Download Return On A Portfolio

Chapter 10
Index Models And The
Arbitrage Pricing Theory
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Index Models and APT
Provide Potential Solutions To:
– Estimation problems in implementing
MPT
– Shortcomings of CAPM
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Single Index Model (SIM)
• Stock’s Rate of Return
– Percentage change in the index (I)
• Common factor
– Changes related to firm-specific events (ei)
• On average = 0
• Any given period, it can be + or -
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SIM Calculations
• Ri = Constant + Common-Factor + Firm-Specific
News
News
• Ri = i + iI + ei
• I = Ri - iI
• Note: CAPM is a specific form of SIM
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The Return Components
Ri
Realized Return
Firm-Specific News
Average Return With
Common-Factor News
i
i
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I
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Why Does SIM Reduce
Computations?
• It Decreases the Number of Calculations of
Covariances
• From
Cov(Ri, Rj) = Cov(i+ iI + ei,j + jI + ej)
• To
Cov (Ri, Rj) = i j2I
• Because (by assumption)
Cov(ei, ej) = 0 and Cov(I, ei) = 0
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Estimation Issues
• Results of portfolio allocation depend on accurate
statistical inputs
• Estimates of
– Expected returns
– Standard deviation
– Correlation coefficient
• Among entire set of assets
• With 100 assets, 4,950 correlation estimates
• Estimation risk refers to potential errors
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Estimation Issues
• With the assumption that stock returns can be described by
a single market model, the number of estimated inputs
required for the covariances reduces to the number of
assets (one beta for each) plus one (the variance of the
Index’s return)
• No. of estimated inputs required for covariances:
No. of stocks
W/o simplification
Under SIM
5
10
6
10
45
11
100
4,950
101
500
124,750
501
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Estimation Issues
• Benefits of using SIM for estimating Eff. Frontier
– Fewer estimated inputs implies less chance for
estimation error (true Eff. Frontier is more likely to be
near where it is estimated to be)
• Costs of using SIM
– Potentially unrealistic, oversimplified assumptions
– Ignores the potential for “industry effects”
– different stocks in the same industry will tend to move
together in ways that are separate from what the market
as a whole is doing
– But, market effects are still the strongest
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Systematic Risk
Common-Factor Risk
Undiversifiable
Portfolio Risk
Unsystematic Risk
Investors are not
rewarded for
unsystematic risk
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Firm-Specific Risk
Diversifiable
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• Systematic Risk
– Inflation rate
– Unemployment rate
– Interest rate
• Unsystematic Risk
– Resignation of the president
– Change in dividends
– New discovery
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Return On A Portfolio
Portfolio + Portfolio Return + Portfolio Return
Intercept Due to
Due to
Market Factor
Firm-Specific
Factors
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Risk And E(R) With SIM
Start With
E(Ri) = I + i  E(I) + E(ei)
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2
With Diversification
Unsystematic Risk
Systematic Risk
30
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n
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Risk And E(R) With SIM
E(ei) = 0
Ri = E(Ri) + I[I - E(I)] + ei
Just Like APT
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APT
• Linear Risk-Return Relationship
• No Arbitrage Opportunities
• Equilibrium Model
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What Is Arbitrage?
• Borrow at 5% and Save at 6%
• Simultaneously buying stock cheaply on
one market and selling it short on another
higher-quoted market
• What Causes the Arbitrage?
– Zero out-of-pocket investment
– Return is always nonnegative
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Arbitrage Pricing Theory (APT)
• Arbitrage is a process of buying a lower priced
asset and selling a higher priced asset, both of
similar risk, and capturing the difference in
arbitrage profits
• The general arbitrage principle states that two
identical securities will sell at identical prices
• Price differences will immediately disappear as
arbitrage takes place
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Arbitrage Pricing Theory
(APT)
• CAPM is criticized because of the
difficulties in selecting a proxy for the
market portfolio as a benchmark
• An alternative pricing theory with fewer
assumptions was developed:
• Arbitrage Pricing Theory
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Assumptions of
Arbitrage Pricing Theory
(APT)
1. Capital markets are perfectly competitive
2. Investors always prefer more wealth to less wealth with
certainty
3. The stochastic process generating asset returns can be
presented as K factor model
– Common factors plus some noise
– Describes the risk-return relationship
• Other Assumptions:
– Large number of assets in the economy
– Short sales allowed with proceeds
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Assumptions of CAPM
That Were Not Required by
APT
APT does not assume
• A market portfolio that contains all risky
assets, and is mean-variance efficient
• Normally distributed security returns
• Quadratic utility function
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Arbitrage Pricing Theory
(APT)
Ri  Ei  bi1 1  bi 2 2  ...  bik  k   i
For i = 1 to N where:
Ri = return on asset i during a specified time period
Ei = expected return for asset i
bik = reaction in asset i’s returns to movements in the kth common factor
δk = a common factor with a zero mean that influences the returns on all assets
εi = a unique effect on asset i’s return that, by assumption, is completely
diversifiable in large portfolios and has a mean of zero
N = number of assets
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Arbitrage Pricing Theory
(APT)
 k Multiple factors expected to have an
impact on all assets:
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Arbitrage Pricing Theory
(APT)
 k Multiple factors expected to have an
impact on all assets:
– Inflation
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Arbitrage Pricing Theory
(APT)
 k Multiple factors expected to have an
impact on all assets:
– Inflation
– Growth in GNP
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Arbitrage Pricing Theory
(APT)
 k Multiple factors expected to have an
impact on all assets:
– Inflation
– Growth in GNP
– Major political upheavals
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Arbitrage Pricing Theory
(APT)
 k Multiple factors expected to have an
impact on all assets:
– Inflation
– Growth in GNP
– Major political upheavals
– Changes in interest rates
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Arbitrage Pricing Theory
(APT)
 k Multiple factors expected to have an
impact on all assets:
– Inflation
– Growth in GNP
– Major political upheavals
– Changes in interest rates
– And many more….
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Arbitrage Pricing Theory
(APT)
 kMultiple factors expected to have an impact on
all assets:
– Inflation
– Growth in GNP
– Major political upheavals
– Changes in interest rates
– And many more….
Contrast with CAPM insistence that only beta is
relevant
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Arbitrage Pricing Theory
(APT)
bik (beta(i,k)) determines how each asset
reacts to this common factor
Each asset may be affected by growth in GNP,
but the effects will differ
In application of the theory, the factors are not
identified
Similar to the CAPM, the unique effects ( εi )
are independent and will be diversified
away in a large portfolio
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APT’s Common Factors
• Used I - E(I) Instead of Just I
• Called the Surprise Factor
• Measures the Difference Between
– Expectations
– Actual outcomes
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Firm’s Beta
The Larger the Beta
The Larger the Effect of the Surprise
On the Firm’s Return
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Arbitrage Pricing Theory
(APT)
• APT assumes that, in equilibrium, the return
on a zero-investment, zero-systematic-risk
portfolio is zero when the unique effects are
diversified away
• The expected return on any asset i (Ei) can
be expressed as:
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Arbitrage Pricing Theory
(APT)
Ei  0  1bi1  2bi 2  ...  k bik
where:
0 = the expected return on an asset with zero systematic
risk:
0
0
1 = the risk premium related to each of the common
factors - for example the risk premium related to
interest rate risk
1
i
0
bik = the pricing relationship between the risk premium
and asset i - that is how responsive asset i is to this
common factor k
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

 E
 E E
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Example of Two Stocks
and a Two-Factor Model
λ0 = the rate of return on a zero-systematic-risk asset (zero beta:
bik=0) is 3 percent
( λ0 = 0.03 )
λ1 = changes in the rate of inflation. The risk premium related to
this factor is 1 percent for every 1 percent change in the rate
( λ1 = 0.01 )
λ2 = percent growth in real GNP. The average risk premium
related to this factor is 2 percent for every 1 percent change in
the rate
( λ2 = 0.02 )
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Example of Two Stocks
and a Two-Factor Model
bx1= the response of asset X to changes in the rate of inflation
is 0.50
(bx1  .50)
by1= the response of asset Y to changes in the rate of inflation
is 2.00
(by1  .50)
bx 2 = the response of asset X to changes in the growth rate of
real GNP is 1.50
(bx 2  1.50)
b y 2= the response of asset Y to changes in the growth rate of
real GNP is 1.75
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(by 2  1.75)
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Example of Two Stocks
and a Two-Factor Model
Ei  0  1bi1  2bi 2
= .03 + (.01)bi1
+ (.02)bi2
Ex = .03 + (.01)(0.50) + (.02)(1.50)
= .065 = 6.5%
Ey = .03 + (.01)(2.00) + (.02)(1.75)
= .085 = 8.5%
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APT vs. CAPM
• Similar Results
– Both yield a linear risk-return relationship
• Advantage of APT
– More realistic, less restrictive assumptions
– Allows for multiple risk factors (e.g., industry effects)
• Disadvantage of APT
– Fails to identify common factors
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Multifactor APT
• Suggested Factors
– Default premium
– Term structure
– Inflation
– Corporate profits
– Market risk
• E(Ri) = λ0 + λ1i1 + λ2i2 + … + λkik
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Empirical Tests of the APT
• Studies by Roll and Ross and by Chen
support APT by explaining different rates of
return with some better results than CAPM
• Reinganum’s study did not explain smallfirm results
• Dhrymes and Shanken question the
usefulness of APT because it was not
possible to identify the factors
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Roll-Ross Study
1. Estimate the expected returns and the factor
coefficients from time-series data on
individual asset returns
2. Use these estimates to test the basic crosssectional pricing conclusion implied by the
APT
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Extensions of the
Roll-Ross Study
• Cho, Elton, and Gruber examined the
number of factors in the return-generating
process that were priced
• Dhrymes, Friend, and Gultekin (DFG)
reexamined techniques and their limitations
and found the number of factors varies with
the size of the portfolio
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The APT and Anomalies
• Small-firm effect
Reinganum - results inconsistent with the APT
Chen - supported the APT model over CAPM
• January anomaly
Gultekin - APT not better than CAPM
Burmeister and McElroy - effect not captured by model,
but still rejected CAPM in favor of APT
• APT and inflation
Elton, Gruber, and Rentzler - analyzed real returns
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The Shanken Challenge to
Testability of the APT
• If returns are not explained by a model, it is not considered
rejection of a model; however if the factors do explain
returns, it is considered support
• APT has no advantage because the factors need not be
observable, so equivalent sets may conform to different
factor structures
• Empirical formulation of the APT may yield different
implications regarding the expected returns for a given set
of securities
• Thus, the theory cannot explain differential returns between
securities because it cannot identify the relevant factor
structure that explains the differential returns
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Alternative Testing Techniques
• Jobson proposes APT testing with a
multivariate linear regression model
• Brown and Weinstein propose using a
bilinear paradigm
• Others propose new methodologies
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