(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

The numerator simpli¬es to 1, which enables us to simplify the entire

expression to:

1 e

post-transaction value of the firm”

1 (1 t)pDE

type 1 dilution 0 (13-3n)

The dilution to the seller is the pre-transaction FMV of shares sold minus

the price paid, or:

1 e

pDE (13-3o)

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 (13-3n) 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

cost of the payment to the owner (which is the identical to the

after-tax cost of the ESOP loan) of $143,951 (B39).

PART 5 Special Topics

444

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 (13-3n). Multiplying by pDE (0.3 0.98

0.297) would lead to the same result as in B30, which proves the

accuracy of (13-3j).

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 (13-3o) of 5.4082% (C31) the pre-transaction price of $1 million

$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 (13-3f) from

the cash to the owner (13-3a), 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). Therefore,

Actual Dilution to ESOP

k , or

Default Dilution to ESOP

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 2D 2

x [pDE(1 e) (1 t)pDEx] k[(1 pDE e] (13-4)

E

Collecting terms, we get:

t)p 2D 2

x[1 (1 t)pDE] pDE(1 e) k[(1 pDE e]

E

Dividing both sides by [1 (1 t)pDE], we solve to:

t)p 2D E

2

pDE(1 e) k[(1 pDE e]

x (13-4a)

1 (1 t)pDE

In other words, equation (13-4a) is the formula for the amount of

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 445

payment to the owner when the ESOP retains the fraction k of the default

dilution. If we let k 0, (13-4a) reduces to (13-3j), the post-transaction

FMV of the ESOP when all dilution goes to the owner. When k 1,

(13-4a) reduces to (13-1a), 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 (13-4a) for x, we get:

t)p 2D 2

pDE(1 e) k[(1 pDE e]

E

D2 pDE (13-4b)

1 (1 t)pDE

Tables 13-3 and 13-3A:

Adjusting Dilution to Desired Levels

Table 13-3 is a numerical example using equation (13-4a). 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 (13-4a)”

which is 27.6%. B11 is the value of the ESOP post-transaction, which we

calculate according to equation (13-3f),10 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 (13-1g), or 6.36%

E

(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 (13-4a). By des-

T A B L E 13-3

Adjusting Dilution to Desired Levels

A B

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 66.67%

8 t tax rate 40.00%

9 e % ESOP costs 4.00%

t)(p2D2

10 x % to owner pDE(1 e) k[(1 pDEe)]/[1 (1 t)pDE] (equation [13-4a]) 27.60%

E

11 ESOP post-trans pDE[1 e (1 t)x] (equation [13-3f]) 23.36%

12 Actual dilution to ESOP B10 B11 4.24%

t)D2 p2

13 Default dilution to ESOP : (1 pDEe (equation [13-1g]) 6.36%

E

14 Actual/default dilution: [12]/[13] k [7] 66.67%

15 Dilution to owner (B5*B6) B10 1.80%

t)*D2 *p2

16 Dilution to owner p*DE ((p*DE)*(1 e) k*((1 p*DE*e))/(1 (1 t)*p*DE) 1.80%

E

10. With pDE factored out.

PART 5 Special Topics

446

T A B L E 13-3A

Adjusting Dilution to Desired Levels”All Dilution to Owner

A B