ńņš. 17 |

The Company cannot obtain VC ļ¬nancing without restructuring its debt.

CHAPTER 12 Valuing Startups 427

58ā“60 are zero in this case. There are 1,200,000 shares (C62) in this sce-

nario before issuing the 1.3 million, and 2,500,000 (C65) shares after doing

so. Dividing $391,202 by 2,500,000 shares, we come to a FMV of this

scenario of $0.156 (C66) per share. Adding the per share values together,

we come to $10.235 $0.156 $10.391 (B66 C66 D66) as the

weighted average conditional FMV of the restructure scenario.

No-Restructure Scenario. The name of this scenario is somewhat of

a misnomer. It means that the Company does not restructure its debt with

the parent. At the onset of this assignment there was no way to know

this, but restructuring of debt would eventually be required. The dis-

counted cash ļ¬‚ow analysis leads to the conclusion that the Company is

unlikely to be able to generate enough cash to pay off the parentā™s note

by its due date of December 31, 200016 ā”even though the forecast shows

proļ¬ts. Therefore, the Company has two choices: become insolvent and

undergo liquidation or restructure later, and undergo a distress sale of

equity approximately one year before the note becomes due.

The second choice obviously leads to a higher value for the share-

holders, as it preserves the cash ļ¬‚ows, even though some of them will

be diverted to the new investor. Accordingly, we ran a DCF analysis to

the ļ¬scal year ending closest to the due date of the note. That value is

$8,000,000 and appears in C44.

The subtotal number of shares is 1,200,000 (F62) before the new in-

vestor. Since there is no restructure with the parent in this scenario, the

shares issued to the president is zero here (F63). In section 4 we calculate

that the new investor will demand one-third of the Company post-

transaction (see description below). That implies the investor will demand

600,000 shares (F64), which will bring the total shares to 1,800,000 (F65).

Dividing $2,753,938 (K41, repeated in F53) by 1,800,000 shares leads to a

value of $1.530 (F66) per share for the no-restructure scenario (this should

more appropriately be called ā˜ā˜restructure laterā™ā™).

Conclusion

Thus, the restructure is preferable by a FMV per share of $10.391 $1.530

$8.861 per share ( D66 F66).

Section 4: Year 2000 Investor Percentage

A future restructure would be a more distressed one than the current one.

The discounted cash ļ¬‚ow analysis indicates that the Company would be

short of cash to pay off the note. With two years gone by, the Company

is more likely to lose the possibility of becoming the market leader and

more likely to be an also ran. Also, it would be a far more highly lever-

aged ļ¬rm without the restructure. Therefore, it would be a higher-risk

ļ¬rm in the year 2000, which dictates using a higher discount rate than

the other scenarios. The result is a value of $8,000,000 (C44, repeated as

B71) before the minority interest discount.

16

The analysis was done in 1996.

PART 5 Special Topics

428

Subtracting the $2 million (B73) minority interest discount leaves us

with an FMV of $6 million (B74). In the DCF we determined the Company

would need a $2 million investment by a new investor, who would re-

quire taking one-third (B75) of the Company. This percentage is used in

section 3, F52 in the no-restructure calculations, as discussed above.

EXPONENTIALLY DECLINING SALES GROWTH MODEL

When forecasting yearly sales for a startup, the appraiser ideally has a

bottom-up forecast based on a combination of market data and reasonable

assumptions. Sometimes those data are not available to us, and even

when they are available, it is often beneļ¬cial to use a top-down approach

based on reasonable assumptions of sales growth rates. In this section we

present a model for forecasting sales of a startup or early-stage company

that semiautomates the process of forecasting sales and can easily be ma-

nipulated for sensitivity analysis. The other choice is to insert sales

growth rates manually for, say, 10 years, print out the spreadsheet with

that scenario, change all 10 growth rates, and repeat the process for val-

uation of multiple scenarios. Life is too short.

One such sales model that has intuitive appeal is the exponentially

declining sales growth rate model, presented in Table 12-4. In the model

we have a peak growth rate (P), which decays with a decay rate constant

(k) to a ļ¬nal growth rate (G). The mathematics may look a little difļ¬cult,

but it is not necessary to understand the math in order to beneļ¬t from

using the model.

The top of Table 12-4 is a list of the parameters of the model. In the

example the ļ¬nal sales growth rate (G) is set at 6% (E6), and the addi-

tional growth rate (A) is calculated to be 294% (E7). The additional growth

rate (A) is the difference between the peak growth rate (P), which is set

at 300% (E8), and the ļ¬nal sales growth rate of 6%. Next we have the

decay rate constant (k), which is set at 0.50 (E9). The larger the decay rate

constant, the faster the sales growth rate will decline to the ļ¬nal growth

rate. Finally, we have Year 1 forecast sales of 100 (E10). All the variables

are speciļ¬ed by the model user with the exception of the additional

growth rate (A), which depends on P and G.

Example #1 shows the forecast sales growth rates (row 17) and sales

(row 18) using the previously speciļ¬ed variables for a case where the

sales growth rate declines after Year 2. We have no sales growth rate in

Year 1 because we assume there are no prior year sales. The expression

Ae k(t 2), for all t greater than or equal

for the sales growth rate G

to 2, where t is expressed in years. For Year 2 the sales growth rate is G

Ae k(2 2) G A 6% 294% 300% (C17), which is our speciļ¬ed

Ae k(3 2)

peak growth rate P. Year 3 growth is G 6% (294%

0.5 1 k(4 2)

e ) 184% (D17). Year 4 growth is G Ae 6% 294%

0.5 2

e 114% (E17), etc. To calculate yearly sales, we simply multiply

the previous year sales by one plus the forecast growth rate.

Example #1A is identical to example #1, except that we have changed

the decay rate constant (k) from 0.50 to 0.30. Notice how reducing k slows

the decay in the sales growth rate. In example #2 we present a case of

the peak growth rate (P) occurring in a general future year f, where we

CHAPTER 12 Valuing Startups 429

T A B L E 12-4

Sales Model with Exponentially Declining Growth Rate Assumption

A B C D E F G H I J K

5 Variable Name Symbol Value Speciļ¬ed/Calculated

6 Final growth rate G 6% Speciļ¬ed

7 Additional growth rate A 294% Calculated

8 Peak growth rate P 300% Speciļ¬ed

9 Decay rate k 0.50 Speciļ¬ed

10 First yearā™s sales Sales1 100 Speciļ¬ed

13 Example # 1 - Sales growth rate declines after year 2

k(t 2)

14 Yearly growth G Ae for all t greater than or equal to 2

16 Year 1 2 3 4 5 6 7 8 9 10

17 Growth N/A 300% 184% 114% 72% 46% 30% 21% 15% 11%

18 Sales 100 400 1,137 2,436 4,179 6,093 7,929 9,566 10,989 12,240

21 Example # 1A - Changing the decay rate (k) from 0.50 to 0.30 slows the decline in the sales growth rate

23 Year 1 2 3 4 5 6 7 8 9 10

24 Growth N/A 300% 224% 167% 126% 95% 72% 55% 42% 33%

25 Sales 100 400 1,295 3,463 7,810 15,194 26,072 40,307 57,237 75,937

28 Example # 2 - Sales growth rate declines after future year f

Ae k(t f), for all t greater than or equal to f, where sales growth rate declines after future year f and

29 Sales growth rate G

30 the peak sales growth (P) occurs in year f. Growth through year f is to be speciļ¬ed by model user. The following is an

31 example with year f 4, and decay rate k 0.5

33 Year 1 2 3 4 5 6 7 8 9 10

34 Growth N/A 100% 200% 300% 184% 114% 72% 46% 30% 21%

35 Sales 100 200 600 2,400 6,824 14,613 25,077 36,559 47,575 57,393

Formula in Cell C17: G A*EXP( k*(C16 2))

F I G U R E 12-3

Sales Forecast (Decay Rate 0.5)

40,000

35,000

30,000

25,000

Sales

20,000

15,000

10,000

5,000

0

1 3 5 7 9 11 13 15 17 19 21 23 25 27

Year

PART 5 Special Topics

430

F I G U R E 12-3A

Sales Forecast (Decay Rate 0.3)

450,000

400,000

350,000

300,000

Sales

250,000

200,000

150,000

100,000

50,000

0

1 3 5 7 9 11 13 15 17 19 21 23 25 27

Year

have chosen the future year to be Year 4. The model user speciļ¬es the

growth rates prior to Year f (we have chosen 100% and 200% in Years 2

and 3, respectively). The growth rates for year f and later are G Ae k(t f).

As you can see, the growth rates from Years 4 through 10 in this example

are identical to the growth rates from Years 2 through 8 in example #1.

Figures 12-3 and 12-3A are graphs that show the sales forecasts from

examples #1 and #1A extended to 28 years. The slower decay rate of 0.3

in Figure 12-3A (versus 0.5 in Figure 12-3) leads to much faster growth.

After 28 years, sales are close to $450,000 versus $38,000. Changing one

single parameter can give the analyst a great deal of control over the sales

forecast. When sensitivity analysis is important, we can control the de-

cline in sales growth simply by using different numbers in cell E9, the

decay rate. This is not only a nice time saver, but it can lead to more

accurate forecasts, as many phenomena in life have exponential decay (or

growth), e.g., the decay of radiation, population of bacteria, etc.

BIBLIOGRAPHY

Fowler, Bradley A. 1989. ā˜ā˜What Do Venture Capital Pricing Methods Tell About Valuation

of Closely Held Firms?ā™ā™ Business Valuation Review (June): 73ā“79.

ā” ā”. 1990. ā˜ā˜Valuation of Venture Capital Portfolio Companiesā”and Other Moving Tar-

ā”

gets.ā™ā™ Business Valuation Review (March): 13ā“17.

ā” ā”. 1996. ā˜ā˜Venture Capital Rates of Return Revisited.ā™ā™ Business Valuation Review

ā”

(March): 13ā“16.

Golder, Stanley C. 1986. ā˜ā˜Structuring and Pricing the Financing.ā™ā™ In Prattā™s Guide to Venture

Capital Sources, 10th ed., ed. Stanley E. Pratt and Jane K. Morris. Wellesley Hills,

Mass.: Venture Economics.

Morris, Jane K. 1988. In Prattā™s Guide to Venture Capital Sources, 12th ed., ed. Jane K. Morris.

Wellesley Hills, Mass.: Venture Economics.

Pacelle, Mitchell. 1999. ā˜ā˜Venture Firms Dethroning Buyout Kings.ā™ā™ Wall Street Journal, 7,

June 1999. p, C1.

Plummer, James L. 1987. QED Report on Venture Capital Financial Analysis. Palo Alto, Calif.:

QED Research, Inc. [See especially 2-7ā“2-10 and 6-2ā“6-13.]

Pratt, Stanley E. and Jane K. Morris, Guide to Venture Capital Sources, Venture Economics,

1986.

CHAPTER 12 Valuing Startups 431

CHAPTER 13

ESOPs: Measuring and

Apportioning Dilution1

INTRODUCTION

What Can Be Skipped

DEFINITIONS OF DILUTION

Dilution to the ESOP (Type 1 Dilution)

Dilution to the Selling Owner (Type 2 Dilution)

Deļ¬ning Terms

TABLE 13-1: CALCULATION OF LIFETIME ESOP COSTS

THE DIRECT APPROACH

FMV Equationsā”All Dilution to the ESOP (Type 1 Dilution; No Type

2 Dilution)

Table 13-2, Sections 1 and 2: Post-transaction FMV with All Dilution

to the ESOP

The Post-transaction Value Is a Parabola

FMV Equationsā”All Dilution to the Owner (Type 2 Dilution)

Table 13-2, Section 3: FMV Calculationsā”All Dilution to the Seller

Sharing the Dilution

Equation to Calculate Type 2 Dilution

Tables 13-3 and 13-3A: Adjusting Dilution to Desired Levels

Table 13-3B: Summary of Dilution Tradeoffs

THE ITERATIVE APPROACH

Iteration #1

Iteration #2

Iteration #3

Iteration #n

SUMMARY

1. Adapted and reprinted with permission from Valuation (June 1997): 3ā“25 and (January 1993): 76ā“

103, American Society of Appraisers, Herndon, Virginia.

433

Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

Advantages of Results

Function of ESOP Loan

Common Sense Is Required

To Whom Should the Dilution Belong?

Deļ¬nitions

The Mathematics of the Post-transaction Fair Market Value Balance

Sheet

Analyzing a Simple Sale

Dilution to Non-selling Owners

Legal issues

Charity

APPENDIX A: MATHEMATICAL APPENDIX

APPENDIX B: SHORTER VERSION OF CHAPTER 13

PART 5 Special Topics

434

This chapter is the result of further thought and research on my treatment

of valuing ESOPs (Abrams 1993 and 1997). It not only simpliļ¬es those

articles, but it goes far beyond them. Reading them is not necessary for

understanding this chapter.

INTRODUCTION

Leveraged ESOPs have confused many ļ¬rms due to their failure to un-

derstand the phenomenon of dilution and inability to quantify it. Many

ESOPs have soured because employees paid appraised fair market value

of the stock being sold to the ESOP, only to watch the fair market value

signiļ¬cantly decline at the next valuation because the ESOP loan was not

included in the pre-transaction fair market value. As a result, employees

have felt cheated. Lawsuits have sometimes followed, further lowering

the value of the ļ¬rm and the ESOP.

There are several types of problems relating to the dilution phenom-

enon:

1. The technical problem of deļ¬ning and measuring the dilution in

value to the ESOP before it happens.

2. The business problem of getting the ESOP Trustee, participants,

and selling owner(s) to agree on how to share the dilution.

3. The technical problem of how to engineer the price to

accomplish the desired goals in 2.

4. The problem of how to communicate each of the foregoing to all

of the participants so that all parties can enter the transaction

with both eyes open and come away feeling the transaction was

winā“win instead of winā“lose.

This chapter provides the analytical solutions to problems 1 and 3

that are necessary for resolving the business and communication prob-

lems of 2 and 4. The appraiser will be able to include the dilution in his

or her initial valuation report so that employees will not be negatively

surprised when the value drops at the next annual valuation. Addition-

ally, the appraiser can provide the technical expertise to enable the parties

to share the dilution, solving problem 3. Both parties will then be fully

informed beforehand, facilitating a winā“win transaction.

What Can Be Skipped

This chapter contains much tedious algebra. For readers who wish to skip

all of the mathematics and optional sections and simply get the bottom

line can read the ā˜ā˜quick-and-dirtyā™ā™ version of this chapter in Appendix

B. The section on the iterative approach can be safely skipped, as it en-

hances the understanding of dilution but contains no additional formulas

of practical signiļ¬cance.

DEFINITIONS OF DILUTION

Two potential parties can experience dilution in stock values in ESOP

transactions: the ESOP and the owner. The dilution that each experiences

differs and can be easily confused.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 435

Additionally, each party can experience two types of dilution: abso-

lute and relative. Absolute dilution is deļ¬ned in the section immediately

below. Relative dilution is more complicated because we can calculate

dilution relative to more than one base. Several formulas can be devel-

oped to calculate relative dilution, but they are beyond the scope of this

book. Thus, for the remainder of this chapter, dilution will mean absolute

dilution.

Dilution to the ESOP (Type 1 Dilution)

We deļ¬ne type 1 dilution as the payment to the selling owner less the

post-transaction fair market value of the ESOP. This can be stated either

in dollars or as a percentage of the pre-transaction value of the ļ¬rm. By

law, the ESOP may not pay more than fair market value to the company

or to a large shareholder, though it is nowhere deļ¬ned in the applicable

statute whether this is pre- or post-transaction value. Case law and De-

partment of Labor proposed regulations indicate that the pre-transaction

value should be used.2

Dilution to the Selling Owner (Type 2 Dilution)

We deļ¬ne Type 2 dilution as the difference in the pre-transaction fair

market value of the shares sold and the price paid to the seller. Again,

this can be in dollars or as a percentage of the ļ¬rmā™s pre-transaction value.

Since it is standard industry practice for the ESOP to pay the owner the

pre-transaction price, Type 2 Dilution is virtually unknown. Those sellers

who wish to reduce or eliminate dilution to the ESOP can choose to sell

for less than the pre-transaction fair market value.

When the ESOP bears all of the dilution, we have only type 1 dilu-

tion. When the owner removes all dilution from the ESOP by absorbing

it himself, then the selling price and post-transaction values are equal and

we have only type 2 dilution. If the owner absorbs only part of the di-

lution from the ESOP, then the dilution is shared, and we have both type

1 and type 2 dilution.

As we will show in Table 13-3B and the Mathematical Appendix,

when the seller takes on a speciļ¬c level of type 2 dilution, the decrease

in type 1 dilution is greater than the corresponding increase in type 2

dilution.

The seller also should consider the effects of dilution on his or her

remaining stock in the ļ¬rm, but that is beyond the scope of this book.

Deļ¬ning Terms

We ļ¬rst deļ¬ne some of terms appearing in the various equations.

Let:

p percentage of ļ¬rm sold to the ESOP, assumed at 30%

t combined federal and state corporate income tax rate, assumed

at 40%

2. Donovan v. Cunningham, 716 F.2d 1467. 29 CFR 2510.3-18(b).

PART 5 Special Topics

436

r the annual loan interest rate, assumed at 10%

i the monthly loan interest rate r/12 0.8333% monthly

V1B the pre-transaction value of 100% of the stock of the ļ¬rm

after discounts and premiums at the ļ¬rm level but before those at

the ESOP level,3 assumed at $1,000,000, as shown in Table 13-2.

The B subscript means before considering the lifetime cost of

initiating and maintaining the ESOP (see E, e, and VjA below). V1B

does not consider the cost of the loan. This differs from VjB, as

described below.

V1A Same as V1B, except this is the pre-transaction value after

deducting the lifetime cost of initiating and maintaining the ESOP

(see E, e, and VjA below) but before considering the loan. Note this

differs from VjA, where j 1, where we do subtract the cost of the

ESOP loan as of iteration j 1.

VjB the value of the ļ¬rm at the jth iteration before deducting the

lifetime ESOP costs (see E below) but after subtracting the net

present value of the ESOP loan (see NPLV) as calculated in

iteration j 1 (for j 1).

VjA the value of the ļ¬rm at the jth iteration after deducting the

lifetime ESOP costs (see immediately below) and the ESOP loan as

of the ( j 1)st iteration.

Vn the ļ¬nal post-transaction value of the ļ¬rm, i.e., at the nth

iteration

E the lifetime costs of initiating and running the ESOP. These

are generally legal fees, appraisal fees, ESOP administration fees,

and internal administration costs. We assume initial costs of

$20,000 and annual costs of $10,000 growing at 6% each year. Table

13-1 shows a sample calculation of the lifetime costs of the ESOP

as $40,000.4

e lifetime ESOP costs as a percentage of the pre-transaction

value E/V1B $40,000/$1 million 4%.

DE one minus net Discounts (or plus net premiums) at the ESOP

level. This factor converts the fair market value of the entire ļ¬rm

on an illiquid control level (V1B) to a fair market value (on a 100%

basis) at the ESOPā™s level of marketability and control (DEV1B). If

we assume that the ESOP provides complete marketability (which

normally one should not, but we are doing so here for didactic

purposes), then to calculate DE we must merely reverse out the

control premium that was applied to the entire ļ¬rm (in the

calculation of V1B), which we will assume was 43%, and reverse

out the discount for lack of marketability that was applied, which

we will assume was 29%.5 The result is: DE [1/(1 43%)]

[1/(1 29%)] 0.7 1.4 0.98. In other words, the net effect

of reversing out the assumed discount and premium is a 2% net

3. In Abrams (1993) the discounts and premiums at the ļ¬rm level are a separate variable. This

treatment is equally as accurate and is simpler.

4. How to calculate the pre-transaction value of the ļ¬rm is outside the scope of this article.

5. These are arbitrary assumptions chosen for mathematical ease.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 437

discount. It could also be a net premium if the minority discount

were less or the premium for marketability were higher. Also, if

we were to assume that the ESOP shares were not at a marketable

minority level, other adjustments would be required.

Lj the amount of the ESOP Loan in iteration j, which equals the

payment to the owner. That equals the FMV of the ļ¬rm in

iteration j multiplied by pDE, the percentage of the ļ¬rm being sold

to the ESOP, multiplied again by the factor for discounts or

premiums at the ESOP level. Mathematically, Lj pDE VjA. Note:

this deļ¬nition only applies in the Iterative Approach where we are

eliminating type 1 dilution.

NPVLj the after-tax, net present value of the ESOP loan as

calculated in iteration j. The formula is NPVLj (1 t)Lj, as

explained below.

n The number of iterations

D1 type 1 dilution (dilution to the ESOP)

D2 type 2 dilution (dilution to the seller)

FMV fair market value

TABLE 13-1: CALCULATION OF LIFETIME ESOP COSTS

We begin by calculating the lifetime cost of the ESOP, including the legal,

appraisal, and administration costs, which are collectively referred to

throughout this chapter as the administration costs or as the lifetime

ESOP costs.

The estimated annual operating costs of the ESOP in Table 13-1 are

$10,000 pretax (B5), or $6,000 after-tax (B6). We assume an annual re-

quired rate of return of 25% (B7). Letā™s further assume ESOP administra-

tion costs will rise by 5% a year (B8). We can then calculate the lifetime

value of the annual cost by multiplying the ļ¬rst yearā™s cost by a Gordon

Model multiple (GM) using an end-of-year assumption. The GM formula

is 1/(r g), or 1/(0.25 0.05) 5.000 (B9). Multiplying 5.000 by $6,000,

we obtain a value of $30,000 (B10).

T A B L E 13-1

Calculation of Lifetime ESOP Costs

A B

5 Pre-tax annual ESOP costs $10,000

6 After-tax annual ESOP costs (1 t) * pre-tax 6,000

7 Required rate of return r 25%

8 Perpetual growth of ESOP costs g 5%

9 Gordon model multiple (end year) 1/(r g) 5.000

10 Capitalized annual costs 30,000

11 Initial outlay-pre-tax 20,000

12 Initial outlay-after-tax (1 t) * pre-tax 12,000

13 Lifetime ESOP costs 42,000

14 Lifetime ESOP costs-rounded to (used in Table 13-2, B9) $40,000

PART 5 Special Topics

438

We next calculate the immediate costs of initiating the ESOP at time

zero, which we will assume are $20,000 (B11), or $12,000 after-tax (B12).

Adding $30,000 plus 12,000, we arrive at a lifetime cost of $42,000 for

running the ESOP (B13), which for simplicity we round off to $40,000

(B14), or 4% of the pre-transaction value of $1 million.6 Adopting the

previous deļ¬nitions, E $40,000 and e 4%.

The previous example presumes that the ESOP is not replacing an-

other pension plan. If the ESOP is replacing another pension plan, then

it is only the incremental lifetime cost of the ESOP that we would cal-

culate here.

THE DIRECT APPROACH

Using the direct approach, we calculate all valuation formulas directly

through algebraic substitution. We will develop post-transaction valua-

tion formulas for the following situations:

1. All dilution remains with the ESOP.

2. All dilution goes to the owner.

3. The ESOP and the owner share the dilution.

We will begin with 1. The owner will be paid pre-transaction price, leav-

ing the ESOP with all of the dilution in value. The following series of

equations will enable us to quantify the dilution. All values are stated as

a fraction of each $1 of pre-transaction value.

FMV Equationsā”All Dilution to the ESOP

(Type 1 Dilution; No Type 2 Dilution)

1 pre-transaction value (13-1)

We pay the owner the p% he or she sells to the ESOP reduced or increased

by DE, the net discounts or premiums at the ESOP level. For every $1 of

pre-transaction value, the payment to the owner is thus:

pDE paid to owner in cash ESOP loan (13-1a)

tpDE tax savings on ESOP loan (13-1b)

The after-tax cost of the loan is the amount paid to the owner less the tax

savings of the loan, or equations (13-1a) and (13-1b).

(1 t)pDE after-tax cost of the ESOP loan (13-1c)

e after-tax lifetime cost of the ESOP (13-1d)

When we subtract (13-1c) plus (13-1d) from (13-1), we obtain the

remaining value of the ļ¬rm:

6. For simplicity, we do not add a control premium and deduct a discount for lack of marketability

at the ļ¬rm level and then reverse that procedure at the ESOP level, as I did in Abrams

(1993).

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 439

1 (1 t)pDE e post-transaction value of the firm (13-1e)

Since the ESOP owns p% of the ļ¬rm, the post-transaction value of the

ESOP is p DE (13-1e):

t)p 2D 2

pDE (1 pDE e post-transaction value of the ESOP

E

(13-1f)

The dilution to the ESOP (type 1 dilution) is the amount paid to

the owner minus the value of the ESOPā™s p% of the ļ¬rm, or (13-1a)

(13-1f):

t)p 2D 2

pDE [pDE (1 pDE e]

E

t)p 2 DE2

(1 pDE e dilution to ESOP (13-1g)

Table 13-2, Sections 1 and 2: Post-transaction FMV with

All Dilution to the ESOP

Now that we have established the formulas for calculating the FMV of

the ļ¬rm when all dilution goes to the ESOP, letā™s look at a concrete ex-

ample in Table 13-2. The table consists of three sections. Section 1, rows

5ā“10, is the operating parameters of the model. Section 2 shows the cal-

culation of the post-transaction values of the ļ¬rm, ESOP, and the dilution

to the ESOP according to equations (13-1e), (13-1f), and (13-1g), respec-

tively, in rows 12ā“18. Rows 21ā“26 prove the accuracy of the results, as

explained below.

Section 3 shows the calculation of the post-transaction values of the

ļ¬rm and the ESOP when there is no dilution to the ESOP. We will cover

that part of the table later. In the meantime, letā™s review the numerical

example in section 2.

B13 contains the results of applying equation (13-1e) using section 1

parameters to calculate the post-transaction value of the ļ¬rm, which is

$0.783600 per $1 of pre-transaction value. We multiply the $0.783600 by

the $1 million pre-transaction value (B5) to calculate the post-transaction

value of the ļ¬rm $783,100 (B14). The post-transaction value of the ESOP

according to equation (13-1f) is $0.2303787 (B15) $1 million pre-

transaction value (B5) $230,378 (B16).

We calculate dilution to the ESOP according to equation (13-1g) as

0.32 0.982

(1 0.4) 0.3 0.98 0.04 0.063622 (B17). When we

multiply the dilution as a percentage by the pre-transaction value of $1

million, we get dilution of $63,622 (B18, B26).

We now prove these results and the formulas in rows 21ā“26. The

payment to the owner is $300,000 0.98 (net of ESOP discounts/pre-

miums) $294,000 (B22). The ESOP takes out a $294,000 loan to pay the

owner, which the company will have to pay. The after-tax cost of the loan

is (1 t) multiplied by the amount of the loan, or 0.6 $294,000

$176,400 (B23). Subtracting the after tax cost of the loan and the $40,000

lifetime ESOP costs from the pre-transaction value, we come to a post-

7. Which itself is equal to pDE the post-transaction value of the ļ¬rm, or B6 B7 B14.

PART 5 Special Topics

440

T A B L E 13-2

FMV Calculations: Firm, ESOP, and Dilution

A B C

4 Section 1: Parameters

5 V1B pre-transaction value $1,000,000

6 p percentage of stock sold to ESOP 30%

7 DE net ESOP discounts/premiums 98%

8 t tax rate 40%

9 E ESOP costs (lifetime costs capitalized; Table 13-1, B14 ) $40,000

10 e ESOP costs/pre-transaction value E/V1B 4%

12 Section 2: All Dilution To ESOP

13 (1 e) (1 t) pDE post-trans FMV-ļ¬rm (equation [13-1e]) 0.783600

14 Multiply by pre-trans FMV B5*B13 B24 $783,600

t)p2D2

15 pDE (1 pDEe post-trans FMV-ESOP (equation [13-1f]) 0.230378

E

16 Multiply by pre-trans FMV B5*B15 B25 $230,378

22

17 (1 t)p DE pDEe dilution to the ESOP (equation [13-1g]) 0.063622

18 Multiply by pre-trans FMV B5*B17 B26 $63,622

20 Proof of Section 2 Calculations:

21 Pre-trans FMV B5 $1,000,000

22 Payment to owner B6*B7*B21 294,000

23 After tax cost of loan (1 B8) * B22 176,400

24 Post-trans FMV-ļ¬rm B21 B23 B9 B14 783,600

25 Post-transaction FMV of ESOP B6*B7*B24 B16 230,378

26 Dilution to the ESOP B22 B25 B18 $63,622

28 Section 3: All Dilution To Seller Multiple V1B FMV

29 Vn (1 e)/[1 (1 t)pDE] post-trans FMVā”ļ¬rm B40 (equation [13-3n]) 0.816049 $816,049

30 Ln p * DE * Vn post-trans FMV-ESOP (equation [13-3j]) 0.239918 $239,918

31 Dilution to seller (B6*B7) B30 (equation [13-3o]) 5.4082%

32 Dilution to seller B5*C31 $54,082

33 Dilution to seller B22 C30 $54,082

35 Proof of Calculation in C29:

36 Pre-trans FMV B5 $1,000,000

37 Payment to owner C30 239,918

38 Tax shield t * B37 95,967

39 After tax cost of ESOP loan B37 B38 143,951

40 Post-trans FMV-ļ¬rm B36 B39 B9 C29 $816,049

transaction value of the ļ¬rm of $783,600 (B24), which is identical to the

value obtained by direct calculation using formula (13-1e) in B14. The

post-transaction value of the ESOP is pDE post-transaction FMVā”ļ¬rm,

or 0.3 0.98 $783,600 $230,378 (B25, B16). The dilution to the ESOP

is the payment to the owner minus the post-transaction value of the ESOP,

or $294,000 (B22) $230,378 (B25) $63,622 (B26, B18). We have now

proved the direct calculations in rows 14, 16, and 18.

The Post-Transaction Value is a Parabola

Equation (13-1f), the formula for the post-transaction value of the ESOP,

is a parabola. We can see this more easily by rewriting (13-1f) as

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 441

D 2 (1 t)p 2

V DE(1 e)p

E

where V is the post-transaction value of the ESOP. Figure 13-1 shows this

function graphically. The straight line, pDE, is a slight modiļ¬cation of a

simple 45 line y x (or in this case V p), except multiplied by DE

98%. This line is the payment to the owner when the ESOP bears all of

the dilution. The vertical distance of the parabola (equation [13-1f]) from

the straight line is the dilution of the ESOP, deļ¬ned by equation (13-1g),

which is itself a parabola. Figure 13-1 should actually stop where p

100%, but it has been extended merely to show the completion of the

parabola, since there is no economic meaning for p 100%.

We can calculate the high point of the parabola, which is the maxi-

mum post-transaction value of the ESOP, by taking the ļ¬rst partial deriv-

ative of equation (13-1f) with respect to p and setting the equation to zero:

V

t)D 2 p

2(1 DE(1 e) 0 (13-2)

E

p

This solves to

(1 e)

p (13-1f)

2(1 t)DE

or p 81.63265%. Substituting this number into equation (13-1f) gives us

38.4%.8 This means that if the

the maximum value of the ESOP of V

owner sells any greater portion than 81.63265% of the ļ¬rm to the ESOP,

F I G U R E 13-1

Post-Transaction Value of the ESOP Vs. % Sold

8. We can verify this is a maximum rather than minimum value by taking the second partial

derivative, 2V/ p 2 t)D 2

2(1 0, which conļ¬rms the maximum.

E

PART 5 Special Topics

442

he actually decreases the value of the ESOP, assuming a 40% tax rate and

no outside capital infusions into the sale. The lower the tax rate, the more

the parabola shifts to the left of the vertical line, until at t 0, where

9

most of the parabola is completed before the line.

FMV Equationsā”All Dilution to the Owner (Type 2 Dilution)

Letā™s now assume that instead of paying the owner pDE, the ESOP pays

him some unspeciļ¬ed amount, x. Accordingly, we rederive (13-1)ā“(13-1g)

with that single change and label our new equations (13-3)ā“(13-3j).

1 pre-transaction value (13-3)

x paid to owner in cash ESOP loan (13-3a)

tx tax savings on ESOP loan (13-3b)

(1 t)x after-tax cost of the ESOP loan (13-3c)

e after-tax ESOP cost (13-3d)

When we subtract (13-3c) plus (13-3d) from (13-3), we come to the re-

maining value of the ļ¬rm of:

(1 e) (1 t)x post-transaction value of the firm (13-3e)

Since the ESOP owns p% of the ļ¬rm and the ESOP bears its net

discount, the post-transaction value of the ESOP is p DE (13-3e), or:

pDE(1 e) (1 t)pDEx post-transaction value of the ESOP (13-3f)

We can eliminate dilution to the ESOP entirely by specifying that the

payment to the owner, x, equals the post-transaction value of the ESOP

(13-3f), or:

x pDE(1 e) (1 t)pDEx (13-3g)

Moving the right term to the left side,

x (1 t)pDEx pDE(1 e) (13-3h)

Factoring out x,

x[1 (1 t)pDE] pDE(1 e) (13-3i)

Dividing through by 1 (1 t)pDE,

pDE(1 e)

x

1 (1 t)pDE

post-transaction FMV of ESOP, all dilution to owner (13-3j)

D 2p 2

9. This is because equation (13-1f) becomes V DE(1 e)p. Given our DE and e, V is

E

2

then approximately equal to 0.92 (p p). If t 0, e 0, and there were no discounts

and premiums at the ESOP level, i.e., DE 1, then the owner would be paid p, the post-

transaction value of the ļ¬rm would be 1 p, and the post-transaction value of the ESOP

p), or p 2

would be p(1 p. This parabola would ļ¬nish at p 1. The maximum post-

transaction ESOP value would be 25% at p 50%.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 443

Substituting equation (13-3j) into the x term in (13-3e), the post-

transaction value of the ļ¬rm is:

pDE(1 e)

(1 e) (1 t) (13-3k)

1 (1 t)pDE

Factoring out the (1 e) from both terms, we get:

(1 t)pDE

(1 e) 1 (13-3l)

1 (1 tpDE

Rewriting the 1 in the brackets as

1 (1 t)pDE

1 (1 t)pDE

we obtain:

1 (1 t)pDE (1 t)pDE

(1 e) (13-3m)

1 (1 t)pDE

The numerator simpliļ¬es to 1, which enables us to simplify the entire

expression to:

1 e

post-transaction value of the firmā”

1 (1 t)pDE

type 1 dilution 0 (13-3n)

The dilution to the seller is the pre-transaction FMV of shares sold minus

the price paid, or:

1 e

pDE (13-3o)

1 (1 t)pDE

Table 13-2, Section 3: FMV Calculationsā”

All Dilution to the Seller

In section 3 we quantify the engineered price that eliminates all dilution

to the ESOP, which according to equation (13-3n) is:

(1 0.04)

$1 million

[1 (0.6) (0.3) (0.98)]

$1 million 0.816049 (B29) $816,049 (C29)

Similarly, the value of the ESOP is: 0.3 0.98 0.816049 $1,000,000

$239,918 (C30) which is also the same amount that the owner is paid

in cash. We can prove this correct as follows:

1. The ESOP borrows $239,918 (B37) to pay the owner and takes

out a loan for the same amount, which the ļ¬rm pays.

2. The ļ¬rm gets a tax deduction, which has a net present value of

its marginal tax rate multiplied by the principal of the ESOP

loan, or 40% $239,918, or $95,967 (B38), which after being

subtracted from the payment to the owner leaves an after-tax

cost of the payment to the owner (which is the identical to the

after-tax cost of the ESOP loan) of $143,951 (B39).

PART 5 Special Topics

444

3. We subtract the after-tax cost of the ESOP loan of $143,951 and

the $40,000 lifetime ESOP costs from the pre-transaction value of

$1 million to arrive at the ļ¬nal value of the ļ¬rm of $816,049

(B40). This is the same result as the direct calculation by formula

in B29, which proves (13-3n). Multiplying by pDE (0.3 0.98

0.297) would lead to the same result as in B30, which proves the

accuracy of (13-3j).

We can also prove the dilution formulas in section 3. The seller ex-

periences dilution equal to the normative price he or she would have

received if he or she were not willing to reduce the sales price, i.e.,

$294,000 (B22) less the engineered selling price of $239,918 (C30), or

$54,082 (C33). This is the same result as using a direct calculation from

equation (13-3o) of 5.4082% (C31) the pre-transaction price of $1 million

$54,082 (C32).

The net result of this approach is that the owner has shifted the entire

dilution from the ESOP to himself. Thus, the ESOP no longer experiences

any dilution in value. While this action is very noble on the part of the

owner, in reality few owners are willing and able to do so.

Sharing the Dilution

The direct approach also allows us to address the question of how to

share the dilution. If the owner does not wish to place all the dilution on

the ESOP or absorb it personally, he or she can assign a portion to both

parties. By subtracting the post-transaction value of the ESOP (13-3f) from

the cash to the owner (13-3a), we obtain the amount of dilution. We can

then specify that this dilution should be equal to a fraction k of the default

dilution, i.e., the dilution to the ESOP when the ESOP bears all of the

dilution. In our nomenclature, the post-transaction value of the ESOP

dilution to the ESOP k (default dilution to the ESOP). Therefore,

Actual Dilution to ESOP

k , or

Default Dilution to ESOP

k the % dilution remaining with the ESOP

The reduction in dilution to the ESOP is (1 k). For example, if k

33%, the ESOP bears 33% of the dilution; the reduction in the amount of

dilution borne by ESOP is 67% (from the default ļ¬gure of 100%).

The formula used to calculate the payment to the owner when di-

lution is shared by both parties is:

t)p 2D 2

x [pDE(1 e) (1 t)pDEx] k[(1 pDE e] (13-4)

E

Collecting terms, we get:

t)p 2D 2

x[1 (1 t)pDE] pDE(1 e) k[(1 pDE e]

E

Dividing both sides by [1 (1 t)pDE], we solve to:

t)p 2D E

2

pDE(1 e) k[(1 pDE e]

x (13-4a)

1 (1 t)pDE

In other words, equation (13-4a) is the formula for the amount of

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 445

payment to the owner when the ESOP retains the fraction k of the default

dilution. If we let k 0, (13-4a) reduces to (13-3j), the post-transaction

FMV of the ESOP when all dilution goes to the owner. When k 1,

(13-4a) reduces to (13-1a), the payment to the owner when all dilution

goes to the ESOP.

Equation to Calculate Type 2 Dilution

Type 2 dilution is equal to pDE, the pre-transaction selling price adjusted

for control and marketability, minus the engineered selling price, x. Sub-

stituting equation (13-4a) for x, we get:

t)p 2D 2

pDE(1 e) k[(1 pDE e]

E

D2 pDE (13-4b)

1 (1 t)pDE

Tables 13-3 and 13-3A:

Adjusting Dilution to Desired Levels

Table 13-3 is a numerical example using equation (13-4a). We let p 30%

(B5), DE 98% (B6), k 2/3 (B7), t 40% (B8), and e 4% (B9). B10 is

the calculation of x, the payment to the sellerā”as in equation (13-4a)ā”

which is 27.6%. B11 is the value of the ESOP post-transaction, which we

calculate according to equation (13-3f),10 at 23.36%. Subtracting the post-

transaction value of the ESOP from the payment to the owner (27.60%

23.36%) 4.24% (B12) gives us the amount of type 1 dilution.

The default type 1 dilution, where the ESOP bears all of the dilution,

t)p2D 2

would be (1 pDEe, according to equation (13-1g), or 6.36%

E

(B13). Finally, we calculate the actual dilution divided by the default di-

lution, or 4.24%/6.36% to arrive at a ratio of 66.67% (B14), or 2/3, which

is the same as k, which proves the accuracy of equation (13-4a). By des-

T A B L E 13-3

Adjusting Dilution to Desired Levels

A B

5 p percentage sold to ESOP 30.00%

6 DE net discounts at the ESOP level 98.00%

7 k Arbitrary fraction of remaining dilution to ESOP 66.67%

8 t tax rate 40.00%

9 e % ESOP costs 4.00%

t)(p2D2

10 x % to owner pDE(1 e) k[(1 pDEe)]/[1 (1 t)pDE] (equation [13-4a]) 27.60%

E

11 ESOP post-trans pDE[1 e (1 t)x] (equation [13-3f]) 23.36%

12 Actual dilution to ESOP B10 B11 4.24%

t)D2 p2

13 Default dilution to ESOP : (1 pDEe (equation [13-1g]) 6.36%

E

14 Actual/default dilution: [12]/[13] k [7] 66.67%

15 Dilution to owner (B5*B6) B10 1.80%

t)*D2 *p2

16 Dilution to owner p*DE ((p*DE)*(1 e) k*((1 p*DE*e))/(1 (1 t)*p*DE) 1.80%

E

10. With pDE factored out.

PART 5 Special Topics

446

T A B L E 13-3A

Adjusting Dilution to Desired Levelsā”All Dilution to Owner

A B

5 p percentage sold to ESOP 30.00%

6 DE net discounts at the ESOP level 98.00%

7 k Arbitrary fraction of remaining dilution to ESOP 0.00%

8 t tax rate 40.00%

9 e % ESOP costs 4.00%

t)(p2D2

10 x % to owner pDE(1 e) k[(1 pDEe)]/1 (1 t)pDE (equation [13-4a]) 23.99%

E

11 ESOP post-trans pDE[1 e (1 t)x] (equation [13-3f]) 23.99%

12 Actual dilution to ESOP [10] [11] 0.00%

t)D2 p2

13 Default dilution to ESOP : (1 pDEe (equation [13-1g]) 6.36%

E

14 Actual/default dilution: [12]/[13] k [3] 0.00%

15 Dilution to owner (B5*B6) B10 5.41%

t)*DE*p2

2

16 Dilution to owner p*DE ((p*DE)*(1 e) k*((1 p*DE*e))/(1 (1 t)*p*DE) 5.41%

ignating the desired level of dilution to be 2/3 of the original dilution,

we have reduced the dilution by 1/3, or (1 k).

If we desire dilution to the ESOP to be zero, then we substitute k

0 in (13-4a), and the equation reduces to

pDE(1 e)

x

[1 (1 t)pDE]

which is identical to equation (13-3j), the post-transaction value of the

ESOP when the owner bears all of the dilution. You can see that in Table

13-3A, which is identical to Table 13-3 except that we have let k 0 (B7),

which leads to the zero dilution, as seen in B14.

Type 2 dilution appears in Table 13-3, rows 15 and 16. The owner is

paid 27.6% (B10) of the pre-transaction value for 30% of the stock of the

company. He normally would have been paid 29.4% of the pre-transaction

value (B5 B6 0.3 0.98 29.4%). Type 2 dilution is 29.4% 27.60%

1.80% (B15). In B16 we calculate type 2 dilution directly using equation

(13-4b). Both calculations produce identical results, conļ¬rming the accu-

racy of (13-4b). In Table 13-3A, where we let k 0, type 2 dilution is

5.41% (B15 and B16).

T A B L E 13-3B

Summary of Dilution Tradeoffs

A B C D E

5 Scenario: Assignment of Dilution

6 100% to 2/3 to 100% to

7 Dilution Type ESOP ESOP Difference Owner

8 1 (ESOP) 6.36% 4.24% 2.12% 0.00%

9 2 (seller) 0.00% 1.80% 1.80% 5.41%

10 Source table 13-2 13-3 13-3A

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 447

Table 13-3B: Summary of Dilution Tradeoffs

In Table 13-3B we summarize the dilution options that we have seen in

Tables 13-2, 13-3, and 13-3A to get a feel for the tradeoffs between type

1 and type 2 dilution. In Table 13-2, where we allowed the ESOP to bear

all dilution, the ESOP experienced dilution of 6.36%. In Table 13-3, by

apportioning one-third of the dilution to him or herself, the seller reduced

type 1 dilution by 6.36% 4.24% 2.12% (Table 13-3B, D8) and under-

took type 2 dilution of 1.80% (D9). The result is that the ESOP bears

dilution of 4.24% (C8) and the owner bears 1.8% (C9). In Table 13-3A we

allowed the seller to bear all dilution rather than the ESOP. The seller

thereby eliminated the 6.36% type 1 dilution and accepted 5.41% type 2

dilution.

Judging by the results seen in Table 13-3B, it appears that when the

seller takes on a speciļ¬c level of type 2 dilution, the decrease in type 1

dilution is greater than the corresponding increase in type 2 dilution. This

turns out to be correct in all cases, as proven in the Appendix A, the

Mathematical Appendix.

As mentioned in the introduction, the reader may wish to skip to the

conclusion section. The following material aids in understanding dilution,

but it does not contain any new formulas of practical signiļ¬cance.

THE ITERATIVE APPROACH

We now proceed to develop formulas to measure the engineered value

per share that, when paid by the ESOP, will eliminate dilution to the

ESOP. We accomplish this by performing several iterations of calculations.

Using iteration, we will calculate the payment to the owner, which be-

comes the ESOP loan, and the post-transaction fair market values of the

ļ¬rm and the ESOP.

In our ļ¬rst iteration the seller pays the ESOP the pre-transaction FMV

without regard for the ESOP loan. The existence of the ESOP loan then

causes the post-transaction values of the ļ¬rm and the ESOP to decline,

which means the post-transaction value of the ESOP is lower than the

pre-transaction value paid to the owner.

In our second iteration we calculate an engineered payment to the

owner that will attempt to equal the post-transaction value at the end of

the ļ¬rst iteration. In the second iteration the payment to the owner is less

than the pre-transaction price because we have considered the ESOP loan

from the ļ¬rst iteration in our second iteration valuation. Because the pay-

ment is lower in this iteration, the ESOP loan is lower than it is in the

ļ¬rst iteration. We follow through with several iterations until we arrive

at a steady-state value, where the engineered payment to the owner ex-

actly equals the post-transaction value of the ESOP. This enables us to

eliminate all type 1 dilution to the ESOP and shift it to the owner as type

2 dilution.

Iteration #1

We denote the pre-transaction value of the ļ¬rm before considering the

lifetime ESOP administration cost as V1B.

PART 5 Special Topics

448

V1B pre-transaction value (13-5)

The value of the ļ¬rm after deducting the lifetime ESOP costs but before

considering the ESOP loan is:11

V1A V1B E V1B V1B e V1B(1 e) (13-5a)

The owner sells p% of the stock to the ESOP, so the ESOP would pay

p times the value of the ļ¬rm. However, we also need to adjust the pay-

ment for the degree of marketability and control of the ESOP. Therefore,

the ESOP pays the owner V1A multiplied by p DE , or:

L1 pDEV1A pDEV1B(1 e) (13-5b)

Our next step is to compute the net present value of the loan. In this

chapter we greatly simplify this procedure over the more complex cal-

culation in my original article (Abrams 1993).12

The net present value of the payments of any loan discounted at the

loan rate is the principal of the loan. Since both the interest and principal

payments on ESOP loans are tax deductible, the after-tax cost of the ESOP

loan is simply the principal of the loan multiplied by one minus the tax

rate.13 Therefore:

NPVL1 (1 t)pDEV1B(1 e) (13-5c)

Iteration #2

We have now ļ¬nished the ļ¬rst iteration and are ready to begin iteration

#2. We begin by subtracting equation (13-5c), the net present value of the

ESOP loan, from the pre-transaction value, or:

V2B V1B (1 t)pDEV1B(1 e)

V1B[1 pDE(1 t)(1 e)] (13-6)

We again subtract the lifetime ESOP costs to arrive at V2A.

V2A V2B E (13-6a)

V2A V1B[1 pDE(1 t)(1 e)] V1Be (13-6b)

Factoring out the V1B, we get:

11. V1A is the only iteration of VjA where we do not consider the cost of the loan. For j 1, we do

consider the after-tax cost of the ESOP loan.

12. You do not need to read that article to understand this chapter.

13. One might speculate that perhaps the appraiser should discount the loan by a rate other than

the nominal rate of the loan. To do so would implicitly be saying that the ļ¬rm is at a

suboptimal D/E (debt/equity) ratio before the ESOP loan and that increasing debt lowers

the overall cost of capital. This is closer to a matter of faith than science, as there are those

that argue on each side of the fence. The opposite side of the fence is covered by two Nobel

Prize winners, Merton Miller and Franco Modigliani (MM), in a seminal article (Miller and

Modigliani 1958). MMā™s famous Proposition I states that in perfect capital markets, i.e., in

the absence of taxes and transactions costs, one cannot raise the value of the ļ¬rm with debt.

They acknowledge a secondary tax effect of debt, which I use here literally and no further,

i.e., adding debt increases the value of the equity only to the extent of the tax shield. Also,

even if there is an optimal D/E ratio and the subject company is below it, it does not need

an ESOP to borrow to achieve the optimal ratio.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 449

V2A V1B[(1 e) pDE(1 t)(1 e)] (13-6c)

Factoring out the (1 e), we then come to the post-transaction value

of the ļ¬rm in iteration #2 of:

V2A V1B(1 e)[1 pDE(1 t)] (13-6d)

It is important to recognize that we are not double-counting E, i.e.,

subtracting it twice. In equation (13-6) we calculate the value of the ļ¬rm

as its pre-transaction value minus the net present value of the loan against

the ļ¬rm. The latter is indirectly affected by E, but in each new iteration,

we must subtract E directly in order to count it in the post-transaction

value.

The post-transaction value of the ESOP loan in iteration #2 is p

DE (13-6d), or:

L2 pDEV1B(1 e)[1 pDE(1 t)] (13-6e)

The net present value of the loan is:

NPVL2 (1 t)pDEV1B(1 e)[1 (1 t)pDE] (13-6f)

Iteration #3

We now begin the third iteration of value. The third iteration FMV before

lifetime ESOP costs is V1B NPVL2, or:

V3B V1B (1 t)pDEV1B (1 e)[1 (1 t)pDE] (13-7)

Factoring out V1B, we have:

V3B V1B{1 pDE(1 t)(1 e)[1 (1 t)pDE]} (13-7a)

Multiplying terms, we get:

p 2D 2 (1 t)2(1

V3B V1B[1 pDE(1 t)(1 e) e)] (13-7b)

E

V3A V3B E (13-7c)

p 2D 2 (1 t)2(1

V3A V1B[1 pDE(1 t)(1 e) e) e] (13-7d)

E

Moving the e at the right immediately after the 1:

V3A V1B[(1 e) pDE(1 t)(1 e)

(13-7e)

p 2D E(1

2

t)2(1 e)]

Factoring out the (1 e):

p 2D 2 (1

V3A V1B(1 e)[1 pDE(1 t) t)] (13-7f)

E

p0 D E(1

0

t)0

Note that the 1 in the square brackets

Iteration #n

Continuing this pattern, it is clear that the nth iteration leads to the fol-

lowing formula:

n1

1) j p j D jE(1 t)j

VnA V1B (1 e) ( (13-8)

j0

PART 5 Special Topics

450

This is an oscillating geometric sequence,14 which leads to the following

solutions. The ultimate post-transaction value of the ļ¬rm is:

1 e

VnA V1B

1 [ pDE(1 t)]

or, dropping the subscript A and simplifying: (13-8a)

post-transaction value of the firmā”

015

with type 1 dilution

1 e

Vn V1B (13-9)

1 (1 t)pDE

Note that this is the same equation as (13-3n). We arrive at the same result

from two different approaches.

The post-transaction value of the ESOP is p DE the value of the

ļ¬rm, or:

pDE(1 e)

Ln V1B

1 (1 t)pDE

post-transaction value of the ESOPā”

with type 1 dilution 0 (13-10)

This is the same solution as equation (13-3j), after multiplying by V1B. The

iterative approach solutions in equations (13-9) and (13-10) conļ¬rm the

direct approach solutions of equations (13-3n) and (13-3j).

SUMMARY

In this chapter we developed formulas to calculate the post-transaction

values of the ļ¬rm, ESOP, and the payment to the owner, both pre-

transaction and post-transaction, as well as the related dilution. We also

derived formulas for eliminating the dilution in both scenarios, as well

as for specifying any desired level of dilution. Additionally, we explored

the trade-offs between type 1 and type 2 dilution.

Advantages of Results

The big advantages of these results are:

1. If the owner insists on being paid at the pre-transaction value,

as most will, the appraiser can now immediately calculate the

dilutive effects on the value of the ESOP and report that in the

initial valuation report.16 Therefore, the employees will be

14. For the geometric sequence to work, pDE(1 t) 1 , which will almost always be the case.

15. The reason the e term is in the numerator and not the denominator like the other terms is that

the lifetime cost of the ESOP is ļ¬xed, i.e., it does not vary as a proportion of the value of

the ļ¬rm (or the ESOP), as that changes in each iteration.

16. Many ESOP trustees prefer this information to remain as supplementary information outside of

the report.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 451

entering the transaction with both eyes open and will not be

disgruntled or suspicious as to why the value, on average,

declines at the next valuation. This will also provide a real

benchmark to assess the impact of the ESOP itself on

proļ¬tability.

2. For owners who are willing to eliminate the dilution to the

ESOP or at least reduce it, this chapter provides the formulas to

do so and the ability to calculate the trade-offs between type 1

and type 2 dilution.

Function of ESOP Loan

An important byproduct of this analysis is that it answers the question

of what is the function of the ESOP loan. Obviously it functions as a

ļ¬nancing vehicle, but suppose you were advising a very cash rich ļ¬rm

that could fund the payment to the owner in cash. Is there any other

function of the ESOP loan? The answer is yes. The ESOP loan can increase

the value of the ļ¬rm in two ways:

1. It can be used to shield income at the ļ¬rmā™s highest income tax

rate. To the extent that the ESOP payment is large enough to

cause pre-tax income to drop to lower tax brackets, that portion

shields income at lower than the marginal rate and lowers the

value of the ļ¬rm and the ESOP.

2. If the ESOP payment in the ļ¬rst year is larger than pre-tax

income, the ļ¬rm cannot make immediate use of the entire tax

deduction in the ļ¬rst year. The unused deduction will remain as

a carryover, but it will suffer from a present value effect.

Common Sense Is Required

A certain amount of common sense is required in applying these for-

mulas. In extreme transactions such as those approaching a 100% sale to

the ESOP, we need to realize that not only can tax rates change, but

payments on the ESOP loan may entirely eliminate net income and reduce

the present value of the tax beneļ¬t of the ESOP loan payments. In ad-

dition, the viability of the ļ¬rm itself may be seriously in question, and it

is likely that the appraiser will have to increase the discount rate for a

post-transaction valuation. Therefore, one must use these formulas with

at least two dashes of common sense.

To Whom Should the Dilution Belong?

Appraisers almost unanimously consider the pre-transaction value ap-

propriate, yet there has been considerable controversy on this topic. The

problem is the apparent ļ¬nancial sleight of hand that occurs when the

post-transaction value of the ļ¬rm and the ESOP precipitously declines

immediately after doing the transaction. On the surface, it somehow

seems unfair to the ESOP. In this section we will explore that question.

Deļ¬nitions

Letā™s begin to address this issue by assessing the post-transaction fair

market value balance sheet. We will use the following deļ¬nitions:

PART 5 Special Topics

452

Pre-Transaction Post-Transaction

A1 assets A2 assets A1 (assets have not changed)

L1 liabilities L2 liabilities

C1 capital C2 capital

Note that the subscript 1 refers to pre-transaction and the subscript 2

refers to post-transaction.

The Mathematics of the Post-Transaction Fair Market Value

Balance Sheet

The nonmathematical reader may wish to skip or skim this section. It is

more theoretical and does not result in any usable formulas.

The fundamental accounting equation representing the pre-

transaction balance sheet is:

A1 L1 C1 pre-transaction FMV balance sheet (13-11)

Assuming the ESOP bears all of the dilution, after the sale liabilities

increase and capital decreases by the sum of the after-tax cost of the ESOP

loan and the lifetime ESOP costs,17 or:

C1 [(13-1c) (13-1d)]

increase in liabilities and decrease in debt (13-12)

As noted in the deļ¬nitions, assets have not changed. Only liabilities

and capital have changed.18 Thus the post-transaction balance sheet is:

A2 {L1 C1[(1 t)pDE e]} {C1 C1[(1 t)pDE e]} (13-13)

The ļ¬rst term in braces equals L2, the post-transaction liabilities, and the

second term in braces equals C2, the post-transaction capital. Note that

A2 A1. Equation (13-13) simpliļ¬es to:

A2 {L1 C1[(1 t)pDE e]} {C1[1 (1 t)pDE e]}

post-transaction balance sheet (13-14)

Equation (13-14) gives us an algebraic expression for the post-

transaction fair market value balance sheet when the ESOP bears all of

the dilution.

Analyzing a Simple Sale

Only two aspects relevant to this discussion are unique about a sale to

an ESOP: (1) tax deductibility of the loan principal, and (2) forgiveness

of the ESOPā™s debt. Letā™s analyze a simple sale to a non-ESOP buyer and

later to an ESOP buyer. For simplicity we will ignore tax beneļ¬ts of all

loans throughout this example.

17. Again, these should only be the incremental costs if the ESOP is replacing another pension

plan.

18. For simplicity, we are assuming the company hasnā™t yet paid any of the ESOPā™s lifetime costs.

If it has, then that amount is a reduction in assets rather than an increase in liabilities.

Additionally, the tax shield on the ESOP loan could have been treated as an asset rather

than a contraliability, as we have done for simplicity. This is not intended to be an

exhaustive treatise on ESOP accounting.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 453

Suppose the fair market value of all assets is $10 million before and

after the sale. Pre-transaction liabilities are zero, so capital is worth $10

million, pre-transaction. If a buyer pays the seller personally $5 million

for one-half of the capital stock of the Company, the transaction does not

impact the value of the ļ¬rmā”ignoring adjustments for control and mar-

ketability. If the buyer takes out a personal loan for the $5 million and

pays the seller, there is also no impact on the value of the company. In

both cases the buyer owns one-half of a $10 million ļ¬rm, and it was a

fair transaction.

If the corporation takes out the loan on behalf of the buyer but the

buyer ultimately has to repay the corporation, then the real liability is to

the buyer, not the corporation, and there is no impact on the value of the

stockā”it is still worth $5 million. The corporation is a mere conduit for

the loan to the buyer.

What happens to the ļ¬rmā™s value if the corporation takes out and

eventually repays the loan? The assets are still worth $10 million post-

transaction.19 Now there are $5 million in liabilities, so the equity is worth

$5 million. The buyer owns one-half of a ļ¬rm worth $5 million, so his or

her stock is only worth $2.5 million. Was the buyer hoodwinked?

The possible confusion over value clearly arises because it is the cor-

poration itself that is taking out the loan to fund the buyerā™s purchase of

stock, and the corporationā”not the buyerā”ultimately repays the loan.

By having the corporation repay the loan, the other shareholder is for-

giving his or her half of a $5 million loan and thus gifting $2.5 million

to the buyer.20 Thus, the ā˜ā˜buyerā™ā™ ultimately receives a gift of $2.5 million

in the form of company stock. This is true whether the buyer is an in-

dividual or an ESOP.21

Dilution to Non-Selling Owners

When there are additional business owners who do not sell to the ESOP,

they experience dilution of their interests without the beneļ¬t of getting

paid. Conceptually, these owners have participated in giving the ESOP a

gift by having the Company repay the debt on behalf of the ESOP.

To calculate the dilution to other owners, we begin with the post-

transaction value of the ļ¬rm in equation (13-1e) and repeat the equation

as (13-1e*). Then we will calculate the equivalent equations for the non-

selling owner as we did for the ESOP in equations (13-1f) and (13-1g),

and we will relabel those equations by adding an asterisk.

1 (1 t)pDE e

post-transaction value of the firm (repeated) (13-1e*)

If the nonselling shareholder owns the fraction q of the outstanding stock,

then his or her post-transaction value is:

19. There is a second-order effect of the ļ¬rm being more highly leveraged and thus riskier that

may affect value (and which we are ignoring here). See Chapter 14.

20. The other half of the forgiveness is a washā”the buyer forgiving it to himself or herself.

21. This does not mean that an ESOP brings nothing to the table in a transaction. It does bring tax

deductibility of the loan principal as well as the Section 1042 rollover.

PART 5 Special Topics

454

q q(1 t)pDE qe

post-transaction value of nonselling shareholderā™s stock (13-1f*)

Finally, we calculate dilution to the nonselling shareholder as his or her

pre-transaction value of q minus the pre-transaction value in equation

(13-1f*), or:

q[(1 t)pDE e]

dilution to nonselling shareholderā™s stock22 (13-1g*)

The dilution formula (13-1g*) tells us that the dilution to the non-

selling shareholder is simply his or her ownership, q, multiplied by the

dilution in value to the ļ¬rm itself, which is the sum of the after-tax cost

of the ESOP loan and the lifetime costs. Here, because we are not mul-

tiplying by the ESOPā™s ownership modiļ¬ed for its unique marketability

and control attributes, we do not get the squared terms that we did in

equation (13-1f) and (13-1g).

It is also important to note that equations (13-1f*) and (13-1g*) do

not account for any possible increase in value the owner might experience

as a result of having greater relative control of the ļ¬rm. For example, if

there were two 50% owners pre-transaction and one sells 30% to the

ESOP, post-transaction the remaining 50% owner has relatively more con-

trol than he or she had before the transaction. To the extent that we might

ascribe additional value to that increase in relative control, we would

adjust the valuation formulas. This would mitigate the dilution in equa-

tion (13-1g*).

Legal Issues

As mentioned above, appraisers almost unanimously consider the pre-

transaction value appropriate. Also mentioned earlier in the chapter, case

law and Department of Labor proposed regulations indicate the pre-

transaction value is the one to be used. Nevertheless, there is ongoing

controversy going back to Farnum, a case in which the Department of

Labor withdrew before going to court, that the post-transaction value may

the most appropriate price to pay the seller.

In the previous section we demonstrated that the ESOP is receiving

a gift, not really paying anything for its stock. Therefore, there is no ec-

onomic justiļ¬cation for reducing the payment to the owner below the

pre-transaction fair market value, which is the price that the seller would

receive from any other buyer. If the ESOP (or any party on its behalf)

demands that it ā˜ā˜payā™ā™ no more than post-transaction value, it is tanta-

mount to saying, ā˜ā˜The gift you are giving me is not big enough.ā™ā™

While the dilution may belong to the ESOP, it is nevertheless an

important consideration in determining the fairness of the transaction for

22. One would also need to consider adjusting for each nonselling shareholderā™s control and

marketability attributes. To do so, we would have to add a term in equation (13-1g*)

immediately after the q. The term would be the ownerā™s equivalent of DE, except

ńņš. 17 |