Part 4. Common Probability Distributions

Question 1

Probability distribution

Answer 1

Specifies the probabilities associated with the possible outcomes of a random variable.

Question 2

Random variable

Answer 2

A quantity whose future outcomes are uncertain.

Question 3

Types of random variables:

Answer 3

1. 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

Question 4

Continuous random variable (Z)

Answer 4

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.

Question 5

Probability Function

Answer 5

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).

Question 6

Cumulative distribution function

Answer 6

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.

Question 7

Discrete uniform distribution

Answer 7

The distribution has a finite number of specified outcomes, and each outcome is equally likely.

Question 8

CDF characteristics:

Answer 8

• 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.

Question 9

Continuous uniform distribution

Answer 9

An appropriate probability model to represent a particular kind of uncertainty in beliefs in which all outcomes appear equally likely.

Question 10

Binomial distribution

Answer 10

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.

Question 11

Bernoulli trial

Answer 11

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.

Question 12

Binomial distribution assumptions:

Answer 12

• The probability p of success is constant for all trials
• The trials are independent.
  ie: X - B(n, p)

Question 13

Up transitory probability

Answer 13

The stock moves up with constant probability p, u is 1 plus the rate of return for an up move.

Question 14

Down transitory probability

Answer 14

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.

Question 15

Binomial tree

Answer 15

We now associate each of the n=4 stock price moves with time indexed by t.

Question 16

Node

Answer 16

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.

Question 17

Final stock price distribution

Answer 17

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.

Question 18

Central Limit Theorem

Answer 18

States that the sum (and mean) of a large number of independent random variables (with finite variance) is approximately normally distributed.

Question 19

Univariate distribution

Answer 19

A single random variable

Question 20

Multi variate normal distribution

Answer 20

Specifies the probabilities for a group of related random variables.

Used for security returns

Question 21

Multivariate normal distribution for returns on n stocks defined by 3 parameters:

Answer 21

1. the list of the mean returns on individual securities (n means in total).
2. The list of the securities variances of return (n variances in total).
3. The list of all the distinct pairwise return correlations: n(n-1)/2 distinct correlations in total.

Question 22

Standard normal distribution

Answer 22

the normal density with u=0 and theta = 1

Question 23

Fama (1976), and Campbell, Lo and MacKinlay (1997):

Answer 23

• 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).

Question 24

Mean - variance analysis

Answer 24

Holds exactly when investors are risk averse, when they choose investments so as to maximise expected utility or satisfaction.

When:

1. returns are normally distributed
2. 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.

Question 25

Safety first rules

Answer 25

Only evaluates downside risk.

Question 26

Shortfall risk

Answer 26

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.

Question 27

Sharpe Ratio

Answer 27

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).

Question 28

2 mainstays in managing financial risk:

Answer 28

1. Value at Risk (VaR)

2. Stress testing/scenario analysis

Question 29

VaR

Answer 29

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).

Question 30

Methods to estimate VaR:

Answer 30

1. Variance-covariance
2. Analytical method

assuming returns follow a normal distribution

Question 31

Lognormal distribution

Answer 31

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.

Question 32

Black-Scholes-Merton model

Answer 32

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.

Question 33

Properties of lognormal distributions

Answer 33

• Bounded below by 0

- It is skewed to the right (long right tail).

Question 34

Price Relative (S1/S0)

Answer 34

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.

Question 35

Continuously compounded return

Answer 35

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

Question 36

CCR Key assumptions

Answer 36

• The returns are independently and identically distributed (i.i.d).

Question 37

Independence

Answer 37

Captures the proposition that investors cannot predict future returns using past returns.

Question 38

Identical distribution

Answer 38

Captures the assumption of stationarity, a property implying that mean and variance of return do not change from period to period.

Question 39

Volatility

Answer 39

This measures the standard deviation of continuously compounded returns on the underlying asset, by convention stated as an annualised measure.

Question 40

Sample variance/std dev

Answer 40

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.

Question 41

Standard t-distribution

Answer 41

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.

Question 42

Chi square distribution

Answer 42

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.

Question 43

F-distribution

Answer 43

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.

Question 44

Monte Carlo Simulation

Answer 44

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.

Question 45

Uses of Monte Carlo Simulation

Answer 45

1. 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.

2. A tool for valuing complex securities for which no analytic pricing formula is available.
3. 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.

Question 46

Random number generator

Answer 46

Refers to an algorithm that produces uniformly distributed random numbers between 0 and 1.

Question 47

Random number

Answer 47

An observation drawn from an uniform distribution.

Question 48

Inverse transformation method

Answer 48

The technique for producing random observations.