CHAPTER 6 An Iterative Valuation Approach 187

present value of EBIBAT, using the calculated WACC as the discount rate

and a midyear assumption.

The valuation section begins in cell D15 with the sum of the present

value of the ¬rst ¬ve years of EBIBAT. The next seven rows are the same

intermediate calculations as in Tables 6-1A, 6-1B, and 6-1C, using a Gor-

don model with an 8% constant growth rate and the midyear assumption

(D16“D21). Our ¬nal iteration of the FMV of the equity plus debt (enter-

prise value, or enterprise FMV) is $6,448,957 (D22). From this we subtract

the FMV of the debt of $2,000,000 to arrive at the ¬nal iteration of FMV

of equity of $4,448,957 (D24).

Let™s look at the calculation of WACC for the ¬rst iteration. For this

¬rm, we assume the FMV of interest-bearing debt is $2,000,000 (C43). We

further temporarily assume the FMV of the equity is its book value of

$800,000 (D43). Using these two initial values as our ¬rst approximation,

debt is 71.4% (F43) of the invested capital and equity is 28.6% (G43). We

calculate the ¬rst iteration of equity discount rate of 30% in cell H43 in

the same way as in the previous tables. We calculate the WACC to be:

WACC [(1 Tax Rate) Debt Discount Rate % Debt]

[Equity Discount Rate % Equity]

or

WACC [(1 0.4) 0.10 71.4%] (.30 28.6%]

12.857% (I43)8

We discount EBIBAT at this WACC to get the FMV of equity of $7,776,091

in cell J43. This iteration of equity is then transferred to cell D44, and the

process is repeated. After 12 iterations we arrive at a FMV of equity of

$4,448,957 (J54). We then con¬rm this value by iterating once more in

Row 55.

Table 6-2B: Initial Choice of Equity Doesn™t Matter

Tables 6-2A and 6-2B demonstrate that the initial choice of equity doesn™t

matter. Instead of choosing book equity as the starting point, in Table

6-2B we make an arbitrary guess of $10,000,000 (D43) as a starting point.

Table 6-2B is identical to Table 6-2A, except in the initial choice of value

of the equity and the intermediate iterations. The ¬nal result is identical.

Note that it does not matter whether your initial guess is too low or too

high: Table 6-2A is too low and Table 6-2B is too high, but they both lead

to the same result.

Convergence of the Invested Capital Approach

As with the equity valuation method, if the method described above does

not converge, an alternative is to take the average of the resulting FMV

of equity and the previously assumed value as your input into column D

8. There is an apparent rounding error, as the percentages of debt and equity to six decimal places

are 0.714286 and 0.285714.

PART 2 Calculating Discount Rates

188

T A B L E 6-2B

WACC Approach with Iterations Beginning with Arbitrary Guess of Equity Value

A B C D E F G H I J

4 1998 1999 2000 2001 2002

5 EBIT 690,000 779,700 865,467 943,359 1,018,828

6 Growth rate in EBIT 15% 13% 11% 9% 8%

7 Income taxes (276,000) (311,880) (346,187) (377,344) (407,531)

8 EBIBAT 414,000 467,820 519,280 566,015 611,297

9 Growth rate-EBIBAT 15% 13% 11% 9% 8%

10 Present value factor 0.9308 0.8064 0.6986 0.6052 0.5243

11 Pres value-EBIBAT $385,341 $377,237 $362,767 $342,566 $320,523

14 Final Valuation:

15 PV 1998“2002 EBIBAT $1,788,434

16 Constant growth rate in EBIBAT 8%

17 Forecast EBIBAT-2003 660,200

18 Gordon model mult SQRT(1 R)/(R G) 14.4646

19 PV-EBIBAT after 2002 as of 1-1-2003 9,549,547

20 Present value factor-5 years 0.488036

21 PV-EBIBAT after 2002 4,660,523

22 Enterprise FMV-100% interest $6,448,957

23 Less FMV of debt (2,000,000)

24 FMV of equity-100% interest $4,448,957

27 Assumptions:

28 EBIT-1997 600,000

29 Income tax rate 40%

30 Discount rate-debt: pre-tax 10%

31 Discount rate-debt: after-tax 6%

32 Unlevered beta 1.05

33 Risk free rate 6%

34 Equity premium 8%

35 Small company premium 3%

36 Wtd avg cost of capital (WACC) 15.428%

38 Capital Structure & Iterations

40 Interest- Interest-

41 Bearing Bearing Equity FMV

42 t Debt Equity Total Debt Equity Disc. Rate WACC Equity

43 FMV debt, eqty at t 1 1 2,000,000 10,000,000 12,000,000 16.7% 83.3% 18.408% 16.340% 3,761,117

44 FMV debt, eqty at t 1 2 2,000,000 3,761,117 5,761,117 34.7% 65.3% 20.080% 15.192% 4,654,820

45 FMV debt, eqty at t 1 3 2,000,000 4,654,820 6,654,820 30.1% 69.9% 19.565% 15.489% 4,397,731

46 FMV debt, eqty at t 1 4 2,000,000 4,397,731 6,397,731 31.3% 68.7% 19.692% 15.412% 4,462,354

47 FMV debt, eqty at t 1 5 2,000,000 4,462,354 6,462,354 30.9% 69.1% 19.659% 15.432% 4,445,498

48 FMV debt, eqty at t 1 6 2,000,000 4,445,498 6,445,498 31.0% 69.0% 19.667% 15.427% 4,449,853

49 FMV debt, eqty at t 1 7 2,000,000 4,449,853 6,449,853 31.0% 69.0% 19.665% 15.428% 4,448,725

50 FMV debt, eqty at t 1 8 2,000,000 4,448,725 6,448,725 31.0% 69.0% 19.666% 15.428% 4,449,017

51 FMV debt, eqty at t 1 9 2,000,000 4,449,017 6,449,017 31.0% 69.0% 19.666% 15.428% 4,448,942

52 FMV debt, eqty at t 1 10 2,000,000 4,448,942 6,448,942 31.0% 69.0% 19.666% 15.428% 4,448,961

53 FMV debt, eqty at t 1 11 2,000,000 4,448,961 6,448,961 31.0% 69.0% 19.666% 15.428% 4,448,956

54 FMV debt, eqty at t 1 12 2,000,000 4,448,956 6,448,956 31.0% 69.0% 19.666% 15.428% 4,448,958

55 FMV debt, eqty at t 1 13 2,000,000 4,448,958 6,448,958 31.0% 69.0% 19.666% 15.428% 4,448,957

56 FMV debt, eqty at t 1 14 2,000,000 4,448,957 6,448,957 31.0% 69.0% 19.666% 15.428% 4,448,957

CHAPTER 6 An Iterative Valuation Approach 189

when starting the next iteration as opposed to just using the latest itera-

tion of equity. This can be done by making a small alteration to the

spreadsheet.

LOG SIZE

The log size model converges far faster than the CAPM versions of the

invested capital approach or the equity valuation method. The reason is

that when we use logarithms to calculate the discount rate, large absolute

changes in equity value cause fairly small changes in the discount rate,

which is not true of CAPM.

SUMMARY

When using CAPM, using this iterative approach will improve appraisal

accuracy and eliminate arguments over the proper leverage. One look at

the difference between the beginning guess of the FMV of equity and the

¬nal FMV will show how much more accuracy can be gained. While it

is true that had we guessed a number based on industry average capi-

talization we would have been closer, the advantage of this approach is

that it obviates the need for precise initial guesses.

The iterative approach should give us the ability to get much closer

answers from both the invested capital and the direct capital approaches,

as long as the subject ¬rm is suf¬ciently pro¬table. The iterative approach

does not seem to work for very small ¬rms with little pro¬tability, but

those are the ¬rms for which you are least likely to want to bother with

the extra work involved in the iterations.

BIBLIOGRAPHY

Abrams, Jay B. 1995. ˜˜An Iterative Valuation Approach.™™ Business Valuation Review

(March): 26“35.

Hamada, R. S. 1972. ˜˜The Effects of the Firm™s Capital Structure on the Systematic Risk

of Common Stocks.™™ Journal of Finance 27: 435“52.

PART 2 Calculating Discount Rates

190

PART THREE

Adjusting for Control and

Marketability

INTRODUCTION

Part 3 of this book, consisting of Chapters 7, 8, and 9, deals with calcu-

lating control premiums, the discount for lack of control (DLOC), and

discount for lack of marketability (DLOM). These topics correspond to

the third and fourth steps in valuing businesses. These are practical,

˜˜how-to™™ chapters.

Adjusting for levels of control and marketability is probably the most

controversial topic in business valuation. As such, Chapter 7 is almost a

book unto itself. It is the longest chapter in this book, and it probably has

the most startling research results of any chapter.

Chapter 7 is divided into two parts: the ¬rst part primarily dealing

with control and the second primarily with marketability. I chose that

order because of the one-way relationship”control affects marketability,

but marketability does not affect control. The chapter begins with a com-

prehensive overview of the major professional articles on the topic and

then proceeds to review a number of academic articles that provide in-

sight into the issue of control.

In part 2 of Chapter 7 we review two quantitative models (other than

my own): Mercer™s quantitative marketability discount model (QMDM)

and Kasper™s bid-ask spread model. We then analyze restricted stock dis-

counts with multiple regression analysis for two reasons. The ¬rst reason

is that this is intrinsically useful in restricted stock discount studies. The

second, more important, reason is that restricted stock discounts serve as

one of the components of my economic components model of DLOM,

which makes up the majority of part 2. At the end of the chapter, Z.

Christopher Mercer provides a rebuttal to my critique of the quantitative

marketability discount model, and we go back and forth with arguments

that the profession should ¬nd interesting and enlightening, and possibly

somewhat confusing and frustrating as well.

Economic Components Model

The heart of Chapter 7 is my own economic components model for

DLOM, which consists of four components:

191

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1. The economic consequences of the delay to sale experienced by

all privately-held ¬rms. I model this component using a

regression analysis of restricted stock discount data published

by Management Planning, Inc. in Mercer™s book.1

2. Extra bargaining power (˜˜monospony power™™) to the buyer

arising from thin markets. The academic article by Schwert

contains a key ¬nding that enables us to estimate this

component of DLOM reliably.

3. Buyer™s transactions costs in excess of transactions costs for

publicly held stocks.

4. Seller™s transactions costs in excess of transactions costs for

publicly held stocks.

We present research on the magnitude of transactions costs for both

buyers and sellers with different business sizes as well as regression anal-

yses of each. This enables us to calculate transactions costs for any busi-

ness size for both buyer and seller.

Items 3 and 4 above, which we label components #3A and #3B in the

chapter, occur every time the business is sold. Those fees and costs leave

the system by being paid to outsiders such as business brokers, account-

ants, attorneys, and appraisers. Thus, we need to be able to calculate the

present value effect of the in¬nite continuum of periodic transactions

costs, which we do in the form of one formula for buyers™ excess trans-

actions costs and another formula for sellers™ excess transactions costs.2

This process is now vastly simpli¬ed over the process in my original

Business Valuation Review article on the topic. We also give an example of

how to calculate DLOM.

A very important test that we perform in Chapter 7 is a comparison

of several models in their ability to explain the restricted stock discounts

from the Management Planning, Inc. data: the Black“Scholes options pric-

ing model (BSOPM) put formula using speci¬cally calculated standard

deviations of returns (volatility) of the public stocks, the BSOPM put us-

ing indirectly calculated (through log size equations) standard deviations,

the quantitative marketability discount model (QMDM), a regression

equation, and the mean discount. The regression equation was the best

forecast of restricted stock discounts, with the BSOPM with directly cal-

culated volatility a very close second. Both the BSOPM using indirectly

calculated volatility and the QMDM were worse than the mean in fore-

casting discounts, with QMDM being farthest out of the money. This is

signi¬cant because it is the ¬rst empirical test of any model to calculate

restricted stock discounts.

Chapters 8 and 9 are practical applications of the work in Chapter 7

in the form of sample reports. Chapter 8 is a sample restricted stock dis-

count report, and Chapter 9 is a sample fractional interest discount study

for a Limited Liability Company interest in real property. Chapter 8 is

1. The data have been corrected since publication in Mercer™s book, and Management Planning,

Inc. provided us with additional data.

2. That is because the seller™s costs on the ¬rst sale do not count in calculating DLOM, whereas

buyer™s costs do. In all subsequent sales of the business, both count.

PART 3 Adjusting for Control and Marketability

192

purely an application of Chapter 7 and contains no research that is not

already in Chapter 7, while Chapter 9 does contain two types of new

research:

1. My own regression analysis of discounts from net asset value