lecture 12 arbitrage pricing theory. pure arbitrage a pure (or risk-free) arbitrage opportunity...
TRANSCRIPT
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Lecture 12
Arbitrage Pricing Theory
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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.
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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.
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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.
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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.
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Single-Factor APT
The return on security i is
ri = E(ri) + iF + ei.
E(ri) is the expected return. F is the factor. i measures the sensitivity of ri to F.
ei is the firm specific return.
E(ei) = 0 and E(F) = 0.
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Well Diversified Portfolios
rP = E(rP) + PF + eP.
P = wii
eP = wiei ’ 0
2(eP) = wi2 2(ei) ’ 0
P2 = P
2F2 + 2(eP) ’ P
2F2
P ’ PF
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Single-Factor APT
Diversified Portfolio
F F
Security i
r rP i
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Single-Factor APT
Two well diversified portfolios with the same beta must have the same expected return.
rp
Factor Realization
A
B
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Single-Factor APT
The expected return on a well diversified portfolio is a linear function of the portfolio’s beta.
E(rP ) = rf + [RP]P
RP is the risk premium.
rf is the risk-free rate.
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Single-Factor APTExpected Return
5%
10%
15%
20%
0.5 1.0 1.5 Beta
A
B
C
D
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Single-Factor APT Let P be a well diversified portfolio.
E(rP ) = rf + [RP]P
RP is the risk premium = E*- rf
E* is the expected return on any well diversified portfolio with * = 1.0.
rf is the risk-free rate or return on a zero beta portfolio.
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Single-Factor APT
1.0 P
E[r ]P
E*
rf
RP = E - r* f
*
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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]i.
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Single-Factor APT Consider the following portfolio which
is part of a well diversified portfolio.
Amount Security Invested E(ri) i
A $20,000 8% 0.6B $40,000 10% 1.2 C $40,000 13% 1.6
E(rP) = .2x8+.4x10+.4x13 = 10.8%
P = .2x0.6+.4x1.2+.4x1.6 = 1.24
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Single-Factor APT
Sell B and purchase $16,000 of A and $24,000 of C.
Amount Security Invested E(ri) i
A $36,000 8% 0.6C $64,000 13% 1.6
E(rP) = .36x8 + .64x13 = 11.2%
P = .36x0.6 + .64x1.6 = 1.24
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Multi-Factor APT
The return on security i is ri = E(ri) + 1iF1+ ... + kiFk+ ei.
E(ri) is the expected return.
Fj is factor j, (j = 1,...,k).
ji measures the sensitivity of ri to factor j, (j = 1,...,k).
ei is the firm specific return.
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Multi-Factor APT The return on a well diversified
portfolio is rP = E(rP) + 1PF1+ ... + kPFk.
E(rP) is the expected return. Fj is factor j, (j = 1,...,k). jP measures the sensitivity of rP to
factor j, (j = 1,...,k). eP = wiei 0.
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Multi-Factor APT
Diversified Portfolio
F
rP
j
The relationship between the return on a well diversified portfolio and factor j, holding other factors equal to zero.
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Multi-Factor APT
Arbitrage causes the expected return on a well diversified portfolio to be
E[rP] = rf + [RP1]1P +...+ [RPk]kP
jP is the sensitivity of portfolio P to
unexpected changes in factor j.
RPj is the risk premium on factor j.
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Multi-Factor APT
1.0 j
E[r ]P
E j
rf
RP = E - rj j f
Relationship when all other betas are zero.
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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]1i +...+ [RPk]ki
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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.
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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.
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Portfolio Sensitivities
1.0
1.0 U
B
S
Z
Portfolios
S - Stocks
B – Bonds
U – Unit Beta
Z – Zero Beta
Inflation Beta
Productivity Beta
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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.
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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.
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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.
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Salomon Brothers
Growth rate in GNP.
Rate of inflation.
Rate of interest.
Rate of change in oil prices.
Rate of growth in defense spending.