Suppose you manage a $1.4 million portfolio. You believe that the alpha of the portfolio
, 0. You would
The return on portfolio over the next month may be described by Equation 26.1, which
states that the portfolio return will equal its “fair” CAPM return (the first two terms on the
that the stock entails because you are worried about a market decline. So you choose to
Because the S&P contracts have a multiplier of $250, and the portfolio has a beta of
1.20, your stock position can be hedged for 1 month by selling five futures contracts:
this case, 1 month. If the contract maturity were longer, one would have to slightly reduce the hedge ratio in a
932
P A R T V I I
Applied Portfolio Management
A warning: Even market-neutral positions are still bets, and they can go wrong. This
is not true arbitrage because your profits still depend on whether your analysis (your per-
ceived alpha) is correct. Moreover, you can be done in by simple bad luck, that is, your
analysis may be correct but a bad realization of idiosyncratic risk (negative values of e in
Equation 26.1 or 26.2) can still result in losses.
The dollar value of the stock portfolio after 1 month will be
$1,400,000
3 (1 1 r
portfolio
)
5 $1,400,000 31 1 .01 1 1.20 (r
M
2 .01) 1 .02 1 e4
5 $1,425,200 1 $1,680,000 3 r
M
1 $1,400,000 3 e
The dollar proceeds from your futures position will be:
5 3 $250 3 (F
0
2 F
1
)
Mark to market on 5 contracts sold
5 $1,250 3 [
S
0
(1.01) 2 S
1
]
Substitute for futures prices from parity relationship
5 $1,250 3
S
0
[1.01 2 (1 1 r
M
)] Because
S
1
5 S
0
(1 1 r
M
) when no dividends are paid
5 $1,250 3 [S
0
(.01 2 r
M
)] Simplify
5 $16,800 2 $1,680,000 3 r
M
Because
S
0
5 1,344
The total value of the stock plus futures position at month’s end will be the sum of the
portfolio value plus the futures proceeds, which equals
Hedged
proceeds
5 $1,442,000 1 $1,400,000 3 e
(26.2)
Notice that the exposure to the market from your futures position precisely offsets your
exposure from the stock portfolio. In other words, you have reduced beta to zero. Your
investment is $1.4 million, so your total monthly rate of return is 3% plus the remain-
ing nonsystematic risk (the second term of Equation 26.2). The fair or equilibrium
expected rate of return on such a zero-beta position is the risk-free rate, 1%, so you
have preserved your alpha of 2%, while eliminating the market exposure of the stock
portfolio.
This is an idealized example of a pure play. In particular, it simplifies by assuming a
known and fixed portfolio beta, but it illustrates that the goal is to speculate on the stock
while hedging out the undesired market exposure. Once this is accomplished, you can
establish any desired exposure to other sources of systematic risk by buying indexes or
entering index futures contracts in those markets. Thus, you have made alpha portable.
Figure 26.1 is a graphical analysis of this pure play. Panel A shows the excess returns to
betting on a positive-alpha stock portfolio “naked,” that is, unhedged. Your expected return
is better than an equilibrium return given your risk, but because of your market exposure
you still can lose if the market declines. Panel B shows the characteristic line for the posi-
tion with systematic risk hedged out. There is no market exposure.
What would be the dollar value and rate of return on the market-neutral posi-
tion if the value of the residual turns out to be 2 4%? If the market return in that
month is 5%, where would the plot of the strategy return lie in each panel of
Figure 26.1 ?
CONCEPT CHECK
26.2
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Final PDF to printer
C H A P T E R
2 6
Hedge
Funds
933
Even market-neutral bets can result in considerable volatility because most hedge funds
use considerable leverage. Most incidents of relative mispricing are fairly minor, and
the hedged nature of long-short strategies makes overall volatility low. The hedge funds
respond by scaling up their bets. This amplifies gains when their bets work out, but also
amplifies losses. In the end, the volatility of the funds is not small.
Return on Positive Alpha Portfolio
Return for Fairly Priced Assets
Alpha = 2%
Excess Market Return, r
M
− r
f
Excess Rate of Return, r
p
− r
f
α = 2%
Characteristic line of your hedged (
β = 0) portfolio is flat
Characteristic line of fairly priced zero-beta asset
Total Market Return, r
M
Total Return on Hedged Portfolio
← α = 2%
r
f
= 1%
3%
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