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

Chapter 7.

Nevertheless, as mentioned before, there are too much missing data

and resulting guesswork to come to solid conclusions. The estimates are

all reasonable, but one could make different reasonable estimates and

come to very different results. Thus, this analysis is worthwhile evidence,

but it proves nothing.

In the remainder of the chapter we will describe the DLOM calcu-

lations in Tables 10-4, 10-6, and their variations as 10-4A, 10-6A, etc.

CALCULATION OF DLOM

As discussed in Chapter 7, there are three components in the economic

components model to the calculation of DLOM. Components #1 and #3,

the delay to sale and transactions costs components, require unique anal-

ysis for each IBA size category. Therefore, we have one spreadsheet for

each of the two components for each IBA size category. Tables 10-4 and

10-6 are the calculations of components #1 and #3, respectively, for the

$25,000 mean selling price ¬rm. Additionally, Table 10-6 contains the

DLOM calculations. We will describe these tables in detail. Tables 10-4A

and 10-6A are identical to Tables 10-4 and 10-6, the only difference being

that these are calculations for the $75,000 mean selling price ¬rms. This

series continues all the way through Tables 10-4F and 10-6F for the

$750,000 mean selling price IBA category. Table 10-5 contains the calcu-

lations of the buyer and seller transactions costs for all size categories.

Table 10-4: Computation of the Delay-to-Sale

Component”$25,000 Firm

Table 10-4 is identical to Table 7-10, except that we are customizing the

calculation for this IBA category of ¬rm. We begin by inserting the selling

6. This excludes the $75,000 mean selling price errors, as that is likely due to the sale being priced

on an asset rather than an income basis. We also exclude this category in the other measures

of mean error.

PART 4 Putting It All Together

366

T A B L E 10-4

Calculation of Component #1”Delay to Sale”$25,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2

6 5.33E 18 5.625E 09 0.0%

7 Value of block-post-discount [2] 4.26E 09 $ 25,000 0.0%

8 FMV-marketable minority 100% interest 5.97E 10 $ 25,000 0.0%

9 Earnings stability [3] 0.1376 0.4200 5.8%

10 Revenue stability [3] 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.2500 3.3%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $ 25,000

16 Selling price $ 25,000

17 Adjusted net income $ (5,934)

18 Assumed pre-tax margin NA

19 Sales $ 75,000

Sales2

20 5.625E 09

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

price in B16 and adjusted net income in B17. For the larger IBA categories,

net income (owner™s discretionary income) is positive, and we divide that

by an assumed pretax margin of 5% in B18 to estimate sales in B19. We

cannot do that for the $25,000 sales category only, because of net losses.

We estimate sales at three times the selling price, or $75,000 (B19). The

square of sales is then $5.625 109, which is calculated in B20 and trans-

ferred to C6.7

We insert the $25,000 mean selling price in C8, C14, and C16. Here

we are calculating the value of 100% of the stock, so the block value and

the value of the entire ¬rm will be identical, which is not true in the

restricted stock calculations in Table 7-10.8

Cell C7 is the post-discount value of the block. However, both C7

and C14 equal $25,000. This is because the discount calculation came to

zero (D12). Normally, C7 would be lower than C14.

A correlation analysis of the Management Planning data, not shown

in the book, revealed that ¬rm size and earnings and revenue stability

are uncorrelated. Thus, we use the averages from Table 7-5, G60 and H60

of 0.42 (C9) and 0.69 (C10), respectively.

7. The calculations in B16 to B20 did not appear in Table 7-10, as they were unnecessary there.

8. Technically, we should be using the marketable minority FMV rather than the illiquid control

FMV in Table 10-4 (and its variants 10-4A, etc.), cell C14 (which also affects C7 and C8).

However, we do not yet know the marketable minority FMV, as that is the point of the

exercise. To even attempt to calculate it would require multiple iterations, which would

greatly complicate the analysis and add nothing, as the regression coef¬cients in B7 and B8

are so small that the difference is immaterial. Therefore, we use the illiquid control values.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 367

Finally we assume that a $25,000 ¬rm takes only three months, or

0.25 (C11) years to sell. Summing D5 through D11 actually results in a

slightly negative discount, which does not make sense. Therefore, we use

a spreadsheet formula to calculate D12 as the maximum of the sum of

D5:D11 and zero. The delay to sale component is zero for all size cate-

gories except $375,000 and $750,000. The calculations of component #1 of

DLOM for those two categories appear in Tables 10-4E and 10-4F. The

main reason for this is that we assume is takes either 0.25 years or 0.33

years to sell ¬rms under the $375,000 category, while we assume that it

takes 0.5 years and 1.0 years to sell in the $375,000 and $750,000 catego-

ries, respectively (Tables 10-4E and 10-F, C11). The resulting discounts are

still small in magnitude. In Table 10-4E, D12, we calculate component #1

as 1.9%, and in Table 10-4F, D12, we calculate component #1 as 8.4%.

Though we did not elect to do so here, it would be a reasonable

approach to rely on our ¬ndings in Chapter 7 that the regression analysis

does not work well for delays to sale of much less than a year. That being

the case, it would make sense to use a different model”even something

so simple as a present value”to calculate the delay to sale component

for under one year. For example, if we assume a 25% discount rate, a

three-month delay to sale implies a 5% discount as component #1, and a

four-month delay to sale implies a 7% discount as component #1. It is

important to recognize that not all models work well across all ranges of

data, and sometimes circumstances force us to use different models. For

simplicity in this analysis, we did not elect to use another model.

Table 10-5: Calculation of Transactions Costs

Table 10-5 contains our calculations of transactions costs for both buyer

and seller for all of the IBA size categories. Column A denotes whether

the transactions costs are for buyer or sellers. Column B is the mean

selling price of the IBA study. Column C is the base 10 logarithm of

column B.

Columns D and F contain, respectively, the x-coef¬cient and the con-

stant from the regression in Table 7-11. In column E we multiply column

C by column D. We add columns E and F together to obtain column G,

which is the regression forecast of all transactions costs except for the

business broker (or investment banker). Column H contains the business

broker fees, which we assume at 10% for sellers and zero for buyers.

Finally, column I is the grand total forecast of transactions costs for buyers

and sellers by size category. Note that both buyer and seller transactions

costs decline as ¬rm size grows.

While the $10 million ¬rm in rows 20 and 21 are outside of the scope

of the IBA study, we use them later on in our own analysis to extrapolate

the results that we derive from our analysis of the IBA study.

Table 10-6: Calculation of DLOM

Table 10-6 is exactly the same format and logic as Table 7-14, which we

already described in Chapter 7. B9 through B12 contain the pure dis-

counts for the four economic components. B9, the pure discount for com-

PART 4 Putting It All Together

368

T A B L E 10-5

Calculation of Transaction Costs for Firms of All Sizes in the IBA Study

A B C D E F G H I

5 FMV log10 FMV X-Coeff. log FMV Coeff. Regr. Constant Forecast Subtotal Bus. Broker Forecast Total

6 Buyer $ 25,000 4.39794 0.01727 0.07596 0.15310 7.7% 0.0% 7.7%

7 Seller $ 25,000 4.39794 0.01599 0.07034 0.14139 7.1% 10.0% 17.1%

8 Buyer $ 75,000 4.87506 0.01727 0.08420 0.15310 6.9% 0.0% 6.9%

9 Seller $ 75,000 4.87506 0.01599 0.07797 0.14139 6.3% 10.0% 16.3%

10 Buyer $ 125,000 5.09691 0.01727 0.08804 0.15310 6.5% 0.0% 6.5%

11 Seller $ 125,000 5.09691 0.01599 0.08152 0.14139 6.0% 10.0% 16.0%

12 Buyer $ 175,000 5.24304 0.01727 0.09056 0.15310 6.3% 0.0% 6.3%

13 Seller $ 175,000 5.24304 0.01599 0.08386 0.14139 5.8% 10.0% 15.8%

14 Buyer $ 225,000 5.35218 0.01727 0.09245 0.15310 6.1% 0.0% 6.1%

15 Seller $ 225,000 5.35218 0.01599 0.08561 0.14139 5.6% 10.0% 15.6%

16 Buyer $ 375,000 5.57403 0.01727 0.09628 0.15310 5.7% 0.0% 5.7%

17 Seller $ 375,000 5.57403 0.01599 0.08915 0.14139 5.2% 10.0% 15.2%

18 Buyer $ 750,000 5.87506 0.01727 0.10148 0.15310 5.2% 0.0% 5.2%

19 Seller $ 750,000 5.87506 0.01599 0.09397 0.14139 4.7% 10.0% 14.7%

20 Buyer $10,000,000 7.00000 0.01727 0.12091 0.15310 3.2% 0.0% 3.2%

21 Seller $10,000,000 7.00000 0.01599 0.11196 0.14139 2.9% 2.0% 4.9%

Note: Regression constants and x-coef¬cients come from Table 7-11. The $10 million ¬rm, using a Lehman Bros. Formula, has a 2% investment banker fee instead of a 10% business broker™s fee.

369

T A B L E 10-6

Calculation of DLOM

A B C D E F G

4 Section 1: Calculation of the Discount For Lack of Marketability

6 1 Col. [C]

7 Pure Discount PV of Perpetual Remaining

8 Component z [1] Discount [2] Value

9 1 0.0% 0.0% 100.0% Delay to sale

10 2 9.0% 9.0% 91.0% Buyer™s monopsony power”thin markets

11 3A 5.7% 6.1% 93.9% Transactions costs”buyers

12 3B 15.1% 1.0% 99.0% Transactions costs”sellers

13 Percent remaining 90.1% Total % remaining components 1 2 3A 3B

14 Final discount 9.9% Discount 1 total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $ 25,000

19 Discount rate r [3] 34.7%

20 Constant growth rate g 2.0%

21 Intermediate calculation: x (1 g)/(1 r) 0.7574

22 Avg # years between sales j 10

[1] Pure Discounts: For Component #1, Table 10-4, cell D12; For Component #2, 9% per Schwert article. For Component #3A and #3B, Table 10-5, cells I6 and I7 2% for public

brokerage costs.

[2] PV of Perpetual Discount Formula: 1 (1 x j)/((1 (1 z)*x j)), per equation [7-9], used for Component #3B.

PV of Perpetual Discount Formula: 1 (1 z)*(1 x j)/((1 (1 z)*x j)), per equation [7-9a], used for Component #3A.

Components #1 and #2 simply transfer the pure discount.

[3] The formula is: 0.5352 (.0186 ln FMV), based on Table 10-1, B34 and B35.

ponent #1, equals zero, and that comes from our calculation in Table

10-4, D12. B10, the pure discount for component #2, equals 9%. That is

the same as it was in Table 7-14, and it comes from the Schwert article.

Components 3A and 3B come from Table 10-5, cells I6 and I7, respectively,

less a 2% brokerage cost for publicly traded stock. These two components

are equal to 5.7% (B11) and 15.1% (B12), respectively.

As in Table 7-12, the ¬rst two components transfer from B9 and B10

to C9 and C10 directly. However, as discussed in the commentary to Table

7-12, transactions costs ˜˜leave the system™™ with every sale. Thus, we must

present value a perpetuity of transactions costs that occur every j 10

years. We do so using the formulas in note [2] to the spreadsheet, which

are equations (7“9) and (7“9a) from Chapter 7. The present value of all

buyers™ transactions costs is 6.1% (C11), and the present value of all sell-

ers™ transactions costs is 1.0% (C12). The ¬nal calculation of DLOM is

11.9% (D14)

Tables 10-6A“10-6F: Calculations of DLOM for

Larger Firms

Tables 10-6A“10-6F are structured and calculated identically to Table

10-6. There are ¬ve differences in the parameters, the ¬rst four of which

tend to increase DLOM as ¬rm size increases, and the last to decrease

DLOM as ¬rm size increases.

PART 4 Putting It All Together

370

T A B L E 10-4A

Calculation of Component #1”Delay to Sale”$75,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2

6 5.33E 18 7.952E 10 0.0%

7 Value of block-post-discount [2] 4.26E 09 $ 75,000 0.0%

8 FMV-marketable minority 100% interest 5.97E 10 $ 75,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.2500 3.3%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $ 75,000

16 Selling price $ 75,000

17 Adjusted net income $ 14,100

18 Assumed pre-tax margin 5%

19 Sales $281,991

Sales2

20 7.95E 10

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

T A B L E 10-4B

Calculation of Component #1”Delay to Sale”$125,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2 [2]

6 5.33E 18 2.778E 11 0.0%

7 Value of block-post-discount [2] 4.26E 09 $125,000 0.0%

8 FMV-marketable minority 100% interest 5.97E 10 $125,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.3330 4.5%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $125,000

16 Selling price $125,000

17 Adjusted net income $ 26,352

18 Assumed pre-tax margin 5%

19 Sales $527,049

Sales2

20 2.78E 11

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 371

T A B L E 10-4C

Calculation of Component #1”Delay to Sale”$175,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2 [2]

6 5.33E 18 4.972E 11 0.0%

7 Value of block-post-discount [2] 4.26E 09 $175,000 0.0%

8 FMV-marketable minority 100% interest 5.97E 10 $175,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.3330 4.5%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $175,000

16 Selling price $175,000

17 Adjusted net income $ 35,255

18 Assumed pre-tax margin 5%

19 Sales $705,109

Sales2

20 4.97E 11

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10

[2] Equal to Pre-Discount Shares Sold in dollars * (1-Discount). B7 equals B14 only when the discount 0%.

[3] Earnings and Revenue stability are assumed at the averages from Table 7-5, G60 and H60, respectively, for all FMVs. In the

Management Planning data, a correlation analysis revealed that ¬rm size and the stability measures are uncorrelated. Therefore, we

assume the same levels for all FMVs.

[4] Total Discount max(discount, 0), because Disc 0 indicates the model is outside of its range of reasonability.

[5] In our regression of the Management Planning, Inc. data, this was a marketable minority interest value. This is an illiquid control

value and is higher by 12% to 25% than the marketable minority value. The regression coef¬cient relating to market capitalization in

B8 is so small that the difference is immaterial, and it is easier to work with the value available.

T A B L E 10-4D

Calculation of Component #1”Delay to Sale”$225,000 Firm [1]

A B C D

4 Coef¬cients Co. Data Discount

5 Intercept 0.1342 NA 13.4%

Revenues2 [2]

6 5.33E 18 6.685E 11 0.0%

7 Value of block-post-discount [2] 4.26E 09 $225,000 0.1%

8 FMV-marketable minority 100% interest 5.97E 10 $225,000 0.0%

9 Earnings stability (assumed) 0.1376 0.4200 5.8%

10 Revenue stability (assumed) 0.1789 0.6900 12.3%

11 Average years to sell 0.1339 0.3330 4.5%

12 Total Discount [4] 0.0%

14 Value of block-pre-discount [5] $225,000

16 Selling price $225,000

17 Adjusted net income $ 40,882

18 Assumed pre-tax margin 5%

19 Sales $817,647

Sales2

20 6.69E 11

[1] Based on Abrams regression of Management Planning, Inc. data-Regression #2, Table 7-10