Index Models and APT

Index Models and APT (Text reference: Chapter 8) Index models Parameter estimation Multifactor models Arbitrage Single factor APT Multifactor APT Index models predate CAPM, originally proposed as a simplification of mean-variance analysis a portfolio with securities requires: 2 2 expected returns variances 2 covariances 3 2 parameter estimates index models rely on an assumption that returns are affected by either macroeconomic factors or firm-specific events 2

in a single factor model: r i E r i β i F e i where r i actual return observed on security i E r i forecasted return on security i β i security i s sensitivity to the macro factor F unanticipated component of the macro factor e i unexpected part of i s return due to firm-specific events if we assume that the rate of return on a broad stock index is a suitable proxy for the macroeconomic factor, we get a single index model: r i r f α i β i r M r f e i or, in excess return form: R i α i β i R M e i 3 two sources of risk: systematic (due to R M ) and firm-specific (due to e i ) the variance of returns for security i is: σ 2 i Var R i Var α i β i R M e i Var β i R M e i β 2 i VarR M Var e i 2β i Cov R M e i β 2 i σ2 M σ 2 e i the covariance between returns for securities i and j is: σ i j Cov R i R j Cov α i β i R M e i α j β j R M e j Cov β i R M e i β j R M e j β i β j Cov R M R M β i Cov R M e j β j Cov e i R M Cov e i e j β i β j σ 2 M 4

to use this model, we need α s, β s, σ 2 e i s, and σ 2 M 3 parameter estimates permits analysts to specialize by industry since they only have to calculate the relationship with the market as a whole, not every other industry individually standard diversification result: consider an equally-weighted portfolio: R p i w i R i σ 2 p i R i α i β i R M e i i i α i R M β i i α p β p R M e p β 2 pσ 2 M σ 2 e p i e i 5 what happens to portfolio variance as gets large? σ 2 e p i 2 σ2 e i σ2 e i 0 as σ 2 P diversifiable risk β 2 p σ2 M systematic risk 6

Parameter estimation parameters can be estimated using simple linear regression on historical data example: monthly data for 999 Excess Returns Month T-Bill Market CIBC Market CIBC Jan.004006.03834.0358.03428.00952 Feb.004069 -.060927 -.076623 -.064996 -.080692 Mar.003504.047488.094233.043984.090729 Apr.004085.063920 -.028497.059835 -.032582 May.0049 -.023455 -.034667 -.027574 -.038786 Jun.004054.026502 -.02200.022448 -.02654 Jul.003270.0025.05670.007755.02400 Aug.003746 -.04482 -.089762 -.08228 -.093508 Sep.003452.000036 -.8644 -.00346 -.22096 Oct.00486.043685.204.039499.5955 ov.00355.037903.03323.034352.029572 Dec.00403.9444.062595.543.058582 X Y 7 regression formulas and calculations: s.e. ˆα ˆβ X t X Y t Y X 2 t X ˆα Y ˆβX s 2 R 2 0 02764 et 2 003438 T 2 e 2 t Y t Y 2 02862 0 4863 e t Y t Ŷ t Y t ˆα ˆβXt s 2 T X 2 X t X 2 0 32720 s.e. ˆβ s 2 X t X 2 0 380805 t α ˆα s.e. ˆα 0 204669 t β ˆβ s.e. ˆβ 2 683427 8

Y (CIBC) X (Market) 9 use e s (i.e. observed deviations from regression line) to compute firm-specific variances σ 2 e i R 2 is the fraction of total variation explained by the macroeconomic factor another formula for it is: R 2 β 2 σ 2 M σ 2 rewrite the index model and take expectations: r i r f α i β i r M r f e i ote that CAPM α i 0 E r i r f α i β i E r M r f there are many more sophisticated ways to estimate β s (see text section 8.2 for examples) 0

Multifactor models may not be able to summarize systematic risk by a single factor (such as a stock index) since β s seem to vary over the business cycle, reasonable candidates for additional factors are variables related to it one possibility (Chen, Roll, and Ross (986)): R i where IP EI UI CG GB α i β IP i IP β EI i EI β UI i UI β CG i % change in industrial production % change in expected inflation % change in unexpected inflation CG β GB i GB e i spread of long term corporate bonds over long term gov t. bonds spread of long term gov t. bonds over T-bills parameters can be estimated using multiple regression alternatively, we can specify models using firm characteristics which seem to be empirically related to systematic risk exposures one model (Fama and French (996)): R i where M SMB HML α i β M i R M β SMB i excess return on market index SMB β HML HML e i spread of small stock portfolio over large stock portfolio spread of high book-to-market portfolio over low book-to-market portfolio empirical evidence suggests that SMB and HML help to predict stock returns, thus proxy in some way for systematic risk i 2

Arbitrage arbitrage arises if a zero investment portfolio can be constructed which yields a sure profit since no investment is needed, large positions can be taken ( profits can be made) large in efficient markets, arbitrage opportunities will be rare and will not last long example: suppose the domestic year risk free interest rate is 4%, the spot exchange rate from CAD into is 2.25, the U.K. year risk free interest rate is 5%, and the year forward exchange rate is 2.20 3 Single Factor APT very similar to index models, but we don t need to assume that a market index proxies for the macroeconomic factor r i E r i β i F e i F could be unexpected changes in GP, interest rates, exchange rates (or all of these in a multifactor version) we have the usual distinction between systematic and diversifiable risk: σ 2 p β 2 p σ2 F σ 2 e p in a well-diversified portfolio σ 2 e p 0 as 4

Single Stock r Well-diversified Portfolio r F F 5 if two well-diversified portfolios have the same β s they must have the same expected returns or else there is an arbitrage opportunity: r F 6

more generally, well-diversified portfolios with different β s must have risk premia proportional to β or else there is an arbitrage opportunity: E r β 7 if we take the well-diversified portfolio to be the market portfolio, and we let F be the unexpected return on the market portfolio, we get the usual CAPM relationship (but only using arbitrage and the return factor assumptions, not other CAPM assumptions) since APT does not require the well-diversified portfolio to be the market portfolio, it provides a justification for using the index model as an implementation of CAPM suppose that two well-diversified portfolios, U and V, are combined into a zero-β portfolio Z as follows: E r Z β Z w U β U w V β V β V β V β U β U β U β V β U β V 0 β U w U E r U w V E r V β V E r U E r V β V β U β V β U r f β V β U β V E r U β U E r V β V E r U r f β U E r V r f r f 8

the last expression on the preceding slide can be rewritten to obtain the APT result for well-diversified portfolios: E r U r f E r V r f β U β V note that if we let one of the two well-diversified portfolios above be the market portfolio (i.e. set V M), then E r U r f E r M r f β U β M E r U r f β U E r M r f it is possible to show using a lot of mathematics that this relationship must also hold for almost all individual securities (see section 8.6 of the text for an informal argument) 9 Multifactor APT for simplicity, consider the two factor case (the extension to k factors is obvious): r i E r i β i F β i2 F 2 e i we now need the concept of a factor portfolio : a well-diversified portfolio with a β with respect to one factor of and β s with respect to all other factors of 0 it can be shown that for any well-diversified portfolio: E r p r f β p E r r f β p2 E r 2 r f again holds for almost all individual securities as well to implement, we can either use statistical methods to uncover the number of factors or economic arguments to pre-specify them see section 8.7 of the text for an overall comparison of CAPM, index models, and APT 20