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

448

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

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 ¬rm is:

1 e

VnA V1B

1 [ pDE(1 t)]

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

post-transaction value of the firm”

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

¬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).

SUMMARY

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

pro¬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

¬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.

De¬nitions

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

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 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.