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1Financial Markets And Empirical RegularitiesAn Introduction to Financial Econometrics
SAMSI Workshop 11/18/05Mike Aguilar UNC at Chapel Hill
www.unc.edu/~maguilar
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2Outline
I. Historical Perspective on Asset Prices
II. Predictability of Asset Returns
III. Asset Pricing Models
IV. Volatility Models
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3Essential Sources
Campbell, Lo, & MacKinlay, The Econometrics of Financial Markets
Cochrane, Asset Pricing Huang & Litzenberger, Foundations for
Financial Economics Hamilton, Time Series Analysis Greene, Econometric Analysis
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4Types of Financial Assets:
Common: Stocks, Bonds, Commodities, Foreign Exchange,
Exotic: Derivatives; Options, Futures,
I. Historical Perspective on Asset Prices
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5I. Historical Perspective on Asset Prices
Source: Campbell Harvey
A History of Asset Prices
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6Definitions: Simple Gross Return:
Compound Gross Return:
Annualized Return:
Continuously Compounded
Return:
t. time at asset of price ==+
tt
tt PP
PR ,11
I. Historical Perspective on Asset Prices
)1(*...*)1(*)1(1)( 11 + +++=+ ktttt RRRkR
=
=
+
1
0
/11
0
11)1(k
jjt
kk
jjt Rk
R
1loglog)1log( =+= tttt PPRr
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7Stationary: The joint distribution between two returns
I. Historical Perspective on Asset Prices
Why Returns?..Because returns tend to be stationary.
htt xx , depends only on h and NOT on t.
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8I. Historical Perspective on Asset Prices
A History of Asset Returns
Source: Campbell Harvey
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9I. Historical Perspective on Asset Prices
Source: Campbell Harvey
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I. Historical Perspective on Asset Prices
Source: Campbell Harvey
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I. Historical Perspective on Asset Prices
Observation:
There exists a Risk / Return Tradeoff
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I. Historical Perspective on Asset Prices
Stats Reminder:Consider a random variable
Mean:
Variance:
Skew:
Kurtosis:
=
T
ttT x
1
1
=
T
ttT x
1
212 )( 3
1
1 )( 3 = T
ttT
xS
4
1
1 )( 4 =
T
ttT
xK
tx
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I. Historical Perspective on Asset Prices
What is the Distribution of Returns?
Normal?
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I. Historical Perspective on Asset Prices
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I. Historical Perspective on Asset Prices
Stylized Facts on the Distribution of Returns:
1. Index Volatility < Stock Volatility
2. Negative Skewness
3. Excess Kurtosis
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II. Predictability of Returns
Can We Predict Returns? Day Traders Say Yes
Efficient Market Hypothesis Says No
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II. Predictability of Returns
Efficient Market Hypothesis (EMH):
Fama (1970): A market in which prices always fully reflect available information is called efficient.
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II. Predictability of Returns
Efficient Market Hypothesis (EMH):
Fama (1970): A market in which prices always fully reflect available information is called efficient.
Malkiel (1992): the market is said to be efficient with respect to an information set
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II. Predictability of Returns
Probability Theory Reminder:
The Information Set tF is a sigma field said to contain all of the relevant information up to time t .
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II. Predictability of Returns
Probability Theory Reminder:
The Information Set tF is a sigma field said to contain all of the relevant information up to time t .
Martingale: ttt xFxE =+ ]|[ 1
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II. Predictability of Returns
EMH:Weak Form Efficiency:
Information Set: Assets own historyTest via Random Walk
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II. Predictability of Returns
EMH:Weak Form Efficiency:
Information Set: Assets own historyTest via Random Walk
Semistrong Efficiency: Information Set: Weak Form + Publicly available dataTest via Event Studies
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II. Predictability of Returns
EMH:Weak Form Efficiency:
Information Set: Assets own historyTest via Random Walk
Semistrong Efficiency: Information Set: Weak Form + Publicly available dataTest via Event Studies
Strong Form Efficiency: Information Set: Weak + Semi + Private InfoTest via Performance Evaluation
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II. Predictability of Returns
Joint hypothesis problem: The EMH cant be tested directly. Even if we reject the hypothesis of efficiency, this could either be because the market is truly inefficient, or because we have assumed an incorrect equilibrium model.
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II. Predictability of Returns
Implication of Weak Form EMH:
Expected Returns follow a martingale.
Random Walk Hypothesis
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II. Predictability of Returns
Random Walk 1: i.i.d Increments
Random Walk 2: Independent, Not Indentical
Random Walk 3: Uncorrelated Increments
ttt PP ++= 1
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II. Predictability of Returns
Time Series Econometrics Reminder: Auto Correlation
Correlation between two observations of the same series at different dates
)0()(
),(
k
xxCov ktt
(k) :ationAutoCorrel
(k) :anceAutoCovari
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II. Predictability of Returns
Time Series Econometrics Reminder: Auto Correlation
Correlation between two observations of the same series at different dates
=
+=
m
kkTk
ktt
TT
kxxCov
1
)(2)2(
)0()(
),(
mQ :Statistic Q PierceBox
(k) :ationAutoCorrel
(k) :anceAutoCovari
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II. Predictability of Returns
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III. Asset Pricing Models
Capital Asset Pricing Model (CAPM)Sharpe (1964) and Lintner (1965)
Consumption Based-CAPM
Inter-temporal-CAPM
Arbitrage Pricing Theory Ross (1976)
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III. Asset Pricing Models - CAPM
CAPM: Differences in excess returns across assets are due to differences in the riskiness of each asset.
Beta: Measure of riskiness
uRRRR fmfi += )()( :Model Market
)()](),[(
fm
fmfi
RRVarRRRRCov
= OLSTradeoff dRisk/Rewar linear a exists There
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CAPM Pricing Equation:
III. Asset Pricing Models - CAPM
)][(][ fmimfi RRERRE +=
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CAPM Pricing Equation:
Testable Implications:
1) Only beta is required to price assets.
2) There is a linear risk/return relationship.
III. Asset Pricing Models - CAPM
)][(][ fmimfi RRERRE +=
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Testing the CAPM:Fama & MacBeth (1973)
III. Asset Pricing Models - CAPM
.
,)( 32
210
ususRR iiimimfi
of deviation standard the is where
:regression sectional-cross the estimate period time each For 1: Step
++++=
3,2,1,0][ == kE kk thatsuch time over estimates parameter Aggregate:2 Step
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35
III. Asset Pricing Models - CAPM
estimated. be first must s' thebecause problem (EIV) variables-in-errors an is There
Fama & MacBeth contd
Fama & MacBeth solution: Sort stocks into portfolios.
Shanken solution: correct variance of estimators post estimation.
tradeoff. return-risk linear positive, a is there &assets price tonecessary factoronly the is
==== 0: 32100H
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III. Asset Pricing Models - CAPM
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III. Asset Pricing Models - CAPM
CAPM Anomalies: Small Firm effect (Keim `81)
P/E Ratio Effect (Ball `78 and Basu `83)
Book-to-Market Effect (Stattman `80 and Rosenberg `85)
Momentum Effect (Jegadeesh and Titman `93)
Dead? Beta Is
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III. Asset Pricing Models - CAPM
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III. Asset Pricing Models Consumption Based Models
01])([
)(
1)(
1
0 01
0
1
1 =++
=
+
= =
=
+
tC
Ct
N
it
N
iititititt
tt
t
tt
RE
WQPQPCCUMax
CCU
t
t
to subject
Income Labor Realeconomy the in assets of Number owned assets of Amount
asset of ValueonSubstituti of Rate ralIntertempo :
AversionRisk of tCoefficien : nConsumptio
FunctionUtility
::::
::
WNQP
CU
Consumption Based Asset Pricing Model
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III. Asset Pricing Models Consumption Based Models
eZmz
Re
T
t
tCC
t tt
')(:
1])([
],[
1
1
=
==
t time at sinstrument of Vector
GMM Estimation
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41
III. Asset Pricing Models Consumption Based Models
)dim( p & )dim( q ;
where Min
t time at sinstrument of Vector
zeTJ
zzeeS
SWWmmJ
eZmz
Re
pqt
T
tttttT
t
T
t
tCC
t tt
===
===
==
=
21
1
1
1
][
)()'(
')(:
])([
],[
1
GMM Estimation
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IV. Volatility Models
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IV. Volatility Models
ARCHAutoregressive Conditional HeteroscedasticityEngle (`82) & French, Schwert, Stambaugh (`87)
=
=
=
++=
+=p
jjtj
q
iitit
q
iitit
qpGARCH
qARCH
1
2
1
22
1
22
:),(
:)(
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IV. Volatility Models
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Financial Econometrics is the Application ofEconometric Methods to Financial Markets
Econometrics: Time Series,GMM,ARCH,.. Finance:
Returns not Predictable? Volatility is Predictable?
Financial Markets And Empirical RegularitiesAn Introduction to Financial EconometricsOutline Essential Sources
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