Probability concepts
Introduction
Fundamental Concepts
• A variable is random if its outcome is uncertain, where an outcome
is an observable future value of the variable.
• An event is the specified set of possible outcomes of a random
variable.
• Events are mutually exclusive when the possible future outcomes can
only occur one at a time and exhaustive when the set of outcomes
includes every possible value the variable could take in the future.
• Example: The future size of a dividend can be stated as a mutually
exclusive and exhaustive event wherein dividends increase, decrease, or
remain unchanged.
• When the occurrence of one event does not affect the probability of the
occurrence of another event, we say the events are independent.
• Events that are not independent are dependent. 2
Probability
Probability is the fundamental building block of statistics.
• Probability is a number between 0 and 1 that describes the chance
that a stated event from the set of possible outcomes will occur.
• A probability distribution is the set of probabilities and their
associated outcomes that describes all possible outcomes and their
associated probabilities.
• We typically use P(E) to denote the probability of event E.
• Properties of probability
1. All probabilities must lie between 0 and 1:
2. For n mutually exclusive and exhaustive events,
the sum of all probabilities must equal 1:
3
Types of probability
Sources of probabilities
• In practice, we observe a number of different types of probability.
• A subjective probability is a personal assessment of the likelihood of an
event or set of events occurring in the future and is so named because it
relies on the subjective judgment of the person making the assessment.
• An empirical probability is one that is estimated from observed data,
typically using the relative frequency at which an event or set of events
has occurred in the past.
• An a priori probability is one whose values are obtained from
mathematical or logical analysis.
4
Probability as odds
Probabilities are often stated as odds for or against a given event occurring.
• Tire industry analysts are interested in the odds of a removal of import
restrictions on Chinese tires. Your analyst has assessed the probability of
removal as 60%. What are the odds for and against removal?
5
Odds for: Odds against:
Odds for: Odds against:
Mutually Inconsistent
ProbabilitiesThe Dutch Book Theorem describes how to profit from inconsistent
probabilities.
• Stock prices and other forward-looking financial variables often
contain probabilities that we cannot observe directly.
• We can often infer something about the probability of an event(s) that
will affect more than one firm from information about the set of firms
that will be affected.
• When this process reveals inferred probabilities for two or more firms
that are different and, therefore, inconsistent, there may be a possibility
for profitable investment strategies.
6
Mutually Inconsistent
probabilities
Focus on: Profiting from Inconsistent Probabilities
• Cleveland Corp. and High Noon Inc. are both U.S. tire makers and are
fundamentally similar. The sales of both companies stand to be harmed
substantially from the removal of import restrictions on inexpensive
imported tires.
• The price of Cleveland Corp. shares reflects a probability of 0.60 that the
restrictions will be removed within the year. The price of High Noon Inc.
stock, however, reflects a 0.70 probability that the restrictions will be
removed within that time frame.
• By all other information related to valuation, the two stocks appear
comparably valued.
How would you characterize the implied probabilities reflected in share
prices?
Which stock is relatively overvalued compared with the other?
7
Mutually Inconsistent
probabilities
Focus on: Profiting from Inconsistent Probabilities
• The implied probabilities would be characterized as inconsistent
probabilities:
8
True Probability of Removing Restrictions
0.60 0.70
Cleveland Corp. Shares fairly valued Shares overvalued
High Noon Inc. Shares undervalued Shares fairly valued
Conditionalandunconditionalprobability
• When we observe the probability that an event (A) occurs without
taking into account whether it is necessarily preceded by any other
specific events, it is known as an unconditional probability.
• Also known as a marginal probability.
• Notation:
• When we observe the probability of a given event after taking into
account that another event has already occurred (A occurs given that
B has occurred), it is known as a conditional probability.
• Notation:
• Language: the probability of A given B
9
Joint Probability
• When two events occur, the combined probability of their
occurrence is known as the joint probability.
• Notation:
• Language: the probability of A and B
• The calculation of a joint probability is governed by a set of probability
rules, as are the calculations of the probability of other combinations of
events.
10
P(A)
P(B)
P(AB)
Working with probabilities
• We use the multiplication rule to assess the joint probability of A
and B occurring.
• Note that from this equation, we can calculate the conditional
probability of A given B, as long as the probability of B isn’t zero.
• We use the addition rule to assess the probability that A or B or both
occur:
11
Multiplication rule
Focus on: Calculations
• Recall that our probability of relaxed trade restrictions has been
estimated at 60%.
• If the probability of reduced sales, given that the trade restrictions
are relaxed, is 80%, then the probability of relaxed trade restrictions
and reduced sales is
12
Addition rule
Focus on: Calculations
• If the probability of relaxed import restrictions is 0.60 and the
probability of a trade war is 0.10, then the probability of relaxed
trade restrictions or a trade war is 0.65 when the joint probability of
a trade war and relaxed trade restrictions is 0.05.
13
P(A) = 0.6
P(B) = 0.1
P(AB) =
0.05
Dependent and Independent
events
• When events are independent, the occurrence of one does not affect
the probability of the other. In other words, P(A|B = 0) and P(B|A) =
0.
• To determine the joint probability, we use the multiplication rule:
or, more generally,
14
P(A) P(B) P(C)
Jointprobabilityofindependentevents
Focus on: Calculations
• Recall our analyst’s estimated probability of relaxed import
restrictions (60%).
• If A is the probability of a heavy winter snowfall (40%) and
B is the independent probability of relaxed restrictions:
• Then the joint probability of relaxed import restrictions and a heavy
winter snowfall is:
15
Total probability rule
Focus on: Calculations
• Recall that we have a 60% (40%) chance of trade restrictions being
relaxed (maintained).
• P(RR) = 0.6, where RR is relaxed restrictions.
• P(NR) = 0.4, where NR is no relaxation.
• Our analysts believe that the stock price will decrease if trade
restrictions are relaxed with a probability of 30% and that they will
decrease if trade restrictions are not relaxed with a probability of
15%.
• P(DS|RR) = 0.30
• P(DS|NR) = 0.15
• What is the probability of a decrease in stock prices?
16
Expected value
Random Variable
• The expected value of a random variable is the probability-weighted
average of the possible outcomes for that variable.
• We anticipate that there is a 15% chance that next year’s return on
holding Cleveland Corp will be 4%, a 60% chance it will be 6%, and a
25% chance it will be 8%. What is the expected return on Cleveland
Corp stock?
17
Variance
Random Variable
• The variance of a random value is the sum of the squared deviations from
the expected value weighted by their associated probabilities.
• This value is a measure of the dispersion of possible values.
• Because it has units that are squared, it is not easy to interpret. Accordingly,
we use its positive square root, standard deviation, more often because it
also measures dispersion but has the same units as expected value.
• The standard deviation of returns for Cleveland Corp. is then:
σ2 𝑋 = 0.15 0.04 − 0.062 2 + 0.6 0.06 − 0.062 2 + 0.25(0.08 − 0.062)2
σ 𝑅 = 0.01249
18
σ2
𝑋 = 𝑋𝑖 − 𝐸 𝑋 2 𝑃 𝑋𝑖 = 𝐸 𝑋 − 𝐸(𝑋) 2
𝑛
𝑖=1
Conditional Expectations
Focus on: Calculations
• The total probability rule applies to expected values just as it does
any mutually exclusive and exhaustive set of possible outcomes
across a set of states.
• This equation allows us to calculate the expected value of a random
variable (X) as a function of the probabilities of future possible
states, P(S), and the conditional value of the expected value of X in
those states, E(X|S).
19
Conditional Expected value
Focus on: Calculations
Example: Recall that we have a 60% chance of relaxed trade
restrictions and, therefore, a 40% of maintaining them. If we expect
the Cleveland Corp. stock to return 6% if trade restrictions are
maintained and lose 11% if they are relaxed, what is the expected
change in return for Cleveland Corp.?
20
Tree diagram
A visual representation of the future possible outcomes and associated
probabilities of a random variable.
• Recall that we have a 60% (40%) chance of trade restrictions being relaxed
(maintained) and a 40% (60%) chance of a heavy (light) winter snowfall.
• Our analyst projects that if trade restrictions are relaxed and there is a light
snowfall, Cleveland Corp. will have a stock return of 15%, but if restrictions
are relaxed and there is a heavy snowfall, there will be a stock return of 5%.
If trade restrictions are not relaxed and there is a light snowfall, we will
have a stock return of 2%, but if we have a heavy snowfall, we will have
stock return of 12%.
How do we represent the possible impact of snowfall and trade restrictions
on the return of Cleveland Corp. stock?
What is our overall expected stock return on Cleveland Corp.?
21
Return  –15%
P(RL) = 0.36
Tree diagram
Focus on: Calculations
22
P(Relaxed restrictions) = 0.6
P(Maintained restrictions)=0.4
P(Heavy snow) = 0.4
P(Heavy snow) = 0.4
P(Light snow) = 0.6
P(Light snow) = 0.6
Return –5%
P(RH) = 0.24
Return  12%
P(MH) = 0.16
Return  2%
P(ML) = 0.24
E(Return) = –4.2%
Covariance
Covariance and correlation are both measures of the extent to which two
random variables move together.
• Covariance is the expected value of the product of each variable’s
deviation from its respective mean.
σ 𝑋,𝑌 = 𝐸{ 𝑋 − 𝐸 𝑋 𝑌 − 𝐸 𝑌 }
σ 𝑅 𝑖,𝑅 𝑗
= 𝑃 𝑅𝑖 [𝑅𝑖 − 𝐸(
𝑛
𝑖=1
𝑅𝑖)][𝑅𝑗 − 𝐸 𝑅𝑗 ]
23
Correlation
Covariance and correlation are both measures of the extent to which two
random variables move together.
• Correlation is a scaled transformation of covariance wherein the extent of
comovement is measured along a scale from exactly the same movement in
the same direction to exactly the same movement in opposite directions.
• When two variables move the same degree in opposing directions, they are
said to be perfectly negatively correlated.
• When two variables move the same degree in the same direction, they are
said to be perfectly positively correlated.
• When there is absolutely no commonality of movement, the variables are
said to be uncorrelated.
ρ 𝑋,𝑌 =
σ 𝑋,𝑌
σ 𝑋σ 𝑌
−1 ≤ ρ 𝑋,𝑌 ≤ 1 24
Covariance and Correlation
Focus On: Calculations
• We anticipate that there is a 15% chance that next year’s stock returns for
Cleveland Corp. will be 4%, a 60% chance they will be 6%, and a 25% chance
they will be 8%. The standard deviation of returns is 1.249%, and the
expected value is 6.2%.
• We anticipate that the same probabilities and states are associated with a 2%
return for High Noon Inc., a 3% return, and a 3.5% return. The standard
deviation of High Noon Inc. returns is then 0.46%, and its expected value is
2.975%.
• What is the covariance between Cleveland and High Noon returns?
• What is the correlation between Cleveland and High Noon returns?
25
σ 𝑅 𝑖,𝑅 𝑗
= 0.15 0.04 − 0.062 (0.02
− 0.02975)
+0.6 0.06 − 0.062 0.03 − 0.02975
+0.25 0.08 − 0.062 0.035 − 0.02975
= 0.0000555
ρ 𝑅 𝑖,𝑅 𝑗
=
0.0000555
0.0046(0.01249)
= 0.966
Portfolio returns
Portfolio expected return, variance, and standard deviation are functions of the
weights invested in each asset, much like probabilities function as weights.
• The expected return to a portfolio is the sum of each of the individual
assets expected returns multiplied by its associated weight.
• The variance of a portfolio’s return is the sum of the squared deviations
from the mean multiplied by the associated weights.
σ2
𝑅 𝑝 = 𝑤𝑖 𝑤𝑗Cov(𝑅𝑖, 𝑅 𝑗
𝑛
𝑗=1
)
𝑛
𝑖=1 26
Portfolio returns
Focus On: Calculations
• Consider a portfolio equally invested in High Noon and Cleveland
Corp. What will be its expected return and standard deviation of
returns? Recall that the expected returns are 2.975% and 6.2%,
respectively. Likewise, the standard deviations are 0.46% and
1.249%, respectively, and the covariance between the two is
0.0000555.
27
σ 𝑅 𝑝 = 0.5 2
(0.01249)2
+(0.5)2
(0.0046)2
+2 0.5 0.5 0.0000555
σ 𝑅 𝑝 = 0.000072 = 0.0084876
Bayes’ Rule
Bayes’ Rule is often used to determine how a subjective belief should change
given new evidence.
• First term: updated probability given new information, also known as
posterior probability.
• Second term: probability of the new information given the event
over the probability of the new information.
• Third term: prior probability of the event.
𝑃 Event Info =
𝑃 Info Event
𝑃 Info
𝑃(Event)
28
Bayesian updating
Focus On: Calculations
• Suppose you have the following prior probabilities:
• P(EPS exceeded consensus) = 0.45
• P(EPS met consensus) = 0.30
• P(EPS fell short of consensus) = 0.25
• Given that DriveMed announces an expansion, what is the
probability that prior quarter EPS (unreleased) exceeds
consensus?
29
Bayesian updating
Focus On: Calculations
• You believe the conditional probabilities are
• P(DriveMed expands | EPS exceeded consensus) = 0.75
• P(DriveMed expands | EPS met consensus) = 0.20
• P(DriveMed expands | EPS fell short of consensus) = 0.05
1. Use the total probability rule.
2. Apply Bayes’ formula.
𝑃 Exceeds Expands =
𝑃 Expands Exceeds
𝑃 Expands
𝑃(Exceeds)
𝑃 Exceeds Expands =
0.75
0.41
0.45 = 0.823171 30
Bayesian updating
Focus On: Calculations
Recall that you updated the probability that last quarter EPS
exceeded the consensus from .45 to .823 after the expansion
announcement.
31
Prior Probabilities Conditional Probabilities
P(EPS exceeded consensus) = 0.45 P(DriveMed expands | EPS exceeded consensus) = 0.75
P(EPS met consensus) = 0.30 P(DriveMed expands | EPS met consensus) = 0.20
P(EPS fell short of consensus) = 0.25 P(DriveMed expands | EPS fell short of consensus) = 0.05
𝑃 Exceeds Expands =
0.75
0.41
0.45 = 0.823171
Bayesian updating
Focus On: Calculations
• Update the prior probability that DriveMed’s EPS met
consensus.
• Result is 0.146341, up from 0.3.
• Update the prior probability that DriveMed’s EPS fell short of
consensus.
• Result is 0.030488, down from 0.25.
• Show that the three updated probabilities sum to 1.
• Result: 0.030488 + 0.146341 + 0.823171 = 1
32
Summary
• The concepts of probability and random variables lie at the center of
inferential and descriptive statistics, and thus form an important basis for
the chapters to come.
• In the descriptive statistics area, measures of expected return, variance,
and covariance play a central role in portfolio applications in
investments.
• We can use a variety of probability tools and rules to determine the
likelihood of a given event or set of events in the future.
• We can use counting rules to determine the probability of a given event or
set of events in combination by enumerating the possible potential
outcomes.
33