Subject 1. Introduction

An experiment is the act of making an observation or taking a measurement. For example, tossing dice and observing the numbers on top is an experiment.

A particular result of an experiment is called an outcome. It is the value assigned to a random variable. For example, there are six possible outcomes to the experiment of tossing dice.

A random variable is a quantity whose future outcomes are uncertain. For example, when you toss dice, the number on top is a random variable; you are unsure which number will come up.

Any outcome or specified set of outcomes of a random variable is called an event. For example, one event in the dice-tossing experiment is observing an odd number (three possible outcomes: 1, 3, 5).

Mutually exclusive events are events that only occurs one at a time. In the above example, event A (observe an odd number) and event B (observe an even number) are mutually exclusive because they cannot occur simultaneously; a number can be either odd or even, but cannot be both.

Exhaustive events are events that cover all the distinct possible outcomes. In the above example, event A and event B are exhaustive because they cover all six possible outcomes of the dice-tossing experiment.

As a general rule (and it is one you can use with confidence), if probabilities of two or more events add up to 1 and the events share no common outcomes at all, those events will always be mutually exclusive and exhaustive. However, it is important to remember that events can be mutually exclusive and exhaustive (as you've just seen), or can be one but not the other, or can be neither mutually exclusive nor exhaustive.

The two defining properties of probability

In general, probability is the likelihood that an event will happen. P(E) stands for "the probability of event E." The two defining properties of probability are:

  • 0 ≤ P(E) ≤1: the probability of any event E is a number between 0 and 1. The probability of 0 means that the event can never happen and the probability of 1 means that the event is certain to happen.

  • The sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1. For example, the sum of the possibilities of two events (observing an odd number event and observing an even number event in the dice-tossing example) is 1, as these events are mutually exclusive and exhaustive.

Empirical probability

Empirical probability is a probability based on relative frequency of occurrence. It is estimated on the basis of historical data. For example, based on historical data over a 10-year period, the probability of default for real estate mortgage loans is 7%. We cannot estimate the empirical probability for an event without historical data. For empirical probabilities to be accurate, relationships must be stable over time.

Priori probability

Priori probability is a probability based on logical analysis rather than observation or personal judgment. For example, when you toss dice fairly, the probability of rolling an even number is 50%.

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

Subjective probability

Subjective probability is a probability based on personal or subjective judgment. For example, based on his own judgment, Bill believes that the probability that IBM's revenue will increase in 2005 is 60%.

According to the Dutch Book Theorem, one of the most important probability results theories for investments, inconsistent probabilities create profit opportunities. Investors should eliminate the profit opportunity and inconsistency through buy and sell decisions exploiting inconsistent probabilities.

Suppose that:

  • If event E occurs, the values of two assets, A and B, will both rise.
  • The price of asset A reflects a higher probability of event E than the price of asset B and thus inconsistent probabilities exist.

If all other factors are equal, asset A is overvalued compared with asset B.

  • If event E does occur, the price of asset A will not rise as much as the price of asset B. This is because the occurrence of event E is mostly incorporated into the price of asset A.
  • If event E does not occur, the prices of both assets will fall, but the price of asset A will decline more than the price of asset B. This is because, compared with asset B, the price of asset A understates the probability that event E may not occur.

Therefore, investors can profit by buying an undervalued asset (i.e., B) and selling an overvalued asset (i.e., A). Conservative investors will buy asset B and reduce or fully liquidate their position in asset A. Aggressive investors will buy asset B and short asset A. This strategy is known as the pairs arbitrage trade, which involves using the proceeds from the short sale of one stock to purchase another.

Note that the above discussion is based on the assumption that the occurrence of event E will increase the values of the two assets (A and B). If the occurrence of event E will reduce the value of assets A and B, asset B is overvalued if compared with asset A. To profit from inconsistent probabilities, investors should buy asset A and sell asset B.

Example

Suppose that if a hike in oil price occurs, the stock prices of American Airlines (AA) and British Airways (BA) will decline. The stock price of AA reflects a 0.7 probability of a hike in oil price, whereas the stock price of BA reflects only a 0.4 probability. In this situation, the stock of AA is undervalued when compared with the stock of BA. A conservative investor can profit by buying the stock of AA and reducing or eliminating his holdings in the stock of BA. An aggressive investor can profit by buying the stock of AA and shorting the stock of BA.

User Contributed Comments 9

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teddajr: Dutch Book Theorem review.
achu: pairs arbitrage trade.
gasetlhobogwe: This is wondeful information,its worth reading
Galt2012: Is it me or are the undervalued/overvalued distinctions inverted in the last example Wouldn't AA be overvalued since it reflects .7 probability? Compare this last example to the explanantion in question #16 in the accompanying questions for this section.
endurance: The example above is stated correct. If stock AA reflects a 0.7 probability of a hike in oil price, the share price is lagging the stock price of BA which has a 0.4 probability.

Stated differently, the possibility of hike in oil prices is priced more into stock AA than stock BA. That means a better relative performance in AA, when the hike is effective.
gulfa99: easy to understand if you have options background
tjmcd86: Alcoa and Boeing
2014: See text book EOC problem number 4 .... Higher probability overvalued and low probability undervalued. See Analyst notes example pair trading apply here.
Paul7059: Comical and timely example in January 2015