(R8) Quantitative Methods: Probability Concepts

Question 1

Random Variable

Answer 1

A quantity whose outcomes are unknown

Question 2

Outcome

Answer 2

Possible values

Question 3

Event

Answer 3

A specified set of outcomes (point or range); denoted by capital letters

Question 4

The two defining properties of a probability are as follows:

Answer 4

= P (A)

1. P (A) is less than or equal to 1 or greater than or equal to zero
2. The sum of all probabilities sums up to 1

Question 5

Mutually exclusive events

Answer 5

If one event happens, another event can’t (Event A or B)

Question 6

Exhaustive events

Answer 6

Covers all possible outcomes

Question 7

The two elements needed to solve a probability problem are:

Answer 7

1. Set of all distinct possible outcomes

2. The probability distribution

Question 8

Empirical Probability

Answer 8

• Based on historical observations
• Past is assumed to be representative of the future
• Historical period must include occurrences of the event

Question 9

Subjective probability

Answer 9

• 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

Question 10

A priori probability

Answer 10

arrived at based on deductive reasoning

Question 11

Odds for E. What is the formula and meaning?

Answer 11

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)

Question 12

Odds against E. What is the formula and meaning?

Answer 12

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)

Question 13

Probability: Terminology (5)

Answer 13

1. Random Variable: Uncertain number
2. Outcome: Realization of random variables
3. Event: Set of one or more outcomes
4. Mutually exclusive: cannot both happen
   5: Exhaustive: Set of events includes all possible outcomes

Question 14

Probability: Types (3)

Answer 14

1. Empirical: Based on analysis of data
   
   2. Subjective: Based on personal perception
   3. A priori: Based on reasoning, not experience

Question 15

Probability: Conditional vs Unconditional

Answer 15

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)

Question 16

Joint Probability (Multiplication rule of probability)

Answer 16

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%

Question 17

Addition Rule for Probabilities

Answer 17

• 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

Question 18

Formula for the probability of A given B; P (A|B) =

Answer 18

P(AB) / P(B)

Question 19

Multiplication rule for independent events

Answer 19

When two events are independent of each other, the joint probability of A and B = P(A) x P(B)

Question 20

Total probability rule formula is

Answer 20

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

Question 21

Portfolio expected return

Answer 21

Weighted average of the expected returns on the different securities (weight of asset x return on the asset)

Question 22

Covariance of returns is negative if these two conditions are met

Answer 22

1. The return on one asset is above its expected value
2. The return on the other asset tends to be below its expected value
   This is an inverse relationship

Question 23

Covariance of returns is equal to 0 when

Answer 23

The return on the assets are unrelated

Question 24

Covariance of returns is positive the following is met

Answer 24

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

Question 25

Bayes’ formula

Answer 25

Rational method for adjusting our viewpoints as we confront new information

P (A|B) = [P(B|A) / P(B) ] x P (A)

Question 26

4 Principles of counting

Answer 26

1. Multiplication rule
2. Multinomials
3. Combinations
4. Permutations

Question 27

Calculate the correlation of returns using a covariance matrix

Answer 27

= covariance / [square root of variance 1 x square root of variance 2]

Question 28

What is permutation used for?

Answer 28

Is a listing in which the order of the listed items does matter

Question 29

What is combination used for?

Answer 29

Is a listing in which the order of the listed items does not matter

Question 30

Another way to calculate covariance if given standard deviation and correlation

Answer 30

= standard deviation of one bond x standard deviation of second bond x the correlation between two bonds

Question 31

Which of the following has the weakest linear relationship? - .85, -.15, or .43

Answer 31

-.15 because its closest to zero

Question 32

A correlation will always be between what two numbers?

Answer 32

-1 and 1

the closer the value is to 1 or -1, the stronger the linear relationship

Question 33

Given a portfolio of 5 stocks, how many unique covariance terms, excluding variances, are required to calculate the portfolio return variance?

Answer 33

Formula: (N^2 - n) / 2

Question 34

Expected Value formula

Answer 34

P(Xi)(Xi); probability of instance times the instance

Question 35

Variance of expected value formula

Answer 35

P(Xi)[Xi - E(X)]^2; probability of instance x (instance - expected value) squared