﻿ Deriving Probability Distributions

Deriving Probability Distributions

Probability distributions can be derived using history, logic, judgment, or a combination of the three.

History-- Observing the past to extrapolate the probability of what will occur in the future is an obvious, but only if the future is expected to look like the past.  If you wanted to know the probability that a thumbtack would land point up or point down, you might flip the thumbtack many times and use the percentage of times it lands point up and point down as estimates of the probability that a thumbtack tossed in the future will land point up or point down.  Resampling techniques rely on such an approach and are becoming more popular now that it is easy to store and process digital data easily.

Logic -- Probability distributions can be derived from logic alone or logic combined with history.  For example, when flipping a coin, one might reasonably assume that there are only two possible outcomes (dismissing the possibilities that the coin won't be lost or land on its edge) and that the two possibilities, heads and tails, are equally likely.  These assumptions, together with the laws of probability result in a probability distribution of .5 for a head and .5 for a tail.

Statisticians have developed formulas for many 'families' of probability distributions based on different sets of assumptions.  Each 'member' of the family is based on a common set of assumptions but is differentiated from other members of the family by one or more parameters.  Some common families of distributions include:

For example, the normal or bell shaped curve is one such family of distributions and depends on the assumptions that the random variable of interest is the sum of other random variables that are independent.  The central limit theorem states that sum (or average) will approach a normal probability distribution even if the random variables being summed are not themselves normal, provided a large enough number are in the sum.  Large enough can be as little as 1 if the random variable being summed is itself normal, or might need to be 10, 20, 30 ... depending on the probability distribution of the random variables being added and how close an approximation to the normal is desired.

When assuming that a random variable is from a well known and tabulated family of distributions, one still needs to estimate the values of the parameters in order for it to be useful.  Parameter estimation is typically achieved from historical observations or by taking samples.

The Black-Scholes Model assumes that at any given moment, the price of a stock is as likely to move up as move down (in a process called a random walk).  It assumes that the amount of movement, either up or down, is represented by a measure called volatility and that the volatility is constant over time.  We believe that these assumptions are too simplistic and that judgment (see below) about the relative likelihood of a stock moving up or down by different amounts can identify investment opportunities because so many people follow strategies based on the Black-Sholes model.

Judgment-- Probability distributions derived from logic and or history, as described above, are not very appealing for evaluating the performance of stocks and stock options because they are based solely on what occurred in the past and future seldom looks like the past.  That isn't to say that we can't use history in making judgments about the future, but these judgments can and should be based on many other considerations as well.

A variety of methods have been used to derive probability distributions from judgment.  These have tended to be rather simplistic and not very useful for deriving probability distributions for stock prices.  For example, one could just ask what a respondent thinks the probability is for each possible outcome -- or for each stock price in a range of prices a likely range.  A probability wheel, consisting of a circle divided into two or more segments is shown to someone to elicit their judgment about the probability of events occurring, each event represented by the area of one of the segments of the circle.  There is little evidence to believe that these approaches are effective.  We will, however, do something similar, but much more robust in our planned approach

Many other ways for gathering judgments related to investing in stocks have been employed -- but not necessarily to derive probability distributions.  Rather, these approaches use some (often times proprietary) formula to combine the judgments and then make recommendations as to which stocks to buy and which to avoid.  For example, the Motley Fool Caps Community in addition to asking for judgment as to whether a stock will outperform or underperform the S&P 500 for a specified time frame.  They also allow participants to enter a 'limit order' consisting of a start limit (the price at which you would buy the stock), and upper and lower close limits (the prices at which you would then sell the stock if you had bought it and it went down or up to the limits respectively).

Our planned approach is to collect judgments according to the principles from Wisdom of Crowds -- that is from a diverse, independent, and decentralized population of individuals --  using pairwise relative comparisons with redundancy as practiced in the Analytic Hierarchy Process.  A powerful mathematical computation, called an eigenvector computation, is applied to these pairwise comparisons in order to derive the probability distribution.  The AHP uses an eigenvector computation which derives priorities/probabilities that are more accurate than any one pairwise comparison.

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