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MATHEMATICAL APPENDIX
DEVELOPING THE DISCOUNT FORMULAS
Initially we assume the current business owner will operate the business
for 10 years, sell it, and pay transaction costs of z.77 The next owner will
run the business another 10 years, sell it, and pay transaction costs. We
assume this pattern occurs ad in¬nitum. Of course, there will be variations
from the sale every 10 years”some will sell after 1 year, others after 30
years. In the meantime, in the absence of prior knowledge, we assume
every 10 years to be a reasonable estimate of the average of what will
occur.



NPV of Cash Flows with Periodic Transaction
Costs Removed
The net present value (NPV) of cash ¬‚ows to the existing business owner
with periodic transaction costs removed is the full amount of the ¬rst 10
years™ cash ¬‚ows, plus (1 z) times the next 10 years™ cash ¬‚ows, where
z)2 times the next 10 years™
z is the periodic transaction cost, plus (1
cash ¬‚ows, etc. We will denote the NPV net of transaction costs, i.e., with
transaction costs removed from the stream of cash ¬‚ows, as NPVTC .

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 g)29
(1 (1
2
(1 z) (A7-1)
r)20.5 r)29.5
(1 (1

Multiplying each term in equation (A7-1) by (1 g)/(1 r), we get:



77. As explained in the body of the chapter, z is an incremental transaction cost. For example,
when we value a small fractional ownership in a privately owned business, often our
preliminary value is on a marketable minority basis. In this case z would be the difference
in transaction cost (expressed as a percentage) between selling a private business interest
and selling publicly traded stock through a stockbroker.




PART 3 Adjusting for Control and Marketability
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)

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