You recall from class our discussion of the exponential function as a way of continuously multiplying a quantity by some constant factor. This is of course the way banks and other financial institutions compound interest on loans or savings accounts, so we'll investigate that in this lab.

Retrieve the Excel data file in the usual way from the other Lab 6 link.

Abraham Lincoln was born February 12, 1809, almost 200 years ago. Suppose Abe's parents had invested $20 (a large but not unattainable sum at the time) in a savings account with an interest rate of 5% (typical of bank CD's not long ago) compounded annually.

What would that savings account be worth today?

Set up the appropriate formulas in columns A and B to calculate this quantity, and be sure to write it down on a separate sheet to hand in. (You should also hand in the first sheet of your Excel worksheet, the answers to the questions below, and the graph mentioned in question 1 below.)

Now calculate the annual percentage rate (APR), often quoted by banks along with the annual percentage yield (APY, in this case 5% or 0.05). The APR is defined as the "rate" that goes in the exponential to give you the same growth. In other words, adding 5% (APY) per year is equivalent to calculating the growth of the account with the formula $20 e ^APR·t where $20 is the amount you start with, APR is the annual percentage rate (expressed as a decimal, so 4.72% = 0.0472), and t is the number of years that have passed. In general, the APR will be slightly less than the APY. Excel's canned function for the exponential function is "exp()". Put your guess for the APR in cell D4, and reference it in your formula for column C as $D$4 so that you can change it to get the right value. To check how close you come, go to cell D5 and calculate the difference between the last figures in columns B and C. When this difference becomes less than $500, you are pretty close to the correct APR.

Now let's finance a house. Suppose the price is $100,000 and you make a 10% down payment.

1) If you finance the balance at an annual percentage rate of 7% compounded monthly, what monthly payment must you pay to pay off the loan in 30 years (360 months)? Remember, to calculate the new balance each month, you first add the interest, then you subtract the payment. In the column marked "Interest", keep track of the amount of interest contained in your payment each month. Make a graph of that interest amount as a function of the number of months from beginning of the loan until you pay it off.

2)  How much are you really paying for the house, i.e., what is sum of all payments including down payment?

3) How much earlier could you pay off the loan if you made a $20,000 down payment but the same monthly payment?

4) How much earlier could you pay off the loan if you made a $10,000 down payment but paid an extra $100 each month?

Turn in the answers to these questions on a separate sheet, and turn in the first page of your Excel sheet, just as you left it after the last calculation. Don't forget the graph.