but we would still have come to the counterintuitive conclusion that

DLOM increases with ¬rm size.

Calculation of DLOM for Large Firms

The preceding result begs the question of what happens to DLOM beyond

the realm of small ¬rms. To answer this question, we extend our analysis

to Tables 10-4G and 10-6G.

Table 10-4G is otherwise identical to its predecessor, Table 10-4F.

Since we do not have the bene¬t of the IBA data at this size level, we

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

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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

1. In the second part of the chapter we will explore the valuation impact of appraiser error in

forecasting cash ¬‚ows.

2. While Chapter 4 was updated to include the Ibbotson 1999 SBBI Yearbook results, this chapter

has not. Therefore, this chapter does not contain the 1998 stock market results, which were

very poor for the log size model. As noted in Chapter 4, large ¬rms outperformed small

¬rms. Therefore, the con¬dence intervals calculated in this chapter would be wider if we

were to include the 1998 results, which are reported in the 1999 SBBI Yearbook.

3. Calculation of the log size discount rate is in rows 35“38. The regression equation in these rows

is based on the 1998 SBBI Yearbook and therefore does not match the equation in Table 4-1.

CHAPTER 11 Measuring Valuation Uncertainty and Error 385

#2 in Table 4-1 has 10 observations. The number of degrees freedom is n

k 1, where n is the number of observations and k is the number of

independent variables; thus we have eight degrees of freedom. Using a

t-distribution with eight degrees of freedom, we add and subtract 2.306

standard errors to form a 95% con¬dence interval. The standard error of

the log size equation through SBBI 1998 was 0.76% (B48), which when

multiplied by 2.306 equals 1.75%. The upper bound of the discount rate

calculated by log size is 13% 1.75% 14.75% (B11), and the lower

4

bound is 13% 1.75% 11.25% (B13).

For purposes of comparison, we assume that CAPM also arrives at

a 13% discount rate (B16). We multiply the CAPM standard error of 2.42%

(B49) by 2.306 standard errors, yielding 5.58% for our 95% con¬dence

interval. In cell B15 we add 5.58% to the 12% discount rate, and in cell

B17 we subtract 5.58% from the 12% rate, arriving at upper and lower

bounds of 18.58% and 7.42%, respectively.

Rows 19 to 21 show the calculations of the midyear Gordon model

multiples (GM) (1 r)/(r g). For r 13% 1.75% and g 8%,

GM 21.2603 (B20), which we multiply by the $324 million cash ¬‚ow

(B8) to come to an FMV (ignoring discounts and premiums) of $6.89 bil-

lion (B24).

We repeat the process using 14.75%, the upper bound of the 95%

con¬dence interval for the discount rate (B11) in the GM formula, to come

to a lower bound of the GM of 15.8640 (B19). Similarly, using a discount

rate of 11.25% (the lower bound of the con¬dence interval, B13) the cor-

responding upper bound GM formula is 32.4791 (B21). The FMVs asso-

ciated with the lower and upper bound GMMs are $5.14 billion (B23) and

$10.52 billion (B25), or 74.6% (C23) and 152.8% (C25), respectively, of our

best estimate of $6.89 billion.

Cell C39 shows the average size of the 95% con¬dence interval

around the valuation estimate. It is 39%, which is equal to 1„2 [(1

74.6%) (152.8% 1)]. It is not literally true that the 95% con¬dence

interval is the same above and below the estimate, but it is easier to speak

in terms of a single number.

Row 28 shows the Gordon model multiple using a CAPM discount

rate, which we assume is identical to the log size model discount rate.

Using the CAPM upper and lower bound discount rates in B15 and B17,

the lower and upper bounds of the 95% con¬dence interval for the CAPM

Gordon model are 10.2920 (B27) and 178.5324 (B29), respectively. Ob-

viously, the latter is an explosive, nonsense result, and the average 95%

con¬dence interval is in¬nite in this case.

4. This is an approximation. The exact formula is:

x2

1 0

Y0 ˆ0 t0.025s 1

x2

n i

i

where ˆ 0 is the regression-determined discount rate for our subject company, xi are the

deviations of the natural logarithm of each decile™s market capitalization from the mean log

of the 10 Ibbotson decile average market capitalizations, t0.025 is the two-tailed, 95% t-

statistic, s is the standard error of the y-estimate as calculated by the regression, n 10, the

number of deciles in the regression sample, and x0 is the deviation of the log of the FMV of

the subject company from the mean of the regression sample.

Part 4 Putting It All Together

386

We obtain the same estimate of FMV for CAPM as the log size model

(B32, B24), but look at the lower bound estimate in B31. It is $3.33 billion

(rounded), or 48.4% (C31) of the best estimate, versus 74.6% (C23) for the

same in the log size model. The CAPM standard error being more than

three times larger creates a huge con¬dence interval and often leads to

explosive results for very large ¬rms.

Valuation Error in the Other-Size Firms

The remaining columns in Table 11-1 have the same formulas and logic

as columns B and C. The only difference is that the size of the ¬rm varies,

which implies a different discount rate and therefore different 95% con-

¬dence intervals. In column D we assume the large ¬rm had cash ¬‚ows

of $15 million last year (D5), which will grow at 7% (D7). We see that the

log size model has an average 95% con¬dence interval of 14% (E39)

and CAPM has an average 95% con¬dence interval of 56% (E40).

Columns F and H are successively smaller ¬rms. Note how the min-

imum valuation uncertainty declines with ¬rm size.

The approximate 95% con¬dence intervals for log size are 39%, 14%,

9%, and 7% (row 39) for the huge, large, medium, and small ¬rm, re-

spectively. The CAPM con¬dence intervals also decline with ¬rm size,

but are much larger than the log size con¬dence intervals. For example,

the CAPM small ¬rm 95% con¬dence interval is 23% (I40)”much

larger than the 7% (I39) interval for the Log Size Model.

The Exact 95% Con¬dence Intervals

As mentioned earlier, rows 39 and 40 are a simpli¬ed approximation of

the 95% con¬dence intervals around the discount rates, used to minimize

the complexity of an already intricate series of calculations and related

explanations.

Row 42 contains the exact 95% con¬dence intervals for log size. Note

that the exact 95% con¬dence intervals are larger than their approxima-

tions in Rows 39 to 40. There are no actual 95% con¬dence intervals for

CAPM.5

Aside from the direct effect of size on the calculation of the discount

rate, there is a secondary, indirect effect of size on the con¬dence inter-

vals. All other things being equal, con¬dence intervals are at their mini-

mum at the mean of the data set, which is over $4 billion for the NYSE,

and increase the further we move away from the mean. The huge ¬rm

in column B”and to a lesser extent the large ¬rm in column D”are close

to the mean of the NYSE market capitalization. Therefore, we have two

opposing forces operating on the con¬dence intervals. The mathematics

of the log size equation and Gordon model multiple are such that the

smaller the ¬rm, the smaller the con¬dence interval for the FMV. How-

ever, the smaller ¬rms are far below the mean of the NYSE sample, so

that tends to increase the actual 95% con¬dence interval.

Thus, the direct effect and the indirect effect on the con¬dence inter-

vals work in opposite directions. Jumping ahead of ourselves for a mo-

5. The reason for this is that the CAPM calculations in the SBBI Yearbook are not a pure

regression, because the y-intercept is forced to the risk-free rate.