Download Probability Concepts

PROBABILITY CONCEPTS
 Key concepts are described
 Probability rules are introduced
 Expected values, standard deviation,
covariance and correlation for individual
portfolio returns are explained
Probability Terminology
Random 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. Unconditional
Two types of probability:
Unconditional: P(A), the probability of an event
regardless 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 Probability
The 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 Occur
P(A or B) = P(A) + P(B) – P(AB)
We must subtract the joint probability P(AB)
Don’t double
count P(AB)
Addition Rule Example
P(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 Events
Dependent 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 Probability
Given:
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
An Investment Tree
Prob of good stock performance 30%
Prob of good economy
Expected
EPS = $1.51
70%
60%
40%
Prob of poor economy
Prob of poor stock performance
60%
40%
EPS = $1.80
Prob = 18%
EPS = $1.70
Prob = 42%
EPS = $1.30
Prob = 24%
EPS = $1.00
Prob = 16%
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
Covariance
Covariance: 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
Correlation
Example: 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 Function
Returns
RB= 40%
RA= 20%
0.15
RA= 15%
RA= 4%
RB= 20%
RB= 0%
E(RB)=18%
Probabilities
0.60
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
Bayes’ Formula
Prob. of interest
rate cut (C)
60%
70%
Good earnings (G) 42%
30%
Poor earnings (P) 18%
20%
40%
Prob. of no interest
rate cut
80%
Prob (C|G) = 42/(42 + 8) = 84%
Prob (C|G) = [Prob(G|C) × Prob(C)]/Prob(G)
Prob (C/G) = (70%
*
60% )/ (42% +8%)
Good earnings (G) 8%
Poor earnings (P) 32%
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!
 2,520
5! 3! 2!
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 Distribution
Confidence 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
90% confidence interval = X ± 1.65s
95% confidence interval = X ± 1.96s
99% confidence interval = X ± 2.58s
 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%
Standard Normal Distribution
 A normal distribution that has been standardized
so that mean = 0 and standard deviation = 1
 To standardize a random variable, calculate the
z-value
 Subtract the mean (so mean = 0) and divide by
standard deviation (so σ = 1)
Calculating Probabilities Using
the Standard Normal Distribution
Example 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 Distribution
Example 1 cont.: Here we need to find the area
under the curve to the left of the z-value of –0.20.
Calculating Probabilities Using the
Standard Normal Distribution
For negative z-value
calculate 1 – table value
Excerpt from a Table of
Cumulative Probabilities for a
Standard Normal Distribution
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