WEB EXTENSION
7A
Derivation of Valuation
Equations
T
he derivation of the formulas for the values of a perpetual preferred stock and
the values of a constant growth stock are presented in this extension.
VALUES
OF A
PERPETUAL PREFERRED STOCK
The value of a perpetual preferred stock is given by
Vps ¼
Dps
ð1 þ rps Þ
1
þ
Dps
ð1 þ rps Þ
2
þ…þ
Dps
ð1 þ rps Þ∞
(7A-1)
For a partial sum up to N, Equation 7A-1 may be rewritten as follows:
"
Vps ¼ Dps
1
1
1
…þ
1þ
2þ
ð1 þ rps Þ
ð1 þ rps Þ
ð1 þ rps ÞN
#
(7A-2)
Multiply both sides of Equation 7A-2 by (1 + rps), which yields
"
Vps ð1 þ rps Þ ¼ Dps
1
1
1
…þ
1þ
1þ
2þ
ð1 þ rps Þ
ð1 þ rps Þ
ð1 þ rps ÞN − 1
#
(7A-3)
Subtract Equation 7A-2 from Equation 7A-3 to obtain
"
Vps ð1 þ rps − 1Þ ¼ Dps 1 −
1
#
ð1 þ rps ÞN
(7A-4)
As N → ∞, 1/(1 + rps)N → 0 and Equation 7A-4 approaches
Vps ðrps Þ ¼ Dps
Thus we obtain Equation 7-8 from the chapter:
Vps ¼
Dps
rps
1
2
Web Extension 7A: Derivation of Valuation Equations
Although we used preferred stock notation, Equations 7-3 and 7-8 are valid for
any perpetuity.
VALUES
OF A
CONSTANT GROWTH STOCK
The proof of Equation 7-2, the chapter’s formula for the value of a constant growth
^
stock, P0 = D1/(rs − g), is developed as follows. Rewrite Equation 7A-1 as
1
2
3
N
^
P0 ¼ D0 ð1 þ gÞ þ D0 ð1 þ gÞ þ D0 ð1 þ gÞ þ … þ D0 ð1 þ gÞ
ð1 þ rs Þ1
ð1 þ rs Þ2
ð1 þ rs Þ3
ð1 þ rs ÞN
"
#
ð1 þ gÞ1 ð1 þ gÞ2 ð1 þ gÞ3 … ð1 þ gÞN
¼ D0
þ
þ
þ þ
ð1 þ rs Þ1 ð1 þ rs Þ2 ð1 þ rs Þ3
ð1 þ rs ÞN
(7A-5)
Multiply both sides of Equation 7A-5 by (1 + rs)/(1 + g):
"
#
^
ð1 þ rs Þ P
ð1 þ gÞ1 ð1 þ gÞ2 … ð1 þ gÞN−1
þ
þ þ
0 ¼ D0 1 þ
ð1 þ gÞ
ð1 þ rs Þ1 ð1 þ rs Þ2
ð1 þ rs ÞN−1
(7A-6)
Subtracting Equation 7A-5 from Equation 7A-6, we obtain
"
#
N
^
ð1 þ rs Þ
ð1
þ
gÞ
−1 P0 ¼ D0 1−
ð1 þ gÞ
ð1 þ rs ÞN
(7A-6)
Assume that rs > g. Then, as N approaches infinity, the term in brackets on the righthand side of the equation approaches unity, leaving
^
ð1 þ rs Þ
−1 P0 ¼ D0
ð1 þ gÞ
This can be rearranged as
^
ð1 þ rs Þ − ð1 þ gÞ P
0 ¼ D0
ð1 þ gÞ
which simplifies to Equation 7-2 in the chapter:
^
ðrs − gÞP0 ¼ D0 ð1 þ gÞ ¼ D1
^
P0 ¼ D1
rs − g