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While the dilution technically belongs to the ESOP, I consider it my duty
to inform the seller of the dilution phenomenon and how it works. While
af¬rming the seller™s right to receive fair market value undiminished by
dilution, I do mention that if the seller has any charitable motivations to
his or her employees”which a minority do”then voluntarily accepting
some of the dilution will leave the Company and the ESOP in better
shape. Of course, in a partial sale it also leaves the remainder of the
owner™s stock at a higher value than it would have had with the ESOP
bearing all of the dilution.

PART 5 Special Topics

Buyouts of Partners and

The Solution
First-Order Impact of Buyout on Post-transaction Valuation
Secondary Impact of Buyout on Post-transaction Valuation
ESOP Dilution Formula as a Benchmark


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Buying out a partner or shareholder is intellectually related to the prob-
lem of measuring dilution in employee stock ownership plans (ESOPs),
which is covered in the previous chapter. There is no substantive differ-
ence in the post-transaction effects of buying out partners versus share-
holders, so for ease of exposition we will use the term partners to cover
both situations.

Suppose you have already valued the drapery manufacturer owned by
the Roth family, the Drapes of Roth. Its FMV on an illiquid minority
interest basis is $1 million pre-buyout. There are four partners, each with
a 25% share of the business: I. M. Roth, U. R. Roth, Izzy Roth, and B.
Roth. There are 1 million shares issued and outstanding, so the per share
FMV is $1 million FMV/1 million shares $1.00 per share. The problem
is the impact on the post-transaction FMV if the three other Roths become
wroth with Izzy Roth and want to buy him out.

The Solution
The solution to the problem ¬rst depends whether the three Roths have
enough money to buy out Izzy with their personal assets. If so, then there
is no impact on the value of the ¬rm. If not then the ¬rm typically will
take out a loan to buy out Izzy.1

First-Order Impact of Buyout on Post-transaction Valuation
To a ¬rst approximation, there should be no impact on the FMV per share.
For simplicity of discussion, we ignore the subtleties of differentials in
the discount for lack of control of 25% versus 33 1/3% interests, although
in actuality the appraiser must consider that issue. The FMV of the ¬rm
has declined by the amount of the loan to $750,000. The shareholders
bought 250,000 shares, leaving $750,000 shares. Our ¬rst approximation
of the post-transaction value is $750,000/750,000 shares $1.00 per share,
or no change.

Secondary Impact of Buyout on Post-transaction Valuation
The $250,000 has increased the debt-to-equity ratio of the ¬rm. The ¬rm
has increased its ¬nancial risk, which raises the overall risk of the ¬rm.2
It is probably appropriate to raise the discount rate 1“2% to re¬‚ect the
additional risk and rerun the pre-transaction discounted cash ¬‚ows to
come to a potential post-transaction valuation. Suppose that value is $0.92

1. It is possible for the shareholders to take out the loan individually and the ¬rm would pay it
indirectly by bonusing out suf¬ciently large salaries to cover the personally loans above and
beyond their normal draw. This has no impact on the solution, as both the direct and
indirect approaches will come to the same result.
2. In the context of the capital asset pricing model, the stock beta rises with additional ¬nancial

PART 5 Special Topics
per share. Is that reasonable? What if the tentative post-transaction value
were $0.78 per share? Is that reasonable?

ESOP Dilution Formula as a Benchmark
A benchmark would be very helpful to determine reasonability. Let™s set
up a hypothetical ESOP with tax attributes similar to the partner to be
bought out. A loan to fund this purchase would have no tax advantages.
While the interest is tax deductible, the ¬rm does not need to engage in
this buyout transaction in order to achieve its optimal debt to equity ratio
in order to have the minimum possible weighted average cost of capital
(WACC). The ¬rm can borrow optimally without a buyout. Therefore, it
is reasonable to consider the after-tax cost of the loan to be the same as
its pre-tax amount, which is the payment to the partner.
The following is a listing and calculation of the various values per-
tinent to this transaction. All values are a fraction of a starting pre-
transaction value of $1.
1 pre-buyout FMV (14-1)
x payment to the partner (14-2)
1 x post-transaction FMV”Firm (14-3)
The hypothetical ESOP owns p% of the ¬rm, where p is the portion
of the partnership bought from the selling partner. Its post-transaction
value is:
p(1 x) post-transaction FMV”Hypothetical ESOP (14-4)
The ¬rst four formulas tell us that for every $1 of pre-transaction value,
the company pays the selling partner x, which leaves a post-transaction
value of the ¬rm of 1 x and post-transaction of the ESOP™s interest in
the partnership of p(1 x).
The company should pay the partner the amount that equates the
payment to the partner with the post-transaction value of the hypothetical
ESOP, or:
x p(1 x) Payment Post-Trans. FMV- Hypothetical ESOP (14-5)
Collecting terms,
x px p (14-5a)
x(1 p) p (14-5b)
Dividing through by 1 p, we come to a ¬nal solution of:
x (14-6)
1 p
Note that equation (14-6) is identical to equation (13-3j) when e 0,
t 0, and DE 1. This makes sense for the following reasons:
1. This is a buyout of a partner. The ESOP is hypothetical only.
There are no lifetime ESOP costs, which means e 0.

CHAPTER 14 Buyouts of Partners and Shareholders 473
2. There are no tax bene¬ts of the loan to buy out the partner.
Therefore, tax savings on the hypothetical ESOP loan are zero
and t 0.
3. There are no ESOP level marketability attributes of marketability
and control in the buyout of the partner, therefore DE
Substituting p 25% into equation (14-6), x 20%. Let™s check the re-
1. The Company pays 20% of the pre-transaction value to the
2. The post-transaction value is the remaining 80%.
3. There are three real partners remaining plus the hypothetical
ESOP, for a total of four partners
4. Each remaining partner has a 1„4 share of the 80%, or 20%,
which is equal to the payment to the ¬rst partner. This
demonstrates that equation (14-6) works.
Thus, for every $1.00 of pre-transaction value, this hypothetical ESOP
benchmark leaves us with $0.80 per share post-transaction value.

If the transaction would not increase ¬nancial risk, the post-transaction
value of the ¬rm would be the same as the pre-transaction value, or $1.00
per share. Incorporating the leverage into the valuation, we have results
of $0.92 per share and $0.78 per share using two different additions to
the discount rate in our discounted cash ¬‚ow analysis. Our hypothetical
ESOP benchmark value is $0.80 per share. What is reasonable?
It is clear that the post-transaction value cannot be more than the
pre-transaction value, so the latter is a ceiling value. It is also clear that
the hypothetical ESOP approach is a ¬‚oor value, because the ESOP really
does not exist and the 250,000 shares are really not outstanding. The hy-
pothetical ESOP approach assumes the shares are outstanding. Therefore,
the post-transaction value must be higher than the hypothetical ESOP
Now we know the post-transaction value of the ¬rm should be less
than $1.00 per share and greater than $0.80 per share. The $0.92 per share
post-transaction value looks quite reasonable, while the $.78 per share
value is obviously wrong. If we had added 1% to the discount rate to
arrive at the $0.92 per share and 2% to the discount rate to produce the
$0.78 per share result, the 1% addition would appear to be the right one.

3. However, this is where the differences mentioned earlier, i.e., differences in the discount for lack
of control of a 25% partner versus a 1/3 partner, would come into play.

PART 5 Special Topics

ADF (annuity discount factor) the present value of a ¬nite stream of
cash ¬‚ows for every beginning $1 of cash ¬‚ow. See Chapter 3.
control premium the additional value inherent in the control interest as
contrasted to a minority interest, which re¬‚ects its power of control1
CARs (cumulative abnormal returns) a measure used in academic ¬-
nance articles to measure the excess returns an investor would have re-
ceived over a particular time period if he or she were invested in a par-
ticular stock. This is typically used in control and takeover studies, where
stockholders are paid a premium for being taken over. Starting some time
period before the takeover (often ¬ve days before the ¬rst announced bid,
but sometimes a longer period), the researchers calculate the actual daily
stock returns for the target ¬rm and subtract out the expected market
returns (usually calculated using the ¬rm™s beta and applying it to overall
market movements during the time period under observation). The excess
actual return over the capital asset pricing model-determined expected
return market is called an ˜˜abnormal return.™™ The cumulation of the daily
abnormal returns over the time period under observation is the CAR. The
term CAR( 5, 0) means the CAR calculated from ¬ve days before the
announcement to the day of announcement. The CAR( 1, 0) is a control
premium, although Mergerstat generally uses the stock price ¬ve days
before announcement rather than one day before announcement as the
denominator in its control premium calculation. However, the CAR for
any period other than ( 1, 0) is not mathematically equivalent to a con-
trol premium.
DLOC (discount for lack of control) an amount or percentage deducted
from a pro rata share of the value of 100% of an equity interest in a
business, to re¬‚ect the absence of some or all of the powers of control.2
DLOM (discount for lack of marketability) an amount or percentage
deducted from an equity interest to re¬‚ect lack of marketability.3

1. Business Valuation Standards, De¬nitions, American Society of Appraisers.
2. Ibid.
3. Ibid.


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economic components model Abrams™ model for calculating DLOM
based on the interaction of discounts from four economic components.
This model consists of four components: the measure of the economic
impact of the delay-to-sale, monopsony power to buyers, and incremental
transactions costs to both buyers and sellers. See the second half of Chap-
ter 7.
discount rate the rate of return on investment that would be required
by a prudent investor to invest in an asset with a speci¬c level risk. Also,
a rate of return used to convert a monetary sum, payable or receivable
in the future, into present value.4
fractional interest discount the combined discounts for lack of control
and marketability.
g the constant growth rate in cash ¬‚ows or net income used in the ADF,
Gordon model, or present value factor.
Gordon model present value of a perpetuity with growth. The end-of-
year Gordon model formula is 1/(r g), and the midyear formula is
1 r/(r g). See Chapter 3.
log size model Abrams™ model to calculate discount rates as a function
of the logarithm of the value of the ¬rm. See Chapter 4.
markup the period after an announcement of a takeover bid in which
stock prices typically rise until a merger or acquisition is made (or until
it falls through).
Ordinary least squares (OLS) regression analysis a statistical technique
that minimizes the sum of the squared deviations between a dependent
variable and one or more independent variables and provides the user
with a y-intercept and x-coef¬cients, as well as feedback such as R2 (ex-
plained variation/total variation) t-statistics, p-values, etc. See Chapter 2.
NPV (net present value of cash ¬‚ows) Same as PV, but usually includes
a subtraction for an initial cash outlay.
PPF (periodic perpetuity factor) a generalization formula invented by
Abrams that is the present value of regular but noncontiguous cash ¬‚ows
that have constant growth to perpetuity. The end-of-year PPF is equal to:
r) j g) j
(1 (1
and the midyear PPF is equal to
1 r (1
r) j g) j
(1 (1
where r is the discount rate, b is the number of years (before) since the
last occurrence of the cash ¬‚ow, and j is the number of years between
cash ¬‚ows. See Chapter 3.
PV (present value of cash ¬‚ows) the value in today™s dollars of cash
¬‚ows that occur in different time periods.

4. Ibid.

r)n, where n is the
present value factor equal to the formula 1/(1
number of years from the valuation date to the cash ¬‚ow and r is the
discount rate. For business valuation, n should usually be midyear, i.e.,
n 0.5, 1.5, . . .
QMDM (quantitative marketability discount model) model for calcu-
lating DLOM for minority interests.5
r the discount rate
runup the period before a formal announcement of a takeover bid in
which one or more bidders are either preparing to make an announce-
ment or speculating that someone else will.

5. Z. Christopher, Mercer, Quantifying Marketability Discounts: Developing and Supporting Marketability
Discounts in the Appraisal of Closely Held Business Interests (Memphis, Tenn: Peabody, 1997)

Glossary 477

Amihud, Y., 232, 282, 379, 381 Freeman, Neill, 233“234, 283
Andersson, Thomas, 219, 283 French, Kenneth R., 119, 146, 155
Annin, Michael, 148, 155

Gilbert, Gregory A., 146, 155, 167
Glass, Carla, 208, 224, 226
Banz, Rolf, 119, 155
Golder, Stanley C., 410, 431
Barca, F., 220, 282
Gordon, M.J., 59, 90n
Bergstrom, C., 282
Gordon model, 25, 50, 59“60, 63“79, 87“90, 93“
Berkovitch, E., 221, 282
97, 140, 153, 157, 175“176, 207, 230, 263“264,
Bhattacharyya, Gouri K., 22, 52
287, 385“387, 392, 394, 396, 398“399, 403
Black, Fisher, 303
Grabowski, Roger, 113, 119, 126, 144, 146, 148“
Black-Scholes options pricing model (BSOPM),
151, 155, 166, 241
192, 235, 246, 251“254, 256, 281, 303, 305“306
Gregory, Gordon, 258n
Black-Scholes put option, 233, 243“246, 298, 306
Guideline Company Method, 46“52, 59, 114,
Boatwright, David, 258n
153, 167“168
Bolotsky, Michael J., 198, 200“206, 230“231, 282
Bradley, M.A., 210, 220, 224“225, 233, 282
Brealey, R.A., 175, 177
Hall, Lance, 236, 298
Hamada, R.S., 183, 190
Harris, Ellie G., 222, 282
Center for Research in Security Prices (CRSP),
Harrison, Paul, 113, 131, 133“135, 155
Hayes, Richard, 134, 155
Chaffe, David B.H., 241“242, 251, 282, 307, 317
Hiatt, R.K., 246n, 262n, 287n, 405n
Copeland, Tom, 176
Hogarth, Robin M., 250, 282
Crow, Matthew R., 249
Horner, M.R., 220, 282
Houlihan Lokey Howard & Zukin (HLHZ)
studies, 198, 206, 210, 212“213, 217, 226, 329n
Desai, A., 210, 220, 224“225, 233, 282 Hull, John C., 241n

Eckbo, B.E., 220“221, 282 Ibbotson & Associates, 120, 134, 147n, 148, 155,
Einhorn, Hillel J., 250, 282 162, 170, 176“177, 385, 387, 404
Ellsberg, Daniel, 250, 282 Ibbotson, Roger G., 139n, 147, 151, 154“155, 207
Euler™s constant, 49, 51 Indro Daniel C., 175, 177
Excel, 2, 44, 51, 115, 124, 136 Institute of Business Appraisers (IBA), 272“273

Fagan, Timothy J., 214, 217, 283 Jacobs, Bruce I., 119, 152“153, 155, 167
Fama, Eugene F., 119, 146, 155 Jankowske, Wayne C., 200“201, 204“206, 231,
Fama-French Cost of Equity Model, 147“148 282
Fowler, Bradley, 405, 410“411, 414“415, 431 Johnson, Bruce A., 274, 276, 282
Franks, J.R., 221, 282 Johnson, Richard A., 22, 52


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Joyce, Allyn A., 170, 177 Neal, L., 134, 155
Julius, J. Michael, 249 Newton, Isaac, 156

Obenshain, Douglas, 258n
Kahneman, Daniel, 247, 284
Kaplan, Paul D., 147, 155
Kasper, Larry J., 88n, 222, 234“235, 281“282
Pacelle, Mitchell, 411, 431
Kasper bid-ask spread model, 191, 222, 234“235
Paudyal, Krishna, 210, 222, 235, 283
Kasper discounted time to market model, 232
Peterson, James D., 147, 155
Kim, E.H., 210, 220, 224“225, 233, 282

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