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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 ESOP™s debt. Let™s analyze a simple sale to a non-ESOP buyer and
later to an ESOP buyer. For simplicity we will ignore tax bene¬ts of all
loans throughout this example.

17. Again, these should only be the incremental costs if the ESOP is replacing another pension
18. For simplicity, we are assuming the company hasn™t yet paid any of the ESOP™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 ¬rm”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 ¬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
stock”it is still worth $5 million. The corporation is a mere conduit for
the loan to the buyer.
What happens to the ¬rm™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 ¬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 buyer™s purchase of
stock, and the corporation”not the buyer”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 ˜˜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
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.

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.

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.

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.
1 (1 t)pDE 1 (A13-3)
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
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.
1 (A13-5)
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.

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.

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-
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.

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


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