˜˜red™™ or ˜˜black™™ before the draw and, if he or she calls it right, wins $100;

otherwise, he or she gets nothing. In gamble B, the subject draws one ball

from an urn that has 50 red balls and 50 black balls. Again, if the subject

forecasts the correct draw, he or she wins $100 and otherwise wins noth-

ing.

Most people are indifferent between choosing red or black in both

gambles. When asked which gamble they prefer, the majority of people

had an interesting response (before we proceed, ask yourself which gam-

ble you would prefer and why). Most people prefer to draw from urn #2.

This is contrary to risk-neutral logic. The ¬nding of Ellsberg and Einhorn

and Hogarth is that people dislike ambiguity and will pay to avoid it.

Ambiguity is a second-order uncertainty. It is ˜˜uncertainty about un-

certainties,™™ and it exists pervasively in our lives. Gamble B has uncer-

tainty, but it does not have ambiguity. The return-generating process is

well understood. It is a clear 50“50 gamble. Gamble A, on the other hand,

is fuzzier. The return-generating process is not well understood. People

feel uncomfortable with that and will pay to avoid it.

It is my opinion that ambiguity aversion probably explains much of

shareholder level discounts. As mentioned earlier in the chapter, Jan-

kowske mentions wealth transfer opportunities and the protection of in-

vestment as economic bene¬ts of control. Many minority investors are

exposed to the harsh reality of having their wealth transferred away.

Many of those who do not experience that still have to worry about it

occurring in the future. The minority investor is always in a more am-

biguous position than a control shareholder.

In our regressions of the partnership pro¬les database that tracks the

results of trading in the secondary limited partnership markets (see Chap-

ter 9), we ¬nd that regular cash distributions are the primary determinant

of discounts from net asset value. Why would this be so? After all, there

have already been appraisals of the underlying properties, and those ap-

praisals certainly included a discounted cash ¬‚ow approach to valua-

tion.52 If the appraisal of the properties already considered cash ¬‚ow, then

52. In the regression we included a dummy variable to determine whether the discount from net

asset value depended on whether the properties were appraised by the general partner or

by independent appraiser. The dummy variable was statistically insigni¬cant, meaning that

the market trusts the appraisals of the general partners as much as the independent

appraisers.

PART 3 Adjusting for Control and Marketability

250

why would we consider cash ¬‚ow again in determining discounts? I

would speculate the following reasons:

1. If the general partner (GP) takes greater than arm™s-length fees

for managing the property, that would not be included in the

appraisal of the whole properties and would reduce the value of

the limited partner (LP) interest. It is a transfer of wealth from

the LP to the GP.

2. Even if the GP takes an arm™s-length management fee, he or she

still determines the magnitude and the timing of the

distributions, which may or may not be convenient for the

individual LPs.

3. LPs may fear potential actions of the GP, even if he or she never

takes those actions. The LP only knows that information about

the investment that the GP discloses and may fear what the GP

does not divulge”which, of course, he or she won™t know. The

LPs may hear rumors of good or bad news and not know what

to do with it or about it.

The bottom line is that investors don™t like ignorance, and they will

pay less for investments that are ambiguous than for ones that are not”

or that are, at least, less ambiguous”even if both have the same expected

value.

Our paradigm for valuation is the two-parameter normal distribu-

tion, where everything depends only on expected return and expected

risk. Appraisers are used to thinking of risk only as either systematic risk,

measured by , or total risk in the form of , the historical standard

deviation of returns. The research on ambiguity avoidance adds another

dimension to our concept of risk, which makes our task more dif¬cult

but affords the possibility of being more realistic.

It is also noteworthy that the magnitude of special distributions, i.e.,

those coming from a sale or re¬nancing or property, was statistically in-

signi¬cant. Investors care only about what they feel they can count on,

the regular distributions.

Black“Scholes Options Pricing Model. One method of modeling

the economic disadvantage of the period of illiquidity is to use the Black“

Scholes options pricing model (BSOPM) to calculate the value of a put

on the stock for the period of illiquidity. A European put, the simplest

type, is the right to sell the stock at a speci¬c price on a speci¬c day. An

American put is the right to sell the stock on or before the speci¬c day.

We will be using the European put.

The origins of using this method go back to David Chaffe (Chaffe

1993), who ¬rst proposed using the BSOPM for calculating restricted

stock discounts for SEC Rule 144 restricted stock. The restricted stock

discounts are for minority interests of publicly held ¬rms. There is no

admixture of minority interest discount in this number, as the restricted

stock studies in Pratt™s Chapter 15 (Pratt, Reilly, and Schweihs 1996) are

minority interests both pre- and posttransaction.

Then Abrams (1994a) suggested that owning a privately held busi-

ness is similar to owning restricted stock in that it is very dif¬cult to sell

CHAPTER 7 Adjusting for Levels of Control and Marketability 251

a private ¬rm in less than the normal due diligence time discussed above.

The BSOPM is a reasonable model with which to calculate Component

#1 of DLOM, the delay to sale discount.

There is disagreement in the profession about using BSOPM for this

purpose. Chapter 14 of Mercer™s book (Mercer 1997) is entitled, ˜˜Why

Not the Black“Scholes Options Pricing Model Rather Than the QMDM?™™

Mercer™s key objections to the BSOPM are:53

1. It requires the standard deviation of returns as an input to the

model. This input is not observable in privately held companies.

2. It is too abstract and complex to meaningfully represent the

thinking of the hypothetical willing investor.

Argument 2 does not matter, as the success of the model is an em-

pirical question. Argument 1, however, turned out to be more true than

I would have imagined. It is true that we cannot see or measure return

volatility in privately held ¬rms. However, there are two ways that we

indirectly measured it. We combined the regression equations from re-

gressions #1 and #2 in Table 4-1 to develop an expression for return vol-

atility as a function of log size, and we performed a regression of the

same data to directly develop an expression for the same. We tried using

both indirect estimates of volatility as inputs to the BSOPM to forecast

the restricted stock discounts in the Management Planning, Inc. data, and

both approaches performed worse than using the average discount. Thus,

argument 1 was an assertion that turned out to be correct.

When volatility can be directly calculated, the BSOPM is superior to

using the mean and the QMDM. So, BSOPM is a competent model for

forecasting when we have ¬rm-speci¬c volatility data, which we will not

have for privately-held ¬rms.

Other Models of Component #1. The regression equation developed

from the Management Planning, Inc. data is superior to both the non-

¬rm-speci¬c BSOPM and the QMDM. Thus, it is, so far, the best model

to measure component #1, the delay to sale component, as long as the

expected delay to sale is one to ¬ve (or possibly as high as six) years.

The QMDM is pure present value analysis. It has no ability to quan-

tify volatility”other than the analyst guessing at the premium to add to

the discount rate. It also suffers from being highly subjective. None of the

components of the risk premium at the shareholder level can be empiri-

cally measured in any way.

Is the QMDM useless? No. It may be the best model in some sce-

narios. As mentioned before, one of the limitations of my restricted stock

discount regression is that because the restricted stocks had so little range

in time to marketability, the regression equation performs poorly when

the time to marketability is substantially outside that range”above ¬ve

to six years. Not all models work in all situations. The QMDM has its

place in the toolbox of the valuation professional. It is important to un-

53. Actually, Chapter 14 is co-authored by J. Michael Julius and Matthew R. Crow, employees at

Mercer Capital.

PART 3 Adjusting for Control and Marketability

252

derstand its limitations in addition to its strengths, which are ¬‚exibility

and simplicity.

The BSOPM is based on present value analysis, but contains far more

heavy-duty mathematics to quantify the probable effects of volatility on

investor™s potential gains or losses. While the general BSOPM did not

perform well when volatility was measured indirectly, we can see by

looking at the regression results that Black“Scholes has the essence of the

right idea. Two of the variables in the regression analysis are earnings

stability and revenue stability. They are the R2 from regressions of earn-

ings and revenues as dependent variables against time as the independent

variable. In other words, the more stabile the growth of revenues and

earnings throughout time, the higher the earnings and revenue stability.

These are measures of volatility of earnings and revenues, which are the

volatilities underlying the volatility of returns. Price stability is another

of the independent variables, and that is the standard deviation of stock

price divided by the mean of returns (which is the coef¬cient of variation

of price) and then multiplied by 100.

Thus, the regression results demonstrate that using volatility to mea-

sure restricted stock discounts is empirically sound. The failure of the

non-¬rm-speci¬c BSOPM to quantify restricted stock discounts is a mea-

surement problem, not a theoretical problem.54

An important observation regarding the MPI data is that MPI ex-

cluded startup and developmental ¬rms from its study. There were no

¬rms that had negative net income in the latest ¬scal year. That may

possibly explain the difference in results between the average 35% dis-

counts in most of the other studies cited in Pratt™s Chapter 15 (Pratt,

Reilly, and Schweihs 1996) and MPI™s results. When using my regression

of the MPI data to calculate component #1 for a ¬rm without positive

earnings, I would make a subjective adjustment to increase the discount.

As to magnitude, we have to make an assumption. If we assume that the

other studies did contain restricted stock sales of ¬rms with negative

earnings in the latest ¬scal year, then it would seem that those ¬rms

should have a higher discount than the average of that study. With the

average of all of them being around 33“35%, let™s say for the moment

that the ¬rms with losses may have averaged 38“40% discounts, all other

things being equal (see the paragraph below for the rationale). Then 38“

40% minus 27% in the MPI study would lead to an upward adjustment

to component #1 of 11% to 13%. That all rests on an assumption that this

is the only cause of the difference in the results of the two studies. Further

research is needed on this topic.

We can see the reason that ¬rms with losses would have averaged

higher discounts than those who did not in the x-coef¬cient for earnings

stability in Table 7-10, cell B9, which is 0.1381. This regression tells us

the market does not like volatility in earnings, which implies that the

54. There is a signi¬cant difference between forecasting volatility and forecasting returns. Returns

do not exhibit statistically signi¬cant trends over time, while volatility does (see Chapter 4).

Therefore, it is not surprising that using long-term averages to forecast volatility fail in the

BSOPM. The market is obviously more concerned about recent than historical volatility in

pricing restricted stock. That is not true about returns.

CHAPTER 7 Adjusting for Levels of Control and Marketability 253

market likes stability in earnings. Logically, the market would not like

earnings to be stable and negative, so investors obviously prefer stable,

positive earnings. Thus, we can infer from the regression in Table 7-10

that, all other things being equal, the discount for ¬rms with negative

earnings in the prior year must be higher than for ¬rms with positive

earnings. Ideally, we will eventually have restricted stock data on ¬rms

that have negative earnings, and we can control for that by including

earnings as a regression variable.

It is also worth noting that the regression analysis results are based

on the database of transactions from which we developed the regression,

while the BSOPM did not have that advantage. Thus, the regression had

an inherent advantage in this data set over all other models.

Abrams™ Regression of the Management Planning, Inc. Data. As

mentioned earlier in the chapter, there are two regression equations in

our analysis of the MPI data. The ¬rst one includes price stability as an

independent variable. This is ¬ne for doing restricted stock studies. How-

ever, it does not work for calculating Component #1 in a DLOM calcu-

lation for the valuation of a privately held ¬rm, whether a business or a

family limited partnership with real estate. In both cases there is no ob-

jective market stock price with which to calculate the price stability.

Therefore, in those types of assignments, we use the less accurate second

regression equation that excludes price stability.

Table 7-10 is an example of using regression #2 to calculate compo-

nent #1, the delay to sale of DLOM, for a privately held ¬rm. Note that

˜˜Value of Block”Post Discount™™ (Table 7-10, A7) is analogous to ˜˜Shares

Sold”$™™ (Table 7-5, A50), and ˜˜FMV“100% Marketable Minority Inter-

est™™ (Table 7-10, B8) is analogous to ˜˜Market Capitalization™™ (Table 7-5,

A51). The regression coef¬cients are in B5“B11. We insert the subject com-

T A B L E 7-10

Calculation of Component #1”Delay To Sale [1]

A B C D

4 Coef¬cients Subject Co. Data Discount

5 Intercept 0.1292 NA 12.9%

Revenues2 [2]

6 5.39E 18 3.600E 13 0.0%

7 Value of block-post-discount [3] 4.39E 09 $4,331,435 1.9%

8 FMV-100% marketable minority interest 6.10E 10 $5,000,000 0.3%

9 Earnings stability 0.1381 0.4500 6.2%

10 Revenue stability 0.1800 0.3000 5.4%

11 Average years to sell 0.1368 1.0000 13.7%

12 Total Discount 13.4%

14 Value of block”pre-discount [4] $5,000,000

[1] Based on Abrams™ Regression #2 of Management Planning, Inc. data

Revenues2 $6,000,0002 (6 106)2 1013

[2] 3.6

[3] Equal to (value of block pre-discount) * (1 discount).

[4] Marketable minority interest FMV

PART 3 Adjusting for Control and Marketability

254

pany data in C6“C11, except for row 7, which we will discuss below. Our

subject company has $5 million in revenues (which, squared, equals 3.6

1013, per (C6), 100% marketable minority interest FMV of $5 million

(C8, analogous to market capitalization for the public companies in the

Management Planning, Inc. data), and earnings and revenue stability of

0.45 (C9) and 0.30 (C10), respectively.55 We estimate it will take one year

to sell the interest (C11).

Since we are valuing 100% of the capital stock of the ¬rm, the value

of the block of stock also has an FMV of $5 million (B14) before DLOM.56

The regression calls for the postdiscount FMV, which means we must

subtract the discount. The formula in cell C7 is: B14*(1 D12), i.e., the

postdiscount FMV equals the prediscount FMV (1 Discount). How-

ever, this is a simultaneous equation since the discount and the shares

sold in dollars each depend on the other. In order to be able to calculate

this, your spreadsheet should be set to allow recalculation with multiple

iterations. Otherwise you will get an error message with a circular ref-

erence.57 Column D is equal to column B column C, except for the y-

intercept in D5, which transfers directly from B5. Adding each of the

components in column D, we obtain a forecast discount of 13.4% (D12).

Limitations of the Regression. There may be combinations of subject

company data that can lead to strange results. This is especially true be-

cause:

1. The subject company data are near the end or outside of the

ranges of data in the regression of the MPI data.

2. There is very little variation in the range of the ˜˜average time to

sale™™ variable in our set. Most all of the restricted stock could be

sold between two and three years from the transaction date,

which is very little variation. Only 4 of the 53 sales were

expected to take less than two years (see below).

3. The R 2 is low.

4. The standard error of the y-estimate is fairly high”10%.

Regarding number 1, 47 of the 53 restricted stock sales in the MPI

database took place before the SEC circulated its Exposure Draft on June

27, 1995,58 to amend Rule 144(d) and (k) to shorten the waiting period

55. We do not explicitly show the detail of the calculations of earnings and revenue stability. Our

sample Restricted Stock Discount Study in Chapter 8, Table 8-1, shows these calculations.

56. Had we been valuing a 10% block of stock, B14 would have been $500,000.

57. If you create your own spreadsheet and make changes to the data, the simultaneous equation

is fragile, and it can easily happen that you may get error messages. When that happens,

you must put in a simple number in C7, e.g., $200,000, allow the spreadsheet to

˜˜recalibrate™™ and come back to equilibrium, then put in the correct formula. We do not have

this iterative problem with the other components of DLOM.

58. Revision of Holding Period Requirements in Rule 144; Section 16(a) Reporting of Equity Swaps

and Other Derivative Securities. File No. S7-17-95, SEC Release Nos. 33-7187; 34-35896; 17

CFR Parts 230 and 241; RIN 3235-AG53. The author expresses his gratitude to John Watson,

Jr., Esq., of Latham & Watkins in Washington, D.C., for providing him with a copy of the

exposure draft.

CHAPTER 7 Adjusting for Levels of Control and Marketability 255

for selling restricted stock to one year from two years and for nonaf¬l-

iated shareholders to sell shares without restriction after two years in-

stead of three.

Two sales took place in 1995 (Esmor Correctional Services, Inc. and

Chantal Pharmaceuticals Corp.) after the SEC Exposure Draft, and four

sales took place in 1996 (ARC Capital, Dense Pac Microsystems, Inc., No-

bel Education Dynamics, Inc., and Unimed Pharmaceuticals). That means

the market knew there was some probability that this would become law

and might shorten the waiting period to sell the restricted stock it was

issuing, and the later the sale, the more likely it was at the time that the

Exposure Draft would become law and provide relief to the buyer of the

restricted stock.

Thus, we should expect that those sales would carry lower discounts

than earlier sales”and that is correct. The discounts on the 1996 sales

were signi¬cantly lower than discounts on the earlier sales, all other

things being equal. The discounts ranged from 16“23% on the 1996 sales.

However, the two post-Exposure Draft 1995 sales had higher-than-

average discounts, which is somewhat counterintuitive. It is true that the

1996 sales would be more affected because the relief from restrictions for

the 1995 sales were more likely to have lapsed from the passage of time

than the 1996 sales, if it would take a long time for the Exposure Draft

to become law. Nevertheless, the two 1995 sales remain anomalies.

The average years needed to sell the stock ranged from a low of 1.2

years for Dense Pac Microsystems to 2.96 years for Sudbury Holdings,

Inc., with the vast majority being between 2 and 3 years. Extrapolating

this model to forecast a restricted stock discount for a sale with a restric-

tion of 10 years, for example, leads to ridiculous results, and even more

than 4 years is very questionable.

The coef¬cient for average years to sell is 0.1368 (B11), which means

that for each year more (less) than the forecast we made for this subject

company of 1 year, the discount increases (decreases) by 13.68%, holding

all else constant. Thus, if we were to forecast for a 10-year restriction, we

would get a discount of 136.8%”a nonsense result.

Thus, the appraiser must exercise good judgment and common sense

in using these results. Mechanically using these regression formulas to all

situations can be dangerous. It may be necessary to run other regressions

with the same data, i.e., using different independent variables or different

transformations of the data, to accommodate valuation assignments with

facts that vary considerably with those underlying these data. Another

possible solution is to assume, for example, that when a particular subject

company™s R 2 is beyond the maximum in the MPI database, that it is

equal to the maximum in the MPI database. It may be necessary to use

the other models, i.e., BSOPM with inferred rather than explicit standard

deviations or the QMDM, for more extreme situations where the regres-

sion equation is strained by extreme data. Hopefully we will soon have

much more data, as there will be increasingly more transactions subject

to the relaxed Rule 144 restrictions.

Component #2: Buyer Monopsony Power

The control stockholder of a privately held ¬rm has no guarantee at all

that he or she can sell his or her ¬rm. The market for privately held

PART 3 Adjusting for Control and Marketability

256

businesses is very thin. Most small and medium-size ¬rms are unlikely

to attract more than a small handful of buyers”and even then probably

not more than one or two every several months”while the seller of pub-

licly traded stock has millions of potential buyers. Just as a monopolist

is a single seller who can drive up price by withholding production, a

single buyer”a monopsonist”can drive price down by withholding pur-

chase.

The presence of 100 or even 10 interested buyers is likely to drive

the selling price of a business to its theoretical maximum, i.e., ˜˜the right

price.™™ The absence of enough buyers may confer monopsony power on

the few who are interested. Therefore, a small, unexciting business will

have an additional component of the discount for lack of marketability

for the additional bargaining power accruing to the buyers in thin mar-

kets.

It is easy to think that component #2 may already be included in

component #1, i.e., they both derive from the long time to sell an illiquid

asset. To demonstrate that they are indeed distinct components and that

we are not double counting, it is helpful to consider the hypothetical case

of a very exciting privately held ¬rm that has just discovered the cure for

cancer. Such a ¬rm would have no lack of interested buyers, yet it still is

very unlikely to be sold in less than one year. In that year other things

could happen. Congress could pass legislation regulating the medical

breakthrough, and the value could decrease signi¬cantly. Therefore, it

would still be necessary to have a signi¬cant discount for component #1,

while component #2 would be zero. It may not take longer to sell the

corner dry-cleaning store, but while the ¬rst ¬rm is virtually guaranteed

to be able to sell at the highest price after its required marketing time,

the dry-cleaning store will have the additional uncertainty of sale, and its

few buyers would have more negotiating power than the buyers of the

¬rm with the cure for cancer.

The results from Schwert, described earlier in the chapter, are rele-

vant here. He found that the presence of multiple bidders for control of