08Lecture_a_FactorPricing modesl with theoires

08Lecture_a_FactorPricing modesl with theories

1. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Lecture 08: Factor PricingLecture 08: Factor Pricing
Prof. Markus K. Brunnermeier

2. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
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 Bounds Factor 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
Multi-period outlookMulti-period outlook
•Empirical Factor Pricing ModelsEmpirical Factor Pricing Models
Arbitrage Pricing Theory (APT) FactorsArbitrage Pricing Theory (APT) Factors
The Fama-French Factor Model + MomentumThe Fama-French Factor Model + Momentum
Factor Models from the StreetFactor Models from the Street
•Salomon Smith Barney’s and Morgan Stanley’s ModelSalomon Smith Barney’s and Morgan Stanley’s Model

3. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
The Merits of Factor ModelsThe Merits of Factor Models
•Without any structure one has to estimate
J expected returns E[R
j
] (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 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

4. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Factor Pricing Setup …Factor Pricing Setup …
•K factors f
1
, f
2
, …, f
K
E[f
k]=0
K is small relative to dimension of M
f
k are not necessarily in M
•F space spanned by f
1
,…,f
K
,e
•in payoffs
b
j,k factor loading of payoff x
j

5. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
……Factor Pricing SetupFactor Pricing Setup
•in returns
•Remarks:
One can always choose orthogonal factors Cov[f
k
, f
k’
]=0
Factors can be observable or unobservable

6. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Factor StructureFactor Structure
•Definition of “factor structure:”
• ) risk can be split in systematic risk and
idiosyncratic (diversifiable) risk

7. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Exact vs. Approximate Factor PricingExact vs. Approximate Factor Pricing
•Multiplying (1) by k
q
and taking expectations
•Rearranging
•Exact factor pricing:
error: 
j
= 0 (i.e. 
j
s orthogonal to k
q
)
e.g. if k
q
2 F

8. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
•Recall error
Note, if 9 risk-free asset and all f
k 2 M, then …
•If k
q
2 F, then factor pricing is exact
•If k
q  F, then
Let’s make use of the Cauchy-Schwarz inequality
(which olds for any two random variables z
1 and z
)
Error-bound
Bound on Factor Pricing Error…Bound on Factor Pricing Error…

9. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Error-Bound if Factor Structure HoldsError-Bound if Factor Structure Holds
•Factor structure ) split idiosyncratic from systematic risk
• ) all idiosyncratic risk 
j
are linearly independent and
span space orthogonal to F. Hence,
•Note
•Error
•Pythagorean Thm: If {z
1, …, z
n} is orthogonal system in
Hilbert space, then
Follows from def. of inner product and orthogonality

10. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Applying Pythagorean Thm to
implies
Multiply by …… and making
use 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
Error-Bound if Factor Structure HoldsError-Bound if Factor Structure Holds

11. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
One Factor Beta Model…One Factor Beta Model…
•Let r be a risky frontier return and set
f = 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 k
q 2 F
) Exact Factor Pricing
_
_
_

12. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
……One Factor Beta ModelOne Factor Beta Model
•Recall
 E[r
j] = r - 
j r q(f)
 E[r
j] = r - 
j {E[r] - r}
•Recall
 
j
= Cov[r
j
, f] / Var[f] = Cov[r
j
, r] / Var[r]
•If r
m
2 E then CAPM
__
_ _

13. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Mimicking Portfolios…Mimicking 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, …, 
J with 
j
J

j =1.
•Sensitivity of portfolio w.r.t. to factor f
k is 
k = 
j 
j 
jk
•Idiosyncratic risk of portfolio is  = 
j  
j
 
2
() = 
j 
2
(
j)
 diversification

14. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
……Mimicking PortfoliosMimicking Portfolios
•Portfolio is only sensitive to factor k
0 (and
idiosyncratic risks) if for each k  k
0 
k= 
j 
jk=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 k
0 if it is the portfolio with
smallest idiosyncratic risk among portfolios that are
sensitive only to k
0.

15. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Observable vs. Unobservable Factors…Observable 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[f
k
, f
k’
]=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)

16. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
……Unobservable Factors…Unobservable Factors…
•Empirical content of factors
Cov[r
i,r
j] = 
k 
ik
jk
2
(f
k)
 
2
(r
j) =
k 
jk
jk
2
(f
k)+
2
(
j)
 (f
k)=1 for k=1,,K. (normalization)
In matrix notation
•Cov[r,r‘] = 
k
k’
k
2
(f
k) + D,
–where 
k = (
1k,…,
Jk).
• = B B’ + D,
–where B
jk
=
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)

17. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
……Unobservable Factors…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 ¸…¸
lambda
J ¸ 0 and non-zero vectors y
1, …, y
J such that
y
j is an eigenvector of A assoc. w/ eigenvalue 
j, that is A y
j = 
j y
j
 
j
J
y
i
j y
i
j’ = 0 for j  j’
 
j
J
y
i
j
y
i
j
= 1
rank (A) = number of non-zero  ‘s
The y
j ‘s are unique (except for sign) if the 
i ‘s are distinct
•Let Y be the matrix with columns (y
1,…,y
J), and
let  the diagonal matrix with entries 
i then

18. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
……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
,…,
k
and associated
eigenvectors y
1,…,y
K
Y
K
the matrix with columns given by y
1
,…,y
K

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 
k is small for k>K.

19. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Why more than ONE mimicking Why more than ONE mimicking
portfolio?portfolio?
•Mimic (un)observable factors with portfolios
[Projection of factor on asset span]
•Isn’t 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)

20. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
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 Bounds Factor 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
Multi-period outlookMulti-period outlook
•Empirical Factor Pricing ModelsEmpirical Factor Pricing Models
Arbitrage Pricing Theory (APT) FactorsArbitrage Pricing Theory (APT) Factors
The Fama-French Factor Model + MomentumThe Fama-French Factor Model + Momentum
Factor Models from the StreetFactor Models from the Street
•Salomon Smith Barney’s and Morgan Stanley’s ModelSalomon Smith Barney’s and Morgan Stanley’s Model

21. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
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 inflation
5.Expected inflation (less important)
Note: The factors replicate market portfolio.

22. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Fama-MacBeth 2 Stage MethodFama-MacBeth 2 Stage Method
•Stage 1: Use time series data to obtain estimates for
each individual stock’s 
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 to
estimate SML
b=market risk premium

23. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
CAPM CAPM esting Fama French (1992)esting Fama 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
s
i
z
e

24. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Book to Market and SizeBook to Market and Size

25. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Fama French Three Factor ModelFama French Three Factor Model
•Form 2x3 portfolios
Size factor (SMB)
•Return of small minus big
Book/Market factor (HML)
•Return of high 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 R
2
.
s
i
z
e
book/market

26. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Fama French Three Factor ModelFama French Three Factor Model
•Form 2x3 portfolios
Size factor (SMB)
•Return of small minus big
Book/Market factor (HML)
•Return of high 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 R
2
.
s
i
z
e
book/market

27. 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%
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1010998877665544332211
High Book/Market Low Book/MarketHigh Book/Market Low Book/Market

28. Book to Market Equity of Portfolios Ranked by Beta Book 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
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29. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
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

30. Monthly Difference Between Winner and Monthly Difference Between Winner and
Loser Portfolios at Announcement Dates Loser Portfolios at Announcement Dates
11335577991113151719212325272929
313335
Months Following 6 Month Performance PeriodMonths Following 6 Month Performance Period
M
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n
d

-1.5%-1.5%
-1.0%-1.0%
-0.5%-0.5%
0.0%0.0%
0.5%0.5%
1.0%1.0%
L
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31. -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
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Cumulative Difference Between Winner andCumulative Difference Between Winner and
Loser Portfolios at Announcement DatesLoser Portfolios at Announcement Dates
11335577991111131315151717191921212323252527272929313133333535

32. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Morgan Stanley’s Macro Proxy ModelMorgan Stanley’s Macro Proxy Model
•Factors
GDP growth
Long-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 factors
Stage 2: Create long-short portfolios of most and least sensitive
stocks [5 quintiles]
•Macro Proxy return predicts macro factor

33. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
Salomon Smith Barney Factor ModelSalomon Smith Barney Factor Model
•Factors
Market trend (drift)
Economic growth
Credit quality
Interest rates
Inflation shocks
Small cap premium

34. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
1930’s40’s 50’s60’s 70’s 80’s 90’sbeyond
The Old Finance
Modern Finance
Modern FinanceModern Finance
Theme: Valuation Based on Rational Economic Behavior
Paradigms: Optimization Irrelevance CAPM EMH
(Markowitz) (Modigliani & Miller) (Sharpe, Lintner & Mossen) (Fama)
Foundation:Financial Economics
Haugen’s view: The Evolution of Academic FinanceHaugen’s view: The Evolution of Academic Finance

35. Fin 501: Asset PricingFin 501: Asset Pricing
17:1617:16 Lecture 08 Lecture 08 Factor PricingFactor Pricing
1930’s40’s 50’s60’s 70’s 80’s 90’sbeyond
The Old Finance
Modern Finance
The New Finance
The New FinanceThe New Finance
Theme: Inefficient Markets
Paradigms:Inductive ad hoc Factor Models Behavioral Models
Expected Return Risk

Foundation:Statistics, Econometrics, and Psychology
Haugen’s view: The Evolution of Academic FinanceHaugen’s view: The Evolution of Academic Finance