Common Probability Distributions

Question 1

Discrete random variable

Answer 1

May be at most a countable number of outcomes. Uniform if each outcome equally likely.

Question 2

continuous random variable

Answer 2

cannot count number of outcomes

Question 3

Interpret cumulative distribution function

Answer 3

Probability that random variable less than or equal to particular value. Sum up all outcomes less than or equal to value.

Question 4

Bernoulli random variable

Answer 4

Random variable w/value of 1 is success, 0 is failure

Question 5

Binomial random variable

Answer 5

number of successes in n Bernoulli trials

Question 6

Construct binomial tree for stock price

Answer 6

Each period have a new split of two nodes for up and down transition probability.

Question 7

Calculate and interpret tracking error

Answer 7

Total return on portfolio (gross of fees) minus total return on benchmark index. Use binomial random distribution to figure out probability that certain tracking error met certain percent of time.

Question 8

continuous uniform distribution

Answer 8

All outcomes equally likely

Question 9

Calculate and interpret probabilities given continuous uniform distribution

Answer 9

Probability that continuous random variable is any particular fixed value is 0, but it’s possible to estimate probability that continuous random variable is more or less than certain amount

Question 10

Explain key properties of normal distribution

Answer 10

• Mean and variance completely describe it
• Skewness of zero, kurtosis of 3
• Linear combination of two or more normal random variables is also normally distributed

Question 11

Univariate vs. multivariate distribution

Answer 11

Univariate describes single random variable

Multivariate specifies probabilities for group of related random variables

Question 12

Role of correlation in multivariate distribution

Answer 12

All distinct pairwise correlations is third parameter that describes multivariate normal distribution

Question 13

Determine probability normally distributed random variable lies inside given interval

Answer 13

68% within 1 s
95% within 2 s
99% within 3 s

Question 14

Define standard normal distribution

Answer 14

Normal density with mean of 0 and standard deviation of 1

Question 15

How to standardize random variable

Answer 15

Z = (X - μ) / σ

Question 16

Calculate and interpret probabilities using standard normal distribution

Answer 16

N(x) is notation - use table for positive x

N(-x) = 1 - N(x)

Question 17

Shortfall risk

Answer 17

Risk that portfolio will fall below a minimum acceptable level over some time horizon

Question 18

Calculate safety-first ratio

Answer 18

SFRatio = [E(Rp) - RL)] / σ

RL = threshold return level

Question 19

Select optimal portfolio using Roy’s safety first criterion

Answer 19

Determine SFRatios and Choose portfolio with highest SFRatio

Question 20

Relationship between normal and lognormal distributions.

Answer 20

The log is normal

Variable follows lognormal distribution if its natural logarithm is normally distributed. Distribution bounded by zero with long right tail.

Question 21

Distinguish between discretely and continuously compounded rates of returns

Answer 21

Finite intervals vs. unbroken compounding

Question 22

Calculate and interpret continuously compounded rate of return

Answer 22

ln(1 + r)

i.e. natural logarithm of 1 plus holding period return

Question 23

Explain monte carlo simulation

Answer 23

generate large number of random samples from specified distribution to represent role of risk in system.

Question 24

Compare monte carlo simulation and historical simulation

Answer 24

Historical info provides the most direct evidence on distributions, but excludes any info on risks not shown in data and can’t be used for “what if” analysis

Question 25

Binomial distribution function

Answer 25

p(x) = n!/(n-x)!x! * p^x * (1-p) ^ n-x

x is number of successes, n is number of trials, p is probability of the event

Question 26

Why lognormal use to model asset prices?

Answer 26

Because it is bounded by zero and is skewed to right (long tail). Asset prices can’t be below zero.

Question 27

Major uses of monte carlo simulation

Answer 27

• Planning
• Risk management
• Value complex securities