IFT Notes for Level I CFA^{®} Program

Since many investment decisions are made in an environment of uncertainty, it is essential for portfolio managers and investment managers to have a fundamental grasp of probability concepts. In this reading, we will focus on:

- Definitions and rules related to probability
- Expected value and variance
- Covariance and correlation

A** random variable** is an uncertain quantity/number. For example, when you roll a die, the result is a random variable.

An **outcome** is the observed value of a random variable. For example, if you roll a 2, it is an outcome.

An **event** can be a single outcome or a set of outcomes. For example, you can define an event as rolling a 2 or rolling an even number.

**Mutually exclusive events** are events that cannot happen at the same time. For example, rolling a 2 and rolling a 3 are examples of mutually exclusive events. They cannot happen at the same time.

**Exhaustive events** are those that cover all possible outcomes. For example, ‘rolling an even number’ or’ rolling an odd number’ are exhaustive events. They cover all possible outcomes.

The **two defining properties of probability** are:

- The probability of any event has to be between 0 and 1.
- The sum of the probabilities of mutually exclusive and exhaustive events is equal to 1.

The methods of estimating probabilities are:

**Empirical probability**: Based on analyzing the frequency of an event’s occurrence in the past.**A priori probability**: Based on formal reasoning and inspection rather than personal judgment.**Subjective probability**: Informed guess based on personal judgment.

Empirical and a priori probabilities are often grouped as objective probabilities because they do not vary from person to person.

**Probability Stated as Odds**

**Odds for an event** are defined as the probability of the event occurring to the probability of the event not occurring. Odds for E = P(E) / [1 – P(E)].

Given odds for E of “a to b”, the implied probability of E is a / (a + b).

**Example**

If the probability of an event is 0.2, what are the odds of it occurring? Alternatively, if the odds are 1 to 4, what is the probability of this event?

**Solution: **

The odds of the event occurring are = = 1/4. This is stated as odds of 1 to 4.

Given the odds, the probability of the event occurring is = = 0.20.

**Odds against an event** are defined as the probability of the event not occurring to the probability of the event occurring. Odds against E = [1 – P(E)] / P(E).

Give odds against E of “a to b”, the implied probability of E is b / (a + b).

**Example**

If P(E) = 0.2, what are the odds against the event occurring? If the odds against an event are 4 to 1, what is the probability of the event?

**Solution:**

P(E) = , Hence the odds against E are 4 to 1.

Given the odds against an event, the probability of the event is

**Unconditional probability** is the probability of an event occurring irrespective of the occurrence of other events. It is denoted as P(A). Unconditional probability is also called ‘marginal’ probability.

**Conditional probability** is the probability of an event occurring given that another event has occurred. It is denoted as P(A|B), which is the probability of event A given that event B has occurred.

Multiplication rule is used to determine the joint probability of two events. It is expressed as:

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

Rearranging the equation we get the formula for computing conditional probabilities:

P(A|B) = P(AB) / P(B)

**Example**

P(interest rates will decrease) = P(D) = 40%

P(stock price increases) = P(S)

P(stock price will increase given interest rates decrease) = P(S|D) = 70%

Compute probability of a stock price increase **and** an interest rate decrease.

**Solution:**

P(SD) = P(S|D) x P(D) = 0.7 x 0.4 = 0.28 = 28%

Addition rule is used to determine the probability that at least one of the events will occur. It is expressed as:

P(A or B) = P(A) + P(B) – P(AB)

P(AB) represents the joint probability that both A and B will occur. It is subtracted from the sum of the unconditional probabilities: P(A) + P(B), to avoid double counting.

If the two events are mutually exclusive, the joint probability: P(AB) is zero and the probability that either A or B will occur is simply the sum of the unconditional probabilities for each event:

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

**Example**

P(price of A increases) = P(A) = 0.5

P(price of B increases) = P(B) = 0.7

P(price of A and B increases) = P(AB) = 0.3

Compute the probability that the price of stock A **or** the price of stock B increases.

**Solution**

P(A or B) = 0.5 + 0.7 – 0.3 = 0.9