N( 0.594)) N(0.594) 0.724, which means there is a 72.4% probability

of being less than or equal to 0.594 standard deviations above the mean.

For perspective, it is useful to note that since the normal distribution is

symmetric, N(0) 0.5000, i.e., there is a 50% probability of being less

than or equal to the mean, which implies there is a 50% probability of

being above the mean.

In B15, we calculate the value of the put option, which is $3.73 (B15),

or 42.0% (B16) of the stock price of $8.875 (B5). Thus, our calculation of

the restricted stock discount for the Chantal block using the Black“Scholes

model is 42.0% (B16).

Table 7-8: Black“Scholes Put Model Results. The stock symbols

in Table 7-8, column A, relate to restricted stock sale numbers 8, 11, 15,

17, 23, 31, 32, 38, and 49“53 in Table 7-5, column A. Cells B6 through B18

show the discounts calculated using the Black“Scholes put model for the

47. One standardizes a normal distribution by subtracting the mean from each value and dividing

by the standard deviation.

PART 3 Adjusting for Control and Marketability

244

T A B L E 7-8

Put Model Results

A B C D E F

4

Black-Scholes

Error2

5 Company Put Calculation Actual Error Absolute Error

6 BLYH 32.3% 31.4% 0.9% 0.0% 0.9%

7 CHTL 42.0% 44.8% 2.8% 0.1% 2.8%

8 DAVX 47.5% 46.3% 1.2% 0.0% 1.2%

9 EDMK 11.9% 16.0% 4.1% 0.2% 4.1%

10 ILT 38.3% 41.1% 2.8% 0.1% 2.8%

11 PLFE 23.7% 15.9% 7.8% 0.6% 7.8%

12 PRDE 13.3% 24.5% 11.2% 1.2% 11.2%

13 RENT 41.5% 32.5% 9.0% 0.8% 9.0%

14 FOFF 27.2% 12.5% 14.7% 2.2% 14.7%

15 ARCCA 36.1% 18.8% 17.3% 3.0% 17.3%

16 DPAC 18.3% 23.1% 4.8% 0.2% 4.8%

17 NEDI 24.6% 19.3% 5.3% 0.3% 5.3%

18 UMED 12.9% 15.8% 2.9% 0.1% 2.9%

19 Mean 28.4% 26.3% 2.1% 0.67% 6.5%

22 Comparison with the Mean as the Discount

Error2

24 Company Mean Discount Actual Error Absolute Error

25 BLYH 27.1% 31.4% 4.3% 0.2% 4.3%

26 CHTL 27.1% 44.8% 17.7% 3.1% 17.7%

27 DAVX 27.1% 46.3% 19.2% 3.7% 19.2%

28 EDMK 27.1% 16.0% 11.1% 1.2% 11.1%

29 ILT 27.1% 41.1% 14.0% 2.0% 14.0%

30 PLFE 27.1% 15.9% 11.2% 1.3% 11.2%

31 PRDE 27.1% 24.5% 2.6% 0.1% 2.6%

32 RENT 27.1% 32.5% 5.4% 0.3% 5.4%

33 FOFF 27.1% 12.5% 14.6% 2.1% 14.6%

34 ARCCA 27.1% 18.8% 8.3% 0.7% 8.3%

35 DPAC 27.1% 23.1% 4.0% 0.2% 4.0%

36 NEDI 27.1% 19.3% 7.8% 0.6% 7.8%

37 UMED 27.1% 15.8% 11.3% 1.3% 11.3%

38 Mean 27.1% 26.3% 0.8% 1.28% 10.1%

13 stocks. The actual discounts are in column C, and the error in the put

model estimate is in column D.48 Columns E and F are the squared error

and the absolute error. Row 19 is the mean of each column. The bottom

half of the table is identical to the top half, except that we use the mean

discount of 27.1% as the estimated discount instead of the Black“Scholes

put model.

A comparison of the top and bottom of Table 7-8 reveals that the put

option model performs much better than the mean discount of 27.1% for

the 13 stocks. The put model™s mean absolute error of 6.5% (F19) and

mean squared error of 0.67% (E19) are much smaller than the mean ab-

solute error of 10.1% (F38) and mean squared error of 1.28% (E38) using

48. The error is equal to the estimated discount minus the actual discount, or column B minus

column C.

CHAPTER 7 Adjusting for Levels of Control and Marketability 245

the MPI data mean discount as the forecast. The mean errors in cells D19

and D38 are not indicative of relative predictive power, since low values

could be obtained even though the individual errors are high due to neg-

ative and positive errors canceling out.

Comparison of the Put Model and the Regression Model

In order to compare the put model discount results with the regression

model, we will analyze Table 7-9, which shows the calculation of dis-

counts, using regression #1 in Table 7-5, on the 13 stocks for which price

data was available.

The intercept of the regression is in cell B6, and the coef¬cients for

the independent variables are in cells B7 through B13. The variables for

each stock are in columns C through O, Rows 7 through 13. Multiplying

the variables for each stock by their respective coef¬cients and then add-

ing them together with the y-intercept results in the regression estimated

discounts in C14 through O14.

The errors in row 16 equal the actual discounts in row 15 minus the

estimated discounts in Row 14. We then calculate the error squared and

absolute error in Rows 17 and 18.

The mean squared error of 0.57% (C20) and the mean absolute error

of 6.33% (C21) are comparable but slightly better than the put model

results of 0.67% and 6.5% in Table 7-8, E19 and F19, respectively. Having

only been able to test the put model on 13 stocks and not the entire

database of 53 reduces our ability to distinguish which model is better.

At this point it is probably best to use an average of the results of both

models when determining a discount in a restricted stock valuation.

Empirical versus Theoretical Black“Scholes. It is important to un-

derstand that in using the BSOPM put for calculating restricted stock

discounts, we are using it as an empirical model, not as a theoretical

model. That is because buying a put on a publicly traded stock does not

˜˜buy marketability™™ for the restricted stock.49 Rather, it locks in a mini-

mum price for the restricted shares once they become marketable, while

allowing for theoretically unlimited price appreciation. Therefore, issuing

a hypothetical put on the freely tradable stock does not accomplish the

same task as providing marketability for the restricted stock, but it does

compensate for the downside risk on the restricted stock during its hold-

ing period.

BSOPM has some attributes that make it a successful predictor of

restricted stock discounts, i.e., it is a better forecaster than the mean dis-

count and did almost as well as the regression of the MPI data.

The reason for BSOPM™s success is that its mathematics is compatible

with the underlying variable”primarily volatility”that would tend to

drive restricted stock discounts. It is logical that the more volatile the

restricted stock, the larger the discount, and that volatility is the single

most important determinant of BSOPM results. Therefore, BSOPM is a

good candidate for empirically explaining restricted stock discounts, even

49. I thank R. K. Hiatt for this observation

PART 3 Adjusting for Control and Marketability

246

T A B L E 7-9

Calculation of Restricted Stock Discounts for 13 Stocks Using Regression from Table 7-5

A B C D E F G H I J K L M N O

5 Coef¬cients BLYH CHTL DAVX EDMK ITL PLFE PRDE RENT FOFF ARCCA DPAC NEDI UMED

6 Intercept 0.0673

7 Rev2 4.629E 18 8.62E 13 5.21E 13 1.14E 15 3.56E 13 1.02E 13 4.37E 16 4.34E 15 1.15E 15 6.10E 15 3.76E 14 3.24E 14 1.95E 15 5.49E 13

8 Shares 3.619E 09 4,452,000 $4,900,000 $999,000 $2,000,000 $975,000 $38,063,000 $21,500,000 $20,650,000 $5,670,000 $2,275,000 $4,500,000 $12,000,000 $8,400,000

sold-$

9 Mkt cap 4.789E 10 98,053,000 149,286,000 18,942,000 12,275,000 10,046,000 246,787,000 74,028,000 61,482,000 43,024,000 18,846,000 108,862,000 60,913,000 44,681,000

10 Earn stab 0.1038 0.04 0.70 0.01 0.57 0.71 0.00 0.31 0.60 0.80 0.03 0.08 0.34 0.09

11 Rev stabil 0.1824 0.64 0.23 0.65 0.92 0.92 0.00 0.26 0.70 0.87 0.74 0.70 0.76 0.74

12 Avg yrs to 0.1722 2.125 2.125 2.750 2.868 2.844 2.861 2.833 2.950 2.375 1.633 1.167 1.738 1.898

sell

13 Price 0.0037 58.6 51.0 24.6 10.5 22.0 17.0 18.0 30.0 23.7 35.0 42.4 32.1 21.0

stability

14 Calculated discount 42.22% 42.37% 37.67% 23.65% 26.25% 26.57% 34.43% 30.97% 15.83% 20.27% 18.68% 15.20% 18.27%

15 Actual discount 31.40% 44.80% 46.30% 16.00% 41.10% 15.90% 24.50% 32.50% 12.50% 18.80% 23.10% 19.30% 15.80%

16 Error (actual calculated) 10.82% 2.43% 8.63% 7.65% 14.85% 10.67% 9.93% 1.53% 3.33% 1.47% 4.42% 4.10% 2.47%

17 Error squared 1.17% 0.06% 0.75% 0.59% 2.21% 1.14% 0.99% 0.02% 0.11% 0.02% 0.20% 0.17% 0.06%

18 Absolute error 10.82% 2.43% 8.63% 7.65% 14.85% 10.67% 9.93% 1.53% 3.33% 1.47% 4.42% 4.10% 2.47%

19 Mean error 0.80%

20 Mean squared error 0.57%

21 Mean absolute error 6.33%

247

though that is not the original intended use of the model, nor is this

scenario part of the assumptions of the model.

Comparison to the Quantitative Marketability Discount

Model (QMDM)

Mercer shows various examples of investment risk premium calculations

Mercer 1997, chapter 10). When he adds this premium to the required

return on a marketable minority basis, he gets the required holding period

return for a nonmarketable minority interest. Judging from his example

calculations of the risk premium for other types of illiquid interests, the

investment speci¬c risk premium for restricted stocks should be some-

where in the range of 1.5“5% or less.50 This is because restricted stocks

have short and well-de¬ned holding periods. Also, the payoff at the end

of the holding period is almost sure to be at the marketable minority level.

To test the applicability of QMDM to restricted stocks, we ¬rst esti-

mate a typical marketable minority level required return. The MPI data-

base average market capitalization is approximately $78 million. This puts

the MPI stocks in the mid-cap to small-cap category, given the dates of

the transactions in the database. A reasonable expected rate of return for

stocks of this size is 15% or so on a marketable minority basis.

We will assume that the stocks, given their size, were probably not

paying any signi¬cant dividends. Therefore, the expected growth rate

equals the expected rate of return at the marketable minority level of 15%.

Given the average years to liquidity of approximately 2.5 years in the

data set, we can calculate a typical restricted stock discount using QMDM.

Assuming a 1.5% investment risk premium, and therefore a required

holding period return of 16.5%, QMDM would predict the following re-

stricted stock discount:

1

1.152.5

Min Discount 1 (FV PVF) 1 3.2%

1.1652.5

where FV future value of the investment and PVF the present value

factor. With a 5% investment risk premium, we have:

1

1.152.5

Max Discount 1 (FV PVF) 1 10.1%

1.202.5

The QMDM forecast of restricted stock discounts thus range from 3“10%,

with the lower end of the range appearing most appropriate, considering

the examples in Mercer™s Chapter 10.51 These calculated discounts are

50. Actually, the lower end of the range”1.5%”appears most appropriate.

51. The QMDM restricted stock discount is insensitive to the absolute level of the discount rate. It

is only sensitive to the premium above the discount rate. For example, changing the

minimum discount formula to

1

1.202.5)

(1

1.2152.5

has little impact on the QMDM result. It is the 1.5% premium that is the difference between

the 20% growth and the 21.5% required return that constitutes the bulk of the QMDM

discount”and, of course, the holding period.

PART 3 Adjusting for Control and Marketability

248

nowhere near the average discount of 27.1% in the MPI database. This

sheds doubt on the applicability of QMDM for restricted stocks and the

applicability of the model in general. At least it shows that the model

does not work well for small holding periods.

I invited Chris Mercer to write a rebuttal to my analysis of the

QMDM results. His rebuttal is at the end of this chapter, just before the

conclusion, after which I provide my comments, as I disagree with some

of his methodology.

Abrams™ Economic Components Model

The remainder of this chapter will be spent on Abrams™ economic com-

ponents model (ECM). The origins of this model appear in Abrams

(1994a) (the ˜˜original article™™). While the basic structure of the model is

the same, this chapter contains major revisions of that article. One of the

revisions is that for greater clarity and ease of exposition, components #2

and #3 have switched places. In the original article, transactions costs was

component #2 and monopsony power to the buyer due to thin markets

was component #3, but in this chapter they are reversed.

We will be assuming that we are applying DLOM to a valuation

determined either directly or indirectly by comparison to publicly traded

¬rms. This could be a guideline company method or a discounted cash

¬‚ow method, with discount rates determined by data on publicly traded

¬rms. The ECM is not meant to be used as described on data coming

from sales of privately held businesses.

Component #1: The Delay to Sale