1 CHAPTER TWELVE ARBITRAGE PRICING THEORY

2 Background Estimating expected return with the Asset Pricing Models of Modern FinanceEstimating expected return with the Asset Pricing Models of Modern Finance –CAPM Strong assumption - strong prediction Strong assumption - strong prediction

Expected Return Risk (Return Variability) Market Index on Efficient Set Market Index A B C Market Beta Expected Return Corresponding Security Market Line x x x x x x x x x x x x x x x x x x x x x x x x

Market Index Expected Return Risk (Return Variability) Market Index Inside Efficient Set Corresponding Security Market Cloud Expected Return Market Beta

5 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 Beta -15% -5% 5% 15% 25% 35% Expected Return -313 Interest Rate Beta A B C D E F

Two stocks l A: E(r) = 4%; Interest-rate beta = l B: E(r) = 26%; Interest-rate beta = 1.83 l Invest 54.54% in E and 45.46% in A l Portfolio E(r) =.5454 * 26% * 4% = 16% l Portfolio beta =.5454 * * = 0 l With many combinations like this, you can create a risk-free portfolio with a 16% expected return. The Arbitrage Pricing Theory

Two different stocks l C: E(r) = 15%; Interest-rate beta = l D: E(r) = 25%; Interest-rate beta = 1.00 l Invest 50.00% in E and 50.00% in A l Portfolio E(r) =.5000 * 25% * 15% = 20% l Portfolio beta =.5000 * * = 0 l 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.

9 No-arbitrage condition for asset pricing l If risk-return relationship is non-linear, you can arbitrage. l Attempts to arbitrage will force linearity in relationship between risk and return. The Arbitrage Pricing Theory

APT Relationship Between Expected Return and Interest Rate Beta -15% -5% 5% 15% 25% 35% Expected Return -313 Interest Rate Beta A B C D E F

11 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

12 FACTOR MODELS MULTIPLE-FACTOR MODELS –FORMULA r i = a i + b i1 F 1 + b i2 F b iK F K + e i where r is the return on security i b is the coefficient of the factor F is the factor e is the error term

13 FACTOR MODELS SECURITY PRICING FORMULA: r i = b b K b K where r i = r RF +(  1  r RF  b i1  2  r RF )b i2 +   r RF  b iK

14 FACTOR MODELS where r is the return on security i  is the risk free rate b is the factor e is the error term

15 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