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CHAPTER TWELVE

ARBITRAGE PRICING THEORY

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BackgroundBackground

• Estimating expected return with the Asset Estimating expected return with the Asset Pricing Models of Modern FinancePricing Models of Modern Finance– CAPMCAPM

• Strong assumption - strong predictionStrong assumption - strong prediction

Expected Return

Risk(Return

Variability)

Market Index on Efficient Set

MarketIndex

A

BC

Market Beta

Expected Return

Corresponding Security Market Line

xxx

xxxx

xxxx

xxxxxxx

xxx

xxx

MarketIndex

Expected Return

Risk(Return Variability)

Market Index Inside Efficient Set

Corresponding Security Market Cloud

Expected Return

Market Beta

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FACTOR MODELS

• ARBITRAGE PRICING THEORY (APT)– is an equilibrium factor model of security returns

– Principle of Arbitrage• the earning of riskless profit by taking advantage of

differentiated pricing for the same physical asset or security

– Arbitrage Portfolio• requires no additional investor funds

• no factor sensitivity

• has positive expected returns

– Example …

Curved Relationship Between Expected Return and Interest Rate BetaCurved Relationship Between Expected Return and Interest Rate Beta

-15%-15%

-5%-5%

5%5%

15%15%

25%25%

35%35%

Expected ReturnExpected Return

-3-3 -1-1 11 33Interest Rate BetaInterest Rate Beta

AABB

CC

DD EE FF

• Two stocks A: E(r) = 4%; Interest-rate beta = -2.20

B: E(r) = 26%; Interest-rate beta = 1.83

Invest 54.54% in E and 45.46% in A

Portfolio E(r) = .5454 * 26% + .4546 * 4% = 16%

Portfolio beta = .5454 * 1.83 + .4546 * -2.20 = 0

With many combinations like this, you can create a risk-free portfolio with a 16% expected return.

The Arbitrage Pricing TheoryThe Arbitrage Pricing Theory

The Arbitrage Pricing TheoryThe Arbitrage Pricing Theory

• Two different stocks C: E(r) = 15%; Interest-rate beta = -1.00 D: E(r) = 25%; Interest-rate beta = 1.00 Invest 50.00% in E and 50.00% in A Portfolio E(r) = .5000 * 25% + .4546 * 15% = 20% Portfolio beta = .5000 * 1.00 + .5000 * -1.00 = 0 With many combinations like this, you can create a risk-

free portfolio with a 20% expected return. Then sell-short the 16% and invest the proceeds in the 20% to arbitrage.

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• No-arbitrage condition for asset pricing If risk-return relationship is non-linear, you

can arbitrage. Attempts to arbitrage will force linearity in

relationship between risk and return.

The Arbitrage Pricing TheoryThe Arbitrage Pricing Theory

APT Relationship Between Expected Return and Interest Rate Beta APT Relationship Between Expected Return and Interest Rate Beta

-15%

-5%

5%

15%

25%

35%

Expected ReturnExpected Return

-3 -1 1 3Interest Rate Beta

A B

C

D

EF

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FACTOR MODELS

• ARBITRAGE PRICING THEORY (APT)– Three Major Assumptions:

• capital markets are perfectly competitive

• investors always prefer more to less wealth

• price-generating process is a K factor model

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FACTOR MODELS

• MULTIPLE-FACTOR MODELS– FORMULA

ri = ai + bi1 F1 + bi2 F2 +. . .

+ biKF K+ ei

where r is the return on security ib is the coefficient of the factorF is the factore is the error term

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FACTOR MODELS

• SECURITY PRICINGFORMULA:

ri = 0 + 1 b1 + 2 b2 +. . .+ KbK

where

ri = rRF +(1rRFbi12rRF)bi2+

rRFbiK

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FACTOR MODELS

where r is the return on security i

is the risk free rate

b is the factor

e is the error term

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FACTOR MODELS

• hence– a stock’s expected return is equal to the risk

free rate plus k risk premiums based on the stock’s sensitivities to the k factors