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Common Probability Distributions in Finance 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 X ~ N , 2 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 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 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 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 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) 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) 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 Fx x0 P X x0 • This probability is given by the area under the normal probability function to the left of x0 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 Pa X b Fx b Fx a 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 Z ~ N 0,1 • The cumulative distribution function of the standard normal distribution is well documented and can be used to find probabilities of normal random variables 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 X Z ~ N 0,1 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 a X b b a Pa X b P F F Z Z 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 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) = 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. 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 FZ(.36) - FZ(0) = .64 - .50 = .14 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 .08 .12 X .20 .12 P P .18 Z .36 .22 .22 • 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) 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