Have you ever used a probability distribution in an appraisal?  If so, this post is for you!

If you haven’t, are you sure?

Really sure?

Methinks you have actually used one; in fact, you have used them in almost every single appraisal you have ever done!  You just did not realize it.  Here’s why.

Under the Income Approach, we forecast some measure of ownership benefits, say cash flow to equity.  Even if we prepare just one forecast, every assumption we make (sales, gross profit margin, etc.) is really our estimate of the most probable level given our economic, industry, and company analysis.  Whether conscious of it or not, we consider a reasonable range of what future sales (etc.) will be and pick out one number.  That reasonable range is a probability distribution.

The same applies to the Market Approach.  It also applies to the Asset Approach (liquidation premise of value) when we rely on fixed asset appraisers’ estimates of the most probable value of real estate and equipment, for example.

Most of the time, the process of reducing the range to a single number happens so fast that we do not think explicitly about probabilities.  Other times, we get at probability by “playing around” with the forecast, still staying within the reasonable range for each assumption.  Occasionally, as with option-based models, we explicitly use probabilities when we introduce the notion of standard deviation (variability).  CAPM and beta also involve normal probability distributions because beta (volatility) is defined as the standard deviation of historical prices.

When we think, for example, that next year’s sales should be somewhere between $1.0 and $1.5 million, we are specifying a uniform probability distribution, one in which sales could be anywhere between and including those limits with equal probability,

When we think about “best, most likely, and worst” cases, we are moving toward a non-uniform probability distribution, with (say) sales of $1.2 to $1.4 million being much more likely than sales of $1.0 to $1.2 million or $1.4 to $1.5 million.  Now our probability distribution has a sort of hump or point in the middle.

When we use means and standard deviations, we are using the so-called “normal” probability distribution, the classic bell-shaped curve.

So congratulations, I hope have convinced you that you are using probability distributions quite frequently.  Now you ask, so what?

Each of the distributions I mentioned – uniform, humped, and normal – reflects a  MAJOR assumption about our knowledge of the future, in particular, the likelihood of specific future outcomes and how wide and humped the range (of reasonability) might be.  As an example, many of the valuation models used by Wall Street up to the Crash of 2008 were based on normal probability distributions (and some other assumptions about whether different inputs were or were not correlated).  It turned out that the probability distributions were not normal, but in fact were much flatter and wider than assumed.  The current buzzword for this is “fat tails”.  The statistical word for that is kurtosis.  A very narrow, pointed normal distribution is leptokurtotic, and a very wide, flat one is platykurtotic, two super terms with which to dazzle at cocktail parties.  Platykurtosis blew up Wall Street.

The reason academics and Wall Street used normal distributions is because the mathematics associated with them is (in their words) “tractable”; that is, you can solve the underlying equations (if you are a math jockey of high order).  This was very important in the days before computing power was cheap and readily available. Other probability distributions, even though they might fit the data better, are less tractable. They can be handled by a technique called Monte Carlo simulation. This lets you input probability distributions of any type.  Then you run the model hundreds of times and combine the results into a new probability distribution.  This lets you conclude things like “there is a 10% chance that sales will be above $1.4 million”.

Now I still hear you asking “So what?”  My point is, think long and hard about whether the probability distribution you are explicitly or implicitly using makes sense.  To just slap a normal distribution on next year’s sales assumes much more confidence than saying “sales could be anywhere from $1.0 to $1.5 million with equal probability.”

I will close with two real-life examples of the abuse of probability distributions that I encountered.

The first was when I worked for Standard Oil of Ohio back in the 1980’s.  The oil exploration people wanted to spend (gulp) $600 million to drill in Alaska adjacent to the existing Prudhoe Bay field.  They provided a probability distribution of potential oil reserves and production based on geological data.  This was standard industry practice, state of the art, and sound.  They made it very clear that there were big risks involved, mainly of there being dry holes.  They got and spent the money.  The holes were indeed dry.  The head of exploration and many others were fired because top management felt they had been misled about the risks.    

The second example happened several years ago. I arbitrated a dispute involving a large public company that lent money to several small businesses to foster economic growth.  Some of the businesses went bankrupt, and acrimonious settlement negotiations ensued.  The large company furnished their financial analysis, in which totally uninformed analysts had assigned normal probability distributions to the forecasts of results for the small companies.  I showed that these distributions made no sense based on other data presented, and this helped the parties settle the case.

The name of the large public company? Enron.  Draw your own conclusions from that.