protected they are in private ¬rms. We therefore reduce this premium for

illiquidity from the premium in Mercer™s example of between 1 and 2%

(H18 and I18) to 0% (K18, L18) for the restricted stock. While it is possible

that the restricted stocks should have a positive premium for this factor,

they are nevertheless far more liquid than all of the private ¬rms in Mer-

cer™s examples. If we should increase K18 and L18 to, say, 1%, then we

should increase H18 and I18 to at least 2“3%, respectively, or probably

higher yet.

Relative to the private C corporation shares, the expected holding

period for the restricted stock is short and certain. We therefore reduce

the premium for holding period uncertainty from between 0 and 1% (H19

and I19) for Example 1 to 0 (K19, L19) for the restricted shares. As both

PART 3 Adjusting for Control and Marketability

278

T A B L E 7-19

QMDM Comparison of Restricted Stock Discount Rate versus Mercer Example 1

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

5 Mercer Example Restricted Stock

1

6 Range of Range of

Returns Returns

7 Components of the Required Holding Period Return Lower Higher Lower Higher

8 Base equity discount rate (adjusted capital asset pricing model)

9 Current yield-to-maturity composite long term treasuries 6.7% 6.7% 6.7% 6.7%

10 Adjusted Ibbotson large stock premium 6.5%

11 applicable beta statistic 1

12 Beta adjusted large stock premium 6.5% 6.5% 6.5% 6.5%

13 Adjusted Ibbotson small stock premium 3.5% 3.5% 3.5% 3.5%

14 Base equity discount rate 16.7% 16.7% 16.7% 16.7%

17 Investment Speci¬c Risk Premiums

18 General illiquidity of the investment [1] 1.0% 2.0% 0.0% 0.0%

19 Uncertainties related to length of expected holding period [2] 0.0% 1.0% 0.0% 0.0%

20 Lack of expected interim cash ¬‚ows [3] 0.5% 1.0% 0.5% 1.0%

21 Small shareholder base [4] 0.0% 1.0% 0.0% 0.0%

22 Range of speci¬c risk premiums for the investment 1.5% 5.0% 0.5% 1.0%

24 Initial range of required returns 18.2% 21.7% 17.2% 17.7%

26 Concluded range of required holding period returns (rounded) 18.0% 22.0% 17.0% 18.0%

[1] The restricted stock should be much easier to sell than a minority stake in a private closely held C corporation, since public minority shareholder rights are generally better protected.

While it is possible that the restricted stocks should have a positive premium for this factor, they are nevertheless far more liquid than all of the private ¬rms in Mercer™s examples. If we

should increase K18 and L18 to 1%, then we should increase H18 and I18 to at least 2% to 3% or probably higher yet.

[2] Relative to the private shares, the expected holding period for the restricted stock is short and certain.

[3] We assume a non dividend paying restricted stock. The example also concerned a non dividend paying C corporation. We therefore assign the same risk premium for this factor.

[4] The restricted stock shares are shares of public corporations, which in general have large shareholder bases.

investments are expected to pay no dividends, there is no difference in

the premium for lack of expected interim cash ¬‚ows (Row 20), although

the latter experiences that lack of dividends for a far shorter and much

more certain time period, which could well justify a lower premium than

the former.

At this point I can digress to pose my objections to the ¬rst two

factors. General illiquidity of the investment is a very fuzzy term. It can mean

almost anything. There is no empirical measure of it. Therefore, it can be

almost anything that one wants it to be”which I admit has its advan-

tages in practical application, but it™s not good science. It is also unclear

where general illiquidity stops and uncertainties in the holding period

begin. Do they overlap? How does one prevent him- or herself from

double-counting them? That is a problem with loosely-de¬ned terms.

Returning to the main train of thought, the private, closely held C

corporation would have a much smaller shareholder base than the re-

stricted stock corporations. We therefore reduce the premium for a small

shareholder base from between 0 and 1% (H21 and I21) for Example 1 to

0 (K21, L21) for the restricted stock. The total speci¬c risk premium for

CHAPTER 7 Adjusting for Levels of Control and Marketability 279

the restricted stock comes to 0.5% (K22) to 1.0% (L22) versus the 1.5%

(H22) to 5% (I22) for the private shares. After adding the base equity

discount rates and rounding, we arrive at a concluded range of required

holding period returns of 18“22% and 17“18% (Row 26) for Mercer™s

Example 1 and the restricted stock, respectively.

Next we need to determine the expected growth rate in value of the

unrestricted marketable minority shares. Since there are no dividends, the

expected growth rate must be equal to the discount rate”by de¬nition.76

In this example the equity discount rate of the unrestricted marketable

shares or the ˜˜base equity discount rate™™ is 16.7%.

Let™s now calculate the QMDM discount on the restricted stock with

the following assumptions:

1. A midrange (of K26 and L26) required holding period return of

17.5%.

2. The 2.5-year average holding period.

3. The growth rate in value of 16.7%.

The calculation is as follows:

1

1.1672.5

DLOM 1 (FV PVF) 1 1.7%

1.1752.5

Assuming the correct discount is 30%, the QMDM is almost 95% too low!

Mercer™s Response

After reviewing Mr. Abrams™ response to my rebuttal of his criticism of

the QMDM, it is apparent that he and I continue to disagree over how

the QMDM is applied in practice. The average marketability discounts in

the 10 examples cited in my rebuttal of his criticism was 37%, and the

median discount was 30%, not 1.7%. Mr. Abrams™ mistake is in assuming

that the discount rate embedded in the pricing of a publicly traded stock

is the required return of restricted stock investors. The fact that the av-

erage restricted stock discount is 30% or so indicates that investors have

extracted a signi¬cant premium in return relative to the expected returns

of the counterpart publicly traded securities.

What may be true ˜˜by de¬nition™™ in a perpetuity calculation may

well not be true for shorter holding periods. The QMDM deals, not with

perpetuity calculations, but with investor assessments of expected cash

¬‚ows over ¬nite time horizons. And it makes explicit the assumptions

made about the relationship between the expected growth in value of

investments and the required returns of investors in those investments. I

maintain that the model does indeed provide an excellent tool for esti-

mating marketability discounts (from an estimated freely traded value)

for minority interests in closely held companies.

76. This is the discount rate applicable to marketable minority shares, not the higher discount rate

applicable to illiquid shares, i.e., the required holding period return.

PART 3 Adjusting for Control and Marketability

280

Conclusion

We have reviewed the professional and some of the academic literature

dealing with control premiums and DLOM. My opinion is that with our

current information set, we should use control premiums in the 21“28%

range. We developed this as being three to four times the value of the

voting rights premium adjusted to U.S. laws and for liquidity differences

between voting and nonvoting stock. This measure is consistent with the

median going private premium of 24.1% (Table 7-1, E21), although it is

preferable to make a clean separation of expected performance improve-

ments, which increase the ˜˜top line,™™ i.e., cash ¬‚ows, versus the pure

value of control, which is represented by a reduction in the discount rate.

We reviewed three quantitative models of DLOM: Mercer™s, Kas-

per™s, and Abrams™. The QMDM was unable to provide any meaningful

restricted stock discounts for the Management Planning, Inc. data, as dis-

counting modest risk premiums for two to three years provides little var-

iation in discount. Abrams™ non-company-speci¬c Black-Scholes options

pricing model performed worse at explaining restricted stock discounts

than the mean, while using BSOPM with ¬rm-speci¬c calculations of stan-

dard deviations was superior to the mean. While that makes Black“

Scholes a viable candidate for restricted stock studies, it is not a possible

model for valuing the delay-to-sale component of DLOM, and we must

use the regression of the MPI data.

We quanti¬ed component #2, monopsony power to the buyer, as 9%,

according to Schwert™s ¬ndings of a 12.2% greater premium in takeovers

when there are multiple buyers than when there is only one buyer.

Finally, we quanti¬ed transactions costs separately for the buyer and

the seller. The premise of fair market value is such that we ask, ˜˜What

would a hypothetical buyer be willing to pay for this interest,™™ which

means that we are presuming a ¬rst sale immediately. Buyers care about

their own transactions costs, but they do not care about sellers™ transac-

tions cost on the immediate transaction. However, buyers do care that in

10 years or so they become the sellers. They therefore care about all sub-

sequent sellers™ (and buyers™) transactions costs. We presented two dis-

count formulas”equations (7-9) and (7-9a), which are appropriate for

seller and buyer, respectively, to translate the pure discount that applies

to each transaction into a discount based on the present value of the

in¬nite continuum of periodic transactions.

In Table 7-14 we applied our DLOM model to a control interest in a

hypothetical private company. The result was a DLOM of 23.1%, which

is a reasonable result.

Of course, the economic components model is merely a model. It is

certainly imperfect, and it must be used with common sense. It is possible

to obtain strange or nonsensical results, and if the appraiser is asleep at

the wheel, he or she may not realize it. There is plenty of room for ad-

ditional research to improve our modeling and results. Nevertheless, in

my opinion this is the most realistic and comprehensive model to date

for calculating DLOM.

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MATHEMATICAL APPENDIX

DEVELOPING THE DISCOUNT FORMULAS

Initially we assume the current business owner will operate the business

for 10 years, sell it, and pay transaction costs of z.77 The next owner will

run the business another 10 years, sell it, and pay transaction costs. We

assume this pattern occurs ad in¬nitum. Of course, there will be variations

from the sale every 10 years”some will sell after 1 year, others after 30

years. In the meantime, in the absence of prior knowledge, we assume

every 10 years to be a reasonable estimate of the average of what will

occur.

NPV of Cash Flows with Periodic Transaction

Costs Removed

The net present value (NPV) of cash ¬‚ows to the existing business owner

with periodic transaction costs removed is the full amount of the ¬rst 10

years™ cash ¬‚ows, plus (1 z) times the next 10 years™ cash ¬‚ows, where

z)2 times the next 10 years™

z is the periodic transaction cost, plus (1

cash ¬‚ows, etc. We will denote the NPV net of transaction costs, i.e., with

transaction costs removed from the stream of cash ¬‚ows, as NPVTC .

g)9

(1 g) (1

1

NPVTC

r)0.5 r)1.5 r)9.5

(1 (1 (1

g)10 g)19

(1 (1

(1 z)

r)10.5 r)19.5

(1 (1

g)20 g)29

(1 (1

2

(1 z) (A7-1)

r)20.5 r)29.5

(1 (1

Multiplying each term in equation (A7-1) by (1 g)/(1 r), we get:

77. As explained in the body of the chapter, z is an incremental transaction cost. For example,

when we value a small fractional ownership in a privately owned business, often our

preliminary value is on a marketable minority basis. In this case z would be the difference

in transaction cost (expressed as a percentage) between selling a private business interest

and selling publicly traded stock through a stockbroker.

PART 3 Adjusting for Control and Marketability