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PART 2 Calculating Discount Rates
T A B L E 4-2A (continued)

Regression Comparison [1]


4 Standard Errors

Adj R2 (Regr #2) [4]
5 Years Regr #1 [2] Regr #2 [3] Total

6 30 0.90% 0.90% 1.80% 12.50%
7 40 0.75% 0.61% 1.36% 76.31%
8 50 0.54% 0.36% 0.90% 88.30%
9 60 0.42% 0.34% 0.76% 96.95%
10 73 0.27% 0.82% 1.09% 89.17%

[1] Summary Regression Statistics from Table 4-2
[2] Table 4-2: I9, I19, ...
[3] Table 4-2: I14, I24, ...
[4] Table 4-2: I13, I23, ...

34.58% and 40.43% (Table 4-2, F43, G43) for the y-intercept”a range of
5.8%”and 1.18% to 0.90% (F44, G44) for the slope”a range of 0.28%.
For 73 years of data, the range is 14% for the y-intercept (G53“F53) and
0.69% (G54“F54) for the slope, which is 21„2 times larger than the 60-year
data. Thus, the past 60 years data are a more ef¬cient estimator of stock
market returns than other time periods, as measured by the size of con-
¬dence intervals around the regression estimates for the log size ap-

18th Century Stock Market Returns
Paul Harrison™s article (Harrison 1998) is a fascinating econometric study
which is very advanced and extremely mathematical. The data for this
study came primarily from biweekly Amsterdam stock prices published
from July 1723 to December 1794 for the Dutch East India Company and
a select group of English stocks that were traded in Amsterdam: the Bank
of England, the English East India Company, and the South Sea Company.
Harrison also examined stock prices from London spanning the 18th cen-
Harrison found the shape of the distribution of stock price returns
in the 18th and 20th centuries to be very similar, although their means
and standard deviations are different. The 18th century returns were
lower”but less volatile”than 20th century returns. He found the distri-
butions to be symmetric, like a normal curve, but leptokurtic (fat tailed),
which means there are more extreme events occurring than would be
predicted by a normal curve. The same fundamental pattern exists in both
1725 and 1995.
Harrison remarks that clearly much has changed over the last 300
years, but, interestingly, such changes do not seem to matter in his anal-
ysis. He comments that the distribution of prices is not driven by infor-
mation technology, regulatory oversight, or by the specialist”none of
these existed in the 18th century markets. However, what did exist in the
18th century bears resemblance to what exists today.
Harrison describes the following as some of the evidence for simi-
larities in the market:

CHAPTER 4 Discount Rates as a Function of Log Size 133
—Stock traders in the 18th century reacted to and affected market
prices like traders today. They competed vigorously for
information,9 and the 18th century markets followed a near
random walk”so much so that an entire pamphlet literature
sprang up in the early 18th century lamenting the
unpredictability of the market. Harrison says that
unpredictability is a theoretical result of competition in the
— Eighteenth century stock markets were informationally ef¬cient,
as shown econometrically by Neal (1990).
— The practices of 18th century brokers were sophisticated.
Investors early in the 18th century valued stocks according to
their discounted stream of future dividends. Tables were
published (such as Hayes 1726) showing the appropriate
discount for different interest rates and time horizons. Traders
engaged in cash contracts, futures contracts, and options; they
sold short, issued credit, and used ˜˜modern™™ investment
strategies, such as forming portfolios, diversi¬cation, and
To all of the foregoing, I would add an observation by King
Solomon, who said, ˜˜There is nothing new under the sun.™™ (Ecclesiastes
1:9) Also in keeping with the theme in our chapter, King Solomon
became the inventor of portfolio theory when he wrote, ˜˜Divide your
wealth into seven, even eight parts, for you cannot know what
misfortune may occur on earth™™ (Ecclesiastes, 11:2).

Conclusion on Data Set
To return to the 20th century, Ibbotson (Ibbotson Associates 1998, p. 27)
enunciated the principle that over the very long run there are very few
events that are truly outliers. Paul Harrison™s research seems to corrob-
orate this. It is in the nature of the stock market for there to be periodic
booms and crashes, indicating that we should use all 73 years of the
NYSE data. On the other hand, the statistical feedback in Table 4-2A
shows that eliminating the 1926“1938 data provides the most statistically
reliable log size relationship. Similarly, Figure 4-4 shows a ¬‚attening of
the regression curve when the decade ending 1935 is eliminated. Paul
Harrison said that even with 300 years of history showing similarity in
the distribution of returns, he would be inclined to label the years in
question as an outlier that should probably be excluded from the regres-

9. A fascinating story that I remember from an economic history course is that Baron Rothschild,
having placed men with carrier pigeons at the Battle of Waterloo, was the ¬rst
nonparticipant to know the results of the battle. He ¬rst paid a visit to inform the King of
the British victory, and then he proceeded to the stock market to make 100 million
pounds”many billions of dollars in today™s money”a tidy sum for having insider
information. He struck a blow for market ef¬ciency. Even his method of making a fortune in
the market that day is a paradigm of the extent of market ef¬ciency then. He knew that he
was being observed. He began selling, and others followed him in a panic. Later, he sent his
employees to do a huge amount of buying anonymously. The markets were indeed
ef¬cient”at least they were by the end of the day!

PART 2 Calculating Discount Rates
sion.10 Thus, we eliminate the years 1926“1937 from the ¬nal regression.
The superior adjusted R 2 and 95% con¬dence intervals of the past 60
years, coupled with Harrison™s results and Ibbotson™s general principle of
using more rather than less data, lead us to conclude that the past 60
years provide the best guide for the future.

Recalculation of the Log Size Model Based on 60 Years
Based on our previous discussion, NYSE data from the past 60 years are
likely to be the most relevant for use in forecasting the future. This time
frame contains numerous data points but excludes the decade of highest
volatility, attributed to nonrecurring historical events, i.e., the Roaring
Twenties and the Depression years. Therefore, we repeat all three regres-
sions for the 60-year time period from 1939“1998, as shown in Column
E of Table 4-1. Regression #1 for this time period is:
r 8.90% (30.79% S) (4-3)
where S is the standard deviation. The adjusted R 2 in this case falls to
95.31% (E30) from the 98.82% (D30) obtained from the 73-year equation,
but is still indicative of a strong relationship. On average, returns were
exceptionally high and volatile during the ¬rst 13 years of the NYSE,
especially in the small ¬rms. It appears that including those years im-
proves the relationship of returns to standard deviation of returns, even
as it worsens the relationship between returns and log size.
The log size equation (regression #2) for the 60-year period is:
r 37.50% [1.039% ln (FMV)] (E42, E48) (4-4)
The regression statistics indicate an excellent ¬t, with an adjusted R 2 of
96.95% (E45).11

Equation (4-4) is the most appropriate for calculating current discount
rates and will be used for the remainder of the book. In the next sections
we will use it to calculate discount rates for various ¬rm sizes and dem-
onstrate its use in a simpli¬ed discounted cash ¬‚ow analysis.

Discount Rates Based on the Log Size Model
Table 4-3 shows the implied equity discount rate for ¬rms of various sizes
using the log size model (regression equation #2) for the past 60 years.
The implied equity discount rate for a $10 billion ¬rm is 13.6% (B7), and
for a $50 million ¬rm it is 19.1% (B10), based on 60-year average market
returns for deciles #1“#10. While those values and all values in between
are interpolations based on the model, the discount rates for ¬rm val-

10. Related in a personal conversation.
11. For 1938“1997 data, adjusted R 2 was 99.54%. The ˜˜perverse™™ results of 1998 caused a
deterioration in the relationship.

CHAPTER 4 Discount Rates as a Function of Log Size 135
T A B L E 4-3

Table of Stock Market Returns Based on FMV”60-Year Model


5 Regression Results Implied Discount

6 Mktable Min FMV Rate (R)
7 $10,000,000,000 13.6%
8 $1,000,000,000 16.0%
9 $100,000,000 18.4%
10 $50,000,000 19.1%
11 $10,000,000 20.8%
12 $5,000,000 21.5%
13 $3,000,000 22.0%
14 $1,000,000 23.2%
15 $750,000 23.5%
16 $500,000 23.9%
17 $400,000 24.1%
18 $300,000 24.4%
19 $200,000 24.8%
20 $150,000 25.1%
21 $100,000 25.5%
22 $50,000 26.3%
23 $30,000 26.8%
24 $10,000 27.9%
25 $1,000 30.3%
26 $1 37.5%

ues below that are extrapolations because they lie outside the original
data set.
Using equation (4-4), the Excel formula for cell B7 is: 0.3750
(0.01039 * ln(A7)). In Lotus 123, the formula would be: 0.3750
(0.01039 * @ ln(A7)).
Regression #2 (equation [4-4]) tells us that the discount rate is a con-
stant minus another constant multiplied by ln (FMV). Since ln (FMV) has
a characteristic upwardly sloping shape, as seen in Figure 4-6, subtracting
a curve of that shape from a constant leads to a discount rate function
that is a mirror image of Figure 4-6. Figure 4-7 is the graph of that rela-
tionship, and the reader can see that the result is a downward sloping
curve. Again, this curve depicts the rate of return, i.e., the discount rate,
as a function of the absolute dollar value of the ¬rm. Note that this is not
on a log scale. Since the regression equation is r 37.50% [1.0309%
ln (FMV)], we begin at the extreme left with a return of 37.5% for a ¬rm
worth $1 and subtract the fraction of the ln FMV dictated by the equation.
ln y.12
An important property of logarithms is that ln xy ln x
Since regression equation #2 has the form r a b ln FMV, where a
0.3750 and b 0.01039, we can ask how the discount rate varies with
differing orders of magnitude in value. First, however, we will work

12. That is because e x ey e x y. Taking logs of both sides of that equation is the proof.

PART 2 Calculating Discount Rates
F I G U R E 4-6

The Natural Logarithm











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