College Algebra Project Saving for the Future

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College Algebra Project

Saving for the Future In this project you will investigate compound interest, specifically how it applies to the typical retirement plan. For instance, many retirement plans deduct a set amount out of an employee’s paycheck. Thus, each year you would invest an additional amount on top of all previous investments including all previously earned interest. If you invest P dollars every year for t years in an account with an interest rate of r (expressed as a decimal) compounded n times per year, then you will have accumulated C dollars as a function of time, given by the following formula.

Compound Interest Formula, with Annual Investments:

C t

P r

n nt

n ( ) =

+ F HG I KJ − + F HG I KJ

L N MM

O Q PP

− + F HG I KJ

1 1 1

1 1

On the following page I will derive this formula to give you a broader

understanding of where it came from and how it is based upon the

single deposit compound interest formula.

If you invest P dollars every year for t years at an interest rate r, expressed as a decimal, compounded n times per year, then you will have accumulated the cumulative amount of C dollars given by the formula derived bellow: Each annual investment would grow according to the compound interest formula:

A t P r

( ) = + F HG I KJ1

Here A is the amount that each deposit would be worth at the end of t years. Thus the first deposit of P dollars would draw interest for the full t years, the second deposit would only draw interest for t-1 years, the third deposit would only draw interest for t-2 years… and the last deposit would only draw interest for a single year. Thus we need to add up each deposit and their respectively gained interest values, resulting in the following:

C t P r

n P

nt n t n t n t n n

( ) = + F HG I KJ + + F HG I KJ + +

F HG I KJ + +

F HG I KJ + + +

F HG I KJ + +

F HG I KJ

− − −

1 1 1 1 1 1

1 2 3 2 1b g b g b g b g b g …

= + F HG I KJ + + F HG I KJ + +

F HG I KJ + +

F HG I KJ + + +

F HG I KJ + +

F HG I KJ

− − −

P r

n P

nt nt n nt n nt n n n

1 1 1 1 1 1

2 3 2 1

…

b g b g

= + F HG I KJ + F HG I KJ + + F HG I KJ + + F HG I KJ + + +

F HG I KJ + +

F HG I KJ

L N M M

O Q P P

− − − − −

P r

n nt n nt n nt n n n n n

1 1 1 1 1 1

2 3 2 1

…

b g b g

= + F HG I KJ + F HG I KJ + + F HG I KJ + +

F HG I KJ + + +

F HG I KJ +

L N MM

O Q PP

+ F HG I KJ =

− − −

P r

n n t n t n t n

1 1 1 1 1 1 1 1

1 2 3 0( ) ( ) ( )

… , note:

= + F HG I KJ + + F HG I KJ + + F HG I KJ + + F HG I KJ + + +

F HG I KJ

L N M M

O Q P P

−

P r

n n n n n t

1 1 1 1 1 1

1 2 3 1b g b g b g b g …

= + F HG I KJ + F HG I KJ

L N MM

O Q PP=

= −

∑P r

n n

k t k

1 1 1

= + F HG I KJ + F HG I KJ

L N MM

O Q PP=

= −

∑P r

n n

k t k

1 1 1

A geometric series of the form: , with , and S ar a

r r a P

n r

n t

k t t

n n

= = −

− = + F HG I KJ = +

F HG I KJ

−

∑ 1

1 1 1 1 1c h

Thus C t

P r

n t

n nt

n : ( ) =

+ F HG I KJ

− + F HG I KJ

L N MM

O Q PP

F H GG

I K JJ =

+ F HG I KJ − + F HG I KJ

L N MM

O Q PP

− + F HG I KJ

1 1

1 1 1

1 1

EXAMPLE: If you invest $1200 every year (P = 1200) for 3 years (t=3) at an interest rate of 5% (r = 0.05) compounded weekly (n = 52), then the first year’s investment of $1200 would earn interest for 3 years, but then the next year, the next investment of $1200 would only earn interest for 2 years, and then the final investment of $1200 would only earn interest for 1 year. This lends itself to the following:

C t P r

n P

nt n t n t

( )

( ) . . .

( )

= + F HG I KJ + +

F HG I KJ + +

F HG I KJ

= + F HG

I KJ + +

F HG

I KJ + +

F HG

I KJ

− −

1 1 1

3 1200 1 0 05

52 1200 1

0 05

52 1200 1

0 05

1 2

52 3 52 2 52 1

1st Investment 2nd Investment 3rd Investment

= $1,394.10 +$1,326.14 +$1,261.50

= $3,981.74

b g b g

Now to verify the previously derived formula:

C t

P r

C t

n nt

n ( )

( )

. .

( )

+ F HG I KJ − + F HG I KJ

L N MM

O Q PP

− + F HG I KJ

+ F HG

I KJ − + F HG

I KJ

L N MM

O Q PP

− + F HG

I KJ

1 1 1

1 1

1200 1 0 05

52 1 1

0 05

1 1 0 05

52 52 3

$3,981.74

As you can see we got the same answer. Now although it was not difficult to do the problem the long way with only three years, when t gets large, the formula simplifies the work quite a bit. However, you have to be sure not to round until the very end where you round to the nearest cent. (So be sure to keep as many decimal places as possible until the end, as you will be taken off for rounding before then.) The entire project deals with annual deposits so you will be using this formula to answer the following questions:

The number in parenthesis is how much each part is worth for a total of 70 points. (10) 1) How much will you have accumulated over a period of 20 years if, in an IRA which has a 10% interest rate compounded monthly, you annually invest:

a. $1 b. $100 c. $20,000 d. Part (a) is called the effective yield of an account. How could Part (a) be used to determine Parts (b) and (c)? (Your answer should be in complete sentences free of grammar, spelling, and punctuation mistakes. )

(10) 2) How much will you have accumulated, if you annually invest $3200 into an IRA at 12% interest compounded quarterly for:

a. 1 year b. 10 years c. 30 years d. How long will it take to earn your first million dollars?

(40) 3) Now you will plan for your retirement. To do this we need to first determine a couple of values. How much will you invest each year? Even $50 a month is a start ($600 a year), you’ll be surprised at how much it will earn. The typical example of a retirement investment is an I.R.A., an Individual Retirement Account, although other options are available. However, for this example, we will assume that you are investing in an I.R.A. (for more information see: http://en.wikipedia.org/wiki/Individual_Retirement_Account ) earning 8% interest compounded annually. (This is a good estimate, basically, hope for 10%, but expect 8%. But again this is just one example; I would see a financial advisor before investing, as there is some risk involved, which explains the higher interest rates.)

b. Determine the formula for the accumulated amount that you will have saved for retirement as a function of time and be sure to simplify it as much as possible. c. Graph this function from t = 0 to t = 50. d. When do you want to retire? Use this to determine how many years you will be investing. (65 years old is a good retirement-age estimate) e. Determine how much you will have at retirement using the values you decided upon above. f. How much of that is interest?

g. Now let’s say you wait just 5 years before you start saving for retirement, how much will that cost you in interest? How about 10 years? How about just 1 year? Now you need to consider if that is enough. If you live to be 90 years old, well above average, then from the time you retire, to the time you are 90, you will have to live on what you have in retirement (not including social security). So if you retired at 65, you will have another 25 years where your retirement funds have to last. h. Determine how much you will have to live on each year. Note, we are neither taking into account taxes nor inflation (which is about 2% a year). Let’s look at this from the other direction then, supposing that you wanted to have $75,000 a year after retirement. i. How much would you need to have accumulated before retirement? j. How much would you need to start investing each year, beginning right

now, to accumulate this amount? A “short-cut” to doing this is to first compute the effective yield at your retirement age, then divide this amount into Part (i). This is the amount you well need to invest each year.

k. That was just using $75,000, how much would you want to have each year

to live on? Now using that value, repeat parts (i) and (j) again. Your answer to (k) would work, if you withdrew all of your retirement funds at once and divided it up. However, if you left the money in the account and let it draw interest, it is possible that the interest itself would be enough to live on, or at the very least if you had to withdraw some of the principle, the remaining portion would still continue to earn interest. Essentially, what you have found is the upper bound for the amount of money that you well need to invest each year to attain your financial goals. Finish by summarizing what you have learned in the entire project and consider setting a goal towards saving for retirement. (10 points)(Your answer should be in complete sentences free of grammar, spelling, and punctuation mistakes.)