Business valuation analysts have been frustrated for years trying to arrive at a reasonable, plausible, defensible and supportable discount for lack of marketability (DLOM). After spending hours studying and adjusting financials, analyzing market data, calculating ratios, writing text, and calculating valuation methods, it is highly unsatisfying for the analyst to simply deduct some more or less arbitrary percentage for lack of marketability.

There are several commonly used methods for estimating an appropriate discount, none of which are wholly satisfactory: Judges often don't like them, the Internal Revenue Service doesn't like them, and clients don't like them. Appraisers, of course, have widely varying opinions as to which is best.

The new method proposed here has the advantage of being based upon quantitative, empirical market data. It is simple to use and, in our testing so far, gives sensible results. It remains to be seen how the approach will stand up in court, but I believe its chances are good.

The Data

The underlying market data for this new approach is taken from the FMV Opinions Restricted Stock Database (the FMV database). We augmented that data by matching it with the Chicago Board Options Exchange volatility index (VIX) value on each transaction date, based on a suggestion by Lance Hall, president of FMV Opinions, Inc., in New York. The VIX is a measure of uncertainty in the public market,1 and has turned out to be quite significant in our analysis as a proxy for general investor sentiment.

In its raw form, the FMV database consists of 516 records describing sales of restricted stock in companies whose common stock is otherwise freely traded in the public markets, from 1980 through early 2005. The difference between the public market price and the restricted stock price is taken to indicate a discount demanded by the private investors, to compensate them for the lack of marketability of their stock.

Unfortunately, the VIX data we could locate only goes back to 1990, which forced us to eliminate restricted stock records before the first VIX date. We were left with 447 records from March 1, 1990, through March 2, 2005.

The Approach

In summary, a simple linear regression was performed on certain data taken from the FMV database. The variables2 were selected based on prior work done by FMV Opinions, and by me with several colleagues working on a number of engagements during the past six years.

In previous attempts to estimate the DLOM using the FMV database, we used the transaction discount percentage as the dependent variable to directly predict the discount. But regression results were very weak. Similarly, an earlier study of limited partnership (LP) data was unsuccessful in directly estimating the minority/marketability discount in sales of LP interests.3

There was some good news: We had much better success by estimating the discounted market price as the dependent variable, then calculating the discount percentage as a derivative metric. This approach seems to work remarkably well.

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Methodology

We ran several regressions using various subsets of the data for two reasons: first, because patterns in the data suggested we should, and second, to test how much difference we would see in the resulting discounts. In all, we ran eight analyses on various subsets (for example, sets with market value less than $3 billion, market value less than $340 million, market value less than $90 million and so on). We ran these same sets with and without calculating a Y-intercept. We looked at the main regression statistics (R2, p-value, t statistic), residual plots and line fit plots, and calculated several other measures. We calculated results for the regression formulas for 20 different test cases, with and without the Y-intercept, to see if the predicted discounts made sense, and to see how much difference there was in the discounts predicted by each of the data subsets.

For each test case, the predicted discounts were very similar for all of the data subsets with market data below $90 million. For databases containing higher market values, however, the predicted discounts were sometimes considerably different. Regressions using a zero intercept generally gave better results.

Dependent/Independent Variables

The dependent variable in this analysis is the discounted market value, which was calculated by applying the adjusted discount percentage to the adjusted market value on the transaction date. This discounted market value variable is analogous to the discounted value an appraiser would derive after applying a DLOM, while the adjusted market value corresponds to the undiscounted value before applying the DLOM.

We identified six key independent variables in the regression:

  1. the adjusted market value at the date of the transaction;
  2. the size of the restricted block purchased, as a percentage of the outstanding stock after the sale;
  3. the adjusted book value of the company issuing the restricted stock as of the previous fiscal year-end;
  4. the annual revenue of the company as of the previous fiscal year-end;
  5. earnings before tax (EBT); and
  6. the VIX index on the transaction date.

Adjusted Market Value

The FMV database refers to “market value,” which is the public stock market value at the end of the month prior to the transaction. But the discount percentage is based on the market price per share on the transaction date. As one might expect, the market price at the end of the prior month is often very different from the market price at the transaction date. So, to keep consistency between the discount percentage and the market price, we calculated an adjusted market value variable, using the market price/share on the transaction date, multiplied by the beginning shares outstanding (before the transaction), plus the offer price. (Offer price is the dollar amount of the private placement.)

The result produces an approximation of the “post-money” price, which is what we believe an investor would have in mind. This result is consistent with the block size variable, that is to say, the “post-money” percentage bought by the investor. This adjusted market value also corresponds more closely to the undiscounted market value that would be the starting point in an appraisal assignment.

Because of how we calculated the adjusted market value, we can include the amount of the offer price without introducing the discounted pricing into the independent variables. The discount percentage then has to be recalculated, to reflect the “post-money” adjusted market value, giving us an adjusted discount percentage. This was calculated as [ 1 - (Offer Price per offering share) / (Adjusted Market Value / shares outstanding after the offering) ].

Book Value, Revenue, EBT

The book value, revenue and EBT variables are from the most recent year-end prior to the transaction date; therefore, they are not necessarily current. In fact, these variables could in some instances be almost a year old. It's therefore probable the investors had access to more recent data.

In an attempt to partially adjust for this discrepancy, we added the offer amount to the book value, before running our regressions. This gives us an approximation to the real “post-money” book value.

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VIX/Block Size

In a prior version of the analysis, we included the VIX at each transaction date as an independent variable. That regression calculation assigned a coefficient of -183 to the VIX variable, which meant that a fixed dollar amount based solely on the VIX variable was deducted in the regression formula, independently of any other values. So, for example, if the VIX was 40, an amount of -$7,320 (40 multiplied by -183) was deducted regardless of whether the company under consideration had a market value of $1 million or $1 billion.

This deduction drove the resulting discounted value negative for companies at the smaller end of the scale — a clearly unrealistic result. Companies at the high end of the market value scale realized almost no effect from the VIX. It seemed apparent, however, that the VIX should affect the discounted market value somewhat in proportion to the market value of a company.

Accordingly, in the analysis presented here, we represented the effect of the VIX using the product of the VIX and the market value as an independent variable.

Similarly, block size was subject to the same effect as the VIX. That is to say, the regression calculations assigned a fixed dollar amount in the earlier attempt to each unit of block size, which had exactly the same dollar effect on the discounted market value of every transaction with the same block size, regardless of the size of the transaction market value. As with the VIX, this was a clearly unrealistic result. In our new analysis, we therefore used the product of the block size and the market value as the independent variable. This made the effect of the block size proportionate to the size of the company.

Editing Data

There were some records in the database that were incomplete for the variables being used, and others that were extreme outliers. So, we eliminated 11 records with incomplete data and those with a market-to-book ratio greater than 100. We also removed six records with market values greater than $5 billion. We were left with a data set of 430 records. The first regression analysis described here was performed on this dataset.

Holding Period

The FMV database not only reports the discount, but also reports the expected holding period restriction based on the requirements of SEC Rule 144. For most of the records, the holding period is two years. For the remainder, after June 1997, the holding period is one year. It's likely that the length of the holding period would have an effect on the magnitude of the discount required by an investor. That is to say, a longer period should result in a higher discount.

We performed a simple calculation to normalize the discounts to a one-year holding period. We assumed the discount for a two-year hold would be the compounded discount for a one-year hold. So our calculation was as follows: If the one-year discount was d1, the two-year discount d2 would be 1 - (1-d1)2. For example, if d1=25 percent, the two-year discount would be 1 - (1.00 - 0.25) × (1.00 - 0.25) = 1 - (0.75 × 0.75) = 43.75 percent. Conversely, the one-year equivalent discount can be calculated from the two-year discount as d1 = 1 - (1 - d2)½.

We tested the regression using the normalized one-year discounted market price in one analysis, and the discounted market price using the reported one- or two-year discount in a second analysis. The difference was not dramatic, but the normalized one-year discounted market price improved the R2 noticeably, and so the normalized one-year discounted market price was used throughout all of the subsequent analyses. (R2 is a measure of how well the regression statistics “explain” the data, or alternatively, a measure of how well the regression fits the data.) Not surprisingly, data adjusted to a one-year hold produced somewhat lower discounts than the mixed one- and two-year holds.

The fact that the difference was not great suggests that many investors were already expecting a longer hold than the nominal holding period. There would be time required to get the stock registered for public sale, and Rule 144 has dribble-out provisions, both of which would have the effect of stretching out the holding period in somewhat complex ways.

However, there seemed to be some benefit from a consistent treatment, with a nominal one-year hold. A more precise analysis of the effect of a one-year hold versus a two-year hold would be interesting. But that's for another article!

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Intercept

A regression analysis calculates the best-fit straight-line through the data points. The calculation can be set to determine the point at which that straight line would intersect the dependent variable axis (that is, the discounted market value). Alternatively, the calculation can be set to run the straight line through the origin, so that the regression line intersects the dependent variable axis at zero. There were only minor differences in the regression statistics with and without the calculated intercept. In most of our analyses, the intercept was statistically insignificant (p greater than 0.20 in most cases). However, the Y-intercept constant has a major effect for values near the low end of the market value range, overriding the effect of the other variables and forcing the discounted value negative. This problem is eliminated using a zero intercept.

The zero intercept option makes real-world sense, given that a zero adjusted market value would surely indicate a zero discounted market value. So, all of the regressions in our analysis were run using the zero intercept option. Note that regression theory states that the R2 is meaningless when the intercept is forced through zero. Some statistical analysis programs (such as, Minitab) don't typically report R2 for regressions run through the origin.4

First, Second Regression

Market values reported in the edited dataset ranged from $3.8 million to $3.2 billion. Revenues ranged from $0 to $1.3 billion. Block sizes ranged from 0.1 percent to almost 44 percent. Discounts ranged from just under 89 percent to a premium of almost 71 percent, for a range of 160 points.

The first regression on the full database resulted in an adjusted R2 of 0.896 when run with a Y-intercept. This is quite high, reflecting a high correlation between the adjusted market value and the discounted market value, enhanced by the very wide range of the data and some extreme influential outliers. Based on the p-value, by far the most statistically significant independent variable was the adjusted market value, followed distantly by the VIX, the block size and book value. Note that this ranking varied with the subset of the data used in the regression. Our analysis of the residuals from the first regression suggests that the data for market values above about $340 million were skewing the results. A second breakpoint in the data occurred at about $90 million. Because most of the companies we value are much smaller, we extracted a subset with adjusted market values less than $90 million, leaving 199 records in the dataset.

Market values reported in this subset ranged from $3.8 million to $87 million. Revenues ranged from $0 to $973 million. Block sizes ranged from 0.1 percent to almost 44 percent. Discounts ranged from just over 78 percent to a premium of almost 23 percent, for a range of 101 points.

The regression on this dataset resulted in an adjusted R2 of 0.896 when run with a Y-intercept. The Y-intercept p-value was 0.39, suggesting that using a zero intercept was justified. Once again, adjusted market value was the dominant independent variable, accounting for most of the statistical correlation. The second most significant variable was the VIX, followed closely by book value, possibly because it's a rough measure of a company's survivability. These variables were followed distantly by revenue, block size and EBT.

We plotted our data with the adjusted market value along the X axis, and the discounted market value on the vertical Y axis. (See “Adjusted vs. Discounted Market Value, p. 30.) We noted a nearly straight line along which most of the data lies — this reflects the high correlation between the two variables and accounts for the high R2 in this regression. The dispersion of data points above and below the line is what we hope to explain with the secondary variables.

We also analyzed the regression coefficients, the t statistics and the p-values for this regression. (See “Which Independent Variables Impact a Discount?” p. 31.) A small p-value indicates a stronger relationship, and similarly, a higher absolute value of the t statistic indicates the variable is more significant. The p-values suggest that block, revenue and EBT contribute very little to the model, but as will be discussed later on, there's a noticeable effect on the discount from changes in these variables.

Practical Effects

Even though the variables other than adjusted market value have small statistical value, they do help account for the differences between the simple average discount and the actual discount observed. And, their effect can be substantial.

Setting all the variables to their median in the subset predicted a DLOM of 21.2 percent. This is approximately the discount predicted when we use just market value as the only independent variable, eliminating the effect of the other variables. Using this result in an appraisal would be similar in concept to using the average discount from any of the discount studies frequently cited, although the DLOM from this data is considerably lower than seen in some other studies.

Setting the variables to the highest or lowest values in the database gives a measure of the effect of the other independent variables. Keeping the adjusted market value variable at the subset average, and setting the other variables to their lowest values (highest in the case of the VIX), predicted a discount of 64 percent. Setting the other variables to their highest values (lowest for the VIX) predicted a premium of 44 percent, for a total range of 108 points. Of course, in the normal course of events, it would be unlikely to have a company whose parameters matched the high or low values in the database. Using variables in more typical ranges gave discounts in the range of roughly 15 percent to 45 percent.

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Test Case

We used the medians from the subset as the client variables. Remember, block size and VIX are each multiplied by market value. (See “Test Case,” p. 32.)

We also looked closely at the VIX, which historically ranged from about 10 in the mid-1990s to 80 in late 2008. (See “How the VIX Has Behaved,” p. 34.) Throughout its history, the VIX has crossed 40 only three times, until late 2009, when it went up to 80. Since May 2009, however, the VIX has dropped back to the upper 20s. (See “A Closer Look at the VIX,” p. 34.)

So what happens when we test a VIX higher than where it is now, say at 35, in the regression formula? We find that with all the other variables unchanged, the discount increases by 20 points. (See “What Happens With a Higher VIX?” this page.)5 And what happens when the block size increases, say from 11.3 percent to 50 percent? (See “When Block Size Increases …” p. 36.) The discount decreases by 4.5 percent.

However, although the block size in our test case is 50 percent, the highest block size in the actual data subset is 44 percent. So, the test case is outside the bounds of the data subset. In actual practice, we would probably ignore a small transgression like this.

Note that the higher block size decreased the discount. In regressions on data subsets that contained higher market values, for example greater than $100 million, the sign of the block size coefficient was negative, meaning that it would have increased the discount. One might speculate that for larger transactions involving larger companies, the investors drove a harder bargain. Also note that the discounts predicted by the regression are for a one-year hold. An adjustment may be necessary for an expected hold longer than one year.

Other Considerations

The lowest adjusted market value in the second regression database is about $3.8 million. This means that application of these regression results might be questionable for valuation client companies with an undiscounted market value much below this amount. However, experimenting with the regression formula using market values near and below the low end of the range gives what appear to be sensible results for the DLOM.

Different subsets of the database give different results. We tested six subsets in addition to the one we discussed throughout this article. Interestingly, most of the lower market value subsets (less than $90 million) yielded very similar results, even when the client data fell well outside the range of the subset. The p-value and t stat variables changed with different subsets, of course, and some of the coefficients changed sign or significance rank, but the indicated discounts tended to be very similar among subsets in almost all cases. Datasets including higher market values or including just the very low end transactions gave substantially different results for some of the test cases.

A smaller subset of the data could be used. As a rule of thumb, with six independent variables, roughly 120 or more transaction records (20 multiplied by the number of variables) might be required to obtain reliable results. Eliminating some of the least significant variables like EBT, revenue and block size would allow the analyst to use a smaller dataset and still get statistically significant results.

The companies included in the FMV database were probably projecting an optimistic future, with strong sales growth and increased profits as an important part of the investors' expectations. At the end of 2008, the economy was in a deep recession and the outlook was highly uncertain; although the outlook had improved by August 2009, a great deal of uncertainty remained. A valuation client with flat or declining prospects might deserve a greater discount than may be indicated by this analysis.

Application

In our practice, we usually use a variation of the quantitative marketability discount model analysis described by Christopher Mercer of Memphis, Tenn.'s Mercer Capital, and a Mandelbaum analysis,6 with occasional use of the FMV database. We intend to supplement those analyses with the results of this analysis, where feasible. In some cases we will customize the regression analysis to better match our subject. For example, for a smaller company with an undiscounted market value of $10 million, we might run the regression using companies in the $4 million to $40 million market cap range. However, we expect that the analysis that we reported here will be adequate for most of our clients.

The author thanks Dr. Neil Schwertman and Andy Mark for their assistance with the statistical analysis; Steve Kam, Michael Bilich, Patrick O'Connell and Craig Lunsman for their assistance in the practical application of the regression model in real-world cases; FMV Opinions, Inc. for their restricted stock database; and Lance Hall for suggesting the importance of the VIX.

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Endnotes

  1. We included the volatility index (VIX) based on a presentation given in Berkeley, Calif., on May 30, 2009, to the San Francisco Valuation Roundtable by Lance Hall of FMV Opinions.
  2. We selected variables based on prior work done from 2002 to 2008 among FMV Opinions, this author, Michael Bilich, and Patrick O'Connell. During that six-year period, we attempted to identify variables that would be predictive of marketability.
  3. In 1996, this author, Steve Kam (principal, Cogent Valuations, LLC), and Curt Smith (Strategic Investment Solutions, Inc.), performed a study of limited partnership data.
  4. Although some statistical analysis programs (such as Minitab) do not report R2 for regressions that run through the origin, we have shown them because the results with and without the zero intercept were so similar.
  5. Because the VIX has its ups and downs, several of our associates have suggested that a 3-month or 6-month average should be used, rather than the VIX on a certain date.
  6. Mandelbaum v. Commissioner, T.C. Memo 1995-255, 69 T.C.M. (CCH) 2852 (June 12, 1995), affirmed 91 F.3d 124 (3d Cir. 1996).

Hans P. Schroeder is the founder, president and a valuation analyst at BEAR, Inc., which is based in Chico, Calif.

Which Independent Variables Impact a Discount?

A look at the T stat and P-value shows the adjusted market value is the primary driver

We analyzed regression coefficients, T stats and P-values to determine which of the six independent variables had the strongest relationship to the size of the discount. The adjusted market value was clearly strongest, with the VIX and the adjusted book value a distant second and third, respectively. Note, however, that although the variables other than the adjusted market value had small statistical value, they are important and help account for the difference between the simple average discount and the actual discount observed.


Coefficients Standard Error T Stat P-value
Adjusted Market Value 0.9907 0.04620 21.4442 5.7570E-53
Block (x Market Value) 0.1167 0.13141 0.8880 3.7563E-01
Adjusted Book Value 0.2573 0.05146 5.0002 1.2821E-06
Revenue 0.0044 0.00658 0.6707 5.0318E-01
EBT 0.0743 0.07481 0.9939 3.2153E-01
Volatility (VIX) x Market Value -0.0127 0.00202 -6.2944 2.0307E-09
— Hans P. Schroeder

Test Case

Using the median value for each of the six independent variables predicts a discount of 21.2 percent

We performed a regression analysis on 199 records from the FMV Opinions Restricted Stock Database, testing the median values for each of the independent variables. Setting the variables to their median produces a discount for lack of marketability of 21.2 percent. In later analyses, we manipulated the variables to determine how their changes impact on the DLOM. (See “What Happens With a Higher VIX?” p. 35, and “When Block Size Increases …” p. 36).

Independent Variables Client Independent Variable Coefficients Product
Market Value Before DLOM 39,195 39,195 0.9907 38,830
Block (x Market Value) 11.3% 4,421.16 0.1167 516
Book Value 5,068 5,068 0.2573 1,304
Revenue 5,588 5,588 0.0044 25
EBT (1,610) (1,610) 0.0743 (120
Volatility (VIX) × Market Value 19.5 762,728 -0.0127 (9,684)
Discounted Value 30,871
Market Value Before DLOM 39,195
Discount Percentage -21.2%
— Hans P. Schroeder

What Happens With a Higher VIX?

With all other variables the same in our test case, a higher volatility index produces a higher discount

Independent Variables Client Independent Variable Coefficients Product
Market Value Before DLOM 39,195 39,195 0.9907 38,830
Block (x Market Value) 11.3% 4,421.16 0.1167 516
Book Value 5,068 5,068 0.2573 1,304
Revenue 5,588 5,588 0.0044 25
EBT (1,610) (1,610) 0.0743 (120
Volatility (VIX) × Market Value 35.0 1,371,813 -0.0127 (17,418
Discounted Value 23,137
Market Value Before DLOM 39,195
Discount Percentage -41.0%
— Hans P. Schroeder

When Block Size Increases …

The discount decreases relative to our test case


Independent Variables Client Independent Variable
Coefficients Product
Market Value Before DLOM 39,195 39,195
0.9907 38,830
Block (× Market Value) 50.0% 19,597.32 0.1167 2,287
Book Value 5,068 5,068
0.2573 1,304
Revenue 5,588 5,588
0.0044 25
EBT (1,610) (1,610)
0.0743 (120
Volatility (VIX) × Market Value 19.5 762,728 -0.0127 (9,684
Discounted Value 32,642
Market Value Before DLOM 39,195
Discount Percentage -16.7%
— Hans P. Schroeder