spectively, 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 (A13-7e) 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 (A13-7f) is $0.23037827 (B15) $1 million pre-

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

We calculate dilution to the ESOP according to equation (A13-7g) 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 $1 million 30% 0.98 (net of ESOP discounts/

premiums) $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-

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

value obtained by direct calculation using formula (A13-7e) 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 (A13-7f), the formula for the post-transaction value of the ESOP,

is a parabola. We can see this more easily by rewriting (A13-7f) as

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

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

PART 5 Special Topics

462

(1 e) (1 t)x post-transaction value of the firm (A13-8e)

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

discount, the post-transaction value of the ESOP is p DEx (A13-8e), or:

pDE(1 e) (1 t)pDEx

post-transaction value of the ESOP (A13-8f)

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

(A13-8f), or:

x pDE(1 e) (1 t)pDEx (A13-8g)

which solves to:

pDE (1 e)

x

1 (1 t)pDE

post-transaction FMV of ESOP, all dilution to owner (A13-8j)

Substituting equation (A13-8j) into the x term in equation (A13-8e), the

post-transaction value of the ¬rm is:

1 e

post-transaction value of the firm”

1 (1 t)pDE

type 1 dilution 0 (A13-8n)

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

the price paid, or:

1 e

pDE (A13-8o)

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 (A13-8n) 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

PART 5 Special Topics

464

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

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

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 (A13-8n). Multiplying by pDE (0.3 0.98

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

accuracy of (A13-8j).

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 (A13-8o) of 5.4082% (C31) the pre-transaction price of $1 mil-

lion $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 (A13-8f)

from the cash to the owner (A13-8a), 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). There-

fore,

Actual Dilution to ESOP

k ,

Default Dilution to ESOP

or 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 2 D 2

x [pDE(1 e) (1 t)pDEx] k[(1 pDE e] (A13-9)

E

which solves to:

t)p 2 D 2

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

E

x (A13-9a)

1 (1 t)pDE

In other words, equation (A13-8a) is the formula for the amount of

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 465

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

dilution. If we let k 0, (A13-8a) reduces to (A13-8j), the post-transaction

FMV of the ESOP when all dilution goes to the owner. When k 1, (A13-

9a) reduces to (A13-7a), 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 (A13-9a) for x, we get:

t)p 2 D E

2

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

D2 pDE (A13-9b)

1 (1 t)pDE

Tables 13-3 and 13-3A: Adjusting Dilution to

Desired Levels

Table 13-3 is a numerical example using equation (A13-9a). 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 (A13-

9a)”which is 27.6%. B11 is the value of the ESOP post-transaction, which

we calculate according to equation (A13-8f),30 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 (A13-7g), 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 (A13-9a). By

designating 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 equation (A13-9a), and the equation reduces to

pDE(1 e)

x

[1 (1 t)pDE]

which is identical to equation (A13-8j), 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

30. With pDE factored out.

PART 5 Special Topics

466

(A13-9b). Both calculations produce identical results, con¬rming the ac-

curacy of (A13-9b). In Table 13-3A, where we let k 0, type 2 dilution is

5.41% (B15 and B16).

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 Appendix A, the Math-

ematical Appendix.

SUMMARY

In this mini-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 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.31 Therefore, the employees will be

entering the transaction with both eyes open and will not be

disgruntled and/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.

31. Many ESOP trustees prefer this information to remain as supplementary information outside of

the report.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 467

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, then 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.

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.

PART 5 Special Topics

468

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.32 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.33 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.34

Dilution to Nonselling 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.

Assuming the nonselling owner has the fraction q of the outstanding

stock of the ¬rm, his or her dilution is equal to:

q[(1 t) pDE e]

dilution to nonselling shareholder™s stock35 (A13-1g*)

The dilution formula (A13-1g*) tells us that the dilution to the non-

selling shareholder is simply his or her ownership, q, multiplied by the

32. 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.

33. The other half of the forgiveness is a wash”the buyer forgiving it to himself or herself.

34. 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.

35. 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

customized for his or her ownership attributes. The details of such a calculation are beyond

the scope of this chapter.

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 469

dilution in value to the ¬rm itself, which is the sum of the after-tax cost

of the ESOP loan and the lifetime costs.

It is also important to note that equation (A13-1g*) does 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 control 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 equation (A13-

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

be 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

purposes of a fairness opinion. If a bank loans $10 million to the ESOP

for a 100% sale, with no recourse or personal guarantees of the owner,

we may likely decide it is not a fair transaction to the ESOP and its

participants. We would have serious questions about the ESOP™s proba-

bility of becoming a long-range retirement program, given the huge debt

load of the Company post-transaction.

Charity