13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
Intermediate Financial Theory
Chapter XIII. The Arbitrage Pricing Theory
July 11, 2006
Intermediate Financial Theory
13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
13.1 Introduction
Equilibrium vs. No Arbitrage
At equilibrium there can be no arbitrage
Usefulness of A-D prices depend on no arbitrage
hypothesis
APT: parallel with A-D pricing
Doing away with states of nature
Replaced by the concept of risk factor
Primitive security = security whose risk is exclusively
determined by its association with one specific factor, totally
immune form association with other risk factors
Intermediate Financial Theory
13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
13.2 Factor Models
r̃j = αj + βj r̃M + ε̃j ,
(1)
E ε̃j = 0, cõv r̃M ,ε̃j = 0, ∀j, and cov ε̃j ,ε̃k = 0, ∀j 6= k
All return characteristics common to different assets are
subsumed in their link with the market.
APT: many, ex ante unidentified factors.
Intermediate Financial Theory
13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
A Quasi-Complete Market Hypothesis
Property 1
N
X
xi = 0 = x T ·1,
i=1
Property 2
N
X
xi βi = 0 = x T · β.
i
Property 2
N
X
2
xi2 σ εi ∼
= 0.
i
Intermediate Financial Theory
13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
Proof of ATP
rP = 0 = x
T
·r
(2)
r
=
λ0 · 1 + λ1 β, or
ri
=
λ0 + λ1 βi for all assets i
(3)
Geometric representation: x orthogonal to 1 and β
x
b
1
Intermediate Financial Theory
13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
Meaning of λ0 and λ1
r̄f = rf = λ0 .
r̄Q = rf + λ1 · 1
r̄i = rf + βi (r̄Q −rf ) .
r̄i = rf + βi (r̄M −rf ) .
Intermediate Financial Theory
(4)
13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
r̃j = aj + bj1 F̃1 + bj2 F̃2 + ẽj
with E ẽj = 0, cov F̃1 , ε̃j = cov F̃2 , ε̃j =
0, ∀j, and cov ε̃j ,ε̃k = 0, ∀j 6= k .
r j = λ0 + λ1 bj1 + λ2 bj2 .
λ0 = rf
λ 1 = r P1 − r f
λ 2 = r P2 − r f
Intermediate Financial Theory
(5)
13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
Let us illustrate. In our two-factor examples, a security j with,
say, bj1 = 0.8 and bj2 = 0.4 is like a portfolio with proportions of
0.8 of the pure portfolio P1 , 0.4 of pure portfolio P2 , and
consequently proportion −0.2 in the riskless asset. By our
usual (no-arbitrage) argument, the expected rate of return on
that security must be:
rj
= −0.2rf + 0.8r P1 + 0.4r P2
= −0.2rf + 0.8rf + 0.4rf + 0.8 r P1 − rf + 0.4 r P2 − rf
= rf + 0.8 r P1 − rf + 0.4 r P2 − rf
= λ0 + bj1 λ1 + bj2 λ2
Intermediate Financial Theory
13.1 Introduction
13.2 Factor Models
13.3 The APT: Statement and Proof
13.4 Multi-factor Models and APT
Advantage of the APT for the portfolio selection
Helps identify the sources of risk/split the systematic risks
into more detailed components and thus permits
modulating one’s risk exposure
E.g. two stocks with same beta could have different
sensitivities to specific risk factors; useful in managing risk
exposure or helping refine conditional expectations on
returns.
Intermediate Financial Theory