1 pre-transaction value (13-1)

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 (13-1a)

tpDE tax savings on ESOP loan (13-1b)

The after-tax cost of the loan is the amount paid to the owner less the tax

savings of the loan, or equations (13-1a) and (13-1b).

(1 t)pDE after-tax cost of the ESOP loan (13-1c)

e after-tax lifetime cost of the ESOP (13-1d)

When we subtract (13-1c) plus (13-1d) from (13-1), we obtain the

remaining value of the ¬rm:

6. 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).

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 439

1 (1 t)pDE e post-transaction value of the firm (13-1e)

Since the ESOP owns p% of the ¬rm, the post-transaction value of the

ESOP is p DE (13-1e):

t)p 2D 2

pDE (1 pDE e post-transaction value of the ESOP

E

(13-1f)

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 (13-1a)

(13-1f):

t)p 2D 2

pDE [pDE (1 pDE e]

E

t)p 2 DE2

(1 pDE e dilution to ESOP (13-1g)

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 (13-1e), (13-1f), and (13-1g), respec-

tively, 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 (13-1e) 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 (13-1f) is $0.2303787 (B15) $1 million pre-

transaction value (B5) $230,378 (B16).

We calculate dilution to the ESOP according to equation (13-1g) 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 $300,000 0.98 (net of ESOP discounts/pre-

miums) $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-

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

PART 5 Special Topics

440

T A B L E 13-2

FMV Calculations: Firm, ESOP, and Dilution

A B C

4 Section 1: Parameters

5 V1B pre-transaction value $1,000,000

6 p percentage of stock sold to ESOP 30%

7 DE net ESOP discounts/premiums 98%

8 t tax rate 40%

9 E ESOP costs (lifetime costs capitalized; Table 13-1, B14 ) $40,000

10 e ESOP costs/pre-transaction value E/V1B 4%

12 Section 2: All Dilution To ESOP

13 (1 e) (1 t) pDE post-trans FMV-¬rm (equation [13-1e]) 0.783600

14 Multiply by pre-trans FMV B5*B13 B24 $783,600

t)p2D2

15 pDE (1 pDEe post-trans FMV-ESOP (equation [13-1f]) 0.230378

E

16 Multiply by pre-trans FMV B5*B15 B25 $230,378

22

17 (1 t)p DE pDEe dilution to the ESOP (equation [13-1g]) 0.063622

18 Multiply by pre-trans FMV B5*B17 B26 $63,622

20 Proof of Section 2 Calculations:

21 Pre-trans FMV B5 $1,000,000

22 Payment to owner B6*B7*B21 294,000

23 After tax cost of loan (1 B8) * B22 176,400

24 Post-trans FMV-¬rm B21 B23 B9 B14 783,600

25 Post-transaction FMV of ESOP B6*B7*B24 B16 230,378

26 Dilution to the ESOP B22 B25 B18 $63,622

28 Section 3: All Dilution To Seller Multiple V1B FMV

29 Vn (1 e)/[1 (1 t)pDE] post-trans FMV”¬rm B40 (equation [13-3n]) 0.816049 $816,049

30 Ln p * DE * Vn post-trans FMV-ESOP (equation [13-3j]) 0.239918 $239,918

31 Dilution to seller (B6*B7) B30 (equation [13-3o]) 5.4082%

32 Dilution to seller B5*C31 $54,082

33 Dilution to seller B22 C30 $54,082

35 Proof of Calculation in C29:

36 Pre-trans FMV B5 $1,000,000

37 Payment to owner C30 239,918

38 Tax shield t * B37 95,967

39 After tax cost of ESOP loan B37 B38 143,951

40 Post-trans FMV-¬rm B36 B39 B9 C29 $816,049

transaction value of the ¬rm of $783,600 (B24), which is identical to the

value obtained by direct calculation using formula (13-1e) 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 (13-1f), the formula for the post-transaction value of the ESOP,

is a parabola. We can see this more easily by rewriting (13-1f) as

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 441

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

function graphically. The straight line, pDE, is a slight modi¬cation of a

simple 45 line y x (or in this case V p), except multiplied by DE

98%. This line is the payment to the owner when the ESOP bears all of

the dilution. The vertical distance of the parabola (equation [13-1f]) from

the straight line is the dilution of the ESOP, de¬ned by equation (13-1g),

which is itself a parabola. Figure 13-1 should actually stop where p

100%, but it has been extended merely to show the completion of the

parabola, since there is no economic meaning for p 100%.

We can calculate the high point of the parabola, which is the maxi-

mum post-transaction value of the ESOP, by taking the ¬rst partial deriv-

ative of equation (13-1f) with respect to p and setting the equation to zero:

V

t)D 2 p

2(1 DE(1 e) 0 (13-2)

E

p

This solves to

(1 e)

p (13-1f)

2(1 t)DE

or p 81.63265%. Substituting this number into equation (13-1f) gives us

38.4%.8 This means that if the

the maximum value of the ESOP of V

owner sells any greater portion than 81.63265% of the ¬rm to the ESOP,

F I G U R E 13-1

Post-Transaction Value of the ESOP Vs. % Sold

8. We can verify this is a maximum rather than minimum value by taking the second partial

derivative, 2V/ p 2 t)D 2

2(1 0, which con¬rms the maximum.

E

PART 5 Special Topics

442

he actually decreases the value of the ESOP, assuming a 40% tax rate and

no outside capital infusions into the sale. The lower the tax rate, the more

the parabola shifts to the left of the vertical line, until at t 0, where

9

most of the parabola is completed before the line.

FMV Equations”All Dilution to the Owner (Type 2 Dilution)

Let™s now assume that instead of paying the owner pDE, the ESOP pays

him some unspeci¬ed amount, x. Accordingly, we rederive (13-1)“(13-1g)

with that single change and label our new equations (13-3)“(13-3j).

1 pre-transaction value (13-3)

x paid to owner in cash ESOP loan (13-3a)

tx tax savings on ESOP loan (13-3b)

(1 t)x after-tax cost of the ESOP loan (13-3c)

e after-tax ESOP cost (13-3d)

When we subtract (13-3c) plus (13-3d) from (13-3), we come to the re-

maining value of the ¬rm of:

(1 e) (1 t)x post-transaction value of the firm (13-3e)

Since the ESOP owns p% of the ¬rm and the ESOP bears its net

discount, the post-transaction value of the ESOP is p DE (13-3e), or:

pDE(1 e) (1 t)pDEx post-transaction value of the ESOP (13-3f)

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

(13-3f), or:

x pDE(1 e) (1 t)pDEx (13-3g)

Moving the right term to the left side,

x (1 t)pDEx pDE(1 e) (13-3h)

Factoring out x,

x[1 (1 t)pDE] pDE(1 e) (13-3i)

Dividing through by 1 (1 t)pDE,

pDE(1 e)

x

1 (1 t)pDE

post-transaction FMV of ESOP, all dilution to owner (13-3j)

D 2p 2

9. This is because equation (13-1f) becomes V DE(1 e)p. Given our DE and e, V is

E

2

then approximately equal to 0.92 (p p). If t 0, e 0, and there were no discounts

and premiums at the ESOP level, i.e., DE 1, then the owner would be paid p, the post-

transaction value of the ¬rm would be 1 p, and the post-transaction value of the ESOP

p), or p 2

would be p(1 p. This parabola would ¬nish at p 1. The maximum post-

transaction ESOP value would be 25% at p 50%.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 443

Substituting equation (13-3j) into the x term in (13-3e), the post-

transaction value of the ¬rm is:

pDE(1 e)

(1 e) (1 t) (13-3k)

1 (1 t)pDE

Factoring out the (1 e) from both terms, we get:

(1 t)pDE

(1 e) 1 (13-3l)

1 (1 tpDE

Rewriting the 1 in the brackets as

1 (1 t)pDE

1 (1 t)pDE

we obtain:

1 (1 t)pDE (1 t)pDE

(1 e) (13-3m)

1 (1 t)pDE