Chapter 10 Arbitrage Pricing Theory & Multifactor Models of Risk and Return PDF

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Taught by Brad Gibbs...

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Chapter 10: Arbitrage Pricing Theory & Multifactor Models of Risk and Return 10.1 Multifactor Models: An Overview Each risk that can be hedged is included as a factor in the SML Factor Models of Security Returns  You can decompose variability into o Systematic/market risk  GDP growth  Interest rates o Idiosyncratic/firm-specific risk  Product launches  Manufacturing glitch Factors=types of systemic risk. Expected value always 0. Single-factor model of excess returns= Ri=E(Ri)+ Ei Ei= can be decomposed into M+ ei M=macro surprise ei= firm specific shock E(m)=0 E(ei)=0 Each has a standard deviation E(m,ei)=0 [we assume they are not correlated] Single factor model= ri=E(ri)+Bim+ei  Bim implies that a macro surprise can affect each security differently  VARi=B*VARm+VARe BKM (10.1) ri= E(ri)+Bi*F+ei Bi*F=economy wide effect Ei=firm specific events Multifactor models: allow stocks to exhibit different sensitivities to the various components of systemic risk Factor loadings/factor betas: coefficients of factors measuring sensitivity of share returns to that factor. BKM 10.2 ri=E(ri)+Bi(GDP)*GDP+Bir*IR+ei These betas measure sensitivity of returns to their respective factors Example: Ri=0.133+1.2GDP-0.3IR+ei This means expected return of 13.3% and one percent point increase in GDP will increase returns by 1.2 % We can see that it is possible to use the negative and positive Betas to hedge a source of risk

Suppose: Risk premium GDP=6% RP IR= -7% Rf= 4% E(R)= 4%+1.2(6%)-.3(-7%)+ei =13.3%

10.2 Arbitrage Pricing Theory (APT) Alternative model to CAPM Key differences o Factor model o No-arbitrage condition o CAPM is microeconomic ‘bottom up’ model whereas APT is macroeconomic “top down”  Intention: derive a “fair” E(r) in relation to underlying risk (ie a SML relationship) Arbitrage, Risk Arbitrage, and Equilibrium Predicts a security market line linking expected returns to risk, but different than SML. Means to obtain a fair E(r) in “SML relationship” to risk 3 Propositions 1. Security returns can be described by a factor model 2. There are enough securities to diversify away idiosyncratic (firm-specific) risk 3. Well-functioning security markets don’t allow for persistence of arbitrage opportunities “Arbitrage opportunity”: when an investor can earn riskless profits w/out making a net investment The Law of One Price: if 2 assets are equivalent in all economically relevant respects, they should have the same market price. Arbitrage activity enforces this. Difference between arbitrage and risk-return dominance Arbitrage: a few investors will go wild and drive up or down the price. Irrespective of wealth or risk aversion, investors will want an infinite position. Risk-return dominance: a lot of investors make a small change which then causes price to adjust Risk (as opposed to Pure) Arbitrage: professional searching for mispriced securities in specific areas, rather than one who seeks strict, risk-free arbitrage opportunities Example: See Aleafia and Emblem example in slides from March 11. Multiple acquirer share price times exchange rate and you see price is below that of current acquired company. You would short .8377 ALEAF/Go long 1 EMMBF. In doing so, you become an Emblem shareholder and thus receive the .8377 shares from ALEAF once the deal goes through  

Well-Diversified Portfolios You can divide portfolio variance into systematic and nonsystematic

Nonsystematic Variance of a portfolio is weighted average of each asset, but you square each weight Well-diversified portfolio= one w/ each weight w that is small enough that the nonsystematic variance is negligible. Rp=E(Rp)+BpF+ep (10.3 BKM) Where Bp is a weighted average of individual betas, same w ep VARp=Bp^2VARF+VAR(ep) This represents both systematic and nonsystematic sources Thus, well-diversified portfolio is diversified over big enough number of securities w weights small enough that VAR(ep) is negligible So we can simplify equations to Rp=E(Rp)+BpF VARp=Bp^2*VAR(F) In single factor world, all portfolios are perfectly correlated; diversification Diversification and Residual Risk in Practice IRL portfolios are limited in terms of # of stocks Largest thousands of stocks (Wilshire 5000) SD decreases when stocks are evenly balanced and when there are more stocks. Increasing number of stocks most effective way to decrease SD. Executing Arbitrage Imagine single factor market. There is a market index portfolio M and a zero-residual portfolio P, w/ a positive alpha. Since Bm=1, you can eliminate risk of P completely as there is np residual risk. You construct Portfolio Z w/ Wp=1/(1-Bp) and Wm=1-Wp Alpha(z)=Wp*Alpha(p) Risk of Z=0. The risk premium must be equal to 0 because both Alpha(Z) and Beta(Z)=0 Risk premium= E(Rz)=1/(1-B) *Alpha

The No-Arbitrage Equation of the APT B/c of arbitrage, risk premium for zero beta portfolios should always=0. Therefore, alpha on any well-diversified portfolio=0 For well-diversified portfolios E(Rp)=Bp*E(Rm). Thus, risk premium is a function of beta and market return– matches the SML of CAPM If portfolios w/same beta have different returns, you can short the lower return one and long the higher return one, making a riskless profit. Can also be true for different Betas, if they have different relationships between their beta and return

Betas and Market Returns  We can illustrate graphically that only macroeconomic factor should command a risk premium in the market

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See fig. 10.1 in BKM– shows return for the well diversified portfolio is exactly determined by the systematic factor F- no firm-specific error The single stock is subject to non-systematic risks, seen in the scatter of points around the line Example: Portfolio A and B. o A: E(r)=10% o B: E®=8% o Both have Beta=1 o Arbitrage opp: Short B and long same amount of A o Payoff:  $1mil*(.1*1F) from long  -1mil*(.08*1F) from short  Net: $20k Thus, well-diversified portfolios w equal betas must have equal E(r)s in market equilibrium (or arbitrage opportunities would exist) BKM 10.9 E(Rp)=Bp*E(Rm)

10.3 The APT, the CAPM, and the Index Model 1. Does APT apply to less than well-diversified portfolios? 2. Is APT a better model than CAPM or do we need both? 3. Can we use Treynor-Black Framework in place of APT? The APT and the CAPM  Key Differences: o APT based on factor model o No arbitrage condition  If residual risk sufficiently high and can’t further diversify a portfolio, can’t fully trust APT  APT frees us of CAPM’s assumption that all investors are mean-variance optimizers  APT can be anchored by the observable market index, whereas CAPM is untestable  Still need CAPM when looking at theoretical grand scale o We can interpret SML as saying that investors are rewarded for exposure to macroeconomic risk but not firm specific risk  Thus multi-factor model shows us how they are rewarded for both types of risk  CAPM implies that many small investors will make small changes and the sum of these changes represents how equilibrium is reached  APT has no reference to risk aversion The APT and Portfolio Optimization in a Single-Index Market APT ignores that if arbitrage position not perfectly diversified, scaling your arbitrage position increases risk In case of mispriced asset w/ positive alpha: Follow Treynor-Black Procedure: 1. Estimate EPm and SDm (benchmarks from index)

2. Place mispriced assets into active portfolio (see textbook p.336) a. Calculate W0A then W*A then W*M 3. Maximize Sharpe ratio by maximizing information ratio IR=AlphaA/SD(eA) When residual risk=0, W0A=infinity b/c risk-free arbitrage. T-B does what APT does while also accounting for non-zero residual risk

10.4 A Multifactor APT A two factor APT: Ri=E(Ri)+Bi1*F1+Bi2*F2+ei  Each F has E(F)=0 b/c we are looking at deviation from expected level, not at the actual level Factor Portfolio: a well-diversified “tracking portfolio” w/ all factors’ betas=0 except for a factor with a beta of 1. It’s tracking the evolution of particular sources of macroeconomic risk which are uncorrelated to other sources of risk. Let’s you focus on the evolution of one macroeconomic factor.  We can do this b/c there are thousands of securities and not many factors  These serve as benchmark portfolios Example: 2 Factor portfolios: Portfolio 1 E(r1)=10% Port 2 E(r2)=12% Rf=4% RP1=6% RP2=8% Suppose we have well diversified Portfolio A with BA1=.5 BA2= .75 What is fair E(r)? Use BKM 10.11 E(r)=rf+BA1[RP1]+BA2[RP2] Just plug and chug Any well diversified portfolio with equal Betas must have equal E(r)s APT tells us fair returns but doesn’t tell us what the factors should be – you are looking to find factors that area sources of systemic risk.

10.5 The Fama-French Three-Factor Model The equation: Rit=alpha+BiM*RMt+Bsmb*smb+Bhml*hml+e SMB=small minus big. Return of portfolio of small stocks minus the return on a portfolio of large stocks HML=high minus low. Return on a portfolio of high book-to-market ratio of stocks minus return on a portfolio of low book-to-market ratio stocks. “value stocks” vs “growth stocks”

These variables can capture sensitivity to risk factors in macroeconomy-they are proxies for variables that are unknown. Explains 90% of diversified portfolio returns vs 70% in CAPM Holds up empirically over time and across markets Yet none of these factors are clearly hedging a source of macro uncertainty– they may be representative of unknown factors...