Is Table 4-3 the last word in calculating discount rates? No, but it is the

best starting point based on the available data. Table 4-3 is an extrapo-

lation of NYSE data to privately held ¬rms. While the results appear very

reasonable to me, it would be preferable to perform a similar regression

for NASD data. Unfortunately, the data are not readily obtainable.

Privately held ¬rms are generally owned by people who are not well

diversi¬ed. Table 4-3 was derived from portfolios of stocks that were di-

versi¬ed in every sense except for size, as size itself was the method of

sorting the deciles. In contrast, the owner of the local bar is probably not

well diversi¬ed, nor is the probable buyer. The appraiser may want to

add a speci¬c company adjustment of, say, 2% to 5% to the discount rate

implied by Table 4-3 to account for that. On the other hand, a $100 million

FMV ¬rm is likely to be bought by a well diversi¬ed buyer and may not

merit increasing the discount rate.

Another common adjustment to Table 4-3 discount rates would be

for the depth and breadth of management of the subject company com-

pared to other ¬rms of the same size. In general, Table 4-3 already incor-

porates the size effect. No one expects a $100,000 FMV ¬rm to have three

Harvard MBAs running it, but there is still a difference between a com-

plete one-man show and a ¬rm with two talented people. In general, this

methodology of calculating discount rates will increase the importance of

comparing the subject company to its peers via RMA Associates or similar

data. Differences in leverage between the subject company and its RMA

peers could well be another common adjustment.

Discounted Cash Flow or Net Income?

Since the market returns are based on the cash dividends and the market

price at which one can sell one™s stock, the discount rates obtained with

the log size model should be properly applied to cash ¬‚ow, not to net

income. We appraisers, however, sometimes work with clients who want

a ˜˜quick and dirty valuation,™™ and we often don™t want to bother esti-

mating cash ¬‚ow. I have seen suggestions in Business Valuation Review

(Gilbert 1990, for example) that we can increase the discount rate and

thereby apply it to net income, and that will often lead to reasonable

results. Nevertheless, it is better to make an adjustment from net income

based on judgment to estimate cash ¬‚ow to preserve the accuracy of the

discount rate.

DISCUSSION OF MODELS AND SIZE EFFECTS

The size effects described by Fama and French (1993), Abrams (1994,

1997), and Grabowski and King (1995) strongly suggest that the tradi-

tional one-factor CAPM model is obsolete. As Fama and French (1993, p.

54) say, ˜˜Many continue to use the one-factor Sharpe“Lintner model to

evaluate portfolio performance and to estimate the cost of capital, despite

the lack of evidence that it is relevant. At a minimum, these results here

and in Fama and French (1992) should help to break this common habit.™™

PART 2 Calculating Discount Rates

146

CAPM

Consider the usual way we calculate discount rates using CAPM. We

average the betas of many different ¬rms in the industry, which vary

considerably in size, and apply the resulting beta to a ¬rm that is prob-

ably 0.1% to 1% of the industry average, without correction for size, and

hence risk. Ignoring the size effect corrupts the CAPM results.

This ¬‚aw also applies to the guideline public company method. The

usual approach is to average price earnings multiples (and/or price cash

¬‚ow multiples, etc.) for the various ¬rms in the industry without cor-

recting for size and apply the multiple to a small private ¬rm. A better

method is to perform a regression analysis of market capitalization

(value) as a function of earnings (or cash ¬‚ow) and forecast growth, when

available. I also recommend using another form of the regression with

P/E or P/CF as the dependent variable and market capitalization and

forecast growth as the independent variables.

The beta used in CAPM is usually calculated by running a regression

of the equity premium for an individual company versus the market pre-

mium. As previously discussed, the inability of the resulting beta to ex-

plain the size effect has called into question the validity of CAPM. An

alternative method of calculating beta has been proposed which attempts

to capture the size effect and better correlate with market equity returns,

possibly ameliorating this problem.

Sum Beta

Ibbotson et al. (Peterson, Kaplan, and Ibbotson 1997) postulated that con-

ventional estimates of beta are too low for small stocks due to the higher

degree of auto-correlation in returns exhibited by smaller ¬rms. They cal-

culated a beta using a multiple regression model for both the current and

the prior period, which they call ˜˜sum beta.™™ These adjusted estimates of

beta helped to account for the size effect and showed positive correlation

with future returns.

This improved method of calculating betas will reduce will reduce

some of the downward bias in CAPM discount rates, but it still will not

account for the size effect differences between the large ¬rms in the

NYSE”where even the smallest ¬rms are large”and the smaller pri-

vately held ¬rms that many appraisers are called upon to value. Size

should be an explicit variable in the model to accomplish that.

It may be possible to combine the models. One could use the log size

model to calculate a size premium over the average market return and

add that to a CAPM calculation of the discount rate using Ibbotson™s sum

betas. It will take more research to determine whether than is a worth-

while improvement in methodology.

The Fama“French Cost of Equity Model18

The Fama“French cost of equity model is a multivariable regression

model that uses size (˜˜small minus big™™ premium SMB) and book to

18. The precise method of calculating beta, SMB, and HML using the three-factor model, along

with the regression equation, is more fully explained in Ibbotson Associates™ Beta Book.

CHAPTER 4 Discount Rates as a Function of Log Size 147

market equity (˜˜high minus low™™ premium HML) in addition to beta

as variables that affect market returns. Michael Annin (1997) examined

the model in detail and found that it does appear to correct for size, both

in the long term and short term, over the 30-year time period tested.

The cost of equity model, however, is neither generally accepted nor

easy to use (Annin 1997), and using it to determine discount rates for

privately held ¬rms is particularly problematic. Market returns are not

available for these ¬rms, rendering direct use of the model impossible.19

Discount rates based on using the three-factor model are published by

Ibbotson Associates in the Cost of Capital Quarterly by industry SIC code,

with companies in each industry sorted from highest to lowest. Deter-

mining the appropriate percentile grouping for a privately held ¬rm is a

major obstacle, however. The Fama“French model is a superior model for

calculating discount rates of publicly held ¬rms. It is not practical for

privately held ¬rms.

Log Size Models

The log size model is a superior approach because it better correlates with

historical equity returns. Therefore, it enables business appraisers to dis-

pense with CAPM altogether and use ¬rm size as the basis for deriving

a discount rate before adjustments for qualitative factors different from

the norm for similarly sized companies.

In another study on stock market returns, analysts at an investment

banking ¬rm regressed P/E ratios against long-term growth rate and mar-

ket capitalization. The R 2 values produced by the regressions were 89%

for the December 1989 data and 73% for the November 1990 data. Sub-

stituting the natural logarithm of market capitalization in place of market

capitalization, the same data yields an R 2 value of 91% for each data set,

a marginal increase in explanatory power for the ¬rst regression but a

signi¬cant increase in explanatory power for the second regression.

From Chapter 3, equation (3-28), the PE multiple is equal to

1 r

PE (1 b)(1 g1)

r g

Using a log size model to determine r, the PE multiple is equal to:

1 a b ln (FMV)

PE (1 RR)(1 g1) (4-18)

a b ln(FMV) g

where g1 is expected growth in the ¬rst forecast year, RR is the retention

ratio,20 a and b are the log size regression coef¬cients, and g is the long-

term growth rate. Looking at equation (4-18), it is clear why using the

log of market capitalization improved the R 2 of the above regression.

Grabowski and King (1995) applied a ¬ner breakdown of portfolio

returns than was previously used to relate size to equity premiums. When

19. Based on a conversation with Michael Annin.

20. Equation (3-28) uses the more conventional term b instead of RR to denote the retention ratio.

Here we have changed the notation in order to eliminate confusion, as we use the term b

for the regression x-coef¬cient.

PART 2 Calculating Discount Rates

148

they performed regressions with 31-year data for 25 and 100 portfolios

(as compared to our 10), they found results similar to the equity premium

form of log size model, i.e., the equity premium is a function of the neg-

ative of the log of the average market value of equity, further supporting

this relationship.21

Grabowski and King (1996) in an update article also used other prox-

ies for ¬rm size in their log size discount rate model, including sales, ¬ve-

year average net income, and EBITDA. Following is a summary of their

regression results sorted ¬rst by R 2 in descending order, then by the stan-

dard error of the y-estimate in ascending order. Overall, we are attempt-

ing to present their best results ¬rst.

R2

Measure of Size Standard Error of Y-Estimate

1. Mkt cap”common equity 93% 0.862%

2. Five-year average net income 90% 0.868%

3. Market value of invested capital 90% 1.000%

4. Five-year average EBITDA 87% 0.928%

5. Book value”invested capital 87% 0.989%

6. Book value”equity 87% 0.954%

7. Number of employees 83% 0.726%

8. Sales 73% 1.166%

Note that the market value of common equity, i.e., market capitali-

zation of common equity, has the highest R 2 of all the measures. This is

the measure that we have used in our log size model. The ¬ve-year av-

erage net income, with an R 2 of 90%, is the next-best independent vari-

able, superior to the market value of invested capital by virtue of its lower

standard error.

This is a very important result. It tells us that the majority of the

information conveyed in the market price of the stock is contained in net

income. When we use a log size model based on equity in valuing a

privately held ¬rm, we do not have the bene¬t of using a market-

determined equity. The value will be determined primarily by the mag-

nitude and timing of the forecast cash ¬‚ows, the primary component of

which is forecast net income. If we did not know that the log of net

income was the primary causative variable of the log size effect, it is

possible that other variables such as leverage, sales, book value, etc. sig-

ni¬cantly impact the log size effect. If we failed to take those variables

into account and our subject company™s leverage varied materially from

the average of the market (in each decile) as it is impounded into the log

size equation, our model would be inaccurate. Grabowski and King™s

research eliminates this problem. Thus, we can be reasonably con¬dent

that the log size model as presented is accurate and is not missing any

signi¬cant variable.

Of Grabowski and King™s eight different measures of size, only mar-

ket capitalization (#1) and the market value of invested capital (#3) have

21. Grabowski and King actually used base 10 logarithms.

CHAPTER 4 Discount Rates as a Function of Log Size 149

the circular reasoning problem of our log size model. The other measures

of size have the advantage in a log size model of eliminating the need

for iteration since the discount rate equation does not depend on the

market value of equity, the determination of which is the ultimate pur-

pose of the discount rate calculation. For example, if we were to use #2,

net income, we would simply insert the subject company™s ¬ve-year av-

erage net income into Grabowski and King™s regression equation and it

would determine the discount rate. This is problematic, however, for de-

termining discount rates for high-growth ¬rms, due to the inability to

adequately capture signi¬cant future growth in sales, net income, and so

on. Start-up ¬rms in high technology industries frequently have negative

net income for the ¬rst several years due to their investment in research

and development. Sales may subsequently rise dramatically once prod-

ucts reach the market. Therefore, ¬ve-year averages are not suitable in

this situation.

Another problem with Grabowski and King™s results is that their data

only encompass 1963“1994, 31 years”the years for which Compustat

data were available for all companies. Thus, their equations suffer from

the same wide con¬dence intervals that our 30-year regressions have.

Their standard error of the y-estimate is 0.862% (Exhibit A, p. 106), which

is six times larger than our 1938“1997 con¬dence intervals.22 Thus, their

95% con¬dence intervals will also be approximately six times wider

around the regression estimate.

As mentioned in the introduction, in their latest article (Grabowski

and King 1999) they demonstrate a negative logarithmic relationship be-

tween returns and operating margin and a positive logarithmic relation-

ship between returns and the coef¬cient of variation of operating margin

and accounting return on equity.

This is their most important result so far because it relates returns to

fundamental measures of risk. Actually, it appears to me that operating

margin in itself works because of its strong correlation of 0.97 to market

capitalization, i.e., value. However, the coef¬cient of variations (CV) of

operating margin and return on equity seem to be more fundamental

measures of risk than size itself. In other words, it appears that size itself

is a proxy for the volatility of operating margin, return on equity, and

possibly other measures. Thus, we must pay serious attention to their

results.

Below is a summary of their statistical results.

R2

Measure of Risk Standard Error of Y-Estimate

1. Log of ¬ve-year operating margin 76% 1.185%

2. Log CV(operating margin) 54% 0.957%

3. Log CV (return on equity) 54% 0.957%