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Profit Distribution Function [Bayesian Inference
Posted on June 3, 2013 @ 09:06:00 AM by Paul Meagher

One factor that an investor takes into account when deciding whether or not to invest in a company is the expected profit that company might make in the near and the longer term.

So how should we represent the expected profit of a company?

One approach that I think might be useful involves diagramming the expected profit distribution of the company. The profit distribution graph would consist of a subjective estimate of the probability that the company will make a given amount of profit over a specified time frame. The Y axis of the graph is labelled "Probability". The X axis of the graph is labelled "Profit". To construct the graph involves estimating the probability that the company will make specific amounts of profit (e.g., 10k to 20k, 20k to 30k, 30k to 40k, 40k to 50k, 50k to 60k, 60k to 70k). So we assign a probability to the event that a company will make 10k to 20k in profit next year. Then we assign a probability to the event that a company will make between 20k and 30k and so on up to our 70k limit (the range and intervals chosen will vary by company). In this manner we can construct a profit distribution.

The profit distribution that is constructed should be constrained so that the mass of the probability distribution sums to 1. If you constrain it in this manner than you can potentially do bayesian inference upon the profit distribution. This could be in the form of conditionalizations that involve saying that given some factor A (e.g., money invested) the profit distribution function will shift - the mean of the profit distribution would ideally go up by an amount greater than the money invested.

So far in my discussions of Bayesian Angel Investing, I have used Bayesian techniques in an objective manner. The inputs into Bayes formula were objectively measurable entities. In the case of generating the profit distribution function for a company, we are subjectively assigning probabilities to possible outcomes. There is no set of trials we can rerun to establish an objective probability function for the profit distribution of a company (i.e., the relative frequency of different profit levels for the same company repeated many times with profit levels measured). The probability that is assigned to a particular profit level should reflect your best estimate of how likely a given profit level is for the compaany within a particular timeframe. So, what is the probabiity that Google will make between X1 billion and X2 billion next year (e.g., .10)? What is the probability that Google will make between X2 and X3 (e.g., .40). Assign mass to the intervals in such a way that the probability mass of all the intervals sums to 1. Then you will meet all the technical requirements for a distribution to be considered a probability distribution. All the probability axioms are satisfied.

Why go through all this bother to estimate the how profitable a company might be? Why not just ball-park a value that you think is most likely and leave it at that.

One reason is because one number does not adequately represent your state of uncertaintly about the outcome.

Another reason has to do with modelling risk. Usually when you model risk you don't use one number to do so. Those modelling risk usually like to work with probability distributions, not simple point estimates of the most likely outcome. It provides a more informative model of the uncertainty associated with a forecast.

Also, if you are constructing a profit distribution function for a company there is no reason to hide that information from the company you want to invest in or from co-investors. The profit distribution function, because it is inspectable, can be updated with new information from the company and other investors who might offer strategic capabilities. So the transparency and inspectability of the uncertainty model are also useful features of this approach.

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