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3. Arbitrage Pricing Model 3. Arbitrage Pricing Model Dr. Youchang Wu WS 2007 Asset Management Youchang Wu 1 Single factor model Assump tions: Asse t ret urns are correl ltdONLYated ONLY because they all respond to one common risk factor, which is usually the market return. rjt = α j + β jrmt + ε jt ,COV(εi ,ε j ) = 0 if i ≠ j Macro events affect returns of all assets Micro events affect only individual assets After controlling for the common risk factor, there is no correlations between returns of different assets. 2 It follows that COV (ri , rj ) = βi β jσ (rm ) Asset Management Youchang Wu 2 Portfolio variance Total variance = Systematic risk + idiosyncratic risk 2 2 2 2 σ (rj ) = β j σ (rm ) + σ (ε j ) 2 2 2 2 σ (rp ) = β p σ (rm ) + σ (ε p ) n n 2 2 2 2 = (∑ wi βi ) σ (rm ) + ∑ wi σ (ε i ) i=1 i=1 n 2 2 → (∑ wi βi ) σ (rm ) as n → +∞ i=1 Factor structure greatly simplies the calculation of portfolio variance. This is the main motivation for Sharpe (1963) to introduce the factor model. Asset Management Youchang Wu 3 Number of parameters To estimate the variance of a portfolio with N assets, originally we would need to estimate – N variances – N*(N-1)/2 covariances With the factor structure, we only need to estimate – N idiosyncratic variances – N betas – 1 factor variance Asset Management Youchang Wu 4 Multifactor models Security returns are driven by multiple common risk factors rjt =αj + β1jF1t + β2 jF2t +...+ βLjFLt +ε jt If th e common ri s k fact ors are uncorrelated with each other, then n n 2 2 2 2 2 σ (rp ) = (∑ wi β1i ) σ (F1 ) + (∑ wi β 2i ) σ (F2 ) + ... i=1 i=1 n n 2 2 2 2 + (∑ wi β Li ) σ (FL ) + ∑ wi σ (ε i ) i=1 i=1 Asset Management Youchang Wu 5 Multifactor model: example El2ftExample: 2 factors Residual β1 β2 variance Security I 0.6 0.2 0.05 Security II 0.9 0.1 0.02 Assume that σF1²=0.12 and σF2²=0.10 and COV(F1, F2)=0. Compute the variance of the following portfolio wI=wII=0.5 Asset Management Youchang Wu 6 Constructing zero-beta portfolio ElExample: Security β1 β2 A-0.41.75 B 1.6 -0.75 C 0.667 -0.25 How can you construct a zero-beta portfolio? Asset Management Youchang Wu 7 Rewriting the factor models For s imp lic ity, de fine fi as the unexpec te d component of Fi, fi ≡ Fi − E(Fi ) By definition , E(fi)=0 Rewrite the factor models as rjt = E(rj ) + β1 j f1t + β2 j f 2t + ... + β Lj f Lt + ε jt Asset return = expected return + response to unexpected factor realization + idiosyncratic return Asset Management Youchang Wu 8 APT Ross (1976) argue d tha t to preven t arbitrage, we must have L E(rj ) = rf + ∑ βljλl l=1 where λl is the risk premium of factor l, i.e., the expected excess return of an asset with βl=1 and βi=0 for all i≠l. =>Expected returns increase linearly with the loadings on common risk factors Asset Management Youchang Wu 9 A simple derivation of APT For simplicity, consider just the single factor model without idiosyncratic risk rj = E(rj ) + β j f The return of a portfolio with two assets is rp = wri + (1 − w)rj = [wE(ri ) + (1 − w)E(rj )] + [wβi + (1 − w)β j ] f If we set w = −β j /(βi − β j ) then this portfolio is risk free, therefore its expected return must be rf. Otherwise there will be arbitrage opportunity Asset Management Youchang Wu 10 A simple derivation of APT It follows that β j β j − E(ri ) + (1 + )E(rj ) = rf βi − β j βi − β j ⇒ βi [E(rj ) − rf ] = β j [E(ri ) − rf ] E(r ) − r E(r ) − r ⇒ i f = j f ≡ λ βi β j The derivation is similar for the case of multiple factors without idiosyncratic risk When they are idiosyncratic risk, APT hold only approximately. Keyyg argument: If APT does not hold, then arbitra geurs can create a well-diversified zero beta portfolios with an expected return different from the risk free rate. This would make arbitrage possible! Asset Management Youchang Wu 11 Arbitrage in expectations Asset Management Youchang Wu 12 Arbitrage in expectations (2) Asset Management Youchang Wu 13 APT Example Security Expected return β1 β2 A 1.0 1.6 16.2 B 3.0 2.8 21.6 rf 000.0 000.0 10. 0 Suppose APT holds What is the risk premium for factor 1? What is the risk premium for factor 2? Asset Management Youchang Wu 14 What are the risk factors APT does no t spec ify w ha t the fac tors st and f or The factor structure is ambiguous. If securities’ returns can be written using the k- factor vector f and the n x k beta matrix B, they can also be wr itten us ing a fac tor vec tor f* and a beta vector B* as long as f * = L−1 f and B* = BL, where L is a non-singular k × k matrix. Asset Management Youchang Wu 15 Chen, Roll and Ross (1986) Chen, R oll an d Ross (1986, “Econom ic forces and the stock market”, Journal of Business) find the following macroeconomic factors are relevant – Unanticipated changes in industrial production – Unanticipated changes in inflation – Unanticipated changes in risk premiums (as measured by the return difference between low grade and high grade bonds – Unanticipated changes in the term structure of interest rates Asset Management Youchang Wu 16 Why are those factors chosen Stocks prices in a discounted cash flow framework E[C] P = r – E[C] = expected dividends – r = discount rate Relevant factors are those that influence either the discount rate or the expected dividends: – Discount rate: • Change in risk premium • Changes in Term Structure – Dividends: • Changes in inflation rate • Changes in industrial production Asset Management Youchang Wu 17 Construction of the factors Industrial production growth: MP(t) = ln I P(t) − ln I P(t −1) Inflation: – Unexpected inflation rate U I(t) = I(t) − E[I(t) t −1] – Changgpes in expected inflation rate DEI (t) = E[I(t + 1) t] − E[I (t) t − 1] Asset Management Youchang Wu 18 Construction of the factors (2) Risk premium: UPR (t) = " Baa and under " bond return − LG B (t) note that UPR(t) is negatively related to unexpected changes in aggregate risk aversion Term Structure: UT S (t) = LGB(t) − T B(t −1) Additionally examined variables: – EWNY(t): ret urn on equall y we ig hte d NYSE in dex – VWNY(t): return on value-weighted NYSE index Asset Management Youchang Wu 19 Empirical Results Model equations: R = a + β MP M P + β DEI DEI + βUI U I + βUPRU PR + βUTS UT S + e R = α + bMP β MP + bDEI β DEI + bUI βUI + bUPR βUPR + bUTS βUTS + ε Results (see the table): •bMP > 0 • bDEI < 0 •bUI < 0 •bUPR > 0 •bUTS<0 Asset Management Youchang Wu 20 Results (Panel C of table 4) Asset Management Youchang Wu 21 Results (Panel D of table 4) Asset Management Youchang Wu 22 Interpretation bMP>0: stocks less sensitive to production risk are more valuable (therefore has a lower return) bUPR>0: stocks less sensitive to the return of low- gg(rade bond returns (relative to that of hi gh-grade bonds) are more valuable bUI<0,bDEI<0: stocks doing well when inflation is unexpectedly high are more valuable bUTS<0: stocks doing well when long -term real rate is unexpectedly low are more valuable. Asset Management Youchang Wu 23 Further remarks Market return as a factor: – The macroeconomic factors don’t lose their significance, even if market return is included as a factor. – The influence of market return is not significant – CAPM would suggest the opposite – There fore the resu lts suppor t APT agai nst CAPM Asset Management Youchang Wu 24 APT and portfolio management See Ro ll, R. an d S. A. Ross, 1984, The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning, Financial Analyst Journal, p. 14-26. The central focus of portfolio strategy is the c ho ices of an appr opri ate pattern of factor sensitivities This choice should depend on the economic characteristics of the investors Asset Management Youchang Wu 25 Example Two factors: inflation and industrial production. Is zero exposure to both factors desirable? – No hedge against inflation – No risk premium Asset Management Youchang Wu 26 Analyzing portfolio strategies Analysis of the investors’ economic situation – What is the investor’s expenditure structure? – What is the investor’ s income structure? – E.g. Institutional investors: Expenditures are lifldbfdibtthiless influenced by food prices; but their travelling expenses are likely to be higher than those o f an average inves tor. Compppare with the market portfolio Asset Management Youchang Wu 27 Strategic portfolio decisions If an institution does not need to hedge so much against inflation due to low sensitivity to food price ⇒ choose a low βI ((g)relative to the average). If an institution does not need to hedge so much against production fluctuations ⇒ choose a high βIP. By bearing such additional systematic risks the institution can earn a higher expected return Asset Management Youchang Wu 28 Tactical portfolio decisions If the firm’s idiosyncratic risk depends on energy cost => heavily invest in the stocks of energy sector. If the firm is unconcerned about inflation in agricultural price => skew portfolio holdings out of this sector Common mistake: pension funds investing in the stock of the sponsoring firm Asset Management Youchang Wu 29 APT and CAPM APT a llows for mu ltip le r is k fac tors bu t requ ires zero correlation among idiosyncratic returns CAPM considers only one factor but does not required zero correlation among idiosyncratic returns CAPM is derived under the assumption that investors hold mean-variance efficient portfolios, while APT is derived under the assumption of no arbitraged in expectations CAPM recommends investors to hold the market ppggportfolio, while APT suggests investors to optimize expoures to risk factors acording to their own profiles.
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