Drawing probability distribution
Almost regardless of your opinion about the predictability or efficiency of the markets, you will probably agree that for most assets, guaranteed returns are uncertain or risky. If we ignore the mathematics underlying probability distributions, we can see that they are pictures that describe a particular view of uncertainty. The probability distribution is a statistical calculation that describes the probability that a given variable will be between or within a specific range on a graph.
Uncertainty refers to randomness. It is different from the lack of predictability or the inefficiency of the market. An emerging research point of view holds that financial markets are uncertain and predictable. Furthermore, markets can be efficient but also uncertain.
In finance, we use probability distributions to draw pictures that illustrate our view of the sensitivity of an asset’s return when we think that the asset’s return can be considered a random variable. In this article, we’ll go over some of the most popular probability distributions and show you how to calculate them.
The distributions can be classified as discrete or continuous, and according to whether it is a probability density function (PDF) or a cumulative distribution.
Discrete vs. Continuous Distributions
Discrete refers to a random variable drawn from a finite set of possible outcomes. A six-sided die, for example, has six discrete outcomes. A continuous distribution refers to a random variable drawn from an infinite set. Examples of continuous random variables include speed, distance, and some asset returns. A discrete random variable is typically illustrated with dots or dashes, while a continuous variable is illustrated with a solid line. The following figure shows discrete and continuous distributions for a normal distribution with mean (expected value) of 50 and a standard deviation of 10:
The distribution is an attempt to plot the uncertainty. In this case, a result of 50 is most likely, but it will only happen about 4% of the time; a result of 40 is one standard deviation below the mean and will occur just under 2.5% of the time.
Probability density vs. cumulative distribution
The other distinction is between the probability density function (PDF) and the cumulative distribution function. The PDF is the probability that our random variable will reach a specific value (or in the case of a continuous variable, fall between an interval). We show that by indicating the probability that a random variable X will be equal to a real value X:
The cumulative distribution is the probability that the random variable X will be less than or equal to the actual value X:
or, for example, if his height is a random variable with an expected value of 5’10 “inches (the average height of his parents), then the question in the PDF is,” What is the probability that he will reach a height of 5’4 “?” The corresponding cumulative distribution function question is: “What is the probability that you are less than 5’4”?
The previous figure showed two normal distributions. Now you can see that these are probability density function graphs (PDF). If we redraw exactly the same distribution as a cumulative distribution, we will get the following:
The cumulative distribution should eventually reach 1.0 or 100% on the y-axis. If we raise the bar high enough, at some point, practically all the results will fall below that bar (we could say that the distribution is typically asymptotic at 1.0).
Finance, a social science, is not as clean as the physical sciences. Gravity, for example, has an elegant formula that we can rely on over and over again. Returns on financial assets, on the other hand, cannot be replicated as consistently. A staggering amount of money has been lost over the years by smart people who have confused precise distributions (that is, as derived from the physical sciences) with confusing and confusing approximations that attempt to represent financial returns. In finance, probability distributions are little more than crude pictorial representations.
The simplest and most popular distribution is the uniform distribution, in which all outcomes are equally likely to occur. A six-sided die has a uniform distribution. Each result has a probability of approximately 16.67% (1/6). Our graph below shows the solid line (so you can see it better), but keep in mind that this is a discrete distribution; can’t roll 2.5 or 2.11:
Now roll two dice together, as shown in the figure below, and the distribution will no longer be uniform. It reaches a maximum of seven, which has a probability of 16.67%. In this case, all other outcomes are less likely:
Now roll three dice together, as shown in the following figure. We begin to see the effects of an amazing theorem: the central limit theorem. The central limit theorem boldly promises that the sum or average of a series of independent variables will tend to be normally distributed, regardless of your own distribution. Our dice are individually uniform, but combine them and as we add more dice, almost magically their sum will tend toward the familiar normal distribution.
The binomial distribution reflects a series of “one or the other” attempts, such as a series of coin tosses. These are called Bernoulli trials, which refer to events that have only two outcomes, but do not need equal odds (50/50). The binomial distribution below plots a series of 10 coin tosses in which the probability of heads is 50% (p-0.5). You can see from the figure below that the probability of throwing exactly five heads and five tails (the order doesn’t matter) is just 25%:
If the binomial distribution seems normal to you, you are correct. As the number of trials increases, the binomial tends toward the normal distribution.
The log normal distribution is very important in finance because many of the more popular models assume that stock prices are logarithmically normal. It’s easy to confuse asset returns with price levels.
Asset returns are often treated as normal – a stock can go up 10% or down 10%. Price levels are often treated as lognormal: a $ 10 stock can go up to $ 30 but cannot go down to – $ 10. The logarithmic normal distribution is not zero and is skewed to the right (again, a stock is not may fall below zero but does not have a theoretical limit to the upside):
The Poisson distribution is used to describe the probabilities of a certain event (for example, a daily portfolio loss below 5%) occurring over a time interval. So in the example below, we assume that some operational process has a 3% error rate. In addition we assume 100 randomized trials; The Poisson distribution describes the probability of obtaining a certain number of errors during a period of time, such as a single day.
The Student’s t distribution is also very popular because it has a “thicker tail” than the normal distribution. Student’s T is typically used when our sample size is small (that is, less than 30). In finance, the left tail represents losses. Therefore, if the sample size is small, we dare to underestimate the probabilities of a large loss. The thicker tail on the student’s T will help us here. Even so, it happens that the thick tail of this distribution is often not thick enough. Financial returns tend to exhibit, on rare occasions catastrophic, really big losses (that is, bigger than predicted by the distributions). Large sums of money have been lost in 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 recovery rates in bond portfolios. The beta distribution is the utility of the distributions. Like normal, you only need two parameters (alpha and beta), but they can be combined for remarkable flexibility. Here are four possible beta distributions:
The bottom line
Like so many shoes in our statistical shoe closet, we try to pick the one that best suits the occasion, but we don’t really know what time is in store for us. We can choose a normal distribution and then find that it underestimated the left-tail losses; so we switch to a skewed distribution, only to find that the data appear more “normal” in the next period. The elegant underlying mathematics may lead you to think that these distributions reveal a deeper truth, but they are more likely to be mere human artifacts. For example, all the distributions we review are fairly fluid, but some asset returns increase discontinuously.
The normal distribution is ubiquitous and elegant, requiring only two parameters (mean and distribution). Many other distributions converge towards the normal (for example, binomial and Poisson). However, many situations, such as returns from hedge funds, loan portfolios, and serious loss events, do not warrant normal distributions.