Part 3. Probability Concepts

Question 1

Random Variable

Answer 1

A quantity whose future outcomes are uncertain.

Question 2

Outcome

Answer 2

A possible value of a random variable.

Question 3

Event

Answer 3

A specified set of outcomes.

Question 4

Probability

Answer 4

A number between 0 and 1 that measures the chance/likelihood that a stated event will occur.

i.e. if there is a probability of 0.65 that the portfolio earns a return below 10%, then there is a 65% chance of that event happening.

Question 5

Properties of probability

Answer 5

1. The probability of any event E is a number between 0 and 1: 0

Question 6

Mutually Exclusive

Answer 6

This means that only one event can occur at a time.

Question 7

Exhaustive

Answer 7

This means that the events cover all possible outcomes.

Question 8

Empirical Probability

Answer 8

Estimating the probability of an event as a relative frequency of occurrence based on historical data.

e.g. suppose you noted 51 out of 60 stocks have a large-cap equity index pay dividends.

The empirical probability of stocks in index paying dividends is P(stock is dividend-paying) = 51/60 = 0.85

An objective probabilty.

Question 9

Subjective probability

Answer 9

Probabilities are drawn from a personal or subjective judgment.

e.g. investors in making buy and sell decisions that determine asset prices.

Question 10

Dutch Book Theorem

Answer 10

Inconsistent probabilities create profit opportunities, where the investors buy and sell decisions exploit inconsistent probabilities to eliminate profit opportunity and inconsistency.

Question 11

Unconditional Probability

Answer 11

Denoted P(A); what is the probability of this event A?

ie: What is the probability that the stock earns a return above the risk-free rate (event A)?

Ans: The unconditional probability that can be viewed as the ratio of two quantities, supposing sum is 0.7, and denominator 1 sum of probabilities of all possible returns; so, P(A) = 0.7.

Question 12

Conditional Probability

Answer 12

“What is the probability of A?”
“What is the probability of A, given that B has occurred?”

Denoted: P(A/B) - “the probability of A given B”

e.g. Suppose the probability that the stock earns a return above the risk-free rate (A), given that the stock earns a positive return (event B).

A (numerator) = the sum of probabilities of stock returns above risk free rate = 0.7.

B (denominator) = the sum of probabilities for all outcomes (returns) above 0% = 0.8

P(A/B) = 0.7/0.8 = 0.875 - positive return, so probability of return is above the risk free rate.

Question 13

Joint Probability

Answer 13

Denoted: P(AB)

The sum of the probabilities of the outcomes they have in common.

e.g. The stock earns a return above the risk-free rate (A), and the stock earns a positive return (B), the outcomes of A are contained within (subset of) the outcomes of B, so P(AB) = P(A).

Question 14

Addition Rule for Probabilities

Answer 14

Given events A and B, the probability that A or B occurs or both occur is equal to the probability that A occurs, plus the probability that B occurs minus the probability that both A and B occur.

Question 15

Independence

Answer 15

Means knowing B tells you nothing about A.

i.e. if P(A/B) = P (A) or P(B/A) = P(B)

Question 16

Multiplication Rule for Independent Events

Answer 16

When two events are independent, the joint probability of A and B equals the product of the individual probabilities of A and B.

P(AB) = P(A)P(B)

Question 17

Total probability rule

Answer 17

This rule explains the unconditional probability of the event in terms of probabilities conditional on the scenarios.

Question 18

Complement (of S)

Answer 18

If we have an event or scenario S, the event not S, denoted by Sc.

With P(S) + P(Sc) = 1

Question 19

Expected Value (of random variable)

Answer 19

The probability-weighted average of the possible outcomes of the random variable.

For random variable X, the expected value of X is denoted as E(X).

Application in forecasts or true value of mean (population mean).

Question 20

Variance (of random variable)

Answer 20

The expected value (probability weighted average) of squared deviations from random variable’s expected value.

Question 21

Standard variation

Answer 21

The positive square root of variance.

Easier to interpret than variance as in same units as random variable.

Question 22

Conditional expected values

Answer 22

The expected value of X conditional on S is the first outcome X1, times the probability of the first outcome given S, P(X1/S) plus the second outcome X2, times the probability of the second outcome given S, P(X2/S), and so forth.

Question 23

Total Probability Rule for the expected value

Answer 23

The principle for stating (unconditional) expected values in terms of conditional expected values.

Question 24

Conditional variance

Answer 24

The variance of EPS given declining interest rate environment and variance of EPS given stable interest rate environment.

1. Has a conditional counterpart to the unconditional concept.
2. We use conditional variance to assess risk given a particular scenario.

Question 25

Expected return on the portfolio E(Rp):

Answer 25

A weighted average of expected returns (R1 to Rn) on the component securities in the portfolio using their respective weights (w1 to wn).

Question 26

Covariance (population):

Answer 26

The probability-weighted average of the cross products of the deviation of each random variable from its own expected value.

Question 27

Interpreting covariance:

Answer 27

If returns are negative, when the return of one asset is above its expected value, the return on the other asset tends to be below its expected value.

Question 28

Joint probability (of 2 random variables, X and Y)

Answer 28

Denoted: P(X,Y)

Give the probability of joint occurrences of values of X and Y.

Question 29

Independence for Random Variables

Answer 29

X and Y are independent if and only if P(X,Y) = P(X)P(Y)

A stronger property than uncorrelatedness as correlation addresses only linear relationships.

Condition holds for independent random variables and also holds for uncorrelated random variables.

Question 30

Multiplication Rule for EV for Product of Uncorrelated Random Variables

Answer 30

The expected value of the product of uncorrelated random variables is the product of their expected values.

E(XY) = E(X)E(Y) if X and Y are uncorrelated

Question 31

Covariances and Correlations of Security Returns (Vasquez)

Answer 31

• The two equity classes had much greater variances and covariance than 2 bond classes, the correlation between equity is lower than bond.
• Long term bonds were more volatile (higher variance) than intermediate term bonds, they both had high correlation.

Question 32

Bayes Formula

Answer 32

A rational method for adjusting our viewpoints as we confront new information.

The rule of total probability expressed the probability of an event as a weighted average of probabilities of the event, given set of scenarios but reverses the given that information.

Uses the occurrence of the event to infer the probability of the scenario generating it.

Question 33

posterior probability

Answer 33

An updated probability reflecting/comes after the new information.

Question 34

Multiplication rule for counting

Answer 34

If one task can be done in n1 ways and a second task given the first can be done in n2 ways and third task given first two can be done in n3 ways, and so on for k tasks the the number of ways the k tasks can be done is (n1)(n2)(n3)…(nk).

Question 35

Investment descision process

Answer 35

1. Stocks are classified in 2 ways, as domestic or foreign. (2)
2. Stocks are assigned to one of 4 industries in our investment universe: consumer, energy, financial or technology. (4)
3. Stocks are classified 3 ways by size: small-cap, mid-cap and large-cap. (3)

Multiplication rule: carry 3 steps in (2)(4)(3) = 24 different ways.

Question 36

N factorial

Answer 36

If we had n analysts, the number of ways we could assign them to n tasks would be:

n! = n(n-1)(n-2)(n-3)….1

By convention 0! = 1

Question 37

Labelling problems

Answer 37

Mutual fund guide ranked 18 bond mutual funds by total returns for the last year.

Assigned each fund to 1 of 5 risk labels: high risk (4 funds), above average risk (4 funds), average risk (3 funds), below average risk (4 funds) and low risk (3 funds) - 18 funds accounted for.

18! possible sequences; to eliminate redundancies from 18! total:

18!/(4!)(4!)(3!)(4!)(3!) = 18!/(24)(24)(6)(24)(6) = 12,864,852,000

Question 38

Multinomial Formula (for labelling problems)

Answer 38

The number of ways that n objects can be labelled with k different labels, with n1 of the first type, n2 of the second type, and so on with n1 + n2 + ….. + nk = n is given by:

n!/n1!n2!….nk!

Question 39

Combination

Answer 39

The number of ways that we can choose r objects from a total of n objects, when the order in which the r objects are listed does not matter.

The formula notation being:

r=n1, n-r = n2 etc