Part 4. Common Probability Distributions Flashcards
Probability distribution
Specifies the probabilities associated with the possible outcomes of a random variable.
Random variable
A quantity whose future outcomes are uncertain.
Types of random variables:
- Discrete random variable - take on at most a countable (possible infinite) number of possible values.
e. g. a discrete random variable X can take on a limited number of outcomes x1, x2,….,xn (n possible outcomes), or a discrete random variable Y can take on an unlimited number of outcomes y1, y2,… (without end).
Countable & finite - the number of yes votes at a corporate board meeting.
Countable & infinite - no. of trades by the market participants.
X - random variable
x - outcome of X
Continuous random variable (Z)
We cannot count the outcomes, so cannot describe possible outcomes of Z with a list z1, z2,….as the outcome (z1 + z2)/2 is not on the list, would always be possible.
e.g. volume of water in a glass, number of central bank board members voting for rate hike.
Probability Function
Specifies the probability that the random variable takes on a specific value: P(X=x) is the probability that the random variable X takes on value x.
For discrete random variable (prob. mass function) is p(X) = P(X=x).
For continuous random variables (prob. density function) denoted as f(x).
Cumulative distribution function
Gives the probability that a random variable X is less than or equal to a particular value x, P(X less than equal to x).
For both discrete and continuous random variables: F(x) = P(X less than equal to x).
Parallel to cumulative relative frequency.
Discrete uniform distribution
The distribution has a finite number of specified outcomes, and each outcome is equally likely.
CDF characteristics:
- the cdf lies between 0 and 1 for any x: 0 to F(x) to 1
- as x increases, the cdf either increases or remains constant.
Continuous uniform distribution
An appropriate probability model to represent a particular kind of uncertainty in beliefs in which all outcomes appear equally likely.
Binomial distribution
Used when we make probability statements about a record of success and failures/anything with binary outcomes.
Built from the Bernoulli random variable.
- Symmetric when the probability of success on trial is 0.5, but asymmetric/skewed otherwise.
Bernoulli trial
If we let Y = 1 when the outcome is successful, and Y = 0 when the outcome is failure, then the probability function of Bernoulli random variable Y is:
p(1) = P(Y=1) = p p(0) = P(Y=0) = 1-p
where p is the probability that the trial is a success.
In n Bernoulli trials, we can have 0 to n successes, and if outcome of individual trial is random, the total number of successes in n trials is also random.
Binomial distribution assumptions:
- The probability p of success is constant for all trials
- The trials are independent.
ie: X - B(n, p)
Up transitory probability
The stock moves up with constant probability p, u is 1 plus the rate of return for an up move.
Down transitory probability
The stock moves down with constant probability 1-p, if it moves down, d is 1 plus the rate of return for a down move.
Binomial tree
We now associate each of the n=4 stock price moves with time indexed by t.
Node
Each boxed value from which successive moves or outcomes branch out in the tree.
The initial node at t=0, shows the beginning stock price S, subsequent nodes represent a potential value for stock price at a specified future time.
Final stock price distribution
A function of the initial stock price, the number of up moves, and the size of up and down moves.
The stock price is a function of a binomial random variable, as well as of u and d, and initial price S.
The standard formula describes a process in which stock volatility is constant over time, but binomial trees can be used to model changing volatility overtime.
Central Limit Theorem
States that the sum (and mean) of a large number of independent random variables (with finite variance) is approximately normally distributed.
Univariate distribution
A single random variable
Multi variate normal distribution
Specifies the probabilities for a group of related random variables.
Used for security returns
Multivariate normal distribution for returns on n stocks defined by 3 parameters:
- the list of the mean returns on individual securities (n means in total).
- The list of the securities variances of return (n variances in total).
- The list of all the distinct pairwise return correlations: n(n-1)/2 distinct correlations in total.
Standard normal distribution
the normal density with u=0 and theta = 1
Fama (1976), and Campbell, Lo and MacKinlay (1997):
- The normal distribution is a closer fit for quarterly and yearly holding period returns on diversified equity portfolio than for daily or weekly returns.
- Persistent departure from normality in most equity return series is kurtosis greater than 3 (fat tails problem).
- Approximating equity return distribution with normal, normal tends to underestimate the probability of extreme returns.
- Normal is less suitable for asset prices than model for returns, as asset can only drop to 0 (worthless).
Mean - variance analysis
Holds exactly when investors are risk averse, when they choose investments so as to maximise expected utility or satisfaction.
When:
- returns are normally distributed
- investors have quadratic utility functions
A concept used in economics for mathematical representation of attitudes towards risk and return.
Generally, considers risk symmetrically, where standard deviation captures variability both above and below mean.
Safety first rules
Only evaluates downside risk.
Shortfall risk
The risk that portfolio value (or portfolio return) will fall below some minimum acceptable level over some time horizon.
e.g. the risk that the assets in a defined benefit plan will fall below plan liabilities.
Sharpe Ratio
If we sub risk free rate, Rf, for the critical level Rl, the SFRatio becomes the Sharpe ratio.
Evaluating portfolios using Sharpe ratio, the portfolio with the highest Sharpe ratio is the one that minimises the probability that portfolio return will be less than risk free rate (given normality assumption).
2 mainstays in managing financial risk:
- Value at Risk (VaR)
2. Stress testing/scenario analysis
VaR
A money measure of the minimum value of losses expected over a specified time period, at a given level of probability.
e.g. 1 day time horizon and given level of probability 0.05 is 95% one day VaR.
If VaR = £5million for a portfolio, there would be a 0.05 probability the portfolio would lose £5m or more in a single day (assuming correct assumptions).
Methods to estimate VaR:
- Variance-covariance
- Analytical method
assuming returns follow a normal distribution
Lognormal distribution
This is widely used for modelling the probability distribution of share and other asset prices.
e.g. Black Scholes Merton option pricing model.
It has been found to be a usefully accurate description of the distribution of prices for many financial assets.
2 parameters are:
- mean
- standard deviation (variance) of its associated normal distribution:
the mean and variance of ln Y, given Y is lognormal.
Black-Scholes-Merton model
Assumes the price of asset underlying the option is lognormally distributed.
A random variable Y follows a lognormal distribution if its natural logarithm, where U is normally distributed.
If natural logarithm of random variable Y, ln Y is normally distributed, then Y follows a lognormal distribution.
Properties of lognormal distributions
- Bounded below by 0
- It is skewed to the right (long right tail).
Price Relative (S1/S0)
This is an ending price, S1, over a beginning price, S0, it is equal to 1 plus the holding period return on the stock from t=0 to t=1;
S1/S0 = 1 + R0,1
General form:
St+1/St = 1 + Rt,t+1,
where Rt,t+1 is the rate of return from t to t+1.
Continuously compounded return
Associated with a holding period return is the natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price (the price relative).
CCR from t to t+1 is:
rt,t+1 = ln(St+1/St) = ln(1 + Rt,t+1)
Used in:
- Asset pricing models
- Risk Management
CCR Key assumptions
- The returns are independently and identically distributed (i.i.d).
Independence
Captures the proposition that investors cannot predict future returns using past returns.
Identical distribution
Captures the assumption of stationarity, a property implying that mean and variance of return do not change from period to period.
Volatility
This measures the standard deviation of continuously compounded returns on the underlying asset, by convention stated as an annualised measure.
Sample variance/std dev
To compute standard deviation of a set of n returns, we sum squared deviation of each return from mean return, then divide sum by n-1.
Square root of sample variance gives sample std dev.
e.g. completed on basis of 250 days markets open for trading; if daily volatility were 0.01, volatility (on annual basis) as 0.01root250 = 0.1581. - shares of Astra Int.
Standard t-distribution
This is a symmetrical probability distribution defined by a single parameter known as degrees of freedom (df), the number of independent variables used in defining sample statistics such as variance, and the probability distributions they measure.
Chi square distribution
This distribution is asymmetrical, and is a family of distributions.
This distribution with k degrees of freedom is the distribution of the sum of the squares of k independent standard normally distributed random variables, hence distribution does not take any negative values.
The df increase, the shape of its pdf becomes more similar to a bell curve.
F-distribution
This is a family of asymmetrical distributions bounded from below by 0, with domain of pdf are positive.
Each F-dist. is defined by 2 values of degrees of freedom, called the numerator and denominator degrees of freedom.
Both numerator (df1) and denominator (df2) degrees of freedom increase, the density function will also become more bell curve like.
Monte Carlo Simulation
The generation of a large number of random samples from a specified probability distribution or distributions to represent the role of risk in the system.
A complement to analytical methods, providing statistical estimates, not exact results.
Its application in investment management continue to grow.
Uses of Monte Carlo Simulation
- To estimate risk and return in investment applications.
- We simulate the portfolios profit and loss performance for a specified time horizon.
- Repeated trials within the simulation (each trial involving a draw of random observations from a probability dist.) produces a simulated frequency distribution of portfolio returns which performance and risk measures are derived.
- A tool for valuing complex securities for which no analytic pricing formula is available.
- An important modelling resource
- The control of assumptions when carrying out a simulation, means we can run a model for valuing such securities through Monte Carlo simulation to examine models sensitivity to change in key assumptions.
Random number generator
Refers to an algorithm that produces uniformly distributed random numbers between 0 and 1.
Random number
An observation drawn from an uniform distribution.
Inverse transformation method
The technique for producing random observations.
