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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--11
Lecture 06: Factor PricingLecture 06: Factor Pricing
Prof. Markus K. Brunnermeier
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--22
OverviewOverview
Theory of Factor Pricing (APT)Theory of Factor Pricing (APT)
Merits of Factor PricingMerits of Factor Pricing
Exact Factor Pricing and Factor Pricing ErrorsExact Factor Pricing and Factor Pricing Errors
Factor Structure and Pricing Error BoundsFactor Structure and Pricing Error Bounds
Single Factor and Beta Pricing (and CAPM)Single Factor and Beta Pricing (and CAPM)
(Factor) Mimicking Portfolios(Factor) Mimicking Portfolios
Unobserved Factor ModelsUnobserved Factor Models
MultiMulti--period outlookperiod outlook
Empirical Factor Pricing ModelsEmpirical Factor Pricing ModelsArbitrage Pricing Theory (APT) FactorsArbitrage Pricing Theory (APT) Factors
TheThe FamaFama--French Factor Model + MomentumFrench Factor Model + Momentum
Factor Models from the StreetFactor Models from the Street Salomon Smith BarneySalomon Smith Barneys and Morgan Stanleys and Morgan Stanleys Models Model
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--33
The Merits of Factor ModelsThe 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 isexplained by systematic components/factors, one can
restrict attention to only K (non-diversifiable) factorsAdvantages: Drastically reduces number of input variables
Models 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 stationarity
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--44
Factor Pricing SetupFactor Pricing Setup K factors f1, f2, , fKE[fk]=0
K is small relative to dimension ofM
fkare not necessarily in M
F
space spanned by f1,,fK,e in payoffs
bj,kfactor loading of payoff xj
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--55
Factor Pricing SetupFactor Pricing Setup
in returns
Remarks:One can always choose orthogonal factors Cov[fk, fk]=0
Factors can be observable or unobservable
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--66
Factor StructureFactor Structure
Definition of factor structure:
risk can be split in systematic risk andidiosyncratic (diversifiable) risk
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--77
Exact vs. Approximate Factor PricingExact vs. Approximate Factor Pricing
Multiplying (1) by kq
and taking expectations
Rearranging
Exact factor pricing:
error: j = 0 (i.e. j s orthogonal to kq ) e.g. if kq F
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--88
Recall error
Note, if risk-free asset and all fkM, then
If k q F, then factor pricing is exact
If k q F, thenLets make use of the Cauchy-Schwarz inequality
(which holds for any two random variables z1 and z2)
Error-bound
Bound on Factor Pricing ErrorBound on Factor Pricing Error
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--99
ErrorError--Bound if Factor Structure HoldsBound if Factor Structure Holds Factor structure split idiosyncratic from systematic risk
all idiosyncratic riskj are 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 orthogonality
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1010
Applying Pythagorean Thm toimplies
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 away
ErrorError--Bound if Factor Structure HoldsBound if Factor Structure Holds
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1111
One Factor Beta ModelOne 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 Eand hence kq F
Exact Factor Pricing
_
_
_
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1212
One Factor Beta ModelOne Factor Beta Model
Recall
E[rj] = r - j r q(f)
E[rj] = r - j {E[r] - r}
Recall
j = Cov[rj, f] / Var[f] = Cov[rj, r] / Var[r]
If rm Ethen CAPM
__
_ _
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1313
Mimicking PortfoliosMimicking Portfolios Regress on factor directly or on portfolio that mimics factor
Theoretical justification: project factor onM Advantage: portfolios have smaller measurement error
Suppose portfolio contains shares 1, , J with jJ j = 1.
Sensitivity of portfolio w.r.t. to factor fkis k= j j jk Idiosyncratic risk of portfolio is = j j
2 () = j 2 (j) diversification
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1414
Mimicking PortfoliosMimicking Portfolios
Portfolio is only sensitive to factor k0 (andidiosyncratic risks) if for each k k0 k= jjk=0, and k0= j jk0 0.
The dimension of the space of portfolios
sensitive to a particular factor is J-(K-1).
A portfolio mimics factor k0 if it is the portfoliowith smallest idiosyncratic risk among portfolios
that are sensitive only to k0.
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1515
Observable vs. Unobservable FactorsObservable vs. Unobservable Factors
Observable factors: GDP, inflation etc.
Unobservable factors:
Let data determine abstract factors
Mimic these factors with mimicking portfoliosCan 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)
E 525 Fi i l E i I
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1616
Unobservable FactorsUnobservable Factors Empirical content of factors
Cov[ri,rj] = kikjk
2
(fk) 2(rj) =kjkjk
2(fk)+2(j)
(fk)=1 for k=1,L,K. (normalization)
In matrix notation Cov[r,r] = kkk
2(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)
E 525 Fi i l E i IE 525 Fi i l E i I
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1717
Unobservable FactorsUnobservable Factors For any symmetric JxJ matrix A (like BB), which is semi-
positive definite, i.e. yAy 0, there exist numbers 1 2 lambda
J 0 and non-zero vectors y
1, , y
Jsuch that
yj is an eigenvector of A assoc. w/ eigenvalue j, that is A yj = j yj
jJ yij y
ij = 0 for j j
jJ yij yij = 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), and
let the diagonal matrix with entries i then
E 525 Fi i l E i IE 525 Fi i l E i I
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1818
Unobservable FactorsUnobservable 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,,yKYK the matrix with columns given by y1,,yK K the 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.
Eco525: Financial Economics IEco525: Financial Economics I
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--1919
Why more than ONE mimickingWhy more than ONE mimicking
portfolio?portfolio?
Mimic (un)observable factors with portfolios[Projection of factor on asset span]
Isnt a single portfolio which mimics pricing kernel
sufficient ONE factor
So why multiple factors?
Not all assets are included (real estate, human capital )
Other factors capture dynamic effects
[since e.g. conditional unconditional. CAPM]
(more later to this topic)
Eco525: Financial Economics IEco525: Financial Economics I
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--2020
OverviewOverview Theory of Factor Pricing (APT)Theory of Factor Pricing (APT)Merits of Factor PricingMerits of Factor Pricing
Exact Factor Pricing and Factor Pricing ErrorsExact Factor Pricing and Factor Pricing Errors
Factor Structure and Pricing Error BoundsFactor Structure and Pricing Error Bounds
Single Factor and Beta Pricing (and CAPM)Single Factor and Beta Pricing (and CAPM)
(Factor) Mimicking Portfolios(Factor) Mimicking Portfolios
Unobserved Factor ModelsUnobserved Factor Models
MultiMulti--period outlookperiod outlook
Empirical Factor Pricing ModelsEmpirical Factor Pricing Models
Arbitrage Pricing Theory (APT) FactorsArbitrage Pricing Theory (APT) Factors
TheThe FamaFama--French Factor Model + MomentumFrench Factor Model + Momentum
Factor Models from the StreetFactor Models from the Street
Salomon Smith BarneySalomon Smith Barneys and Morgan Stanleys and Morgan Stanleys Models Model
Eco525: Financial Economics IEco525: Financial Economics I
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--2121
APT Factors of Chen, Roll and Ross (1986)APT Factors of Chen, Roll and Ross (1986)
1. Industrial production(reflects changes in cash flow expectations)
2. Yield spread btw high risk and low risk corporate bonds(reflects changes in risk preferences)
3. Difference between short- and long-term interest rate(reflects shifts in time preferences)
4. Unanticipated inflation5. Expected inflation (less important)
Note: The factors replicate market portfolio.
Eco525: Financial Economics IEco525: Financial Economics I
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Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--2222
FamaFama--MacBethMacBeth 2 Stage Method2 Stage Method
Stage 1: Use time series data to obtain estimates for
each individual stocks j
(e.g. use monthly data for last 5 years)
Note: is just an estimate [around true j ]
Stage 2: Use cross sectional data and estimated j
s toestimate SML
b=market risk premium
Eco525: Financial Economics IEco525: Financial Economics I
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing
Eco525: Financial Economics IEco525: Financial Economics I
Slide 06Slide 06--2323
CAPMCAPM estingesting FamaFama French (1992)French (1992) Using newer data slope of SML b is not significant (adding size and B/M)
Dealing with econometrics problem: s are only noisy estimates, hence estimate of b is biased
Solution:
Standard Answer: Find instrumental variable
Answer in Finance: Derive estimates for portfolios Group stocks in 10 x 10 groups
sorted to size and estimated j
Conduct Stage 1 of Fama-MacBeth for portfolios
Assign all stocks in same portfolio same Problem: Does not resolve insignificance
CAPM predictions: b is significant, all other variables insignificant
Regressions: size and B/M are significant, b becomes insignificant Rejects CAPM
Portfolio
size
Eco525: Financial Economics IEco525: Financial Economics I
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Eco525: Financial Economics Ia a
Slide 06Slide 06--2424
Book to Market and SizeBook to Market and Size
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing Slide 06Slide 06--2525
FamaFama French Three Factor ModelFrench Three Factor Model Form 2x3 portfolios
Size factor (SMB) Return ofsmall minus big
Book/Market factor (HML)
Return ofhigh minus low For
s are big and s do not vary much
For (for each portfolio p using time series data)
s are zero, coefficients significant, high R2
.
size
book/market
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing Slide 06Slide 06--2626
FamaFama French Three Factor ModelFrench Three Factor Model Form 2x3 portfolios
Size factor (SMB) Return ofsmall minus big
Book/Market factor (HML)
Return ofhigh minus low For
s are big and s do not vary much
For (for each portfolio p using time series data)
p
s are zero, coefficients significant, high R2
.
size
book/market
Book to Market as a Predictor of ReturnBook to Market as a Predictor of Return
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Book to Market as a Predictor of ReturnBook to Market as a Predictor of Return
ValueValue
GrowthGrowth
0%0%
5%5%
10%10%
15%15%
20%20%
25%25%
AnnualizedRateofRetu
rn
AnnualizedRateofRetu
rn
1010998877665544332211
High Book/MarketHigh Book/Market Low Book/MarketLow Book/Market
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Book to Market Equity of Portfolios Ranked by BetaBook to Market Equity of Portfolios Ranked by Beta
0.60.6 0.80.8 11 1.21.2 1.41.4 1.61.6 1.81.8
BetaBeta
0.50.5
0.60.6
0.70.7
0.80.8
0.90.9
11
B
ooktoMark
etEquity
B
ooktoMark
etEquity
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing Slide 06Slide 06--2929
Adding Momentum FactorAdding Momentum Factor
5x5x5 portfolios Jegadeesh & Titman 1993 JF rank stocks
according to performance to past 6 months
Momentum Factor
Top Winner minus Bottom Losers Portfolios
Monthly Difference Between Winner andMonthly Difference Between Winner and
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Loser Portfolios at Announcement DatesLoser Portfolios at Announcement Dates
11 33 55 77 99 11 13 15 17 19 21 23 25 27 2929 31 33 35
Months Following 6 Month Performance PeriodMonths Following 6 Month Performance Period
MonthlyDifference
BetweenWin
nerand
--1.5%1.5%
--1.0%1.0%
--0.5%0.5%
0.0%0.0%
0.5%0.5%
1.0%1.0%
LoserP
ortfolios
Cumulative Difference Between Winner andCumulative Difference Between Winner and
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--5%5%
--4%4%
--3%3%
--2%2%
--1%1%
0%0%
1%1%
2%2%
3%3%4%4%
5%5%
Months Following 6 Month Performance PeriodMonths Following 6 Month Performance Period
Cum
ulativeDiffe
renceBetween
Cum
ulativeDiffe
renceBetween
Win
nerandLoserPortfolios
Win
nerandLoserPortfolios
Loser Portfolios at Announcement DatesLoser Portfolios at Announcement Dates
11 33 55 77 99 1111 1313 1515 1717 1919 2121 2323 2525 2727 2929 3131 3333 3535
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Morgan StanleyMorgan Stanleys Macro Proxy Models Macro Proxy Model Factors
GDP growthLong-term interest rates
Foreign exchange (Yen, Euro, Pound basket)
Market Factor
Commodities or oil price index
Factor-mimicking portfolios (Macro Proxy)
Stage 1: Regress individual stocks on macro factorsStage 2: Create long-short portfolios of most and least
sensitive stocks [5 quintiles] Macro Proxy return predicts macro factor
Eco525: Financial Economics IEco525: Financial Economics I
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09:55 Lecture 0609:55 Lecture 06 Factor PricingFactor Pricing Slide 06Slide 06--3333
Salomon Smith Barney Factor ModelSalomon Smith Barney Factor Model
Factors
Market trend (drift)
Economic growth
Credit quality
Interest rates
Inflation shocksSmall cap premium
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1930s 40s 50s 60s 70s 80s 90s beyond
The Old Finance
Modern Finance
Modern FinanceModern FinanceTheme: Valuation Based on Rational Economic BehaviorParadigms: Optimization Irrelevance CAPM EMH
(Markowitz) (Modigliani & Miller) (Sharpe, Lint ner & Mossen) (Fama)
Foundation: Financial Economics
HaugenHaugens view: The Evolution of Academic Finances view: The Evolution of Academic Finance
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1930s 40s 50s 60s 70s 80s 90s beyond
The Old Finance
Modern Finance
The New Finance
The New FinanceThe New Finance
Theme: Inefficient MarketsParadigms: Inductive ad hocFactor Models Behavioral Models
Expected Return Risk
Foundation: Statistics, Econometrics, and Psychology
HaugenHaugens view: The Evolution of Academic Finances view: The Evolution of Academic Finance