|
The APT and Portfolio Optimization in a Single-Index Market
The APT is couched in a single-factor market
5
and applies with perfect accuracy to well-
diversified portfolios. It shows arbitrageurs how to generate infinite profits if the risk pre-
mium of a well-diversified portfolio deviates from Equation 10.9. The trades executed by
these arbitrageurs are the enforcers of the accuracy of this equation.
In effect, the APT shows how to take advantage of security mispricing when diversifica-
tion opportunities are abundant. When you lock in and scale up an arbitrage opportunity
you’re sure to be rich as Croesus regardless of the composition of the rest of your portfolio,
but only if the arbitrage portfolio is truly risk-free! However, if the arbitrage position is not
perfectly well diversified, an increase in its scale (borrowing cash, or borrowing shares to
go short) will increase the risk of the arbitrage position, potentially without bound. The
APT ignores this complication.
Now consider an investor who confronts the same single factor market, and whose
security analysis reveals an underpriced asset (or portfolio), that is, one whose risk
premium implies a positive alpha. This investor can follow the advice weaved throughout
Chapters 6–8 to construct an optimal risky portfolio. The optimization process will
consider both the potential profit from a position in the mispriced asset, as well as the risk
of the overall portfolio and efficient diversification. As we saw in Chapter 8, the Treynor-
Black (T-B) procedure can be summarized as follows.
6
1. Estimate the risk premium and standard deviation of the benchmark (index) portfo-
lio, RP
M
and s
M
.
2. Place all the assets that are mispriced into an active portfolio. Call the alpha of the
active portfolio a
A
, its systematic-risk coefficient b
A
, and its residual risk s ( e
A
).
Your optimal risky portfolio will allocate to the active portfolio a weight, w
A
*
:
w
A
*
w
A
0
1
1 w
A
0
(1
2 b
A
)
w
e
E R
A
A
A
M
M
0
2
(
)
(
)
;
2
The allocation to the passive portfolio is then, w
M
*
5 1 2 w
A
*
. With this alloca-
tion, the increase in the Sharpe ratio of the optimal portfolio, S
P
, over that of the
passive portfolio, S
M
, depends on the size of the information ratio of the active
portfolio, IR
A
5 a
A
/ s ( e
A
). The optimized portfolio can attain a Sharpe ratio of
S
P
5 "S
M
2
1 IR
A
2
.
3. To maximize the Sharpe ratio of the risky portfolio, you maximize the IR of the
active portfolio. This is achieved by allocating to each asset in the active portfolio
a portfolio weight proportional to: w
Ai
5 a
i
/ s
2
( e
i
). When this is done, the square
of the information ratio of the active portfolio will be the sum of the squared indi-
vidual information ratios: IR
A
2
5 aIR
i
2
.
Now see what happens in the T-B model when the residual risk of the active portfolio
is zero. This is essentially the assumption of the APT, that a well-diversified portfolio
(with zero residual risk) can be formed. When the residual risk of the active portfolio goes
to zero, the position in it goes to infinity. This is precisely the same implication as the
APT: When portfolios are well-diversified, you will scale up an arbitrage position without
5
The APT is easily extended to a multifactor market as we show later.
6
The tediousness of some of the expressions involved in the T-B method should not deter anyone. The calcula-
tions are pretty straightforward, especially in a spreadsheet. The estimation of the risk parameters also is a rela-
tively straightforward statistical task. The real difficulty is to uncover security alphas and the macro-factor risk
premium, RP
M
.
bod61671_ch10_324-348.indd 336
bod61671_ch10_324-348.indd 336
6/21/13 3:43 PM
6/21/13 3:43 PM
Final PDF to printer
C H A P T E R
1 0
Arbitrage Pricing Theory and Multifactor Models of Risk and Return
337
Table 10.2 summarizes a rudimentary experiment that compares the prescriptions and
predictions of the APT and T-B model in the presence of realistic values of residual risk.
We use relatively small alpha values (1 and 3%), three levels of residual risk consistent
with values in Table 10.1 (2, 3, and 4%), and two levels of beta (0.5 and 2) to span the
likely range of reasonable parameters.
The first set of columns in Table 10.2 , titled Active Portfolio, show the parameter
values in each example. The second set of columns, titled Zero-Net-Investment, Arbitrage
(Zero-Beta), shows the weight in the active portfolio and resultant information ratio of
the active portfolio. This would be the Sharpe ratio if the arbitrage position (the positive-
alpha, zero-beta portfolio) made up the entire risky portfolio (as would be prescribed
Do'stlaringiz bilan baham: | 
