Compound Interest

1
Compound Interest
2
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
  youve made on your investment after one year?

3
An Example

• How much money would you have in your IRA
  account?
• How much interest would you get after two years?

4
An Example

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

5
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.

6
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.

9
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.

10
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

11
Yield

• To find the effective annual yield, y, notice
  that we gained 280.72 on interest after a year
  compounded quarterly. That interest represents a
  gain of 5.61 on 5000

12
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

14
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. Heres why
• As n gets really large, m also becomes really
  large, and

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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

18
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 0.00105 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 1.00105

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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
  e0.055. As we would expect, this corresponds
  to the annual rate of 0.055.

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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 ? 0.0007692. The weekly ratio
  corresponding to this weekly rate is e0.04/52.
• We call Rrf e0.04/52 ? 1.0007695 the risk-free
  weekly ratio for the Walt Disney option.

23
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.

24
Preliminary Reports

• You will deliver your preliminary report on
  Monday March 28th, 2005.
• Download your teams 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 teams
  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.