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

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

E

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

462

(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)

x

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: