Reading 8

Two properties of probability are

(1) The sum of the probabilities of all possible mutually exclusive events is 1. (2) The probability of any event cannot be greater than 1 or less than 0.

What changes to portfolio variance formula if covariance is mentioned

(W1)^2(stddeviation1)^2+(W2)^2(stddeviation1)^2+2(w1)(w2)(covariance)

Portfolio variance formula

(W1)^2(stddeviation1)^2+(W2)^2(stddeviation1)^2+2(w1)(w2)(std1)(std2)(correlation)

How can expected values or returns be calculated?

- Conditional Probabilities Conditional expepcted values are relient upon the outcome of some other event Using the total probability rule we can estimate the unconditional exppected return on the stock s the sum of the expected return given no tariff times the probability a tariff will not be enacted plus the expected return given a tariff times the probability a tariff will be enacted

Covariance

A measure of linear association between two variables. Positive values indicate a positive relationship; negative values indicate a negative relationship Meaasure of how two assets move together negative - they move in opposite directions

Likelihood

Conditional probability of an occurrence is also called a likelihood

Labeling Formula

Applies to three or more subgroups of predetermined size, each element of the entire group must be assigned a place, or label in one of the three or more subgroups

How is the variance of EPS calculated? Std Deviation?

As the pprobability weighted sum of the squared differences between each possible outcome and expected EPS Variance = sum of P(EPS - exp value) ^2 Std Deviation - Squareroot

What is the probability of occurrence of any event?

Between 0 and 1

A priori probability

Comes from a formal reasoning and inspection process; an objective probability

What should you avoid if events aren't mutually exclusive

Double counting by subtracting the joint probability that both And B will occur from the sum of the unconditional probabilities

What does variance and standard deviation measure?

Dsipersion or valitility of only one variable

Mutually exclusive Events

Events that cannot both happen at the same time

Symbol ! , meaning of 4!

Factorial = 4! = 4 * 3 * 2 * 1 = 24

(k=n)

If there are n labels n!/n = n The number of ways to assign n different labels to n items is simply n!

What is the expected value of a random variable?

Is the weighted average of the possible outcomes for the variable Add formula sum of P(EPS)

If a probability of an event sums to 1 then what is it?

Mutually exclusive and exhaustive

(k=2)

Number of labels = 2, the n items can only be in one of two groups, and n1 + n2 = n , let r =n1 and n2 = n - r. Since there aare only two catergories we usually talk about choosing r items. Then (n-r) items are not chosen Combination formula

How to describe the odds of an event happening?

Odds for the event occurring are one to seven and the odds against the event happening are the reciprocal of 1/7 which is seven to one.

Permutation formula

Only applies to two groups of predetermined size, look for the word order

Total Probability Rule

P(A) = P(A l B₁)(P(B₁) + P(A l B₂)P(B₂) +....P(A l Bn)P(Bn) Used to determine the unconditional probability of event, given conditional probabilities. Where B1, B2, is a mutually exclusive and exhaustive set of outcomes

Unconditional Probability

P(B)

Covariance of Asset Returns

P(Ra1,Rb1)(Ra1-ExpValue)(Rb1-ExpValue)

How can the correlation between two random variables be efined?

P(Ri,Rj) - no units - measures the strength of the linear relation between two random variables -ranges. from -1 to + 1 - if = 1, perfect positive correlation this means that movement in one random variable results in a proportional positive movement in the other relative to its mean if = -1 perfect negative correlation if = 0 no linear relationship, indicating that prediction of Ri cannot be made on the basis of Rj using linear methods

Example of random variable, outcome, event, mutually exclusive event, and exhaustive events

Random variable - rolling a number Roll a 4 - outcome rolling 4 - event rolling a 4 and a 6 are mutually exclusive event rolling an even and rolling an odd number is a set of mutually exclusive and exhaustive events

Labeling

Refers to the situation where there are n items that can each receive one of k different labels. The number of items that receives label 1 is n1 and the number that receive label 2 is n2

Permutataion Formula

Specific ordering of a group of objects, the question of how many different groups of size r in specific order can be chosen from n objects is answered by the permutation formula The number of permutations of r objecjts from n objects = n!/(n-r)!

Portfolio Exp Value Formula

Sum of w1(R1)

subjective probability

The least formal method of developing probabilities and involves the use of personal judgement.

P(A|B)

The probability of A given the occurrence of B

Unconditional probability

The probability of an event regardless of the past or future occurrence of other events; marginal pprobability

What does the total probability rule highlight?

The relationship between unconditional and conditional pprobabilities of mutually excllusive and exhaustive events.

What happens if the joint probability for mutually exclusive events is zero

Then do not add P(AB)

correlation coefficient

To make the covariance of two random variables eaasier to interpret, it may be divided by the product of the random variables standard deviations. The resulting value is called the correlation coefficient Cov(Ra,Rb)/(std of Ra)(std of Rb)

Multiplication rule of counting

Two or more groups, only one item can be selected from each group

Computing covariance of returns from joint probability model

Use probability weighted average of the products of the random variables deviations from their means for each possible outcome Sum of (P)(Ra-EV1)(Rb-Ev2)

Multiplication rule of probability

Used to determine the joint probability of two events P(AB) = P(A | B) x P(B)

Addition Rule of pprobability

Used to determine the probability that at least one of two events will occur P(A or B) = P(A) + P(B) - P(AB)

Tree Diagram

Used to show the probabilities of various outcomes shows the probabilities of two events and the conditional probabilities of two subswequent events

Bayes Formula

Used to update a given set of prior probabilities for a given event in response to the arrival of new information Updated probability = (Probability of new information for a given event/Unconditional probability of new information) * prior probability of event

How can the exp value and variance for a portfolio of asset be determined?

Using the properties of the individual assets in the portfolio. Establish the portfolio weight for each asset.

If P(E) = 0 then what is the outcome of the event

Will never happen

What would the covariance of the returns of a stock and a riskless asset be?

Zero because the riskless assets returns never move, regardless of movements in the stocks return

Outcome

an observed value of a random variable

When dealing with independent events, what. does add and or stand for?

and - multiplication or - addition

Empirical probability

established by analyzing old data; objective probability

Exhaustive events

events are those that include all possible outcomes

Event

is a single outcome or a set of outcomes

Properties of Covariance

may range from negative infinity to positive infinity

Combination Formula

n!/(n-r)!r! where nCr is the number of possible ways of selecting r items from a set of n items when the order of selection is not important, (nr) and read n choose r

General expression for n factorial is

n!= n * (n-1) * (n-2) * (n-3) * ..... * 1, where by defnition, 0! = 1

Permutation Formula

nPr = n!/(n-r)! where nPr is the number of ppossible ways to select r items from a set of n items when the order of selection is important

Joint Probability

of two events is the probability that they will both occur; use the multiplication rule of prob. to find out

If P(E) = 1 then what is the outcome of the event

the event is certain to occur and the outcome is not random can be exhaustive and mutually exhaustive

Independent Events, example

the occurrence of one event does not affect the probability of the other; can be defined in conditional probabilities P(A|B) = P(A) A is independent PRIORI. PROBABILITIES

Conditional Probability

the probability of an event ( A ), given that another ( B ) has already occurred. The occurrence of one event affects the probability of the occurrence of another event

Random Variable

uncertain quantity/number

Factorial

used by itself when there aare no groups, we are only arranging a given set of n items

How to calculate the portfolio weight of each asset?

wi = market value of investment in asset i / market value of the portfolio