<< . .

. 61
( : 66)



. . >>

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
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

<< . .

. 61
( : 66)



. . >>