l pch11
TRANSCRIPT
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.1
Investments
Chapter 11: The Arbitrage Pricing Theory
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.2
Factor Risk Models
• Relate the common movements of asset prices to a series of common risk factors.
• Examined here:
SIM (Single Index Model)
MIM (Multiple Index Model)
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.3
The Single Index Model
Assumes two factors are responsible for a given asset’s rate of return:
1. Changes in a common risk factor.
2. Changes related to asset-specific events.
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Slide 11.4
The Single Index Model
Linear Relationship:
Where:
Ri is the rate of return on asset i, I is the percentage change in the common risk factor, ei is asset-specific component, βi measures the sensitivity of the i-th asset’s return to changes in the common risk factor.
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.5
The Single Index Model
- Assumptions on ei:• E(ei) = 0• Independence of specific news, E(ei, es) = 0• Independence of common factor, E((I-E(I)ei) = 0- The model simplifies to: E(Ri) = ai + i E(I)- The risk is: 2222
ieIii
risksystematicIi :22
riskicunsystematie :2
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.6
SIM versus Mean-Variance Efficient Set
- SIM simplifies calculations• Only betas need to be estimated; covariances are
then estimated by definition:
- Beta estimates on the other hand are very sensitive, due to structural changes and the choice of “Index”.
2, Isisi
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Slide 11.7
The Multiple Index Model
• Allows for several common factors to influence a given asset’s rate of return.
• Relationship:
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Slide 11.8
Example with 3 factors1. Inflation (Its market price, or risk premium)2. GDP growth ( “ )3. The $/€ spot exchange rate, S, ( “ )
• Our model is:
risk icunsystemat theis
)"( beta rate exchangespot theis
)" ( beta GDP theis
risk) systematic (its betainflation theis
ε
β
β
β
εFβFβFβRR
S
GDP
I
SSGDPGDPII
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.9
Example
• Suppose we have made the following estimates:
1. I = -2.30
2. GDP = 1.50
3. S = 0.50.
• Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contributes 1% to the return.
εFβFβFβRR SSGDPGDPII
%1ε
%150.050.130.2 SGDPI FFFRR
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.10
Example
We must decide what surprises took place in the systematic
factors. If it was the case that the inflation rate was expected
to be 3%, but in fact was 8% during the time period, then
FI = Surprise in the inflation rate
= actual – expected
= 8% - 3%
= 5%
%150.050.130.2 SGDPI FFFRR
%150.050.1%530.2 SGDP FFRR
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.11
Example
If it was the case that the rate of GDP growth was expected
to be 4%, but in fact was 1%, then
FGDP = Surprise in the rate of GDP growth
= actual – expected
= 1% - 4%
= -3%
%150.050.1%530.2 SGDP FFRR
%150.0%)3(50.1%530.2 SFRR
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.12
Example
If it was the case that $/€ spot exchange rate was expected to
increase by 10%, but in fact remained stable during the time
period, then
FS = Surprise in the exchange rate
= actual – expected
= 0% - 10%
= -10%
%150.0%)3(50.1%530.2 SFRR
%1%)10(50.0%)3(50.1%530.2 RR
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.13
Example
Finally, if it was the case that the expected return on the
stock was 8%, then
%150.0%)3(50.1%530.2 SFRR
%12
%1%)10(50.0%)3(50.1%530.2%8
R
R
%8R
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.14
The Arbitrage Pricing Theory
Arbitrage:• A strategy that makes a positive return without
requiring an initial investment.• In other words: arbitrage opportunities exist when
two items that are the same sell at different prices.• In efficient markets, profitable arbitrage
opportunities will quickly disappear.
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Slide 11.15Arbitrage Pricing Theory (True example)
Assume you bet on UEFA match Sakhim-Newcastle (sept 20, 2004), with the following odds:
1 gives 8, X gives 3.9 and 2 gives 1.7 times your money.
The following strategy is a money machine!Result: 1 X 2Odds: 8 3.9 1.7Invest 1000: 129 264.5 606.5
Return: 1032 1031 1031
This is an arbitrage free strategy since it generates arisk free profit of SEK 31, no matter the result!
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Slide 11.16Arbitrage Pricing Theory (True Example)
Formulate the following linear programming to find the allocation of your capital:Min kSubject to:8x <= k3.9y <= k1.7z <= kx + y + z = kx >= 0, y >= 0, z >= 0
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Slide 11.17
The Arbitrage Pricing Theory
• The APT investigates the market equilibrium prices when all arbitrage opportunities are eliminated.
• The APT implies a linear equilibrium relationship between expected return and the factor sensitivities (betas)
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Slide 11.18
The APT: Assumptions
• Perfect competitive capital market• All investors have homogeneous expectations,
regarding mean, variance and covariance• More wealth is preferred to less (but no need to
know for risk attitudes)• Large number of capital assets exist• Short sales are allowed
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Slide 11.19
The Arbitrage Pricing Theory
• The expected return on a security under the APT with a single factor is given (precisely as by SML) by:
• The expected return on a security under the APT with multiple factors is given by:
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Slide 11.20 Relationship Between the Return on the Common Factor & Excess Return
Excess return
The return on the factor F
i
iiii εFβRR
If F = 0, then i > 0
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Slide 11.21 Relationship Between the Return on the Common Factor & Excess Return
Excess return
The return on the factor F
If we assume that there is no
unsystematic risk, then i = 0
FβRR iii
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Slide 11.22
Arbitrage Portfolios and Factor Models• Now let us consider what happens to portfolios of stocks
when each of the stocks follows a one-factor (F) model.• We will create portfolios from a list of N stocks and will
capture the systematic risk with a 1-factor model.• The ith stock in the list have returns:
iiii εFβRR
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Slide 11.23
Arbitrage Portfolios and Factor Models• We know that the portfolio return is the weighted average
of the returns on the individual assets in the portfolio:
NNiiP RXRXRXRXR 2211
)(
)()( 22221111
NNNN
P
εFβRX
εFβRXεFβRXR
NNNNNN
P
εXFβXRX
εXFβXRXεXFβXRXR
222222111111
iiii εFβRR
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.24
Arbitrage Portfolios and Factor ModelsThe return on any portfolio is determined by three sets of
parameters:
In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.
NNP RXRXRXR 2211
1. The weighed average of expected returns.
FβXβXβX NN )( 2211
2. The weighted average of the betas times the factor.
NN εXεXεX 2211
3. The weighted average of the unsystematic risks.
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Slide 11.25
Arbitrage Portfolios and Factor Models
So the return on a diversified portfolio is determined by two sets of parameters:
1. The weighed average of expected returns.
2. The weighted average of the betas times the factor F.
FβXβXβX
RXRXRXR
NN
NNP
)( 2211
2211
In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.
Levy and Post, Investments © Pearson Education Limited 2005
Slide 11.26
Arbitrage Portfolios and Factor Models
The return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor.
FβXβXRXRXR NNNNP )( 1111
FβRR PPP
NNP RXRXR 11
that Recall
NNP βXβXβ 11
and
PR Pβ
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Slide 11.27
APT vs. CAPM• Both models are based on completely different
sets of assumptions.• None the less, both models can predict the same
risk-return relationship: when the asset returns obey the SIM the APT relationship is identical to 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.
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Slide 11.28
Empirical Tests of the APT
• Strong indications that risk factors other than the market portfolio affect expected returns.
• Important to remember that tests of the APT are joint tests of the validity of the APT, the research methodology and the quality of the data.
• Empirical methods are based less on theory and more on looking for some regularities in the historical record.