have to forecast sales in a different fashion. The calculation of component

#1 is still not sensitive at this level to the square of revenues, so we can

afford to be imprecise. Assuming an average P/E multiple of 12.5, we

PART 4 Putting It All Together

378

divide the assumed $10 million selling price by the P/E multiple to arrive

at net income of $800,000. Dividing that by an assumed pretax margin of

1014 (B20, trans-

5% leads to sales of $16 million (B19), which is $2.56

ferred to C6) when squared. That contributes only “0.1% (D6) to the cal-

culation of the pure discount from the delay to sale component (it was

0.0% in Table 10-4F, D6).

The really signi¬cant difference in the calculation comes from cell

D7, which is 4.0% in Table 10-4G and zero in Table 10-4F. The ¬nal

calculation of component #1 is 5.1% (D12) for the $10 million ¬rm, com-

pared to 8.4% for the $750,000 ¬rm. Thus it seems that component #1

rises sharply somewhere between $375,000 and $750,000 ¬rms, but then

begins to decline as the size effect dominates and causes transactions costs

to decline, while not adding any additional time to sell the ¬rm.

Table 10-6G is our calculation of DLOM for the $10 million ¬rm.

Comparing it to Table 10-6F, the DLOM calculation for the $750,000 ¬rm,

the ¬nal result is 15.0% (Table 10-6G, D14) versus 20.4% (Table 10-6F,

D14). Thus, it appears that DLOM should continue to decline with size.

Thus it appears that DLOM rises with size up to about $1 million in

selling price and declines thereafter. Another factor we did not consider

here that also would contribute to a declining DLOM with size is that

the number of interested buyers would tend to increase with larger size,

which should lower component #2”buyer™s monopsony power”below

the 9% from the Schwert article cited in Chapter 7.

INTERPRETATION OF THE ERROR

As mentioned earlier, the magnitude of the error in Table 10-2 is fairly

small. The ¬ve right columns average a 0.4% error (I29) and a 4.2% (I30)

mean absolute error. We can interpret this as a victory for the log size

and economic components models”and I do interpret it that way, to

some degree. However, the many assumptions that we had to make ren-

der our calculations too speculative for us to place much con¬dence in

them. They are evidence that we are probably not way off the mark, but

certainly fall short of proving that we are right.

An assumption not speci¬cally discussed yet is the assumption that

the simple means of Raymond Miles™s categories is the actual mean of

the transactions in each category. Perhaps the mean of transactions in the

$500,000 to $1 million category is really $900,000, not $750,000. Our results

would be inaccurate to that extent and that would be another source of

error in reconciling between the IBA P/E multiples and my P/CF mul-

tiples. It does appear, though, that Table 10-2 provides some evidence of

the reasonableness of the log size and economic components models.

Amihud and Mendelson (1986) show that there is a clientele effect

in investing in publicly held securities. Investors with longer investment

horizons can amortize their transactions costs, which are primarily the

bid“ask spread and secondarily the broker™s fees,9 over a longer period,

thus reducing the transactions cost per period. Investors will thus select

9. Because broker™s fees are relatively insigni¬cant in publicly held securities, we will ignore them

in this analysis. That is not true of business broker™s fees for selling privately held ¬rms.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 379

their investments by their investment horizons, and each security will

have two components to its return: that of a zero bid“ask spread asset

and a component that rewards the investor for the illiquidity that he is

taking on in the form of the bid“ask spread.

Thus, investors with shorter investment horizons will choose secu-

rities with low bid“ask spreads, which also have smaller gross returns,

and investors with longer time horizons will choose securities with larger

bid“ask spreads and larger gross returns. Their net returns will be higher

on average than those of short-term investors because the long-term in-

vestor™s securities choices will have higher gross returns to compensate

them for the high bid“ask spread, which they amortize over a suf¬ciently

long investment horizon to reduce its impact on net returns. A short-term

investment in a high bid“ask spread stock would lose the bene¬t of the

higher gross return by losing the bid“ask spread in the sale with little

time over which to amortize the spread.

Investors in privately held ¬rms usually have a very long time ho-

rizon, and the transactions costs are considerable compared to the bid“

ask spreads of NYSE ¬rms. In the economic components model I assumed

investors in privately held ¬rms have the same estimate of j, the average

time between sales, in addition to the other variables, growth (g), discount

rate, (r), and buyers™ and sellers™ transactions costs, z. There may be size-

based, systematic differences in investor time horizons; if so, that would

be a source of error in Table 10-2.

Suf¬ciently long time horizons may also predispose the buyer to

forgo some of the DLOM he or she is entitled to. If DLOM should be,

say, 25%, what is the likelihood of the buyer caving in and settling for

20% instead? If time horizons are j 10 years, then the buyer amortizes

the 5% ˜˜loss™™ over 10 years, which equals 0.5% per year. If j 20, then

the loss is only 0.25% per year. Thus, long time horizons should tend to

reduce DLOM, and that is not a part of the economic components

model”at least not yet. It would require further research to determine

if there are systematic relationships between ¬rm size and buyers™ time

horizons.

CONCLUSION

It does seem, then, that we are on our way as a profession to developing

a ˜˜uni¬ed valuation theory,™™ one with one or two major principles that

govern all valuation situations. Of course, there are numerous subprin-

ciples and details, but we are moving in the direction of a true science

when we can see the underlying principles that unify all the various

phenomena in our discipline.

Of course, if one asks if valuation is a science or an art, the answer

is valuation is an art that sits on top of a science. A good scientist has to

be a good artist, and valuation art without science is reckless fortune

telling.

PART 4 Putting It All Together

380

BIBLIOGRAPHY

Amihud, Yakov, and Haim Mendelson. 1986. ˜˜Asset Pricing and the Bid“Ask Spread.™™

Journal of Financial Economics 17:223“249.

Miles, Raymond C. 1992. ˜˜Price/Earnings Ratios and Company Size Data for Small Busi-

nesses.™™ Business Valuation Review (September): 135“139.

Pratt, Shannon P. 1993. Valuing Small Businesses and Professional Practices, 2d ed. Burr Ridge,

Ill.: McGraw-Hill.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 381

CHAPTER 11

Measuring Valuation Uncertainty

and Error

INTRODUCTION

Differences Between Uncertainty and Error

Sources of Uncertainty and Error

MEASURING VALUATION UNCERTAINTY

Table 11-1: 95% Con¬dence Intervals

Valuing the Huge Firm

Valuation Errors in the Others Size Firms

The Exact 95% Con¬dence Intervals

Table 11-2: 60-Year Log Size Model

Summary of Valuation Implications of Statistical Uncertainity in the

Discount Rate

MEASURING THE EFFECTS OF VALUATION ERROR

De¬ning Absolute and Relative Error

The Valuation Model

Dollar Effects of Absolute Errors in Forecastng Year 1 Cash Flow

Relative Effects of Absolute Errors in Forecasting Year 1 Cash Flow

Absolute and Relative Effects of Relative Errors in Forecasting Year 1

Cash Flow

Absolute Errors in Forecasting Growth and the Discount Rate

De¬nitions

The Mathematics

Example Using the Error Formula

Relative Effects of Absolute Error in r and g

Example of Relative Valuation Error

Valuation Effects on Large Versus Small Firms

Relative Effect of Relative Error in Forecasting Growth and

Discount Rates

Tables 11-4“12-4b: Examples Showing Effects on Large vs. Small

Firms

Table 11-5: Summary of Effects of Valuation Errors

SUMMARY AND CONCLUSIONS

383

Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

INTRODUCTION

This chapter describes the impact of various sources of valuation uncer-

tainty and error on valuing large and small ¬rms. It will also provide the

reader with a greater understanding of where our analysis is most vul-

nerable to the effects of errors and demonstrate where appraisers need to

focus the majority of their efforts.

Differences between Uncertainty and Error

It is worthwhile to explain the differences between uncertainty and error.

I developed the log size equation in Chapter 4 by regression analysis.

Because the R2 is less than 100%, size does not explain all of the differ-

ences in historical rates of return. Unknown variables and/or random

variation explain the rest. When we calculate a 95% con¬dence interval,

it means that we are 95% sure that the true value of the dependent vari-

able is within the interval and 5% sure it is outside of the interval. That

is the uncertainty. One does not need to make an error to have uncertainty

in the valuation.

Let™s suppose that for a ¬rm of a particular size, the regression-

determined discount rate is 20% and the 95% con¬dence interval is be-

tween 18% and 23%. It may be that the true and unobservable discount

rate is also 20%, in which case we have uncertainty, but not error. On the

other hand, if the true discount rate is anything other than 20%, then we

have both uncertainty and error”even though we have used the model

correctly. Since the true discount rate is unobservable and unknowable

for privately held ¬rms, we will never be certain that our model will

calculate the correct discount rate”even when we use it properly. If one

makes a mistake in using the model, that is what we mean by appraiser

error. For the remainder of this chapter, we will use the simpler term,

error, to mean appraiser-generated error. The ¬rst part of the chapter deals

with valuation uncertainty, and the second part deals with valuation

error.

Sources of Uncertainty and Error

We need only look at the valuation process in order to see the various

sources of valuation uncertainty and error. As mentioned in the Intro-

duction to this book, the overall valuation process is:

— Forecast cash ¬‚ows.

— Discount cash ¬‚ows to present value.

— Calculate valuation premiums and discounts for degree of

control and marketability.

Uncertainty is always present, and error can creep into our results at each

stage of the valuation process.

MEASURING VALUATION UNCERTAINTY

In forecasting cash ¬‚ows, even when regression analysis is a valid tool

for forecasting both sales and costs and expenses, it is common to have

Part 4 Putting It All Together

384

fairly wide 95% con¬dence intervals around our sales forecasts, as we

discovered in Chapter 2. Thus, we usually have a substantial degree of

uncertainty surrounding the sales forecast and a typically smaller, though

material, degree of uncertainty around the forecast of ¬xed and variable

costs. As each company™s results are unique, we will not focus on a quan-

titative measure of uncertainty around our forecast of cash ¬‚ows in this

chapter.1 Instead, we will focus on quantitative measures of uncertainty

around the discount rate, as that is generic.

For illustration, we use a midyear Gordon Model formula,

(1 r)/(r g), as our valuation formula. Although a Gordon model

is appropriate for most ¬rms near or at maturity, this method is inappli-

cable to startups and other high-growth ¬rms, as it presupposes that the

company being valued has constant perpetual growth.

Table 11-1: 95% Con¬dence Intervals

Table 11-1 contains calculations of 95% con¬dence intervals around the

valuation that results from our calculation of discount rate. We use the

72-year regression equation for the log size model. It is the relevant time

frame for comparison with CAPM, since the CAPM results in the SBBI

1998 Yearbook (Ibbotson Associates 1998) are for 72 years.2 Later, in Table

11-2, we examine the 60-year log size model for comparison. For purposes

of this exercise, we will assume the forecast cash ¬‚ows and perpetual

growth rate are correct, so we can isolate the impact of the statistical

uncertainty of the discount rate.

The exact procedure for calculating the 95% con¬dence intervals is

mathematically complex and would strain the patience of most readers.

Therefore, we will use a simpler approximation in our explanation and

merely present the ¬nal results of the exact calculation in row 42.

Valuing the Huge Firm

Because the log size model produces a mathematical relationship between

return and size, our exploration of 95% con¬dence intervals around a

valuation result necessitates separate calculations for different-size ¬rms.

We begin with the largest ¬rms and work our way down.

In Table 11-1, cell B5 we show last year™s cash ¬‚ow as $300 million.

Using the log size model, the discount rate is 13%3 (B6), and we assume

a perpetual growth rate of 8% (B7). We apply the perpetual growth rate

to calculate cash ¬‚ows for the ¬rst forecast year. Thus, forecast cash ¬‚ow

$300 million 1.08 $324 million (B8).

In B12 we repeat the 13% discount rate. Next we form a 95% con¬-

dence interval around the 13% rate in the following manner. Regression