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 within$42,200 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 x 0

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 < 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 >.363). But, P(Z >.363) = 1 – P(Z .363) = F Z (.363) From the table of the cumulative standard normal distribution, we find that F Z (.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 < 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 F Z (.36) - F Z (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 F Z (Z  -.18), we subtract from 1 the cumulative normal distribution for its symmetric value, i.e., 1 - F Z (Z .18)

18. Finding Areas Under the Normal Distribution From the table of the standard normal distribution, F Z (Z .18) =.57 Thus, F Z (Z  -.18) = 1 - F Z (Z .18) =.43 Finally, P(-.18  Z .36) = F Z (Z .36) - F Z (Z  -.18) =.64 -.43 =.21