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 addins, 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 6A1 shows two
uniform distributions.
Distribution A of Figure 6A1 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 6A1. 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 standalone risk.