive projections based on 12-18% rates of return.  Needless to say, in these days of 3% savings accounts, a lot of these invest-
ments have shown some disappointing results.  So, if you really want a realistics projection of how much money you will have, start with some realistic assumptions!

Your IRA

Let's make our first projection of your IRA.  We'll need some assumptions.  Let's assume that you partied all through your twenties, and spent every dime you earned.  How much did you party?  A LOT.  Let's just say that you did your share to fuel the nation's economic recovery.  But you are thirty now, and it's time to get serious.  Let's assume that you will manage to make the maximum allowable contribution, $2000 for an individual.  What will you invest in?  A good choice for many people is a stock market mutual fund that attempts to match the performance of the stock market.  The average annual rate of return of the stock market for the last fifty years has been 10%.  So let's be conservative and assume an annual rate of return of 8%.  How much money will you have when you retire at the age of 65?

Answer

Think about what is happening.  The first year, you put in $2000.  That $2000 will give an 8% rate of return, compounded annually for 35 years.  So, according to our formula, the total amount A for that first $2000 will be:

2000(1.08^35)

The next year (age 31) we put in $2000, and that second $2000 will return 8%, compounded annually, for 34 years.

2000(1.08^34)

So the total return of the second $2000 will be added to the total return of the first $2000, and the total return of the third $2000 will be added to that, and so on.  Well, we can automate that quite easily with Mathematica.

Clear[f,x,v,t]
f[x_]:=2000*1.08^x
v=Sum[f[x], {x,1,n}];
t=Table[{30+n, v},{n,1,35}]//Round

The first line cleared our variables.  The second line defined our function.  The third line added all of my totals for n years.  Line four specifies n to be 35 (age 30 to age 65) and will display our annual cumulative totals.  The following line will display the output in a pretty table:

t//TableForm

This may be pretty, but we don't need it to plot our results.  Let's plot t and give our plot a name.

a=ListPlot[t,Prolog->PointSize[.01],
AxesLabel->{"Age","Dollars"},
Epilog->Text["[a]",{62,350000}]]

This is a pleasant looking graph that shows a nice, steady growth.  So it looks like you will have a good deal of money when you hit 65.  Let's compare this growth with the amount of money you actually contributed.

Clear[v]
g[v_]:=2000+2000*v
s=Table[{30+v, g[v]},{v,0,34}]

b=ListPlot[s, AxesLabel->{"Age", "Dollars"}, 
	Epilog->Text["[b]",{60,100000}],
	PlotRange->{0,360000}]

A different kind of growth.  Let's look at a and b together.

Show[a,b]

Well! Put in a little, get a lot!

What Do You Think?

Write a few paragraphs about the growth of your retirement fund.  Some of the points you should address are:
-What kind of growth does your retirement fund have?  What kind of growth does the plot of your annual contributions have?
-During what years do the plots seem to run together?  About what year do the plots begin to diverge and your IRA really begins to take off?  What is the reason for this?
-What were your total annual contributions?  What was the total value of your IRA at age 65?  What is the difference between these two figures?  What is the source of that difference?  Are you suprised at the amount of money that you will have at age 65?
-Comment about the relative importance of the annual contributions versus the annual interest earned through the years.  For example, was your annual contribution at age 60 as important as your annual contribution at age 30?  Why?  
-What conclusions can you draw from this example?
-Finally, at age 65, do you think that your relatives would be nicer to you if they knew that you had this much money?

Starting Early

Does it make a big difference if we start early?  For instance, does it matter a whole lot if we start our IRA at age 21 instead of age 30?  Try it out.  Assume that you will begin your IRA at age 21, that you will contribute $2000 each year to age 64 (By the way, is this a reasonable assumption?), and that your annual rate of return will be 8%.  Write a program and produce a table that shows the growth of your IRA each year until retirement.  Plot your results and call your plot "c".  Make another plot comparing the results starting at age 21 and starting at age 30.  Finally, write a paragraph in which you describe your conclusions.

Answer:
First, clear some variables.  Next, some make some minor changes to the function we used before.

Your code should look something like this:

Clear[f,x,c,p,w,n]
f[x_]:=2000*1.08^x
p=Sum[f[x], {x,1,n}];
w=Table[{21+n, p}, {n,1,44}]//Round

c=ListPlot[w, Prolog->PointSize[.015], 
	AxesLabel->{"Age", "Dollars"}, 
	Epilog->Text["[c]", {60,650000}], 
	AxesOrigin->{21,0}]

Show[c,a]

Well, what do you think?  Is the difference between the two totals at age 65 enough to make you want to start early?  Why or why not?

The Power Of Compound Interest

Here's an interesting assumption to make.  Suppose that you began your IRA at age 21.  You make annual contributions of $2000.00 each year, with a rate of return of 8%.  You make your annual contributions for ten years, and then stop.  You never put another cent into your IRA for the rest of your life.  However, you leave your IRA alone, and allow your balance to grow to age 65.  How much money will you have when you retire?  

Make a table and a plot that describes the growth of your account for the first ten years.  Call you plot "d".  Next, make a table and plot that describes the growth of your IRA from 31 to 65.  Call this plot "e".  Show[d,e] and name this plot "f".  Finally, Show[f,a].  Compare your results.  Write a few paragraphs describing your conclusions from this example.

Answer
First, use your function to calculate your growth during the ten years you will make an annual contribution.  Then, multiply that balance by 1.08 to the 34th power.

Your code looks like this:

Clear[f,x,d,e,f,p,q,r,s,n]
f[x_]:=2000*1.08^x
q=Sum[f[x], {x,1,n}];
r=Table[{21+n, q}, {n,1,10}]//Round

d=ListPlot[r, Prolog->PointSize[.015], 
	AxesLabel->{"Age", "Dollars"}, 
	Epilog->Text["[d]", {30,35000}], 
	AxesOrigin->{21,0}]

s=Table[{31+p,31291*1.08^p},{p,1,34}]//Round

e=ListPlot[s, Prolog->PointSize[.015], 
	AxesLabel->{"Age", "Dollars"}, 
	Epilog->Text["[e]", {60,400000}], 
	AxesOrigin->{21,0}]

f=Show[e,d, Epilog->Text["[f]", {60,400000}]]

Show[f,a]

Hey!  What's going on here?  In our scenario beginning with age thirty, we made total annual contributions of $70,000.  In our new scenario, what was the total of our annual contributions?  Which scenario ourperformed the other?  How could this be?  Please write a few paragraphs describing the two scenarios, tell me what happened, and tell me what you think about all this.

Your Retirement Plan

For your final project you will write a plan for your own retirement.  Begin by writing down your assumptions, and describing your rational for those assumptions.  For example, you don't have to limit yourself to a $2000.00 annual contribution to an IRA.  You may decide that you would be able to save more (or less) than that, and there are other investments besides an IRA.  You may feel that you might not be able to save very much fresh out of high school or college, but you would be able to increase your annual contributions in your thirties or forties.  On the other hand, what about paying for a house, or supporting a family?  Describe how you think you would balance the need to start saving early with the desire to buy things now!  What about rates of return?  Do you think you are an investment whiz who could do better (or worse!) than eight percent?

For your projection, produce a table showing the value of your account each year until age 65, and a plot of those values.  Do a comparison plot of your annual contributions only.  

Finally, write a few final paragraphs describing any final thoughts or conclusions you have about this lesson.  In particular, talk about the likelihood that you will actually follow your plan.