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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.
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 e0.055/52or 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:
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.
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.
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.
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.
Histograms—Exam 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.
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
Histogram—Exams Example • According to our table the maximum value is 98 and the minimum value is 10 • Let’s 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 we’ll use a bin width of 10.
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 you’ve created 10 numbers (since we wanted 10 bins) • Let’s 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
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
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!
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 Disney’s stock is growing or falling between each week. • With this information, we can get a sense of the stock’s volatility.
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
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 can’t 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)
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
