CFA: Quantitative Methods: Conditional and Joint Probabilities
Correlation Coefficient
(correlation)- To make the covariance of two random variables easier to interpret, it may be divided by the product of random variables standard deviations. The resulting value is the correlation coefficient. Corr(Ri, Rj)= Cov (Ri, Rj)/ (σ(Ri) σ(Rj)) Cov (Ri, Rj)= Corr (Ri, Rj) x ((σ(Ri) σ(Rj))
What are the properties of covariance
1. Is a general representation of the same concept as the variance. That is, the variance measures how a random variable moves with itself, and the covariance measures how one variable moves with another random variable. 2. The covariance of Ra with itself is equal to the variance of Ra: that is Cov (Ra, Ra)= Var(Ra) 3. The covariance may range from negative infinity to postive infinity.
What are the two defining properties of probability
1. the probability of occurence of any event Ei is between 0 and 1 (0<P(Ei)<1) 2. if a set of events E1, E2..... are mutually exclusive and exhaustive, the probabilities of those events sum to one.
Consider a portfolio consisting of eight stocks. Your goal is to designate four of the stocks as long term holds, three of the stocks as short term holds, and one stock as well. How many ways can these eight stocks be labeld?
8! / (4! x 3! x 1!) =
How can correlation be forward looking or backward looking?
Correlation can be forward looking if it uses covariance from a probability model, or backward looking if it uses sample covariance from historical data.
The joint probability of getting two 4's is calculated as.....
P(4 on the first die and 4 on the second die)= 1/6 x 1/6= .0278
How can you calculate covariance?
Probability model or Historical Data. Probability model= Cov (Ra, Rb)= E {P(S) x (Ra- E(Ra))(Rb- E(Rb)) Historical Data: example sample covariance Cov1,2= (Sum {Rt,1 - R1)(Rt,2 - R2]}/ (n-1)
Expected Value
The expected value of a random variable is the weighted average of the possible outcomes for the variable.
Read this ... P(AB)= P(A/B) x P(B)
The joint probability of A and B, P(AB), is equal to the conditional probability of A given B, P(A/B), times the unconditional probablity of B, P(B) Rearranged to P(A/B) = P(AB)/P(B)
Multiplication of Probability
Used to determine the joint probability of two events: P(A/B)= P(A/B) x P (B)
Portfolio variance (assuming two risky assets)
Var(Rp)= Wa²σ² (Ra) + Wb² σ² (Rb) + 2WaWb σ(Ra)σ(Rb) ρ(Ra,Rb) - If provided w/correlation Var(Rp)= Wa²σ² (Ra) + Wb² σ² (Rb) + 2WaWb Cov(Ra, Rb) - If provided w/covariance Cov (Ra, Rb)= Corr (Ra, Rb) x ((σ(Ra) σ(Rb))
How do you determine the weight of the portfolio of an asset i?
Wi= (market value of investment in asset i)/market value of the portfolio). In order to find the expected value and variance of a portfolio of assets this is the first step.
Scatterplots
a graphed cluster of dots, each of which represents the values of two variables. The slope of the points suggests the direction of the relationship between the two variables. The amount of scatter suggests the strength of the correlation (little scatter indicates high correlation). even if there is no linear relationship between two variables the scatterplot can show that they are related in a predictable way
Permutation formula
a permutation is a specfic 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. The number of permutations of r objects from n objects = n! (n-r)!
Explain when to use labeling formula
applies to 3 or more subgroups of predetermined size. each element of the entire group must be assigned a plcae or label in one of the three or more subgroups
Explain when to use permutation formula
applies to only two groups of predetermined size. Look for a specific reference to ORDER being important
Explain when to use combination formula
applies to only two groups of predetermined size. Look for the word CHOOSE or COMBINATION
Mutually exclusive events
are events that cannot happen at the same time
If there are only two categories (k=2) then what does that equation look like and what is it called?
called binomial or combination formula. ∩Cᵣ = n!/((n-r)!r! ∩Cᵣ= number of possible ways (combinations) of selecting r items from a set of n items when the order of selection is not important. or you can say (n r) or n choose r.
What does correlation measure?
correlation measures the strength of the linear relationship between two random variables. it has no units. Ranges from -1 to 1. That is -1 ≤ Corr(Ri, Rj) ≤+1
Empirical probability
established by analyzing past data. Objective probability
Covariance
how two assets move together (for example S&P 500 and the automotive industry or Stock A and Stock B. It is the expected value of the product of the deviations of the two random variables from their respective expected values. Cov (Ri, Rj)= E {[Ri-E(Ri)][Rj- E(Rj)]}
an event
is a single outcome or a set of outcomes
An outcome
is an observed value of a random variable
Random Variable
is an uncertain quantity / number
A priori probability
is determined using a formal reasoning and inspection process. objective probability
Subjective probability
is the least formal method of developing probabilities and involves the use of personal judgement
Explain when to use factorial
is used by itself when there are no groups. we are only arranges a given set of n items. Given n items, there are n! ways of arranging them
Total Probability Rule
is used to determine the unconditional probability of an event, given conditional probabilities P(A)= P(A/B1)P(B1) + P(A/B2)P(b2)........ where B1, B2 are mutually exclusive and exhaustive events
Bayes Formula
is 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) x prior probability of event
Explain when to use the multiplication rule of counting
it is used when there are two or more groups. the key is that one item may be selected from each group. if there are k steps required to complete a task and each step can be done in n ways, the number of different ways to complete the task is n1 x n1 x n3.....
If there are n labels where (k=n) then the different way to label is what?
n!
independent events
refer to events for which the occurrence of one has no inlfuence on the occurence of others. Expressed in conditional probabilities. Events A and B are independent if and only if : P(A/B)=P(A) or P(B/A) = P(B) P(AB)= P(A)P(B)
Spurious correlation
refers to correlation that is either the result of chance or present due to changes in both variables over time that is caused by their association with a third variable.
Unconditional probability
refers to the probability of an event regardless of the past or future occurrence of other events. (marginal probability).
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, and so on.... n1 + n2 + n3= n The total number of ways that the labels can be assigned is : n! /(n₁!) x(n₂!) x .........
Portfolio expected value
the expected value of a portfolio composed of n assets with weights Wi, and expected values Ri, , can be determined using the following formula, E(Rp)= ∑ WiE(Ri)= W₁E(R₁) + W₂ E(R₂) + ..........Wn E(Rn)
conditional probability
the occurance of one event afffects the probability of the occurance of another event. The key is the word "given". for example we might be concerned with the probability of a recession given that the monetary authority increases interest rates. P(recession/increase in interest rates. P (A/B)
What does Corr(Ri, Rj) = -1.0 mean?
the random variables have perfect negative correlation. this means that a movement in one random variable results in an exact opposite proportional movement in the other relative to its mean.
What does Corr(Ri, Rj) = 1.0 mean?
the random variables have perfect positive correlation. the movement in one random variable results in a proportional positive movement in the other relative to its mean.
What does Corr(Ri, Rj) = 0 mean?
there is no linear relationship between the variables, indicating that prediction of Ri cannot be made on the basis of Rj using linear methods
exhaustive events
those that include all possible outcomes
Addition rule of probability
used to determine the probability that at least one of two events will occur P(A or B)= P(A) + P(B)- P(AB) IF a and b are mutually exclusive then P(AB) is 0 so the new equation is P(A or B)= P(A) + P(B)
What is the expected value of a random variable?
weighted average of the possible outcomes for the variable. E(X)= P(x1)x1 + P(x2)s2 +.............
