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Compound Interest

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An Example Suppose that you were going to invest $5000 in an IRA earning interest at an annual rate of 5.5% How would you determine the amount of interest you’ve made on your investment after one year?

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**An Example How much money would you have in your IRA account?**

How much interest would you get after two years?

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An Example How much money would you have in your IRA account after two years? What about 10 years?

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Compound Interest Notice that the interest in our account was paid at regular intervals, in this case every year, while our money remained in the account. This is called compounding annually or one time per year.

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Compound Interest Suppose that instead of collecting interest at the end of each year, we decided to collect interest at the end of each quarter, so our interest is paid four times each year. What would happen to our investment? Since our account has an interest rate of 5.5% annually, we need to adjust this rate so that we get interest on a quarterly basis. The quarterly rate is:

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Compound Interest So for our IRA account of $5000 at the end of a year looks like: After 10 years, we have:

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**Compound Interest Formula**

P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years. Think of P as the present value, and F as the future value of the deposit.

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Yield One may compare investments with different interest rates and different frequencies of compounding by looking at the values of P dollars at the end of one year, and then computing the annual rates that would produce these amounts without compounding.

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Yield Such a rate is called the effective annual yield, annual percentage yield, or simply yield. In our previous example when we compounded quarterly, after one year we had:

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Yield To find the effective annual yield, y, notice that we gained $ on interest after a year compounded quarterly. That interest represents a gain of 5.61% on $5000:

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Finding the Yield To find the effective annual yield, find the difference between our money after one year and our initial investment and divided by the initial investment. Therefore, interest at an annual rate r, compounded n times per year has yield y:

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More on Yield There may be times when we need to find the annual rate that would produce a given yield at a specified frequency of compounding. In other words, we need to solve for r :

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**Compound Interest Formula**

Notice that when we collected our interest more times during each year, i.e. we compounded more frequently, the amount of money in our account was actually greater than if only collected interest one time a year. What would happen to our money if we compounded a really large number of times?

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**Continuous Compounding**

As n increases, approaches a constant value in the Frequency Worksheet. Here’s why: As n gets really large, m also becomes really large, and: n=mr

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**Continuous Compounding**

The value of P dollars after t years, when compounded continuously at an annual rate r , is On the calculator use the button (on TI-83: 2nd + LN ) In Excel, use the function EXP(x )

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Yield The effective annual yield, y, for compounding continuously at an annual interest rate of r is: To find the annual interest rate r if we know the yield, y, we would have to solve for r in the above equation. To do this you would use logarithms:

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Present Values If we are given the future value, F, the annual interest rate r, the number of times compounded per year n, and the length of time invested t, we may solve the present value P :

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Ratios Sometimes we are not interested in the percentage that an investment increases by. Rather, we would like to know by what factor the investment increased or decreased. Such factors are computed by find the ratio of the future value to the present value. This ratio, R, for continuous compounding is: This allows us to convert the interest rate for a given period to a ratio of future to present value for the same period.

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Example Ratios Suppose that in our IRA example, the annual interest rate of 5.5% is compounded continuously. If we wanted to know the weekly rate our investment would increase, we would simply have 0.055/52 or or 0.105%. This would mean that the ratio of the future value to the present value between consecutive weeks compounded continuously would be e0.055/52 or

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Example Ratios Multiplying by this weekly ratio 52 times yields a yearly ratio of (e0.055/52)52 = e(0.055/52)52 = e As we would expect, this corresponds to the annual rate of

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**The Project How can compound interest help us price a stock option?**

Our annual risk-free rate of 4%, compounded continuously, gives a weekly risk-free rate of rrf = 0.04/52 The weekly ratio corresponding to this weekly rate is e0.04/52. We call Rrf = e0.04/52 the risk-free weekly ratio for the Walt Disney option.

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The Project Compound interest can help us with option pricing in a second way. Suppose that we know a future value F for our 20 week option at the end of the 20 weeks. We suppose that money will earn at the risk-free annual interest rate or 4% compounded continuously. This can be used to find the present value, P, of the option.

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Preliminary Reports You will deliver your preliminary report on Monday March 28th, 2005. Download your team’s historical data today and start looking at the behavior of your companies particular stock. For the report you want to determine what a reason able price will be for investors considering buying an option for your team’s particular company and an explanation as to how you determined that. The Chicago Board of Options Exchange is a good resource for understanding how options are priced. I will need two hard copies of your powerpoint slides. Because this is a new project, you will need to draft new team contracts. Those will be collected in class on Wednesday, March 30th, 2005.

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Chapter 9: Mathematics of Finance

Chapter 9: Mathematics of Finance

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