Quantitative Methods - Common Probability Distributions

Start - 1:45 pm

Common Probability Distributions

I'm going to be using Schweser for this section as well.

Introduction

• Probability distribution - specifies the probabilities of possible outcomes of a random variable
• We will explore 4 distributions and their uses
• Definitions
• Random variable - we know this
• "Discrete" - countable
• Continuous - noncountable (e.g. rate of return)
• Here you usually talk about the probability of falling in a range, because any given discrete outcome has 0 probability
• Must understand whether a variable is continuous or discrete, and sometimes can choose
• Usually guided by which distribution is most efficient for the task
• Every random variable is associated with a prob distribution that describes it completely

Probability function

• p(x) is the probability that a random variable is equal to a specific value
• That is, p(x) is the probability that X = x
• p(x) must be from 0 to 1 and the sum of all probabilities must = 1
• Probability density function - used to calculate probability of an outcome between two values
• Cumulative distribution function - defines probability that X takes a value less than or equal to x - represents the sum or cumulative value of the probabilities
• F(x) = P(X<=x)

Discrete uniform random variable

• Probabilities of all possible outcomes are equal
• Assume X = [2, 4, 6, 8, 10]
• p(6) = 0.2
• F(6) = np(x) = 3(0.2) = 0.6, the probability that X<= 6.  Note that 6 is the third number.

Binomial distribution

• Binomial random variable is the number of successes in a given number of trials
• Outcome is either success or failure
• Probability of success p is constant for each trial, trials are independent
• "Bernoulli" random variable is a variable for which there is only 1 trial
• Trial is a mini experiment
• Final outcome: number of successes in a series of n trials
• Binomial probability function defines probability of x successes in n trials
• probability of exactly x successes in n trials = (number of ways to choose x from n) * p^x * (1-p)^(n-x)
• Number of ways = n! / ((n-x)!*x!))
• p = probability of success on each trial
• Combined formula:

• Sidenote - expected value
• Expected value simply equals np, the number of trials times p(success)

Binomial trees

• Shows all possible combinations of up and down moves for a number of periods
• Each possible value along a tree is a node
• Using the price changes (%) each time and the cumulative probabilities of up and down moves you can assign probabilities and find expected value - see below

Continuous Uniform Distribution

• A range from a to b - outcomes can only occur from a to b
• P(X is between x1 and x2) = (x2 - x1) / (b - a)
• This is just the selected area divided by the total area
• Note that the distribution is a rectangle
• CDF is therefore a straight line going up from a and then horizontal after b

The Normal Distribution

• Completely described by mean and variance
• Skewness = 0, so mean = median = mode
• Kurtosis = 3, therefore excess kurtosis = 0 (all kurtosis is measured relative to normal)
• A linear combination of normally distributed random variables is also normally distributed
• Probabilities get smaller at tails but never go to 0

Univariate and Multivariate Distributions

• Univariate - distribution of a single variable
• But sometimes the relationship between two or more random variables is relevant
• Multivariate Distributions
• Specifies probabilities associated with a group of random variables
• Only meaningful when behavior of a random variable somehow depends on that of the others
• Can apply both to discrete and continuous
• For two discrete variables, described by joint probability tables
• For two continuous variables, can make a multivariate normal distribution if the individual variables follow a normal distribution
• Correlation and Multivariate Normal Distributions
• Similar to normal, a Multivariate can be described by the means and variances of the individual random variables
• One must also specify the correlation between the pair of variables
• Correlation is the strength of the linear relation between a pair variables
• To describe a Multivariate in a portfolio of n assets, you need n means, n variances, and 0.5*n*(n-1) correlations.  Ex, for 4 assets, you need 4 means, 4 variances, and 0.5*4*3=6 correlations
• When building a portfolio, all else equal you want lower correlation because this means lower variance

Confidence Intervals

• A 95% confidence interval is the range we expect the variable to be in 95% of the time
• Based on expected value and standard deviation(s)
• 68% fall within 1 std dev and 95% fall within 2 std devs
• 90% CI = Xbar +/- 1.65 std devs
• 95% CI = Xbar +/- 1.96 std devs
• 99% CI = Xbar +/- 2.58 std devs

Standard Normal Distribution

• Has been standardized so it has a mean of 0 and sdev of 1
• To standardize any random variable:
• z = (observation - population mean) / std dev
• Can use this z-score and the table to calculate probabilities of falling above/below a certain range

Shortfall Risk/Safety First Ratio/Roy's Safety First Criterion

• Shortfall risk - probability that a portfolio value (or return) will fall below a particular (target) value or return over a given period of time
• Roy's Safety First Criterion (similar to Sharpe Ratio but uses benchmark instead of riskfree rate)
• Minimizes the shortfall risk by maximizing the Safety First Ratio
• Safety First Ratio:
• SF = [E(Rp) - Rl] / sdevportfolio
• This ratio gives the number of sdevs below the mean

Normal and Lognormal Distributions

• Lognormal is generated by function e^x where x is normally distributed
• Lognormal is skewed right and bounded by 0 on the left side - useful for modeling asset prices which never take negative values
• If we used normal, we would admit possibility of returns less than -100%
• This allows us to model 'price relatives' (i.e. 1+HPR)

Discrete versus Continuous Compounding

• Discrete goes period by period
• For an annual rate R, the continuously compounded EAR = e^R - 1
• Can reverse this calculation by taking the ln of 1+EAR
• Continuously compounded rates are additive for multiple periods

Monte Carlo Simulation

• Technique based on repeated generation of one or more risk factors that affect security values
• Uses this it generates a distribution of security values
• Must first make probability distributions for each risk factor
• Computer then generates random variables based on assumed probabilities
• These then spit out security values
• Repeated thousands of times to generate an idea of mean and perhaps variance of security values
• Applications:
• Value complex securities
• Simulate a trading strategy
• Determine risk of a portfolio
• Simulate pension fund assets/liabilities
• Value portfolios of assets that have non-normal returns distributions
• Limitations
• Complex and output only as good as input
• Not analytic but statistical, cannot provide insights that analytic methods can

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

• Rather than model distribution of risk factors, a prior period is used
• Each iteration randomly selects one of these past changes and calculates value of portfolio in question based on the changes in risk factors
• Advantage is that it uses actual distributions so you don't have to estimate
• But past changes in risk factors don't necessarily predict future changes
• Infrequent events might not be reflected unless captured in the timeframe
• Another disadvantage is it is not as good at 'what if' analysis as Monte Carlo - i.e. in Monte Carlo you can increase the variance of one of the risk factors by 20%, in historical you cannot do this

End of reading.

3:45 pm

2 hours