This post compares and contrasts different ways of applying the Direct Market Data Method (DMDM) in our appraisals. The DMDM is part of the Market Approach, in which we develop value indications from sales of guideline companies drawn from sources like the IBA Market Data File, BizComps, and / or Pratt’s Stats.
I write about this because I just finished a series of valuations using the DMDM in which I benefited from unusually large samples (over 50 guidelines) that allowed me to apply statistical analysis techniques (normalization and regression). See my post dated November 2, 2009 (below) which discusses sample size requirements.
Statistical analysis is increasingly prevalent in business valuations for many reasons:
- In tax valuations, the burden of proof is on the taxpayer (usually our client) to justify conclusions. Statistics help us do that.
- Daubert rules hold us to a high scientific standard of proof. Statistics help.
- Distinguished jurists like David Laro are demanding more thorough analyses.
- The Internet has made more quantitative data available at reasonable cost.
- Personal computers have made statistical analysis easier.
- Professional standards prohibit us from “proclaiming without proof” our valuation assumptions.
- Many people entering our profession are trained to use statistical methods.
- Much relevant academic research also relies on statistics.
Before I go further, I am not advocating that we become statistical experts. I have a working knowledge of the subject, having studied it in college and used its methods throughout my career, but I am by no means an expert. Much of statistics relies on deep, important mathematical premises (some of which I will point out later) and on advanced calculus and other tools which I simply accept on faith. I believe, however, that with a rudimentary knowledge of statistics, we can choose to use some of these methods to improve our work product and meet the increasing demands cited above. How intensively we use them depends on our knowledge, the quality of the data, its quantity, and how well we can explain and defend ourselves.
Let’s begin the discussion with some assumptions:
- We are using the DMDM as a valuation method. It will have some weight in our final value conclusion along with indications developed by other approaches and methods. We are not just using it to provide a reasonability test.
- The standard of value is anything but investment value (the value to a specific buyer or seller based on their unique attributes or synergies). Investment value rules out comparisons with “any willing buyer or seller” and thus prevents us from relying on guidelines that do not have the same unique features. The standard of value could be fair market value, fair value (for financial reporting) or fair value (for dissenting shareholder matters).
- Sample sizes are adequate or better. We have enough guideline transactions (after eliminating outliers, dated ones, etc.) and also comparative industry financial information (e.g. RMA data), itself based on an adequate sample size.
- We have to develop a point estimate of value; a range is not acceptable.
- There are no extraordinarily complicating facts and circumstances (such as an environmental liability, a distressed business, or a company undergoing major changes) associated with the valuation that would call into question the relevance of the guideline transactions or industry financial data.
- All I am discussing here is how to develop the appropriate price to sales ratio for our subject company. (The same considerations apply if we use discretionary earnings.) I will not get into “packaging adjustments” that convert the resulting estimate of operating asset value to equity value.
With all that said we can now get into the comparison / contrasting process. I can think of four ways to go about developing the price / sales ratio (PSR) for our subject based on the guideline and industry data:
- Make a subjective judgment.
- Tie the PSR to the raw guideline data and industry financial data.
- Normalize the guideline PSRs, and then tie them to industry data.
- Perform a regression analysis (predicting selling price as a function of sales; the regression results will tell us the PSR).
Method 1 obviously fails to satisfy any of the prerequisites mentioned at the beginning of this post. Brokers can do this when they price businesses for sale, but we appraisers cannot when we have to satisfy requirements involving support for our conclusions.
Method 2 is what most of us do most of the time. We take the raw data (computed PSRs) and rank them (in descending order). Based on the rank order, we compute percentiles. For example, at the median, half the PSRs are above and half are below it. At the top quartile, 25% are above and 75% are below it. Now, based on how our subject compares to the industry financial data, we decide what percentile is appropriate and calculate the PSR from that. I look at every single (RMA) ratio over the last five years for the subject and the industry, average each over the period, and compute the ratio of company to industry for each one. I then compute the average of all the company to industry ratios. If the average is 1, then I apply the median PSR. If the average is 1.5, I apply 1.5 times the median, and so forth. I have found this to be a nice, simple, easily explained way to develop the PSR.
Now the water is going to start to get deeper, but gradually! Method 2 implicitly assumes that the PSRs are UNIFORMLY distributed. In other words, the probability that any given PSR is any given value (in the range) is THE SAME. If you made a graph of the PSR on the horizontal (X) axis and the probability of occurrence on the vertical (Y) axis, it would be a straight line (segment) parallel to the X axis between the lowest and highest PSR.
Is this assumption correct? WE DON’T KNOW! The only way we could is if we had data on ALL transactions (in the industry), and we obviously do not. We rely on our SAMPLE to be representative of the whole. In Method 2 we assume the whole distribution (and the sample) are both uniformly distributed. The more important question is whether this assumption is reasonable. THAT we can assess by looking at the X-Y graph and seeing if the data points are spaced relatively uniformly or not.
If this is as far as you go with the DMDM, I think that is fine. What follows gets harder and into deeper water, and to my mind is optional for most of us.
Many of us go a little further and use what I will call Method 3. We calculate summary descriptive statistics (the mean and standard deviation, and maybe the coefficient of variation). We now discard the Method 2 assumption of uniform distribution and replace it with the Method 3 assumption of normal distribution – the bell-shaped curve. If the bell curve is a perfect fit for the data (a big if), then 95% of the observed PSRs fall within plus or minus three standard deviations (three sigma, in current parlance) of the mean. The XY graph of the distribution will resemble a bell shaped curve.
If you use Method 3, you need to investigate how closely the data fit the bell curve. Here I refer you to a good book on statistics, but I warn you that the water gets pretty deep, deeper than I personally like to swim in or put in a valuation report. I do my investigating via a graph. The key point is that the percentiles using a bell curve will be different that those using the uniform distribution. To use the bell curve, you have to “normalize” the PSRs. This is not the same meaning we apply to normalizing earnings. Here, we mean taking each PSR, subtracting the mean of all of them from it, and dividing the difference by the standard deviation. The resulting PSRs are normalized in that they have a mean of 0 and a standard deviation of 1. Now, when we do our percentile analysis, we use the normalized PSRs and a statistical table to look up the right values.
And now the deepest water of all, at least for me, is that when you do this, the statistical table to use is not the normal distribution but something called “Student’s t” or the “t distribution”. The reason for this is that we are using the SAMPLE mean and standard deviation as proxies for the unknown “real” or population mean and standard deviation. This introduces error. The “t distribution” looks like the normal distribution but it is a little flatter and wider because of the error. (I am DROWNING here, sinking well under the X axis.)
OK, safely back on land, you can see the problems associated with using the bell curve. Remember, we do not know how the PSRs are distributed; a guess of uniformity is just as reasonable as a guess of normality, and a picture is worth a thousand words.
(A sadistic professor friend of mine reviewed this post and added that I have not addressed whether the RMA industry data is normally distributed or not. All the same issues apply to that as well as to the PSRs. I am never talking to her again.)
All of this brings me to Method 4, regression. Here, rather than working with those infernal percentiles and wrestling with distribution issues, what we do is graph the data (sales on the horizontal axis, price on the vertical axis) and find the line which best fits them. The best fit is determined by minimizing the squared distances between the line and the data points. (Squaring is necessary because points below the line have negative distances.) If you question the math of regression, see a good statistics book, I guarantee you enough deep water to find whales! In regression we come up with a simple relationship: Price = A + (B X Sales) where A and B are constants. Microsoft Excel has a regression package. You can fool with it to do things like force the regression constant (A) to be zero (so you get an equation that says Price = B X Sales or do fancy transformations of the price and sales data to model non-linear relationships. (Major water depth warning.) You must know your stuff to use regressions; there are all sorts of things to check and question. Pictures are worth a thousand words, again, and regression technicalities are hard to explain and to defend against a real expert.