Probability Applications

Question 0

Discrete random variable

Answer 0

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

Question 1

Random Variable

Answer 1

a quantity whose future outcomes are uncertain

Question 2

Continuous random variable

Answer 2

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 3

Probability function

Answer 3

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

Question 4

Probability density function

Answer 4

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

Question 5

Cumulative distribution function

Answer 5

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

Question 6

Bernoulli random variable

Answer 6

A random variable having the outcomes 0 and 1.

n = 1: Y ~ B(1, p).

Question 7

Binomial random variable

Answer 7

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

a binomial random variable is completely described by two parameters, n and p. We write

X ~ B(n,p) 

Question 8

Probability function for a binomial random variable

Answer 8

x n-x

=n!/(n−x)!x! [p (1−p) ]

Question 9

Variance of Bernoulli random variable

Answer 9

p(1 − p)

Question 10

Variance of binomial random variable

Answer 10

np(1 − p)

Question 11

Mean of a continuous random variable

Answer 11

μ = (a + b)/2

Question 12

Variance of a continuous random variable

Answer 12

σ2 = (b − a)2/12.

Question 13

Defining characteristics of a normal distribution

Answer 13

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 14

Normal density function

Answer 14

f(x)=( 1/σ√2π ) exp( −(x−μ)2 / 2σ2 ) for −∞<+ ∞

Question 15

Standard normal distribution

Answer 15

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

Question 16

Standardization formula

Answer 16

Z = (X – μ)/σ  

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

Question 17

Value at Risk (VAR)

Answer 17

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

Question 18

Scenario analysis

Answer 18

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 19

Stress testing

Answer 19

A set of techniques for estimating losses in extremely unfavorable combinations of events or scenarios.

Question 20

Monte Carlo simulation

Answer 20

An approach to estimating a probability distribution of outcomes to examine what might happen if particular risks are faced. This method is widely used in the sciences as well as in business to study a variety of problems.

Monte Carlo simulation involves the use of a computer to represent the operation of a complex financial system. A characteristic feature of Monte Carlo simulation is the generation of a large number of random samples from specified probability distribution(s) to represent the operation of risk in the system. Monte Carlo simulation is used in planning, in financial risk management, and in valuing complex securities. Monte Carlo simulation is a complement to analytical methods but provides only statistical estimates, not exact results.

Question 21

Asian call option

Answer 21

A European-style option with a value at maturity equal to the difference between the stock price at maturity and the average stock price during the life of the option, or $0, whichever is greater.

Question 22

Simulation trial

Answer 22

A complete pass through the steps of a simulation.

Question 23

Historical simulation

Answer 23

Another term for the historical method of estimating VAR. This term is somewhat misleading in that the method involves not a simulation of the past but rather what actually happened in the past, sometimes adjusted to reflect the fact that a different portfolio may have existed in the past than is planned for the future.

Historical simulation is an established alternative to Monte Carlo simulation that in one implementation involves repeated sampling from a historical data series. Historical simulation is grounded in actual data but can reflect only risks represented in the sample historical data. Compared with Monte Carlo simulation, historical simulation does not lend itself to “what if” analyses.

Question 24

Relationship between normal and lognormal distributions

Answer 24

If continuously compounded returns are normally distributed, asset prices are lognormally distributed. This relationship is used to move back and forth between the distributions for return and price. Because of the central limit theorem, continuously compounded returns need not be normally distributed for asset prices to be reasonably well described by a lognormal distribution.

Question 25

Roy’s safety first criterion

Answer 25

Roy’s safety-first criterion, addressing shortfall risk, asserts that the optimal portfolio is the one that minimizes the probability that portfolio return falls below a threshold level. According to Roy’s safety-first criterion, if returns are normally distributed, the safety-first optimal portfolio P is the one that maximizes the quantity [E(RP) − RL]/σP, where RL is the minimum acceptable level of return.

Question 26

Standard normal random variable

Answer 26

The standard normal random variable, denoted Z, has a mean equal to 0 and variance equal to 1. All questions about any normal random variable can be answered by referring to the cumulative distribution function of a standard normal random variable, denoted N(x) or N(z).

Question 27

Normal random variable

Answer 27

For a normal random variable, approximately 68 percent of all possible outcomes are within a one standard deviation interval about the mean, approximately 95 percent are within a two standard deviation interval about the mean, and approximately 99 percent are within a three standard deviation interval about the mean.

A normal random variable, X, is standardized using the expression Z = (X − μ)/σ, where μ and σ are the mean and standard deviation of X. Generally, we use the sample mean X⎯⎯⎯ as an estimate of μ and the sample standard deviation s as an estimate of σ in this expression.

Question 28

Normal distribution

Answer 28

The normal distribution is a continuous symmetric probability distribution that is completely described by two parameters: its mean, μ, and its variance, σ2.

Question 29

Univariate and multivariate distribution

Answer 29

A univariate distribution specifies the probabilities for a single random variable. A multivariate distribution specifies the probabilities for a group of related random variables.

Question 30

Normal distribution standard deviation intervals

Answer 30

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 31

Frequently referenced values in the standard normal table

Answer 31

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 32

Negative numbers in the standard normal table

Answer 32

For a non-negative number x, use N(x) from the table. Note that for the probability to the right of x, we have P(Z ≥ x) = 1.0 − N(x).

For a negative number −x, N(−x) = 1.0 − N(x): Find N(x) and subtract it from 1. All the area under the normal curve to the left of x is N(x). The balance, 1.0 − N(x), is the area and probability to the right of x. By the symmetry of the normal distribution around its mean, the area and the probability to the right of x are equal to the area and the probability to the left of −x, N(−x).

For the probability to the right of −x, P(Z ≥ −x) = N(x).

Question 33

Price relative

Answer 33

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

Question 34

Continuously compounded return

Answer 34

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

r t,t+1 = ln(St+1/St) = ln(1 + Rt,t+1)