Common Probability Distributions in Finance

1
Common Probability Distributions in Finance
2
The Normal Distribution

• The normal distribution is a continuous,
  bell-shaped distribution that is completely
  characterized by two parameters its mean and
  standard deviation
• If a random variable X follows a normal
  distribution with mean ? and variance ?2, we
  write

3
Properties of Normal Distribution

• Since the normal distribution can be completely
  characterized by its mean and variance, any
  probability question about a normal random
  variable can be answered if these two parameters
  are known
• The normal distribution is symmetric
• Skewness is zero
• Mean, median and mode are the same

4
Properties of Normal Distribution

• Due to the symmetric nature of the normal
  distribution, we can derive the following
  statements
• Approximately 68 of the values of a normal
  variable fall within the interval ? ? ?
• Approximately 95 of the values of a normal
  variable fall within the interval ? ? 2?
• Approximately 99 of the values of a normal
  variable fall within the interval ? ? 3?

5
Properties of Normal Distribution

• To be more precise, the following intervals with
  their corresponding cutoffs are frequently used
  in association with a sample from a normal
  distribution
• 90 of the values of a normal variable lie within
  ? 1.65 sample standard deviations from the sample
  mean
• 95 of the values of a normal variable lie within
  ? 1.96 sample standard deviations from the sample
  mean
• 99 of the values of a normal variable lie within
  ? 2.58 sample standard deviations from the sample
  mean

6
Properties of Normal Distribution

• Example Suppose that the variable approved
  mortgage amount follows a normal distribution
• Taking a sample of 200 loan approvals from a
  bank, it is found that the sample mean is
  150,000 and the sample standard deviation is
  55,000
• In this case, 95 of approved mortgages will be
  within42,200 and 257,800

7
Normal Distribution and Portfolio Returns

• One potentially interesting application of the
  normal distribution is in describing data on
  asset returns
• The normal distribution is a good fit for
  quarterly or annual holding period returns on a
  diversified equity portfolio
• However, it does not fit equally well monthly,
  weekly or daily period returns
• In general, the normal distribution tends to
  underestimate the probability of extreme returns
  (the fat tails problem)

8
Normal Distribution and Portfolio Returns

• Relative to the normal distribution, the actual
  distribution of the data may contain more
  observations in the center and in the tails
• This implies that the actual distribution
  compared to the normal distribution has
• More observations clustered near the mean
• A higher probability of observing extreme values
  on both tails of the distribution (fat tails)

9
The Cumulative Distribution Function of a Normal
Distribution

• If a random variable X follows a normal
  distribution with mean ? and variance ?2 , the
  cumulative distribution function is
• This probability is given by the area under the
  normal probability function to the left of x0

10
The Cumulative Distribution Function of a Normal
Distribution

• Similarly, if a and b are two possible values of
  the normal random variable X, with a lt b, then
  the probability that X will take values in
  between those two cutoffs is given by

11
The Standard Normal Distribution

• The standard normal distribution is a normal
  distribution with mean 0 and variance 1
• We denote a standard normal variable with Z and
  write
• The cumulative distribution function of the
  standard normal distribution is well documented
  and can be used to find probabilities of normal
  random variables

12
Finding Areas Under the Normal Distribution

• We say that a normal random variable X is
  standardized if we subtract from it its mean and
  divide by its standard deviation
• Thus, the new variable Z follows the standard
  normal distribution

13
Finding Areas Under the Normal Distribution

• Using the above transformation of a normal into a
  standard normal variable, we rewrite the result
  of the probability that a normal variable takes
  values between two cutoffs as follows

14
Finding Areas Under the Normal Distribution

• Example Suppose that portfolio returns follow a
  normal distribution, which we have estimated to
  have a mean return of 12 and standard deviation
  of return of 22 per year
• What is the probability that portfolio return
  will exceed 20? What is the probability that
  portfolio returns will be between 12 and 20?
• If X is portfolio return, the variable (X -
  .12)/.22 follows the standard normal distribution

15
Finding Areas Under the Normal Distribution

• For X .2, Z (.2 - .12)/.22 .363.
• We need to find P(Z gt .363). But, P(Z gt .363) 1
  P(Z ? .363) FZ(.363)
• From the table of the cumulative standard normal
  distribution, we find that FZ(.363) is equal to
  .64 and, thus, the probability of a return above
  20 is 1 - .64 .36.

16
Finding Areas Under the Normal Distribution

• For the second part, note that 12 is the mean of
  the distribution, meaning that P(X lt 12) .5
  and the same will be true for the corresponding
  value of the standard normal variable
• Thus, P(.12 ? X ? .20) is the same as P (0 ? Z ?
  .36), which is equal to FZ(.36) - FZ(0) .64 -
  .50 .14

17
Finding Areas Under the Normal Distribution

• To expand upon the last question, what if we were
  interested in the probability that portfolio
  returns are between 8 and 20?
• Following the above steps and transforming the
  normal variable into a standard normal, P(.08 ? X
  ? .20) is equal to
• To find the cumulative standard normal
  distribution for -.18, which is FZ(Z ? -.18), we
  subtract from 1 the cumulative normal
  distribution for its symmetric value, i.e., 1 -
  FZ(Z ? .18)

18
Finding Areas Under the Normal Distribution

• From the table of the standard normal
  distribution, FZ(Z ? .18) .57
• Thus, FZ(Z ? -.18) 1 - FZ(Z ? .18) .43
• Finally, P(-.18 ? Z ? .36) FZ(Z ? .36) - FZ(Z ?
  -.18) .64 - .43 .21