Perhaps the most common probability distribution is the normal distribution, or " bell curve ," although several distributions exist that are commonly used. Typically, the data generating process of some phenomenon will dictate its probability distribution. This process is called the probability density function. Academics, financial analysts and fund managers alike may determine a particular stock's probability distribution to evaluate the possible expected returns that the stock may yield in the future.
The stock's history of returns, which can be measured from any time interval, will likely be composed of only a fraction of the stock's returns, which will subject the analysis to sampling error.
By increasing the sample size, this error can be dramatically reduced. There are many different classifications of probability distributions. Some of them include the normal distribution, chi square distribution, binomial distribution , and Poisson distribution. The different probability distributions serve different purposes and represent different data generation processes. The binomial distribution, for example, evaluates the probability of an event occurring several times over a given number of trials and given the event's probability in each trial.
Another typical example would be to use a fair coin and figuring the probability of that coin coming up heads in 10 straight flips. A binomial distribution is discrete , as opposed to continuous, since only 1 or 0 is a valid response. The most commonly used distribution is the normal distribution, which is used frequently in finance, investing, science, and engineering.
The normal distribution is fully characterized by its mean and standard deviation, meaning the distribution is not skewed and does exhibit kurtosis. This makes the distribution symmetric and it is depicted as a bell-shaped curve when plotted.
A normal distribution is defined by a mean average of zero and a standard deviation of 1. Unlike the binomial distribution, the normal distribution is continuous, meaning that all possible values are represented as opposed to just 0 and 1 with nothing in between. Stock returns are often assumed to be normally distributed but in reality, they exhibit kurtosis with large negative and positive returns seeming to occur more than would be predicted by a normal distribution.
In fact, because stock prices are bounded by zero but offer a potential unlimited upside, the distribution of stock returns has been described as log-normal. This shows up on a plot of stock returns with the tails of the distribution having greater thickness. Probability distributions are often used in risk management as well to evaluate the probability and amount of losses that an investment portfolio would incur based on a distribution of historical returns.
One popular risk management metric used in investing is value-at-risk VaR. VaR yields the minimum loss that can occur given a probability and time frame for a portfolio. Alternatively, an investor can get a probability of loss for an amount of loss and time frame using VaR. Misuse and overeliance on VaR has been implicated as one of the major causes of the financial crisis. As a simple example of a probability distribution, let us look at the number observed when rolling two standard six-sided dice.
Securities and Exchange Commission. In finance, the left tail represents the losses. Therefore, if the sample size is small, we dare underestimate the odds of a big loss. The fatter tail on the student's T will help us out here. Even so, it happens that this distribution's fat tail is often not fat enough.
Financial returns tend to exhibit, on rare catastrophic occasion, really fat-tail losses i. Large sums of money have been lost making this point. Finally, the beta distribution not to be confused with the beta parameter in the capital asset pricing model is popular with models that estimate the recovery rates on bond portfolios.
The beta distribution is the utility player of distributions. Like the normal, it needs only two parameters alpha and beta , but they can be combined for remarkable flexibility. Four possible beta distributions are illustrated below:. Like so many shoes in our statistical shoe closet, we try to choose the best fit for the occasion, but we don't really know what the weather holds for us.
We may choose a normal distribution then find out it underestimated left-tail losses; so we switch to a skewed distribution, only to find the data looks more "normal" in the next period. The elegant math underneath may seduce you into thinking these distributions reveal a deeper truth, but it is more likely that they are mere human artifacts. For example, all of the distributions we reviewed are quite smooth, but some asset returns jump discontinuously.
The normal distribution is omnipresent and elegant and it only requires two parameters mean and distribution. Many other distributions converge toward the normal e. However, many situations, such as hedge fund returns, credit portfolios, and severe loss events, don't deserve the normal distributions. Portfolio Management. Risk Management. Financial Ratios. Trading Psychology. Tools for Fundamental Analysis. Your Privacy Rights. To change or withdraw your consent choices for Investopedia.
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Popular Courses. Fundamental Analysis Tools for Fundamental Analysis. Table of Contents Expand. Drawing Probability Distribution. Discrete vs. Continuous Distributions. Probability Density vs. Cumulative Distribution. Uniform Distribution.
Binomial Distribution. For each trial, only two outcomes are possible: success or failure. The trials are independent. What happens on the first trial does not affect the second trial, and so on. Now check the "patients cured" variable in Tutorial 2 in the Crystal Ball Getting Started Guide against the conditions of the binomial distribution:.
There are two possible outcomes: the patient is either cured or not cured. The trials are independent of each other. What happens to the first patient does not affect the second patient. The probability of curing a patient 0. Since the conditions of the variable match the conditions of the binomial distribution, the binomial distribution would be the correct distribution type for the variable in question. If historical data are available, use distribution fitting to select the distribution that best describes your data.
The feature is described in detail in Fitting Distributions to Data. You can also populate a custom distribution with your historical data. After you select a distribution type, determine the parameter values for the distribution. Each distribution type has its own set of parameters.
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