Suppose you have $1,000 to invest for a period of up to 20 years, and you are trying to decide whether to put your money in money market, bond, or stock mutual funds. Over the past 10 years money market funds have had average returns of 5% per year, bond funds have had average returns around 8% per year and stock funds have had average returns of 16% per year. Of course there are varying degrees of risk connected with each of these types of funds. In order to weigh risk versus return over the long run, we want to estimate and compare the average return for each of these investments, assuming similar growth rates, over the next 20 years.

Part 1: Deriving an interest rate formula

Part 2: Simulating the Growth of Money

Estimate the worth of a $1,000 investment earning 5% interest, compounded yearly, after 20 years.

Do the same for $1,000 earning 8% and 16%

Derive equations for the value of $1,000 invested in accounts paying 5%, 8% and 16% interest rate compounded yearly.

Use the parametric graphing capabilites of your graphing calculator to simulate the growth of these investments simultaneously. Use your estimated total worth to set the y-viewing window. Adjust as needed. Label your graphs (if your calculator is capable).

After you have constructed these graphs, estimate: (1) the amount earned at each of these investment rates at the end of 10 years, and (2) the number of years of growth needed to reach an amount equal to half the total worth achieved after 20 years.

Trace each of your graphs, first to the half-way points in both time and in accumulation, and then all the way to 240 months to determine the accuracy of your above estimations. Discuss your observations, as well as the methods and accuracy of your estimations. What can you conclude?

Part 3: Changing the Compounding Period

Estimate the value of the above investments if the interest was compounded monthly instead of yearly. Discuss how you made these estimations

Derive equations for the growth of $1,000 at 5%, 8% and 16% compounded monthly.

Again, use parametric graphing to simulate the growth of this money on your graphing calculator.

Notice that the graphs do not appear to be "growing" at first. Explain why.

Trace your graphs to determine the accuracy of your estimations.

Discuss the effect of compounding on growth.

Extension 1: Extreme Compounding Cases

Christopher Columbus, upon arriving in 1492, put one dollar into an account earning 5% interest per year in the New World bank. In 1992, 500 years later, his descendants found documentation of this account and went to the bank to claim their inheritance. Unfortunately, the documentation did not contain information about how the money was to be compounded. The bank argued that the old accounts compounded money every 500 years. Some of Columbus' relatives claimed their forefather would have only opened an account with yearly compounding, others believed he would have argued for monthly compounding, yet others asserted that he would have negotiated for daily compounding.

Estimate how much the account would be worth after growing at 5% yearly for 500 years at each of these three compounding periods.

Alter the expressions derived in Part 1 of this activity to fit these three situations, and use your calculator to determine the correct amounts. Compare your estimations with the correct amounts. Discuss your results.

Extension 2: The General and Limiting Cases

Use your previously derived equations to derive a general equation for the amount of accumulation (or return) R, of an initial amount A, invested at a yearly interest rate i, for t years, compounded C times per year. Find the limit of this expression as C increases without bound; that is find the return for an investment with continuous compounding.

Extension 3: Comparing Graph Types

The activities above asked you to simulate the growth of money by graphing functions in parametric form. Now enter the monthly compounding functions in your calculator in y = form, and draw their graphs. Compare and contrast the two forms and their graphs.