78106_06_Web_Ch06A_p01-05 - WEB EXTENSION 6A Continuous...

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WEB EXTENSION 6A Continuous Probability Distributions I n Chapter 6, we illustrated risk/return concepts using discrete distributions, and we assumed that only three states of the economy could exist. In reality, however, the state of the economy can range from a deep recession to a fantastic boom, and there is an infinite number of possibilities in between. It is inconvenient to work with a large number of outcomes using discrete distributions, but it is relatively easy to deal with such situations with continuous distributions because many such distribu- tions can be completely specified by only two or three summary statistics such as the mean (or expected value), standard deviation, and a measure of skewness. In the past, financial managers did not have the tools necessary to use continuous distribu- tions in practical risk analyses. Now, however, firms have access to computers and powerful software packages, including spreadsheet add-ins, that can process continu- ous distributions. So if financial risk analysis is computerized, as is increasingly the case, it is often preferable to use continuous distributions to express the distribution of outcomes. 6.1 U NIFORM D ISTRIBUTION One continuous distribution that is often used in financial models is the uniform distribution , in which each possible outcome has the same probability of occurrence as any other outcome; hence, there is no clustering of values. Figure 6A-1 shows two uniform distributions. Distribution A of Figure 6A-1 has a range of - 5 to +15%. Therefore, the absolute size of the range is 20 units. Since the entire area under the density function must equal 1.00, the height of the distribution, h, must be 0.05: 20h = 1.0, so h = 1/20 = 0.05. We can use this information to find the probability of different outcomes. For example, suppose we want to find the probability that the rate of return will be less than zero. The probability is the shaded area under the density function from - 5to0% : Area = (Right point - Left point)(Height of distribution) =[0 - ( - 5)][0.05] = 0.25 = 25% Similarly, the probability of a rate of return between 5 and 15 is 50%: Probability = Area = (15 - 5)(0.05) = 0.50 = 50% The expected rate of return is the midpoint of the range, or 5%, for both distribu- tions in Figure 6A-1. Since there is a smaller probability of the actual return falling very far below the expected return in Distribution B, Distribution B depicts a less risky situation in the sense of stand-alone risk.
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78106_06_Web_Ch06A_p01-05 - WEB EXTENSION 6A Continuous...

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