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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

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 0.00%
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]) 23.99%
E
11 ESOP post-trans pDE[1 e (1 t)x] (equation [13-3f]) 23.99%
12 Actual dilution to ESOP [10] [11] 0.00%
t)D2 p2
13 Default dilution to ESOP : (1 pDEe (equation [13-1g]) 6.36%
E
14 Actual/default dilution: [12]/[13] k [3] 0.00%
15 Dilution to owner (B5*B6) B10 5.41%
t)*DE*p2
2
16 Dilution to owner p*DE ((p*DE)*(1 e) k*((1 p*DE*e))/(1 (1 t)*p*DE) 5.41%




ignating the desired level of dilution to be 2/3 of the original dilution,
we have reduced the dilution by 1/3, or (1 k).
If we desire dilution to the ESOP to be zero, then we substitute k
0 in (13-4a), and the equation reduces to
pDE(1 e)
x
[1 (1 t)pDE]
which is identical to equation (13-3j), the post-transaction value of the
ESOP when the owner bears all of the dilution. You can see that in Table
13-3A, which is identical to Table 13-3 except that we have let k 0 (B7),
which leads to the zero dilution, as seen in B14.
Type 2 dilution appears in Table 13-3, rows 15 and 16. The owner is
paid 27.6% (B10) of the pre-transaction value for 30% of the stock of the
company. He normally would have been paid 29.4% of the pre-transaction
value (B5 B6 0.3 0.98 29.4%). Type 2 dilution is 29.4% 27.60%
1.80% (B15). In B16 we calculate type 2 dilution directly using equation
(13-4b). Both calculations produce identical results, con¬rming the accu-
racy of (13-4b). In Table 13-3A, where we let k 0, type 2 dilution is
5.41% (B15 and B16).


T A B L E 13-3B

Summary of Dilution Tradeoffs


A B C D E

5 Scenario: Assignment of Dilution

6 100% to 2/3 to 100% to
7 Dilution Type ESOP ESOP Difference Owner

8 1 (ESOP) 6.36% 4.24% 2.12% 0.00%
9 2 (seller) 0.00% 1.80% 1.80% 5.41%
10 Source table 13-2 13-3 13-3A




CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 447
Table 13-3B: Summary of Dilution Tradeoffs
In Table 13-3B we summarize the dilution options that we have seen in
Tables 13-2, 13-3, and 13-3A to get a feel for the tradeoffs between type
1 and type 2 dilution. In Table 13-2, where we allowed the ESOP to bear
all dilution, the ESOP experienced dilution of 6.36%. In Table 13-3, by
apportioning one-third of the dilution to him or herself, the seller reduced
type 1 dilution by 6.36% 4.24% 2.12% (Table 13-3B, D8) and under-
took type 2 dilution of 1.80% (D9). The result is that the ESOP bears
dilution of 4.24% (C8) and the owner bears 1.8% (C9). In Table 13-3A we
allowed the seller to bear all dilution rather than the ESOP. The seller
thereby eliminated the 6.36% type 1 dilution and accepted 5.41% type 2
dilution.
Judging by the results seen in Table 13-3B, it appears that 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. This
turns out to be correct in all cases, as proven in the Appendix A, the
Mathematical Appendix.
As mentioned in the introduction, the reader may wish to skip to the
conclusion section. The following material aids in understanding dilution,
but it does not contain any new formulas of practical signi¬cance.


THE ITERATIVE APPROACH
We now proceed to develop formulas to measure the engineered value
per share that, when paid by the ESOP, will eliminate dilution to the
ESOP. We accomplish this by performing several iterations of calculations.
Using iteration, we will calculate the payment to the owner, which be-
comes the ESOP loan, and the post-transaction fair market values of the
¬rm and the ESOP.
In our ¬rst iteration the seller pays the ESOP the pre-transaction FMV
without regard for the ESOP loan. The existence of the ESOP loan then
causes the post-transaction values of the ¬rm and the ESOP to decline,
which means the post-transaction value of the ESOP is lower than the
pre-transaction value paid to the owner.
In our second iteration we calculate an engineered payment to the
owner that will attempt to equal the post-transaction value at the end of

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