4106 Advanced Investment Management
Asset Pricing Models
session 3
Andrei Simonov
Asset Pricing Models
1
3/23/2016
Agenda
CAPM & C-CAPM.
 Testing CAPM(s)
 Fama & French evidence.
 APT & multifactor models

Asset Pricing Models
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Assumptions:
Single holding period
 Investors are risk-averse
 Investors are ”small”
 The information about asset payoffs is
common knowledge
 Assets are in unlimited supply
 Assets are perfectly divisible
 No transaction cost
 Wealth W is invested in assets

Asset Pricing Models
3
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Reminder on Optimal diversification at
individual investor level: condition of
optimality
How can you tell whether a portfolio p is well
diversified or efficient?
 For each security i, E(Ri) - r must be lined up
with cov(Ri,Rp) or, equivalently, with:
E(Ri) - r
i = cov(Ri,Rp)/var(Rp)

i
Asset Pricing Models
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Market level: syllogism
port 1
All weighted averages of E(Ri) - r
avg 1+2
efficient portfolios
port 2
are efficient
 Assume each person holds an efficient cov(Ri,Rp)
portfolio
 At equilibrium, the market portfolio, m, is
an average of individually held portfolios
 Therefore, the market portfolio must be
efficient

Asset Pricing Models
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Market level: Math (1)
Conditions of optimality (Remember Microeconomics?)

~ ~
W  W0 1  rf    j w j ~
rj  rf ,
~
~ ~
~
~ ~
~


E
u
(
W
)

E
u
'
(
W
)
r

r

0
,

j

E
u
'
(
W
)
E
r

r


Cov
u
'
(
W
), rj
j
f
j
f
max
a j 




  
~
~ bi ~ 2  ai
~
ui (W )  aiW  W    E Wi
2
 bi


i
i
i
j
 
 a
~
E~
rj  rf    i  E M
 i bi


1






  


1



 Wm 0Cov~
rM , ~
rj     i1  Wm 0Cov~
rM , ~
rj 
 i


Risk Aversion
Asset Pricing Models

 ~
~
 E rj  rf  Cov Wi , ~
rj

 ~
 ai
~  ~
 E rj  rf     E M  E rj  rf

 i bi

 Cov ij wij ~
rj  rf , ~
rj  Wm 0Cov~
rM , ~
rj 
   
  
 ai
~

LHS


E
W
i
i  b
i
 i
~
RHS 
Cov W , ~
r

6
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Market level: Math (2)

But Market Portfolio is a legidimate asset,
1



1
~
~
E rM  rf    i  Wm 0Var rM  
Cov~
rM , ~
rj  ~

 i

~

E
r

r

E rM  rf

j
f
1
~
Var rM 

1 

~
~
~
E rj  rf    i  Wm 0CovrM , rj 

 i














~
~
E r  r  E r r
 j 
f
j  M
f 

Market price of risk:
 
 


 
 
E~
r r E ~
r r
M
f
M
f
~
,
,
E
r r
~
~
M
f
Var r
Std r
M
M
Asset Pricing Models
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Market equilibrium

For each security i, E(Ri) - r must be lined
up with cov(Ri,Rm) or, equivalently, with: i
= cov(Ri,Rm)/var(Rm)
E(Ri) - r
m = m var(Rm) = E(Rm) - r
i

CAPM can be extended to the case in which
there exists no risk-less asset.
Asset Pricing Models
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Extensions
No risk-free assets
 No short sales
 Different lending and borrowing rate
 Restricted opportunity sets (Hietala, JF89)
 Personal taxes
 Multi-period extensions
 Liquidity
The simple form of CAPM is rather robust. Modification
of basic assumptions leads to changes in existing terms
and appearance of new terms (”induced” factors)
However, sumultaneous modification of multiple
assumptions leads to SERIOUS departure from
standard CAPM.

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What is expected risk premium?
Expected risk premium= Var(rM)/E[W/]
 Plays central role in any discussion about
the market
 What is that? How to measure it? What
will it tell us about mankind and economy
(or Asset Pricing Model)?

– Historical perspective
– Equilibrium perspective
Asset Pricing Models
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Asset Classes Returns: US History
Asset Pricing Models
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Asset Classes Returns: Swedish
History
1 SEK invested in 1900
1000000
100000
10000
DMS Global Sweden Bill TR
DMS Global Sweden Inflation
DMS Global Sweden Equity TR
DMS Global Sweden Bond TR
1000
100
10
1
1900
0.1
Asset Pricing Models
1920
1940
1960
1980
2000
12
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Goetzmann&Jorion: International Evidence
Asset Pricing Models
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Goetzmann&Jorion: International Evidence
Median Market RR 0.75%
GDP-weighted RR 4%
Asset Pricing Models
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Estimates from historical data
Ibbotson & Sinquefield (76): Real ERP= 5%
 Ibbotson & Chen (2000)
4%
 Fama & French (2002) longer period 4.4%
 Jagannathan, McGrattan,Scherbina (2000)

– 1926-70 ERP=7%
– 1971-99 ERP=0.7%
Asset Pricing Models
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Equilibrium Approach: C-CAPM
Individuals have preferences over consumption
C described by CRRA u=-C1-g
 Certainty case: marginal utility of consumption
today =discounted marginal utility of
consumption tomorrow times teturn of asset ri:
C-gt[(1+ri)/(1+r)] C-gt+1
 In case of uncertainty
C-gtE[(1+ri)/(1+r)] C-gt+1
 Introducing consumption growth
g=C(t+1)/C(t)-1

Asset Pricing Models
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C-CAPM(2)
E 1  r 1  g    1  r
g
i
Taylor expansion : 1  ri 1  g   1  ri  gg  ggri 
g
g (1  g )
2
g2
Applying Expectatio ns operator :
r  Eri   gE g   g E ri E g   covri , g  
g (1  g )
Eg   Varg 
2
2
If t is small, then quadratic terms can be disregarde d :
r  Eri   gE g   g covri , g  
g (1  g )
Varg 
2
This eq. should be true for both risky and riskless assets :
g (1  g )
r  rf  gE g  
Varg   Eri   rf  g covri , g 
2
Asset Pricing Models
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C-CAPM: Equity premium puzzle

Mehra&Prescott(85); Mankiv &Zeldes(91):
–
–
–
–
1890-1979[1948-88]: Risk premium=0.06 [.08]
Std of consumption growth =0.036 [0.014]
Std of market returns=0.167 [0.14]
Correlation between consumption growth and
market returns = 0.40 [0.45]
– 0.06=g*0.40*0.167*.036 => g=25 [90]
Asset Pricing Models
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Equity premium puzzle
Gamble: take 20% paycut if state of the
world is ”bad” (prob=1/2) and stay at your
current salary in good state, or agree on
permanent cut of X%:
 0.5*(0.81g+1)=x1g

Asset Pricing Models
19
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Equity premium puzzle
Gamble: take 20% paycut if state of the
world is ”bad” (prob=1/2) and stay at your
current salary in good state, or agree on
permanent cut of X%:
 0.5*(0.81g+1)=x1g
 If g=25 then x=17.7%
 Realistic estimate for gamma=3

Asset Pricing Models
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How can we solve it?

Habit formation u=-(Ct-Ct-1)1-g
– Increases demand for bonds, lower Rf
– “Keeping up with the Joneses”: instead C(t-1)
there is AVERAGE consumption in the
reference group.
Idiosynchratic labor risk
 Disaster states and survivorship bias.
 Liquidity premium
 Limited Participation

Asset Pricing Models
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Is CAPM right?

Content: cross-sectional relationship
– when comparing securities to each other, linear,
positive-slope relationship of mean excess return
(risk premium) with beta
– zero intercept
– no variable, other than beta, matters as a measure of
risk
Asset Pricing Models
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Is CAPM right?


How can you tell?
Two-pass approach
– for each security, measurement of mean excess return
and beta using history of returns (time series)
– relate mean excess return to beta (cross section)

First pass has no economic meaning, just a
measurement. Second pass is embodiment of
CAPM.
Asset Pricing Models
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Example
Excess returns %
Market
Year
index
A
B
C
D
E
F
G
H
I
1
2
3
4
5
6
7
8
9
10
11
12
29.65%
-11.91%
14.73%
27.68%
5.18%
25.97%
10.64%
1.02%
18.82%
23.92%
-41.61%
-6.64%
33.88%
-49.87%
65.14%
14.46%
15.67%
-32.17%
-31.55%
-23.79%
-4.59%
-8.03%
78.22%
4.75%
-25.20%
24.70%
-25.04%
-38.64%
61.93%
44.94%
-74.65%
47.02%
28.69%
48.61%
-85.02%
42.95%
36.48%
-25.11%
18.91%
-23.31%
63.95%
-19.56%
50.18%
-42.28%
-0.54%
23.65%
-0.79%
-48.60%
42.89%
-54.39%
-39.86%
-0.72%
-32.82%
69.42%
74.52%
28.61%
2.32%
26.26%
-68.70%
26.27%
-39.89%
44.92%
-3.91%
-3.21%
44.26%
90.43%
15.38%
-17.64%
42.36%
-3.65%
-85.71%
13.24%
39.67%
-54.33%
-5.69%
92.39%
-42.96%
76.72%
21.95%
28.83%
18.93%
23.31%
-45.64%
-34.34%
74.57%
-79.76%
26.73%
-3.82%
101.67%
1.72%
-43.95%
98.01%
-2.45%
15.36%
2.27%
-54.47%
40.22%
-71.58%
14.49%
13.74%
24.24%
77.22%
-13.40%
28.12%
37.65%
80.59%
-72.47%
-1.50%
90.19%
-26.64%
18.14%
0.09%
8.98%
72.38%
28.95%
39.41%
94.67%
52.51%
-80.26%
-24.46%
First pass
Mean
8.12%
5.18%
4.19%
2.75%
6.15%
8.05%
9.90%
11.32%
13.11%
22.83%
1
-0.470
0.594
0.417
1.379
0.901
1.777
0.663
1.911
2.083
8.997%
-0.631%
-0.636%
-5.051%
0.729%
-4.528%
5.936%
-2.410%
5.918%

a
Asset Pricing Models
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First pass: Security A
100.00%
= -0.4704
a= 8.997%
80.00%
excess return on security A
60.00%
40.00%
20.00%
-50.00%
-40.00%
-30.00%
-20.00%
-10.00%
0.00%
0.00%
10.00%
20.00%
30.00%
40.00%
-20.00%
-40.00%
-60.00%
excess return on market
Asset Pricing Models
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Second pass: CAPM line
25.00%
Intercept = 3.82%
Slope m = 5.21%
Adjusted R2 = 0.44
mean excess return of each security
20.00%
I
15.00%
H
aG
10.00%
G
F
E
market
A
D
5.00%
B
C
0.00%
-1
-0.5
0
0.5
1
1.5
2
2.5
beta of each security
Asset Pricing Models
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Discussion: CAPM may not be
testable
1. the “market” is not
observable (Roll
critique)
2. should use timevarying version, based
on the information set
of the investors. The
latter is not observable
(Hansen and Richard
critique).
30.00%
E(Ri) - r
date 2
20.00%
B
15.00%
A
B
B
10.00%
date 1
A
5.00%
i
0.00%
0
Asset Pricing Models
A
25.00%
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
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There are deviations from CAPM
(or a’s)
Fama and French (1992) investigate 100 NYSE
portfolios for the period 1963-1990
 The portfolios are grouped into 10 size classes
and 10 beta classes
 They find that return differential (risk premium)
on  is negative (and non significant)
 whereas return differential on size is large and
significant.

Asset Pricing Models
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“beta is dead ?”
Average Returns (in %/year) for portfolios
Sorted on Size and Betas
All firms Low Beta High Beta
All firms
15.0%
16.1%
13.7%
Small cap
18.2%
20.5%
17.0%
Large cap
10.7%
12.1%
6.7%
Asset Pricing Models
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Book/market also
Average Returns (in %/year) for portfolios
Sorted on Size and B/M Ratios
All firms Low B/M High B/M
(“Growth”) (“Value”)
All firms
14.8%
7.7%
19.6%
Small cap
17.6%
8.4%
23.0%
Large cap
10.7%
11.2%
14.2%
Asset Pricing Models
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Recent thinking





Question: is Fama-French evidence reliable?
Return differential may come in “waves”.
Perhaps, CAPM is right at each point in time
But CAPM line moves about
When “indicator variables” are used to track these
changes over time (such as variables we listed in
lecture on TAA), size and B/M no longer show up
in CAPM
So, these variables were showing up in the FamaFrench analysis, not because CAPM was wrong,
but only because movements in the line had not
been properly accounted for
Asset Pricing Models
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Other criticism of CAPM

No account of re-investment risk (multiperiod aspects)
– inter-temporal hedging

No account of investors’ non traded wealth
(similar to Roll critique)
– when human capital included, revised CAPM
holds up better
Asset Pricing Models
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Conclusion on CAPM
If CAPM were right every mean-variance
investor could just hold the market portfolio
(“index fund”), adjusting the level of risk by
mixing it with riskless asset.
 If CAPM is not right, there is room for “tilted”
index funds. See Dimensional Fund Advisors
case.
 If CAPM is not right is it because risk is multidimensional?

Asset Pricing Models
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Arbitrage Pricing Theory



Large number of securities, finite number of factors
Residuals become irrelevant. Only exposures b to
common factors matter for pricing
Dichotomy of risk variables:
– some (factors, in finite number) affect all securities
– others (residuals, in large number) affect only one security
each

Pricing equation: there must exist premia :

Otherwise, could work out “approximate” arbitrage
E ( Ri )  0  b1,i 1b2,i  2..
Asset Pricing Models
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Math: Derivation of APT(1)


Main requirement: No possibility to create something out of
nothing
Certainty case: n+1 assets, K factors, n>K, ”0” asset is risk-free
one
K
~
rj  a j  k 1  jk Fk , j  1,2,..., n

Consider portfolio of yj0=(1-Skjk) units of riskless asset and
yjk=jk of risky asset
K
K
~
rp  1  k 1  jk rf  k 1  jk Fk
Looks similar to returns of asset j! Let us exploit it by buying $1
worth of this portfolio and shorting $1 of asset j. No-arbitrage
requirement tells that the return on such portfolio should be 0:


1  



K
K
K
~

 a j  1  k 1  jk rf
Asset Pricing Models
~
 r  k 1  jk Fk  a j  k 1  jk Fk  0
k 1 jk f
K
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Math: Derivation of APT(2)

Uncertainty case:
~
rj  a j  k 1  jk Fk   j , j  1,2,..., n
K

For any small e, there exists at most N assets, N<n,
for which arbitrage condition is violated:


a j  1  k 1  jk rf  e , j  1,2,..., N (n)

K
The no-arbitrage condition requires that N/n0 as
n. Moreover, as Dybvig(1983) shows, the error
also decreases as the number of players in economy
increases
Asset Pricing Models
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APT & Ideas that have survived

A statistical concept: exposure
– beta is an “exposure” to risk


need to keep track of exposures, when constructing a portfolio
use of “exposures” to classify securities and systematize portfolio
construction process
– beta measures each asset’s contribution to total portfolio risk

idea useful for risk management: need to develop accounting of risk (or
breakdown of risks)
– but beta needs a generalization: find more common factors

A pricing concept:
– In pricing, only common risk factors matter. Other risks can be
diversified away.
Asset Pricing Models
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Statistical models: streamlining
the investment process

Covariance matrix is huge (many entries)
– Leads to imprecisely computed portfolios

Let us impose structure on the matrix
– We may have 10000 securities to choose from
– In fact, there may be only 20 basic sources of
risks
– A particular security can be seen as a portfolio
of these basic risks and must be identified as
such
Asset Pricing Models
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Example: one factor model

One-factor model ( b’s called “loadings” or
“exposures”):
Ri,t  ai  bi  Ft  e i,t
where residuals ei,t are independent
 Then:
cov( Ri , R j )  bi  b j  var F ; i  j
var( Ri )  bi 2  var F   vare i 
Asset Pricing Models
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Example: construct common factor
10000
Small stocks
Large stocks
1000
Long-term T-Bonds
Intermediate-term T-Bonds
T-Bills
Inflation
100
Re-scaled First factor
10
1
1920
1930
1940
1950
1960
1970
1980
1990
2000
0.1
Asset Pricing Models
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How to build a factor?
Eigenvectors of covariance matrix represent the
dimensions of risk.
 Procedure: solve det(S-I)=0
 If there dimension of S is k (=4 in our case), then
there are up to k unique solutions. Rank them in
descending order: 1>2>3>4
 In our case 0.2>>0.014>>0.007>>0.001
 Find eigenvector  that is the solution of S1
 In our case it is  =(0.92,0.40,0.02,0.01) those are
non-normalized weights!

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Example: compare correlations
Correlation original
Small stocks - Tbills
Large stocks - Tbills
Long-term T-Bonds - Tbills
Intermediate-term T-Bonds - Tbills
Small stock s - Tbills Large stock s - Tbills Long-term T-Bonds - Tbills Intermediate-term T-Bonds - Tbills
1.0000000
0.8091950
1.0000000
0.1052384
0.2364890
1.0000000
0.0885737
0.1944899
0.7678739
1.0000000
Correlation reconstructed
Small stocks - Tbills
Large stocks - Tbills
Long-term T-Bonds - Tbills
Intermediate-term T-Bonds - Tbills
Small stock s - Tbills Large stock s - Tbills Long-term T-Bonds - Tbills Intermediate-term T-Bonds - Tbills
1.0000000
0.8644608
1.0000000
0.1378009
0.1206442
1.0000000
0.1155091
0.1011278
0.0161205
1.0000000
Variances of securities
0.169297376
Asset Pricing Models
0.043319362
0.006016743
0.003450979
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Estimating beta of a security

Use model:
cov( Ri , Rm )
i 
 bi   F  e i
var( Rm )
negligible
original betas
beta of factor
loadings
beta from model
IntermediateSmall stocks Large stocks Long-term Tterm T-Bonds Tbills
Tbills
Bonds - Tbills
Tbills
1.3587
0.6413
0.0409
0.0257
x
=
0.9954
1.3798
0.9954
0.6111
0.9954
0.0363
0.9954
0.0230
1.3735
0.6083
0.0361
0.0229
(example: “market” = 50% large stocks + 50% small stocks)
Asset Pricing Models
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Extension: multi-factor
statistical models

Multi-factor model:
Ri,t  ai  b1,i  F1,t  b2,i  F2,t  ..  e i,t
When I hold security i, I am truly holding
b1,i of risk #1, b2,i of risk #2 etc..
 Then:

cov( Ri , Rm )
i 
 b1,i   F1  b2,i   F2  ..  e i
var( Rm )
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Estimating Statistical Factor
Models: three approaches
Capture the way in which securities returns
move together:
 Factor analysis
Factor analysis constructs a set of abstract factors
that best explain the estimated covariances
 Throws no light on underlying economic
determinants of the covariances

Use of macroeconomic variables
 Use of firm specific variables

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Estimating Statistical Factor
Models:macroeconomic variables

Business cycle risk
– unanticipated growth in industrial production

Confidence risk
– default spread (Baa - Aaa) which is a proxy for
unanticipated changes in risk premia

Term premium risk
– return on long bonds minus short bonds, which is a
proxy for unanticipated shifts in slope of yield curve


Other: Oil prices
Get exposures (loadings) of each stock by multiple
regression
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Estimating Statistical Factor Models:
use of firm-specific information

Form factor-mimicking portfolios which capture
factors:
– Market: RM - r
– B/M: High-minus-low (HML): RHML = RH - RL
– Size: Small-minus-big (SMB): RSMB = RS - RB

Estimate exposures by regression
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Example calculation of multifactor statistical model
Excess returns %
Market
index
Year
1
2
3
4
5
6
7
8
9
10
11
12
First pass (time series)
Mean
Exposure to market risk
Exposure to I - A risk
29.65%
-11.91%
14.73%
27.68%
5.18%
25.97%
10.64%
1.02%
18.82%
23.92%
-41.61%
-6.64%
8.12%
1
0
Asset Pricing Models
I - A risk
56.31%
23.23%
-47.00%
-14.37%
-6.69%
104.55%
60.50%
63.20%
99.26%
60.54%
-158.48%
-29.21%
A
B
33.88%
-49.87%
65.14%
14.46%
15.67%
-32.17%
-31.55%
-23.79%
-4.59%
-8.03%
78.22%
4.75%
C
-25.20%
24.70%
-25.04%
-38.64%
61.93%
44.94%
-74.65%
47.02%
28.69%
48.61%
-85.02%
42.95%
D
36.48%
-25.11%
18.91%
-23.31%
63.95%
-19.56%
50.18%
-42.28%
-0.54%
23.65%
-0.79%
-48.60%
E
42.89%
-54.39%
-39.86%
-0.72%
-32.82%
69.42%
74.52%
28.61%
2.32%
26.26%
-68.70%
26.27%
F
-39.89%
44.92%
-3.91%
-3.21%
44.26%
90.43%
15.38%
-17.64%
42.36%
-3.65%
-85.71%
13.24%
G
39.67%
-54.33%
-5.69%
92.39%
-42.96%
76.72%
21.95%
28.83%
18.93%
23.31%
-45.64%
-34.34%
74.57%
-79.76%
26.73%
-3.82%
101.67%
1.72%
-43.95%
98.01%
-2.45%
15.36%
2.27%
-54.47%
H
I
40.22%
-71.58%
14.49%
13.74%
24.24%
77.22%
-13.40%
28.12%
37.65%
80.59%
-72.47%
-1.50%
90.19%
-26.64%
18.14%
0.09%
8.98%
72.38%
28.95%
39.41%
94.67%
52.51%
-80.26%
-24.46%
17.65%
5.18%
4.19%
2.75%
6.15%
8.05%
9.90%
11.32%
13.11%
22.83%
0 1.1033751 -0.57441 0.750926 0.449096 -0.18206 1.833844 1.034249 1.662516 1.103375
1 -0.616466 0.457581 -0.13089 0.364342 0.424359 -0.02226 -0.14529 0.097397 0.383534
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BARRA and other “quant” shops

Similar approach:
– measure exposures onto any number of risk
categories (country, industry, small vs. big etc..).
– get value of factor return for each category, each
time period, by comparing across firms
– interpret this month’s factor return as a way of
pinpointing the category of firms that are
currently most profitable

Up to 80 factors!
Asset Pricing Models
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Some of
BARRA
variables
(due to
RosenbergMcKibben
73)
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Predicted
BETA
look at
www.barra.com
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Statistical factor models in the standard
CAPM vs. Multi-factor pricing models
Multi-factor statistical models can be used
to estimate parameters of the standard
CAPM. E.g., securities’ betas from
exposures times factor betas
 One may still use standard CAPM: still
one risk premium (single-factor pricing
model)
 Opposite: several risk premia. “Multifactor pricing models”

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Multi-factor pricing models:
state-dependent preferences
Investor cares about portfolio variance, and
also about performance in a recession:
– investors try to buy stocks that do well in a
recession
– this drives down expected return of those stocks
beyond the market beta effect (recession < 0):
E Ri   r  m  bi,m  recession  bi,recession
– Might be a way to price political risk
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Numerical example
Exposures obtained by multiple regression
of individual security on the factors, as in
statistical models
 Prices of risk obtained by second-pass
cross-sectional regression like in CAPM.

Asset Pricing Models
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Example calculation: multifactor pricing model
Excess returns %
Market
index
Year
First pass (time series) cf. slide 15
Mean
8.12%
Exposure to market risk
1
Exposure to I - A risk
0
I - A risk
A
B
C
D
E
F
G
H
I
17.65%
5.18%
4.19%
2.75%
6.15%
8.05%
9.90%
11.32%
13.11%
22.83%
0 1.1033751 -0.57441 0.750926 0.449096 -0.18206 1.833844 1.034249 1.662516 1.103375
1 -0.616466 0.457581 -0.13089 0.364342 0.424359 -0.02226 -0.14529 0.097397 0.383534
Second pass (cross-section)
Adjusted
R Square
0.50857
Intercept lambda /mkt
lambda /I-A
3.07%
6.29%
13.26%
Line
9.36%
Compare to:
CAPM line
9.03%
Asset Pricing Models
16.33%
1.84%
5.53%
6.06%
10.73%
7.55%
14.31%
7.65%
14.82%
15.09%
1.37%
6.91%
5.99%
11.00%
8.51%
13.08%
7.27%
13.77%
14.67%
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Example: E(R) on security B
Risk factor
Exposures b
Contribution
Prices of risk 
Intercept (should be
3.07%
zero really)
Market risk
-0.57441 
6.29% =
-3.61%
I - A risk
0.457581 
13.26% =
6.07%
Expected excess return
=
5.53%
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“Other” justification for this sort of
pricing: Arbitrage Pricing Theory



Large number of securities, finite number of factors
Residuals become irrelevant. Only exposures b to
common factors matter for pricing
Dichotomy of risk variables:
– some (factors, in finite number) affect all securities
– others (residuals, in large number) affect only one security
each

Pricing equation: there must exist premia :

Otherwise, could work out “approximate” arbitrage
E ( Ri )  0  b1,i 1b2,i  2..
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Conclusions on Asset Pricing
Models:



Securities are analyzed as portfolios of underlying risks
Make clear distinction between risk and exposure to
risk!
Less restrictive approach to asset pricing: several
dimensions of risk are priced
– general multi-factor model based on incompletely specified
investor behavior
– APT relies on absence of “approximate” arbitrage
Asset Pricing Models
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Practicality: Estimation Risk

Óptimization results are usually
suffering from:
– Huge short positions in many assets in
no-constraint case.
– “Corner” solutions with zero positions in
may assets if constraints are imposed
(most portfolios have < 3 assets).
– Huge positions in obscure markets with
small cap
– Large shifts in positions when exp.
returns or covariances changes just a
bit…

All of those are coming from one
common cause: difficulties in
estimation of expected returns.
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Another example (GS 2003):

Here are forecasts
given by “Wall Street
Protagonist”
(Columns 1 and 2)
and set of portfolio
weights that are
generated by it.
RETURN STD DEV
Japanese Gov Bonds
4.7%
4.2%
EU Gov Bonds
5.1%
3.6%
US Gov Bonds
5.2%
4.6%
US Equities
5.4%
15.5%
Global Fixed income
6.0%
3.6%
EU Equities
6.1%
16.6%
US Inv Grade Corp Bonds 6.3%
5.4%
EAFE
8.0%
15.3%
Hedge Fund Portfolio
8.0%
5.2%
US High Yield
8.9%
7.3%
Private Equity
9.0%
28.9%
Emerging Debt
9.0%
17.6%
REITs
9.0%
13.0%
Japanese Equity
9.5%
19.6%
Emerging Equities
11.8%
23.4%
Portfolio ER
Portfolio Volatility
Asset Pricing Models
Unconstrained No Short
weights
Sales
-2.02
0.00
-3.21
0.00
-4.84
0.00
-0.11
0.00
14.93
0.00
-2.58
0.00
-3.86
0.00
3.14
0.00
0.58
0.55
-0.10
0.36
0.01
0.00
-0.29
0.00
0.04
0.08
-0.72
0.01
0.03
0.00
4.90%
18.20%
5.10%
8.40%
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Equilibrium and individual
asset’ expected return
From previous section one can expect that
ERP is between 4 and 6% and is fairly
stable with time
 One can make forecast for individual
assets that are different from long term
premium.
 But by forecasting one asset class, we are
implicitly making forecast for other asset
classes as well.

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Practicality: How to express
views?



Method is due to Black & Litterman (Goldman
Sachs). The core themes: equilibrium returns and
views.
Investors normally have views/preferences. They are
NOT incorporated into optimization process.
Views=mathematically expressed preferences of
individual investors.
Step Action
1
2
3
4
5
6
Calculate equilibrium returns
Define weight for news
Set target tracking error
Set target market exposure
Get portfolio weight
Examine risk distribution
Asset Pricing Models
Purpose
Set neutral reference point
Dampen impact of aggressive news
Control risk wrt benchmark
control directional effect
Find allocation that maximize performance
Is risk diversifies?
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Equilibrium optimal portfolio



Imagine that the investor thin
that US is still in recession.
Thus, stocks will perform badly,
and bonds will perform OK.
Mathematically, it is equivalent
to assuming that bonds will go
up 0.8%, and stocks will drop
2.5%
Result: see Table 8:
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Updating of discrete
probabilities
1. We have a probability estimate for event H:
prior probability P(H)
2. New information D is gained
3. Update the estimate using Bayes’ theorem:
posterior probability P(H|D)
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The Bayes’ theorem
The updating is done using the Bayes’
theorem:
P( D | H ) P( H )
P( H | D) 
P( D)
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Example: Using Bayes’
theorem
1,5 % of the population suffer from schizophrenia

P(S) = 0.015 (prior probability)
Brain atrophy is found in
– 30 % of the schizophrenic  P(A|S) = 0.3
– 2 % of normal people  P(A|S) = 0.02
If a person has brain atrophy, the probability that he is schizophrenic
(posterior probability) is:
P( A | S ) P( S )
P( S | A) 
P( A | S ) P( S )  P( A | S ) P( S )
0.3  0.015

 0.186
0.3  0.015  0.02  0.985
Picture: Clemen s. 250
Figure: Posterior probability with different
prior probabilities.
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Updating of continuous
distributions
Choose a theoretical distribution, P(X=x|),
for the physical process of interest.
Assess uncertainty about parameter :
prior distribution, f()
Note:
Uncertainty about
X has two parts:
Observe data x1
Update using Bayes’ theorem:
posterior distribution of , f(|x1)
Asset Pricing Models
1. Due to the
process itself,
P(X=x|).
2. Uncertainty
about , f(), later
updated to f(|x1).
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Updating of continuous
distributions
Bayes’ theorem for continuous  :
f ( | x1 ) 


f ( x1 |  ) f ( )
f ( x1 |  ) f ( )d



f(x1|) is called the likelihood function of  with a given
observed data x1.
In most cases the posterior distribution can not be
calculated analytically, but must be solved numerically.
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Normal distribution
1. The physical process of interest is normal distributed:
X ~ N(, 2)
( is assumed to be known)
2. Prior distribution for :
 ~ N(m0, 20)
(notation: 20 = 2 / n0)
3. Observe a sample of the physical process:
– sample size: n1
– sample mean: x1
4. The posterior distribution, calculated using the Bayes’
theorem, gets reduced to:
m0 2 n1  x1 02 n0 m0  n1 x1
m* 

2
2
 ~ N(m*, 2*), where
 n1   0
n0  n1
2 2

n1
2
2
0
 * 2

2
 n1   0 n0  n1
Asset Pricing Models
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Expressing Views




Thus, one need to specify set of “views” and “precisions” of
views for each asset f(x|).
“No views” is equivalent to having x For this case
posterior=prior.
Model will deviate further for assets where views are stronger.
All assets are affected:
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Further risk control & results:




Minimize tracking error
Mkt. Exposure of the portfolio () (neutral should be 1)
Look at diversification (are all eggs in one basket?)
Results (according to GS, 95-97): (+103 bp, +83 bp, -26bp)
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Other uses:



Black-Litterman model is essentially Tactical Asset
Allocation model (provided that algorithm of selecting
“views” is specified).
But it can be used effectively in updating priors on the
distribution of the signals.
It can be used to bring in new asset classes for which
the recorded history is short or unreliable (venture
capital funds, hedge funds, emerging markets, etc.)
Asset Pricing Models
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