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to the ESOP according to equations (A13-7e), (A13-7f), and (A13-7g), re-
spectively, in rows 12“18. Rows 21“26 prove the accuracy of the results,
as explained below.
Section 3 shows the calculation of the post-transaction values of the
¬rm and the ESOP when there is no dilution to the ESOP. We will cover
that part of the table later. In the meantime, let™s review the numerical
example in section 2.
B13 contains the results of applying equation (A13-7e) using section
1 parameters to calculate the post-transaction value of the ¬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 ¬rm $783,100 (B14). The post-transaction value of the ESOP
according to equation (A13-7f) is $0.23037827 (B15) $1 million pre-
transaction value (B5) $230,378 (B16).
We calculate dilution to the ESOP according to equation (A13-7g) 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 21“26. The
payment to the owner is $1 million 30% 0.98 (net of ESOP discounts/
premiums) $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-
transaction value of the ¬rm of $783,600 (B24), which is identical to the
value obtained by direct calculation using formula (A13-7e) in B14. The
post-transaction value of the ESOP is pDE post-transaction FMV”¬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 (A13-7f), the formula for the post-transaction value of the ESOP,
is a parabola. We can see this more easily by rewriting (A13-7f) as
D 2 (1 t)p 2
V DE(1 e)p

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

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

PART 5 Special Topics
(1 e) (1 t)x post-transaction value of the firm (A13-8e)
Since the ESOP owns p% of the ¬rm and the ESOP bears its net
discount, the post-transaction value of the ESOP is p DEx (A13-8e), or:
pDE(1 e) (1 t)pDEx
post-transaction value of the ESOP (A13-8f)
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
(A13-8f), or:
x pDE(1 e) (1 t)pDEx (A13-8g)
which solves to:
pDE (1 e)
1 (1 t)pDE
post-transaction FMV of ESOP, all dilution to owner (A13-8j)
Substituting equation (A13-8j) into the x term in equation (A13-8e), the
post-transaction value of the ¬rm is:
1 e
post-transaction value of the firm”
1 (1 t)pDE
type 1 dilution 0 (A13-8n)
The dilution to the seller is the pre-transaction FMV of shares sold minus
the price paid, or:
1 e
pDE (A13-8o)
1 (1 t)pDE

Table 13-2, Section 3: FMV Calculations”All Dilution to
the Seller
In section 3 we quantify the engineered price that eliminates all dilution
to the ESOP, which according to equation (A13-8n) 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 ¬rm pays.
2. The ¬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

PART 5 Special Topics
cost of the payment to the owner (which is identical to the after-
tax cost of the ESOP loan) of $143,951 (B39).
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 ¬nal value of the ¬rm of $816,049
(B40). This is the same result as the direct calculation by formula
in B29, which proves (A13-8n). Multiplying by pDE (0.3 0.98
0.297) would lead to the same result as in B30, which proves the
accuracy of (A13-8j).
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 (A13-8o) of 5.4082% (C31) the pre-transaction price of $1 mil-
lion $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 (A13-8f)
from the cash to the owner (A13-8a), 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). There-
Actual Dilution to ESOP
k ,
Default Dilution to ESOP
or 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 ¬gure of 100%).
The formula used to calculate the payment to the owner when di-
lution is shared by both parties is:
t)p 2 D 2
x [pDE(1 e) (1 t)pDEx] k[(1 pDE e] (A13-9)

which solves to:
t)p 2 D 2
pDE(1 e) k[(1 pDE e]
x (A13-9a)
1 (1 t)pDE
In other words, equation (A13-8a) is the formula for the amount of

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 465
payment to the owner when the ESOP retains the fraction k of the default
dilution. If we let k 0, (A13-8a) reduces to (A13-8j), the post-transaction
FMV of the ESOP when all dilution goes to the owner. When k 1, (A13-
9a) reduces to (A13-7a), 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 (A13-9a) for x, we get:
t)p 2 D E
pDE(1 e) k[(1 pDE e]
D2 pDE (A13-9b)
1 (1 t)pDE

Tables 13-3 and 13-3A: Adjusting Dilution to
Desired Levels
Table 13-3 is a numerical example using equation (A13-9a). 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 seller”as in equation (A13-
9a)”which is 27.6%. B11 is the value of the ESOP post-transaction, which
we calculate according to equation (A13-8f),30 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 (A13-7g), or 6.36%
(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 (A13-9a). By
designating 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 equation (A13-9a), and the equation reduces to
pDE(1 e)
[1 (1 t)pDE]
which is identical to equation (A13-8j), 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

30. With pDE factored out.

PART 5 Special Topics
(A13-9b). Both calculations produce identical results, con¬rming the ac-
curacy of (A13-9b). In Table 13-3A, where we let k 0, type 2 dilution is
5.41% (B15 and B16).

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
Judging by the results seen in Table 13-3B, it appears that 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. This
turns out to be correct in all cases, as proven in Appendix A, the Math-
ematical Appendix.

In this mini-chapter we developed formulas to calculate the post-
transaction values of the ¬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 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.31 Therefore, the employees will be
entering the transaction with both eyes open and will not be
disgruntled and/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

31. Many ESOP trustees prefer this information to remain as supplementary information outside of
the report.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 467
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
¬nancing vehicle, but suppose you were advising a very cash rich ¬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 ¬rm in two ways:
1. It can be used to shield income at the ¬rm™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, then that
portion shields income at lower than the marginal rate and
lowers the value of the ¬rm and the ESOP.
2. If the ESOP payment in the ¬rst year is larger than pre-tax
income, the ¬rm cannot make immediate use of the entire tax
deduction in the ¬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 bene¬t of the ESOP loan payments. In ad-
dition, the viability of the ¬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 ¬nancial sleight of hand that occurs when the
post-transaction value of the ¬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.

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.

PART 5 Special Topics
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.32 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.33 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.34

Dilution to Nonselling 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.
Assuming the nonselling owner has the fraction q of the outstanding
stock of the ¬rm, his or her dilution is equal to:
q[(1 t) pDE e]
dilution to nonselling shareholder™s stock35 (A13-1g*)
The dilution formula (A13-1g*) tells us that the dilution to the non-
selling shareholder is simply his or her ownership, q, multiplied by the

32. 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.
33. The other half of the forgiveness is a wash”the buyer forgiving it to himself or herself.
34. 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.
35. 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 469
dilution in value to the ¬rm itself, which is the sum of the after-tax cost
of the ESOP loan and the lifetime costs.
It is also important to note that equation (A13-1g*) does 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 control 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 equation (A13-

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
be 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
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.


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