(9) Common Probability Distributions

Question 1

LOS 10. a: Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions.

Answer 1

A probability distribution lists all the possible outcomes of an experiment, along with their associated probabilities. A probability distribution completely describes a random variable.

Question 2

LOS 10. a: Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions.

Answer 2

A discrete random variable has positive probabilities associated with a finite number of outcomes (countable, non zero probabilities for each outcome)

A continuous random variable has positive probabilities associated with a range of outcome values-the probability of any single value is zero (Uncountable)

Question 3

LOS 10. b: Describe the set of possible outcomes of a specified discrete random variable.

Answer 3

The set of possible outcomes of a specified discrete random variable is a finite set of values. 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.

Question 4

LOS 10. c: Interpret a cumulative distribution function.

Answer 4

A cumulative distribution function (cdf) gives the probability that a random variable will be less than or equal to specific values. The cumulative distribution function for a random variable X may be expressed as F(x) = P(X =< x). The CDF is more relevant than discrete and continuous probability functions.

Question 5

LOS 10. d: Calculate and interpret probabilities for a random variable, given its cumulative distribution function.

Answer 5

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 the area under the probability distribution to the left of that value.

Question 6

LOS 10. e: Define a discrete uniform distribution, a Bernoulli random variable, and a binomial random variable.

Answer 6

A discrete uniform distribution is one where there are n discrete, equally likely outcomes (each probability is the same for the distribution).

Question 7

LOS 10. e: Define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable.

Answer 7

A Bernoulli random variable is one where there are only 2 outcomes (each outcome is called a trial)

A Binomial random variable measures the number of successes in N bernoulli trials

Question 8

LOS 10. f: Calculate and interpret probabilities given the discrete uniform and the binomial distribution functions.

Answer 8

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

Question 9

LOS 10. f: Calculate and interpret probabilities given the discrete uniform and the binomial distribution functions.

Answer 9

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

Question 10

Explain the Bernoulli equation logic

Answer 10

Question 11

LOS 10. g: Construct a binomial tree to describe stock price movement.

Answer 11

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:

uuS = 1.01² x 50 = 51.01 with probability of (0.6)² = 0.36

udS = 1.01(1/1.01) x 50 = 50 with probability of (0.6)(0.4) = 0.24

duS = (1/1.01)1.01 x 50 = 50 with probability of (0.4)(0.6) = 0.24

uuS = (1/1.01)² x 50 = 49.01 with probability of (0.4)² = 0.16

Question 12

Calculate and interpret tracking error.

Answer 12

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

Tracking error measures how closely a portfolio follows its benchmark

Question 13

LOS 10. h: Define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution.

Answer 13

A continuous uniform distribution is described by a lower limit ‘a’ and an upper limit ‘b’

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 values. Letting a and b be the lower and upper limit of the uniform distribution, respectively. See image below.

If its asking for P(X =< A), then formula is

x - a / b - a

Question 14

LOS 10. j: Explain the key properties of the normal distribution.

Answer 14

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 is always 3.
• A linear combination of normally distributed variables is also normally distributed

Question 15

LOS 10. j: Distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution.

Also what parameters are needed for a univariate and a multivariate distribution?

Answer 15

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

Univariate parameters: mean and variance

Multivariate parameters: mean, variance, and corrleations between each pair of random variables

Question 16

LOS 10. j: Distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution.

Answer 16

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

Question 17

LOS 10. k: Determine the probability that a normally distributed random variable lies inside a given interval.

Answer 17

A confidence interval is a range within which we have a given level of confidence of finding a point estimate (e.g., the 90% confidence interval for X is Xbar – 1.65s to Xbar + 1.65s).

The probability that a normall distributed random variable X will be within A standard deviations of its mean, µ, [i.e., P(µ - Aσ <= X <= µ + Aσ)], may be calculated as F(A) – F(-A), where F(A) is the cumulative standard normal probability of A, or as 1 – 2[F(-A)].

Question 18

LOS 10. l: Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.

Answer 18

The standard normal probability distribution has a mean of 0 and a standard deviation of 1. Standardizing finds the number of standard deviations away from the mean X lies

Question 19

LOS 10. l: Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.

Answer 19

A normally distributed random variable X can be standardized as Z = (x - µ) / σ and Z will be normally distributed with mean = 0 and standard deviation 1. This formula calculated the number of standard deviations away from the mean X lies

µ = population mean

σ = standard deviation

Question 20

LOS 10. l: Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.

Answer 20

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

Question 21

LOS 10. m: Define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety-first criterion.

Answer 21

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

[Equation]

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.

E(Rp) - expected portfolio return

RT - target/threshold return

Question 22

LOS 10. n: Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices.

Answer 22

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.

A normal distribution is unbounded on both ends, while a lognormal distribution is bounded by zero and unbounded on the upper end.

Normal distribution is used to model returns (used as an approximation of returns)

Lognormal distribution = (V_T/V₀), where V_T is ending value asset price and V₀ is beginning value asset price

Normal distribution = Ln (V_T/V₀)

If an assets continuously compounded return is normally distributed then the assets future prices must be lognormally distributed

Question 23

LOS 10. o: 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.

Answer 23

As we decrease the length of discrete compounding periods (e.g., from quarterly to monthly) 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 (HPR) over any period, the equivalent continuously compounded rate over the period is ln(1 + HPR).

Holding period return = (V_T/V₀) - 1; where V_T is the ending asset price and V₀ is the beginning asset price

Continuously compounded return (rT) = (V_T/V₀)

Question 24

LOS 10. p: Explain Monte Carlo simulation and describe its applications and limitations.

Answer 24

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.

Question 25

LOS 10. q: Compare Monte Carlo simulation and historical simulation.

Answer 25

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 values. A limitation of historical simulation is that it cannot consider the effects of significant events that did not occur in the same period. It also does not factor in cause/effect relationships

Question 26

Two types of probability distributions:

Answer 26

1. Probability Function (Possible outcomes + associated probabilities)
   
   1. Probability that the random variable takes on a specific value
   2. P(X=x)
   3. This completely describes the variable
   4. Used for discrete variables
2. Cumulative Distribution Function (probability that a random variable X is less than or equal to a particular value x, P (X=

Question 27

Calculate the Mean and variance of a bernoulli and binomial random variable

Answer 27

1. Bernoulli mean = p
2. Binomial mean = nxp
3. Bernoulli variance = p (1-p)
4. Binomial variance = nxp (1-p)

Where p = probability and n = number of trials

Question 28

Two types of continuous distributions

Answer 28

1. Normal Distribution
2. Lognormal distribution

Question 29

Probability density function is solving for…

Answer 29

The area under the curve

Question 30

Centril Limit Theorem Definition

Answer 30

the sum of a large number of independent random variables is approximately normally distributed

Question 31

Univariate vs. Multivariate distributions

Answer 31

• Univariate: describes the distributions of a single random variable
• Multivariate: specifies probabilities associated with a group of random variables
  
  • Takes into account the interrelationships between the two variables

Question 32

Parameters needed to describe a univariate (2) and multivariate (3) normal distribution:

Answer 32

Multivariate parameters:

1. The means of the individual random variables
2. The variances of the individual random variables
3. The correlation between each possible pair of random variables

Univariate parameters:

1. Mean
2. Standard deviation

Question 33

Confidence interval formulas

Answer 33

• 90% CI = sample mean + or - 1.65 (standard deviation)
• 95% CI = sample mean + or - 1.96 (standard deviation)
• 99% CI = sample mean + or - 2.58 (standard deviation)

Question 34

What does E(R_p) and R_T stand for?

Answer 34

R_p - expected portfolio return

R_T - target/threshold return

Question 35

3 Features of a lognormal distribution

Answer 35

1. Bounded by zero on the lower end
2. Upper end is unbounded
3. skewed to the right (positive skew)

Question 36

Continuously compounded rate of return from HPR (holding period return) =

Answer 36

R_cc = ln (1 + hpr)

Question 37

Volatility definition and formula. How do you annualize volatility?

Answer 37

Volatility is expressed by taking the standard deviation of continuously compounded returns

Continuosly compounded return = e^rT

Where rT is the compounded return

rT = (V_T/V_o)

V_T is the ending asset price

Vo is the beginning asset price

Take the standard deviation of continously compounded returns x square root of 250 = annualized volatility

Question 38

Two major assumptions that are necessary for the probability function of a binomial random variable

Answer 38

1. The probability of an earnings increase is constant from year to year
2. Earnings increases are independent trials

Question 39

Calculate mean/expected value and variance of a cumulative distribution function for a continuous uniform distribution

Answer 39

Mean/expected value = (a+b)/2

Variance = (b-a)/12