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. 17
( 18)



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15
The Company cannot obtain VC ¬nancing without restructuring its debt.




CHAPTER 12 Valuing Startups 427
58“60 are zero in this case. There are 1,200,000 shares (C62) in this sce-
nario before issuing the 1.3 million, and 2,500,000 (C65) shares after doing
so. Dividing $391,202 by 2,500,000 shares, we come to a FMV of this
scenario of $0.156 (C66) per share. Adding the per share values together,
we come to $10.235 $0.156 $10.391 (B66 C66 D66) as the
weighted average conditional FMV of the restructure scenario.

No-Restructure Scenario. The name of this scenario is somewhat of
a misnomer. It means that the Company does not restructure its debt with
the parent. At the onset of this assignment there was no way to know
this, but restructuring of debt would eventually be required. The dis-
counted cash ¬‚ow analysis leads to the conclusion that the Company is
unlikely to be able to generate enough cash to pay off the parent™s note
by its due date of December 31, 200016 ”even though the forecast shows
pro¬ts. Therefore, the Company has two choices: become insolvent and
undergo liquidation or restructure later, and undergo a distress sale of
equity approximately one year before the note becomes due.
The second choice obviously leads to a higher value for the share-
holders, as it preserves the cash ¬‚ows, even though some of them will
be diverted to the new investor. Accordingly, we ran a DCF analysis to
the ¬scal year ending closest to the due date of the note. That value is
$8,000,000 and appears in C44.
The subtotal number of shares is 1,200,000 (F62) before the new in-
vestor. Since there is no restructure with the parent in this scenario, the
shares issued to the president is zero here (F63). In section 4 we calculate
that the new investor will demand one-third of the Company post-
transaction (see description below). That implies the investor will demand
600,000 shares (F64), which will bring the total shares to 1,800,000 (F65).
Dividing $2,753,938 (K41, repeated in F53) by 1,800,000 shares leads to a
value of $1.530 (F66) per share for the no-restructure scenario (this should
more appropriately be called ˜˜restructure later™™).


Conclusion
Thus, the restructure is preferable by a FMV per share of $10.391 $1.530
$8.861 per share ( D66 F66).

Section 4: Year 2000 Investor Percentage
A future restructure would be a more distressed one than the current one.
The discounted cash ¬‚ow analysis indicates that the Company would be
short of cash to pay off the note. With two years gone by, the Company
is more likely to lose the possibility of becoming the market leader and
more likely to be an also ran. Also, it would be a far more highly lever-
aged ¬rm without the restructure. Therefore, it would be a higher-risk
¬rm in the year 2000, which dictates using a higher discount rate than
the other scenarios. The result is a value of $8,000,000 (C44, repeated as
B71) before the minority interest discount.


16
The analysis was done in 1996.




PART 5 Special Topics
428
Subtracting the $2 million (B73) minority interest discount leaves us
with an FMV of $6 million (B74). In the DCF we determined the Company
would need a $2 million investment by a new investor, who would re-
quire taking one-third (B75) of the Company. This percentage is used in
section 3, F52 in the no-restructure calculations, as discussed above.


EXPONENTIALLY DECLINING SALES GROWTH MODEL
When forecasting yearly sales for a startup, the appraiser ideally has a
bottom-up forecast based on a combination of market data and reasonable
assumptions. Sometimes those data are not available to us, and even
when they are available, it is often bene¬cial to use a top-down approach
based on reasonable assumptions of sales growth rates. In this section we
present a model for forecasting sales of a startup or early-stage company
that semiautomates the process of forecasting sales and can easily be ma-
nipulated for sensitivity analysis. The other choice is to insert sales
growth rates manually for, say, 10 years, print out the spreadsheet with
that scenario, change all 10 growth rates, and repeat the process for val-
uation of multiple scenarios. Life is too short.
One such sales model that has intuitive appeal is the exponentially
declining sales growth rate model, presented in Table 12-4. In the model
we have a peak growth rate (P), which decays with a decay rate constant
(k) to a ¬nal growth rate (G). The mathematics may look a little dif¬cult,
but it is not necessary to understand the math in order to bene¬t from
using the model.
The top of Table 12-4 is a list of the parameters of the model. In the
example the ¬nal sales growth rate (G) is set at 6% (E6), and the addi-
tional growth rate (A) is calculated to be 294% (E7). The additional growth
rate (A) is the difference between the peak growth rate (P), which is set
at 300% (E8), and the ¬nal sales growth rate of 6%. Next we have the
decay rate constant (k), which is set at 0.50 (E9). The larger the decay rate
constant, the faster the sales growth rate will decline to the ¬nal growth
rate. Finally, we have Year 1 forecast sales of 100 (E10). All the variables
are speci¬ed by the model user with the exception of the additional
growth rate (A), which depends on P and G.
Example #1 shows the forecast sales growth rates (row 17) and sales
(row 18) using the previously speci¬ed variables for a case where the
sales growth rate declines after Year 2. We have no sales growth rate in
Year 1 because we assume there are no prior year sales. The expression
Ae k(t 2), for all t greater than or equal
for the sales growth rate G
to 2, where t is expressed in years. For Year 2 the sales growth rate is G
Ae k(2 2) G A 6% 294% 300% (C17), which is our speci¬ed
Ae k(3 2)
peak growth rate P. Year 3 growth is G 6% (294%
0.5 1 k(4 2)
e ) 184% (D17). Year 4 growth is G Ae 6% 294%
0.5 2
e 114% (E17), etc. To calculate yearly sales, we simply multiply
the previous year sales by one plus the forecast growth rate.
Example #1A is identical to example #1, except that we have changed
the decay rate constant (k) from 0.50 to 0.30. Notice how reducing k slows
the decay in the sales growth rate. In example #2 we present a case of
the peak growth rate (P) occurring in a general future year f, where we



CHAPTER 12 Valuing Startups 429
T A B L E 12-4

Sales Model with Exponentially Declining Growth Rate Assumption


A B C D E F G H I J K

5 Variable Name Symbol Value Speci¬ed/Calculated

6 Final growth rate G 6% Speci¬ed
7 Additional growth rate A 294% Calculated
8 Peak growth rate P 300% Speci¬ed
9 Decay rate k 0.50 Speci¬ed
10 First year™s sales Sales1 100 Speci¬ed

13 Example # 1 - Sales growth rate declines after year 2
k(t 2)
14 Yearly growth G Ae for all t greater than or equal to 2

16 Year 1 2 3 4 5 6 7 8 9 10

17 Growth N/A 300% 184% 114% 72% 46% 30% 21% 15% 11%
18 Sales 100 400 1,137 2,436 4,179 6,093 7,929 9,566 10,989 12,240

21 Example # 1A - Changing the decay rate (k) from 0.50 to 0.30 slows the decline in the sales growth rate

23 Year 1 2 3 4 5 6 7 8 9 10

24 Growth N/A 300% 224% 167% 126% 95% 72% 55% 42% 33%
25 Sales 100 400 1,295 3,463 7,810 15,194 26,072 40,307 57,237 75,937

28 Example # 2 - Sales growth rate declines after future year f

Ae k(t f), for all t greater than or equal to f, where sales growth rate declines after future year f and
29 Sales growth rate G
30 the peak sales growth (P) occurs in year f. Growth through year f is to be speci¬ed by model user. The following is an
31 example with year f 4, and decay rate k 0.5

33 Year 1 2 3 4 5 6 7 8 9 10

34 Growth N/A 100% 200% 300% 184% 114% 72% 46% 30% 21%
35 Sales 100 200 600 2,400 6,824 14,613 25,077 36,559 47,575 57,393

Formula in Cell C17: G A*EXP( k*(C16 2))




F I G U R E 12-3

Sales Forecast (Decay Rate 0.5)

40,000
35,000
30,000
25,000
Sales




20,000
15,000
10,000
5,000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27
Year




PART 5 Special Topics
430
F I G U R E 12-3A

Sales Forecast (Decay Rate 0.3)

450,000
400,000
350,000
300,000
Sales




250,000
200,000
150,000
100,000
50,000
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27
Year




have chosen the future year to be Year 4. The model user speci¬es the
growth rates prior to Year f (we have chosen 100% and 200% in Years 2
and 3, respectively). The growth rates for year f and later are G Ae k(t f).
As you can see, the growth rates from Years 4 through 10 in this example
are identical to the growth rates from Years 2 through 8 in example #1.
Figures 12-3 and 12-3A are graphs that show the sales forecasts from
examples #1 and #1A extended to 28 years. The slower decay rate of 0.3
in Figure 12-3A (versus 0.5 in Figure 12-3) leads to much faster growth.
After 28 years, sales are close to $450,000 versus $38,000. Changing one
single parameter can give the analyst a great deal of control over the sales
forecast. When sensitivity analysis is important, we can control the de-
cline in sales growth simply by using different numbers in cell E9, the
decay rate. This is not only a nice time saver, but it can lead to more
accurate forecasts, as many phenomena in life have exponential decay (or
growth), e.g., the decay of radiation, population of bacteria, etc.


BIBLIOGRAPHY
Fowler, Bradley A. 1989. ˜˜What Do Venture Capital Pricing Methods Tell About Valuation
of Closely Held Firms?™™ Business Valuation Review (June): 73“79.
” ”. 1990. ˜˜Valuation of Venture Capital Portfolio Companies”and Other Moving Tar-

gets.™™ Business Valuation Review (March): 13“17.
” ”. 1996. ˜˜Venture Capital Rates of Return Revisited.™™ Business Valuation Review

(March): 13“16.
Golder, Stanley C. 1986. ˜˜Structuring and Pricing the Financing.™™ In Pratt™s Guide to Venture
Capital Sources, 10th ed., ed. Stanley E. Pratt and Jane K. Morris. Wellesley Hills,
Mass.: Venture Economics.
Morris, Jane K. 1988. In Pratt™s Guide to Venture Capital Sources, 12th ed., ed. Jane K. Morris.
Wellesley Hills, Mass.: Venture Economics.
Pacelle, Mitchell. 1999. ˜˜Venture Firms Dethroning Buyout Kings.™™ Wall Street Journal, 7,
June 1999. p, C1.
Plummer, James L. 1987. QED Report on Venture Capital Financial Analysis. Palo Alto, Calif.:
QED Research, Inc. [See especially 2-7“2-10 and 6-2“6-13.]
Pratt, Stanley E. and Jane K. Morris, Guide to Venture Capital Sources, Venture Economics,
1986.




CHAPTER 12 Valuing Startups 431
CHAPTER 13


ESOPs: Measuring and
Apportioning Dilution1




INTRODUCTION
What Can Be Skipped
DEFINITIONS OF DILUTION
Dilution to the ESOP (Type 1 Dilution)
Dilution to the Selling Owner (Type 2 Dilution)
De¬ning Terms
TABLE 13-1: CALCULATION OF LIFETIME ESOP COSTS
THE DIRECT APPROACH
FMV Equations”All Dilution to the ESOP (Type 1 Dilution; No Type
2 Dilution)
Table 13-2, Sections 1 and 2: Post-transaction FMV with All Dilution
to the ESOP
The Post-transaction Value Is a Parabola
FMV Equations”All Dilution to the Owner (Type 2 Dilution)
Table 13-2, Section 3: FMV Calculations”All Dilution to the Seller
Sharing the Dilution
Equation to Calculate Type 2 Dilution
Tables 13-3 and 13-3A: Adjusting Dilution to Desired Levels
Table 13-3B: Summary of Dilution Tradeoffs
THE ITERATIVE APPROACH
Iteration #1
Iteration #2
Iteration #3
Iteration #n
SUMMARY


1. Adapted and reprinted with permission from Valuation (June 1997): 3“25 and (January 1993): 76“
103, American Society of Appraisers, Herndon, Virginia.




433




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
Advantages of Results
Function of ESOP Loan
Common Sense Is Required
To Whom Should the Dilution Belong?
De¬nitions
The Mathematics of the Post-transaction Fair Market Value Balance
Sheet
Analyzing a Simple Sale
Dilution to Non-selling Owners
Legal issues
Charity
APPENDIX A: MATHEMATICAL APPENDIX
APPENDIX B: SHORTER VERSION OF CHAPTER 13




PART 5 Special Topics
434
This chapter is the result of further thought and research on my treatment
of valuing ESOPs (Abrams 1993 and 1997). It not only simpli¬es those
articles, but it goes far beyond them. Reading them is not necessary for
understanding this chapter.

INTRODUCTION
Leveraged ESOPs have confused many ¬rms due to their failure to un-
derstand the phenomenon of dilution and inability to quantify it. Many
ESOPs have soured because employees paid appraised fair market value
of the stock being sold to the ESOP, only to watch the fair market value
signi¬cantly decline at the next valuation because the ESOP loan was not
included in the pre-transaction fair market value. As a result, employees
have felt cheated. Lawsuits have sometimes followed, further lowering
the value of the ¬rm and the ESOP.
There are several types of problems relating to the dilution phenom-
enon:
1. The technical problem of de¬ning and measuring the dilution in
value to the ESOP before it happens.
2. The business problem of getting the ESOP Trustee, participants,
and selling owner(s) to agree on how to share the dilution.
3. The technical problem of how to engineer the price to
accomplish the desired goals in 2.
4. The problem of how to communicate each of the foregoing to all
of the participants so that all parties can enter the transaction
with both eyes open and come away feeling the transaction was
win“win instead of win“lose.
This chapter provides the analytical solutions to problems 1 and 3
that are necessary for resolving the business and communication prob-
lems of 2 and 4. The appraiser will be able to include the dilution in his
or her initial valuation report so that employees will not be negatively
surprised when the value drops at the next annual valuation. Addition-
ally, the appraiser can provide the technical expertise to enable the parties
to share the dilution, solving problem 3. Both parties will then be fully
informed beforehand, facilitating a win“win transaction.

What Can Be Skipped
This chapter contains much tedious algebra. For readers who wish to skip
all of the mathematics and optional sections and simply get the bottom
line can read the ˜˜quick-and-dirty™™ version of this chapter in Appendix
B. The section on the iterative approach can be safely skipped, as it en-
hances the understanding of dilution but contains no additional formulas
of practical signi¬cance.

DEFINITIONS OF DILUTION
Two potential parties can experience dilution in stock values in ESOP
transactions: the ESOP and the owner. The dilution that each experiences
differs and can be easily confused.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 435
Additionally, each party can experience two types of dilution: abso-
lute and relative. Absolute dilution is de¬ned in the section immediately
below. Relative dilution is more complicated because we can calculate
dilution relative to more than one base. Several formulas can be devel-
oped to calculate relative dilution, but they are beyond the scope of this
book. Thus, for the remainder of this chapter, dilution will mean absolute
dilution.

Dilution to the ESOP (Type 1 Dilution)
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-
partment of Labor proposed regulations indicate that the pre-transaction
value should be used.2

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%


2. Donovan v. Cunningham, 716 F.2d 1467. 29 CFR 2510.3-18(b).


PART 5 Special Topics
436
r the annual loan interest rate, assumed at 10%
i the monthly loan interest rate r/12 0.8333% monthly
V1B the pre-transaction value of 100% of the stock of the ¬rm
after discounts and premiums at the ¬rm level but before those at
the ESOP level,3 assumed at $1,000,000, as shown in Table 13-2.
The B subscript means before considering the lifetime cost of
initiating and maintaining the ESOP (see E, e, and VjA below). V1B
does not consider the cost of the loan. This differs from VjB, as
described below.
V1A Same as V1B, except this is the pre-transaction value after
deducting the lifetime cost of initiating and maintaining the ESOP
(see E, e, and VjA below) but before considering the loan. Note this
differs from VjA, where j 1, where we do subtract the cost of the
ESOP loan as of iteration j 1.
VjB the value of the ¬rm at the jth iteration before deducting the
lifetime ESOP costs (see E below) but after subtracting the net
present value of the ESOP loan (see NPLV) as calculated in
iteration j 1 (for j 1).
VjA the value of the ¬rm at the jth iteration after deducting the
lifetime ESOP costs (see immediately below) and the ESOP loan as
of the ( j 1)st iteration.
Vn the ¬nal post-transaction value of the ¬rm, i.e., at the nth
iteration
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 $40,000.4
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
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%.5 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



3. In Abrams (1993) the discounts and premiums at the ¬rm level are a separate variable. This
treatment is equally as accurate and is simpler.
4. How to calculate the pre-transaction value of the ¬rm is outside the scope of this article.
5. These are arbitrary assumptions chosen for mathematical ease.


CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 437
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.
Lj the amount of the ESOP Loan in iteration j, which equals the
payment to the owner. That equals the FMV of the ¬rm in
iteration j multiplied by pDE, the percentage of the ¬rm being sold
to the ESOP, multiplied again by the factor for discounts or
premiums at the ESOP level. Mathematically, Lj pDE VjA. Note:
this de¬nition only applies in the Iterative Approach where we are
eliminating type 1 dilution.
NPVLj the after-tax, net present value of the ESOP loan as
calculated in iteration j. The formula is NPVLj (1 t)Lj, as
explained below.
n The number of iterations
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).


T A B L E 13-1

Calculation of Lifetime ESOP Costs


A B

5 Pre-tax annual ESOP costs $10,000
6 After-tax annual ESOP costs (1 t) * pre-tax 6,000
7 Required rate of return r 25%
8 Perpetual growth of ESOP costs g 5%
9 Gordon model multiple (end year) 1/(r g) 5.000
10 Capitalized annual costs 30,000
11 Initial outlay-pre-tax 20,000
12 Initial outlay-after-tax (1 t) * pre-tax 12,000
13 Lifetime ESOP costs 42,000
14 Lifetime ESOP costs-rounded to (used in Table 13-2, B9) $40,000




PART 5 Special Topics
438
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.6 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.


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 (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
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
the ¬rst iteration. In the second iteration the payment to the owner is less
than the pre-transaction price because we have considered the ESOP loan
from the ¬rst iteration in our second iteration valuation. Because the pay-
ment is lower in this iteration, the ESOP loan is lower than it is in the
¬rst iteration. We follow through with several iterations until we arrive
at a steady-state value, where the engineered payment to the owner ex-
actly equals the post-transaction value of the ESOP. This enables us to
eliminate all type 1 dilution to the ESOP and shift it to the owner as type
2 dilution.


Iteration #1
We denote the pre-transaction value of the ¬rm before considering the
lifetime ESOP administration cost as V1B.

PART 5 Special Topics
448
V1B pre-transaction value (13-5)
The value of the ¬rm after deducting the lifetime ESOP costs but before
considering the ESOP loan is:11
V1A V1B E V1B V1B e V1B(1 e) (13-5a)
The owner sells p% of the stock to the ESOP, so the ESOP would pay
p times the value of the ¬rm. However, we also need to adjust the pay-
ment for the degree of marketability and control of the ESOP. Therefore,
the ESOP pays the owner V1A multiplied by p DE , or:
L1 pDEV1A pDEV1B(1 e) (13-5b)
Our next step is to compute the net present value of the loan. In this
chapter we greatly simplify this procedure over the more complex cal-
culation in my original article (Abrams 1993).12
The net present value of the payments of any loan discounted at the
loan rate is the principal of the loan. Since both the interest and principal
payments on ESOP loans are tax deductible, the after-tax cost of the ESOP
loan is simply the principal of the loan multiplied by one minus the tax
rate.13 Therefore:
NPVL1 (1 t)pDEV1B(1 e) (13-5c)


Iteration #2
We have now ¬nished the ¬rst iteration and are ready to begin iteration
#2. We begin by subtracting equation (13-5c), the net present value of the
ESOP loan, from the pre-transaction value, or:
V2B V1B (1 t)pDEV1B(1 e)
V1B[1 pDE(1 t)(1 e)] (13-6)
We again subtract the lifetime ESOP costs to arrive at V2A.
V2A V2B E (13-6a)
V2A V1B[1 pDE(1 t)(1 e)] V1Be (13-6b)
Factoring out the V1B, we get:


11. V1A is the only iteration of VjA where we do not consider the cost of the loan. For j 1, we do
consider the after-tax cost of the ESOP loan.
12. You do not need to read that article to understand this chapter.
13. One might speculate that perhaps the appraiser should discount the loan by a rate other than
the nominal rate of the loan. To do so would implicitly be saying that the ¬rm is at a
suboptimal D/E (debt/equity) ratio before the ESOP loan and that increasing debt lowers
the overall cost of capital. This is closer to a matter of faith than science, as there are those
that argue on each side of the fence. The opposite side of the fence is covered by two Nobel
Prize winners, Merton Miller and Franco Modigliani (MM), in a seminal article (Miller and
Modigliani 1958). MM™s famous Proposition I states that in perfect capital markets, i.e., in
the absence of taxes and transactions costs, one cannot raise the value of the ¬rm with debt.
They acknowledge a secondary tax effect of debt, which I use here literally and no further,
i.e., adding debt increases the value of the equity only to the extent of the tax shield. Also,
even if there is an optimal D/E ratio and the subject company is below it, it does not need
an ESOP to borrow to achieve the optimal ratio.


CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 449
V2A V1B[(1 e) pDE(1 t)(1 e)] (13-6c)
Factoring out the (1 e), we then come to the post-transaction value
of the ¬rm in iteration #2 of:
V2A V1B(1 e)[1 pDE(1 t)] (13-6d)
It is important to recognize that we are not double-counting E, i.e.,
subtracting it twice. In equation (13-6) we calculate the value of the ¬rm
as its pre-transaction value minus the net present value of the loan against
the ¬rm. The latter is indirectly affected by E, but in each new iteration,
we must subtract E directly in order to count it in the post-transaction
value.
The post-transaction value of the ESOP loan in iteration #2 is p
DE (13-6d), or:
L2 pDEV1B(1 e)[1 pDE(1 t)] (13-6e)
The net present value of the loan is:
NPVL2 (1 t)pDEV1B(1 e)[1 (1 t)pDE] (13-6f)


Iteration #3
We now begin the third iteration of value. The third iteration FMV before
lifetime ESOP costs is V1B NPVL2, or:
V3B V1B (1 t)pDEV1B (1 e)[1 (1 t)pDE] (13-7)
Factoring out V1B, we have:
V3B V1B{1 pDE(1 t)(1 e)[1 (1 t)pDE]} (13-7a)
Multiplying terms, we get:
p 2D 2 (1 t)2(1
V3B V1B[1 pDE(1 t)(1 e) e)] (13-7b)
E

V3A V3B E (13-7c)
p 2D 2 (1 t)2(1
V3A V1B[1 pDE(1 t)(1 e) e) e] (13-7d)
E

Moving the e at the right immediately after the 1:
V3A V1B[(1 e) pDE(1 t)(1 e)
(13-7e)
p 2D E(1
2
t)2(1 e)]
Factoring out the (1 e):
p 2D 2 (1
V3A V1B(1 e)[1 pDE(1 t) t)] (13-7f)
E

p0 D E(1
0
t)0
Note that the 1 in the square brackets


Iteration #n
Continuing this pattern, it is clear that the nth iteration leads to the fol-
lowing formula:
n1
1) j p j D jE(1 t)j
VnA V1B (1 e) ( (13-8)
j0




PART 5 Special Topics
450
This is an oscillating geometric sequence,14 which leads to the following
solutions. The ultimate post-transaction value of the ¬rm is:
1 e
VnA V1B
1 [ pDE(1 t)]
or, dropping the subscript A and simplifying: (13-8a)
post-transaction value of the firm”
015
with type 1 dilution
1 e
Vn V1B (13-9)
1 (1 t)pDE
Note that this is the same equation as (13-3n). We arrive at the same result
from two different approaches.
The post-transaction value of the ESOP is p DE the value of the
¬rm, or:
pDE(1 e)
Ln V1B
1 (1 t)pDE
post-transaction value of the ESOP”
with type 1 dilution 0 (13-10)


This is the same solution as equation (13-3j), after multiplying by V1B. The
iterative approach solutions in equations (13-9) and (13-10) con¬rm the
direct approach solutions of equations (13-3n) and (13-3j).


SUMMARY
In this chapter we developed formulas to calculate the post-transaction
values of the ¬rm, ESOP, and the payment to the owner, both pre-
transaction and post-transaction, as well as the related dilution. We also
derived formulas for eliminating the dilution in both scenarios, as well
as for specifying any desired level of dilution. Additionally, we explored
the trade-offs between type 1 and type 2 dilution.


Advantages of Results
The big advantages of these results are:
1. If the owner insists on being paid at the pre-transaction value,
as most will, the appraiser can now immediately calculate the
dilutive effects on the value of the ESOP and report that in the
initial valuation report.16 Therefore, the employees will be


14. For the geometric sequence to work, pDE(1 t) 1 , which will almost always be the case.
15. The reason the e term is in the numerator and not the denominator like the other terms is that
the lifetime cost of the ESOP is ¬xed, i.e., it does not vary as a proportion of the value of
the ¬rm (or the ESOP), as that changes in each iteration.
16. Many ESOP trustees prefer this information to remain as supplementary information outside of
the report.




CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 451
entering the transaction with both eyes open and will not be
disgruntled or suspicious as to why the value, on average,
declines at the next valuation. This will also provide a real
benchmark to assess the impact of the ESOP itself on
pro¬tability.
2. For owners who are willing to eliminate the dilution to the
ESOP or at least reduce it, this chapter provides the formulas to
do so and the ability to calculate the trade-offs between type 1
and type 2 dilution.

Function of ESOP Loan
An important byproduct of this analysis is that it answers the question
of what is the function of the ESOP loan. Obviously it functions as a
¬nancing vehicle, but suppose you were advising a very cash rich ¬rm
that could fund the payment to the owner in cash. Is there any other
function of the ESOP loan? The answer is yes. The ESOP loan can increase
the value of the ¬rm in two ways:
1. It can be used to shield income at the ¬rm™s highest income tax
rate. To the extent that the ESOP payment is large enough to
cause pre-tax income to drop to lower tax brackets, that portion
shields income at lower than the marginal rate and lowers the
value of the ¬rm and the ESOP.
2. If the ESOP payment in the ¬rst year is larger than pre-tax
income, the ¬rm cannot make immediate use of the entire tax
deduction in the ¬rst year. The unused deduction will remain as
a carryover, but it will suffer from a present value effect.

Common Sense Is Required
A certain amount of common sense is required in applying these for-
mulas. In extreme transactions such as those approaching a 100% sale to
the ESOP, we need to realize that not only can tax rates change, but
payments on the ESOP loan may entirely eliminate net income and reduce
the present value of the tax bene¬t of the ESOP loan payments. In ad-
dition, the viability of the ¬rm itself may be seriously in question, and it
is likely that the appraiser will have to increase the discount rate for a
post-transaction valuation. Therefore, one must use these formulas with
at least two dashes of common sense.

To Whom Should the Dilution Belong?
Appraisers almost unanimously consider the pre-transaction value ap-
propriate, yet there has been considerable controversy on this topic. The
problem is the apparent ¬nancial sleight of hand that occurs when the
post-transaction value of the ¬rm and the ESOP precipitously declines
immediately after doing the transaction. On the surface, it somehow
seems unfair to the ESOP. In this section we will explore that question.
De¬nitions
Let™s begin to address this issue by assessing the post-transaction fair
market value balance sheet. We will use the following de¬nitions:

PART 5 Special Topics
452
Pre-Transaction Post-Transaction

A1 assets A2 assets A1 (assets have not changed)
L1 liabilities L2 liabilities
C1 capital C2 capital


Note that the subscript 1 refers to pre-transaction and the subscript 2
refers to post-transaction.

The Mathematics of the Post-Transaction Fair Market Value
Balance Sheet
The nonmathematical reader may wish to skip or skim this section. It is
more theoretical and does not result in any usable formulas.
The fundamental accounting equation representing the pre-
transaction balance sheet is:
A1 L1 C1 pre-transaction FMV balance sheet (13-11)
Assuming the ESOP bears all of the dilution, after the sale liabilities
increase and capital decreases by the sum of the after-tax cost of the ESOP
loan and the lifetime ESOP costs,17 or:
C1 [(13-1c) (13-1d)]
increase in liabilities and decrease in debt (13-12)
As noted in the de¬nitions, assets have not changed. Only liabilities
and capital have changed.18 Thus the post-transaction balance sheet is:
A2 {L1 C1[(1 t)pDE e]} {C1 C1[(1 t)pDE e]} (13-13)
The ¬rst term in braces equals L2, the post-transaction liabilities, and the
second term in braces equals C2, the post-transaction capital. Note that
A2 A1. Equation (13-13) simpli¬es to:
A2 {L1 C1[(1 t)pDE e]} {C1[1 (1 t)pDE e]}
post-transaction balance sheet (13-14)
Equation (13-14) gives us an algebraic expression for the post-
transaction fair market value balance sheet when the ESOP bears all of
the dilution.

Analyzing a Simple Sale
Only two aspects relevant to this discussion are unique about a sale to
an ESOP: (1) tax deductibility of the loan principal, and (2) forgiveness
of the ESOP™s debt. Let™s analyze a simple sale to a non-ESOP buyer and
later to an ESOP buyer. For simplicity we will ignore tax bene¬ts of all
loans throughout this example.


17. Again, these should only be the incremental costs if the ESOP is replacing another pension
plan.
18. For simplicity, we are assuming the company hasn™t yet paid any of the ESOP™s lifetime costs.
If it has, then that amount is a reduction in assets rather than an increase in liabilities.
Additionally, the tax shield on the ESOP loan could have been treated as an asset rather
than a contraliability, as we have done for simplicity. This is not intended to be an
exhaustive treatise on ESOP accounting.


CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 453
Suppose the fair market value of all assets is $10 million before and
after the sale. Pre-transaction liabilities are zero, so capital is worth $10
million, pre-transaction. If a buyer pays the seller personally $5 million
for one-half of the capital stock of the Company, the transaction does not
impact the value of the ¬rm”ignoring adjustments for control and mar-
ketability. If the buyer takes out a personal loan for the $5 million and
pays the seller, there is also no impact on the value of the company. In
both cases the buyer owns one-half of a $10 million ¬rm, and it was a
fair transaction.
If the corporation takes out the loan on behalf of the buyer but the
buyer ultimately has to repay the corporation, then the real liability is to
the buyer, not the corporation, and there is no impact on the value of the
stock”it is still worth $5 million. The corporation is a mere conduit for
the loan to the buyer.
What happens to the ¬rm™s value if the corporation takes out and
eventually repays the loan? The assets are still worth $10 million post-
transaction.19 Now there are $5 million in liabilities, so the equity is worth
$5 million. The buyer owns one-half of a ¬rm worth $5 million, so his or
her stock is only worth $2.5 million. Was the buyer hoodwinked?
The possible confusion over value clearly arises because it is the cor-
poration itself that is taking out the loan to fund the buyer™s purchase of
stock, and the corporation”not the buyer”ultimately repays the loan.
By having the corporation repay the loan, the other shareholder is for-
giving his or her half of a $5 million loan and thus gifting $2.5 million
to the buyer.20 Thus, the ˜˜buyer™™ ultimately receives a gift of $2.5 million
in the form of company stock. This is true whether the buyer is an in-
dividual or an ESOP.21

Dilution to Non-Selling Owners
When there are additional business owners who do not sell to the ESOP,
they experience dilution of their interests without the bene¬t of getting
paid. Conceptually, these owners have participated in giving the ESOP a
gift by having the Company repay the debt on behalf of the ESOP.
To calculate the dilution to other owners, we begin with the post-
transaction value of the ¬rm in equation (13-1e) and repeat the equation
as (13-1e*). Then we will calculate the equivalent equations for the non-
selling owner as we did for the ESOP in equations (13-1f) and (13-1g),
and we will relabel those equations by adding an asterisk.
1 (1 t)pDE e
post-transaction value of the firm (repeated) (13-1e*)
If the nonselling shareholder owns the fraction q of the outstanding stock,
then his or her post-transaction value is:


19. There is a second-order effect of the ¬rm being more highly leveraged and thus riskier that
may affect value (and which we are ignoring here). See Chapter 14.
20. The other half of the forgiveness is a wash”the buyer forgiving it to himself or herself.
21. This does not mean that an ESOP brings nothing to the table in a transaction. It does bring tax
deductibility of the loan principal as well as the Section 1042 rollover.




PART 5 Special Topics
454
q q(1 t)pDE qe
post-transaction value of nonselling shareholder™s stock (13-1f*)
Finally, we calculate dilution to the nonselling shareholder as his or her
pre-transaction value of q minus the pre-transaction value in equation
(13-1f*), or:
q[(1 t)pDE e]
dilution to nonselling shareholder™s stock22 (13-1g*)
The dilution formula (13-1g*) tells us that the dilution to the non-
selling shareholder is simply his or her ownership, q, multiplied by the
dilution in value to the ¬rm itself, which is the sum of the after-tax cost
of the ESOP loan and the lifetime costs. Here, because we are not mul-
tiplying by the ESOP™s ownership modi¬ed for its unique marketability
and control attributes, we do not get the squared terms that we did in
equation (13-1f) and (13-1g).
It is also important to note that equations (13-1f*) and (13-1g*) do
not account for any possible increase in value the owner might experience
as a result of having greater relative control of the ¬rm. For example, if
there were two 50% owners pre-transaction and one sells 30% to the
ESOP, post-transaction the remaining 50% owner has relatively more con-
trol than he or she had before the transaction. To the extent that we might
ascribe additional value to that increase in relative control, we would
adjust the valuation formulas. This would mitigate the dilution in equa-
tion (13-1g*).

Legal Issues
As mentioned above, appraisers almost unanimously consider the pre-
transaction value appropriate. Also mentioned earlier in the chapter, case
law and Department of Labor proposed regulations indicate the pre-
transaction value is the one to be used. Nevertheless, there is ongoing
controversy going back to Farnum, a case in which the Department of
Labor withdrew before going to court, that the post-transaction value may
the most appropriate price to pay the seller.
In the previous section we demonstrated that the ESOP is receiving
a gift, not really paying anything for its stock. Therefore, there is no ec-
onomic justi¬cation for reducing the payment to the owner below the
pre-transaction fair market value, which is the price that the seller would
receive from any other buyer. If the ESOP (or any party on its behalf)
demands that it ˜˜pay™™ no more than post-transaction value, it is tanta-
mount to saying, ˜˜The gift you are giving me is not big enough.™™
While the dilution may belong to the ESOP, it is nevertheless an
important consideration in determining the fairness of the transaction for


22. One would also need to consider adjusting for each nonselling shareholder™s control and
marketability attributes. To do so, we would have to add a term in equation (13-1g*)
immediately after the q. The term would be the owner™s equivalent of DE, except

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