PROBABILITY CONCEPTS

Key concepts are described Probability rules are introduced Expected values, standard deviation,

covariance and correlation for individual portfolio returns are explained

Probability TerminologyRandom variable: uncertain number

Outcome: realization of random variable

Event: set of one or more outcomes

Mutually exclusive: cannot both happen

Exhaustive: set of events includes all possible outcomes

Two Properties of Probability

Probability of an event, P(Ei), is between 0 and 1

0 ≤ P(Ei) ≤ 1

For a set of events that are mutually exclusive and exhaustive, the sum of probabilities is 1

ΣP(Ei) = 1

Types of Probability

Empirical: based on analysis of data

Subjective: based on personal perception

A priori: based on reasoning, not experience

Odds For or Against

Probability that a horse will win a race = 20%

Odds for: 0.20 / (1 – 0.20) = 1/4

= one-to-four

Odds against: (1 – 0.20) / 0.20 = 4/1

= four-to-one

Conditional vs. UnconditionalTwo types of probability:

Unconditional: P(A), the probability of an eventregardless of the outcomes of other events, e.g., probability market will be up for the day

Conditional: P(A|B), the probability of A given that B has occurred, e.g., probability that the market will be up for the day, given that the Fed raises interest rates

Joint ProbabilityThe probability that both of two events will occur is their joint probability

Example using conditional probability:

P (interest rates will increase) = P(I) = 40%

P (recession given a rate increase) = P(R|I) = 70%

Probability of a recession and an increase in rates,

P(RI) = P(R|I) × P(I) = 0.7 × 0.4 = 28%

Probability that at Least One of Two Events Will OccurP(A or B) = P(A) + P(B) – P(AB)

We must subtract the joint probability P(AB)

Don’t doublecount P(AB)

Addition Rule ExampleP(I) = prob. of rising interest rates is 40%

P(R) = prob. of recession is 34%

Joint probability P(RI) = 0.28 (calculated earlier)

Probability of either rising interest rates or recession

= P(R or I) = P(R) + P(I) – P(RI)

= 0.34 + 0.40 – 0.28 = 0.46

For mutually exclusive events the

joint probability P(AB) = 0 so:

P(A or B) = P(A) + P(B)

Joint Probability of any Number of Independent EventsDependent events: knowing the outcome of one

tells you something about the probability of the other

Independent events: occurrence of one event does not influence the occurrence of the other. For the joint probability of independent events, just multiply

Example: Flipping a fair coin, P (heads) = 50% The probability of 3 heads in succession is:

0.5 × 0.5 × 0.5 =0.53 = 0.125 or 12.5%

Calculating Unconditional ProbabilityGiven:

P (interest rate increase) = P(I) = 0.4

P (no interest rate increase) = P(IC) = 1 – 0.4 = 0.6

P (Recession | Increase) = P(R|I) = 0.70

P (Recession | No Increase) = P(R|IC) = 0.10

What is the (unconditional) probability of recession?

P(R) = P(R|I) × P(I) + P(R|IC) × P(IC)

= 0.70 × 0.40 + 0.10 × 0.60 = 0.34

EPS = $1.80Prob = 18%

An Investment Tree

EPS = $1.70Prob = 42%

EPS = $1.30Prob = 24%

EPS = $1.00Prob = 16%

Expected EPS = $1.51

Prob of good economy

60%

Prob of poor economy40%

30%

70%

60%

40%Prob of poor stock performance

Prob of good stock performance

Expected Value using Total Probability

Using the probabilities from the Tree:

Expected(EPS) = $1.51

= .18(1.80) + .42(1.70) + .24(1.30) + .16(1.00)

Conditional Expectations of EPS:

E(EPS)|good economy =

.30(1.80) + .70(1.70) = $1.73

E(EPS)|poor economy =

.60(1.30) + .40(1.00) = $1.18

CovarianceCovariance: A measure of how two variables move

together Values range from minus infinity to positive infinity Units of covariance difficult to interpret Covariance positive when the two variables tend to be

above (below) their expected values at the same time

For each observation, multiply each probability times the product of the two random variables’

deviations from their means and sum them

Correlation Correlation: A standardized measure of the linear

relationship between two variables

Values range from +1, perfect positive correlation

to –1, perfect negative correlation r is sample correlation coefficient ρ is population correlation coefficient

CorrelationExample: The covariance between two assets is 0.0046, σA = 0.0623 and σB = 0.0991. What is the correlation between the two assets (ρA,B)?

Expected Value, Variance, and Standard Deviation (probability model)

Expected Value: E(X) = ΣP(xi)xi

Expected Value, Variance, and Standard Deviation (probability model)

Variance: σ2(X) = ΣP(xi)[xi – E(X)]2

Standard deviation: square root of σ2 = 0.1136

Portfolio Expected Return

Expected return on a portfolio is a weighted average of the expected returns on the assets in the portfolio

Portfolio Variance and Standard Deviation

Portfolio variance also uses the weight of the assets in the portfolio

Portfolio standard deviation is the square root of the variance

Joint Probability FunctionReturns RB= 40% RB= 20% RB= 0% E(RB)=18%

RA= 20% 0.15

RA= 15% 0.60

RA= 4% 0.25

E(RA)=13%

CovAB= 0.15 (.20 - .13) (.40 - .18) +

0.6 (.15 - .13) (.20 - .18) +

0.25 (.04 - .13) (0 - .18) = 0.0066

Probabilities

Good earnings (G)

Bayes’ Formula

Poor earnings (P)

Prob. of interest rate cut (C)

60%

40%

70%

30%

20%

80%

Good earnings (G)

Poor earnings (P)

Prob. of no interest rate cut

42%

18%

8%

32%

Prob (C|G) = 42/(42 + 8) = 84%

Prob (C|G) = [Prob(G|C) × Prob(C)]/Prob(G)

Prob (C/G) = (70% * 60% )/ (42% +8%)

Factorial for Labeling

Out of 10 stocks, 5 will be rated buy, 3 will be rated hold, and 2 will be rated sell. How many ways are there to do this?

10!

5! 3! 2!2,520

Choosing r Objects from n Objects

When order does not matter and with just 2 possible labels, we can use the combination formula (binomial formula)

Example: You have 5 stocks and want to place orders to sell 3 of them. How many different combinations of 3 stocks are there?

Choosing r Objects from n Objects

When order does matter, we use the

permutation formula:

You have 5 stocks and want to sell 3, one at a time. The order of the stock sales matters. How many ways are there to choose the 3 stocks to sell in order?

Calculator Solutions: nCr and nPr

How many ways to choose 3 from 5, order doesn’t matter? 5 → 2nd → nCr → 3 → = 10

How many ways to choose 3 from 5, order does matter? 5 → 2nd → nPr → 3 → = 60

Functions only on BAII Plus (and Professional)

Probability Functions A probability function, p(x), gives the probability

that a discrete random variable will take on the value x

e.g. p(x) = x/15 for X = {1,2,3,4,5}→ p(3) = 20% A probability density function (pdf), f(x) can be

used to evaluate the probability that a continuous random variable with take on a value within a range

A cumulative distribution function (cdf), F(x), gives the probability that a random variable will be less than or equal to a given value

Properties of Normal Distribution

Completely described by mean and variance Symmetric about the mean (skewness = 0) Kurtosis (a measure of peakedness) = 3 Linear combination of normally distributed random

variables is also normally distributed Probabilities decrease further from the mean, but the

tails go on forever

Multivariate normal: more than one r.v., correlation between their outcomes

Confidence Interval: Normal DistributionConfidence interval: a range of values around an expected outcome within which we expect the actual outcome to occur some specified percent of the time.

Confidence Interval: Normal Distribution

Example: The mean annual return (normally distributed) on a portfolio over many years is 11%, and the standard deviation of returns is 8%. A 95% confidence interval on next year’s return is 11% + (1.96)(8%) = –4.7% to 26.7%

90% confidence interval = X ± 1.65s

95% confidence interval = X ± 1.96s

99% confidence interval = X ± 2.58s

Standard Normal DistributionA normal distribution that has been standardized

so that mean = 0 and standard deviation = 1To standardize a random variable, calculate the

z-valueSubtract the mean (so mean = 0) and divide by

standard deviation (so σ = 1)

Calculating Probabilities Using the Standard Normal DistributionExample 1: The EPS for a large sample of firms is normally distributed and has µ = $4.00 and σ = $1.50. Find the probability of a value being lower than $3.70.

3.70 is 0.20 standard deviations below the mean of 4.00.

Calculating Probabilities Using the Standard Normal DistributionExample 1 cont.: Here we need to find the area under the curve to the left of the z-value of –0.20.

Excerpt from a Table of Cumulative Probabilities for a Standard Normal Distribution

Calculating Probabilities Using the Standard Normal Distribution

For negative z-value calculate 1 – table value

Calculating Probabilities Using the Standard Normal Distribution

Find the area to the left of z-value + 0.20: From the table this is 0.5793

Since the distribution is symmetric, for negative values we take 1 minus the table value

Probability of values less than $3.70 is 1 – 0.5793 = 42.07%

With a z-table for negatives, F(-0.20) = 0.4207