Transcript Lecture 12 - University of Washington

Lecture 12
Arbitrage Pricing Theory
Pure Arbitrage

A pure (or risk-free) arbitrage
opportunity exists when an investor
can construct a zero-investment
portfolio that yields a sure profit.

Zero-investment means that the
investor does not have to use any of
his or her own money.
Pure Arbitrage

One obvious case is when a violation
of the law of one price occurs.

Example: The exchange rate is
$1.50/£ in New York and $1.48/£ in
London.
Arbitrage Pricing Theory

The APT is based on the premise
that equilibrium market prices
ought to be rational in the sense
that they rule out risk-free
arbitrage opportunities.
Arbitrage Pricing Theory

The APT assumes that:
1. Security returns are a function of
one or more macroeconomic
factors.
2. All securities can be sold short
and the proceeds can be used to
purchase other securities.
Single-Factor APT

The return on security i is
ri = E(ri) + biF + ei.
E(ri) is the expected return.
 F is the factor.
 bi measures the sensitivity of ri to F.
 ei is the firm specific return.
 E(ei) = 0 and E(F) = 0.

Well Diversified Portfolios

rP = E(rP) + bPF + eP.

bP =

Swibi
eP = Swiei  0
s 2(eP) = Swi2 s 2(ei)  0

sP2 = bP2sF2 + s 2(eP)  bP2sF2

sP  b P sF

Single-Factor APT
Diversified Portfolio
Security i
rP
i
i
ri
i
i
i
i
i
i
F
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
F
Single-Factor APT

Two well
diversified
portfolios
with the
same beta
must have
the same
expected
return.
rp
A
B
Factor
Realization
Single-Factor APT

The expected return on a well
diversified portfolio is a linear
function of the portfolio’s beta.
E(rP ) = rf + [RP]bP

RP is the risk premium.
 rf
is the risk-free rate.
Single-Factor APT
Expected
Return
C
20%
i
B
i
15%
10%
A
i
iD
5%
0.5
1.0
1.5
Beta
Single-Factor APT

Let P be a well diversified portfolio.
E(rP ) = rf + [RP]bP
RP is the risk premium = E*- rf
 E* is the expected return on any
well diversified portfolio with
b* = 1.0.
 rf is the risk-free rate or return on
a zero beta portfolio.

Single-Factor APT
E[rP ]
E
*
*
RP = E - r
*
r
f
1.0
bP
f
Single-Factor APT
Risk-free arbitrage applies only to
well diversified portfolios.
 However, an investor can increase
the expected return on her portfolio
without increasing systematic risk if
individual securities violate the
relationship

ri = E(ri) + [RP]bi.
Single-Factor APT

Consider the following portfolio which
is part of a well diversified portfolio.
Amount
Security Invested
E(ri)
bi
A
$20,000
8%
0.6
B
$40,000
10%
1.2
C
$40,000
13%
1.6

E(rP) = .2x8+.4x10+.4x13 = 10.8%

bP = .2x0.6+.4x1.2+.4x1.6 = 1.24
Single-Factor APT

Sell B and purchase $16,000 of A and
$24,000 of C.
Security
A
C
Amount
Invested
$36,000
$64,000
E(ri)
8%
13%
bi
0.6
1.6

E(rP) = .36x8 + .64x13 = 11.2%

bP = .36x0.6 + .64x1.6 = 1.24
Multi-Factor APT

The return on security i is
ri = E(ri) + b1iF1+ ... + bkiFk+ ei.
E(ri) is the expected return.
 Fj is factor j, (j = 1,...,k).

 bji measures the sensitivity of ri to
factor j, (j = 1,...,k).
 ei is the firm specific return.
Multi-Factor APT

The return on a well diversified
portfolio is
rP = E(rP) + b1PF1+ ... + bkPFk.
E(rP) is the expected return.
 Fj is factor j, (j = 1,...,k).
 bjP measures the sensitivity of rP to
factor j, (j = 1,...,k).


eP =
Swiei g 0.
Multi-Factor APT
The relationship
between the
return on a well
diversified
portfolio and
factor j, holding
other factors
equal to zero.
Diversified Portfolio
rP
i
i
i
i
i
i
F
j
Multi-Factor APT

Arbitrage causes the expected return
on a well diversified portfolio to be
E[rP] = rf + [RP1]b1P +...+ [RPk]bkP

bjP is the sensitivity of portfolio P to
unexpected changes in factor j.

RPj is the risk premium on factor j.
Multi-Factor APT
E[rP ]
Ej
RP j = E j - r
f
r
f
1.0
b
j
Relationship when all other betas are zero.
Multi-Factor APT
Risk-free arbitrage applies only to
well diversified portfolios.
 However, an investor can increase
the expected return on her portfolio
without increasing systematic risk if
individual securities violate the
relationship

E[ri] = rf + [RP1]b1i +...+ [RPk]bki
Portfolio Strategy
Portfolio strategy involves choosing
the optimal risk-return tradeoff.
 The APT can be used to estimate
> security expected returns,
> security variances, and
> covariances between security
returns.

Portfolio Strategy
The APT can also be used to refine
the measure of risk.
 Factor risks can affect investors
differently.
 The appropriate pattern of factor
sensitivities depends upon a
variety of considerations unique
to the investor.

Portfolio Sensitivities
Productivity
Beta
1.0
Bh
Sh
hU
Portfolios
S - Stocks
B – Bonds
U – Unit Beta
Z
h
1.0
Inflation Beta
Z – Zero Beta
Identifying Factors

The biggest problem is identifying
the factors that systematically
affect security returns.

Theory is silent regarding the
factors.

A variety of macroeconomic factors
have been used.
Chen, Roll & Ross

Growth rate in industrial production.

Rate of inflation.

Expected rate of inflation.

Spread between long-term and
short-term interest rates.

Spread between low-grade and
high-grade bonds.
Berry, Burmeister & McElroy

Growth rate in aggregate sales.

Rate of return on the S&P500.

Rate of inflation.

Spread between long-term and
short-term interest rates.

Spread between low-grade and
high-grade bonds.
Salomon Brothers

Growth rate in GNP.

Rate of inflation.

Rate of interest.

Rate of change in oil prices.

Rate of growth in defense spending.