Efficient Markets
Financial Economics
Bruce C. Dieffenbach
Question 1
Consider a stock for which the dividend
D
t
follows a random walk with
a positive upward trend:
D
t
+
1
=
D
t
(
1
+
m
+
e
t
+
1
)
.
Here
t
denotes the time period. The error term
e
t
+
1
has mean zero and
is uncorrelated for different periods; the error term represents the “news”
about the dividend. At time
t
, the expected future dividend is
E
(
D
t
+
n
) =
D
t
(
1
+
m
)
n
.
In market equilibrium, the price of the stock is the present value of ex
pected dividends,
P
t
=
E
(
D
t
+
1
)
1
+
R
+
E
(
D
t
+
2
)
(
1
+
R
)
2
+
E
(
D
t
+
3
)
(
1
+
R
)
3
+
···
.
The interest rate
R
is constant,
R
>
m
.
a) Find the stock price
P
t
as a function of the dividend
D
t
, by calculating
the present value of expected dividends.
b) The rate of return from time
t
to time
t
+
1 is
D
t
+
1
+
P
t
+
1
−
P
t
P
t
.
By substituting your formula from (a) into this expression for the rate of
return, show that the efficientmarket theory applies: show that expected
rate of return equals
R
, that the rate of return is uncorrelated at different
times, and that the news determines the unexpected rate of return.
Answer 1
a) We have
E
(
D
t
+
n
) =
D
t
(
1
+
m
)
n
,
for any
n
. Computing the present value of expected dividends, we have
P
t
=
E
(
D
t
+
1
)
1
+
R
+
E
(
D
t
+
2
)
(
1
+
R
)
2
+
E
(
D
t
+
3
)
(
1
+
R
)
3
+
···
=
D
t
(
1
+
m
)
1
+
R
+
D
t
(
1
+
m
)
2
(
1
+
R
)
2
+
D
t
(
1
+
m
)
3
(
1
+
R
)
3
+
···
(an infinite geometric sum)
=
D
t
(
1
+
m
)
1
+
R
1
−
1
+
m
1
+
R
=
D
t
(
1
+
m
)
R
−
m
.
b) The rate of return comes from the dividend plus the capital gain,
rate of return
=
D
t
+
1
+(
P
t
+
1
−
P
t
)
P
t
.
Using the relationship
P
t
=
D
t
(
1
+
m
)
/
(
R
−
m
)
, we evaluate
rate of return
=
D
t
+
1
+
P
t
+
1
P
t
−
1
=
D
t
+
1
+
D
t
+
1
(
1
+
m
)
R
−
m
D
t
(
1
+
m
)
R
−
m
−
1
=
D
t
+
1
(
R
−
m
)+
D
t
+
1
(
1
+
m
)
D
t
(
1
+
m
)
−
1
= (
1
+
R
)
D
t
+
1
D
t
(
1
+
m
)
−
1
= (
1
+
R
)
D
t
(
1
+
m
+
e
t
+
1
)
D
t
(
1
+
m
)
−
1
=
R
+(
1
+
R
)
e
t
+
1
(
1
+
m
)
.
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The expected rate of return is the first term,
R
, as the second term is zero
on average. The unexpected rate of return is the second term,
(
1
+
R
)
e
t
+
1
(
1
+
m
)
.
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 Spring '08
 Dieffenbach
 Economics, Time Value Of Money, Dividend, Net asset value

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