20 Constant growth rate g (Table 10-2, row 24) 2.5%

21 Intermediate calculation: x (1 g)/(1 r) 0.7728

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 I8 and I9 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.

PART 4 Putting It All Together

374

T A B L E 10-6B

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 4.5% 4.9% 95.1% Transactions costs”buyers

12 3B 14.0% 1.3% 98.7% Transactions costs”sellers

13 Percent remaining 89.8% Total % remaining components 1 2 3A 3B

14 Final discount 10.2% Discount 1 total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $ 125,000

19 Discount rate r [3] 31.7%

20 Constant growth rate g (Table 11-2, row 24) 3.0%

21 Intermediate calculation: x (1 g)/(1 r) 0.7822

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 I10 and I11 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.

T A B L E 10-6C

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 4.3% 4.7% 95.3% Transactions costs”buyers

12 3B 13.8% 1.3% 98.7% Transactions costs”sellers

13 Percent remaining 89.8% Total % remaining components 1 2 3A 3B

14 Final discount 10.2% Discount 1 total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $ 175,000

19 Discount rate r [3] 31.0%

20 Constant growth rate g (Table 10-2, row 24) 3.0%

21 Intermediate calculation: x (1 g)/(1 r) 0.7860

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 I12 and I13 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.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 375

T A B L E 10-6D

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 4.1% 4.5% 95.5% Transactions costs”buyers

12 3B 13.6% 1.6% 98.4% Transactions costs”sellers

13 Percent remaining 89.5% Total % remaining components 1 2 3A 3B

14 Final discount 10.5% Discount 1 total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $ 225,000

19 Discount rate r [3] 30.6%

20 Constant growth rate g (Table 10-2, row 24) 4.5%

21 Intermediate calculation: x (1 g)/(1 r) 0.8003

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 I4 and I5 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.

T A B L E 10-6E

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 1.9% 1.9% 98.1% Delay to sale

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

11 3A 4.7% 5.3% 94.7% Transactions costs”buyers

12 3B 14.2% 1.9% 98.1% Transactions costs”sellers

13 Percent remaining 87.6% Total % remaining components 1 2 3A 3B

14 Final discount 12.4% Discount 1 total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $ 375,000

19 Discount rate r [3] 29.6%

20 Constant growth rate g (Table 11-2, row 24) 5.0%

21 Intermediate calculation: x (1 g)/(1 r) 0.8100

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 1% 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.

PART 4 Putting It All Together

376

T A B L E 10-6F

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 8.4% 8.4% 91.6% Delay to sale

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

11 3A 4.2% 4.8% 95.2% Transactions costs”buyers

12 3B 13.7% 2.3% 97.7% Transactions costs”sellers

13 Percent remaining 81.4% Total % remaining components 1 2 3A 3B

14 Final discount 18.6% Discount 1 total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $ 750,000

19 Discount rate r [3] 28.3%

20 Constant growth rate g (Table 11-2, row 24) 6.0%

21 Intermediate calculation: x (1 g)/(1 r) 0.8259

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 I8 and I9 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.

T A B L E 10-6G

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 5.1% 5.1% 94.9% Delay to sale

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

11 3A 2.7% 3.6% 96.4% Transactions costs”buyers

12 3B 4.4% 1.5% 98.5% Transactions costs”sellers

13 Percent remaining 85.0% Total % remaining components 1 2 3A 3B

14 Final discount 15.0% Discount 1 total % remaining

16 Section 2: Assumptions and Intermediate Calculations:

18 FMV-equity of co. (before discounts) $10,000,000

19 Discount rate r [3] 23.5%

20 Constant growth rate g (Table 10-2, row 24) 8.0%

21 Intermediate calculation: x (1 g)/(1 r) 0.8743

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 I20 and I21 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.

CHAPTER 10 Empirical Testing of Abrams™ Valuation Theory 377

1. As ¬rm size increases, our assumed growth rate, g, increases. By

our analysis of the partial derivatives in the Mathematical

Appendix to Chapter 7, that causes an increase in DLOM.

2. As ¬rm size increases, the log size discount rate, r, decreases. By

our analysis of the partial derivatives in the Mathematical

Appendix to Chapter 7, that also causes an increase in DLOM.

3. As mentioned earlier, for ¬rm sizes under $375,000, we assumed

the delay to sale to be 0.33 years or less, which lead to a zero

discount for component #1. For the $375,000 and $750,000 ¬rms,

we assumed a one-half-year and one-year delay to sale, which

led to a component #1 pure discount of 1.9% (Table 10-6E, B9)

and 8.4% (Table 10-6F, B9), respectively. The latter accounts for

the vast majority of the much higher DLOM for the $750,000

mean selling price ¬rms. Had that been zero, like all of the

others except the $375,000 ¬rm, DLOM for the $750,000 ¬rms

would have been 13.1%”much closer to DLOM for the smaller

¬rms.

4. We assumed a 1% broker™s fee for publicly traded stocks for the

$375,000 and $750,000 ¬rms, while we assumed a 2% fee for the

¬rms under that size. This increased the pure discount for

components #3A and #3B by 1% for those two size categories,

and therefore increased DLOM.

5. Transactions costs decrease as size increases. Buyers™ transactions

costs are 7.7% (Table 10-5, I6) for $25,000 ¬rms and 5.2% for

$750,000 ¬rms (I18), for a difference of 2.5%. Sellers™ transactions

costs are 17.1% (I7) for $25,000 ¬rms and 14.7% (I19) for

$750,000 ¬rms, for a difference of 2.4%.

Items 1 through 4 above cause DLOM to increase with size, while

item 5 causes DLOM to decrease with size. Looking at Table 10-2, it is

clear that the ¬rst four items dominate, which causes DLOM to increase

with size. This is not a result that I would have predicted before. I would

have thought that overall, DLOM decreases with size.

As mentioned earlier in this chapter, had we used a different model,

it would have been possible to assign a pure discount for the delay to

sale of perhaps 3“5% using another model. This would have narrowed

the differences between DLOM for the small ¬rms and the large ones,

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.