Name: ___________________

Math 122 Lab 2: Compound Interest

What to Turn In

Fill in the blanks, providing explanations as needed, for the first two pages. Then write a report on the three problems at the end. Each problem can be written up as a separate paragraph. But, be sure to describe the problem, indicate how you solved it and very clearly state your conclusions. (For the third problem, attach your calculation table but don’t put it in the report.)

Different Types of Compounding

Suppose you invest $1000 at 8% interest. How much will you have at the end of one year? The correct answer is, “It depends.” Different types of compounding yield different amounts of money; this is why loan advertisements are required to list the APY (annual percentage yield). In this lab, we look at different types of interest, see why the number e is so important, and look at a couple of investment decisions.

The first case is simple interest. At the end of each year, you add 8% of what you had at the beginning of the year. For $1000 at 8%, at the end of one year you add 8% of $1000, which is $80. You now have $1080. It is important to understand the following way of writing this calculation:

$1000 + .08(1000) = $1000 ( 1+.08 ) = $(1.08)(1000)

The calculation would be similar if you had a different starting amount. For any starting amount, to add 8% you can just multiply by 1.08 (so you end up with 108% of what you started with). How much would you have after 2 years? Take the $1080 and multiply by 1.08: 1.08 ($1080) = $1166.40. One more piece of algebra will show us how to quickly get more answers:

1166.40 = 1.08 (1080) = 1.08 (1.08 (1000) ) = 1.08² (1000)

How much would you have after ten years? Instead of computing the amounts for each individual year, simply compute 1.08¹⁰ (1000). Write the answer here:

Now, compute how much you would have after 20 years:

Semi-annual compounding: Now that we have done simple interest, let’s try the case where you get paid interest twice a year. For 8% interest, this means that you get 4% at the halfway mark and another 4% at the end of the year. So, at the half-year mark, you have $1000 plus 4% of $1000. You shouldn’t be surprised that we want to write that as (1.04)(1000) = 1040. Then at the end of one year we add another 4% to get (1.04)²(1000) = 1081.60. Notice that this is $1.60 more than you got from simple interest. Explain in terms of interest payments why you have more money this time.

Now, explain why the calculation for ten years is (1.04)²⁰(1000) and write the answer here. Compare this to the ten-year amount for simple interest.

To summarize simple and semi-annual compounding, the amount after t years is

Simple: (1.08)^t(1000) Semi-Annual: (1.04)^2t(1000)

The general question is, how much money do you have if interest is compounded n times per year? We have seen the answers for n = 1 (simple) and n = 2 (semi-annual).

Compounding quarterly means n = 4. Write down the amount after t years here, then evaluate it for

t = 10 years and compare it to the amounts for simple and semi-annual interest.

Compounding monthly means n = 12. Write down the amount after t years (leave the fraction .08/12 unsimplified), then evaluate it for t = 10 years and compare it to the amounts for simple, semi-annual and quarterly interest.

At this stage, it should be clear that the more times interest is added, the more money you have. Is there any limit to how much you can get? To answer this question, we need the general formula for compounding n times per year. Here it is for one year at 8% starting with $1000:

( 1 + .08/n )ⁿ 1000

The question can be restated as asking for the limit of this expression as n→∞. Do the following on your calculator and guess the limit. Again, these calculations are for one year.

n = 100: ___________

n = 1000: ___________

Limit: ___________

Compare your limit to e^.08(1000). The limit case is called continuous compounding (you’re getting more interest all the time). Notice how easy it is to compute this using the exponential function. In the above example, we say that the APY for 8% compounded continuously is 8.329%. Explain where this number comes from, and show that you can compute it using e^.08 − 1.

For each of the following interest rates, find the APY for each type of interest. Round to 3 decimals.

Rate Simple Monthly Continuous

8% 8% 8.300% 8.329%

4%

10%

70%

Based on the first two pages, we can use an important shortcut for money that is compounded continuously. If $C is invested at rate r for t years, the value of the investment is $Ce^rt. That is, money compounded continuously grows exponentially. Now, we’ll look at three investment situations.

1. Person A invests $10,000 in 1990 and person B invests $20,000 in 2000. If both receive 12% interest compounded continuously, what are the values of the investments in 2010?

2. A person bought a set of basketball trading cards in 1985 for $34. In 1995, the “book price” for this set was $9000. Assuming exponential growth, find the rate at which this investment grew. At this rate of return, what would the set be worth in 2005?

3. This is a simplified exploration of a question that many people face. If you have enough money to pay cash for a car, should you pay cash or finance it? Even though the steps below simplify the situation, there is still a lot of work and the conclusion is valid. For this problem, we assume that the car costs $24,000 and you have $40,000 in the bank earning 8% interest compounded continuously.

Option 1: Pay Cash

Pay out the $24,000 and then leave the remaining $16,000 in the bank. Compute how much you have in the bank after 1 year and compute how much interest you earned. Now you must pay taxes equal to 28% of the interest earned. Now, how much is in the bank?

Option 2: Finance It

You work out the following deal on the car: no down payment, monthly payments of $2200 for one year. (This corresponds to a loan at 20%.) Do the following for each of the twelve months:

1. Subtract the $2200 payment from the account.

2. Add 8% interest for one month.

3. Write down how much interest was earned.

Add up all of the interest earned for the year, and compute taxes (again, 28% of the interest). Now, you get a tax break for the portions of the car payments that were interest. Since you paid $26,400 (12 times $2200) total for a $24,000 car, your interest payments equal $2400. You get a tax credit for 28% of this amount. All in all, (account minus taxes plus tax credit) how much is in the bank?

Overall, are you better off financing or paying cash?