category. At the high end, H6 $750,000, which is the mean price in the

$500,000 to $1 million sales price category.

Row 7 is the mean P/E multiple reported in the article. Note that

the P/E multiple constantly rises as the mean selling price rises. Figure

10-1 shows this relationship clearly. Row 8 is owner™s discretionary in-

come, which is row 6 divided by row 7, i.e., P P/E E, where P is

price and E is earnings.

The IBA™s de¬nition of owner™s discretionary income is net income

before income taxes and owner™s salary. It does not conform to the arm™s-

length income that appraisers use in valuing businesses. Therefore, we

subtract our estimate of an arm™s-length salary for owners, which we do

in row 9. This is an educated guess, but Raymond Miles felt my estimates

were reasonable.

In row 10, we add back personal expenses charged to the business.

Unfortunately, no one has any data on this. I have asked many account-

ants for their estimates, and their answers vary wildly. Ultimately, I de-

cided to estimate this at 10% (cell B33) of owner™s discretionary income

(row 8).

4. For public ¬rms, this is market capitalization, i.e., price per share number of shares.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 361

362

T A B L E 10-2

Reconciliation to IBA Database

A B C D E F G H I

4 Part 1: IBA P/CF Multiples

6 Mean selling price: Illiquid 100% Int 25,000 75,000 125,000 175,000 225,000 375,000 750,000 Avg

7 Mean P/E ratio 1.66 2.11 2.44 2.74 3.06 3.44 4.26

8 Owner™s discretionary inc [6]/[7] 15,060 35,545 51,230 63,869 73,529 109,012 176,056

9 Arm™s length salary 22,500 25,000 30,000 35,000 40,000 50,000 75,000

10 Personal exp charged to bus”assume B33* [8] 1,506 3,555 5,123 6,387 7,353 10,901 17,606

11 Adjusted net income [8] [9] [10] (5,934) 14,100 26,352 35,255 40,882 69,913 118,662

12 Effective corp. inc tax rate 0% 0% 0% 0% 0% 0% 0%

13 Adjusted inc taxes 0 0 0 0 0 0 0

14 Adj net inc after tax (5,934) 14,100 26,352 35,255 40,882 69,913 118,662

15 Cash ¬‚ow/net income (assumed) 95% 95% 95% 95% 95% 95% 95%

16 Adj cash ¬‚ow after tax [14] * [15] (5,637) 13,395 25,035 33,493 38,838 66,417 112,729

17 Avg disc to cash equiv value (Table 10-3) 6.7% 6.7% 6.7% 6.7% 6.7% 6.7% 6.7%

18 Adj sell price (illiq 100% int) {1 [17]}*[6] 23,317 69,951 116,585 163,220 209,854 349,756 699,512

19 Adjusted price/cash ¬‚ow multiple [18]/[16] NM 5.2 4.7 4.9 5.4 5.3 6.2

21 Part 2: Log Size P/CF Multiples

22 Control prem-% (1982“1991 Avg) [note 1] 25% 25% 25% 25% 25% 25% 25%

23 DLOM-% (Tables 10-6, 10-6A, 10-6B, etc.) 9.9% 10.1% 10.2% 10.2% 10.5% 12.4% 18.6%

24 Adj sell price (mkt min) [18]/{(1 [22])*(1 [23])} 20,704 62,221 103,838 145,440 187,511 319,458 687,614

25 Discount rate r .5352 .0186 ln (FMVMkt Min) 35.0% 33.0% 32.0% 31.4% 30.9% 29.9% 28.5%

26 Growth rate g (assumed) 2.0% 2.5% 3.0% 4.0% 4.5% 5.0% 6.0%

27 Theoretical P/CF (1 g)*SQRT(1 r)/(r g) 3.6 3.9 4.1 4.4 4.5 4.8 5.3

28 P/CF-Illiquid control [27]*(1 [22])*(1 [23]) 4.0 4.4 4.6 4.9 5.1 5.3 5.4

29 Error {1 [28]/[19]} NM 16.5% 1.7% 0.2% 6.3% 0.2% 12.5% 4.1%

30 Absolute error [note 2] NM 16.5% 1.7% 0.2% 6.3% 0.2% 12.5% 4.2%

31 Squared error [note 2] 2.7% 0.0% 0.0% 0.4% 0.0% 1.6% 0.4%

33 Personal exp % of Owner™s discretionary inc 10%

35 Sensitivity Analysis: How the error varies with Cell B33 Error

personal exp

37 2% 17.3%

38 4% 14.0%

39 6% 10.7%

40 8% 7.4%

41 10% 4.1%

[1] Approximate midpoint of the 21% to 28% control premium estimated in Chapter 7

[2] The averages are for the last 5 columns only, as the sales under $100,000 are mostly likely asset-based, not income based.

F I G U R E 10-1

P/E Ratio as a Function of Size (From the IBA Database)

4.5

4

3.5

3

P/E Multiple

2.5

2

1.5

1

0.5

0

25,000 75,000 125,000 175,000 225,000 375,000 750,000

Average Selling Price

Row 11 is adjusted net income, which is row 8 row 9 row 10.

Row 12 is an estimate of the effective corporate income tax rate. This is

a judgment call. An accountant convinced me that even for the $1 million

sales, the owner™s discretionary income is low enough that it would not

be taxed at all. Any excess remaining over salary would be taken out of

taxable income as a bonus. I acceded to his opinion, though this point is

arguable”especially for the higher dollar sales. It is true that what counts

here is not who the seller is, but who the buyer is. A large corporation

buying a small ¬rm would still impute corporate taxes at the maximum

rate; however, only the last category is at all likely to be bought by a large

¬rm, and even then, most buyers of $0.5 to $1 million ¬rms are probably

single individuals. Therefore, it makes sense to go with no corporate

taxes, with a possible reservation in our minds about the last column.

With this zero income taxes assumption, row 13 equals zero and row

14, adjusted income after taxes, equals row 11.

Next we need to convert from net income to cash ¬‚ow. Again, the

information does not exist, so we need to make reasonable assumptions.

For most businesses, cash ¬‚ow lags behind net income. Most of these are

small businesses that sold for fairly small dollar amounts, which means

that expected growth”another important missing piece of information”

must be low, on average. The lower the growth, the less strain on cash

¬‚ow. We assume cash ¬‚ow is 95% of adjusted net income. It would be

reasonable to assume this ratio is smaller for the higher value businesses,

which presumably have higher growth. We do not vary our cash ¬‚ow

ratio, as none of these are likely to be very high-growth businesses. Thus,

all cells in row 15 equal 95%. In row 16 we multiply row 14 by row 15

to calculate adjusted after-tax cash ¬‚ow.

The next step in adjusting the IBA multiples is to reduce the nominal

selling price to a cash-equivalent selling price, which we calculate in Table

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 363

10-3. Exhibit 33-3 in Pratt (Pratt 1993) shows a summary of sale data from

Bizcomps. Businesses selling for less than $100,000 have a 60% average

cash down, and businesses selling for more than $500,000 have an aver-

age 58% cash down. Using a 60% cash down, we assume the seller ¬-

nances the 40% (Table 10-3, B11) balance for 7 years, which is 84 months

(B8, C8) at 8% (B5) with a market rate of 14% (C5).

The annuity discount factor (ADF), the formula for which is

r)n]

1 [1/(1

ADF

r

is 53.3618 (C9) at the market rate of interest and 64.15926 (B9) at the

nominal rate. One minus the ratio of two equals the discount to cash

equivalent value if the loan is 100% ¬nanced, or

53.3618

1 16.8%

64.15926

(B10). We multiply this by the 40% ¬nanced (B11) to calculate the average

discount to cash equivalent value of 6.7% (B12), which we transfer back

to Table 10-2, row 17.

Multiplying the mean selling price in row 6 by one minus the dis-

count to cash equivalent value in row 17 leads to an adjusted mean selling

price in row 18. For example, $25,000 (1 6.7%) $23,317 [B6

(1 B17) B18].

Finally, we divide row 18 by row 16 to calculate the adjusted price

to cash ¬‚ow (P/CF) multiple for the IBA database. In general, the P/CF

multiple rises as price rises, although not always. There is no meaningful

P/CF multiple in B19, because adjusted cash ¬‚ow in B16 is negative. The

P/CF multiples begin in C19 at 5.2 for a mean selling price of $75,000,

then decline to 4.7 (D19) for a mean selling price of $125,000, and rise

steadily to 6.2 (H19) for a mean selling price of $750,000. The only excep-

tion is that the P/CF is greater at 5.4 for the $225,000 selling price than

at 5.3 for the $375,000 selling price. The ¬rst anomaly is probably not

signi¬cant, because many, if not most, ¬rms selling under $100,000 are

T A B L E 10-3

Proof of Discount Calculation

A B C

4 Nominal Market

5 r 8% 14%

6 i r/12 0.6667% 1.1667%

7 Yrs 7 7

8 n Yrs *12 84 84

9 ADF @ 14%, 84 mos. 64.15926114 53.36176

10 Discount on total prin 16.8%

11 % ¬nanced 40%

12 Discount on % ¬nanced 6.7%

PART 4 Putting It All Together

364

priced based on their assets rather than their earnings capacity. The sec-

ond anomaly, from P/CF of 5.4 to 5.3, is a very small reversal of the

general pattern of rising P/CF multiples in the IBA database.

Part 2: Log Size P/CF Multiples

In this section of Table 10-2 we will calculate ˜˜theoretical™™ P/CF multiples

based on the log size model and the DLOM calculations in Chapter 7.

The term theoretical is somewhat of a misnomer, as the calculation of both

the log size equation and DLOM is empirically based. Nevertheless, we

use the term for convenience.

Before we can apply the log size equation from Table 10-1, we need

a marketable minority interest FMV, while the adjusted selling price

(FMV) in row 18 is a illiquid control value. Therefore, we need to divide

row 18 by one plus the control premium times one minus DLOM, which

we do in row 24. We assume a control premium of 25% (row 22), which

is the approximate midpoint of the 21“28% range of control premiums

discussed in Chapter 7.

The calculation of DLOM is unique for each size category and ap-

pears in Tables 10-6 and 10-6A“10-6F. We will cover those tables later. In

the meantime, DLOM rises steadily from 9.9% (B23) for the $25,000 mean

selling price to 18.6% (H23) for the $750,000 mean selling price category.

Row 24, the marketable minority FMV, is row 18 [(1 row 22)

(1 row 23)]. The marketable minority values are all lower than the

illiquid control values, as the control premium is much greater in mag-

nitude than DLOM.

We calculate the log size discount rate in row 25 using the regression

equation from Table 10-1. It ranges from a high of 35.2% (B25) for the

smallest category to a low of 28.7% (H25) for the largest category.

Next we estimate the constant growth rates that the buyers and sell-

ers collectively implicitly forecast when they agreed on prices. It is un-

fortunate that none of the transactional databases that are publicly avail-

able contain even historical growth rates, let alone forecast growth rates.

Therefore, we must make another estimate. We estimate growth rates to

rise from 2% (B26) to 6% (H26), growing at 0.5% for each category, except

the last one going from 5% to 6%. It is logical that buyers will pay more

for faster growing ¬rms.

In row 27 we calculate a midyear Gordon model:

1 r

(1 g)

r g

with r and g coming from rows 25 and 26, respectively.5 This is a mar-

ketable minority interest P/CF multiple when cash ¬‚ow is expressed as

the trailing year™s cash ¬‚ow. In row 28 we convert this to an illiquid

control P/CF by doing the reverse of the procedure we performed in row

5. The purpose of the (1 g) term is correct for the fact that we are applying it to each dollar of

prior year™s cash ¬‚ow and not to the customary next year™s cash ¬‚ow.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 365

24”we multiply by one plus the control premium and one minus DLOM,

i.e., P/CFIlliq Control P/CFMM (1 CP) (1 DLOM) row 27

(1 row 22) (1 row 23).

In row 29 we calculate the error, which is one minus the ratio of row

28 divided by row 19, or one minus the ratio of the forecast log size-

based P/CF to the IBA™s adjusted P/CF. Row 30 is the absolute value of

the errors in row 29. The absolute values of the errors are most extreme

for the low and high values of the mean selling price, with a 16.5% (C30)

absolute error for the $75,000 mean selling price and a 12.5% (H30) ab-

solute error for the $750,000 selling price, with small absolute errors in

between ranging around 0.2“6.3%. The mean error is 4.1% (I29).6

Conclusion

The mean absolute error is 4.2% (I30). Rounding this to 4%, that is a very

respectable result. It is evidence supporting the log size model in Chapter

4 and control premium and economic components model of DLOM in