28 21 21,247 6,002 15,245 704,974 0.8114 12,370

29 22 21,247 5,875 15,372 689,602 0.8034 12,350

30 23 21,247 5,747 15,500 674,102 0.7954 12,330

31 24 21,247 5,618 15,630 658,472 0.7876 12,309

32 25 21,247 5,487 15,760 642,712 0.7798 12,289

33 26 21,247 5,356 15,891 626,821 0.7720 12,269

34 27 21,247 5,224 16,024 610,798 0.7644 12,248

35 28 21,247 5,090 16,157 594,641 0.7568 12,228

36 29 21,247 4,955 16,292 578,349 0.7493 12,208

37 30 21,247 4,820 16,427 561,922 0.7419 12,188

38 31 21,247 4,683 16,564 545,357 0.7346 12,168

39 32 21,247 4,545 16,702 528,655 0.7273 12,148

40 33 21,247 4,405 16,842 511,813 0.7201 12,128

41 34 21,247 4,265 16,982 494,831 0.7130 12,108

42 35 21,247 4,124 17,123 477,708 0.7059 12,088

43 36 21,247 3,981 17,266 460,442 0.6989 12,068

44 37 21,247 3,837 17,410 443,032 0.6920 12,048

45 38 21,247 3,692 17,555 425,476 0.6852 12,028

46 39 21,247 3,546 17,701 407,775 0.6784 12,008

47 40 21,247 3,398 17,849 389,926 0.6717 11,988

48 41 21,247 3,249 17,998 371,928 0.6650 11,968

49 42 21,247 3,099 18,148 353,781 0.6584 11,949

PART 1 Forecasting Cash Flows

110

T A B L E A3-5 (continued)

Present Value of a Loan at Discount Rate Different than Nominal Rate

A B C D E F G

5 Pmt

6 # Pmt Int Prin Bal PVF (r1) PV(P)

50 43 21,247 2,948 18,299 335,482 0.6519 11,929

51 44 21,247 2,796 18,451 317,031 0.6454 11,909

52 45 21,247 2,642 18,605 298,425 0.6391 11,890

53 46 21,247 2,487 18,760 279,665 0.6327 11,870

54 47 21,247 2,331 18,917 260,749 0.6265 11,850

55 48 21,247 2,173 19,074 241,675 0.6203 11,831

56 49 21,247 2,014 19,233 222,442 0.6141 11,811

57 50 21,247 1,854 19,393 203,048 0.6080 11,792

58 51 21,247 1,692 19,555 183,493 0.6020 11,772

59 52 21,247 1,529 19,718 163,775 0.5961 11,753

60 53 21,247 1,365 19,882 143,893 0.5902 11,734

61 54 21,247 1,199 20,048 123,845 0.5843 11,714

62 55 21,247 1,032 20,215 103,630 0.5785 11,695

63 56 21,247 864 20,383 83,247 0.5728 11,676

64 57 21,247 694 20,553 62,693 0.5671 11,656

65 58 21,247 522 20,725 41,969 0.5615 11,637

66 59 21,247 350 20,897 21,071 0.5560 11,618

67 60 21,247 176 21,071 0 0.5504 11,599

68 Total 1,274,823 274,823 1,000,000 730,970

70 Assumptions:

72 Prin 1,000,000

73 Int 10.0000%

74 Int Mo r 0.8333%

75 Int 12.0000%

76 Int Mo r1 1.0000%

77 Years 5

78 Months n 60

79 Pymt 21,247

80 Start month S 3

81 (1/(r1 r))*((1/(1 r) n) (1/(1 r1) n))*PYMT 730,970

Present Value of the Principal when the Discount Rate is

Different than the Nominal Rate

When valuing a loan at a discount rate, r1, that is different than the nom-

inal rate of interest, r, the present value of principal is as follows:

1 1 1

r)n r)n 1 r)n 2

(1 (1 (1

PV (Amort)

r1)2 r1)3

1 r1 (1 (1

1

1 r

... Pymt (A3-25)

r1)n

(1

We can move the second denominator into the ¬rst to simplify the equa-

tion:

CHAPTER 3 Annuity Discount Factors and the Gordon Model 111

1 1

PV (Amort)

r)n(1 r)n 1(1 r1)2

(1 r1) (1

(A3-26)

1

... Pymt

r1)n

(1 r)(1

Multiplying both sides by (1 r)/(1 r1), we get:

1 r 1 1

PV (Amort) n1 2 n2

r1)3

1 r1 (1 r) (1 r1) (1 r) (1

1

... Pymt (A3-27)

r1)n

(1 r)(1

Subtracting equation (A3-27) from equation (A3-26) and simplifying, we

get:

r1 r 1 1

PV (Amort) Pymt

r)n(1 r1)n

1 r1 (1 r1) (1 r)(1

(A3-28)

This simpli¬es to:

1 1 1

PV (Amort) Pymt (A3-29)

r)n r1)n

r1 r (1 (1

Table A3-5 is almost identical to Section 1 of Table A3-3. We use a

nominal interest rate of 10% per year (B73), which is 0.8333% per month

(B74), and a discount rate of 12% per year (B75), or 1% per month (B76).

We discount the principal amortization at r1, the discount rate of 1%,

in Column F, so that Column G gives us the present value of the principal,

which totals $730,970 (G68). The Excel formula equivalent for equation

(A3-29) appears in cell A81, and the result of that formula appears in

G81, which matches the brute force calculation in G68, thus demonstrat-

ing the accuracy of the formula.

CONCLUSION

In this mathematical appendix to the ADF chapter, we have presented:

— ADFs with stub periods (partial years) for both midyear and

end-of-year.

— Tables to demonstrate their accuracy.

— ADFs to calculate the amortization of principal on a loan.

— A formula for the after-tax PV of a loan.

PART 1 Forecasting Cash Flows

112

PART TWO

Calculating Discount Rates

Part 2 of this book, Chapters 4, 5, and 6, deals with calculating discount

rates; discounting cash ¬‚ows is the second of the four steps in business

valuation.

Chapter 4 is a long chapter, with a signi¬cant amount of empirical

analysis of stock market returns. Our primary ¬nding is that returns are

negatively related to the logarithm of the size of the ¬rm. The most suc-

cessful measure of size in explaining returns of publicly held stocks is

market capitalization, though research by Grabowski and King shows

that many other measures of size also do a fairly good job of explaining

stock market returns.

In their 1999 article, Grabowski and King found the relationship of

return to three underlying variables: operating margin, the logarithm of

the coef¬cient of variation of operating margin, and the logarithm of the

coef¬cient of variation of return on equity. This is a very important re-

search result, and it is very important that professionals read and under-

stand their article. Even so, their methodology is based on Compustat

data, which leaves out the ¬rst 37 years of the New York Stock Exchange

data. As a consequence, their standard errors are higher than my log size

model, and appraisers should be familiar with both.

In this chapter, we:

— Develop the mathematics of potential log size equations.

— Analyze the statistical error in the log size equation for different

time periods and determine that the last 60 years, i.e., 1939“1998,

is the optimal time frame.

— Present research by Harrison that shows that the distribution of

stock market returns in the 18th century is the same as it is in

the 20th century and discuss its implications for which 20th

century data we should use.

— Give practical examples of using the log size equation.

— Compare log size to the capital asset pricing model (CAPM) for

accuracy.

— Discuss industry effects.

113

Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

— Discuss industry effects.

— Present a claim that, with rare exceptions, valuations of small

and medium-sized privately held businesses do not require a

public guideline companies method (developing PE and other

types of multiples), as the log size model satis¬es the intent

behind the Revenue Ruling 59-60 requirement to use that

approach when it is relevant.

The last bullet point is very important; in my opinion, it frees ap-

praisers from wasting countless hours on an approach that is worse than

useless for valuing small ¬rms.1 The log size model itself saves much

time compared to using CAPM. The former literally takes one minute,

while the latter often requires one to two days of research. Log size is

also much more accurate for smaller ¬rms than is either CAPM or the

buildup approach. Using 1939“1998 data, the log size standard error of

the valuation estimate is only 41% as large as CAPM standard error. This

means that the CAPM 95% con¬dence intervals are approximately two

and one half times larger than the log size con¬dence intervals.2

Summarizing, log size has two advantages:

— It saves much time and money for the appraiser.

— It is far more accurate.

For those who prefer not to read through the research that leads to

our conclusions and simply want to learn how to use the log size model,

Appendix C presents a much shorter version of Chapter 4. It also serves

as a useful refresher for those who read Chapter 4 in its entirety but

periodically wish to refresh their skills and understanding.

Chapter 5 discusses arithmetic versus geometric mean returns. There

have been many articles in the professional literature arguing whether

arithmetic or geometric mean returns are most appropriate. For valuing

small businesses, the two measures can easily make a 100% difference in

the valuation, as geometric returns are always lower than arithmetic re-

turns (as long as returns are not identical in every period, which, of

course, they are not). Most of the arguments have centered around Pro-

fessor Ibbotson™s famous two-period example.

The majority of Chapter 5 consists of empirical evidence that arith-

metic mean returns do a better job than geometric means of explaining

log size results. Additionally, we spend some time discussing a very

mathematical article by Indro and Lee that argues for using a time

horizon-weighted average of the arithmetic and geometric means.

For those who use CAPM, whether in a direct equity approach or in

an invested capital approach, there is a trap into which many appraisers

fall, which is producing an answer that is internally inconsistent.

Common practice is to assume a degree of leverage”usually equal

to the subject company™s existing or industry average leverage”

1. When the subject company is close to the size of publicly traded ¬rms, say one half their size,

then the public guideline company approach is reasonable.

2. Using 1938“1997 data, the log size standard error was only 6% as large as CAPM™s standard

error. 1998 was a bad year for the log size model.

PART 2 Calculating Discount Rates

114

assuming book value for equity. This implies an equity for the ¬rm, which

is an ex-ante value of equity. The problem comes when the appraiser

stops after obtaining his or her valuation estimate. This is because the

calculated value of equity will almost always be inconsistent with the

value of equity that is implied in the leverage assumed in the calculation

of the CAPM discount rate.

In Chapter 6 we present an iterative method that solves the problem

by repeating the valuation calculations until the assumed and the calcu-

lated equity are equal.

PART 2 Calculating Discount Rates 115

CHAPTER 4

Discount Rates as a Function of

Log Size1

PRIOR RESEARCH

TABLE 4-1: ANALYSIS OF HISTORICAL STOCK RETURNS

Regression #1: Return versus Standard Deviation of Returns

Regression #2: Return versus Log Size

Regression #3: Return versus Beta

Market Performance

Which Data to Choose?

Tables 4-2 and 4-2A: Regression Results for Different Time Periods

18th Century Stock Market Returns

Conclusion on Data Set

Recalculation of the Log Size Model Based on 60 Years

APPLICATION OF THE LOG SIZE MODEL

Discount Rates Based on the Log Size Model

Need for Annual Updating

Computation of Discount Rate Is an Iterative Process

Practical Illustration of the Log Size Model: Discounted Cash Flow

Valuations

The Second Iteration: Table 4-4B

Consistency in Levels of Value

Adding Speci¬c Company Adjustments to the DCF Analysis: Table

4-4C

Total Return versus Equity Premium

Adjustments to the Discount Rate

Discounted Cash Flow or Net Income?

DISCUSSION OF MODELS AND SIZE EFFECTS

CAPM

1. Adapted and reprinted with permission from Valuation (August 1994): 8“24 and The Valuation

Examiner (February/March 1997): 19“21.

117

Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

Sum Beta

The Fama“French Cost of Equity Model

Log Size Models

Heteroscedasticity

INDUSTRY EFFECTS

SATISFYING REVENUE RULING 59-60 WITHOUT A GUIDELINE

PUBLIC COMPANY METHOD

SUMMARY AND CONCLUSIONS

APPENDIX A: AUTOMATING ITERATION USING

NEWTON™S METHOD

APPENDIX B: MATHEMATICAL APPENDIX

APPENDIX C: ABBREVIATED REVIEW AND USE

PART 2 Calculating Discount Rates

118

PRIOR RESEARCH

Historically, small companies have shown higher rates of return when

compared to large ones, as evidenced by data for the New York Stock

Exchange (NYSE) over the past 73 years of its existence (Ibbotson Asso-

ciates 1999). The relationship between ¬rm size and rate of return was

¬rst published by Rolf Banz in 1981 and is now universally recognized.

Accordingly, company size has been included as a variable in several

models used to determine stock market returns.

Jacobs and Levy (1988) examined small ¬rm size as one of 25 vari-

ables associated with anomalous rates of return on stocks. They found

that small size was statistically signi¬cant both in single-variable and

multivariate form, although size effects appear to change over time, i.e.,

they are nonstationary. They found that the natural logarithm (log) of

market capitalization was negatively related to the rate of return.

Fama and French (1993) found they could explain historical market

returns well with a three-factor multiple regression model using ¬rm size,

the ratio of book equity to market equity (BE/ME), and the overall market

factor Rm Rf , i.e., the equity premium. The latter factor explained overall

returns to stocks across the board, but it did not explain differences from

one stock to another, or more precisely, from one portfolio to another.2

The entire variation in portfolio returns was explained by the ¬rst

two factors. Fama and French found BE/ME to be the more signi¬cant

factor in explaining the cross-sectional difference in returns, with ¬rm size

next; however, they consider both factors as proxies for risk. Furthermore,

they state, ˜˜Without a theory that speci¬es the exact form of the state

variables or common factors in returns, the choice of any particular ver-

sion of the factors is somewhat arbitrary. Thus detailed stories for the

slopes and average premiums associated with particular versions of the

factors are suggestive, but never de¬nitive.™™

Abrams (1994) showed strong statistical evidence that returns are

linearly related to the natural logarithm of the value of the ¬rm, as mea-

sured by market capitalization. He used this relationship to determine the

appropriate discount rate for privately held ¬rms. In a follow-up article,

Abrams (1997) further simpli¬ed the calculations by relating the natural

log of size to total return without splitting the result into the risk-free

rate plus the equity premium.

Grabowski and King (1995) also described the logarithmic relation-

ship between ¬rm size and market return. They later (Grabowski and

King 1996) demonstrated that a similar, but weaker, logarithmic relation-

ship exists for other measures of ¬rm size, including the book value of

common equity, ¬ve-year average net income, market value of invested

capital, ¬ve-year average EBITDA, sales, and number of employees. Their

latest research (Grabowski and King 1999) demonstrates a negative log-

arithmic relationship between returns and operating margin and a posi-

2. The regression coef¬cient is essentially beta controlled for size and BE/ME. After controlling for

the other two systematic variables, this beta is very close to 1 and explains only the market

premium overall. It does not explain any differentials in premiums across ¬rms or

portfolios, as the variation was insigni¬cant.

CHAPTER 4 Discount Rates as a Function of Log Size 119

tive logarithmic relationship between returns and the coef¬cient of vari-

ation of operating margin and accounting return on equity.

The discovery that return (the discount rate) has a negative linear

relationship to the natural logarithm of the value of the ¬rm means that

the value of the ¬rm decays exponentially with increasing rates of return.

We will also show that ¬rm value decays exponentially with the standard

deviation of returns.

TABLE 4-1: ANALYSIS OF HISTORICAL STOCK RETURNS

Columns A“F in Table 4-1 contain the input data from the Stocks, Bonds,

Bills and In¬‚ation 1999 Yearbook (Ibbotson Associates 1999) for all of the

regression analyses as well as the regression results. We use the 73-year

average arithmetic returns in both regressions, from 1926 to 1998. For

simplicity, we have collapsed 730 data points (73 years 10 deciles) into

73 data points by using averages. Thus, the regressions are cross-sectional