Reading 8 LOS's Flashcards
LOS 8a: Define a random variable, an outcome, an event, mutuall exclusive events, and exhaustive events
- A random variable is one whose possible values or results are uncertain
- an outcome is the observed value of a random variable
- an event could be a single outcome or a set of outcomes
- Mutually exclusive events are events that cannot happen simultaneously. the occurrence of one precludes the occurrence of the other
- Exhaustive events cover the range of all possible outcomes of an event
LOS 8b: State the two defining properties of probability and distinguish among empirical, subjective, and a priori probabilities
A probability is a number between 0 and 1 that reflects the chance of a certain event ocurring. There are two basic defining principles of probability:
- The probability of any event E, is a number from 0 to 1, where P(Ei) is the probability of event i happening
- The sum of the probabilities of mutually exclusive exhaustive events equals 1
Methods of Estimating Probabilities
- An empirical probability estimates the probability of an event based on the frequency of its occurrence in the past
- A subjective probability draws on subjective reasoning and personal judgment to estimate probabilities
- A prior probability is based on formal analysis and reasoning rather than personal judgment
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LOS 8c: State the probability of an event in terms of odds for and against the event
The odds for an event are stated as the probability of the event occurring to the probability of the event not occurring.
- odds for event E, are stated as P(E) to [1-P(E)]
- if the odds for are given as “a to b” then
- P(E) = a / (a+b)
The odds against an event are stated as the probability of the event not occurring to the probability of the event occurring.
- Odds against event E, are state in the forom of [1 -P(E)] to P(E)
- If the odds against are given as “a to b” then:
- P(E) = b / (a+b)
For example if Susan thinks the probability of the market rising tomorrow is .40
- Susans odds for the market rising are .4/( 1-.04) = 2 to 3
- Susans odds against market rising ( 1-.4) / .4 = 3 to 2
LOS 8d: Distinguish between unconditional and conditional properties
LOS 8e: Explain the multiplication, addition, and total probability rules
LOS 8f: Calculate and interpret 1) the joint probability of two events, 2) the probability that at least one of two events will occur, given the probability of each and the joint probability of the two events, and 3) a joint probability of any number of independent events
unconditional or marginal probabilities estimate the probability of an event irrespective of the occurrence of other events.
Conditional probabilities express the probability of an event occurring given that another event has occurred.
Joint probability is the probability that both events occurr
- If A and B are mutually exclusive events, the joint probability P(AB) equals 0, because mutually exclusive events cannot occurr simultaneously
- If A is contained within the set of possible outcomes for B, P(AB)= P(A)
The conditional probability P(A|B), is the probability of event A occurring given that event B has occurred divided by the unconditional probability of event B occurring
- P(A|B) = P(AB)/ P(B) given that P(B) does not equal 0
The multiplication rule for probability to calculate joint probabilities can be derived from the conditional probability formula by rearranging it:
- P(AB) = P(A|B) x P(B)
Calculating the Probability that at Least One of Two Events Will Occur
Sometimes we want to determine the probability of event A or event B occurring. Here we need to invoke the addition rule for probabilities, which calculates the probability of at least one of A and B occurring:
- P(A or B) = P(A) + P(B) - P(AB)
we subtract their joint probability to avoid double counting
LOS 8g: Distinguish between dependent and independent events
With two Dependent events, the occurence of one is related to the occurrence of the other.
Two events are independent if the occurrence of one does not have any bearing on the occurrence of the other. When two events are independent:
- P(A|B) = P(A) or equivalently P(B|A) = P(B)
If we are trying to forecaset an event, information about a dependent event may be useful, whereas info about an independent event would be worthless
NOTE with independent events, the word and implies multiplication and the word or implies addition:
- P(A or B) = P(A) + P(B) - P(AB)
- P(A and B) = P(A) X P(B)
LOS 8e part 2: Explain the total probability rule
LOS 8h: Explain the total probability rule. Calculate and interpret an unconditional probability using the total probability rule
The total probability rule expresses the unconditional probability of an event in terms of conditional probabilities for mutuall exclusive and exhaustive events.In the simple case, we will assume that only two outcomes are possible, S and no S, Sc.
- P(S) + P(Sc) = 1 ( this means that the events are exhaustive)
- P(S and Sc) = 0 (this means that the events are mutually exclusive)
In this case:
- P(A) = P(AS) + P(ASc)
Replacing the joint probabilities on the right hand side of the equation with conditional probabilities using the multiplication rule, the probability of A can be reformulated as:
- P(A) = P(A|S) x P(S) + P(A|Sc) x P(Sc)
The probability of event A, P(A), is expressed as a weighted average in the total probability rule. The weights applied to the conditional probabilities are the probabilities of the scenarios.
remember for this rule to apply the scenarios must be mutually exclusive and exhaustive
Total probability rule for n possible scenarios:
- P(A) = P(A|S1) x P(S1) + ……… P(A|Sn) X P(Sn)
Expected Value
the expected value of a random variable is the probability-weighted average of all possible outcomes for the random variable.
- E(X) = ΣP(Xi)Xi
Expected values are forecasts of what we think might happen based on probabilities.
Variance and Standard Deviation
The variance of a random variable around its expectded value is the probability weighted sume of the squared differences between each outcome and the expected value.
- σ2(X) = E{[X -E(X)]2}
- σ2(X) = ΣP(Xi) [Xi-E(X)]2
LOS 8i: Explain the use of conditional expectation in investment applications
Analysts use conditional expected values that are calculated using conditional probabilites. The goal is to calculate the expected value of a random variable given all the possible scenarios that can occur
There is a way to state unconditional expected values in terms of conditional expected values through a principle known as total probability rule for expected value and is given as
- E(X) = E(X|S)P(S) + E(X|Sc)P(Sc)
- E(X) = E(X|S1) x P(S1) + …… E(X|Sn)P(Sn)
Given that there are only two possible scenarios, the expeted value of X equals the expected value of X given scenario A multiplied by the probability of Scenario A, plus the expected value of X given scenario B time the probability of scenario B
LOS 8j: Explain the use of a tree diagram to represent an investment problem
A tree diagram can give you first the state of the economy, either good or bad and the probabilities of the economy being in these states. From these states it can branch off and say that if in this state there is a probability of a good and bad return. This can carry on and on to show outcomes based on conditional events
LOS 8k; Calculate and interpret covariance and correlation
Covariance is a measure of the extent to which two random variable move together. For two variables, X and Y, the covariance is calculated as:
- Cov (XY) = E{[X-E(X)][Y- E(Y)]}
We can also write the covariance formula in terms of the returns on Asset A and Asset B
- Cov(RaRb) = E {[Ra- E(Ra)][Rb-E(Rb)]}
properties of covariance
- Covariance is similar concept to variance. The difference lies in the fact that variance measures how a random variable varies with itself, while covariance measures how a random varialbe varies with another random variable
- Covariance is symmetric Cov(XY)= Cov(YX)
- Covariance can range from positive infinity to negative
- The covariance of X with itself is equal to the variance of X
- When covariance is positive it means they move together
- If negative they move inversely
Limitations to Covariance
- Because the unit that covariance is expressed depends on the unit that the data is presented in, it is difficult to compare covariance across data sets that have different scales
- In practice, it is difficult to interpret covariance as it can take on extreme large values
- Covariance does not tell us anything about the strength of the relationship between two variables
Correlation Coefficient
The correleation coefficient measures the strength and direction of the linear relationship between two randome variables. It is obtained by dividing the covariance of the two random variables by the product of their standard deviations
- Corr(A,B) = Þ (greak roe) (A,B) = Cov(AB)/ (σA)(σB)
properties of the correlation coefficient
- it measures the strength of the relationship between two random variables
- It has no unit
- It lies between -1 and 1
- a correlation coefficient of 1 indicates perfect positive correlation
- -1 means perfect negative correlation
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LOS 8l: Calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio
The first step in calculating portfolio expected returns and variance is to compute the weights of the individual assets compromising the portfolio:
- Weight of asset i = Market value of investment i / market value of portfolio
The expected value of returns on a portfolio is a funtion of the returns on the individual assets and their respective weigths
- E(Rp) = ΣwiE(Ri) = w1E(R1) + w2E(R2) +…… wNE(RN)
Calculating the variance of the portfolio is more complicated. The variance is not only a funtion of individual asset weights and variance, but also of the covariance of the assets with each other
- Var(RP) =ΣΣ wiwjCov(Ri,Rj)
The variance of a 2-asset portfolio that contains only asset A and B is given as:
- var(Rp) = wA2σ2(RA) + wB2σ2(RB) + 2wAwBCov(RA,RB)
For anymore than 2-asset, make sure to take the Cov of every pair. So if there were 3 assets we would have 3 Covariance terms, 4 There would be 6 and so on
LOS 8m: Calculate and interpret covariance given a joint probability function
Steps to solve
- Find asset weights
- Divide asset weight by total weight of portfolio
- Find expected returns on individual assets
- Multiply probabilities of returns by expected returns and sum to get total expected return
- Find the variance of the individual assets
- multiply probabilites of returns my expected returns minus total expected return squared and sum to get variance of asset
- find the Covariance of the asset returns
- Mutliply the probability of returns times the expected returns minus the total expected return of each asset multiplied together. Sum up to get covariance
LOS 8n: Calculate and interpret an updated probability using Bayes’ formula
Bayes’ formula relates the conditional and marginal probabilities of two random events. Lets derive it
- Step 1:
- P(A|B) =P(AB) /P(B) from the conditional probability formula of A given B
- make P(AB) the subject to get:
- P(AB) = P(A|B) X P(B)
- Step 2:
- P(B|A) = P(BA)/ P(A) again move around to get
- P(BA) = P(B|A) x P(A)
- Step 3:
- Since the joint probabilites P(AB) and P(BA) are equal, we can equate expressions:
- P(A|B) x P(B) = P(B|A) x P(A)
- make P(B|A) the subject to get:
- P(B|A) = P(A|B) x P(B)/P(A)
Essentially by using Bayes’ formula we can reverse the “given that probability “ P(A|B) and convert it into P(B|A) using P(A) and P(B)
replacing B with Event and A with Information, Bayes’ formula can be translated to
- P(Event|Information) = P(Information | Event) x P(event) / P(Information)
LOS 8o: Identify the most appropriate method to solve a particular counting problem, and solve counting problems using factorial, combination, and permutation concepts.
Multiplication Rule of Counting
Suppose there are k tasks that must be done. The first one can be done in n1 ways, the second given how the first was done can be done in n2 ways and so on. The number of different ways that the k tasks can be done equals n1xn2x….. nk
Labeling Problems
these refer to situations where there are n items, each of which can receive one of k different lables. WE give each object in the group a label to place it in a category. n1 items can be given the first label, n2 can be given the second label and so on
Combinations
The combination formula is used in a special case of the labeling problem. Specifically the combination formula is used when the number of labels that can be assinged, k, equals 2. In such a situation, any item can only be labeled as one of the other, so n1 + n2 = n. Suppose the number of objects that reveive the first label is r leaving those to receive the second lable as 1-r. The formula for combinations can be stated as:
- nCr = (n r) = n! / (n-r)!(r!)
Permutations
When order in which labels are assigned to two groups is an important consideration the permutation formula is used. The number of permutations of r objects from n items equals:
- nPr= n! /(n-r)!
Tips:
- Factorials are used when there is only one group. Given n items, there are n! ways of arranging them
- The labeling formula is used for three of more groups of predetermined size. Each item must be labeled as a member of one of the groups
- The combination formula is used when there are only two groups of predeteremined size and crucually, the order or rank of labeling is NOT important
- The permutation formula is used when there are only two groups of predetermined size, and the order or rank IS important
