56 FMV Regression Model Geometric Mean Arithmetic Mean Difference

57 $20,000,000 72 year 14.2% 22.1% 7.9%

58 60 year 16.6% 21.5% 4.9%

59 $300,000 72 year 16.3% 28.5% 12.1%

60 60 year 19.0% 26.5% 7.5%

[1] Values from Ibbotson™s 1998 SBBI Yearbook, Table 7-3

[vn / vo]1 / n 1

[2] Geometric mean for 1937-1997 was calculated using Year End Index Values for 1937 (for year starting 1938) and 1997 according to the formula rg

[3] From Table 4-1, Chapter 4

Table 5-4: Log Size Comparison of Discount Rates and

Gordon Model Multiples Using AM versus GM

Table 5-4 illustrates this, where discount rates are calculated using the log

size model, with both the arithmetic and geometric mean regression equa-

tions derived from Tables 4-1 and 5-3, respectively. There is a dramatic

difference in discount rates, especially with small ¬rms. The log size dis-

count rate for a $250,000 ¬rm is 26.76% using the AM (B7) and 19.12%

using the GM (C7). The resulting midyear Gordon model multiples are

5.42 (D7) using the AM and 8.32 (E7) using the GM.

Column F is the ratio of the Gordon model multiples using the ge-

ometric mean to the Gordon model multiples using the arithmetic mean.

Dividing the 8.32 GM multiple by the 5.42 AM multiple gives us a ratio

of 153.41%, i.e., the GM leads to a valuation that is 53.41% higher than

the AM for such a small ¬rm (this is assuming a ¬rm with 6% constant

growth). Notice that the ratio declines continuously as we move down

Column F. The overvaluation of a $10 billion ¬rm using the GM is

12.57%”far less than the overvaluation of the $250,000 ¬rm. The differ-

ences are signi¬cantly greater when using the 72-year log size models, as

including the most volatile years in the regression makes for a greater

difference in the AM versus GM Gordon model multiples. These numer-

ical examples underscore the importance of using the arithmetic mean

when valuing expected future earnings or cash ¬‚ow.

INDRO AND LEE ARTICLE

This article (Indro and Lee 1997) is extremely mathematical, exceedingly

dif¬cult reading. The authors begin by citing (Brealey and Myers 1991),

CHAPTER 5 Arithmetic versus Geometric Means 175

T A B L E 5-4

Comparison of Discount Rates Derived from the Log Size Model Using 60-Year

Arithmetic and Geometric Means

A B C D E F

5 Gordon Model Ratio

Multiples Using

6 Firm Size AM [1] GM [2] AM [3] GM [3] GG / AG [4]

7 $250,000 26.76% 19.12% 5.42 8.32 153.41%

8 $1,000,000 25.09% 18.33% 5.86 8.83 150.61%

9 $25,000,000 21.21% 16.49% 7.24 10.29 142.14%

10 $50,000,000 20.38% 16.10% 7.63 10.67 139.85%

11 $100,000,000 19.54% 15.70% 8.07 11.09 137.34%

12 $500,000,000 17.60% 14.78% 9.35 12.20 130.52%

13 $10,000,000,000 14.00% 13.08% 13.35 15.03 112.57%

Conclusion: The ratio of Gordon Model Multiples decreases with ¬rm size (Column F)

Notes:

[1] Arithmetic Mean (AM) Regression Equation, 60 year model r 41.72% 0.01204 Ln (FMV)

[2] Geometric Mean (GM) Regression Equation, 60 year model. r 26.2% 0.0057 Ln (FMV)

[3] Gordon Model Multiple calculated assuming 6% growth in earnings-midyear assumption. Discount rates are not rounded in these

calculations.

[4] Geometric Gordon Model Multiple / Arithmetic Gordon Model Multiple

who say that if monthly returns are identically and independently dis-

tributed, then the arithmetic average of monthly returns should be used

to estimate the long-run expected return. They then cite empirical evi-

dence that there is signi¬cant negative autocorrelation in long-term equity

returns and that historical monthly returns are not independent draws

from a stationery distribution. This means that high returns in one time

period will tend to mean that on average there will be low returns in the

next period, and vice-versa. Based on this, Copeland, Koller, and Murrin

(1994) argue that the geometric average is a better estimate of the long-

run expected returns.

Indro and Lee show that the arithmetic and geometric means have

upward and downward biases, respectively, and that a horizon-weighted

average of the two is the least biased and most ef¬cient estimator.

If the authors are correct, it would mean that there would no longer

be a single discount rate. Every year would have its own unique

weighted-average discount rate. That would also add complexity to the

use of the Gordon model to calculate a residual value.

Because of the extremely dif¬cult mathematics in the article, it was

necessary to speak to academic sources to evaluate it. Professor Myers,

cited above, did agree that long-term (¬ve-year) returns are negatively

autocorrelated but that there are ˜˜very few data points.™™ He had not fully

read the article, is not sure of its signi¬cance, and did not have an opinion

of it. Ibbotson Associates does not feel the evidence for mean reversion

is that strong, and on that basis is not moved to change its opinion that

the AM is the correct mean. It seems that it will take some time before

this article gets enough academic attention to cause the valuation profes-

sion to make any changes in the way it operates.

PART 2 Calculating Discount Rates

176

BIBLIOGRAPHY

Brealey, R. A., and Stewart C. Myers. 1991. Principles of Corporate Finance. New York:

McGraw-Hill.

Copeland, Tom, Tim Koller, and Jack Murrin. 1994. Valuation: Measuring and Managing

the Value of Companies. John Wiley & Sons, Inc. New York, NY.

Ibbotson Associates. 1998. Stocks, Bills, Bonds and In¬‚ation: 1998 Yearbook. Chicago: The

Associates. 107“08; 153“155.

Indro, Daniel C., and Wayne Y. Lee. 1997. ˜˜Biases in Arithmetic and Geometric Averages

as Estimates of Long-Run Expected Returns and Risk Premia.™™ Financial Management

26, no. 4 (Winter): 81“90.

Joyce, Allyn A. 1995. ˜˜Arithmetic Mean vs. Geometric Mean: The Issue in Rate of Return.™™

Business Valuation Review ( June): 62“68.

CHAPTER 5 Arithmetic versus Geometric Means 177

CHAPTER 6

An Iterative Valuation Approach

INTRODUCTION

EQUITY VALUATION METHOD

Table 6-1A: The First Iteration

Table 6-1B: Subsequent Iterations of the First Scenario

Table 6-1C: Initial Choice of Equity Doesn™t Matter

Convergence of the Equity Valuation Method

INVESTED CAPITAL APPROACH

Table 6-2A: Iterations Beginning with Book Equity

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

Convergence of the Invested Capital Approach

LOG SIZE

SUMMARY

BIBLIOGRAPHY

179

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your best guess of the FMV of equity or you can use the net

book value of equity. Eventually your initial guess will make no

difference.

Steps 1“6 are not repeated. The following steps are iterative.

7. Calculate a relevered beta and equity discount rate using your

initial capital structure and use it to value the ¬rm.

8. Substitute the ¬rst calculated fair market value of equity into a

new capital structure and use the new weights to calculate the

next iteration of beta, equity discount rate, and FMV of equity.

9. Keep repeating 7 and 8 until you reach a steady state value for

beta, equity discount rate, and FMV of equity.

Let™s illustrate this with a couple of examples.

Table 6-1A: The First Iteration

We use a deliberately simple discounted future earnings approach in Ta-

ble 6-1A to illustrate how this process works. Starting with a ¬rm whose

net income before taxes (NIBT) in 1997, the previous year, was $400,000

(cell D28), we assume a declining growth rate in income: 15% (B7) in

1998, 13% (C7) in 1999, ¬nishing with 8% (F7) in 2002. We use these

growth rates to forecast income in 1998“2002. Subtracting 40% for income

taxes, we arrive at net income after taxes (NIAT) of $276,000 in 1998 (B9),

rising to $407,531 in 2002 (F9). The bottom row of the top section is the

present value of NIAT, using the calculated equity discount rate and a

midyear assumption.

The valuation section begins in cell D17 with the sum of the present

value of NIAT for the ¬rst ¬ve years. The next seven rows are interme-

diate calculations using a Gordon model with an 8% constant growth rate

and the midyear assumption (D17“D23). Forecast income in 2003 is the

2002 net income times one plus the growth rate [F9 (1 D18) D19

$440,134]. The midyear Gordon model multiple, D20, is equal to SQRT(1

r)/(r g) SQRT(1 D36)/(D36 D18) 8.1456. Multiplying $440,134

8.1456 $3,585,135 (D21), which is the present value of net income after year

2002 as of December 31, 2002. The present value factor for ¬ve years is

0.377146 (D22). Multiplying $3,585,135 0.377146 $1,352,121 (D23), which

is the present value of income after 2002 as of the valuation date, January 1,

1998.

Adding the present value of the ¬rst ¬ve years™ net income of $1,055,852

(D17) to the present value of the net income after ¬ve years of $1,352,121

(D23), we arrive at our ¬rst approximation of the FMV of the equity of

$2,407,973 (D24).

Rows 28 through 35 contain the assumptions of the model and the data

necessary to lever and unlever industry average betas and calculate equity

discount rates. The discount rate is in cell D36, though it is calculated in G54

and transferred from there.

Rows 42 through 46 detail the calculation of an unlevered beta of 0.91

(F46) from an average of publicly traded guideline companies. In the capital

structure and iterations section, Row 54 shows the market value of debt and

CHAPTER 6 An Iterative Valuation Approach 181

T A B L E 6-1A

Equity Valuation Approach with Iterations Beginning with Book Equity: Iteration #1

A B C D E F G H

5 1998 1999 2000 2001 2002

6 Net inc before taxes 460,000 519,800 576,978 628,906 679,219

7 Growth rate in NIBT 15% 13% 11% 9% 8%

8 Income taxes (184,000) (207,920) (230,791) (251,562) (271,687)

9 Net inc after taxes 276,000 311,880 346,187 377,344 407,531

10 Present value factor 0.9071 0.7464 0.6141 0.5053 0.4158

11 Pres value NIAT $250,357 $232,777 $212,601 $190,675 $169,441

16 Final Valuation:

17 PV 1998“2002 net income $1,055,852

18 Constant growth rate in income G 8%

19 Forecast net income-2003 440,134

20 Gordon model mult SQRT(1 R)/(R G) 8.1456

21 Present value-net inc after 2002 as of 12/31/2002 3,585,135

22 Present value factor-5 years 0.377146

23 Present value of net income after 2002 as of 1/1/98 1,352,121

24 FMV of equity-100% interest $2,407,973

27 Assumptions:

28 Net income before tax-1997 400,000

29 Income tax rate 40%

30 Discount rate-debt: pre-tax 10%

31 Discount rate-debt: after-tax 6%

32 Unlevered beta (from F46) 0.91

33 Risk free rate 6%

34 Equity premium 8%

35 Small company premium 3%

36 Equity discount rate R 21.534%

38 Calculation of Equity Discount Rate Using Comparables

40 Equity Unlevered

41 Beta Debt Equity D/E Beta

42 Guideline Company #1 1.15 454,646 874,464 52.0% 0.88

43 Guideline Company #2 1.20 146,464 546,454 26.8% 1.03

44 Guideline Company #3 0.95 46,464 705,464 6.6% 0.91

45 Guideline Company #4 0.85 52,646 846,467 6.2% 0.82

46 Totals or averages 1.04 700,220 2,972,849 23.55% 0.91

49 Capital Structure & Iterations

51 Interest-

52 Bearing Equity Before Relevered Equity FMV

53 t Debt Iteration D/E Beta Disc. Rate Equity

54 FMV debt, eqty at t 1 1 900,000 750,000 1.20 1.5668 21.534% 2,407,973

55 FMV debt, eqty at t 1 2 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

56 FMV debt, eqty at t 1 3 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

57 FMV debt, eqty at t 1 4 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

58 FMV debt, eqty at t 1 5 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

59 FMV debt, eqty at t 1 6 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

60 FMV debt, eqty at t 1 7 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

61 FMV debt, eqty at t 1 8 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

62 FMV debt, eqty at t 1 9 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

63 FMV debt, eqty at t 1 10 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

64 FMV debt, eqty at t 1 11 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

65 FMV debt, eqty at t 1 12 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

66 FMV debt, eqty at t 1 13 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

67 FMV debt, eqty at t 1 14 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

68 FMV debt, eqty at t 1 15 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

69 FMV debt, eqty at t 1 16 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

70 FMV debt, eqty at t 1 17 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

71 FMV debt, eqty at t 1 18 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

72 FMV debt, eqty at t 1 19 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

73 FMV debt, eqty at t 1 20 900,000 2,407,973 0.37 1.1152 17.921% 2,407,973

PART 2 Calculating Discount Rates

182

the book value of equity (our initial guess of market value) as well as the

implied debt/equityratioandreleveredbetaaccordingtoHamada™sformula

(Hamada 1972):2

Debt

1 (1 Tax Rate)

levered unlevered

Equity

Cell G54 is the discount rate of 21.534% for the ¬rst iteration, calculated ac-

cording to the formula

Disc Rate Risk Free Rate ( Equity Premium)

levered

Small Company Premium

We use this discount rate to calculate the ¬rst iteration of FMV of equity in

cell H54.

Table 6-1B: Subsequent Iterations of the First Scenario

Table 6-1B is identical to Table 6-1A, except that it contains nine iterations

in the capital structure section instead of 1. Also, cell D36 contains the

¬nal equity discount rate from Row 62.3 We denote the iteration number

as t, which appears in Column B, Rows 54“62. When t 1, we obtain

an equity discount rate of 21.534% (G54) and a FMV of the equity of

$2,407,973 (H54), as before. This tells us that our initial guess of the FMV

of the equity, which was the book value of the equity of $750,000 (D54),

is too low.

We substitute the $2,407,973 (H54) ¬rst iteration of equity into the

new capital structure in D55 to get a debt/equity ratio of 0.37 (E55), as

seen in the second iteration of Table 6-1B. This changes the discount rate

to 17.921% (G55). This results in the second iteration of equity value of

$3,245,701 (H55). We use the new equity as the basis for our third itera-

tion, which we calculate in the same fashion as the previous iteration. We

follow these steps until we reach a steady state, which in this case occurs

in the eighth iteration, with a FMV of $3,404,686 (H61). We must carry

out an additional iteration to know for sure that we have reached a steady

state, which is the purpose of iteration #9.

Table 6-1C: Initial Choice of Equity Doesn™t Matter

Tables 6-1B and 6-1C demonstrate that the initial choice of equity doesn™t

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

6-1C we make an arbitrary guess of $5,000,000 (D54) as a starting point.4

2. This equation is most accurate when the ¬rm™s pretax discount rate for debt is close to the risk-

free rate.

3. Actually, D36 takes on the value calculated in each iteration in G54 through G62, so the discount

rate used in all the calculations changes in each iteration of the spreadsheet.

4. For those who buy the electronic spreadsheet from the author, which is not included with this

book, the steps are: (1) input your initial guess of equity in D54; (2) initialize the

spreadsheet by pressing Control-X; (3) press Control-Z for each iteration. Every time you

press Control-Z, the spreadsheet will calculate one iteration of value, as in Rows 54 to 62.

Repeat pressing Control-Z until you have reached a steady state, i.e., the value in Column

H is the same twice in a row.

CHAPTER 6 An Iterative Valuation Approach 183

T A B L E 6-1B

Equity Valuation Approach with Iterations Beginning with Book Equity

A B C D E F G H

5 1998 1999 2000 2001 2002

6 Net inc before taxes 460,000 519,800 576,978 628,906 679,219

7 Growth rate in NIBT 15% 13% 11% 9% 8%

8 Income taxes (184,000) (207,920) (230,791) (251,562) (271,687)

9 Net inc after taxes 276,000 311,880 346,187 377,344 407,531

10 Present value factor 0.9228 0.7857 0.6690 0.5696 0.4850

11 Pres value NIAT $254,680 $245,045 $231,602 $214,952 $197,669

16 Final Valuation:

17 PV 1998“2002 net income $1,143,949

18 Constant growth rate in net income G 8%

19 Forecast net income-2003 440,134

20 Gordon model mult SQRT(1 R)/(R G) 11.4763

21 Present value-net inc after 20002 as of 12/31/2002 5,051,106

22 Present value factor-5 years 0.447573

23 Pres value of net income after 2002 as of 1/1/98 2,260,738

24 FMV of equity-100% interest $3,404,686

27 Assumptions:

28 Net income before tax-1997 400,000

29 Income tax rate 40%

30 Discount rate-debt: pre-tax 10%

31 Discount rate-debt: after-tax 6%

32 Unlevered beta (from F46) 0.91

33 Risk free rate 6%

34 Equity premium 8%

35 Small company premium 3%

36 Equity discount rate R 17.443%

38 Calculation of Equity Discount Rate Using Comparables

40 Equity Unlevered

41 Beta Debt Equity D/E Beta

42 Guideline Company #1 1.15 454,646 874,464 52.0% 0.88

43 Guideline Company #2 1.20 146,464 546,454 26.8% 1.03

44 Guideline Company #3 0.95 46,464 705,464 6.6% 0.91

45 Guideline Company #4 0.85 52,646 846,467 6.2% 0.82

46 Totals or averages 1.04 700,220 2,972,849 23.55% 0.91

49 Capital Structure & Iterations

51 Interest-

52 Bearing Equity Before Relevered Equity FMV

53 t Debt Iteration D/E Beta Disc. Rate Equity

54 FMV debt, eqty at t 1 1 900,000 750,000 1.20 1.5668 21.534% 2,407,973

55 FMV debt, eqty at t 1 2 900,000 2,407,973 0.37 1.1152 17.921% 3,245,701

56 FMV debt, eqty at t 1 3 900,000 3,245,701 0.28 1.0625 17.500% 3,385,037

57 FMV debt, eqty at t 1 4 900,000 3,385,037 0.27 1.0562 17.450% 3,402,345

58 FMV debt, eqty at t 1 5 900,000 3,402,345 0.26 1.0555 17.444% 3,404,409

59 FMV debt, eqty at t 1 6 900,000 3,404,409 0.26 1.0554 17.443% 3,404,653

60 FMV debt, eqty at t 1 7 900,000 3,404,653 0.26 1.0554 17.443% 3,404,682

61 FMV debt, eqty at t 1 8 900,000 3,404,682 0.26 1.0554 17.443% 3,404,686

62 FMV debt, eqty at t 1 9 900,000 3,404,686 0.26 1.0554 17.443% 3,404,686

PART 2 Calculating Discount Rates

184

T A B L E 6-1C

Equity Valuation Approach with Iterations Beginning with Arbitrary Equity

A B C D E F G H

5 1998 1999 2000 2001 2002

6 Net inc before taxes 460,000 519,800 576,978 628,906 679,219

7 Growth rate in NIBT 15% 13% 11% 9% 8%

8 Income taxes (184,000) (207,920) (230,791) (251,562) (271,687)

9 Net inc after taxes 276,000 311,880 346,187 377,344 407,531

10 Present value factor 0.9228 0.7857 0.6690 0.5696 0.4850

11 Pres value NIAT $254,680 $245,045 $231,602 $214,952 $197,669

16 Final Valuation:

17 PV 1998“2002 net income $1,143,949

18 Constant growth rate in net income G 8%

19 Forecast net income-2003 440,134

20 Gordon model mult SQRT(1 R)/(R G) 11.4763

21 Present value-net inc after 20002 as of 12/31/2002 5,051,106

22 Present value factor-5 years 0.447573

23 Pres value of net income after 2002 as of 1/1/98 2,260,738

24 FMV of equity-100% interest $3,404,686

27 Assumptions:

28 Net income before tax-1997 400,000

29 Income tax rate 40%

30 Discount rate-debt: pre-tax 10%

31 Discount rate-debt: after-tax 6%

32 Unlevered beta (from F46) 0.91

33 Risk free rate 6%

34 Equity premium 8%

35 Small company premium 3%

36 Equity discount rate R 17.443%

38 Calculation of Equity Discount Rate Using Comparables

40 Equity Unlevered

41 Beta Debt Equity D/E Beta

42 Guideline Company #1 1.15 454,646 874,464 52.0% 0.88

43 Guideline Company #2 1.20 146,464 546,454 26.8% 1.03

44 Guideline Company #3 0.95 46,464 705,464 6.6% 0.91

45 Guideline Company #4 0.85 52,646 846,467 6.2% 0.82

46 Totals or averages 1.04 700,220 2,972,849 23.55% 0.91

49 Capital Structure & Iterations

51 Interest-

52 Bearing Equity Before Relevered Equity FMV

53 t Debt Iteration D/E Beta Disc. Rate Equity

54 FMV debt, eqty at t 1 1 900,000 5,000,000 1.18 1.0093 17.074% 3,538,676

55 FMV debt, eqty at t 1 2 900,000 3,538,676 0.25 1.0499 17.399% 3,420,038

56 FMV debt, eqty at t 1 3 900,000 3,420,038 0.26 1.0547 17.438% 3,406,499

57 FMV debt, eqty at t 1 4 900,000 3,406,499 0.26 1.0553 17.442% 3,404,901

58 FMV debt, eqty at t 1 5 900,000 3,404,901 0.26 1.0554 17.443% 3,404,712

59 FMV debt, eqty at t 1 6 900,000 3,404,712 0.26 1.0554 17.443% 3,404,689

60 FMV debt, eqty at t 1 7 900,000 3,404,689 0.26 1.0554 17.443% 3,404,687

61 FMV debt, eqty at t 1 8 900,000 3,404,687 0.26 1.0554 17.443% 3,404,686

62 FMV debt, eqty at t 1 9 900,000 3,404,686 0.26 1.0554 17.443% 3,404,686

CHAPTER 6 An Iterative Valuation Approach 185

Table 6-1C is identical to Table 6-1B except in the initial choice of value

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

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

high: as Table 6-1B is too low and Table 6-1C is too high, but they both

lead to the same FMV.

Convergence of the Equity Valuation Method

While rare, it can happen that the FMV diverges instead of converges. 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 when starting the next iteration as

opposed to using just the latest iteration of equity alone. This can be done

by making a small alteration to the spreadsheet.5

INVESTED CAPITAL APPROACH

Tables 6-2A and 6-2B are examples of the invested capital approach. They

are very similar to Table 6-1B for the equity valuation method, with the

following exceptions:

1. We determine earnings before interest but after taxes (EBIBAT)

as the income measure.6 This should be normalized EBIBAT.7

2. We discount EBIBAT using the WACC.

3. We must subtract the market value of debt from the calculated

market value of invested capital to get the market value of

equity.

4. We must calculate a new WACC for every new iteration of FMV

of equity.

5. We do not show the calculation of unlevered beta but will

assume that it has already been calculated to be 1.05.

Let™s illustrate this with a couple of examples.

Table 6-2A: Iterations Beginning with Book Equity

Earnings before interest and taxes (EBIT) in 1997, the previous year, was

$600,000 (cell D28). We assume a declining growth rate in earnings as

before: 15% (B6) in 1998, 13% (C6) in 1999, ¬nishing with 8% (F6) in 2002.

We use these growth rates to forecast EBIT in 1998“2002. Subtracting 40%

for income taxes, we arrive at earnings before interest, but after taxes

(EBIBAT) of $414,000 in 1998 (B8), rising to $611,297 in 2002 (F8). The

growth rates in EBIBAT are identical to those for EBIT because we assume

a constant 40% income tax (D29). The last row of the top section is the

5. Change the formula in D55, which previously was H55, to AVERAGE(D54,H54). Then copy

the formula down Column D.

6. It is better to use cash ¬‚ow (before interest but after taxes), but for simplicity we use EBIBAT.

7. This does not necessarily correspond to the NIBT in Tables 6-1A, 6-1B, and 6-1C, because we are

dealing with a different hypothetical company.

PART 2 Calculating Discount Rates

186

T A B L E 6-2A

WACC Approach with Iterations Beginning with Book Equity

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 2,000,000 800,000 2,800,000 71.4% 28.6% 30.000% 12.857% 7,776,091

44 FMV debt, eqty at t 1 2 2,000,000 7,776,091 9,776,091 20.5% 79.5% 18.696% 16.099% 3,927,835

45 FMV debt, eqty at t 1 3 2,000,000 3,927,835 5,927,835 33.7% 66.3% 19.966% 15.254% 4,599,240

46 FMV debt, eqty at t 1 4 2,000,000 4,599,240 6,599,240 30.3% 69.7% 19.592% 15.473% 4,411,165

47 FMV debt, eqty at t 1 5 2,000,000 4,411,165 6,411,165 31.2% 68.8% 19.685% 15.416% 4,458,814

48 FMV debt, eqty at t 1 6 2,000,000 4,458,814 6,458,814 31.0% 69.0% 19.661% 15.431% 4,446,410

49 FMV debt, eqty at t 1 7 2,000,000 4,446,410 6,446,410 31.0% 69.0% 19.667% 15.427% 4,449,617

50 FMV debt, eqty at t 1 8 2,000,000 4,449,617 6,449,617 31.0% 69.0% 19.665% 15.428% 4,448,787

51 FMV debt, eqty at t 1 9 2,000,000 4,448,787 6,448,787 31.0% 69.0% 19.666% 15.428% 4,449,002

52 FMV debt, eqty at t 1 10 2,000,000 4,449,002 6,449,002 31.0% 69.0% 19.666% 15.428% 4,448,946

53 FMV debt, eqty at t 1 11 2,000,000 4,448,946 6,448,946 31.0% 69.0% 19.666% 15.428% 4,448,960

54 FMV debt, eqty at t 1 12 2,000,000 4,448,960 6,448,960 31.0% 69.0% 19.666% 15.428% 4,448,957

55 FMV debt, eqty at t 1 13 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 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

compiled by Partnership Pro¬les, Ltd.

2. My regression analysis of private fractional interest data.

Thus, in Chapter 10 we use three models for calculating the fractional

interest discount: the economic components model, the partnership pro-

¬les database regression, and the private data regression.

If any chapter may have rough edges to it, Chapter 7 is it. I hope to

be able to smooth those edges in future editions of this book. For now,

however, this chapter will have to remain as it is.

The calculation of the discount for lack of control in Chapter 9 is also

subject to further research and revision. Nevertheless, this is valuable and

novel material well worth the struggle through the quantitative parts.

I caution the reader not to get bogged down in the quantitative parts

of Chapter 7. Read it through lightly ¬rst for understanding the gist, and

do not worry about understanding every statistic in the academic articles.

The most important thing to get out of Chapter 7 in a ¬rst reading is an

understanding of why the acquisition premium data that we have been

using for the past 30 years tell us almost nothing useful about the value

of control of a private ¬rm and why we have to look elsewhere. It is then

worth a second reading to master the technical details.

PART 3 Adjusting for Control and Marketability 193

CHAPTER 7

Adjusting for Levels of Control

and Marketability

INTRODUCTION

THE VALUE OF CONTROL AND ADJUSTING FOR LEVEL OF

CONTROL

Prior Research”Qualitative Professional

Nath

Mercer (1990)

Bolotsky

Jankowske

Roach

Mercer (1998) and (1999)

Summary of Professional Research on Control Premiums

Prior Research”Academic

Schwert (1996)

Lease, McConnell, and Mikkelson (1983)

Megginson (1990)

My Conclusions from the Megginson Results

My Analysis of the Megginson Results

The Houlihan Lokey Howard & Zukin (HLHZ) Study

International Voting Rights Premia

Bradley, Desai, and Kim (1988)

Maquieira, Megginson, and Nail (1998)

Other Corporate Control Research

Menyah and Paudyal

My Synthesis and Analysis

Decomposing the Acquisition Premium

Inferences from the Academic Articles

The Disappearing Control Premium

The Control Premium Reappears

Estimating the Control Premium

DLOC

195

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DISCOUNT FOR LACK OF MARKETABILITY (DLOM)

Mercer™s Quantitative Marketability Discount Model

Kasper™s BAS Model

Restricted Stock Discounts

Regression of MPI Data

Using the Put Option Model to Calculate DLOM of Restricted

Stock

Annualized Standard Deviation of Continuously Compounded

Returns

Calculation of the Discount

Table 7-8: Black“Scholes Put Model Results

Comparison of the Put Model and the Regression Model

Empirical versus Theoretical Black“Scholes

Comparison to the Quantitative Marketability Discount Model

(QMDM)

Abrams™ Economic Components Model

Component #1: The Delay to Sale

Psychology

Black“Scholes Options Pricing Model

Other Models of Component #1

Abrams Regression of the Management Planning, Inc. Data

Limitations of the Regression

Component #2: Buyer Monopsony Power

Component #3: Transactions Costs

Table 7-11: Quantifying Transactions Costs for Buyer and Seller

Component #3 Is Different than #1 and #2

Developing Formulas to Calculate DLOM Component #3

A Simpli¬ed Example of Sellers™ Transactions Costs

Tables 7-12 and 7-13: Proving Formulas (7-9) and (7-9a)

Value Remaining Formula and the Total Discount

Table 7-14: Sample Calculation of DLOM

Evidence from the Institute of Business Appraisers

Mercer™s Rebuttal

Expected Growth and Expected Returns

Conclusion

My Counterpoints

Mercer™s Response

Conclusion

MATHEMATICAL APPENDIX

PART 3 Adjusting for Control and Marketability

196

INTRODUCTION

Adjusting for levels of control and marketability is a complicated and

very important topic. We will be discussing control premiums (CP), their

opposite, discount for lack of control (DLOC), and discount for lack of

marketability (DLOM).

Historically, these valuation adjustments have accounted for substan-

tial adjustments in appraisal reports”often 20“40% of the net present

value of the cash ¬‚ows”and yet valuation analysts may spend little to

no time calculating and explaining these adjustments.

This is a long chapter, with much data and analysis. It will be helpful

to break the discussion into two parts. The ¬rst part will deal with pri-

marily with control and the second part primarily with marketability. I

say primarily, because the two concepts are interrelated. The level of con-

trol of a business interest impacts its level of marketability. Therefore, it

is logical to begin with a discussion of control. Because of the interrela-

tionship, two academic articles that we will discuss in the section on

control relate more to marketability, yet they ¬t in better in the control

discussion.

THE VALUE OF CONTROL AND ADJUSTING FOR LEVEL

OF CONTROL

We will begin our analysis of the effects of control on value by reviewing

prior qualitative professional research and prior academic research. Then

we will present some additional data and come to some conclusions

about the magnitude of control premiums and DLOC.

The top portion of Figure 7-1 shows the traditional level of values

chart.3 The conventional wisdom represented in the traditional levels of

value chart holds that it is appropriate to add a control premium to the

F I G U R E 7-1

Traditional Levels of Value Chart

Level of Value Adjustment up To Adjustment Down To

Control interest Control premium NA

Marketable minority interest Reverse out DLOM DLOC

Private Minority Interest NA DLOM

Mercer™s Mercer (1998) modi¬ed traditional levels of value chart

Strategic value Value of synergies NA

Control value Control premium Eliminate synergies

Marketable minority interesta Reverse out DLOM DLOC

Private minority interest NA DLOM

a

Often referred to in the literature as the ˜˜as-if-freely-traded-value™™ for private ¬rms.

3. The bottom portion shows Chris Mercer™s modi¬ed traditional levels of value chart, which is

identical to the one above, except with the addition of the strategic value. We will cover this

later in the chapter.

CHAPTER 7 Adjusting for Levels of Control and Marketability 197

marketable minority interest value. There are signi¬cant opinions to the

contrary, i.e., that one should not add any control premium whatsoever.

Additionally, there is controversy over the appropriate magnitude of the

control premium among those who do add them to the marketable mi-

nority interest value. We will cover that in greater depth later in the chap-

ter. Of course, if the valuation method is a guideline company approach

using a database of sales of privately held ¬rms, the starting value is a

private control interest, and a control premium is inappropriate.

It is extremely important to understand that the valuation adjust-

ments in Figure 7-1 be appropriate to the valuation method used. If we

are valuing a control interest and we used a discounted cash ¬‚ow analysis

with discount rates calculated using New York Stock Exchange data, the

resulting value is a marketable minority interest, and a control premium

must be considered.4

The alternative levels of value chart is two tiered, i.e., it is a 2 2

chart (2 rows and 2 columns, versus the traditional chart, which is 3

1). It represents the four basic types of ownership interests, which are

combinations of public versus private and control versus minority inter-

est. Obviously, there are shades of gray in between the extremes. Bolotsky

(1991) was the ¬rst to propound this chart, although he used it for slightly

different purposes, which we discuss below. Much later in the chapter,

we will discuss Figure 7-3, which is my own extension of Bolotsky™s levels

of value chart to a 3 2 chart.

The traditional sources of control premiums are the Mergerstat and

the Houlihan Lokey Howard & Zukin (HLHZ) studies.5 Table 7-1, col-

umns B and C show Mergerstat™s compilation of average (mean) and

median ¬ve-day acquisition premiums from 1985“1997. The premiums

were measured as (POffer/P5Day) 1, where the numerator is the offering

price and the denominator is the minority trading price ¬ve days before

the announcement of the offer. Mean acquisition premiums have ranged

from 35“45%, with the average being 39.5% (B21), while median premi-

ums have ranged from 27“35%, with an average of 30.5% (C21).

F I G U R E 7-2

Two-Tiered Levels of Value Charta

Public Private

Control x x

Minority x x

a

Note: these are also the four basic types of ownership interests.

4. It is also important to make sure the measure of income is consistent with the interest valued.

When valuing a control interest, it is appropriate to add back excess salaries of the owners.

When valuing a minority interest that cannot force salaries lower, the add-back is

inappropriate.

5. HLHZ now owns Mergerstat, although the latter was previously owned by Merrill Lynch and

the W. T. Grimm Co.

PART 3 Adjusting for Control and Marketability

198

T A B L E 7-1

Synergies as Measured by Acquisition Minus Going-Private Premiums

A B C D E F G

5 Acquisition Going Private Difference

Premiums [1] Prem [2] Synergy?

6 Mean Median Mean Median Mean Median

7 1985 37.1% 27.7% 30.9% 25.7% 6.2% 2.0%

8 1986 38.2% 29.9% 31.9% 26.1% 6.3% 3.8%

9 1987 38.3% 30.8% 34.8% 30.9% 3.5% 0.1%

10 1988 41.9% 30.9% 33.8% 26.3% 8.1% 4.6%

11 1989 41.0% 29.0% 35.0% 22.7% 6.0% 6.3%

12 1990 42.0% 32.0% 34.3% 31.6% 7.7% 0.4%

13 1991 35.1% 29.4% 23.8% 20.0% 11.3% 9.4%

14 1992 41.0% 34.7% 24.8% 8.1% 16.2% 26.6%

15 1993 38.7% 33.0% 34.7% 20.0% 4.0% 13.0%

16 1994 41.9% 35.0% 41.9% 35.0% 0.0% 0.0%

17 1995 44.7% 29.2% 29.8% 19.2% 14.9% 10.0%

18 1996 36.6% 27.3% 34.8% 26.2% 1.8% 1.1%

19 1997 35.7% 27.5% 30.4% 24.5% 5.3% 3.0%

20 1998 40.7% 30.1% 29.1% 20.4% 11.6% 9.7%

21 Mean 39.5% 30.5% 32.1% 24.1% 7.4% 6.4%

[1] Mergerstat-1999, Chart 1-8, Page 23 (Mergerstat-1994, Figure 41, Page 98 for 1985-1988). Mergerstat is a division of Houlihan

Lokey Howard & Zukin.

[2] Mergerstat-1999, Table 1-39, Page 42. For 1985-1988, Mergerstat-1994, Figure 39, Page 96.

Note that we deliberately use the term acquisition premium instead of

the more common term control premium. Eventually we will distinguish

between the amounts that are paid for control versus the amounts that

are paid for synergies, as the latter are generally part of investment value

and not fair market value.

The mean and median going-private premiums are 32.1% and 24.1%

(D21 and E21), and the difference of means and medians of acquisition

versus going private premiums are 7.4% and 6.4%, respectively. At a ¬rst

glance, it would seem that the 7.4% mean or 6.4% median difference is a

potential measure of synergies, as going private transactions do not have

synergies. Also, the 32.1% mean or 24.1% median going private premium

would be good candidates as a benchmark for measuring control pre-

miums.

Table 7-1A shows additional detail of the premiums as measured by

different points in time. An n-day acquisition premium is equal to

PAnnouncement

1, where the denominator is the stock price n days before

Pn

the announcement date. It is clear that acquisition premiums increase

with an increase in n, as can be seen by moving to the right across any

row. Rows 6 and 7 show median and mean acquisition premiums for

ordinary acquisitions, while columns B through D in rows 10 and 11 show

the same premiums with the potential 7.4% synergies from Table 7-1, F21

subtracted. These are a net acquisition premium for ordinary acquisitions

CHAPTER 7 Adjusting for Levels of Control and Marketability 199

T A B L E 7-1A

Acquisition and Going-Private Transactions Premiums

A B C D F G H J K L

Going Private

4 Ordinary Acquisitions Transactions Difference

5 Gross Acquisition Premiums 1 Day 5 Days 30 Days 1 Day 5 Days 30 Days 1 Day 5 Days 30 Days

6 Median 22.8% 28.4% 36.7%

7 Mean (with max. of 100%) [1] 27.2% 32.9% 42.7%

9 Premiums net of 7.4% synergies

10 Median 15.4% 21.0% 29.3% 20.0% 23.0% 26.9% 4.6% 2.0% 2.4%

11 Mean (with max. of 100%) [1] 19.8% 25.5% 35.3% 25.7% 27.2% 31.7% 5.9% 1.7% 3.6%

[1] All premiums 100% treated as 100%

Data Source: Mergerstat database. This contains going private premiums from 3/89 to 5/98 and ordinary acquisition premiums from 12/83 to 1/99. There are 46 to 69 going private

premiums and 1,175 to 1,430 ordinary premiums.

that is a candidate for the control premium. Columns F through H in

rows 10 and 11 are the going private transactions, and columns J through

L are the difference of the net acquisition premiums for ordinary acqui-

sitions and the going private premiums. Notice that only the one-day

differences at 4.6% and 5.9% (J10 and J11) are signi¬cant in size, while

the ¬ve-day and thirty-day differences are quite small. Again it seems

that going private premiums are a strong contender for the measure of

the value of control.

The traditional calculation of discount for lack of control (DLOC) is

based on the control premium. If the marketable minority FMV is $100

per share and one buys control for $140 per share, the control premium

(CP) is $40 per share. In percentages, the premium is $40 per share di-

vided by the marketable minority price of $100 per share, or $40/$100

40%. Going in the other direction, DLOC is the $40 premium divided by

the control price of $140, or 28.6%. Symbolically, DLOC CP/(1 CP).

The vast majority of valuation assignments for valuation profession-

als call for a fair market value standard of value. Unless a market is

dominated by strategic buyers and the subject company is a reasonable

candidate to be bought by a strategic buyer in the mergers & acquisitions

(M&A) market, it is necessary to remove any synergistic element in ac-

quisition premiums before applying a control premium. The data are con-

fusing, and there are different ideological camps in the valuation profes-

sion. The goal of the control section of this chapter is to present a large

body of professional and academic research, arrive at a coherent expla-

nation of the diverse data, and provide guidance and quantitative bench-

marks for use in the profession.

Prior Research”Qualitative Professional

As mentioned earlier, we examine two types of prior research: profes-

sional and academic. In this section we examine prior professional re-

search on control premiums. The professional research itself is composed

of a long series of articles that develop important valuation theory that

PART 3 Adjusting for Control and Marketability

200

is primarily qualitative. We will now review the main articles, which are

written by Eric Nath, Chris Mercer, Michael Bolotsky, and Wayne Jan-

kowske. Again, because control and marketability are so intertwined,

these articles also contain material relevant to adjusting for marketability.

Nath

The original attack on the traditional position came from Eric Nath (1990).

Nath later clari¬ed and slightly modi¬ed his initial position (Nath 1994

and 1997). Nath argued:

— Fewer than 4% of all public ¬rms are taken over each year.

Using an ef¬cient markets hypothesis argument, Nath said that

the LBO funds, strategic buyers, and their bankers, who

collectively represent hundreds of billions of dollars scouring the

market for deals, keep the market clean. Any good takeover

opportunity will not last long. If there were hidden premiums in

the ¬rms, their stock prices would rapidly be bid up to that

level.

— Minority shares in publicly held ¬rms are liquid. The existence of

liquidity tends to eliminate nonstrategic acquisition premiums if

the companies are well managed and management

communicates effectively with investors (I would add that they

must be benevolent to minority interests, which is the usual case

in publicly held ¬rms and is not usual in privately held ¬rms).

— The previous points lead to the conclusion that the publicly

traded prices are control values and not just minority values. His

major conclusion, which contradicts conventional wisdom of the

three-tiered levels of value chart, is that starting from a public

market derived value, one must take both DLOM and DLOC to

value a privately held minority interest. Apparently in¬‚uenced

by articles from Bolotsky and Jankowske, both discussed below,

Nath later (1997) switched to the two-tiered levels of value

structure. Doing so had no material effect on his conclusions,

merely the presentation.

— Buyers are often strategically motivated, and therefore what they

pay is not equivalent to FMV. Nath™s evidence is that similar

premiums are paid for minority interest acquisitions. More

recently, his position has modi¬ed. Nath is concerned whether

the market of relevant buyers for a subject company is likely to

consist of many strategic buyers who would participate in an

auction for the company. If so, then he contends that strategic

value essentially becomes fair market value. If there are not

many strategic buyers, then he is still concerned that the M&A

multiples may contain a strategic element in the acquisition

premium, leading to overvaluation of the company unless that

element is removed. He determines this by an analysis of three

entities: the company itself, the market for ¬rms in that industry,

and the M&A databases.

— Several problems with the computations of control premiums

cause them to be misleading. Mergerstat™s control premium

statistics exclude acquisition discounts, i.e., some acquisitions

CHAPTER 7 Adjusting for Levels of Control and Marketability 201

occur at lower prices than the minority trading price. Other

problems are that the range of premiums is enormous and

Mergerstat uses simple averages instead of market weighted

averages.

— There are at least two additional problems in using takeover

prices as an indicator of value. The ¬rst is that transactions are

unique and time-speci¬c. Just because a speci¬c buyer pays a

speci¬c premium for a particular ¬rm, that does not mean that

another buyer would pay a similar premium for a comparable

¬rm, let alone for a much smaller and less exciting company. The

second problem, Nath contends, is that some people overpay.

These points are related to the comments above on the strategic

element of acquisition premiums in the M&A market.

Mercer (1990)

In his initial article on the topic, Mercer (1990) disagreed with Nath.6

Mercer was the most notable proponent of the traditional viewpoint and

levels of value chart. He disagreed with Nath™s belief that ¬rms that are

taken over are different than those that are not, which led him to disagree

with Nath™s conclusion that the public minority price is a control value.

Bolotsky (1991) said that that is a matter of opinion and cannot be tested.

Mercer also contested Nath™s statement that most takeovers are fully

or partially motivated by strategic reasons and that this makes the trans-

action prices unsuitable measures of FMV. Essentially, Mercer said that

buyer motivations are irrelevant and that it is not up to us to question

their motivations”just to use the data generated by their actions.

He also wrote that premiums paid for minority interests are premi-

ums paid for creeping control, not for synergies. Bolotsky agreed with

Mercer on this point.

Mercer has since changed his views considerably, and we will cover

his 1998 article separately in this professional review.

Bolotsky7

Michael Bolotsky (1991) and (1995) agrees with many of Nath™s criticisms

of conventional wisdom but disagrees with his conclusions. With regard

to the results of his analysis, he represents a middle position between

Mercer and Nath. With regard to the theoretical underpinnings, his work

is unique in that it is the ¬rst article that abandons the linear levels of

value concept entirely and replaces it with a multifactor, multidimension

matrix of fundamental attributes. For example, he gets rid of the concept

that 100% ownership value must always be somehow higher than or

equal to minority ownership value. He contends that both Mercer and

Nath are still arguing around a linear concept of going from ˜˜up here™™

to ˜˜down there.™™

Bolotsky has a comprehensive, logical framework of analysis that

includes differences in ownership rights, liquidity, information access,

6. He has since changed his views considerably, and we will cover his 1998 article later.

7. I thank Michael Bolotsky for editing this section and helping me to interpret his work correctly.

PART 3 Adjusting for Control and Marketability

202

and information reliability between the four types of ownership interests

listed in Figure 7-2. Bolotsky™s article is important theoretical work and

obviously in¬‚uenced subsequent articles by both Nath and Mercer. Bol-

otsky™s article contains no empirical evidence nor any attempt to quantify

the implications of his framework into an economic model. The practical

signi¬cance of the article is that he disagrees with Nath™s conclusion that

valuing a private minority interest with reference to public minority in-

terests as a starting point requires applying both a DLOM and a DLOC.

It is signi¬cant that Bolotsky did not attempt to squash four levels

of value (public-control, public-minority, private-control, and private-

minority) into three, as both Nath and Mercer did.8 Bolotsky™s assertion

that the more the buy side knows about the seller and the more he or

she can rely on it, the higher the price, is also signi¬cant and logical.

Bolotsky characterized the public markets as a consensus opinion of

value that may occasionally experience an anomalous trade, but that trade

will be quickly bid back to a rational, equilibrium consensus value. He

says:

[T]he purchase of an entire company is typically a one-time purchase of a

unique item; the price that ultimately gets recorded is not the consensus

opinion of the limited group of buyers and sellers for a particular entire

company but is rather the winning bid, which is normally the highest bid.

There is no ˜˜market™™ process going on here in the sense described above

for public minority blocks. It is analogous to a situation where the single

anomalous trade described earlier does not get rapidly bid down to a con-

sensus price; instead, it gets memorialized in Mergerstat Review, to be re-

lied upon by valuation consultants. Clearly, relative to fair market value,

there is an upwards bias in prices that represent either the highest bid, the

only bid, or the bid of a buyer bringing special attributes to the table.

Bolotsky takes a middle position on whether the takeovers are for

typical or atypical public companies, the former position being taken by

Mercer and the latter by Nath. Bolotsky says that it is inappropriate to

insist that unless a subject company is in play, one must assume there is

no control premium. He thus disagrees with Nath on that point.

Bolotsky™s theoretical framework has no concept of sequential levels

of value, with control value at the top, followed by minority marketable

value, and nonmarketable minority value at the bottom. Rather, Bolotsky

advances the concept that the value of various types of ownership inter-

ests is the result of building up the contribution to value of fundamental

ownership attributes, to the degree that each attribute applies to the in-

terest in question. In addition, Bolotsky™s framework implies that rather

than discounts and premiums, there are adjustments for differences in own-

ership attributes and that the adjustment can be positive, negative, or

zero. In this framework there is nothing that mandates that a 100% own-

8. Nath stopped doing that in his December 1997 article. Until fairly recently, Mercer believed that

there are only three levels of value, as he contends that there is no discount for lack of

marketability (DLOM) for private control interests, as the private control interest has control

over cash ¬‚ows (I disagree that control over cash ¬‚ows eliminates DLOM and will cover

that later in the chapter). He has more recently added a strategic level of value, as shown in

the lower section of Figure 7-1, but it is still linear, i.e., a single column of values.

CHAPTER 7 Adjusting for Levels of Control and Marketability 203

ership position will be equal to or greater in value than a public minority

price; indeed, Bolotsky™s framework implies that if investors in a security

value liquidity and the options that liquidity provides to a greater degree

than they value power, then public minority pricing for that security will

exceed 100% ownership pricing.

Thus, Bolotsky says that for those public companies where we would

conclude that the per share 100% ownership value is the same amount

as the public minority value, the two prices might be the same but for

very different reasons. In effect, the same price for different ownership

interests is resulting from the net of the differences in the impact of var-

ious ownership attributes on each interest. Since the concept is of the net

of differences, there is no reason why the net difference in price between

a 100% ownership position and a public minority position cannot be zero

or even negative. Accordingly, when we state that the public minority

value and the 100% ownership value are the same, we are really saying

that we should apply a net value adjustment of zero. I agree with his

position that there is a very important distinction in saying we are ap-

plying a net zero premium versus saying that by de¬nition there is no

premium.

Bolotsky also states that Nath™s conclusion is internally ¬‚awed in that

in valuing private minority interests with reference to public minority

prices as a starting point we need to take both a DLOM and a DLOC. He

argues that if a public minority block of shares happens to have the same

per-share value as a 100% ownership interest, this does not affect the fact

that the block in question is still a minority block of stock having no

attributes of control over the company. He contends it would be illogical

to subtract a DLOC from a block that has no attributes of control.9 Rather,

the oftentimes extreme price differentials between public and private mi-

nority interests must be explained by other ownership attributes besides

control, including but not limited to differentials in relative liquidity, rel-

ative level of information availability, and relative information reliability.

Bolotsky claims”reasonably, in my opinion”that there are many

public ¬rms whose perceived 100% ownership value will be more than

their minority value, but not enough more to make a tender offer worth-

while. In addition, Bolotsky™s theoretical framework is the only one that

can readily accommodate several market features that appear anomalous

when relying on the linear levels of value framework, such as 100% own-

ership pricing at levels considerably below IPO pricing for many com-

panies in today™s markets.

Jankowske

Wayne Jankowske™s article (Jankowske 1991) corrects certain key errors

in the articles by Nath and Bolotsky. He says one does not have to accept

Nath™s assertion that the marketable minority value is a control value to

accept the proposition that DLOC can differ between public and private

9. This is a very logical statement and appears to be self-evident. Nevertheless, I will disagree with

this later in the chapter.

PART 3 Adjusting for Control and Marketability

204

¬rms. He says differences in legal and contractual protection, agency

costs, relative incentives, and differential economic bene¬ts can account

for differences in the public versus private DLOC.

Prevailing wisdom™s assertion is that since public market prices are

minority prices, we can use public guideline company prices to value

private minority shares, with only DLOM necessary. Jankowske says that

for that to be true, it implies that the economic disadvantages of lack of

control associated with public minority shares is equal to that of private

minority shares, which is unrealistic.

Conceptually, the magnitude of DLOC in guideline public prices

makes no difference, whether it is 30%, or, as Nath contended in his ¬rst

two articles, 0%. The difference between the public and subject company™s

DLOC must be recognized to avoid an overvaluation.

He developed the following formula to value a private minority in-

terest:10

FMVMM

Additional DLOC (DLOCSC DLOCGC)

1 DLOCGC

where

FMVMM the marketable minority fair market value

DLOCGC discount for lack of control in the public guideline companies

DLOCSC discount for lack of control in the subject company

He gave the following example: FMVMM, the marketable minority interest

value is $900; DLOCGC, the discount for lack of control implicit in the

public minority stock is 10%; and DLOCSC, the discount for lack of control

appropriate to the subject company, is 40%. His calculation of incremental

discount is:

$900

(40% 10%) $1,000 30% $300

1 10%

He disagrees with Bolotsky that the guideline ¬rms must have iden-

tical shareholder attributes.

In his second article on the topic (Jankowske 1995), he stressed that

it is the economic bene¬ts to which we must look as a justi¬cation for

control premiums, not the powers that come with control. He cites the

following economic bene¬ts of control:

— Company level.

— Performance improvements.

— Synergy.

— Shareholder Level.

— Wealth transfer opportunities”the Machiavellian ability to

transfer wealth from the minority shareholders.

— Protection of investment”the ¬‚ip side of the above point is

that control protects the shareholder from being exploited. This

10. I have changed his notation.

CHAPTER 7 Adjusting for Levels of Control and Marketability 205

motivation is important because it relates to ambiguity

avoidance in the academic literature reviewed in this chapter.

— Liquidity”control enhances liquidity in privately held

businesses.

Of the company level advantages, the extent to which performance

improvements on a standalone basis account for control premiums prop-

erly belongs in our calculations of fair market value. That portion ac-

counted for by synergy is investment value and should not be included

in fair market value.

Roach

George Roach (1998) summarized percentage acquisition premiums from

a database of business sales. The premiums were measured as (PAcq/P5Day)

1, where the numerator is the acquisition price and the denominator

is the minority trading price ¬ve days before the announcement of the

acquisition. He also provided the premium based on the price 30 days

prior. We excerpt from his Exhibit IV to our Table 7-2.

There is no pattern to the ¬rst three premiums listed in Table 7-2.

The 50 SIC code difference level between buyers and sellers has the

highest premium, which is counterintuitive. Roach found similar patterns

in the results for median premiums. Additionally, while the 30-day pre-

miums were higher than the 5-day premiums, the patterns were similar.

Under the assumption that acquisitions of ¬rms in the same or al-

most the same SIC code are more likely to be strategic acquisitions than

¬rms acquired in very different SIC codes, Roach™s analysis appears to be

strong evidence that premiums paid for strategic buys are no larger than