Posterior probability measures the likelihood that an event will occur given that a related event has already occurred. It is a modification of the original probability or the probability without further information, which is called prior probability. Posterior probability is calculated using Bayes’ Theorem. Financial modeling of stock portfolios is a common application of posterior probability in finance. It is sometimes difficult to accurately assign probabilities to events, limiting posterior probability's usefulness.
In order to calculate posterior probability, the conditional probability of two dependent events can be examined. Let A be the target event, then P(A) is the a priori probability. Let B be a second event that is dependent, or is related to the event A, with probability P(B). Furthermore, let the likelihood of event B occurring, given that A occurs, be P(B|A).
Using Bayes’ Theorem, the posterior probability P(A|B) can be calculated. The theory states: P(A|B) = ^{ P(B|A)*P(A)}⁄_{ P(B)}. Note that if events A and B are independent, then their joint probability is P(A|B) = P(A). This means that their posterior and prior probabilities are identical, since the event B has no effect on the event A.
An example from finance is to calculate whether a stock price will rise, given that interest rates have risen. Let A be the event that stock prices rise, and the probability that stocks will rise is 50% or P(A) = 0.50. Let B be the event that interest rates rise and the probability that stocks rise is 75% or P(B) = 0.75. Finally, let the likelihood that interest rates will rise given that stock prices rise be 20% or P(B|A) = 0.20.
The probability that stock prices will rise given that interest rates rise can be determined by plugging these values into Bayes’ Theorem. It gives P(A|B) = ^{ 0.20*0.50}⁄_{ 0.75 } = 0.13 or 13%. This means that if interest rates are rising, stock prices have a 13% chance of rising too, not exactly a safe bet.
Financial analysts use posterior probability to analyze the interrelationships of many different types of events. Foreign exchange rates, changes in economic policies, and consumer spending habits are all examples of events that could affect stock prices. Quantifying the probabilities that these events will occur is very difficult. Also defining the impact that an event will have on a stock price can also be very challenging.
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