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27 20 21,247 6,128 15,119 720,220 0.8195 12,391
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

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