Reading 10- part 1

msk2222's version from 2018-02-18 17:35



The Discrete Uniform Distribution

Question Answer
definitionThe distribution has a finite number of specified outcomes, and each outcome is equally likely.
The cdf has two other characteristic propertiesThe cdf lies between 0 and 1 for any x: 0 ≤ F(x) ≤ 1. As we increase x, the cdf either increases or remains constant.
The Binomial Distribution When we make probability statements about a record of successes and failures, or about anything with binary outcomes, we often use the binomial distribution
binomial random variableX is defined as the number of successes in n Bernoulli trials. A binomial random variable is the sum of Bernoulli random variables Yi, i = 1, 2, …, n:


Now we extend the model to describe stock price movement on three consecutive days. Each day is an independent trial. The stock moves up with constant probability p (the up transition probability); if it moves up, u is 1 plus the rate of return for an up move. The stock moves down with constant probability 1 − p (the down transition probability); if it moves down, d is 1 plus the rate of return for a down move. We graph stock price movement in Figure 2, where we now associate each of the n = 3 stock price moves with time indexed by t. The shape of the graph suggests why it is a called a binomial tree. Each boxed value from which successive moves or outcomes branch in the tree is called a node; in this example, a node is potential value for the stock price at a specified time.


Question Answer
pdf for a uniform random variable isf(x) = 1/(b-a) for x between a and b & f(x)= 0 otherwise
middle line of the expression for the cdf captures this relationshipf(x)=0 for x =< a, = (x-a)/(b-a) for x between a and b, = 1 for x >= b
mathematical operation that corresponds to finding the area under the curve of a pdf f(x) from a to bthe integral of f(x) from a to b
The Normal DistributionThe normal distribution is completely described by two parameters—its mean, μ, and variance, σ2normal 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 0.16 As a consequence of symmetry, the mean, median, and the mode are all equal for a normal random variable
A multivariate distributionthe probabilities for a group of related random variables
There are two steps in standardizing a random variable XZ = (X – μ)/σ

Applications of the Normal Distribution

Question Answer
Modern portfolio theory (MPT)wide use of the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return
choosing among portfolios using Roy’s criterionCalculate each portfolio’s SFRatio. Choose the portfolio with the highest SFRatio.
if returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio (SFRatio)SFRatio = [E(RP) – RL]/σP
Two mainstays in managing financial riskValue at Risk (VaR) and stress testing/scenario analysis
Stress testing/scenario analysisrefers to a set of techniques for estimating losses in extremely unfavorable combinations of events or scenarios
Value at Riskis a money measure of the minimum value of losses expected over a specified time period (for example, a day, a quarter, or a year) at a given level of probability (often 0.05 or 0.01).