INVESTMENTSChapter 7ArbitragePricing Theory
INVESTMENTSArbitrage Pricing TheoryArbitrage -arises if an investor can construct a zero investment portfolio with a sure profitSince no investment is required, an investor can create large positions to secure large levels of profitIn efficient markets, profitable arbitrage opportunities will quickly disappear
INVESTMENTSArbitrage Example from Text pp. 308-310CurrentExpectedStandardStock Price$Return%Dev.%A10 10
INVESTMENTSArbitrage ,B,C
INVESTMENTSArbitrage Action and ReturnsE. Ret.* P* 3 shares of D and buy 1 of A, B & C to form PYou earn a higher rate on the investment than you pay on the short sale
INVESTMENTSFactor model of asset returnsSuppose that asset returns are driven by a few common ~~factors and diversifiable noise:r=E(r)+bf+...bf+uiii11iKKiWhereis the expected return on asset i;Eri~~~are news on common factors driving all asset f,f,...,f12K~=F−E(F)kkkgives how sensitive the return on asset i with respect bikto news on the k-th factor—is called the loading of asset i ~on factor fk~uis the idiosyncratic component in asset i’s return that iis unrelated to other asset returns ~~~f,f,...,f,u12Kihave zero means.
INVESTMENTSexampleCommon factors driving asset returns may include GNP, ~finterest rates, inflation, etc. Let be the news on interest intrate. Before a board meeting of the Fed, the market expect the Fed not to change the interest rate. After the meeting, Greenspan announces that:~f=0There is no change in interest rate---”no news”intThere is a ¼% increase in interest rate—positive surprise~f=%int~What should be the sign of the factor loadings on , fintbe for fixed income securities, stocks, commodity futures?
INVESTMENTSProperties of factor modelsThe following results provide the building blocks of APT.1. Any diversified portfolio p is exposed only to factor risks~~r=E(r)+bf+...bfppp11pKK
INVESTMENTScontinuedA diversified portfolio, that is not exposed to any factor b=b=...=b=0risk( ), must offer risk-free rate;pp2pK1There always exists portfolios that are exposed only to the ~r=r+bfpkpkpkkrisk of a single factor.Example: suppose two well diversified portfolios, both ~~exposed only to the risk of the first two factors f,f12~~~~r=+2f+,r=+f+,212112
INVESTMENTScontinuedA portfolio Pk, that has unitary risk of factor k, offer a expected return with the factor risk:r=rpkfksuch a portfolio is called a factor portfolio for factor k, and r−ris the premium of factor kfkfExample. In the above example, we found portfolio p that bears only the risk of factor 1. Its loading is . Consider following portfolio p1:200% invested in p and –100% invested in the risk-free portfolio P0.
INVESTMENTSAPTSuppose that asset returns are driven by a few common ~~factors and diversifiable noise:r=E(r)+bf+...bf+uiii11iKKiFor an arbitrary asset, its expected return depends only on its factor exposure:Er≈r+b(Er−r)+b(Er−r)+...+b(Er−r)ifi1f1fi2f2fiKfKfWherer−ris the premium on factor k;ffkis asset i’s loading of factor kbik
INVESTMENTSWe illustrate APT with an exampleSuppose that there are two factors:~(1) unanticipated market returnf1~~~~~fr=r+bf+bf+u(2) unanticipated inflation2iii11i22iSuppose thatr=5%,r−r=8%,r−r=−2%Fffff12The returns on factor portfolios are:~~r=(+)+fp11~~r=(−)+fP22
INVESTMENTScontinuedWe first consider assets with only factor risksb=b=For an asset with12~~~r=r+f+fqq12APT requires that its expected rate of return must ber=r+b(r−r)+b(r−r)qf1ff2ff12=+×+×(−)=11%Suppose that was instead 10%. Then there is a free rqlunch.
INVESTMENTScontinuedConsider the following portfolio:(1) buy 100 of portfolio P1,(2) buy 100 of portfolio P2(3) sell $100 of asset q(4) sell $100 of risk-free rateThis portfolio has the following characteristics:Requires zero initial investment (an arbitrage portfolio);Bear no factor risk (and no idiosyncratic risk)Pay (13+3-10-5)=1 surely.This would be an arbitrage.In absence of arbitrage, equation must hold for assets with only factor risks.
INVESTMENTScontinuedWhat if an asset also bears idiosyncratic risks? Since it cannot be replicated by other assets, in particular the factor portfolios, the pricing formula need not hold.However in the presence of idiosyncratic risks, deviations from the pricing equation cannot be pervasive. In other words, for most assets, the pricing formula has to be approximately correct.
INVESTMENTScontinuedSuppose that the pricing formula are violated for many assets. Let us focus on those with the same factor risksForm a diversified portfolio of these assets, q;Portfolio q bears only factor risks, and violate APT.
INVESTMENTSPortfolio &Individual Security ComparisonE(r)%E(r)%FFPortfolioIndividual Security
INVESTMENTSExampleSuppose that assets A,B,C, we =+×+×(−)=7%ArBrC
INVESTMENTSexample-continuedInvestors hold well-diversified portfolios with different exposures to the two factors---depending on how much each investor worries about inflation:Investors who worries more about inflation will seek to hold more of the portfolio that provides a hedge against inflation:(1) start with the market portfolio;(2) sell off assets with negative correlation with factor 2;(3) use the proceeds to buy assets with positive correlation with factor 2.
INVESTMENTSImplementation of APTThe implementation of APT involves three steps:1. Identify the factors;2. Estimate factor loadings of assets3. Estimate factor premia
INVESTMENTS1 factorsSince the theory itself does not specify the factors, we have to construct the factors empirically:(a) using macroeconomic variables;change in GDP growth;change in T-bill yield (proxy for expected inflation)changes in yield spread between T-bonds and T-bills;changes in default premium on corporate bonds;Changes in oil prices(b) using statistical analysis—factor analysis:estimate covariance of asset returnsextract “factors”from the covariance matrix(c ) Data mining: explore different portfolios to find those whose returns can be used as factors
INVESTMENTSFactor loadings, factor premia, APT pricing2 Factor factors, we can regress past asset returns on the factors to estimate factor loadings (bik)~~~~r=r+bf+...+bf+uitii11tikktit3 factor premia. Given the factor loading of individual assets, we can construct factor portfolios. For the k-th factor, we have ~~r=r+fppkktktThe premium of the k-th factor isr−r=r−rffpkfk4 APT pricing. By APT, the return on asset i is given byr=r+br+...+b(r−r)ifi1f1iKffK
INVESTMENTSDisequilibriumExample—single factor exampleE(r)%10AD76CRisk Free for F
INVESTMENTSDisequilibriumExample-continuedShort Portfolio CUse funds to construct an equivalent risk higher return Portfolio D-D is comprised of A & Risk-Free AssetArbitrage profit of 1%
INVESTMENTSAPT with Market Index PortfolioE(r)%M[E(r) -r]MfMarket Risk PremiumRisk Free Beta (Market Index)
INVESTMENTSAPT and CAPM ComparedAPT applies to well diversified portfolios and not necessarily to individual stocksWith APT it is possible for some individual stocks to be mispriced-not lie on the SMLAPT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolioAPT can be extended to multifactor models
