Algebra Project Assignment
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 Investmentsr
C(t) = [P {1+(r/n)}n {1 - (1 + (r/n))nt}] / [1 - {1 + (r/n)}n]
If you would like the derivation of this formula, please send me an email and I will send you the information.
Your answers should be typed and in a single document. If you would like to attach your work, scan it and add it to the end of your typed document. You can either attach a Word file or a pdf file. Many word processing programs will save a document as a pdf file if you select Save as and look for file types.
1) How much will you have accumulated over a period of 30 years if, in an IRA which has a 8% interest rate compounded monthly, you annually invest:
a. $1
b. $1000 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.)
2) How much will you have accumulated, if you annually invest $4000 into an IRA at 10% interest compounded quarterly for:
a. 1 year
b. 15 years
c. 30 years
d. How long will it take to earn your first million dollars?
3) Now you will plan for your retirement. To do this we need to first determine a couple of values.
a. 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. You can choose a number you think you can afford on your life circumstances or you can dream big.
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. 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.) List your P, r, and n to earn points for this question.
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. You need to be able to show me what you used for r, n, and P so that I can calculate your answers. Plug in those values into the formula and simplify the equation.
c. Graph this function from t = 0 to t = 50. Ways to show graphs:
• Excel
• Hand draw, take a pic with phone and import it into your document as a picture.
• Online graphing calculator program (try googling free graphing calculators or use one listed in the Tech websites from Module 1)
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). You need to say how old you are if you are retiring when you are 65 or tell me how long until you retire.
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 $30,000 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 $30,000, how much would you want to have each year to live on? Now using that value, repeat parts (i) and (j) again. You need to state what you would want to live on and it needs to be something besides $30,000.
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 will need to invest each year to attain your financial goals.
l. Finish by summarizing what you have learned in the entire project and consider setting a goal towards saving for retirement. (Your answer should be in complete sentences free of grammar, spelling, and punctuation mistakes.) This should be a paragraph not one sentence.