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284
g)10
1 g 1 g (1
NPVTC
r)1.5 r)10.5
1 r (1 (1
g)11 g)20
(1 (1
(1 z)
r)11.5 r)20.5
(1 (1
g)21 g)20
(1 (1
2
(1 z)
r)21.5 r)30.5
(1 (1
(A7-2)
Subtracting equation (A7-2) from equation (A7-1), we get:
g)10
1 g (1
1
1 NPV
r)0.5 r)10.5
1 r (1 (1
g)10 g)20
(1 (1
(1 z)
r)10.5 r)20.5
(1 (1
g)20 g)30
(1 (1
2
(1 z) (A7-3)
r)20.5 r)30.5
(1 (1
Note that all terms in each sequence drop out except for the ¬rst
terms in equation (A7-1) and the last terms in equation (A7-2). In equation
(A7-4), we collect the positive terms from equation (A7-3) in the ¬rst set
of square brackets and the negative terms from equation (A7-3) in the
second one. Additionally, the left-hand side of equation (A7-3) reduces to
(r g)/(1 r)NPVTC . Multiplying through by (1 r)/(r g), we get:
1 r
NPVTC
r g
g)10 g)20
(1 (1
1 2
(1 z) (1 z)
r)0.5 r)10.5 r)20.5
(1 (1 (1
g)10 g)20 g)30
(1 (1 (1
2
(A7-4)
(1 z) (1 z)
r)10.5 r)20.5 r)30.5
(1 (1 (1
Next we will manipulate the right-hand side of the equation only.
We divide the term (1 r)/(r g) by 1 r, which leaves that term as
(1 r)/(r g) and we multiply all terms inside the brackets by
1 r. The latter action has the effect of reducing the exponents in the
denominators by 0.5 years. Thus, we get:
1 r
NPVTC
r g
10 20
1 g 1 g
2
1 (1 z) (1 z)
1 r 1 r
10 20 30
1 g 1 g 1 g
2
(A7-5)
(1 z) (1 z)
1 r 1 r 1 r
Recognizing that each term in brackets is an in¬nite geometric se-
quence, this solves to:




CHAPTER 7 Adjusting for Levels of Control and Marketability 285
1 r
NPVTC
r g
10
1 g
1 r
1
(A7-6)
g)10
(1 z)(1 g)10
(1 z)(1
1 1
r)10 r)10
(1 (1

Since the denominators are identical, we can combine both terms in
the brackets into a single term by adding the numerators.
10
1 g
1
1 r
1 r
NPVTC (A7-7)
10
r g 1 g
1 (1 z)
1 r

Letting x (1 g)/(1 r), this simpli¬es to:
1 r x10
1
NPVTC (A7-8)
z)x10
r g 1 (1


The Discount Formula
D, the component of the discount for lack of marketability that measures
the periodic transaction costs, is one minus the ratio of the NPV of the
cash ¬‚ows net of transaction costs (NPVTC) to the NPV without removing
transaction costs (NPV). Using a midyear Gordon model formula of
(1 r)/(r g) as the NPV, we come to:
1 r x10
1
z)x10
r g 1 (1
NPVTC
D 1 1 (A7-9)
NPV 1 r
r g
The term ( 1 r)/(r g) cancels out, and the expression simpli¬es
to:
1 g
x10
1
r, ’ 0
D 1 , where x and g x 1
z)x10
1 (1 1 r
(A7-10)
Equation (A7-10) is the formula for the discount assuming a sale
every 10 years. Instead of assuming a business sale every 10 years, now
we let the average years between sale be a random variable, j, which
leads to the generalized equation in equation (A7-11):
xj
1
D 1
z)x j
1 (1
generalized discount formula“sellers™ transaction costs (A7-11)
In determining fair market value, we ask how much would a rational
buyer pay for (and for how much would a rational seller sell) a business
interest. That presumes a hypothetical sale at time zero. Equation (A7-11)
is the formula appropriate for quantifying sellers™ transaction costs, be-

PART 3 Adjusting for Control and Marketability
286
cause the buyer does not care about the seller™s costs, which means he or
she will not raise the price in order to cover the seller. However, the buyer
does care that 10 years down the road, he or she will be a seller, not a
buyer, and the new buyer will reduce the price to cover his or her trans-
action costs, and so on ad in¬nitum. Thus, we want to quantify the dis-
counts due to transaction costs for the continuum of sellers beginning
with the second sale, i.e., in year j. Equation (A7-11) accomplishes that.
Using an end-of-year Gordon model assumption instead of midyear
cash ¬‚ows leads to the identical equation, i.e., (A7-11) holds for both.


Buyer Discounts Begin with the First Transaction
An important variation of equation (A7-11) is to consider what happens
if the ¬rst relevant transaction cost takes place at time zero instead of
t j, which is appropriate for quantifying the discount component due
to buyers™ transaction costs. With this assumption, we would modify the
above analysis by inserting a (1 z) in front of the ¬rst series of bracketed
terms in equation (A7-1) and increasing the exponent of all the other (1
z) terms by one. All the other equations are identical, with the (1 z)
term added. Thus, the buyers™ equivalent formula of equation (A7-8) is:
1 r x10
1
NPVTC (1 z)
z)x10
r g 1 (1
NPV with buyers™ transaction costs removed (A7-8a)
Obviously, equation (A7-8a) is lower than equation (A7-8), because
the ¬rst relevant cost occurs 10 years earlier. The generalized discount
formula equivalent of equation (A7-11) for the buyer scenario is:
x j)
(1 z)(1
D 1
z)x j
1 (1
generalized discount, formula”buyers™ transactions costs
(A7-11a)
We demonstrate the accuracy of equations (A7-11) and (A7-11a),
which are excerpted from here and renumbered in the chapter as equa-
tions (7-9) and (7-9a), in Tables 7-12 and 7-13 in the body of the chapter.


NPV of Cash Flows with Finite Transactions
Costs Removed78
The previous formulas for calculating the present value of the discount
for buyers™ and sellers™ transactions costs are appropriate for business
valuations. However, for calculating that component of DLOM for limited
life entities such as limited partnerships whose document speci¬es a ter-
mination date, the formulas are inexact, although they are often good
approximations. In this section we develop the formulas for components
#3A and #3B of DLOM for limited life entities.79 This section is very math-
ematical and will have practical signi¬cance for most readers only when


78. This section is written by R. K. Hiatt.
79. Even in limited partnerships, it is necessary to question whether the LP is likely to renew, i.e.,
extend its life. If so, then the perpetuity formulas (A7-11) and (A7-11a) may be appropriate.


CHAPTER 7 Adjusting for Levels of Control and Marketability 287
the life of the entity is short (under 30 years) and the growth rate is close
to the discount rate. Some readers may want to skip this section, perhaps
noting the ¬nal equations, (A7-23) and (A7-24). Consider this section as
reference material.
Let™s assume a fractional interest in an entity, such as a limited part-
nership, with a life of 25 years that sells for every j 10 years. Thus,
2 sales80 of the frac-
after the initial hypothetical sale, there will be s
tional interest before dissolution of the entity. Let™s de¬ne n as the number
of years to the last sale before dissolution. We begin by repeating equa-
tions (A7-1) and (A7-2) as (A7-12) and (A7-13), with the difference that
the last incremental transaction cost occurs at n 20 years instead of
going on perpetually.
g)9
(1 g) (1
1
NPVTC
r)0.5 r)1.5 r)9.5
(1 (1 (1
g)10 g)19
(1 (1
(1 z)
r)10.5 r)19.5
(1 (1
g)20
(1
2
(1 z) (A7-12)
r)20.5
(1
g)10
1 g 1 g (1
NPVTC
r)1.5 r)10.5
1 r (1 (1
g)11 g)20
(1 (1
(1 z)
r)11.5 r)20.5
(1 (1
g)21
(1
2
(1 z) (A7-13)
r)21.5
(1
Subtracting equation (A7-13) from equation (A7-12), we get:
g)10
1 g (1
1
1 NPVTC
r)0.5 r)10.5
1 r (1 (1
g)10 g)20
(1 (1
(1 z)
r)10.5 r)20.5
(1 (1
g)20
(1
2
(1 z) (A7-14)
r)20.5
(1
Note that the ¬nal term ˜˜should have™™ a subtraction of (1 g) /
0.5
(1 r) , but that equals zero for g r. Therefore, we leave that term
out. Again, the ¬rst term of the equation reduces to (r g)/(1 r). We
then multiply both sides by its inverse:
g)10
(1
1 r 1
NPVTC
r)0.5 r)10.5
r g (1 (1
g)10 g)20
(1 (1
(1 z)
r)10.5 r)20.5
(1 (1
g)20
(1
2
(1 z) (A7-15)
r)20.5
(1


80. It is important not to include the initial hypothetical sale in the computation of s.


PART 3 Adjusting for Control and Marketability
288
As before, we divide the ¬rst term on the right-hand side of the equation
by 1 r and multiply all terms inside the brackets by the same. This
has the same effect as reducing the exponents in the denominators by 0.5
years.
10
1 r 1 g
NPVTC 1
r g 1 r
10 20
1 g
1 g
(1 z)
1 r 1 r
20
1 g
2
(A7-16)
(1 z)
1 r
Letting y 1 z and x (1 g)/(1 r), equation (A7-16) becomes:
1 r
x10) y(x10 x20) y2x20]
NPVTC [(1 (A7-17)
r g
1 r
yx10 y2x20) (x10 yx20)]
NPVTC [(1 (A7-18)
r g
Within the square brackets in equation (A7-18), there are two sets of
terms set off in parentheses. Each of them is a ¬nite geometric sequence.
The ¬rst sequence solves to
y3x30
1
yx10
1
and the second sequence solves to
x10 y2x30
yx10
1
They both have the same denominator, so we can combine them. Thus,
equation (A7-18) simpli¬es to:
x10 y2x30 y3x30
1 r 1
NPVTC (A7-19)
yx10
r g 1
Note that if we eliminate the two right-hand terms in the square brackets
in the numerator, equation (A7-10) reduces to equation (A7-8). We can
now factor the two right-hand terms and simplify to:
x10 y2x30(1
1 r 1 y)
NPVTC
yx10
r g 1
x10 zy2x30
1 r 1
yx10
r g 1
1 r x10 z)2x30
1 z(1
(A7-20)
z)x10
r g 1 (1
Since j 10, s 2, n 20, and n j 30, we can now generalize this
equation to:
1 r xj z)sxn j
1 z(1
NPVTC (A7-21)
z)x j
r g 1 (1

CHAPTER 7 Adjusting for Levels of Control and Marketability 289
As before, the discount component is D 1 NPVTC/NPV. This comes
to:
1 r xj z)sxn j
1 z(1
z)x j
r g 1 (1
D 1 (A7-22)
1 r
r g
Canceling terms, this simpli¬es to:
xj z)sx n j
1 z(1
D 1 (A7-23)
z)x j
1 (1
discount component”sellers™ costs”finite life
Note that as the life of the entity (or the interest in the entity) that
we are valuing goes to in¬nity, n ’ , so xn j ’ 0 and (A7-23) reduces
to equation (A7-11).
The equivalent expression for buyers™ costs is:
xj z)sxn j]
(1 z)[1 z(1
D 1
z)x j
1 (1

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