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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.




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