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