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

2.00

0.00

0 5 10 15 20 25 30

FMV ($Millions)

through some general equations where we vary the value of the ¬rm by

a factor of K.

Let r1 the discount rate for Firm #1, whose value FMV1

F I G U R E 4-7

Discount Rates as a Function of FMV

40%

35%

30%

25%

Discount Rate

20%

15%

10%

5%

0%

0 20 40 60 80 100 120

Fair Market Value, Marketable Minority ($ Millions)

For scaling reasons, we eliminate values above $100 million

CHAPTER 4 Discount Rates as a Function of Log Size 137

F I G U R E 4-8

1939“1998 Decile Standard Deviations as a Function of Ln(FMV)

100%

90%

80%

70%

Standard Deviation of Returns

60%

50%

40%

10

30% 9

Std Dev = -3.13% x Ln FMV + 87.77% 8

7 3

R2 = 0.9894

4

6

20%

2

5

3

1

10%

0%

0 5 10 15 20 25 30

Ln(FMV)

Standard deviations of yearly returns are derived from the CRSP Deciles. Data labels are decile numbers. The Y intercept is

the regression intercept, not an actual data point.

r2 the discount rate for Firm #2, whose value FMV2 K

FMV1

r1 a b ln FMV1 (4-6)

regression equation #2 applied to Firm #1

r2 a b ln (K FMV1) (4-7)

regression equation #2 applied to Firm #2

r2 a b [ln K ln FMV1] (4-8)

r2 a b ln FMV1 b ln K (4-9)

r2 r1 b ln K (4-10)

In words, the discount rate of a ¬rm K times larger (smaller) than Firm

#1 is always b ln K smaller (larger) than r1.

Let™s illustrate the nature of this relationship with some speci¬c ex-

amples. First, let™s examine what happens with orders of magnitude of

PART 2 Calculating Discount Rates

138

10. Ln 10 2.302535, so b ln 10 0.01039 2.302585 .02391, or

2.4%. This means that if Firm #2 is 10 times larger (smaller) than Firm

#1, its discount rate should be 2.4% lower (higher) than the Firm #1 dis-

count rate. This result can be seen in Table 4-3. The $10 billion ¬rm has

a discount rate of 13.6%, while the $1 billion ¬rm has a discount rate of

16.0%, which is 2.4% higher. The $100 million ¬rm has a discount rate of

18.4%, which is 2.4% higher than the $1 billion ¬rm. Because of the math-

ematical properties of logarithms, the same percentage change in FMV will

always result in the same absolute change in the discount rate. This phe-

nomenon is also seen in graphs containing log scales. Equal distances on

a log scale are equal percentage changes, not absolute changes.

Let™s try one more useful calculation”an order of magnitude 2. Ln

2 0.6931, so that b ln K 0.01039 0.6931 0.72%. Doubling

(halving) the value of the ¬rm reduces (increases) the discount rate by

0.72%. You can see that in going from a $10 million ¬rm to a $5 million

¬rm, the discount rate has increased from 20.8% to 21.5%, a 0.7% differ-

ence (see Table 4-3).

Now it is possible to construct your own table. All you need to know

is your starting FMV and discount rate. The rest follows easily from the

above formulas. Also, we can easily interpolate the table. Suppose you

wanted to know the discount rate for a $25 million ¬rm. Simply start

with the $50 million ¬rm, where r 19.1%, and add 0.7% 19.8%.

Need for Annual Updating

Tables 4-1 through 4-3 should be updated annually, as the Ibbotson av-

erages change, and new regression equations should be generated. This

becomes more crucial when shorter historical time periods are used, be-

cause changes will have a greater impact on the average values.

Additionally, it is important to be careful to match the regression

equation to the year of the valuation. If the valuation assignment is ret-

roactive and the valuation date is 1994, then one should use a regression

equation for 1939“1994.13

Computation of Discount Rate Is an Iterative Process

In spite of the straightforwardness of these relationships, we have a prob-

lem of circular reasoning when it comes to computing of the discount

rate. We need FMV to obtain the discount rate, which is in turn used to

discount cash ¬‚ows or income to calculate the FMV! Hence, it is necessary

to make sure that our initial estimate of FMV is consistent with the ¬nal

result. If it is not, then we have to use the calculated FMV from the end

of iteration #1 as our new assumed FMV in iteration #2. Using either

equation (4-4) or Table 4-3, that will imply a new discount rate, which

we use to value the ¬rm. We keep repeating the process until the results

are consistent.

It is extremely rare to require more than two iterations to achieve

consistency in the ex ante and ex post values. The reason is that even if

13. Alternatively, one could either use the regression equation in the original article, run one™s own

regression on the Ibbotson data, or contact the author to provide the right equation.

CHAPTER 4 Discount Rates as a Function of Log Size 139

we guess the value of the ¬rm incorrectly by a factor of 10, we will only

be 2.4% off in our discount rate. By the time we come to the second

iteration, we usually are consistent. The reason behind this is that the

discount rate is based on the logarithm of the value. As we saw earlier,

there is not much difference between the log of $10 billion and the log of

$10 million, and multiplying that by the x-coef¬cient of 0.01039 further

reduces the effects of an initial incorrect estimate of value. This is a con-

vergent system 99% of the time with any kind of reasonable initial guess

of value and even most unreasonable guesses.

The need for iteration arises because of the mathematical properties

of the equations we use in valuing a ¬rm. The simplest type of valuation

is that of a ¬rm with constant growth to perpetuity, where we simply

apply the Gordon growth model (˜˜Gordon model™™) to our forecast of cash

¬‚ow for the coming year. For simplicity, we will use the end-year Gordon

model formula, although it is not as accurate as the midyear formula.

We use the following de¬nitions:

CF cash ¬‚ow (available to equity) in year t 1 (the ¬rst

forecast year)

a 0.3750, the regression constant from regression #2

b 0.01039, the x-coef¬cient from regression #2

V fair market value (FMV) of the ¬rm

r the discount rate

Using the Gordon model and ignoring valuation discounts and pre-

miums, the FMV of the ¬rm is:

CF

V (4-11)

r g

Per equation (4-6), our log size equation for the discount rate is:

r a b ln V (4-12)

Substituting (4-12) into (4-11), we get:

CF

V (4-13)

a b ln V g

Equation (4-13) is a transcendental equation with no analytic

solution.14 Therefore, successive approximation is the only method of de-

termining an answer. The simple iterative procedure in Tables 4-4A, 4-4B,

and 4-4C is very easy to use and works in almost all situations.

Practical Illustration of the Log Size Model: Discounted

Cash Flow Valuations

Let™s illustrate how the iterative process works with a speci¬c example.

The assumptions in Tables 4-4A, 4-4B and 4-4C are identical, except for

the discount rate. Table 4-4A is a very simple discounted cash ¬‚ow (DCF)

14. I thank my friend William Scott, Jr., a physicist, for the terminology and the de¬nitive word

that there is no analytic solution.