Reading 10: Common Probability Distributions Flashcards
Probability Distribution
PoORV 1
Lists all the possible outcomes of an experiments, along with their associated probabilities
Discrete Random Variable (PPFO)
Positive probabilities associated with a finite number of outcomes
E.g. number of days it will rain in a month
Probability Function (Px) (Definition)
Probability that a random variable is equal to a specific value
E.g. Probability that random variable X = x
Continuous Random Variable
(PoINO)
P(x1 < X < x2)
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)
Distributions
The assignment of probabilities to the possible outcomes
Cumulative Distribution Function (“CDF”) (SoP)
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
Cumulative Distribution Formula
F(x) = P(X < x)
Where:
< = less than or equal to
Discrete uniform distribution (“NDE”)
One where there are n discrete, equally likely outcomes
Cumulative Distribution Function (Fxn) for the “nth” outcome (Formula)
F(x) = n*P(x)
Where:
n = number of possible outcomes
Binomial Random Variable (Definition) (PxNt)
Probability of “x” successes in “n” trials where the outcomes can either be ‘success’ or ‘failure’
Bernoulli Random Variable
A binomial random variable for which the number of trials is 1
Binomial Random Variable (Formula) (CFS)
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!)
Expected Value of X (Formula)
E(X) = np
Where:
n = number of trials
p = probability of success on each trial
Variance of a binomial random variable (Formula)
Variance of X = np(1-p)
Binomial Tree (“Binomial Stock Price Model”) is constructed by…
Showing all the possible combinations of up-moves and down-moves over a number of successive periods
Node
Each of the possible values along a binomial tree
Binomial Tree (“Binomial Stock Price Model”) applications
Pricing Options.
We can make it more realistic by shortening the length of the periods and increasing the number of periods and possible outcomes
Continuous Uniform Distribution (Definition)
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%
Continuous Uniform Distribution (Formulas)
P(x1 < X < x2) = (x2 - x1)/(b-a)
Where:
X’s = probability between these two
A/B = total distribution
Continuous Uniform Distribution (Prob of x’s between a and b expression)
a < x1 < x2 < b
For all x’s between a and b
Continuous Uniform Distribution (Prob of x outside boundaries expression)
P(x < a x > b) = 0
Probability of x outside of the boundaries is zero
Continuous Uniform Distribution (Prob of outcomes between x1 and x2 formula)
= (x2 - x1) / (b-a)
Normal Distribution (Key Properties x5)
- Symmetrical and bell-shaped curve with a single peak at the exact center of the distribution
- Mean = median = mode, and all are in the exact center of the distribution
- Skew = 0, Kurtosis = 3. Can be completely defined by its mean as a result
- Tails get thin but extend infinitely
- Linear combination of normally distributed random variables is normally distributed as well
Univatiate Distribution
Distribution of a single random variable
Multivariate Distribution
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
Correlation (Definition relating to multi/univariates)
Feature that distinguishes a multivariate distribution from a univariate normal
It indicates the strength of the linear relationship between a pair of random variables
Correlation (Formula for multi/univariates)
0.5n(n-1)
Where:
N = Number of assets
Confidence Interval (Definition + example)
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
Confidence Interval (Formula)
Mean + / - (1 + CI)(SD) = x
Where:
CI = Specified confidence interval
+ / - - Use plus for the top of the range and minus for the bottom
Standard Normal Distribution (Definition)
Normal distribution that has been standardized so that it has a mean of zero and a standard deviation of 1
Z-Value (Definition)
Number of standard deviations a given observation is from the population mean
Standardization (Definition)
Process of converting an observed value for a random variable to it z-value
Z Value (Formula)
Z = (observation - population mean) / Standard Deviation
Z Table (Important characteristic)
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
Shortfall Risk (Definition)
Probability that a portfolio value or return will fall below a particular (target) value or return over a given time period
Roy’s Safety-First Criterion (Definition)
States that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level (“threshold level”)
Roy’s Safety-First Criterion (Expression)
Minimize P(Rp < RL) Where: Rp = portfolio return Rl = threshold level return
What other rule is Roy’s Safety-First Criterion similar to?
Sharpe Ratio. Although Sharpe ratio measures excess returns over a risk free rate
Shortfall Risk Ratio (Formula)
SFRatio = [E(Rp) - RL] / SD p
Where:
Rp = portfolio return
RI = threshold level return
Larger SFR ratio the better
Explain the two steps used when choosing portfolio’s with Roy’s safety-first criterion
- Determine the shortfall risk ratio
2. Choose the portfolio with the largest SFR Ratio
Lognormal Distribution (Definition)
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
Lognormal Distribution (Characteristics)
- 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
Price Relative (Definition)
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
Ending Price (Formula)
Price Relative x Beginning Price
Discretely Compounded Returns
Compound returns given some discrete compounding period, such as semiannual or quarterly
Compound Returns (Formula)
[(1+ I/Y)^compounding periods] - 1
EAR based on continuous compounding
[x^Rcc]-1
Where:
Rcc = stated annual rate
EAR to continuous compounding rate
1+EAR + [LN key in the calculator]
HPR to continuous compounding rate
1+HPR + [LN key in the calculator]
Continuous compounding rate to HPR
HPRt = [e^Rcc x t] -1
Where:
t = holding period years
Monte Carlo Simulation (Definition)
Uses randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values
Monte Carlo Simulation (Limitations)
- Fairly Complex
- Answers are no better than the assumptions used
- Not an analytic method, but a statistical one, and cannot provide the insights that analytical methods can
Monte Carlo Simulation (Uses)
- Value complex securities
- Simulate the profits/losses from a trading strategy
- Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets and liabilities
- Simulate pension fund assets/liabilities to examine the variability of the difference between the two
- Value portfolios of assets that have non-normal returns distributions
Historical Simulation (Definition)
Historical simulation is based on actual changes in value or actual changes in risk factors over some prior period
Historical Simulation (Limitations)
- Past changes in risk factors may not be a good indication of future changes
- Infrequent events may not be reflected in historical simulation results
- Cannot address the ‘what of’ scenario that the MC simulation can
Historical Simulation (Advantage)
Distribution of changes in the risk factors does not have to be estimated
90% Confidence Interval for X is…
1.65
95% Confidence Interval for X is…
1.96
99% Confidence Interval for X is…
2.58
