Common Probability Distributions

Question 1

Describes the probabilities of all possible outcomes for a random variable

Answer 1

Probability Distribution

Question 2

The number of possible outcomes that can be counted , and for each possible outcome, there is a measureable and positive probability

Answer 2

Discrete Random Variable

Question 3

A variable which the number of possible outcomes is infinite, even if lower and upper bounds exist.

Answer 3

Continuous Random Variable

Question 4

A function that defines the probability that a random variable, X, takes on a value equal to or less than a specific value, X.

Answer 4

Cumulative Distribution Function

*F(x) == P(X<=x)

Question 5

A variable for which the probabilities of all possible outcomes for a discrete random variable are equal

Answer 5

Discrete Uniform Random Variable

Question 6

The number of “successes” in a given number of trials, where the outcome is either a success or a failure. (The trials must be independent.)

Answer 6

Binomial Random Variable

Question 7

Calculation for the probability of x successes in n trials (in a Bernoulli random variable)

Answer 7

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

Question 8

Expected Value for Bernoulli Random Variable

Answer 8

E(x) = n * p

Question 9

Variance of a Binomial Random Variable

Answer 9

Var(X) = (n*p) * (1-p)

Question 10

A distribution defined over a range that spans between a lower limit, a, and upper limit, b, which serve as parameters of the distribution.

Answer 10

Continuous Uniform Distribution

Question 11

Properties of Continuous Uniform Distribution

Answer 11

*a<= x1<= x2<= b
**P(x<a> b) = 0
*** P(x1 < x < x2) = (x2-x1) / (b-a)
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Question 12

Properties of a Normal Distribution

Answer 12

• X is normally distributed with mean,u, and variance, sigma^2
• • Skewness = 0, Mean = Median = Mode
• ** Kurtosis = 3
• *** A linear combination of normally distributed random variables is also normally distributed.
• ** The probabilities of outcomes further above and below the mean get smaller and smaller, but do not go to zero.

Question 13

A distribution the specifies the probabilities associated with a group of random variables and is meaningful only when the behavior of each random variable in the group is in some way dependent on the behavior of the others.

Answer 13

Multivariate Distribution

Question 14

Confidence Interval Ranges

Answer 14

90% C.I. is x +/- 1.65 sd
95% C.I. is x +/- 1.96 sd
99% C.I. is x +/- 2.58 sd

Question 15

A normal distribution that has been standardized so that it has a u = 0, and sigma = 1.

Answer 15

Standard Normal Distribution

* z = (x - u) /sigma

Question 16

The number of standard deviations above or below the mean

Answer 16

z-score

Question 17

The probability that a portfolio value or return will fall below a particular value or return over a given period of time.

Answer 17

Shortfall Risk

Question 18

The optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level.

Answer 18

Roy’s Safety First Ratio

* min P(Rp -Rl)

Question 19

A distribution that is generated by the function e^x, where x is normally distributed

Answer 19

Lognormal Distribution

• Skewed to the right
• • Bounded from below by zero, so that it is useful for modeling asset prices which never take a negative value

Question 20

Calculation for Continuously Compounded Returns

Answer 20

Rcc = e^Rcc - 1

Question 21

Applications of Rcc

Answer 21

• ln(EAR) = Rcc
• • Rcc = ln(S1/S0) = ln( 1 + HPR)
• ** HPRt = e^Rcc*T - 1

Question 22

A technique based on the repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values.

Answer 22

Monte Carlo Simulation

Question 23

Monte Carlo Simulations are used to:

Answer 23

• Value complex securities
• • Simulate the profits/losses for a trading strategy
• ** Calculate estimates of value at risk(VAR) to determine the riskiness of a portfolio of assets and liabilities
• *** Simulate pension funds assets and liabilities overtime to examine the variability of the difference
• ** Value portfolios of assets that have non-normal returns distributions

Question 24

Limitations of Monte Carlo Simulations

Answer 24

• Fairly Complex
• • Will provide answers that are no better than the assumptions about the distributions of the risk factors and the pricing/valuation model that is used.
• ** Not an analytic method, only statistical method.

Question 25

A simulation based on actual changes in value or risk factors over some prior period.

Answer 25

Historical simulation

Question 26

Advantage to historical simulations

Answer 26

distribution of changes in risk factors don’t have to be estimated.

Question 27

Disadvantages of historical simulations

Answer 27

• risk factors of past may not be good predictors of the future
• • Cannot address “what-if” scenarios like MC simulations can. (i.e. What if we increase variance of risk B by 20%. How much does portfolio change? and in what direction?