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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%
10 x % to owner pDE(1 e) k[(1 pDEe)]/1 (1 t)pDE (equation [13-4a]) 23.99%
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%
14 Actual/default dilution: [12]/[13] k [3] 0.00%
15 Dilution to owner (B5*B6) B10 5.41%
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)
[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, con¬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


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
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 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 signi¬cance.

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
¬rm and the ESOP.
In our ¬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 ¬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 ¬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 ¬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
¬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 ¬rm before considering the
lifetime ESOP administration cost as V1B.

PART 5 Special Topics
V1B pre-transaction value (13-5)
The value of the ¬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 ¬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 ¬nished the ¬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 ¬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). MM™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 ¬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 ¬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 ¬rm
as its pre-transaction value minus the net present value of the loan against
the ¬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
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)

V3A V3B E (13-7c)
p 2D 2 (1 t)2(1
V3A V1B[1 pDE(1 t)(1 e) e) e] (13-7d)

Moving the e at the right immediately after the 1:
V3A V1B[(1 e) pDE(1 t)(1 e)
p 2D E(1
t)2(1 e)]
Factoring out the (1 e):
p 2D 2 (1
V3A V1B(1 e)[1 pDE(1 t) t)] (13-7f)

p0 D E(1
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:
1) j p j D jE(1 t)j
VnA V1B (1 e) ( (13-8)

PART 5 Special Topics
This is an oscillating geometric sequence,14 which leads to the following
solutions. The ultimate post-transaction value of the ¬rm is:
1 e
1 [ pDE(1 t)]
or, dropping the subscript A and simplifying: (13-8a)
post-transaction value of the firm”
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
¬rm, or:
pDE(1 e)
Ln V1B
1 (1 t)pDE
post-transaction value of the ESOP”
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) con¬rm the
direct approach solutions of equations (13-3n) and (13-3j).

In this 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 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 ¬xed, i.e., it does not vary as a proportion of the value of
the ¬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
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, 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.
Let™s begin to address this issue by assessing the post-transaction fair
market value balance sheet. We will use the following de¬nitions:

PART 5 Special Topics
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 de¬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 ¬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) simpli¬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 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

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