# College algebra project saving for the future

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.

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