Chapter 11: Continuous Compounding

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Chapter 11 Continuous Compounding Ratios
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Types of Compounding

1. Discrete when interest is earned every year,
   quarterly, month, day, etc. Will use for
   comparison but we will explore thoroughly later.
2. Continuous interest is earned continuously. We
   will use this for project 3.

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Discrete Compounding Formula

• F future value
• Ppresent value
• iinterest rate per period
• ntotal number of compounding periods
• F P(1 i)n

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Find the future value of 1000 in 3 years with
annual interest of 5 compounded monthly

• P 1000
• i 5/12
• n 36 compounding periods
• Find F

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Continuous Compounding Formula

• F future value
• P present value
• r annual interest rate
• t years
• F Pert

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Example find F

• What is the future value of an investment of 500
  at annual interest of 5.5 for 10 years,
  compounded continuously?
• How to do in Excel?

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Example Find t

• Suppose a couple invest 2500 in an account that
  earns 4.3 compounded continuously. How long will
  it take before they earn 1000 in interest?
• .. 1.4 e 0.043t (will continue later)

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Logarithms

• Recall p logb n means bp n
• Example log10 1000 3 because
• Example log2 16 4 because

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Similarly for base e,

• loge x p because ep x where e is an
  irrational number
• We write loge x ln x

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What is ln 1000?

• ln 1000 some number p such that
• ep 1000.
• Hence, e1 2.7
• e2 ?
• e3?
• etc

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Calculator

• You can use ln key in a scientific calculator
  and find ln 1000 6.908
• This means e 6.908 1000

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Back to a previous problem

• 1.4 e 0.043t
• ln 1.4 ln (e 0.043t)
• ln 1.4 0.043t ln e
• ln 1.4 0.043 t
• ln 1.4/0.043 t
• 7.825 t

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Example

• 18
• 22

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Ratios

• We are interested in comparing stock prices from
  one week to the one preceding it.
• We can do this by finding the ratio of the future
  value to the present value.

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• Ratio R F/P growth ratio. Also called
• Weekly ratio how much the value grow per week.
• Monthly ratio how much the value grow per month
• Etc.

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Example

• A week ago, the stock of a company was 50.43.
  This week, the value is 51.62. Find the weekly
  ratio and explain what it means. Find the
  increase.

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• When the growth ratio is greater than one, the
  stock has increased in value.
• When the growth ratio is smaller than one, the
  stock had decreased in value.
• When the growth ratio is equal to one, nothing
  had changed.

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Example

• A week ago, the stock of a company was 50.43.
  This week, its value is 48.21. What is the
  weekly ratio and what does it mean? The the
  decrease.

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Other ways to find ratios

• For continous compounding
• F/P e rt
• For discrete compounding
• F/P (1 i)n

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Example

• An investment is growing at a monthly rate of
  0.5.
• What is the monthly ratio?
• What is the yearly ratio?
• What is the annual percentage yield? (or
  increase)

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Example

• If a bank account compounds interest continuously
  at an annual interest of 10, what is the monthly
  ratio? (1.008368)

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Focus on the Project

• The first step to pricing our stock option is to
  compute the weekly ratios from the data we
  downloaded.
• This can be easily done by dividing next weeks
  adjusted closing price by the current closing
  price. Use Excel.

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Focus on the Project

• Use the fact that our class project risk-free
  annual interest rate of 4 to compute the weekly
  risk-free ratio. Keep in mind that there are 52
  weeks in a year.
• Hence, R e rt
• e (0.04)(1/52)
• 1.0007695

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Focus on the Project

• 1.0007695 is our weekly risk-free ratio for our
  class project.
• Note that it is a good idea to keep several
  decimal places at this point of our computations.

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• Using the risk-free rate and the fact that we
  have a 20-week option period, we can get a
  preliminary estimate for the price of the stock
  option after this 20-week period.
• Note that the time 20 weeks has to be converted
  in years.

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Focus on the Project

• The closing price of DIS stock at the start of
  the option period was 21.87. Our preliminary
  estimate for the value of the option is
• F Pe rt
• 21.87 e (0.04)(20/52)
• 22.21

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• 4
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