INVESTMENTS
Arbitrage
Pricing Theory
Chapter 7
INVESTMENTS
Arbitrage Pricing Theory
Arbitrage - arises if an investor can construct a zero
investment portfolio with a sure profit
? Since no investment is required, an investor can
create large positions to secure large levels of
profit
? In efficient markets, profitable arbitrage
opportunities will quickly disappear
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Current Expected Standard
Stock Price$ Return% Dev.%
A 10 25.0 29.58
B 10 20.0 33.91
C 10 32.5 48.15
D10 2.5 8.58
Arbitrage Example from Text
pp. 308-310
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Mean S.D. Correlation
Portfolio
A,B,C 25.83 6.40 0.94
D 22.25 8.58
Arbitrage Portfolio
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Arbitrage Action and Returns
E. Ret.
St.Dev.
* P
* D
Short 3 shares of D and buy 1 of A, B & C to
form P
You earn a higher rate on the investment than
you pay on the short sale
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Factor model of asset returns
? Suppose that asset returns are driven by a few common
factors and diversifiable noise:
? Where
? is the expected return on asset i;
? are news on common factors driving all asset
returns.
? gives how sensitive the return on asset i with respect
to news on the k-th factor—is called the loading of asset i
on factor
? is the idiosyncratic component in asset i’s return that
is unrelated to other asset returns
? have zero means.
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example
? Common factors driving asset returns may include GNP,
interest rates, inflation, etc. Let be the news on interest
rate. Before a board meeting of the Fed, the market expect
the Fed not to change the interest rate. After the meeting,
Greenspan announces that:
? There is no change in interest rate---”no news”
? There is a ?% increase in interest rate—positive surprise
? What should be the sign of the factor loadings on ,
be for fixed income securities, stocks, commodity futures?
int
~
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int
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~
int
=f
int
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f
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Properties of factor models
? The following results provide the building
blocks of APT.
? 1. Any diversified portfolio p is exposed
only to factor risks
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~
...
~
)(
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++=
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continued
? A diversified portfolio, that is not exposed to any factor
risk( ), must offer risk-free rate;
? There always exists portfolios that are exposed only to the
risk of a single factor.
? Example: suppose two well diversified portfolios, both
exposed only to the risk of the first two factors
0...
2
1
====
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continued
? A portfolio Pk, that has unitary risk of factor k, offer a
expected return with the factor risk:
?
? such a portfolio is called a factor portfolio for factor k, and
? is the premium of factor k
? Example. In the above example, we found portfolio p that
bears only the risk of factor 1. Its loading is 0.5. Consider
following portfolio p1:
? 200% invested in p and –100% invested in the risk-free
portfolio P0.
fkpk
rr =
ffk
rr ?
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APT
? Suppose that asset returns are driven by a few common
factors and diversifiable noise:
? For an arbitrary asset, its expected return depends only on
its factor exposure:
? Where
? is the premium on factor k;
? is asset i’s loading of factor k
)(...)()(
2211 ffKiKffiffifi
rErbrErbrErbrEr ?++?+?+≈
ff
rr
k
?
ik
b
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)(
11
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We illustrate APT with an example
? Suppose that there are two factors:
? (1) unanticipated market return
? (2) unanticipated inflation
? Suppose that
? The returns on factor portfolios are:
1
~
f
2
~
f
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~
~~
~
2211
+++=
%2%,8%,5
21
?=?=?=
ffffF
rrrrr
2
1
~
)02.005.0(
~
~
)08.005.0(
~
2
1
fr
fr
P
p
+?=
++=
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continued
? We first consider assets with only factor risks
? For an asset with
?
? APT requires that its expected rate of return must be
? Suppose that was instead 10%. Then there is a free
lunch.
0.1
21
==bb
21
~~
~
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qq
++=
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)02.0(0.108.00.105.0
)()(
21
21
=
?×+×+=
?+?+=
fffffq
rrbrrbrr
q
r
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continued
? Consider the following portfolio:
? (1) buy 100 of portfolio P1,
? (2) buy 100 of portfolio P2
? (3) sell $100 of asset q
? (4) sell $100 of risk-free rate
? This portfolio has the following characteristics:
? Requires zero initial investment (an arbitrage portfolio);
? Bear no factor risk (and no idiosyncratic risk)
? Pay (13+3-10-5)=1 surely.
? This would be an arbitrage.
? In absence of arbitrage, equation must hold for assets with
only factor risks.
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continued
? What if an asset also bears idiosyncratic
risks? Since it cannot be replicated by other
assets, in particular the factor portfolios, the
pricing formula need not hold.
? However in the presence of idiosyncratic
risks, deviations from the pricing equation
cannot be pervasive. In other words, for
most assets, the pricing formula has to be
approximately correct.
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continued
? Suppose that the pricing formula are violated for
many assets. Let us focus on those with the same
factor risks
? Form a diversified portfolio of these assets, q;
? Portfolio q bears only factor risks, and violate APT.
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Portfolio &Individual Security
Comparison
F
E(r)%
Portfolio
F
E(r)%
Individual Security
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Example
? Suppose that assets A,B,C, we have
0.61.0C
0.21.5B
1.00.5A
b2b1asset
C
B
A
r
r
r %7)02.0(0.108.05.005.0 =?×+×+=
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example-continued
? Investors hold well-diversified portfolios with different
exposures to the two factors---depending on how much
each investor worries about inflation:
? Investors who worries more about inflation will seek to
hold more of the portfolio that provides a hedge against
inflation:
? (1) start with the market portfolio;
? (2) sell off assets with negative correlation with factor 2;
? (3) use the proceeds to buy assets with positive correlation
with factor 2.
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Implementation of APT
? The implementation of APT involves three
steps:
? 1. Identify the factors;
? 2. Estimate factor loadings of assets
? 3. Estimate factor premia
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1 factors
? Since the theory itself does not specify the factors, we have
to construct the factors empirically:
? (a) using macroeconomic variables;
? change in GDP growth;
? change in T-bill yield (proxy for expected inflation)
? changes in yield spread between T-bonds and T-bills;
? changes in default premium on corporate bonds;
? Changes in oil prices
? (b) using statistical analysis—factor analysis:
? estimate covariance of asset returns
? extract “factors” from the covariance matrix
? (c ) Data mining: explore different portfolios to find those
whose returns can be used as factors
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Factor loadings, factor premia, APT
pricing
? 2 Factor loadings.given factors, we can regress past asset
returns on the factors to estimate factor loadings (bik)
? 3 factor premia. Given the factor loading of individual assets, we can
construct factor portfolios. For the k-th factor, we have
? The premium of the k-th factor is
? 4 APT pricing. By APT, the return on asset i is given by
itktiktiiit
ufbfbrr
~
~
...
~
~
11
++++=
ktpkp
frr
kt
~
~
+=
fpkff
rrrr
k
?=?
)(...
11 f
K
fiKfifi
rrbrbrr ?+++=
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E(r)%
Beta for F
10
7
6
Risk Free 4
A
D
C
.5 1.0
Disequilibrium Example—single factor
example
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Disequilibrium Example-continued
? Short Portfolio C
? Use funds to construct an equivalent risk
higher return Portfolio D
- D is comprised of A & Risk-Free Asset
? Arbitrage profit of 1%
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E(r)%
Beta (Market Index)
Risk Free
M
1.0
[E(r
M
) - r
f
]
Market Risk Premium
APT with Market Index Portfolio
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? APT applies to well diversified portfolios and not
necessarily to individual stocks
? With APT it is possible for some individual stocks to be
mispriced - not lie on the SML
? APT is more general in that it gets to an expected return
and beta relationship without the assumption of the market
portfolio
? APT can be extended to multifactor models
APT and CAPM Compared