compared to takeovers without competitive bidding. Based on the re-

gression in Table 4 of his article, we assumed a typical deal con¬guration

that would apply to a privately held ¬rm.59 The premium without an

auction was 21.5%. Adding 12.2%, the premium with an auction was

33.7%. To calculate the discount for lack of competition, we go in the

other direction, i.e., 12.2% divided by one plus 33.7% 0.122/1.337

9.1%, or approximately 9%. This is a useful benchmark for D 2.

However, it is quite possible that D 2 for any subject interest should

be larger or smaller than 9%. It all depends on the facts and circumstances

of the situation. Using Schwert™s measure of the effect of multiple versus

single bidders as our estimate of D 3 may possibly have a downward bias

in that the markets for the underlying minority interests in the same ¬rms

is very deep. So it is only the market for control of publicly held ¬rms

that is thin. The market for privately held ¬rms is thin for whole ¬rms

and razor thin for minority interests.

59. We assume a successful purchase, a tender offer, and a cash deal.

CHAPTER 7 Adjusting for Levels of Control and Marketability 257

Component #3: Transactions Costs

Transactions costs in selling a privately held business are substantially

more than they are for selling stock in publicly traded ¬rms. Most stock

in publicly traded ¬rms can be sold with a broker™s fee of 1“2%”or less.

Table 7-11: Quantifying Transactions Costs for Buyer and

Seller. Table 7-11 shows estimates of transactions costs for both the

buyer and the seller for the following categories: legal, accounting, and

appraisal fees (the latter split into posttransaction, tax-based appraisal for

allocation of purchase price and/or valuation of in-process R&D and the

pretransaction ˜˜deal appraisal™™ to help buyer and/or seller establish the

right price), the opportunity cost of internal management spending its

time on the sale rather than on other company business, and investment

banking (or, for small sales, business broker) fees. The ¬rst ¬ve of the

categories appear in columns B through F, which we subtotal in column

G, and the investment banking fees appear in column H. The reason for

segregating between the investment banking fees and all the others is

that the others are constantly increasing as the deal size (FMV) decreases,

while investment banking fees reach a maximum of 10% and stop in-

creasing as the deal size decreases.

Rows 6“9 are transactions costs estimates for the buyer, while rows

13“16 are for the seller. Note that the buyer does not pay the investment

banking fees”only the seller pays. Rows 20“23 are total fees for both

sides.

Note that the subtotal transactions costs (column G) are inversely

related to the size of the transaction. For the buyer, they are as low as

0.23% (I6) for a $1 billion transaction and as high as 5.7% (I9) for a $1

million transaction. We summarize the total in Rows 27“30 and include

the base 10 logarithm of the sales price as a variable for regression.60 The

purpose of the regression is to allow the reader to calculate an estimated

transactions costs for any size transaction.

The buyer regression equation is:

Buyer Subtotal Transaction Cost

0.1531 (0.0173 log10 Price)

Price

The regression coef¬cients are in cells B48 and B49. The adjusted R2

is 83% (B37), which is a good result. The standard error of the y-estimate

is 0.9% (B38), so the 95% con¬dence interval around the estimate is ap-

proximately two standard errors, or 1.8%”a very good result.

The seller regression equation is:

Seller Subtotal Transaction Cost

0.1414 (0.01599 log10 Price)

Price

The regression coef¬cients are in cells B67 and B68. The adjusted R2 is

82% (B56), which is a good result. The standard error of the y-estimate is

60. Normally we use the natural logarithm for regression. Here we chose base 10 because the logs

are whole numbers and are easy to understand. Ultimately, it makes no difference which

one we use in the regression. The results are identical either way.

PART 3 Adjusting for Control and Marketability

258

T A B L E 7-11

Estimates of Transaction Costs [1]

A B C D E F G H I

4 Buyer

Tax Deal

5 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

6 $1 billion 0.10% 0.02% 0.02% 0.00% 0.09% 0.23% 0.00% 0.23%

7 $100 million 1.00% 0.10% 0.06% 0.00% 0.16% 1.32% 0.00% 1.32%

8 $10 million 1.50% 0.23% 0.20% 0.00% 0.25% 2.18% 0.00% 2.18%

9 $1 million 4.00% 0.30% 0.70% 0.00% 0.70% 5.70% 0.00% 5.70%

11 Seller

Tax Deal

12 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

13 $1 billion 0.10% 0.01% 0.00% 0.02% 0.05% 0.18% 0.75% 0.93%

14 $100 million 1.00% 0.05% 0.00% 0.05% 0.10% 1.20% 1.10% 2.30%

15 $10 million 1.50% 0.08% 0.00% 0.20% 0.15% 1.93% 2.75% 4.68%

16 $1 million 4.00% 0.10% 0.00% 0.75% 0.42% 5.27% 10.00% 15.27%

18 Total

Tax Deal

19 Deal Size Legal [2] Acctg Appraisal Appraisal [3] Internal Mgt [4] Subtotal Inv Bank Total

20 $1 billion 0.20% 0.03% 0.02% 0.02% 0.14% 0.41% 0.75% 1.16%

21 $100 million 2.00% 0.15% 0.06% 0.05% 0.26% 2.52% 1.10% 3.62%

22 $10 million 3.00% 0.30% 0.20% 0.20% 0.40% 4.10% 2.75% 6.85%

23 $1 million 8.00% 0.40% 0.70% 0.75% 1.12% 10.97% 10.00% 20.97%

25 Summary For Regression Analysis-Buyer Summary For Regression Analysis-Seller

26 Sales Price Log10 Price Subtotal Sales Price Log10 Price Subtotal

27 $1,000,000,000 9.0 0.23% $1,000,000,000 9.0 0.18%

28 $100,000,000 8.0 1.32% $100,000,000 8.0 1.20%

29 $10,000,000 7.0 2.18% $10,000,000 7.0 1.93%

30 $1,000,000 6.0 5.70% $1,000,000 6.0 5.27%

259

T A B L E 7-11 (continued)

Estimates of Transaction Costs [1]

A B C D E F G H

32 SUMMARY OUTPUT: Buyer Subtotal Fees as a Function of Log10 FMV

34 Regression Statistics

35 Multiple R 0.9417624

36 R square 0.88691642

37 Adjusted R square 0.83037464

38 Standard error 0.00975177

39 Observations 4

41 ANOVA

42 df SS MS F Signi¬cance F

43 Regression 1 0.001491696 0.0014917 15.68603437 0.058237596

44 Residual 2 0.000190194 9.5097E 05

45 Total 3 0.00168189

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

48 Intercept 0.1531 0.033069874 4.62959125 0.043626277 0.010811717 0.295388283

49 Log10 price 0.0172725 0.004361126 3.96055986 0.058237596 0.036036923 0.001491923

51 SUMMARY OUTPUT: Seller Subtotal Fees as a Function of Log10 FMV

53 Regression Statistics

54 Multiple R 0.93697224

55 R square 0.87791699

56 Adjusted R square 0.81687548

57 Standard error 0.00943065

58 Observations 4

60 ANOVA

61 df SS MS F Signi¬cance F

62 Regression 1 0.00127912 0.00127912 14.38229564 0.063027755

63 Residual 2 0.000177874 8.8937E 05

64 Total 3 0.001456994

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

67 Intercept 0.14139 0.031980886 4.42107833 0.04754262 0.00378726 0.27899274

68 Log10 price 0.0159945 0.004217514 3.79239972 0.063027755 0.034141012 0.002152012

also 0.9% (B57), which gives us the same con¬dence intervals around the

y-estimate of 1.8%.

Rows 73 and 74 show a sample calculation of transactions costs for

the buyer and seller, respectively. We estimate FMV before discounts for

our subject company of $5 million (B73, B74). The base 10 logarithm of 5

million is 6.69897 (C73, C74).61 In D73 and D74, we insert the x-coef¬cient

from the regression, which is 0.0172725 (from B49) for the buyer and

0.0159945 (from B68) for the seller. We multiply column C column

61. In other words, 106.69897 5 million.

PART 3 Adjusting for Control and Marketability

260

T A B L E 7-11 (continued)

Estimates of Transaction Costs [1]

A B C D E F G H I J

70 Sample Forecast of Transactions Costs For $5 Million Subject Company:

72 FMV log10 FMV X-Coeff. log FMV Coef Regr. Constant Forecast Subtotal Inv Bank [5] Forecast Total

73 Buyer $5,000,000 6.698970004 0.0172725 0.115707959 0.1531 3.7% 0.0% 3.7%

74 Seller $5,000,000 6.698970004 0.0159945 0.107146676 0.14139 3.4% 5.0% 8.4%

Notes:

[1] Based on interviews with investment banker Gordon Gregory, attorney David Boatwright, Esq; and Douglas Obenshain, CPA. Costs include buy and sell side. These are estimates of average costs. Actual costs vary with the complexity

of the transaction.

[2] Legal fees will vary with the complexity of the transaction. An extremely complex $1 billion sale could have legal fees of as much as $5 million each for the buyer and the seller, though this is rare. Complexity increases with: stock deals

(or asset deals with a very large number of assets), seller carries paper , contingent payments, escrow, tax-free (which is treated as a pooling-of-interests), etc.

[3] We are assuming the seller pays for the deal appraisal. Individual sales may vary. Sometimes both sides hire a single appraiser and split the fees, and sometimes each side has its own appraiser.

[4] Internal management costs are the most speculative of all. We estimate 6,000 hours (3 people fulltime for 1 year) at an average $150/hr. internal cost for the $1 billion sale, 2,000 hours @ $80 for the $100 million sale, 500 hours at $50

for the $10 million sale, and 200 hours @$35 for the $1 million sale for the buyer, and 60% of that for the seller. Actual results may vary considerably from these estimates.

[5] Ideally calculated by another regression, but this is sight-estimated. Can often use the Lehman Bros. Formula”5% for 1st $1 million, 4%, for 2nd, etc., leveling off at 1% for each $1 million.

261

D column E. F73 and F74 are repetitions of the regression constants

from B48 and B67, respectively. We then add column E to column F to

obtain the forecast subtotal transactions costs in G73 and G74. Finally, we

add in investment banking fees of 5%62 for the seller (the buyer doesn™t

pay for the investment banker or business broker) to arrive at totals of

3.7% (I73) and 8.4% (I74) for the buyer and seller, respectively.

Component #3 Is Different than #1 and #2. Component #3, trans-

actions costs, is different than the ¬rst two components of DLOM. For

component #3, we need to calculate explicitly the present value of the

occurrence of transactions costs every time the company sells. The reason

is that, unlike the ¬rst two components, transactions costs are actually

out-of-pocket costs that leave the system.63 They are paid to attorneys,

accountants, appraisers, and investment bankers or business brokers. Ad-

ditionally, internal management of both the buyer and the seller spend

signi¬cant time on the sale to make it happen, and they often have to

spend time on failed acquisitions before being successful.

We also need to distinguish between the buyer™s transactions costs

and the seller™s costs. The reason for this is that the buyer™s transactions

costs are always relevant, whereas the seller™s transactions costs for the

immediate transaction reduce the net proceeds to the seller but do not

reduce FMV. However, before the buyer is willing to buy, he or she should

be saying, ˜˜It™s true, I don™t care about the seller™s costs. That™s his or her

problem. However, 10 years or so down the road when it™s my turn to

be the seller, I do care about that. To the extent that seller™s costs exceed

the brokerage cost of selling publicly traded stock, in 10 years my buyer

will pay me less because of those costs, and therefore I must pay my

seller less because of my costs as a seller in Year 10. Additionally, the

process goes on forever, because in Year 20, my buyer becomes a seller

and faces the same problem.™™ Thus, we need to quantify the present value

of a periodic perpetuity of buyer™s transactions costs beginning with the

immediate sale and sellers™ transactions costs that begin with the second

sale of the business.64 In the next section we will develop the mathematics

necessary to do this.

Developing Formulas to Calculate DLOM Component #3. This

section contains some dif¬cult mathematics, but ultimately we will arrive

at some very usable formulas that are not that dif¬cult. It is not necessary

to follow all of the mathematics that gets us there, but it is worthwhile

to skim through the math to get a feel for what it means. In the Mathe-

62. We could run another regression to forecast investment banking fees. This was sight estimated.

One could also use a formula such as the Lehman Brothers formula to forecast investment

banking fees.

63. I thank R. K. Hiatt for the brilliant insight that the ¬rst two components of DLOM do not have

this characteristic and thus do not require this additional present value calculation.

64. One might think that the buyers™ transactions costs are not relevant the ¬rst time, because the

buyer has to put in due diligence time whether or not a transaction results. In individual

instances that is true, but in the aggregate, if buyers would not receive compensation for

their due diligence time, they would cease to buy private ¬rms until the prices declined

enough to compensate them.

PART 3 Adjusting for Control and Marketability

262

matical Appendix we develop the formulas below step by step. In order

to avoid presenting volumes of burdensome math in the body of the

chapter, we present only occasional snapshots of the math”just enough

to present the conclusions and convey some of the logic behind it.

For simplicity, suppose that, on average, business owners hold the

business for 10 years and then sell. Every time an owner sells, he or she

incurs a transactions cost of z. The net present value (NPV) of the cash

¬‚ows to the business owner is:65

NPV NPV1 (1 z)NPV11 (7-1)

10

Equation (7-1) states that the NPV of cash ¬‚ows at Year 0 to the

owner is the sum of the NPV of the ¬rst 10 years™ cash ¬‚ows and (1

z) times the NPV of all cash ¬‚ows from Year 11 to in¬nity. If transactions

costs are 10% every time a business sells, then z 10% and 1 z

66

90%. The ¬rst owner would have 10 years of cash ¬‚ows undiminished

by transactions costs and then pay transactions costs of 10% of the NPV

at Year 10 of all future cash ¬‚ows.

The second owner operates the business for 10 years and then sells

at Year 20. He or she pays transactions costs of z at Year 20. The NPV of

cash ¬‚ows to the second owner is:

NPV11 NPV11 (1 z)NPV21 (7-2)

20

Substituting (7-2) into equation (7-1), the NPV of cash ¬‚ows to the

¬rst owner is:

NPV NPV1 (1 z)[NPV11 (1 z)NPV21 ] (7-3)

10 20

This expression simpli¬es to:

z)2 NPV21

NPV NPV1 (1 z)NPV11 (1 (7-4)

10 20

We can continue on in this fashion ad in¬nitum. The ¬nal expression

for NPV is:

z)i 1

NPV (1 NPV[10(i (7-5)

1) 1] 10i

i1

The NPV is a geometric sequence. Using a Gordon model, i.e., as-

suming constant, perpetual growth, in the Mathematical Appendix, we

show that equation (7-5) solves to:

10

1 g

1

1 r

1 r

NPVTC (7-6)

10

r g 1 g

1 (1 z)

1 r

where NPVTC is the NPV of the cash ¬‚ows with the NPV of the trans-

actions costs that occur every 10 years removed, g is the constant growth

65. Read the hyphen in the following equation™s subscript text as the word ˜˜to,™™ i.e., the NPV from

one time period to another.

66. z is actually an incremental transaction cost, as we will explain later in the chapter.

CHAPTER 7 Adjusting for Levels of Control and Marketability 263

rate of cash ¬‚ows, r is the discount rate, and cash ¬‚ows are midyear.67

The end-of-year formula is the same, replacing the 1 r in the nu-

merator with the number 1.

The NPV of the cash ¬‚ows without removing the NPV of transactions

costs every 10 years is simply the Gordon model multiple of ( 1 r)/

(r g), which is identical with the ¬rst term on the right-hand side of

equation (7-6). The discount for lack of marketability for transactions costs

is equal to:

NPVTC

DLOM 1 (7-7)

NPV

The fraction in equation (7-7) is simply the term in the large braces

in equation (7-6). Thus, DLOM simpli¬es to:

10

1 g

1

1 r x 10

1

D 1 1 (7-8)

10

z)x10

1 (1

1 g

1 (1 z)

1 r

r, ’ 0 1.68

where x (1 g)/(1 r), D is the discount, and g x

Equation (7-8) is the formula for the discount assuming a sale every

10 years. Instead of assuming a business sale every 10 years, now we let

the average years between sale be a random variable, j, which leads to

the generalized equation in (7-9) for sellers™ transactions costs:69

j

1 g

1

1 r xj

1

D3B 1 1

j

z)x j

1 (1

1 g

1 (1 z)

1 r

DLOM formula”sellers™ costs (7-9)

Using an end-of-year Gordon Model assumption instead of midyear

cash ¬‚ows leads to the identical equation, i.e., equation (7-9) holds for

both.

Analysis of partial derivatives in the Mathematical Appendix shows

that the discount, i.e., DLOM, is always increasing with increases in

growth (g) and transactions costs (z) and is always decreasing with in-

creases in the discount rate (r) and the average number of years between

sales ( j). The converse is true as well. Decreases in the independent var-

iables have opposite effects on DLOM as increases do.

67. This appears as equation (A7-7) in the Mathematical Appendix.

68. This is identical with equation (A7-10) in the Mathematical Appendix.

69. This is identical with equation (A7-11) in the Mathematical Appendix. Note that we use the

plural possessive here because we are speaking about an in¬nite continuum of sellers (and

buyers).

PART 3 Adjusting for Control and Marketability