132

T A B L E 4-2A (continued)

Regression Comparison [1]

A B C D E

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-

proach.

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-

tury.

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

market.

— 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

hedging.

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

134

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

APPLICATION OF THE LOG SIZE MODEL

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

A B

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

136

F I G U R E 4-6

The Natural Logarithm

20.00

18.00

16.00

14.00

12.00

Ln(FMV)

10.00

8.00

6.00

4.00