Reading 10 - Common Probability Distribution

Question 1

Reading 10 – Common Probability Distributions – Learning outcomes

Answer 1

The candidate should be able to:

define a probability distribution and distinguish between discrete and continuous random variables and their probability functions;

describe the set of possible outcomes of a specified discrete random variable;

interpret a cumulative distribution function;

calculate and interpret probabilities for a random variable, given its cumulative distribution function;

define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable;

calculate and interpret probabilities given the discrete uniform and the binomial distribution functions;

construct a binomial tree to describe stock price movement;

calculate and interpret tracking error;

define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution;

explain the key properties of the normal distribution;

distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution;

determine the probability that a normally distributed random variable lies inside a given interval;

define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution;

define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety-first criterion;

explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices;

distinguish between discretely and continuously compounded rates of return and calculate and interpret a continuously compounded rate of return, given a specific holding period return;

explain Monte Carlo simulation and describe its applications and limitations;

compare Monte Carlo simulation and historical simulation.

Question 2

Probability distribution

Answer 2

A distribution that specifies the probabilities of a random variable’s possible outcomes.

Question 3

Random variable

Answer 3

A quantity whose future outcomes are uncertain

Question 4

The two basic types of random variables

Answer 4

Are discrete random variables and continuous random variables

Question 5

Discrete random variable

Answer 5

A random variable that can take on at most a countable number of possible values.

Question 6

Continuous random variable

Answer 6

A random variable for which the range of possible outcomes is the real line (all real numbers between −∞ and +∞ or some subset of the real line).

Question 7

Probability function

Answer 7

A function that specifies the probability that the random variable takes on a specific value.

Question 8

We can view a probability distribution in two ways:

Answer 8

Probability function

Probability density function

Question 9

Probability function

Answer 9

A function that specifies the probability that the random variable takes on a specific value.

Question 10

Probability density function

Answer 10

A function with non-negative values such that probability can be described by areas under the curve graphing the function.

For continuous random variables, the probability function is denoted f(x) and called the probability density function (pdf), or just the density.

Question 11

A probability function has two key properties (which we state, without loss of generality, using the notation for a discrete random variable):

Answer 11

• 0 ≤ p(x) ≤ 1, because probability is a number between 0 and 1.
• The sum of the probabilities p(x) over all values of X equals 1. If we add up the probabilities of all the distinct possible outcomes of a random variable, that sum must equal 1.

Question 12

Cumulative distribution function

Answer 12

A function giving the probability that a random variable is less than or equal to a specified value.

For both discrete and continuous random variables, the shorthand notation is F(x) = P(X ≤ x).

Question 13

The cdf has two other characteristic properties:

Answer 13

• The cdf lies between 0 and 1 for any x: 0 ≤ F(x) ≤ 1.
• As we increase x, the cdf either increases or remains constant.

Question 14

Bernoulli random variable

Answer 14

A random variable having the outcomes 0 and 1.

named after the Swiss probabilist Jakob Bernoulli (1654–1704)

Question 15

Bernoulli trial

Answer 15

An experiment that can produce one of two outcomes.

If we let Y equal 1 when the outcome is success and Y equal 0 when the outcome is failure, then the probability function of the 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.

Question 16

What does the hat on the p indicated for bernoullis trial?

Answer 16

The “hat” over p indicates that it is an estimate of p.

Question 17

Binomial random variable

Answer 17

The number of successes in n Bernoulli trials for which the probability of success is constant for all trials and the trials are independent.

In n Bernoulli trials, we can have 0 to n successes. If the outcome of an individual trial is random, the total number of successes in n trials is also random. A binomial random variable X is defined as the number of successes in n Bernoulli trials. A binomial random variable is the sum of Bernoulli random variables Y**_i, i = 1, 2, …, n:

X = Y₁ + Y₂ + … + Y**_n

Question 18

The binomial distribution makes these assumptions:

Answer 18

• The probability, p, of success is constant for all trials.
• The trials are independent.

Question 19

Binomial probability function

Answer 19

Question 20

If the price you trade at is fair, then?

Answer 20

50 percent of the trades you do with a broker should be profitable.

Of course, you need to adjust for the direction of the overall market after the trade (any broker’s record will be helped by a bull market) and perhaps make other risk adjustments. Assume that these adjustments have been made.

Question 21

Tracking error

Answer 21

The standard deviation of the differences between a portfolio’s returns and its benchmark’s returns; a synonym of active risk.

Question 22

Mean and variance of binomial random variables equations

Answer 22

Question 23

Mean and variance of binomial random variables explained

Answer 23

Because a single Bernoulli random variable, Y ~ B(1, p), takes on the value 1 with probability p and the value 0 with probability 1 − p, its mean or weighted-average outcome is p. Its variance is p(1 − p).10 A general binomial random variable, B(n, p), is the sum of n Bernoulli random variables, and so the mean of a B(n, p) random variable is np. Given that a B(1, p) variable has variance p(1 − p), the variance of a B(n, p) random variable is n times that value, or np(1 − p), assuming that all the trials (Bernoulli random variables) are independent.

Question 24

Binomial model

Answer 24

A model for pricing options in which the underlying price can move to only one of two possible new prices.

Question 25

Up transition probability

Answer 25

The probability that an asset’s value moves up.

Question 26

Down transition probability

Answer 26

The probability that an asset’s value moves down in a model of asset price dynamics.

Question 27

Binomial tree

Answer 27

The graphical representation of a model of asset price dynamics in which, at each period, the asset moves up with probability p or down with probability (1 – p).

Question 28

Continuous uniform distribution

Answer 28

The continuous uniform distribution is the simplest continuous probability distribution. The uniform distribution has two main uses. As the basis of techniques for generating random numbers, the uniform distribution plays a role in Monte Carlo simulation. As the probability distribution that describes equally likely outcomes, the uniform distribution is an appropriate probability model to represent a particular kind of uncertainty in beliefs in which all outcomes appear equally likely.

Question 29

The pdf for a uniform random variable is:

Answer 29

Question 30

The mathematical operation that corresponds to finding the area under the curve of a pdf f(x) from a to b is the integral of f(x) from a to b:

Answer 30

Question 31

For a continuous uniform random variable, what is the mean and variance?

Answer 31

The mean is given by μ = (a + b)/2 and the variance is given by σ² = (b − a)²/12.

Question 32

Central limit theorem

Answer 32

The central limit theorem states that the sum (and mean) of a large number of independent random variables is approximately normally distributed.

Question 33

The defining characteristics of a normal distribution are as follows:

Answer 33

• The normal distribution is completely described by two parameters—its mean, μ, and variance, σ². We indicate this as X ~ N(μ, σ²) (read “X follows a normal distribution with mean μ and variance σ²”). We can also define a normal distribution in terms of the mean and the standard deviation, σ (this is often convenient because σ is measured in the same units as X and μ). As a consequence, we can answer any probability question about a normal random variable if we know its mean and variance (or standard deviation).
• The normal distribution has a skewness of 0 (it is symmetric). The normal distribution has a kurtosis (measure of peakedness) of 3; its excess kurtosis (kurtosis − 3.0) equals 0.¹⁷ As a consequence of symmetry, the mean, median, and the mode are all equal for a normal random variable.
• A linear combination of two or more normal random variables is also normally distributed.

Question 34

Univariate distribution

Answer 34

A distribution that specifies the probabilities for a single random variable.

Question 35

Multivariate distribution

Answer 35

A probability distribution that specifies the probabilities for a group of related random variables.

Question 36

Multivariate normal distribution

Answer 36

A probability distribution for a group of random variables that is completely defined by the means and variances of the variables plus all the correlations between pairs of the variables.

Question 37

A multivariate normal distribution for the returns on n stocks is completely defined by three lists of parameters:

Answer 37

• the list of the mean returns on the individual securities (n means in total);
• the list of the securities’ variances of return (n variances in total); and
• the list of all the distinct pairwise return correlations: n(n − 1)/2 distinct correlations in total.

For example, a distribution with two stocks (a bivariate normal distribution) has two means, two variances, and one correlation: 2(2 − 1)/2. A distribution with 30 stocks has 30 means, 30 variances, and 435 distinct correlations: 30(30 − 1)/2. The return correlation of Dow Chemical with American Express stock is the same as the correlation of American Express with Dow Chemical stock, so these are counted as one distinct correlation.

Question 38

In order to specify the normal distribution for portfolio return, we need:

Answer 38

we need the means, variances, and the distinct pairwise correlations of the component securities.

Question 39

The normal density function

Answer 39

Question 40

Standard normal distribution

Answer 40

The normal density with mean (μ) equal to 0 and standard deviation (σ) equal to 1.

Question 41

Also called Unit normal distribution, how does this relate to options?

Answer 41

Option returns are skewed. Because the normal is a symmetrical distribution, we should be cautious in using the normal distribution to model the returns on portfolios containing significant positions in options.

Question 42

Having established that the normal distribution is the appropriate model for a variable of interest, we can use it to make the following probability statements:

Answer 42

• Approximately 50 percent of all observations fall in the interval μ ± (2/3)σ.
• Approximately 68 percent of all observations fall in the interval μ ± σ.
• Approximately 95 percent of all observations fall in the interval μ ± 2σ.
• Approximately 99 percent of all observations fall in the interval μ ± 3σ.

Question 43

Standardizing

Answer 43

A transformation that involves subtracting the mean and dividing the result by the standard deviation.

Question 44

Standard normal random variable

Answer 44

Z = (X – μ)/σ  

REMEMBER: N(−x) = 1 − N(x)

Question 45

The following are some of the most frequently referenced values in the standard normal table:

Answer 45

• The 90th percentile point is 1.282: P(Z ≤ 1.282) = N(1.282) = 0.90 or 90 percent, and 10 percent of values remain in the right tail.
• The 95th percentile point is 1.65: P(Z ≤ 1.65) = N(1.65) = 0.95 or 95 percent, and 5 percent of values remain in the right tail. Note the difference between the use of a percentile point when dealing with one tail rather than two tails. Earlier, we used 1.65 standard deviations for the 90 percent confidence interval, where 5 percent of values lie outside that interval on each of the two sides. Here we use 1.65 because we are concerned with the 5 percent of values that lie only on one side, the right tail.
• The 99th percentile point is 2.327: P(Z ≤ 2.327) = N(2.327) = 0.99 or 99 percent, and 1 percent of values remain in the right tail.

Question 46

Mean-variance analysis and when is it applicable

Answer 46

An approach to portfolio analysis using expected means, variances, and covariances of asset returns.

In economic theory, mean–variance analysis holds exactly when investors are risk averse; when they choose investments so as to maximize expected utility, or satisfaction; and when either 1) returns are normally distributed, or 2) investors have quadratic utility functions.

Question 47

Utility functions

Answer 47

Utility functions are mathematical representations of attitudes toward risk and return.

Question 48

Safety-first rules

Answer 48

Rules for portfolio selection that focus on the risk that portfolio value will fall below some minimum acceptable level over some time horizon.

Question 49

Shortfall risk

Answer 49

The risk that portfolio value will fall below some minimum acceptable level over some time horizon.

Question 50

Explain Roy’s safety-first criterion

Answer 50

Suppose an investor views any return below a level of R_L* as unacceptable. Roy’s safety-first criterion states that the optimal portfolio minimizes the probability that portfolio return, *R_P, falls below the threshold level, R_L*.²⁵ In symbols, the investor’s objective is to choose a portfolio that minimizes *P*(*R_P < R_L*). When portfolio returns are normally distributed, we can calculate *P*(*R_P < R_L*) using the number of standard deviations that *R_L lies below the expected portfolio return, E(R_P*). The portfolio for which *E*(*R_P) − R_L* is largest relative to standard deviation minimizes *P*(*R_P < R**_L). Therefore, if returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio (SFRatio):

Question 51

Roy’s safety-first criterion

Answer 51

SFRatio = [E(R_P*) – *R_L]/σ_P

Question 52

There are two steps in choosing among portfolios using Roy’s criterion (assuming normality):

Answer 52

1) Calculate each portfolio’s SFRatio.
2) Choose the portfolio with the highest SFRatio.

Question 53

Scenario analysis

Answer 53

Analysis that shows the changes in key financial quantities that result from given (economic) events, such as the loss of customers, the loss of a supply source, or a catastrophic event; a risk management technique involving examination of the performance of a portfolio under specified situations. Closely related to stress testing.

Question 54

Value at risk

Answer 54

(VaR) A money measure of the minimum value of losses expected during a specified time period at a given level of probability.

Question 55

The lognormal distribution

Answer 55

Closely related to the normal distribution, the lognormal distribution is widely used for modeling the probability distribution of share and other asset prices. For example, the lognormal appears in the Black–Scholes–Merton option pricing model. The Black–Scholes–Merton model assumes that the price of the asset underlying the option is lognormally distributed.

A random variable Y follows a lognormal distribution if its natural logarithm, ln Y, is normally distributed. The reverse is also true: If the natural logarithm of random variable Y, ln Y, is normally distributed, then Y follows a lognormal distribution. If you think of the term lognormal as “the log is normal,” you will have no trouble remembering this relationship.

Question 56

Two must noteworthy observations about the lognormal distribution:

Answer 56

The two most noteworthy observations about the lognormal distribution are that it is bounded below by 0 and it is skewed to the right (it has a long right tail). Note these two properties in the graphs of the pdfs of two lognormal distributions in Figure 7. Asset prices are bounded from below by 0. In practice, the lognormal distribution has been found to be a usefully accurate description of the distribution of prices for many financial assets. On the other hand, the normal distribution is often a good approximation for returns. For this reason, both distributions are very important for finance professionals.

Question 57

The expressions for the mean and variance of a lognormal variable are? Where μ and σ²are the mean and variance of the associated normal distribution (refer to these expressions as needed, rather than memorizing them):

Answer 57

• Mean (μ_L) of a lognormal random variable = exp(μ + 0.50σ²)
• Variance (σ_L²) of a lognormal random variable = exp(2μ + σ²) × [exp(σ²) − 1]

Question 58

Continuous vs. discrete compounding

Answer 58

Continuous compounding treats time as essentially continuous or unbroken, in contrast to discrete compounding, which treats time as advancing in discrete finite intervals. Continuously compounded returns are the model for returns in so-called continuous time finance models such as the Black–Scholes–Merton option pricing model.

Question 59

Continuous time

Answer 59

Time thought of as advancing in extremely small increments.

Question 60

Price relative

Answer 60

A ratio of an ending price over a beginning price; it is equal to 1 plus the holding period return on the asset.

• S₁/S*₀, 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;
• S_t+1/S_t* = 1 + R**_{t,<em>t</em>+1}

Question 61

Continuously compounded return

Answer 61

The natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price.

In this reading we use lowercase r to refer specifically to continuously compounded returns.

Question 62

The continuously compounded return from t to t + 1 is:

Answer 62

r_t,t+1 = ln(S_t+1/S_t) = ln(1 + R_t,t+1)  

Question 63

Independently and identically distributed (IID)

Answer 63

With respect to random variables, the property of random variables that are independent of each other but follow the identical probability distribution.

A key assumption in many investment applications is that returns are independently and identically distributed (IID).

Independence captures the proposition that investors cannot predict future returns using past returns (i.e., weak-form market efficiency). Identical distribution captures the assumption of stationarity.

Question 64

Stationary

Answer 64

Stationarity implies that the mean and variance of return do not change from period to period.

Assume that the one-period continuously compounded returns (such as r_0,1) are IID random variables with mean μ and variance σ² (but making no normality or other distributional assumption). Then the equation would be:

E(r_0,<em>T</em>) = E(r_T*_{–1,<em>T</em>}) + *E*(*r_{T–2,<em>T</em>–1}) + … + E(r_0,1) = μT  

(we add up μ for a total of T times) and

σ²(r_0,<em>T</em>) = σ²T  

Standard deviation as well.

Question 65

Volatility

Answer 65

As used in option pricing, the standard deviation of the continuously compounded returns on the underlying asset.

Volatility is also called the instantaneous standard deviation, and as such is denoted σ. The underlying asset, or simply the underlying, is the asset underlying the option.

Question 66

Annualizing is often done on the basis of how many days per year?

Answer 66

Annualizing is often done on the basis of 250 days in a year, the approximate number of days markets are open for trading. The 250-day number may lead to a better estimate of volatility than the 365-day number. Thus if daily volatility were 0.01, we would state volatility (on an annual basis) as

0.01*SQRT(250)=0.1581