Adjust for marketability or lack thereof.

3.

Adjust for degree of control.

4.

Applying a Valuation Model to the Steps

The sales described in the article are all $1 million or less. It is a reason-

able assumption that the vast majority of the small ¬rms in the IBA trans-

actional database are mature. The number of high-growth startup ¬rms

in that database is likely to be small. Therefore, it is reasonable to assume

a constant growth rate to perpetuity. Using a Gordon model to apply to

the next year™s forecast cash ¬‚ows should give us a fairly accurate FMV

on a marketable minority level. Using a midyear assumption, the formula

is:

1 r

FMV CFt 1

r g

where r is the discount rate, which we will estimate using the log size

model, and g is the constant growth rate, which we will estimate. That

takes care of the ¬rst two valuation steps.

PART 4 Putting It All Together

358

We will use the economic components model from Chapter 7 for our

calculations of DLOM. We assume a control premium of 25%, which is

the approximate midpoint of the 21“28% range estimated in Chapter 7.

There are only two major principals in steps 2 and 3 of business

valuation: risk and marketability, which are both functions of size. Thus,

size is the overriding principle in steps 2 and 3 of the valuation process,

and step 1 determines size. If value depends only on the forecast cash

¬‚ows, risk, and marketability, and the latter two are in turn dependent

on size, then in essence value depends only on size (and possibly control).

That statement sounds like a tautology, but it is not.

This chapter is an attempt to identify the fewest, most basic princi-

ples underlying the inexact science of valuation. The remainder of this

chapter covers the calculations that test the log size model and DLOM

calculations.

TABLE 10-1: LOG SIZE FOR 1938“1986

In Table 10-1 we develop the log size equation for the years 1938“1986.

We use 1938 as the starting year to eliminate the highly volatile Roaring

Twenties and Depression years 1926“1937. The reason we stop at 1986

has to do with the IBA database. The article is based on sales from 1982“

1991.2 We take 1986 as the midpoint of that range and calculate our log

size equation from 1938“1986.

Cells B7“B16 and C7“C16 contain the mean and standard deviation

of returns for the 10 deciles for the period 1938“1986. We need to be able

to regress the returns against 1986 average market capitalization for each

decile. Unfortunately, those values are unavailable and we must estimate

them.

D7“D16 contain the market capitalization for the average ¬rm in

each decile for 1994, the earliest year for which decile breakdowns are

available. E7“E16 are the 1986 year-end index values in Ibbotson™s Table

7-4. F7“F16 are the 1994 year-end index values, with our estimate of in-

come returns removed.3

Column G is our estimate of 1986 average market capitalization per

¬rm for each decile. We calculate it as Column D Column E Column

F. Thus, the average ¬rm size in decile #1 for 1986 is $7.3 billion (G7),

and for decile #10 it is $32.49 million (G16).

Rows 18“35 contain our regression analysis of arithmetic mean re-

turns as a function of the logarithm of the market capitalization”exactly

2. A footnote in the article states that in relation to Figure 1 (and I con¬rmed this with the author,

Raymond Miles), those dates apply to the rest of the article.

3. SBBI, Table 7-4, approximate income returns have been removed from the 1994 values. The

adjustment was derived by comparing the large company stock total return indices with the

capital appreciation indices for 1994 and 1986 per SBBI Tables B-1 and B-2. It was found

that 77.4% of the total return was due to capital appreciation. There were no capital

appreciation indices for small company stocks. We removed 1 77.4% 22.6% of the gain

in the decile index values for deciles #1 through #5, 22.6%/2 11.3% for deciles #6 through

#8, and made no adjustment for #9 and #10. Larger stocks tend to pay larger dividends.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 359

360

T A B L E 10-1

Log Size Equation for 1938“1986 NYSE Data by Decile and Statistical Analysis: 1938“1986

A B C D E F G H I

5 Year-End Index Values [1] [D] [E]/[F] Ln [G]

6 Decile Mean Std Dev 94 Mkt Cap 1986 1994 1986 Mkt Cap Ln(Mkt Cap)

7 1 11.8% 15.8% 14,847,774,614 198.868 404.436 7,300,897,357 22.7113

2 14.0% 18.3% 3,860,097,544 434.686 920.740 1,822,371,137 21.3234

9 3 15.0% 19.7% 2,025,154,234 550.313 1,248.528 892,625,877 20.6097

10 4 15.8% 22.0% 1,211,090,551 637.197 1,352.924 570,396,575 20.1618

11 5 16.7% 23.0% 820,667,228 856.893 1,979.698 355,217,881 19.6882

12 6 17.1% 23.8% 510,553,019 809.891 1,809.071 228,566,124 19.2473

13 7 17.6% 26.4% 339,831,804 786.298 1,688.878 158,216,901 18.8795

14 8 19.0% 28.5% 208,098,608 1,122.906 2,010.048 116,253,534 18.5713

15 9 19.7% 29.9% 99,534,481 1,586.521 2,455.980 64,297,569 17.9790

16 10 22.7% 38.0% 33,746,259 6,407.216 6,654.508 32,492,195 17.2965

18 SUMMARY OUTPUT

20 Regression Statistics

21 Multiple R 0.9806

22 R square 0.9617

23 Adjusted R square 0.9569

24 Standard error 0.0064

25 Observations 10

27 ANOVA

28 df SS MS F Signi¬cance F

29 Regression 1 0.0082 0.0082 200.6663 0.0000

30 Residual 8 0.0003 0.0000

31 Total 9 0.0085

33 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%

34 Intercept 0.5352 0.0259 20.6710 0.0000 0.4755 0.5949

35 Ln(Mkt Cap) (0.0186) 0.0013 (14.1657) 0.0000 (0.0216) (0.0156)

[1] SBBI, Table 7-3*, approximate income returns have been removed from the 1994 values. The adjustment was derived by comparing the large company stock total return indices with the capital appreciation indices for 1994 and 1986 per

SBBI Tables B-1 and B-2. It was found that 77.4% of the total return was due to capital appreciation. There were no capital appreciation indices for small company stocks. We removed (1-77.4%) of the gain in the decile index values for

deciles 1 through 5, [(1-77.4%)/2] for deciles 6 through 8, and made no adjustment for 9 and 10. Larger stocks tend to pay larger dividends.

*Used with permission. 1998 Ibbotson Associates, Inc. All rights reserved. [Certain portions of this work were derived from copyrighted works of Roger G. Ibbotson and Rex Sinque¬eld.] Source: CRSP University of Chicago, Used with

permission. All rights reserved.

the same as Table 4-1, regression #2. The regression equation is: r 0.5352

“ 0.0186 ln FMV.4 We use this regression equation in Table 10-2.

TABLE 10-2: RECONCILIATION TO THE IBA DATABASE

Table 10-2 is the main table in this chapter. All other tables provide details

that ¬‚ow into this table.

The purpose of the table is to perform two series of calculations,

which make up part 1 and part 2 of the table, respectively. The ¬rst series

calculates adjusted price to cash ¬‚ow (P/CF) multiples for each size cat-

egory of IBA database results described in the article. The second series

is to calculate theoretical P/CF multiples using the log size equation and

the DLOM methodology in Chapter 7. Ultimately we compare them, and

they match reasonably well.

Unfortunately, there are much data that we do not have, which will

force us to make estimates. There are so many estimates in the following

analysis, that we will not be able to make strong conclusions. It would

be easy to manipulate the results in Table 10-2 to support different points

of view. Nevertheless, it is important to proceed with the table, as we

will still gain valuable insights. Additionally, it points out the de¬ciencies

in the information set available. This is not a criticism of the IBA database.

All of the other transactional databases of which I am aware suffer from

the same problems. This analysis highlights the type of information that

would be ideal to have in order to come to stronger conclusions.

Part 1: IBA P/CF Multiples

We begin in row 6. The mean selling prices in row 6 are the means of the

corresponding range of selling prices reported in the article. Thus, B6

$25,000, which is the mean selling price for ¬rms in the $0 to $50,000

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.