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# Practical application of Probability in Finance

February 6, 2013

Probability in Finance is a tool for modeling financial markets their risks and returns. The topics are also covered in many finance certifications such as CFA Program conducted by CFA Institute, FRM conducted by GARP, PRM conducted by PRMIA

A lot of candidates skip this section, finding it hard to understand. However if you see the practical application of probability in finance, you will understand its importance.

There are a lot of real world applications of probability in finance as can be widely seen in portfolio expected returns and variance.

Probability in broad sense is term used to indicate vague possibility that something might happen. It is also used as a synonym with chance.

## Deterministic phenomena

If the result of experiment is certain, then experiment is said to be of deterministic nature.

## Probabilistic Phenomena

If the result is not unique and can be any one of the several possible outcomes, such an experiment is called a random experiment. Random experiment is of probabilistic nature.

e.g when we throw a dice we may get the outcome any integer from 1 to 6. Hence it is called random experiment

## Biased and UnBiased Experiments

When we throw a die, if there is any reason to believe that one of the numbers will turn up more frequently we say it is a biased die. If all the six numbers are equally likely we call it unbiased die.

Similarly a coin also can be termed as an unbiased or biased depending on whether the head or tail are equally likely or not. Mostly questions asked in exams like CFA Program, FRM are biased on unbiased experiments.

## Occurences

The basic outcomes of an experiment are termed as occurrences. Hence, if we throw a die there are 6 occurrences. If we toss a coin there are 2 occurrences.

Events are those that can be defined by us and each event may have one or more occurences. e.g if we throw a dice we can define following events –

getting a 1, getting a 2, getting a 3, getting a 4, getting a 5, getting a 6.

Here each if the events has one occurrence. We can also define following 2 events when a die is thrown

1. Getting an even number

2. Getting an odd number

Each of these 2 events has 3 occurrences which are

• getting even number {2, 4, 6}

• getting odd number {1,3,5}

So in this case we say out of 6 occurrences in the experiment, 3 are favorable to even of “getting an even number” and 3 are favorable to “getting an odd number”.

Equally likely events – 2 events are said to be equally likely when there is no reason to expect one rather than the other.

## Mutually Exclusive Events

Events that can not occur simultaneously are called Mutually exclusive events. e.g Throw a die two events defines as event 1 occurring of odd number on dice and event 2 occurrence of even number on dice, are mutually exclusive events.

Let’s consider a practical example of probability in investment world

If we take a real example for analyst’s work, An investor concerns center on “returns”. i.e what would be the return of a given portfolio. We can define here the return on risky asset as an example of random variable whose outcome is uncertain.

e.g lets assume a portfolio’s target return for next year is 10%. In this case we can define 2 events portfolio

• Event 1 earns a return below 10%

• Event 2 portfolio earns a return of 10%

## Empirical Probability

In investments we often estimate the probability of an event based on historical data, this probability is called Empirical probability. Relationships must be stable through time for empirical probabilities to be accurate.

## Subjective probability

Probability that is based on personal assessment of probability without any reference to any particular data. Investors in making buy and sell decisions that determine asset prices, often draw on subjective probability. It is also used in Bayes theorem.

In a more narrow range of well defined problems we can sometimes deduce probabilities by reasoning about the problem. The resulting probability is a priori probability, one based on logical analysis rather than on personal judgement or observation. The counting methods are particularly important in calculating a priori probabilities. Because a priori and empirical probabilities do not vary from person to person they are often termed as objective probabilities.

Probability stated as Odds. Given a probability P(E)

Odds for E = P(E)/(1-P(E)). The odds of E are the probability of E divided by 1 (minus) probability of E. Given odds for E of “a to b”, the implied probability of E is a/(a+b).

e.g odds for the company’s EPS for FY 2010 beating 4.25\$ are 1 to 7 means that the speaker believes the probability of event is 1/8 = .125

The statement that odds against the company’s EPS for FY 2010 beating 4.25 \$ are 15 to 1 is consistent with a belief that the probability of the event is 1/16. = .0625.

To further explain odds for an event, if P(E) = 1/8, the odds for E are (1/8)/(7/8) = 1/7 which means for each occurrence of E, we expect seven cases of non occurrence

There are lot of other practical examples which we will cover in our later topics. If you have any other doubts or queries, mention it in the comments box below and we shall get back to you at the earliest.

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