Reading 8 Probability Concepts Flashcards
Calculate the expected value of returns on a portfolio.
n∑i=1wiE(Ri)
wi = weights of each security or asset class
E(Ri) = expected return of each security or asset class
Identify the 2 basic defining principles of probability.
The probability of any event, E, has value from 0 to 1. Therefore, 0 ≤ P(Ei) ≤ 1, where P(Ei) is the probability of event, i, occurring.
The sum of the probabilities of mutually exclusive, exhaustive events equals 1. Thus ∑P(Ei) = 1.
Note: an empirical probability is the probability of an event based on the frequency of its occurrence in the past.
Define random variable, outcome, event, and mutually exclusive events.
A random variable is one with uncertain values.
An outcome is an observed value of a random variable.
An event is a single outcome or a set of outcomes.
Mutually exclusive events cannot both occur. The occurrence of one precludes the occurrence of the other.
Give the formula to calculate the variance of a portfolio.
σ2(RP)=E[(RP−ERP)2]=w21σ21+w22σ22+w2
3σ23+2(w1w2COV,2+w1w3COV1,3+w2w3COV2,3)
When is the permutation formula used? Express the formula mathematically.
The permutation formula is used when the order in which labels are assigned to two groups is an important consideration.
nPr=n!(n−r)!
Expresses the number of ways to select groups of r objects from a total number of n objects, when groups of r objects must have unique groups.
What does the correlation coefficient measure, and what are the properties of it.
The correlation coefficient measures the strength and direction of the linear relationship between two random variables.
The correlation coefficient has no units. It lies between −1 and +1. And a correlation coefficient of zero indicates no linear relationship between two random variables.
Note: A correlation coefficient of +1 indicates a perfect positive correlation between two random variables. A ⎼1 indicates a perfect negative correlation. A shortcoming is that it does not specify which factor causes the linear relationship.
Describe Bayes’ formula.
It relates the conditional and marginal probabilities of two random events.
Bayes’ formula works inversely from an occurrence of an event to infer the probability of the scenario that generated it.
Explain the expected value of a random variable.
The probability-weighted average of all possible outcomes for the random variable.
Identify the limitations of covariance.
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 values.
Covariance does not tell us anything about the relative strength of the relationship between the two variables.
Distinguish between unconditional and conditional probabilities.
Unconditional (marginal) probabilities – The probability of an event irrespective of the occurrence of other events.
Conditional probabilities – The probability of an event occurring given that another event has occurred. Mathematically expressed:
P(A|B) = P(AB)P(B) given that P(B)≠0
Give the formula for combinations.
nCr=(nr)=n!(n−r)!r!
Expresses the number of ways to select groups of r objects from a total number of n objects, when groups of r objects can repeat in a different order.
Identify the limitations of correlation analysis.
Two variables can have a strong nonlinear relation and still exhibit low correlation, which only measures linear association.
Correlation may be an unreliable measure when there are outliers in either or both of the series.
Strong linear correlations may highlight misleading relationships due to lack of causation.
Distinguish between subjective probability and a priori probability.
Subjective probability – personal judgment to estimate probabilities; e.g., you think the probability of rain is 80%.
A priori probability – Formal analysis and reasoning rather than personal judgment; e.g., meteorological conditions like these result in rain 70% of the time.
List the properties of covariance.
Measures how a random variable varies with another random variable.
Is symmetric.
Ranges from positive to negative infinity.
The covariance of X is equal to the variance of X.
Covariance of returns is zero if the returns are unrelated.
Differentiate between dependent and independent events.
Dependent – The occurrence of one is related to the occurrence of the other.
Independent – The occurrence of one is not related to the occurrence of the other.
