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Levy and Post, Investments © Pearson Education Limited 2005

Slide 11.1

Investments

Chapter 11: The Arbitrage Pricing Theory


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)


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.


Slide 11.4


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.


Slide 11.5


- 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


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


Slide 11.7

The Multiple Index Model

• Allows for several common factors to influence a given asset’s rate of return.

• Relationship:


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


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


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


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


= 1% - 4%

= -3%

%150.050.1%530.2 SGDP FFRR

%150.0%)3(50.1%530.2 SFRR


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


= 0% - 10%

= -10%

%150.0%)3(50.1%530.2 SFRR

%1%)10(50.0%)3(50.1%530.2 RR


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


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.


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!


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


Slide 11.17


• 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)


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


Slide 11.19


• 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:


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


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


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


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


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.


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.


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


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.


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.

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