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

discount component”buyers™ costs”finite life (A7-24)

Summary of Mathematical Analysis in Remainder

of Appendix

The remainder of the appendix is devoted to calculating partial deriva-

tives necessary to evaluate the behavior of the discount formula (A7-11).

The partial derivatives of D with respect to its underlying independent

variables, g, r, z, and j, give us the slope of the discount as a function of

each variable. The purpose in doing so is to see how D behaves as the

independent variables change.

It turns out that D is a monotonic function with respect to each of

its independent variables. That is analytically convenient, as it means that

an increase in any one of independent variables always affects D in the

same direction. For example, if D is monotonically increasing in g, that

means that an increase in g will always lead to an increase in D, and a

decrease in g leads to a decrease in D. If D is monotonically increasing,

there is no value of g such that an increase in g leads either to no change

in D or a decrease in D.

The results that we develop in the remainder of the appendix are

that the discount, D, is monotonically increasing with g with z and de-

creasing with r and j. The practical reader will probably want to stop

here.

MATHEMATICAL ANALYSIS OF THE DISCOUNT”

CALCULATING PARTIAL DERIVATIVES

We can compute an alternative form of equation (A7-11) by multiplying

the numerator by 1 and changing the minus sign before the fraction to

a plus sign. This will ease the computations of the partial derivatives of

the expression.

PART 3 Adjusting for Control and Marketability

290

xj 1

D 1 (A7-25)

z)x j

1 (1

z)x j]jx j 1} {(x j z)jx j 1]}

{[1 (1 1)[ (1

D

(A7-26)

z)x j]2

x [1 (1

Factoring out jx j 1, we get:

jx j 1{[1 z)x j] (x j

(1 1)(1 z)}

D

(A7-27)

z)x j]2

x [1 (1

jx j 1[1 z)x j z)x j

(1 (1 (1 z)]

D

(A7-28)

z)x j]2

x [1 (1

Note that (1 z)x j and (1 z)x j cancel out in the numerator. Also,

the 1 (1 z) z. This simpli¬es to:

jx j 1z

D

0 (A7-29)

z)x j]2

x [1 (1

Since j, x, and z are all positive, the numerator is positive. Since the

denominator is squared, it is also positive. Therefore, the entire expression

is positive. The means that the discount is monotonically increasing in x.

We begin equation (A7-30) with a repetition of the de¬nition of x in

order to compute its partial derivatives.

1 g

x (A7-30)

1 r

Differentiating equation (A7-30) with respect to g, we get:

x (1 r)(1) 1

0 (A7-31)

r)2

g (1 1 r

Differentiating equation (A7-30) with respect to r, we get:

(1 g)(1) (1 g)

x

0 (A7-32)

r)2 r)2

r (1 (1

Using the chain rule, the partial derivative of D with respect to g is

the partial derivative of D with respect to x multiplied by the partial

derivative of x with respect to g, or:

D Dx

0 (A7-33)

g xg

The ¬rst term on the right-hand side of the equation is positive by

equation (A7-29), and the second term is positive by equation (A7-31).

Therefore, the entire expression is positive and thus the discount is mon-

otonically increasing in g. Using the chain rule again with respect to r,

we get:

D Dx

0 (A7-34)

r xr

Thus, the discount is monotonically decreasing in r. Now we make

an algebraic substitution to simplify the expression for D in order to fa-

cilitate calculating other partial derivatives.

CHAPTER 7 Adjusting for Levels of Control and Marketability 291

Let y 1 z (A7-35)

dy

1 (A7-36)

dz

Substituting equation (A7-35) into equation (A7-25), we get:

xj 1

D 1 (A7-37)

yx j

1

x j)( x j) x j(x j

D (1 1)

(A7-38)

yx j)2 yx j)2

y (1 (1

D dy x j(x j

D 1)( 1)

0 (A7-39)

z)x j]2

z y dz [1 (1

The denominator of (A7-39), being squared, is positive. The numer-

ator is also positive, as x j is positive and less than one, which means that

xj 1 is negative, which when multiplied by 1 results in a positive

number. Thus, the entire partial derivative is positive, which means that

D is monotonically increasing in z, the transaction costs. This result is

intuitive, as it makes sense that the greater the transaction costs, the

greater the discount.

Differentiating equation (A7-37) with respect to j, the average num-

ber of years between sales, we get:

yx j)x j ln x (x j 1)( y)x j ln x

(1

D

(A7-40)

yx j)2

j (1

Factoring out x j ln x, we get:

x j ln x(1 yx j yx j x j ln x(1

y) y)

D

(A7-41)

yx j)2 yx j)2

j (1 (1

x j z ln x

D

0 (A7-42)

z)x j]2

j [1 (1

The denominator is positive. The numerator is negative; since x 1,

ln x 0. Thus, the discount is monotonically decreasing in j, the average

years between sale. That is intuitive, as the less frequently business sell,

the smaller the discount should be.

Summary of Comparative Statics

Summarizing, the discount for periodic transaction costs is related in the

following ways to its independent variables:

Variable Varies with Discount Monotonically

r Negatively Decreasing

g Positively Increasing

z Positively Increasing

j Negatively Decreasing

PART 3 Adjusting for Control and Marketability

292

CHAPTER 8

Sample Restricted Stock

Discount Study

ENCO, INC.

As of AUGUST 11, 1997

The information contained in this report is con¬dential. Neither all nor

any part of the contents shall be conveyed to the public without the prior

written consent and approval of Abrams Valuation Group (AVG). AVG™s

opinion of value in this report is valid only for the stated purpose and

date of the appraisal.

Note: all names are ¬ctional

Note: Because this sample report is in a book, there are slight changes in

the table numbering and appearance of the report to accommodate the

book format.

293

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