This shows the FMV of a ¬rm or portfolio declines exponentially

with the discount rate. This is reminiscent of a continuous time present

value formula; in this case, though, instead of traveling through time we

are traveling though expected rates of return. The same is true of equation

(B4-6), where we are traveling through degree of risk.

What Does the Exponential Relationship Mean?

Let™s try to get an intuitive feel for what an exponential relationship

means and why that makes intuitive sense. Equation (B4-6) shows that

the fair market value of the ¬rm is an exponentially declining function

of risk, as measured by the standard deviation of returns. Repeating equa-

tion (B4-6), FMV Ae k S, k 0. Because we ¬nd that risk itself is primarily

related to the size of the ¬rm, we come to a similar equation for size.

Cemr, m

Repeating equation (B4-10), we see that FMV 0.

In physics, radioactive minerals such as uranium decay exponen-

tially. That means that a constant proportion of uranium decays at every

moment. As the remaining portion of uranium is constantly less over time

due to the radioactive decay, the amount of decay at any moment in time

or during any ¬nite time period is always less than the previous period.

A graph of the amount of uranium remaining over time would be a

downward sloping curve, steep at ¬rst and increasing shallow over time.

Figure 4-3 shows an exponential decay curve.

It appears the same is true of the value of ¬rms. Instead of decaying

over time, their value decays over risk. Because it turns out that risk is

so closely related to size and the rate of return is so closely related to

size, the value also decays exponentially with the market rate of return,

i.e., the discount rate. The graph of exponential decay in value over risk

has the same general shape as the uranium decay curve.

Imagine the largest ship in the world sailing on a moderately stormy

ocean. You as a passenger hardly feel the effects of the storm. If instead

you sailed on a slightly smaller ship, you would feel the storm a bit more.

As we keep switching to increasingly smaller ships, the storm feels in-

PART 2 Calculating Discount Rates

160

creasingly powerful. The smallest ship on the NYSE might be akin to a

35-foot cabin cruiser, while appraisers often have to value little paddle-

boats, the passengers of which would be in danger of their lives while

the passengers of the General Electric boat would hardly feel the turbu-

lence.

That is my understanding of the principle underlying the size effect.

Size offers diversi¬cation of product and service. Size reduces transaction

costs in proportion to the entity, e.g., the proceeds of ¬‚oating a $1 million

stock issue after ¬‚otation costs are far less in percentage terms than ¬‚oat-

ing a $100 million stock issue. Large ¬rms have greater depth and breadth

of management, and greater staying power. Even the chances of beating

a bankruptcy exist for the largest businesses. Remember Chrysler? If it

were not a very big business, the government would never have jumped

in to rescue it. The same is true of the S&Ls. For these and other reasons,

the returns of big businesses ¬‚uctuate less than small businesses, which

means that the smaller the business, the greater the risk, the greater the

return.

The FMV of a ¬rm or portfolio declining exponentially with the dis-

count rate/risk is reminiscent of a continuous time present value formula,

e r t; in this case, though, instead of

where Present Value Principal

traveling through time we are traveling though expected rates of return/

risk.

CHAPTER 4 Discount Rates as a Function of Log Size 161

APPENDIX C

Abbreviated Review and Use

This abbreviated version of the chapter is intended for those who simply

wish to learn the model without the bene¬t of additional background and

explanation, or wish to use it as a quick reference for review.

INTRODUCTION

Historically, small companies have shown higher rates of return than

large ones, as evidenced by New York Stock Exchange (NYSE) data over

the past 73 years (Ibbotson Associates 1999). Further investigation into

this phenomenon has led to the discovery that return (the discount rate)

strongly correlates with the natural logarithm of the value of the ¬rm

(¬rm size), which has the following implications:

— The discount rate is a linear function negatively related to the

natural logarithm of the value of the ¬rm.

— The value of the ¬rm is an exponential decay function, decaying

with the investment rate of return (the discount rate).

Consequently, the value also decays in the same fashion with the

standard deviation of returns.

As we have already described regression analysis in Chapter 3, we

now apply these techniques to examine the statistical relationship be-

tween market returns, risk (measured by the standard deviation of re-

turns) and company size.

REGRESSION #1: RETURN VERSUS STANDARD

DEVIATION OF 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 73-year av-

erage returns in both regressions. 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 rather than time series. In Col-

umn A we list Ibbotson Associates™ (1999) division of the entire NYSE

into 10 different divisions”known as deciles”based on size, with the

largest ¬rms in decile 1 and the smallest in decile 10.36 Columns B through

F contain market data for each decile which is described below.

Note that the 73-year average market return in Column B rises with

each decile, as does the standard deviation of returns (Column C). Col-

umn D shows the 1998 market capitalization of each decile, which is the

price per share times the number of shares. It is also the fair market value

(FMV).

Dividing Column D (FMV) by Column F (the number of ¬rms in the

decile), we obtain Column G, the average capitalization, or the average

36. All of the underlying decile data in Ibbotson originate with the University of Chicago™s Center

for Research in Security Prices (CRSP), which also determines the composition of the deciles.

PART 2 Calculating Discount Rates

162

fair market value of the ¬rms in each decile. Column H, the last column

in the table titled ln (FMV), is the natural logarithm of the average FMV.

Regression of ln (FMV) against standard deviation of returns for the

period 1926“1998 (D26 to D36, Table 4-1), gives rise to the equation:

r 6.56% (31.24% S) (4-1)

where r return and S standard deviation of returns.

The regression statistics of adjusted R 2 of 98.82% (D30) a t-statistic

of the slope of 27.4 (D35), a p-value of less than 0.01% (D36), and the

standard error of the estimate of 0.27% (D28), all indicate a high degree

of con¬dence in the results obtained. Also, the constant of 6.56% (D26) is

the regression estimate of the long-term risk-free rate, which compares

favorably with the 73-year arithmetic mean income return from 1926“

1998 on long-term Treasury Bonds of 5.20%.37

The major problem with direct application of this relationship to the

valuation of small businesses is coming up with a reliable standard de-

viation of returns. Appraisers cannot directly measure the standard de-

viation of returns for privately held ¬rms, since there is no objective stock

price. We can measure the standard deviation of income, and we covered

that in our discussion in the chapter of Grabowski and King (1999).

REGRESSION #2: RETURN VERSUS LOG SIZE

Fortunately, there is a much more practical relationship. Notice that the

returns are negatively related to the market capitalization, i.e., the fair

market value of the ¬rm. The second regression in Table 4-1 (D42“D51)

is the more useful one for valuing privately held ¬rms. Regression #2

shows return as a function of the natural logarithm of the FMV of the

¬rm. The regression equation for the period 1926“1998 is:

r 42.24% [1.284% ln (FMV)] (4-2)

The adjusted R 2 is 92.3% (D45), the t-statistic is 10.4 (D50), and the p-

value is less than 0.01% (D51), meaning that these results are statistically

robust. The standard error for the Y-estimate is 0.82% (D43), which means

that we can be 95% con¬dent that the regression forecast is accurate

within approximately 2 0.82% 1.6.

Recalculation of the Log Size Model Based on 60 Years

NYSE data from the past 60 years are likely to be the most relevant for

use in forecasting the future (see chapter for discussion). This time frame

still contains numerous data points, but it excludes the decade of highest

volatility, attributed to nonrecurring historical events, i.e., the Roaring

Twenties and Depression years. Also, Table 4-2A shows that the 60-year

regression equation has the highest adjusted R 2 and lowest standard error

37. SBBI-1999, p. 140 uses this measure as the risk-free rate for CAPM. Arguably, the average bond

yield is a better measure of the risk-free rate, but the difference is immaterial.

CHAPTER 4 Discount Rates as a Function of Log Size 163

when compared to the other four examined. Therefore, we repeat all three

regressions for the 60-year time period from 1939“1998, as shown in Table

4-1, Column E. Regression #1 for this time period for 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.

The corresponding log size equation (regression #2) for the 60-year

period is:

r 37.50% [1.039% ln (FMV)] (4-4)

The regression statistics indicate a good ¬t, with an adjusted R 2 of 96.95%

(E45).38 Equation (4-4) will be used for the remainder of the book to cal-

culate interest rates, as this time period is the most appropriate for cal-

culating current discount rates.

Need for Annual Updating

Table 4-1 should be updated annually, as the Ibbotson averages change,

and new regression equations should be generated. This becomes more

crucial when shorter time periods are used, because 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 retroactive and the valuation date is 1994,

then don™t use the regression equation for 1939“1998. Instead, either use

the regression equation in the original article, run your own regression

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

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 keep repeating the process until the

results are consistent. Fortunately, discount rates remain virtually con-

stant over large ranges of values, so this should not be much of a problem.

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)

analysis of a hypothetical ¬rm. The basic assumptions appear in Rows

B7 through B12. We assume the ¬rm had $100,000 cash ¬‚ow in 1998. We

38. For 1938“1997 data, adjusted R 2 was 99.54%. The ˜˜perverse™™ results of 1998 caused a

deterioration in the relationship.

PART 2 Calculating Discount Rates

164

forecast annual growth through the year 2003 in B8 through F8 and per-

petual growth at 6% thereafter in B10. In B9 we assume a 20% discount

rate.

The DCF analysis in Rows B22 through B32 is standard and requires

little explanation other than that the present value factors are midyear,

and the value in B28 is a marketable minority interest. It is this value

($943,285) that we use to compare the consistency between the assumed

discount rate (in Row 4) and calculated discount rate according to the log

size model.

We begin calculating the of discount rate using the log size model in

B34, where we compute ln (943,285) 13.7571. This is the natural log of

the marketable minority value of the ¬rm. In B35 we multiply that result

by the x-coef¬cient from the regression, or 0.01039, to come to 0.1429.

We then add that product to the regression constant of 0.3750, which

appears in B36, to obtain an implied discount rate of 23% (rounded, B37).

Comparison of the two discount rates (assumed and calculated) re-

veals that we initially assumed too high a discount rate, meaning that we

undervalued the ¬rm. B29“B31 contain the control premium and discount

for lack of marketability. Because the discount rate is not yet consistent,

ignore these numbers in this table, as they are irrelevant.

In Chapter 7, we discuss the considerable controversy over the ap-

propriate magnitude of control premiums. Nevertheless, it is merely a

parameter in the spreadsheet, and its magnitude does not affect the logic

of the analysis.

The Second Iteration: Table 4-4B

Having determined that a 20% discount rate is too low, we revise our

assumption to a 23% discount rate (B9) in Table 4-4B. In this case, we

arrive at a marketable minority FMV of $ 783,919 (B28). When we perform

the discount rate calculation with this value (B34“B37), we obtain a

matching discount rate of 23%, indicating that no further iterations are

necessary.

Consistency in Levels of Value

In calculating discount rates, it is important to be consistent in the level

of fair market value that we are using. Since the log size model is based

on returns from the NYSE, the corresponding values generated are on a

marketable minority basis. Consequently, it is this level of value that we

should use for the discount rate calculations.

Frequently, however, the marketable minority value is not the ulti-

mate level of fair market value that we are calculating. Therefore, it is

crucial to be aware of the differing levels of FMV that occur as a result