Reading 10: Common Probability Distributions

Question 1

Probability Distribution

PoORV 1

Answer 1

Lists all the possible outcomes of an experiments, along with their associated probabilities

Question 2

Discrete Random Variable (PPFO)

Answer 2

Positive probabilities associated with a finite number of outcomes

E.g. number of days it will rain in a month

Question 3

Probability Function (Px) (Definition)

Answer 3

Probability that a random variable is equal to a specific value

E.g. Probability that random variable X = x

Question 4

Continuous Random Variable
(PoINO)

P(x1 < X < x2)

Answer 4

Positive probabilities associated with an infinite number of outcomes

P(x) = 0

e.g. if a variable can take on any value between two specified values (rain)

Question 5

Distributions

Answer 5

The assignment of probabilities to the possible outcomes

Question 6

Cumulative Distribution Function (“CDF”) (SoP)

Answer 6

Cumulative probability that a random variable will be less than or equal to a specific value e.g. sum of all probabilities up to a specific point

Question 7

Cumulative Distribution Formula

Answer 7

F(x) = P(X < x)
Where:
< = less than or equal to

Question 8

Discrete uniform distribution (“NDE”)

Answer 8

One where there are n discrete, equally likely outcomes

Question 9

Cumulative Distribution Function (Fxn) for the “nth” outcome (Formula)

Answer 9

F(x) = n*P(x)
Where:
n = number of possible outcomes

Question 10

Binomial Random Variable (Definition) (PxNt)

Answer 10

Probability of “x” successes in “n” trials where the outcomes can either be ‘success’ or ‘failure’

Question 11

Bernoulli Random Variable

Answer 11

A binomial random variable for which the number of trials is 1

Question 12

Binomial Random Variable (Formula) (CFS)

Answer 12

P(x) = n! / (n-x)!x! * p^x * (1-p)^n-x
Where:
p = probability of success in each trial (don’t confuse with px!)

Question 13

Expected Value of X (Formula)

Answer 13

E(X) = np
Where:
n = number of trials
p = probability of success on each trial

Question 14

Variance of a binomial random variable (Formula)

Answer 14

Variance of X = np(1-p)

Question 15

Binomial Tree (“Binomial Stock Price Model”) is constructed by…

Answer 15

Showing all the possible combinations of up-moves and down-moves over a number of successive periods

Question 16

Node

Answer 16

Each of the possible values along a binomial tree

Question 17

Binomial Tree (“Binomial Stock Price Model”) applications

Answer 17

Pricing Options.
We can make it more realistic by shortening the length of the periods and increasing the number of periods and possible outcomes

Question 18

Continuous Uniform Distribution (Definition)

Answer 18

Where the probability of X occurring between a and b e.g. if the range is one half of the whole distribution then the probability of x occurring in the range will be 50%

Question 19

Continuous Uniform Distribution (Formulas)

Answer 19

P(x1 < X < x2) = (x2 - x1)/(b-a)
Where:
X’s = probability between these two
A/B = total distribution

Question 20

Continuous Uniform Distribution (Prob of x’s between a and b expression)

Answer 20

a < x1 < x2 < b

For all x’s between a and b

Question 21

Continuous Uniform Distribution (Prob of x outside boundaries expression)

Answer 21

P(x < a x > b) = 0

Probability of x outside of the boundaries is zero

Question 22

Continuous Uniform Distribution (Prob of outcomes between x1 and x2 formula)

Answer 22

= (x2 - x1) / (b-a)

Question 23

Normal Distribution (Key Properties x5)

Answer 23

1. Symmetrical and bell-shaped curve with a single peak at the exact center of the distribution
2. Mean = median = mode, and all are in the exact center of the distribution
3. Skew = 0, Kurtosis = 3. Can be completely defined by its mean as a result
4. Tails get thin but extend infinitely
5. Linear combination of normally distributed random variables is normally distributed as well

Question 24

Univatiate Distribution

Answer 24

Distribution of a single random variable

Question 25

Multivariate Distribution

Answer 25

Probabilities associated with a group of random variables. Only meaningful when the behavior of each variable in the group is dependent upon the behavior of the others

Question 26

Correlation (Definition relating to multi/univariates)

Answer 26

Feature that distinguishes a multivariate distribution from a univariate normal It indicates the strength of the linear relationship between a pair of random variables

Question 27

Correlation (Formula for multi/univariates)

Answer 27

0.5n(n-1) Where: N = Number of assets

Question 28

Confidence Interval (Definition + example)

Answer 28

Range within which we have a given level of confidence of finding a point estimate (mean) e.g. a 95% confidence interval is a range that we expect a random variable to be in 95% of the time

Question 29

Confidence Interval (Formula)

Answer 29

Mean + / - (1 + CI)(SD) = x Where: CI = Specified confidence interval + / - - Use plus for the top of the range and minus for the bottom

Question 30

Standard Normal Distribution (Definition)

Answer 30

Normal distribution that has been standardized so that it has a mean of zero and a standard deviation of 1

Question 31

Z-Value (Definition)

Answer 31

Number of standard deviations a given observation is from the population mean

Question 32

Standardization (Definition)

Answer 32

Process of converting an observed value for a random variable to it z-value

Question 33

Z Value (Formula)

Answer 33

Z = (observation - population mean) / Standard Deviation

Question 34

Z Table (Important characteristic)

Answer 34

Contains values generated using the cumulative density function e.g. values in the z-table are the probabilities of observing a z-value that is LESS THAN a given value

Question 35

Shortfall Risk (Definition)

Answer 35

Probability that a portfolio value or return will fall below a particular (target) value or return over a given time period

Question 36

Roy's Safety-First Criterion (Definition)

Answer 36

States that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level ("threshold level")

Question 37

Roy's Safety-First Criterion (Expression)

Answer 37

``` Minimize P(Rp < RL) Where: Rp = portfolio return Rl = threshold level return ```

Question 38

What other rule is Roy's Safety-First Criterion similar to?

Answer 38

Sharpe Ratio. Although Sharpe ratio measures excess returns over a risk free rate

Question 39

Shortfall Risk Ratio (Formula)

Answer 39

SFRatio = [E(Rp) - RL] / SD p Where: Rp = portfolio return RI = threshold level return Larger SFR ratio the better

Question 40

Explain the two steps used when choosing portfolio's with Roy's safety-first criterion

Answer 40

1. Determine the shortfall risk ratio | 2. Choose the portfolio with the largest SFR Ratio

Question 41

Lognormal Distribution (Definition)

Answer 41

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

Question 42

Lognormal Distribution (Characteristics)

Answer 42

1. Skewed to the right | 2. Bounded from below by zero so that it is useful for modeling asset prices which never take on negative values

Question 43

Price Relative (Definition)

Answer 43

End-of-period price of the asset divided by the beginning price (S1/S0) and is equal to (1+HPR) Price relative of zero indicates a HPR of -100% as the asset has gone down to zero

Question 44

Ending Price (Formula)

Answer 44

Price Relative x Beginning Price

Question 45

Discretely Compounded Returns

Answer 45

Compound returns given some discrete compounding period, such as semiannual or quarterly

Question 46

Compound Returns (Formula)

Answer 46

[(1+ I/Y)^compounding periods] - 1

Question 47

EAR based on continuous compounding

Answer 47

[x^Rcc]-1 Where: Rcc = stated annual rate

Question 48

EAR to continuous compounding rate

Answer 48

1+EAR + [LN key in the calculator]

Question 49

HPR to continuous compounding rate

Answer 49

1+HPR + [LN key in the calculator]

Question 50

Continuous compounding rate to HPR

Answer 50

HPRt = [e^Rcc x t] -1 Where: t = holding period years

Question 51

Monte Carlo Simulation (Definition)

Answer 51

Uses randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values

Question 52

Monte Carlo Simulation (Limitations)

Answer 52

1. Fairly Complex 2. Answers are no better than the assumptions used 3. Not an analytic method, but a statistical one, and cannot provide the insights that analytical methods can

Question 53

Monte Carlo Simulation (Uses)

Answer 53

1. Value complex securities 2. Simulate the profits/losses from a trading strategy 3. Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets and liabilities 4. Simulate pension fund assets/liabilities to examine the variability of the difference between the two 5. Value portfolios of assets that have non-normal returns distributions

Question 54

Historical Simulation (Definition)

Answer 54

Historical simulation is based on actual changes in value or actual changes in risk factors over some prior period

Question 55

Historical Simulation (Limitations)

Answer 55

1. Past changes in risk factors may not be a good indication of future changes 2. Infrequent events may not be reflected in historical simulation results 3. Cannot address the 'what of' scenario that the MC simulation can

Question 56

Historical Simulation (Advantage)

Answer 56

Distribution of changes in the risk factors does not have to be estimated

Question 57

90% Confidence Interval for X is...

Answer 57

1.65

Question 58

95% Confidence Interval for X is...

Answer 58

1.96

Question 59

99% Confidence Interval for X is...

Answer 59

2.58