Ratios

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Ratios Histograms
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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
• This value tells us that for any week, the value
  of our investment will grow by factor of about
  1.00105 by the following week. For example, the
  value of our investment after 1 week would be

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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.

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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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Histograms

• Help to organize large amounts of data into
  groups, called bins. The number of times an
  observation falls into a group or bin is a
  measure of its frequency.
• For example, a useful application of bins would
  be for organizing the scores on an exam.

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HistogramsExam Example

• Go to the class website and find the excel file
  labeled HistogramScoreDemo.xls under the
  Worksheets link.
• Open the file in the web browser.
• Copy and paste the data contained in the Data
  into a blank excel document.

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Histograms Exams Example

• Find the maximum and minimum scores
• Decide on how many bins you would like to have
• A good rule is no more than 20 bins
• Too many makes the graph cluttered
• Bin width (max min)/( bins)
• Use a nice whole number
• No weird decimals
• Should cover the maximum value or exceed
• Mark off each bin width on graph starting with
  the smallest value you want to graph

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HistogramExams Example

• According to our table the maximum value is 98
  and the minimum value is 10
• Lets make 10 bins for our histogram
• The bin width will be approximately 8.8
• We need to find a whole bin width that will
  cover the maximum value of 98
• Suppose the bin width is 9
• The intervals for our bins will be
• 10-19, 19-28, 28-37, 37-46, 46-55, 55-64, 64-73,
  73-82, 82-91,91-100
• Although 9 is an okay bin width for covering
  the maximum value, to make the histogram easier
  to read well use a bin width of 10.

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What you need to do in Excel

• Type in an empty cell Bin Limits
• Start with the minimum value of 10 and then add
  10 each time until youve created 10 numbers
  (since we wanted 10 bins)
• Lets use the Histogram feature in Excel
• Before we do, you need to know that each of the
  numbers in your Bin Limits represents the upper
  most value that an exam score can assume.
• Excel will count all exam scores that are less
  than or equal to the first bin limit. The second
  bin limit is the number of scores that are
  strictly greater than the previous bin limit but
  less than or equal to the second bin limit and so
  on

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What your Excel file should look like?
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Check your bins

• To make sure that all data is accounted for, it
  is a good idea to add up all the frequency values
  and make sure they add to the total number of
  data points.
• Do this now!
• They should total to 50

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Relative Frequency

• Ratio of the frequency for each bin to the total
  number of frequencies
• This can be used to compare the sizes of the bins
  in terms of percentages
• Determine the relative frequency of your bins now!

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What your Excel file should look like
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Bar Graphs--Frequency
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Bar Graph--Relative Frequency
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Class Project

• How can histograms help us to price a stock
  option?
• Because we have the adjusted closing prices of
  Walt Disney stock, we can compute the weekly
  ratio between consecutive closing prices.
• This information can give us a sense of the rate
  at which Walt Disneys stock is growing or
  falling between each week.
• With this information, we can get a sense of the
  stocks volatility.

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Class Project--Bins
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Class ProjectBar Graph
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What Information can we extract?

• According to the ratios of adjusted closing
  prices
• Max 1.1774
• Min 0.7578
• Average 1.0019
• From this we can say that Walt Disney stock went
  up on average by about 0.19 each week during the
  years of our historical data
• Looking at our chart, we can also get a sense of
  how often Walt Disney stock went up for a given
  ratio
• Example 30 of the time the ratio of adjust
  closing prices fell between 0.97 and 1

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Problems with Histogram?

• Looking at our histogram we might be tempted to
  use the weekly ratios to predict the future value
  of our stock when our option expires.
• According to assumption 1 we cant do this
  because we would be basing this on past closing
  prices.
• Our past weekly ratios could be used to predict
  future volatility, but those ratios are too
  reliant upon past ratios for them to be reliable.
  Why?
• If we have two different stocks, their average
  weekly ratios might be higher or lower than the
  risk-free weekly ratio which is based on the rate
  of return by a US treasury bill, called the
  risk-free rate.
• Assumption 3 says that all stocks that can be
  predicted probabilistically are assumed have the
  same rate of return.
• That rate of return is the risk-free rate
  (Assumption 4)

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Problems?

• We need to find a way to bring all stocks to some
  common means of comparison so that the average
  weekly ratios will be the same as the risk-free
  weekly ratio.
• For example, consider two possible savings
  accounts. Account A compounds interest quarterly
  at a rate of 4. Account B compounds interest
  monthly at a rate of 3.9. Which account is more
  likely to acrue more interest?
• To bring these two accounts to some common means
  of comparison, we would need to look at the
  effective annual yield.
• The same idea is analogous for our options project