7.
If the beta of the portfolio were 1.0, she would sell $1 million of the index.
Because beta is 1.25, she should sell $1.25 million of the index.
8.
a.
1,200
×
1.01 = 1,212
b.
$12 million /($250
×
1,200) = 40 contracts short
c.
40
×
250
×
(1,212 – S
T
) = 12,120,000 – 10,000S
T
d.
The expected return on a stock is:
α
+ r
f
+
β
[E(r
M
) – r
f
]
The CAPM predicts that
α
= 0.
In this case, however, if you believe that
α
= 2% (i.e., 0.02), then you forecast a portfolio return of:
r
P
= 0.02 + 0.01 + 1.0
×
(r
M
– 0.01) +
ε
= 0.03 +
[
1
×
(r
M
– 0.01)
]
+
ε
where
ε
is diversifiable risk, with an expected value of zero.
e.
Because the market is assumed to pay no dividends:
r
M
= (S
T
– 1,200)/1,200 = (S
T
/1,200) – 1
The rate of return can also be written as:
r
P
= 0.03 +
(
1
×
[(S
T
/1,200) – 1 – 0.01]
)
+
ε
The
dollar
value of the stock portfolio as a function of the market index
is therefore:
$12 million
×
(1 + r
P
) =
$12 million
×
[0.03 + (S
T
/1,200) – 0.01 +
ε
] =
$240,000 + 10,000S
T
+ ($12 million
×
ε
)
The dollar value of the short futures position will be (from part c):
12,120,000 – 10,000S
T
The total value of the portfolio plus the futures proceeds is therefore:
[240,000 + 10,000S
T
+ (12 million
×
ε
)] + [12,120,000 – 10,000S
T
] =
$12,360,000 + ($12 million
×
ε
)
The payoff is independent of the value of the stock index.
Systematic risk
has been eliminated by hedging (although firm-specific risk remains).
f.
The portfolio-plus-futures position cost $12 million to establish.
The
expected end-of-period value is $12,360,000.
The rate of return is
therefore 3%.
23-4