Common Probability Distribution (Reading 9)

Learning Outcome Statements (LOS)

a

Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions:

     A probability distribution lists all the possible outcomes of an experiment, along with their associated probabilities

     A discrete random variable has positive probabilities associated with a finite number of outcomes

     A continuous random variable has positive probabilities associated with a range of outcome value—the probability of any single value is zero.

b

Describe the set of possible outcomes of a specified discrete random variable:

     The set of possible outcomes of a specific discrete random variable is a finite set of value. An example is the number of days last week on which the value of a particular portfolio increased. For a discrete distribution, p(x)=0 when x cannot occur, or p(x)>0 if it can.

c     

Interpret a cumulative distribution function:

     A probability function specifies the probability that a discrete random variable is equal to a specific value; P(X=x) =p(x).

     The two key properties of a probability function are:

·        

·        

     A probability density function (pdf) is the term for a function for a continuous random variable used to determine the probability that it will fall in a particular range.

     A cumulative distribution function (cdf) gives the probability that a random variable will be less than or equal to specific values.

d    

Calculate and interpret probabilities for a random variable, given its cumulative distribution function:

     Given the cumulative distribution function for a random variable, the probability that an outcome will be less than or equal to a specific value is represented by area under the probability distribution to the left of that value.

e  

Define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable:

     A discrete uniform distribution is one where there are n discrete, equally likely outcomes.

     The binomial distribution is a probability distribution for a binomial (discrete) random variable that has two possible outcomes.       

f 

Calculate and interpret probabilities given the discrete uniform and the binomial distribution functions:

     For a discrete uniform distribution with n possible outcomes, the probability for each outcome equals 1/n.

     For a binomial distribution, if the probability of success is p, the probability of x success in n trials is

g

Construct a binomial tree to describe stock price movement:

     A binomial tree illustrates the probabilities of all the possible values that a variable (such as a stock price) can take on, given the probability of an up-move and the magnitude of an up-move (the up-move factor).

     With an initial stock price S=50, U=1.01. D=1/1.01. and prob(U)=0.6, the possible stock prices after two periods are:

·         with probability

·         with probability

·         with probability

·         with probability

h

Calculate and interpret tracking error:

     Tracking error is calculated as the total return on a portfolio minus the total return on a benchmark or index portfolio.

i  

Define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution:

     A continuous uniform distribution is one where the probability of X occurring in a possible range is the length of the range relative to the total of all possible value. Letting a and b be the lower and upper limit of the uniform distribution, respectively, then for:

j    

Explain the key properties of the normal distribution:

     The normal probability distribution and normal curve have the following characteristics:

·         The normal curve is symmetrical and bell-shaped with a single peak at the exact center of the distribution.

·         Mean = median = mode, and all are in the exact center of the distribution.

·         The normal distribution can be completely defined by its mean and standard deviation because the skew is always zero and kurtosis always 3. 

k

Distinguish between a univariate and a multivariate distribution, and explain the role of correlation in the multivariate normal distribution:

     Multivariate distributions describe the probabilities for more than one random variable, whereas a univariate distribution is for a single random variable.

     The correlation of a multivariate distribution describes the relation between the outcomes of its variables relative to their expected values.

l  

Determine the probability that a normally distributed random variable lies inside a given interval:

      A confidence interval is a range within which we have a given level of confidence of finding a point estimate.

     Confidence intervals for any normally distributed random variable are:

·         90%: μ±1.65 standard deviations.

·         95%: μ±1.96 standard deviations.

·         99%: μ±2.58 standard deviations.

     The probability that a normally distributed random variable X will be within A standard deviation of its mean, μ, is calculated as two times [1-the cumulative left-hand tail probability, F(-A)], or two times {1-right-hand tail probability, [1-F(A)]}, where F(A) is the cumulative standard normal probability of A.

m

Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution:

     The standard normal probability distribution has a mean of 0 and a standard deviation of 1.

     A normally distributed random variable X can be standardized as Z= and Z will be normally distributed with mean = 0 and standard deviation 1.

     The z-table is used to find the probability that X will be less than or equal to a given value.

·         , which is found in the standard normal probability table.

·        

n

Define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety-first criterion:

     The safety-first ratio for portfolio P, based on a target return R_T, is:

     Shortfall risk is the probability that a portfolio’s value ( or return) will fall below a specific value over a given period of time. Greater safety-first ratios are preferred and indicate a smaller shortfall probability. Roy’s safety-first criterion states that the optimal portfolio minimizes shortfall risk. 

o

Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices.

     If x is normally distributed, e^x follows a lognormal distribution. A lognormal distribution is often used to model asset prices, since a lognormal random variable cannot be negative and can take on any positive value.

p

Distinguish between discretely and continuously compounded rates of return, and calculate and interpret a continuously compounded rate of return, given a specific holding period return:

     As we decrease the length of discrete compounding periods the effective annual rate increases. As the length of the compounding period in discrete compounding gets shorter and shorter, the compounding becomes continuous, where the effective annual rate = eⁱ-1.

     For a holding period return over any period, the equivalent continuously compounded rate over the period is ln(1+HPR).

q

Explain Monte Carlo simulation and describe its major application and limitations:

     MONTE Carlo simulation uses randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values. Its limitations are that it is fairly complex and will provide answers that are no better than the assumptions used.

r

Compare Monte Carlo simulation and historical simulation:

     Historical simulation uses randomly selected past changes in risk factors to generate a distribution of possible security values, in contrast to Monte Carlo simulation, which uses randomly generated value. A limitation of historical simulation is that it cannot consider the effects of significant events that did not occur in the sample period.

Formulas:

Exercise Problems:

1.    A random variable with a finite number of equally likely outcomes is best described by a:

A.    Binomial distribution

B.     Discrete uniform distribution

C.     Continuous uniform distribution

Ans: B; the discrete uniform distribution has a finite number of specified outcomes, and each outcome is equally likely.

A is incorrect; we use binomial distribution when we make probability statements about a record of successes and failures, or about anything with binary outcomes.

C is incorrect; continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution’s support are equally probable. It contains infinite numbers.

2.    Assume that a stock’s price over the next two periods is as show below.

The initial value of the stock is $100. The probability of an up move in any given period is 40% and the probability of a down move in any given period is 60%. Using the binomial model, the probability that the stock’s price will be $101.20 at the end of two periods is closest to:

A.    16%

B.     24%

C.     48%

Ans: C; in this binomial tree,

A discrete uniform distribution consists of the following twelve values:

On a single draw from the distribution, the probability of drawing a value between -2.0 and 2.0 from the distribution is closest to:

A.    16.67%

B.     18.04%

C.     27.59%

Ans: A; sort the numbers from smallest to largest:

So there are two numbers between -2.0 and 2.0.

3.    An analyst determines that 60 percent of all U.S. pension funds hold hedge funds. In evaluating this probability, a random sample of 10 U.S. pension funds is taken. Using the binomial probability function, , the probability that exact 6 of the 10 firms in the sample hold hedge fund is closest to:

A.    25.08%

B.     27.99%

C.     60.00%

Ans: A; use the formula in the problem:

4.    An energy analyst forecasts that the price per barrel of crude oil five years from now will range between USD$175 and USD$205. Assume oil prices are a continuous uniform distribution. Recall that the cumulative distribution function for a continuous uniform variable is:

The probability that the price will be less than USD$180 five years from now is closest to:

A.    5.6%

B.     16.7%

C.     44.4%

Ans: B; since $180 is in the range between $175 and $205, so

5.    Which of the following statements about a normal distribution is least accurate? A normal distribution:

A.    Has an excess kurtosis of 3

B.     Is completely described by two parameters

C.     Can be the linear combination of two or more normal random variables.

Ans: A; excess kurtosis is kurtosis minus 3. For all normal distribution, kurtosis is equal to 3, which means excess kurtosis is 0.

B is incorrect; normal distribution is completely described by mean and standard deviation.

C is incorrect; one of the defining characteristics of a normal distribution is that it could be a linear combination of two or more normal random variables is also normal distributed.

6.    An analyst determines that approximately 99 percent of the observations of daily sales for a company are within the interval from $230,000 to$480,000 and that daily sales for the company are normally distributed. The standard deviation of daily sales for the company is closest to:

A.    $41,667

B.     $62,500

C.     $83,333

Ans: A; 99% Confidence intervals for normally distributed random variable is μ±2.58 standard deviations. In this problem: