Slide 12-1Factor PricingLecture 12: Factor PricingProf. Markus K. BrunnermeierSlide 12-2Factor PricingOverview1. ICAPM – multiperiod economic model (last lecture)2. Asset Pricing Theory (APT) – static statistical model Merits of Factor Pricing Exact Factor Pricing and Factor Pricing Errors Factor Structure and Pricing Error Bounds  Single Factor and Beta Pricing (and CAPM) (Factor) Mimicking Portfolios Unobserved Factor Models Multi-period outlook3. Empirical Factor Pricing Models Arbitrage Pricing Theory (APT) Factors The Fama-French Factor Model + MomentumSlide 12-3Factor PricingThe Merits of Factor Models• Without any structure one has to estimate J expected returns E[Rj] (for each asset j) J standard deviations J(J-1)/2 co-variances• Assume that the correlation between any two assets is explained by systematic components/factors, one can restrict attention to only K (non-diversifiable) factors Advantages: Drastically reduces number of input variablesModels expected returns (priced risk)Allows to estimate systematic risk (even if it is not priced, i.e. uncorrelated with SDF)Analysts can specialize along factors  Drawbacks: Purely statistical model (no theory)(does not explain why factor deserves compensation: risk vs. mispricing)relies on past data and assumes stationaritySlide 12-4Factor PricingFactor Pricing Setup …• K factors f1, f2, …, fK E[fk]=0 K is small relative to dimension of M fkare not necessarily in M• F space spanned by f1,…,fK,e• in payoffs bj,kfactor loading of payoff xjSlide 12-5Factor Pricing…Factor Pricing Setup• in returns• Remarks:One can always choose orthogonal factors Cov[fk, fk‟]=0Factors can be observable or unobservableSlide 12-6Factor PricingFactor Structure• Definition of “factor structure:”• ) risk can be split in systematic risk andidiosyncratic (diversifiable) riskSlide 12-7Factor PricingExact vs. Approximate Factor Pricing• Multiplying (1) by kqand taking expectations• Rearranging• Exact factor pricing: error: j= 0 (i.e. js orthogonal to kq ) e.g. if kq2 FSlide 12-8Factor Pricing• Recall error Note, if 9 risk-free asset and all fk2 M, then …• If kq2 F, then factor pricing is exact • If kqF, then Let‟s make use of the Cauchy-Schwarz inequality (which holds for any two random variables z1and z ) Error-boundBound on Factor Pricing Error…Slide 12-9Factor PricingError-Bound if Factor Structure Holds• Factor structure ) split idiosyncratic from systematic risk • ) all idiosyncratic risk jare linearly independent andspan space orthogonal to F. Hence, • Note • Error • Pythagorean Thm: If {z1, …, zn} is orthogonal system in Hilbert space, then  Follows from def. of inner product and orthogonalitySlide 12-10Factor PricingApplying Pythagorean Thm to implies Multiply by …… and makinguse of RHS is constant for constant max[2(j)]. ) For large J, most securities must have small pricing error• Intuition for Approximate Factor Pricing: Idiosyncratic risk can be diversified awayError-Bound if Factor Structure HoldsSlide 12-11Factor PricingOne Factor Beta Model…• Let r be a risky frontier return and setf = r – E[r] (i.e. f has zero mean) q(f) = q(r) – q(E[r]) • Risk free asset exists with gross return of r q(f) = 1 – E[r]/r• f and r span E and hence kq2 F) Exact Factor Pricing___Slide 12-12Factor Pricing…One Factor Beta Model• Recall E[rj] = r -jr q(f) E[rj] = r -j{E[r] - r}• Recall j= Cov[rj, f] / Var[f] = Cov[rj, r] / Var[r]• If rm2 E then CAPM___ _Slide 12-13Factor PricingMimicking Portfolios…• Regress on factor directly or on portfolio that mimics factor  Theoretical justification: project factor on M  Advantage: portfolios have smaller measurement error• Suppose portfolio contains shares 1, …, Jwith jJj= 1.• Sensitivity of portfolio w.r.t. to factor fkis k= j j jk• Idiosyncratic risk of portfolio is = j j2( ) = j2(j) diversificationSlide 12-14Factor Pricing…Mimicking Portfolios• Portfolio is only sensitive to factor k0(and idiosyncratic risks) if for each k k0 k=jjk=0, and k0=j jk00.• The dimension of the space of portfolios sensitive to a particular factor is J-(K-1).• A portfolio mimics factor k0if it is the portfolio with smallest idiosyncratic risk among portfolios that are sensitive only to k0.Slide 12-15Factor PricingObservable vs. Unobservable Factors…• Observable factors: GDP, inflation etc.• Unobservable factors: Let data determine “abstract” factors Mimic these factors with “mimicking portfolios” Can always choose factors such that• factors are orthogonal, Cov[fk, fk‟]=0 for all k k‟• Factors satisfy “factor structure” (systemic & idiosyncratic risk)• Normalize variance of each factor to ONE) pins down factor sensitivity (but not sign, - one can always change sign of factor)Slide 12-16Factor Pricing…Unobservable Factors…• Empirical content of factorsCov[ri,rj] = k ik jk2(fk)2(rj) =k jk jk2(fk)+2(j) (fk)=1 for k=1,L,K. (normalization)In matrix notation• Cov[r,r„] = k k‟k2(fk) + D,– where k= (1k,…,Jk).• = B B‟ + D, – where Bjk=jk, and D diagonal.– For PRINCIPAL COMPONENT ANALYSIS assume D=0(if D contains the same value along the diagonal it does affect eigenvalues but not eigenvectors – which we are after)Slide 12-17Factor Pricing…Unobservable Factors…• For any symmetric JxJ matrix A (like BB‟), which is semi-positive definite, i.e. y‟Ay ¸ 0, there exist numbers 1¸2¸…¸ lambdaJ¸ 0 and non-zero vectors y1, …, yJsuch that yjis an eigenvector of A assoc. w/ eigenvalue j, that is A yj= jyjjJyijyij‟= 0 for j j‟jJyijyij= 1 rank (A) = number of non-zero „s The yj„s are unique (except for sign) if the i„s are distinct• Let Y be the matrix with columns (y1,…,yJ), andlet the diagonal matrix with entries ithenSlide 12-18Factor Pricing…Unobservable Factors• If K-factor model is true, BB' is a symmetric positive semi-definite matrix of rank $K.$ Exactly K non-zero eigenvalues 1,…,kand associated eigenvectors y1,…,yK YKthe matrix with columns given by y1,…,yK Kthe diagonal matrix with entries j, j=1,…, K. BB'= K Hence,• Factors are not identified but sensitivities are (except for sign.)• In practice choose K so that kis small for k>K.Slide 12-19Factor