(8) Quantitative Methods: Probability Concepts (kyle created) Flashcards
Random Variable
A quantity whose outcomes are unknown
Outcome
Possible values
Event
A specified set of outcomes (point or range); denoted by capital letters
The two defining properties of a probability are as follows:
= P (A)
- P (A) is less than or equal to 1 or greater than or equal to zero
- The sum of all probabilities sums up to 1
Mutually exclusive events
If one event happens, another event can’t
Exhaustive events
Covers all possible outcomes
The two elements needed to solve a probability problem are:
- Set of all distinct possible outcomes
2. The probability distribution
Empirical Probability
- Based on historical observations
- Past is assumed to be representative of the future
- Historical period must include occurrences of the event
Subjective probability
- Adjust an empirical probability based on intuition or experience
- This occurs when there is a complete lack of empirical observations or
- to make a personal assessment
A priori probability
arrived at based on deductive reasoning
Odds for E. What is the formula and meaning?
P (E) / 1 - P(E);
Ex: P(E) = 10%
Odds for E = .1 / (1 - .10)
Odds for E = .1/.9 or 1 to 9
This is saying for each occurrence of event E, we should expect 9 events of non-occurrence
If Odds for E = 1 to 9 then to get the probability you take 1 / (1+9)
Odds against E. What is the formula and meaning?
1 - P (E) / P(E);
Ex: P(E) = 10%
Odds against E = .9 / .1 or 9 to 1
This is saying for every 9 non-occurrence of event E, we should expect 1 occurrence of the event
If Odds against E = 9 to 1 then to get the probability you take 1 / (9+1)
Probability: Terminology (5)
- Random Variable: Uncertain number
- Outcome: Realization of random variables
- Event: Set of one or more outcomes
- Mutually exclusive: cannot both happen
5: Exhaustive: Set of events includes all possible outcomes
Probability: Types (3)
- Empirical: Based on analysis of data
- Subjective: Based on personal perception
- A priori: Based on reasoning, not experience
Probability: Odds for or Against (2)
Odd happening/ odds not happening.
Given probability that a horse will win a race= 20%
Odds for: .20/ (1-.2)= .2/.8= 1/4 or 1 to 4 Odds against: (1-.2) .2= .8/.2= 4 or 4 to 1
Probability: Conditional vs Unconditional (2)
Two types of probability:
- Unconditional P (A), the probability of an event regardless of the outcomes of other events
- Conditional P (A|B), the probability of A given that B has occurred (e.g. The probability that Marley will be up, given the red raises interests rates)
Joint Probability (Multiplication rule of probability)
Notation is P (AB) which means the probability of A and B
The probability that both events will occur is their joint probability
Examples using conditional probability:
P (interest rates will increase) = P (A) = 40%
P (recession given a rate increase) = P (B|A) = 70%
Probability of a recession and an increase in rates,
P (BA) =P(B|A) x P(A) =.7 x .4 = 28%
Addition Rule for Probabilities
- P (A or B) = P(A) + P(B) - P(AB)
- We must subtract the joint probability P (AB)
- At least one of two events will occur
Formula for the probability of A given B; P (A|B) =
P(AB) / P(B)
Multiplication rule for independent events
When two events are independent of each other, the joint probability of A and B = P(A) x P(B)
Total probability rule formula is
P(A) = P(A|S1)xP(S1) + P(A|S2)xP(S2)….+ P(A|Sn)xP(Sn)
A is the event
S1 is scenario 1
S2 is scenario 2
Portfolio expected return
Weighted average of the expected returns on the different securities
Covariance of returns is negative if these two conditions are met
- The return on one asset is above its expected value
- The return on the other asset tends to be below its expected value
This is an inverse relationship
Covariance of returns is equal to 0 when
The return on the assets are unrelated
Covariance of returns is positive the following is met
The returns on both assets tend to be on the same side (above or below) their expected values at the same time
This is a positive relationship
Bayes’ formula
Rational method for adjusting our viewpoints as we confront new information
P (event | information) = [P (Information | event) / P (information) ] x P (event)
4 Principles of counting
- Multiplication rule
- Multinomials
- Combinations
- Permutations
Calculate the correlation of returns using a covariance matrix
= covariance / [square root of variance 1 x square root of variance 2]
What is permutation used for?
Is a listing in which the order of the listed items does matter
What is combination used for?
Is a listing in which the order of the listed items does not matter
Another way to calculate covariance if given standard deviation and correlation
= standard deviation of one bond x standard deviation of second bond x the correlation between two bonds
Which of the following has the weakest linear relationship? - .85, -.15, or .43
-.15 because its closest to zero
A correlation will always be between what two numbers?
-1 and 1
the closer the value is to 1 or -1, the stronger the linear relationship
Given a portfolio of 5 stocks, how many unique covariance terms, excluding variances, are required to calculate the portfolio return variance?
Formula: N^2 - n / 2
