INVESTMENTS Arbitrage Pricing Theory Chapter 7 INVESTMENTS Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit ? Since no investment is required, an investor can create large positions to secure large levels of profit ? In efficient markets, profitable arbitrage opportunities will quickly disappear INVESTMENTS Current Expected Standard Stock Price$ Return% Dev.% A 10 25.0 29.58 B 10 20.0 33.91 C 10 32.5 48.15 D10 2.5 8.58 Arbitrage Example from Text pp. 308-310 INVESTMENTS Mean S.D. Correlation Portfolio A,B,C 25.83 6.40 0.94 D 22.25 8.58 Arbitrage Portfolio INVESTMENTS Arbitrage Action and Returns E. Ret. St.Dev. * P * D Short 3 shares of D and buy 1 of A, B & C to form P You earn a higher rate on the investment than you pay on the short sale INVESTMENTS Factor model of asset returns ? Suppose that asset returns are driven by a few common factors and diversifiable noise: ? Where ? is the expected return on asset i; ? are news on common factors driving all asset returns. ? gives how sensitive the return on asset i with respect to news on the k-th factor—is called the loading of asset i on factor ? is the idiosyncratic component in asset i’s return that is unrelated to other asset returns ? have zero means. iKiKiii ufbfbrEr +++= ~ ... ~ )( 11 i Er K fff ~ ,..., ~ , ~ 21 )( ~ kkk FEFf ?= ik b k f ~ i u ~ iK ufff , ~ ,..., ~ , ~ 21 INVESTMENTS example ? Common factors driving asset returns may include GNP, interest rates, inflation, etc. Let be the news on interest rate. Before a board meeting of the Fed, the market expect the Fed not to change the interest rate. After the meeting, Greenspan announces that: ? There is no change in interest rate---”no news” ? There is a ?% increase in interest rate—positive surprise ? What should be the sign of the factor loadings on , be for fixed income securities, stocks, commodity futures? int ~ f 0 ~ int =f %25.0 ~ int =f int ~ f INVESTMENTS Properties of factor models ? The following results provide the building blocks of APT. ? 1. Any diversified portfolio p is exposed only to factor risks KpKppp fbfbrEr ~ ... ~ )( 11 ++= INVESTMENTS continued ? A diversified portfolio, that is not exposed to any factor risk( ), must offer risk-free rate; ? There always exists portfolios that are exposed only to the risk of a single factor. ? Example: suppose two well diversified portfolios, both exposed only to the risk of the first two factors 0... 2 1 ==== pKpp bbb kpkpkpk fbrr ~ += 21 ~ , ~ ff , ~ 5.0 ~ 2.0 211 ffr ++= , ~ 5.1 ~ 23.0 212 ffr ++= INVESTMENTS continued ? A portfolio Pk, that has unitary risk of factor k, offer a expected return with the factor risk: ? ? such a portfolio is called a factor portfolio for factor k, and ? is the premium of factor k ? Example. In the above example, we found portfolio p that bears only the risk of factor 1. Its loading is 0.5. Consider following portfolio p1: ? 200% invested in p and –100% invested in the risk-free portfolio P0. fkpk rr = ffk rr ? INVESTMENTS APT ? Suppose that asset returns are driven by a few common factors and diversifiable noise: ? For an arbitrary asset, its expected return depends only on its factor exposure: ? Where ? is the premium on factor k; ? is asset i’s loading of factor k )(...)()( 2211 ffKiKffiffifi rErbrErbrErbrEr ?++?+?+≈ ff rr k ? ik b iKiKiii ufbfbrEr +++= ~ ... ~ )( 11 INVESTMENTS We illustrate APT with an example ? Suppose that there are two factors: ? (1) unanticipated market return ? (2) unanticipated inflation ? Suppose that ? The returns on factor portfolios are: 1 ~ f 2 ~ f iiiii ufbfbrr ~ ~~ ~ 2211 +++= %2%,8%,5 21 ?=?=?= ffffF rrrrr 2 1 ~ )02.005.0( ~ ~ )08.005.0( ~ 2 1 fr fr P p +?= ++= INVESTMENTS continued ? We first consider assets with only factor risks ? For an asset with ? ? APT requires that its expected rate of return must be ? Suppose that was instead 10%. Then there is a free lunch. 0.1 21 ==bb 21 ~~ ~ ffrr qq ++= %11 )02.0(0.108.00.105.0 )()( 21 21 = ?×+×+= ?+?+= fffffq rrbrrbrr q r INVESTMENTS continued ? Consider 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 rate ? This 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. INVESTMENTS continued ? What 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. INVESTMENTS continued ? Suppose that the pricing formula are violated for many assets. Let us focus on those with the same factor risks ? Form a diversified portfolio of these assets, q; ? Portfolio q bears only factor risks, and violate APT. INVESTMENTS Portfolio &Individual Security Comparison F E(r)% Portfolio F E(r)% Individual Security INVESTMENTS Example ? Suppose that assets A,B,C, we have 0.61.0C 0.21.5B 1.00.5A b2b1asset C B A r r r %7)02.0(0.108.05.005.0 =?×+×+= INVESTMENTS example-continued ? Investors 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. INVESTMENTS Implementation of APT ? The implementation of APT involves three steps: ? 1. Identify the factors; ? 2. Estimate factor loadings of assets ? 3. Estimate factor premia INVESTMENTS 1 factors ? Since 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 returns ? extract “factors” from the covariance matrix ? (c ) Data mining: explore different portfolios to find those whose returns can be used as factors INVESTMENTS Factor loadings, factor premia, APT pricing ? 2 Factor loadings.given factors, we can regress past asset returns on the factors to estimate factor loadings (bik) ? 3 factor premia. Given the factor loading of individual assets, we can construct factor portfolios. For the k-th factor, we have ? The premium of the k-th factor is ? 4 APT pricing. By APT, the return on asset i is given by itktiktiiit ufbfbrr ~ ~ ... ~ ~ 11 ++++= ktpkp frr kt ~ ~ += fpkff rrrr k ?=? )(... 11 f K fiKfifi rrbrbrr ?+++= INVESTMENTS E(r)% Beta for F 10 7 6 Risk Free 4 A D C .5 1.0 Disequilibrium Example—single factor example INVESTMENTS Disequilibrium Example-continued ? Short Portfolio C ? Use funds to construct an equivalent risk higher return Portfolio D - D is comprised of A & Risk-Free Asset ? Arbitrage profit of 1% INVESTMENTS E(r)% Beta (Market Index) Risk Free M 1.0 [E(r M ) - r f ] Market Risk Premium APT with Market Index Portfolio INVESTMENTS ? APT applies to well diversified portfolios and not necessarily to individual stocks ? With APT it is possible for some individual stocks to be mispriced - not lie on 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 APT and CAPM Compared