(Vol 1) RIII
expected return on the portfolio
(E(Rp)) is a weighted average of the expected returns (R1 to Rn) on the component securities using their respective weights (w1 to wn):
Odds for E
= P(E)/[1 − P(E)]. The odds for E are the probability of E divided by 1 minus the probability of E. Given odds for E of "a to b," the implied probability of E is a/(a + b).
Odds against E
= [1 − P(E)]/P(E), the reciprocal of odds for E. Given odds against E of "a to b," the implied probability of E is b/(a + b).
node
A declining interest rate environment points us to the node of the tree that branches off into outcomes of $2.60 and $2.45. We can find expected EPS given a declining interest rate environment as follows, using Equation 10:
Definition of Event
An event is a specified set of outcomes
Outcome
An outcome is a possible value of a random variable.Using Exhibit 1 as an example, a portfolio manager may have a return objective of 10% a year. The portfolio manager's focus at the moment may be on the likelihood of earning a return that is less than 10% over the next year. Ten percent is a particular value or outcome of the random variable "portfolio return." Although we may be concerned about a single outcome, frequently our interest may be in a set of outcomes. The concept of "event" covers both.
likelihoods
Conditional probabilities of an observation (here: DriveMed expands) are sometimes referred to as likelihoods. Again, likelihoods are required for updating the probability. Next, you combine these conditional probabilities or likelihoods with your prior probabilities to get the unconditional probability for DriveMed expanding, P(DriveMed expands), as follows:
conditional probability
Contrast the question "What is the probability of A?" with the question "What is the probability of A, given that B has occurred?" The probability in answer to this last question is a conditional probability, denoted P(A | B) (read: "the probability of A given B").
Independence for Random Variables
Definition of Independence for Random Variables. Two random variables X and Y are independent if and only if P(X,Y) = P(X)P(Y).
Independent Events
Definition of Independent Events. Two events A and B are independent if and only if P(A | B) = P(A) or, equivalently, P(B | A) = P(B).
variance
Definition of Variance. The variance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random vari- able's expected value:
Correlation
However, the division in the definition makes correlation a pure number (without a unit of measurement) and places bounds on its largest and smallest possible values, which are +1 and -1, respectively. If two variables have a strong positive linear relation, then their correlation will be close to +1. If two variables have a strong negative linear relation, then their correlation will be close to -1. If two variables have a weak linear relation, then their correlation will be close to 0. Using the above definition, we can state a correlation matrix from data in the covariance matrix alone. Exhibit 13 shows the correlation matrix.
Multiplication Rule for Counting.
If one task can be done in n1 ways, and a second task, given the first, can be done in n2 ways, and a third task, given the first two tasks, can be done in n3 ways, and so on for k tasks, then the number of ways the k tasks can be done is (n1)(n2)(n3) ... (nk). Exhibit 21 illustrates the multiplication rule where, for example, we have three steps in an investment decision process. In the first step, stocks are classified two ways, as domestic or foreign (represented by dark- and light-shaded circles, respectively). In the second step, stocks are assigned to one of four industries in our investment universe: consumer, energy, financial, or technology (represented by four circles with progressively darker shades, respectively). In the third step, stocks are classified three ways by size: small-cap, mid-cap, and large-cap (represented by light-, medium-, and dark-shaded circles, respectively). Because the first step can be done in two ways, the second in four ways, and the third in three ways, using the multiplication rule, we can carry out the three steps in (2)(4)(3) = 24 different ways.
n factorial
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 or n factorial. (By convention, 0! = 1.) To review, in this application, we repeatedly carry out an operation (here, job assignment) until we use up all members of a group (here, three analysts). With n members in the group, the multiplication formula reduces to n factorial.
empirical probability
In investments, we often estimate the probability of an event as a relative frequency of occurrence based on historical data. For example, suppose you noted that 51 of the 60 stocks in a particular large-cap equity index pay dividends. The empirical probability of the stocks in the index paying a dividend is P(stock is dividend paying) = 51 / 60 = 0.85.
posterior probability
Prior to DriveMed's announcement, you thought the probability that DriveMed would beat consensus expectations was 45%. On the basis of your interpretation of the announcement, you update that probability to 82.3%. This updated probability is called your posterior probability because it reflects or comes after the new information. The Bayes' calculation takes the prior probability, which was 45%, and multiplies it by a ratio—the first term on the right-hand side of the equal sign. The denominator of the ratio is the probability that DriveMed expands, as you view it without consider- ing (conditioning on) anything else. Therefore, this probability is unconditional
expected value
The expected value of a random variable is the probability-weighted average of the possible outcomes of the random variable. For a random variable X, the expected value of X is denoted E(X). Expected value (for example, expected stock return) looks either to the future, as a forecast, or to the "true" value of the mean (the population mean). We should dis- tinguish expected value from the concepts of historical or sample mean. The sample mean also summarizes in a single number a central value. However, the sample mean presents a central value for a particular set of observations as an equally weighted average of those observations. In sum, the contrast is forecast versus historical, or population versus sample.
Multiplication Rule for Expected Value of the Product of Uncorrelated Random Variables.
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. Many financial variables, such as revenue (price times quantity), are the product of random quantities. When applicable, the above rule simplifies calculating expected value of a product of random variables.
Combination Formula (Binomial Formula)
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 combination formula tells us how many ways we can select a group of size r. We can illustrate this formula with the binomial option pricing model. (The binomial pricing model is covered later in the CFA curriculum. The only intuition we are concerned with here is that a number of different pricing paths can end up with the same final stock price.) This model describes the movement of the underlying asset as a series of moves, price up (U) or price down (D). For example, two sequences of five moves containing three up moves, such as UUUDD and UDUUD, result in the same final stock price. At least for an option with a payoff dependent on final stock price, the number but not the order of up moves in a sequence matters. How many sequences of five moves belong to the group with three up moves? The answer is 10, calculated using the combination formula ("5 choose 3"):
unconditional probability
The probability in answer to the straightforward question "What is the probability of this event A?" is an unconditional probability, denoted P(A). Suppose the question is "What is the probability that the stock earns a return above the risk-free rate (event A)?" The answer is an unconditional probability that can be viewed as the ratio of two quantities. The numerator is the sum of the probabilities of stock returns above the risk-free rate. Suppose that sum is 0.70. The denominator is 1, the sum of the probabilities of all possible returns. The answer to the question is P(A) = 0.70.
complement
The total probability rule is stated below for two cases. Equation 5 gives the sim- plest case, in which we have two scenarios. One new notation is introduced: If we have an event or scenario S, the event not-S, called the complement of S, is written SC. Note that P(S) + P(SC) = 1, as either S or not-S must occur. Equation 6 states the rule for the general case of n mutually exclusive and exhaustive events or scenarios.
permutation
To address the first question above, we need to count ordered listings such as first place, New Company; second place, Fir Company; third place, Well Company. An ordered listing is known as a permutation, and the formula that counts the number of permutations is known as the permutation formula. A more formal definition states that a permutation is an ordered subset of n distinct objects. Permutation Formula is 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 matter
joint probability
To state an exact definition of conditional probability, we first need to introduce the concept of joint probability. Suppose we ask the question "What is the probability of both A and B happening?" The answer to this question is a joint probability, denoted P(AB) (read: "the probability of A and B"). If we think of the probability of A and the probability of B as sets built of the outcomes of one or more random variables, the joint probability of A and B is the sum of the probabilities of the outcomes they have in common. For example, consider two events: 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 (a subset of) the outcomes of B, so P(AB) equals P(A). We can now state a formal definition of conditional probability that provides a formula for calculating it.
joint probability function
We can also calculate covariance using the joint probability function of the random variables, if that can be estimated. The joint probability function of two random variables X and Y, denoted P(X,Y), gives the probability of joint occurrences of values of X and Y. For example, P(X=3, Y=2), is the probability that X equals 3 and Y equals 2.
diffuse priors
When the prior probabilities are equal, the probability of information given an event equals the probability of the event given the information. When a decision maker has equal prior probabilities (called diffuse priors), the probability of an event is determined by the information.
Multiplication Rule for Independent Events
When two events are independent, the joint probability of A and B equals the product of the individual probabilities of A and B.
dependent events
When two events are not independent, they are dependent: The probability of occurrence of one is related to the occurrence of the other. If we are trying to forecast one event, information about a dependent event may be useful, but information about an independent event will not be useful. For example, suppose an announcement is released that a biotech company will be acquired at an attractive price by another company. If the prices of pharmaceutical companies increase as a result of this news, the companies' stock prices are not independent of the biotech takeover announcement event. For a different example, if two events are mutually exclusive, then knowledge that one event has occurred gives us information that the other (mutually exclusive) event cannot occur.
addition rule for probabilities.
When we have two events, A and B, that we are interested in, we often want to know the probability that either A or B occurs. Here the word "or" is inclusive, meaning that either A or B occurs or that both A and B occur. Put another way, the probability of A or B is the probability that at least one of the two events occurs. Such probabilities are calculated using the addition rule for probabilities.
probability
a number between 0 and 1 that measures the chance that a stated event will occur.
Dutch Book Theorem
inconsistent probabilities create profit opportunities. In our example, investors' buy and sell decisions exploit the inconsistent probabilities to eliminate the profit opportunity and inconsistency.
A random variable
is a quantity whose future outcomes are uncertain
Bayes' formula
is a rational method for adjusting our viewpoints as we confront new information. Bayes' formula and related concepts have been applied in many business and investment decision-making contexts. Bayes' formula makes use of Equation 6, the total probability rule. To review, that rule expressed the probability of an event as a weighted average of the probabilities of the event, given a set of scenarios. Bayes' formula works in reverse; more precisely, it reverses the "given that" information. Bayes' formula uses the occurrence of the event to infer the probability of the scenario generating it. For that reason, Bayes' formula is sometimes called an inverse probability. In many applications, including those illustrating its use in this section, an individual is updating his/her beliefs concerning the causes that may have produced a new observation.
mutually exclusive
means that only one event can occur at a time
exhaustive
means that the events cover all possible outcomes.
. Do I want to assign every member of a group of size n to one of n slots (or tasks)? If no, then proceed to next question.
n! = n(n - 1)(n - 2)(n - 3)...1
. Do I want to count the number of ways to apply one of three or more labels to each member of a group? If no, then proceed to next question.
n!/ (n! n1!n2!...nk!)
Do I want to count the number of ways I can choose r objects from a total of n, when the order in which I list the r objects does not matter? If no, then proceed to next question.
nCr = n!/((n - r)!r!)
a priori probability
one based on logical analysis rather than on observation or personal judgment. Because a priori and empirical probabilities generally do not vary from person to person, they are often grouped as **objective probabilities.
subjective probability
one drawing on personal or subjective judgment. Subjective probabilities are of great importance in investments. Investors, in making buy and sell decisions that determine asset prices, often draw on subjective probabilities.
standard deviation
the square root of the variance
conditional variances
the variance of EPS given a declining interest rate environment and the variance of EPS given a stable interest rate environment. The relationship between unconditional variance and conditional variance is a relatively advanced topic. The main points are 1) that variance, like expected value, has a con- ditional counterpart to the unconditional concept and 2) that we can use conditional variance to assess risk given a particular scenario.
prior probabil- ities (or priors, for short)
■■ P(EPS exceeded consensus) = 0.45 ■■ P(EPS met consensus) = 0.30 ■■ P(EPS fell short of consensus) = 0.25 These probabilities are "prior" in the sense that they reflect only what you know now, before the arrival of any new information. The next day, DriveMed announces that it is expanding factory capacity in Singapore and Ireland to meet increased sales demand. You assess this new information. The decision to expand capacity relates not only to current demand but probably also to the prior quarter's sales demand. You know that sales demand is positively related to EPS. So now it appears more likely that last quarter's EPS will exceed the consensus. The question you have is, "In light of the new information, what is the updated probability that the prior quarter's EPS exceeded the consensus estimate?"
