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 ˜˜buyer™™ 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 bene¬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 ¬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 ¬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 wash”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 shareholder™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 shareholder™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 ¬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 ESOP™s ownership modi¬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 ¬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 justi¬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 ˜˜pay™™ no more than post-transaction value, it is tanta-

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

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 shareholder™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 owner™s equivalent of DE, except

customized for his or her ownership attributes. The details of such a calculation are beyond

the scope of this chapter.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 455

purposes of a fairness opinion. If a bank loans $10 million to the ESOP

for a 100% sale, with no recourse or personal guarantees of the owner,

we may likely decide it is not a fair transaction to the ESOP and its

participants. We would have serious questions about the ESOP™s proba-

bility of becoming a long-range retirement program, given the huge debt

load of the Company post-transaction.

Charity

While the dilution technically belongs to the ESOP, I consider it my duty

to inform the seller of the dilution phenomenon and how it works. While

af¬rming the seller™s right to receive fair market value undiminished by

dilution, I do mention that if the seller has any charitable motivations to

his or her employees”which a minority do”then voluntarily accepting

some of the dilution will leave the Company and the ESOP in better

shape. Of course, in a partial sale it also leaves the remainder of the

owner™s stock at a higher value than it would have had with the ESOP

bearing all of the dilution.

BIBLIOGRAPHY

Abrams, Jay B. 1993. ˜˜An Iterative Procedure to Value Leveraged ESOPs.™™ Valuation (Jan-

uary): 71“103.

” ”. 1997. ˜˜ESOPs: Measuring and Apportioning Dilution.™™ Valuation (June): 3“25.

”

Miller, Merton, and Franco Modigliani. 1958. ˜˜The Cost of Capital, Corporation Finance,

and the Theory of Investment.™™ American Economic Review 48: 61“97.

APPENDIX A: MATHEMATICAL APPENDIX

The purpose of this appendix is to perform comparative static analysis,

as is commonly done in economics, on the equations for dilution in the

body of the chapter in order to understand the tradeoffs between type 1

and type 2 dilution.

We use the same de¬nitions in the appendix as in the chapter. Type

1 dilution is equal to the payment to the owner less the post-transaction

value of the ESOP, or x (13-3f):

D1 x [pDE(1 e) (1 t)pDEx] (A13-1)

Factoring out the x,

D1 x[1 (1 t)pDE] pDE(1 e) (A13-2)

We can investigate the impact on type 1 dilution for each $1 change

in payment to the owner by taking the partial derivative of (A13-2) with

respect to x.

D1

1 (1 t)pDE 1 (A13-3)

x

Equation (A13-3) tells us that each additional dollar paid to the owner

increases dilution to the ESOP by more than $1.

A full payment to the owner (the default payment) is pDE for $1 of

pre-transaction value. We pay the owner x, and the difference of the two

is D2, the type 2 dilution.

PART 5 Special Topics

456

D2 pDE x (A13-4)

We can investigate the impact on type 2 for each $1 change in payment

to the owner by taking the partial derivative of (A13-4) with respect

to x.

D2

1 (A13-5)

x

Type 2 dilution moves in an equal but opposite direction from the amount

paid to the owner, which must be the case to make any sense. Together,

equations (A13-3) and (A13-5) tell us that each additional dollar paid the

owner increases the dilution to the ESOP more than it reduces the dilution

to the owner. We can also see this by taking the absolute value of the

ratio of the partial derivatives:

D2/ x 1

1 (A13-6)

D1/ x 1 (1 t)pDE

Signi¬cance of the Results

Equation (A13-6) demonstrates that for every $1 of payment forgone by

the owner, the dilution incurred by the owner will always be less than

the dilution eliminated to the ESOP. The reason for this is that every $1

the owner forgoes in payment costs him $1 in type 2 dilution, yet it saves

the ESOP:

1. The $1, plus

2. It reduces the ESOP loan by pDE and saves the ESOP the after-

tax cost of the lowered amount of the loan, or (1 t)pDE.

There appears to be some charity factor inherent in the mathematics.

Finally, we have not dealt with the fact that by the owner taking on

some or all of the dilution from the ESOP loan, he or she increases the

value of his or her (1 p) share of the remaining stock by reducing the

dilution to it. Such an analysis has no impact on the valuation of the

ESOP, but it should be considered in the decision to initiate an ESOP.

APPENDIX B: SHORTER VERSION OF CHAPTER 13

This appendix provides a bare-bones version of Chapter 13, removing all

mathematical analysis and optional sections of the iterative approach and

all of the second part of the chapter. The reader can then see the bottom

line of the chapter without struggling through the voluminous mathe-

matics. It will also serve as a refresher for those who have already read

the chapter.

INTRODUCTION

Leveraged ESOPs have confused many ¬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

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 457

signi¬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 ¬rm and the ESOP.

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

enon:

1. The technical problem of de¬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

win“win instead of win“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 win“win transaction.

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.

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

lute and relative. Absolute dilution is de¬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 de¬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 ¬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 de¬ned in the applicable

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

PART 5 Special Topics

458

partment of Labor proposed regulations indicate that the pre-transaction

value should be used.23

Dilution to the Selling Owner (Type 2 Dilution)

We de¬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 ¬rm™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 speci¬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 ¬rm, but that is beyond the scope of this book.

De¬ning Terms

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

Let:

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

t combined federal and state corporate income tax rate, assumed

at 40%

r the annual loan interest rate, assumed at 10%

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

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 $40,000.24

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 ¬rm

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

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

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 459

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

basis) at the ESOP™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 ¬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%.25 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

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.

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). Let™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 ¬rst year™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).

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.26 Adopting the

previous de¬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.

25. These are arbitrary assumptions chosen for mathematical ease.

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

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

Abrams (1993).

PART 5 Special Topics

460

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 Equations”All Dilution to the ESOP

(Type 1 Dilution; No Type 2 Dilution)

1 pre-transaction value (A13-7)

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 (A13-7a)

tpDE tax savings on ESOP loan (A13-7b)

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

savings of the loan, or equations (A13-7a) and (A13-7b).

(1 t)pDE after-tax cost of the ESOP loan (A13-7c)

e after-tax lifetime cost of the ESOP (A13-7d)

When we subtract (A13-7c) plus (A13-7d) from (A13-7), we obtain

the remaining value of the ¬rm:

1 (1 t)pDE e post-transaction value of the firm (A13-7e)

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

ESOP is p DE (A13-7e):

t)p 2D2

pDE (1 pDE e

E

post-transaction value of the ESOP (A13-7f)

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

the owner minus the value of the ESOP™s p% of the ¬rm, or (A13-7a)

(A13-7f):

t)p 2D2

pDE [pDE (1 pDE e]

E

t)p 2D2

(1 pDE e dilution to ESOP (A13-7g)

E

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 461

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

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

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

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