1Financial Markets And Empirical RegularitiesAn Introduction to Financial Econometrics

SAMSI Workshop 11/18/05Mike Aguilar UNC at Chapel Hill

www.unc.edu/~maguilar

2Outline

I. Historical Perspective on Asset Prices

II. Predictability of Asset Returns

III. Asset Pricing Models

IV. Volatility Models

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

4Types of Financial Assets:

Common: Stocks, Bonds, Commodities, Foreign Exchange,

Exotic: Derivatives; Options, Futures,

I. Historical Perspective on Asset Prices

5I. Historical Perspective on Asset Prices

Source: Campbell Harvey

A History of Asset Prices

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

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.

8I. Historical Perspective on Asset Prices

A History of Asset Returns

Source: Campbell Harvey

9I. Historical Perspective on Asset Prices

Source: Campbell Harvey

10

I. Historical Perspective on Asset Prices

Source: Campbell Harvey

11

I. Historical Perspective on Asset Prices

Observation:

There exists a Risk / Return Tradeoff

12

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

13

I. Historical Perspective on Asset Prices

What is the Distribution of Returns?

Normal?

14

I. Historical Perspective on Asset Prices

15

I. Historical Perspective on Asset Prices

Stylized Facts on the Distribution of Returns:

1. Index Volatility < Stock Volatility

2. Negative Skewness

3. Excess Kurtosis

16

II. Predictability of Returns

Can We Predict Returns? Day Traders Say Yes

Efficient Market Hypothesis Says No

17

II. Predictability of Returns

Efficient Market Hypothesis (EMH):

Fama (1970): A market in which prices always fully reflect available information is called efficient.

18

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

19

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 .

20

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

21

II. Predictability of Returns

EMH:Weak Form Efficiency:

Information Set: Assets own historyTest via Random Walk

22

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

23

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

24

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.

25

II. Predictability of Returns

Implication of Weak Form EMH:

Expected Returns follow a martingale.

Random Walk Hypothesis

26

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

27

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

28

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

29

II. Predictability of Returns

30

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)

31

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

32

CAPM Pricing Equation:

III. Asset Pricing Models - CAPM

)][(][ fmimfi RRERRE +=

33

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 +=

34

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

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

36

III. Asset Pricing Models - CAPM

37

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

38

III. Asset Pricing Models - CAPM

39

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

40

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

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

42

IV. Volatility Models

43

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

:),(

:)(

44

IV. Volatility Models

45

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