Objective:
In this lab we will compare the yield from a savings account under different conditions (different interest rates, compounding at different intervals). Along the way, you will become familiar with recursively-defined functions.
Suppose that your long-lost great-great-aunt Helen had been in Philadelphia on July 4, 1776. She became so enthusiastic about the prospects for a new nation and her hopes for its long existence that she invested $1 with a local banker to gain interest at the rate of 3% per year, compounded annually, until July 4, 2020. At that time the entire sum was to go to one of her descendants. The particular formula for who is to receive this money is intricate, as befits a lawyer, but the upshot is that you are the lucky one. You and your lawyer, who established beyond question that the money is yours, will split the money at that time.
Enter your names, one as heir and one as lawyer. Write a paragraph about how you plan to divide up the money.
F Mathcad tip: If regions start to overlap, you can choose Separate Regions from the Format menu. To insert blank lines into a document, to give yourself more room to write, just press the Enter key.
We'll let k be a variable that designates, on any given July 4, the number of years the account has been in existence. Define k to vary between 0 and an appropriate upper limit.
Finally, enter a formula to describe how to obtain Ak+1, the amount at the end of k+1 years, from Ak, the amount in the fund after k years. Your Mathcad formula should have the form
Ak+1 : = something involving Ak.
Your definition of Ak is an example of a recursive formula, one that allows you to carry out a calculation in a number of steps, one step at a time.
Once you have typed in the formula and pressed Enter, Mathcad computes all those amounts for you. It now knows how much money is in the account after one year, after two years, and so forth -- and you can ask it to display those amounts.
Remember,
- k keeps track of how many times you have computed the interest, and
- Ak keeps track of how much money you have in the bank at that time.
Ask Mathcad to display A1.
Record your answer here in text, so it will not be lost when you make changes
in the worksheet later: When $1 is invested at 3%, the amount in the year 2020 is...
2. Suppose your ancestor had been twice as enthusiastic and had invested $2. Before you perform any calculations, make a conjecture about the amount your account would be worth in the year 2020 and write your guess here:
Modify the worksheet above to check your conjecture. (Just go back up and change the
values of some things.)
3. Now suppose that your ancestor had only invested $1, but had obtained a rate of 6%. Conjecture how much your account would be worth in the year 2020:
Were you surprised? Write a paragraph to explain how and why the answers to Questions 2 and 3 differ.
4. Let's return to the original problem of $1 invested at a rate of 3% per year. Now assume that the interest is compounded twice a year. (So you get 1.5% at the end of six months, and then another 1.5% at the end of the year.) Fill in the values below to find out how much this would produce in the year 2020.
(How many times is the interest computed per year? Call that variable m. So how many times is the interest computed? How much interest do you get each time?)
5. We wonder what will happen if the interest is compounded more and more often --
3 times per year, 4 times per year, etc. We want to make a table of the values of the account in the year 2020 as a function of the number m of times per year the interest is compounded, for m = 1 to 10. For each value of m, go back and modify the worksheet to calculate the corresponding value of the account in the year 2020. In order to avoid roundoff error, you will probably find it helpful to write your interest as 
, for instance, instead of figuring out the interest yourself. For each m, jot down on a sheet of paper what the value of the account is in 2020.
Now, in the space below, define m to be a variable ranging from 1 to 10, and then define a new variable valuem. You will enter the values from your list on your scrap paper into the variable valuem. [Just type valuem : = first value , second value , etc., placing commas between values until you get to the end. ] Mathcad will build a table of values for you.
We would like to investigate compounding rates more frequent than 10 times per year.
However, the computer is performing a large number of computations -- more as m gets
larger --and it is getting tedious to wait for the computation to finish. We will learn soon
that there is a non-recursive formula which will compute these amounts for us:
(Parts of this formula should look familiar.) Check a few values of f(m) and compare
them to the values you obtained earlier.
Experiment with different values of m to find the largest amount you could ever receive
(to the nearest cent) under any compounding scheme. (Beware of computer error if you are using unreasonably large numbers).
