Avramov Doron Financial Econometrics

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Lecture Notes

Financial Econometrics

Professor Doron E. Avramov

Hebrew University of Jerusalem

'Prof. Doron Avramov

Financial Econometrics1



Syllabus: Motivation• Why do we need a course in financial

econometrics?

• The past few decades have been characterized

by an extraordinary growth in the use of

quantitative methods in financial markets inanalyzing various asset classes; be it equities,

fixed income instruments, commodities, or

derivative securities.





Syllabus: MotivationFinancial market participants, both academics

and practitioners, have been routinely using

advanced econometric techniques in a host of

applications including portfolio management,

risk management, modeling volatility,understanding pivotal issues in corporate

finance, asset pricing, interest rate modeling, as

well as regulatory purposes.





Syllabus: Objectives• This course attempts to provide a fairly deep

understanding of topical issues in asset pricing

and deliver econometric methods in which to

develop research agenda in financial economics.

•The course targets advanced master level and

PhD level students in finance and economics.





Syllabus: Prerequisite• I will assume some prior exposure to matrix

algebra, distribution theory, Ordinary Least

Squares, and Maximum Likelihood

Estimation.

• I will also assume you have some skills in

computer programing beyond Excel.

Suggested packages include MATLAB,STATA, SAS, R, etc.





Syllabus: Topics to be CoveredOverview:

Matrix algebra

Regression analysisLaw of iterated expectations

Variance decomposition

Taylor approximationDistribution theory

Hypothesis testing

OLS

MLE





Syllabus: Topics to be CoveredTesting asset pricing models including the CAPM

and multi factor models: Time series analysis.

The econometrics of the mean variance frontier

Estimating expected returns

Estimating the covariance matrix of stock returns.

Forming mean variance efficient portfolio, the

Global Minimum Volatility Portfolio, and the

minimum Tracking Error Volatility Portfolio.'Prof. Doron Avramov




Syllabus: Topics to be CoveredThe Sharpe ratio: estimation and distribution

The Delta method

The Black-Litterman approach for estimating expected

returns.

Principal component analysis.





Syllabus: Topics to be CoveredRisk management and the downside risk measures:

value at risk, shortfall probability, expectedshortfall, target semi-variance, downside beta, and

drawdown.

Option pricing: testing the validity of the B&S

formula

Model verification based on failure rates





Syllabus: Topics to beCoveredVariance ratio tests

Predicting asset returns using time series regressions

The econometrics of back-testing

Understanding time varying volatility models

including ARCH, GARCH, EGARCH, stochastic

volatility, implied volatility, and realized volatility





Syllabus: Grade Components• Assignments (35%): there will be three problem sets

during the term. You can form study groups to prepare

the assignments with up to four students per group.

•Class Participation (15%) - Attending AT LEAST 80%

of the sessions is mandatory.•Take-home final exam (50%): based on class material,

handouts, assignments, and readings.





Let us StartThis session is mostly an overview. Major contents:

• Why do we need a course in financial econometrics?• Normal, Bivariate normal, and multivariate normal

densities

• The Chi-squared, F, and Student t distributions

• Regression analysis

• Basic rules and operations applied to matrices

• Iterated expectations and variance decomposition'Prof. Doron Avramov




Financial EconometricsIn previous courses in finance and economics you

mastered the concept of the efficient frontier.

A portfolio lying on the frontier is the highest

expected return portfolio for a given volatility target.

Or it is the lowest volatility portfolio for a given

expected return target.





Plotting the Efficient Frontier





However, how could you Practically Form

an Efficient Portfolio?

• Problem: there are TOO many parameters to estimate.

For instance, investing in ten assets requires:

which is about ten estimates for expected return, ten for volatility, and 45 for covariances/correlations.

Overall, 65 estimates are required. That is a lot!!!

10

2

1

.

.

µ

µ

µ

2

1010,1

2

22,1

10,1

2

,

1

σ σ

σ σ

σ σ

KKKKK

KKKKKKKK

KKKKK

KKKKK





More generally, if there are N investable assets, you need:

• N estimates for the means,

• N estimates for the volatilities,

• 0.5N(N-1) estimates for correlations.

Overall: 2N+0.5N(N-1) estimates are required!

Mean, volatility, and correlation estimates are noisy

as reflected through their standard errors.





• The volatility estimate is less noisy than the mean.

• I will later show that the standard error of the volatilityestimate is lower than that of mean return.

• We have T asset return observations:

•The sample mean and volatility are given by

T R R R R KKK321 ,,

( )

1

ˆ

2

1

−

−= ∑=

T

R RT

t t σ Less noisy



T

R R

T

t t ∑ == 1

Sample Mean and Volatility



Estimation Methods

• One of the ideas here is to introduce robust methods

in which to estimate the comprehensive set of

parameters.

• We will discuss asset pricing models and the Black

Litterman approach for estimating expected returns.• We will further introduce several methods for

estimating the large-scale covariance matrix of asset

returns.





Mean-Variance vs. Down

Side Risk

•We will comprehensively cover topics in meanvariance analysis.

•We will also depart from the mean variance

paradigm and consider down side risk measuresto form as well as evaluate investment strategies.

• Why do we need to resort to down side risk?





Down Side Risk

• For one, investors often care more about the

down side of investment payoffs than the upside

potential.

•The practice of risk management as well as

regulations of financial institutions are typicallyabout downside risk – such as targeting VaR,

shortfall probability, and expected shortfall.

•Moreover, there is a major weakness embedded

in the mean variance paradigm.

'Prof. Doron AvramovFinancial Econometrics

20



Drawback in the Mean-Variance Setup

• To illustrate, consider two stocks, A and B, with

correlation coefficient smaller than unity.

•There are five states of nature in the economy.

•Returns in the various states are described on the

next page.


21



• Stock A dominates stock B in every state of nature.

• Nevertheless, a mean-variance investor may consider

stock B because it can reduce the portfolio’s volatility.

• Investment criteria based on down size risk measurescould get around this weakness.

S5S4S3S2S1

20.05%10.50%15.00%8.25%5.00%A3.01%3.10%2.95%2.90%3.05%B


22

Drawback in Mean-Variance



The Normal Distribution• In various applications in finance and economics, a

common assumption is that quantities of interests, such

as asset returns, economic growth, dividend growth,

interest rates, etc., are normally (or log-normally)

distributed.

• The normality assumption is primarily done for

analytical tractability.

• The normal distribution is symmetric.

• It is characterized by the mean and the variance.


23



Confidence Level Normality suggests that deviation of 2 SD away from

the mean creates an 80% range (from -32% to 48%) for

the realized return with 95% confidence level.


24



Dispersion around the Mean

• Assuming that x is a zero mean random variable.

•As the distribution takes the form:

• When is small, the distribution is concentrated and

vice versa.

2,0~ σ N x


25



Probability Distribution Function

• The Probability Density Function (pdf) of the normal

distribution is:

• If then:

• The Cumulative Density Function (CDF), which is the

integral of the pdf, is 50% at that point.

( )

−−=2

2

2 2

1exp

2

1)(

σ

µ

πσ

r r pdf

==

r µ

σ 1

π 2

1)( =r pdf


26



Normally Distributed Return•Assume that the US excess rate of return on the

market portfolio is normally distributed with annual

mean (equity premium) and volatility given by 8% and20%, respectively.

•That is to say that with a nontrivial probability theactual excess annual market return can be negative.

See figures on the next page.


27



Confidence Intervals for Annual Excess

Return on the Market

The probability that the realization is negative

%99)2.0308.02.0308.0(Pr

%95)2.0208.02.0208.0(Pr

%68)2.008.02.008.0(Pr

=×+<<×−

=×+<<×−

=+<<−

Rob

Rob

Rob


28

%4.34)4.0(Pr

2.0

08.00

2.0

08.0Pr )0(Pr

=−<=

−<

−=<

z ob

Rob Rob



Higher Moments• Skewness – the third moment - is zero.

• Kurtosis – the fourth moment – is three.

• Odd moments are all zero.

• Even moments are (often complex) functions of themean and the variance.

•In the next slides, skewness and kurtosis are presented

for other distribution functions.


29



Skewness• The skewness can be negative (left tail) or

positive (right tail).


30

Kurtosis



Kurtosis


31

• Mesokurtic - A term used in a statistical context where the

kurtosis of a distribution is similar, or identical, to the

kurtosis of a normally distributed data set.



•

•Positive skewness means non-zero probability for large payoffs.

• Kurtosis is a measure for how tick the distribution’s tails are.

• When is the skewness zero? In symmetric distributions.


32

The Essence of Skewness and

Kurtosis



Bivariate Normal Distribution

• Bivariate normal:

• The marginal densities of x and y are

• What is the distribution of y if x is known?

22

2

,

,,~

y xy

xy x

y

x N

y

x

σ σ

σ σ

µ

µ

(( )[ ]2

2

,~

,~

y y

x x

N y

N x

σ µ

σ µ


33



Conditional Distribution

• Conditional distribution:

always non-negative• When the correlation between x and y is positive

and - then the conditional expectation ishigher than the unconditional one.

( )

−−+ 2

2

22

,~ x

xy y x

x

xy y x N x y

σ σ σ µ

σ σ µ

x x µ >


34



Conditional Moments• If we have no information about x, or if x and y are

uncorrelated, then the conditional and unconditional

expected return of y are identical.• Otherwise, the expected return of y is .

• If , then:

Here, the realization of x does not say anything about y.

0= xyσ

x yµ

y

x

xy

y x y σ σ

σ σ σ =−=

2

2

2


35



Conditional Standard Deviation

• Developing the conditional standard deviation further:

• When goodness of fit is higher the conditional standarddeviation is lower.

( )2

22

2

2

2

2

2 11 R y

y x

xy

y

x

xy

y x y −=

⋅

−=−= σ

σ σ

σ σ

σ

σ σ σ


36

M lti i t N l



• Define Z=AX+B.

( )mmm

m

m N

x

x

x

X ××× ∑

= ,~

.

. 1

2

1

1 µ


37

Multivariate Normal



• Let us make some transformations to end up with

N (0,I):

( )

( )( ) ),0(~

),0(~

),(~

11

2

1'

'11

'

1111

I N B A Z A A

A A N B A Z

A A B A N B X A Z

nmmnnmmmmn

nmmmmnnmmn

nmmmmnnmmnnmmn

×××

−

×××

×××⋅××

×××××××××

−−∑

∑+−

∑++⋅=

µ

µ

µ


38

Multivariate Normal



Multivariate Normal

• Consider an N -vector or returns which is normally

distributed:

• Then:

• What does mean? A few rules:

( ) N N N N N R ××× ∑,~ 11 µ

( )

( ) ( ) I N R

N R

,0~

,0~

2

1

µ

µ

−∑

∑−

−

2

1−

∑

I =∑⋅∑

∑=∑⋅∑

∑=∑⋅∑

−

−−−

2

1

2

1

2

1

2

1

12

1

2

1


39



• If X 1~N(0,1) then

• If X 1~N(0,1), X 2~N (0,1) & then:

( )2~ 222

21 χ X X +

21 X X ⊥


40

The Chi-Squared Distribution

( )1~ 22

1 χ X



• Moreover if

then

( )

( )

21

2

2

2

1

~

~

X X

n X

m X

⊥

χ

χ

( )nm X X ++ 221 ~ χ


41

More about the Chi-Squared

The F Distribution



•Gibbons, Ross & Shanken (GRS) designated a finite

sample asset pricing test that has the F - Distribution.

•If

•Then

),(~2

1

nm F

n X m

X

A =

( )( )

21

2

2

21

~

~

X X

n X

m X

⊥

χ

χ


42

The F Distribution

The Student’s t Distribution



The Student s t Distribution


43

The t Distribution



• The pdf of student-t is given by:

• Where is beta function and is a number of

degrees of freedom

The t Distribution


44

( )( )

2

12

121,2

1,

+−

+⋅

⋅=

v

v x

v Bvv x f

( )ba B , 1≥v

The t Distribution



• As noted earlier, is a sample of length T of

stock returns which are normally distributed.

• The sample mean and variance are

• The resulting t -value which has the t distribution with T-1

d.o.f is

The t Distribution


45

T r r r ,,, 21 K

T

r r

r

T K+

=

1

and ( )∑= −−=

T

t t r r T s 1

22

1

1

1−−=T

sr t µ

The t Distribution



• The t -distribution is the sampling distribution of the t -value

when the sample consist of independent and identically

distributed observations from a normally distributed population.

• It is obtained by dividing a normally distributed random variable

by a square root of a Chi-squared distributed random variable

when both random variables are independent.

• Indeed, later we will show that when returns are normally

distributed the sample mean and variance are independent.

The t Distribution


46

R i A l i



Regression Analysis• Various applications in corporate finance and asset

pricing require the implementation of a regression

analysis.• We will estimate regressions using matrix notation.

• For instance, consider the time series regression

t t t t Z Z R ε β β α +++= 2211 T ,2,1, KK=∀


47

R iti th t i t i f



• Rewriting the system in a matrix form:

yields

• We will derive regression coefficients and their standard

errors using OLS, MLE, and Method of Moments.

=×

21

2221

1211

3

,,1

.

.

,,1

,,1

T T

T

Z Z

Z Z

Z Z

X

=×

2

113

β β

α

γ

11331 ×××× += T T T X R ε γ

=×

T

T

ε

ε

ε

ε ..

2

1

1


48

=×

T

T

R

R

R

R..

2

1

1



Vectors and Matrices: some Rules

• If A is a row vector:

• Then the transpose of A is a column vector:

• The identity matrix satisfies

[ ]t t aaa A 112,111 ,K=×

=×

1

21

11

1

.

.'

t

t

a

a

a

A

B I B B I =⋅=⋅


49

M l i li i f M i



Multiplication of Matrices:

,

The inverse of the matrix A is A-1 which satisfies:

=×

22,21

12,11

22aa

aa A

=×

22,12

21,11

22'aa

aa A

=×

22,21

12,11

22bb

bb B

++

++=××

22221221,21221121

22121211,21121111

2222babababa

babababa B A

==−

×× 1,0

0,11

2222 I A A

111)(

'')'(

−−− =

=

A B AB

A B AB


50



Solving Large Scale Linear Equations:

• Of course, A has to be a square invertible matrix.

b

b X A mmmm

1

11

−

×××

=

=


51

Linear Independence and Norm



Linear Independence and Norm

• The vectors are linearly independent if there

does not exists scalars such that

• The norm of a vector V is


52

N V V ,,1 K

N cc ,,1 K

unless

V cV cV c N N 02211 =+++ K

021 === N ccc K

V V V '=

A Positive Definite Matrix



A Positive Definite Matrix

• An matrix is called positive definite if

for any nonzero vector V.

• A non positive definite matrix cannot be inverted and

its determinant is zero.


53

N N × ∑

0' 11 >⋅∑⋅ ××× N N N N V V

The Trace of a Matrix



The Trace of a Matrix

• Let

• Then

• Trace is the sum of diagonal elements.


54

( ) ( )[ ]111

2

,1,

2

22,1

,1

2

,,,

1

××× =

=∑ N

N

N

N N

N

N N σ σ

σ σ

σ σ

σ σ

K

KKKKK

KKKKKKKK

KKKKK

KKKKK

( )

22

2

2

1 N tr σ σ σ +++=∑K

Matrix Vectorization



Matrix Vectorization

This is the vectorization operator –

it works for both square as well

as non square matrices.


55

( )

( )

( )

( )

=∑×

N

N Vec

σ

σ

σ

M

2

1

12

The VECH Operator



The VECH Operator

•Similar to but takes only the distinct elements

of .


56

( )( )

( )

=∑×

+

2

2

2

2

1

1

12

1

N

N N N

Vech

σ

σ

σ

σ

M

M

( )∑Vec

∑

Partitioned Matrices



•A, a square nonsingular matrix, is partitioned as

•Then the inverse of A is given by

•Check that


57

=

××

××

2212

2111

2221

1211

,

,

mmmm

mmmm

A A

A A

A

( ) ( )

( )

−−−

−−−=

−−−

−−−−

12

1

112122

1

2122121121

1

22

22

1

12

1

21221211

1

212212111

,

,

A A A A A A A A A A

A A A A A A A A A A A

( )21

1

mm I A A +− =⋅

Matrix Differentiation



• Y is an M -vector. X is an N -vector. Then

• Let


58

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=∂∂

×

N

M M M

N

N M

X

Y

X

Y

X

Y

X

Y

X

Y

X

Y

X

Y

,,,

,,,

21

1

2

1

1

1

K

M

K

A X

Y =

∂∂

11 ×××=

N N M M X AY





• Let


59

''

'

A X Y

Z

AY X

Z

=∂∂

=∂∂

11'

×××=

N N M M X AY Z





• Let

• If C is symmetric, then


60

( )'' C C X

X

+=

∂

∂θ

11'

×××=

N N N N X C X θ

C X X

'2=∂∂θ

Kronecker Product:



•It is given by

g pmnmg np B AC ××× ⊗=

=×

Ba Ba

Ba BaC

nmn

m

mg np

,

,

1

111

KKKKK

KKKKKKKKK

KKKKK


61

Kronecker Product:

Kronecker Product:



For A,B square matrices, it follows that

'Prof. Doron AvramovFinancial Econometrics62

( )

( )( ) ( ) BD AC DC B A

B A B A

B A B A N M

N N M M

⊗=⊗⊗

=⊗⊗=⊗

××

−−− 111

Operations on Matrices



Operations on Matrices


[ ][ ] [ ]

)')((),(

)(')'()'(

)'()'(

)()''()()()()(

)()()(

)()()(

µ µ µ

µ µ

−−=Σ=

Σ+===

=

= +=+

+=+=⊗

X X E X E

where

Atr A A XX E tr A XX tr E

AX X tr E AX X E

Avec Bvec ABtr Bvech Avech B Avech

Bvec Avec B Avec

Btr Atr B Atr

Using Matrix Notation in a Portfolio



The expectation and the variance of a portfolio’s rate of

return in the presence of three stocks are formulated as

ω ω

ρ σ σ ω ω ρ σ σ ω ω ρ σ σ ω ω

σ ω σ ω σ ω σ

µ ω µ ω µ ω µ ω µ

∑=

+++++=

=++=

'

222

'

233232133131122121

23

23

22

22

21

21

2

332211

p

p


Choice Context

Using Matrix Notation in a Portfolio



where

=×

3

2

1

13

ω

ω

ω

ω

=∑×

2

33231

23

2

221

1312

2

1

33

,,

,,

,,

σ σ σ

σ σ σ

σ σ σ

=×

3

2

1

13

µ

µ

µ

µ

'Prof. Doron AvramovFinancial Econometrics 65

Choice Context

The Expectation, Variance, and



Covariance of Sum of RandomVariables

)()()()( z cE ybE xaE cz byax E ++=++


),(2),(2),(2

)()()()( 222

z ybcCov z xacCov y xabCov

z Var c yVar b xVar acz byaxVar

+++

++=++

),(),(),(),(),(

w ybdCov z ybcCovw xadCov z xacCovdwcz byaxCov

+++=++

Law of Iterated Expectations (LIE)



Law of Iterated Expectations (LIE)

• The LIE relates the unconditional expectation of a

random variable to its conditional expectation via the

formulation

• Paul Samuelson shows the relation between the LIE andthe notion of market efficiency – which loosely speaking

asserts that the change in asset prices cannot be predicted

using current information.

Prof. Doron AvramovFinancial Econometrics

[ ] [ ] X Y E E Y E x |=

'67

LIE and Market Efficiency



LIE and Market Efficiency

• Under rational expectations, the time t security

price can be written as the rational expectation of

some fundamental value, conditional on informationavailable at time t :

• Similarly,• The conditional expectation of the price change is

• The quantity does not depend on the information.

][| ** P E I P E P t t t ==

Prof. Doron AvramovFinancial Econometrics

][|

*

11

*

1 P E I P E P t t t +++ ==

0][][]|[**

11 =−=− ++ P E P E E I P P E t t t t t t

'68

Variance Decomposition (VD)



Variance Decomposition (VD)

• Let us decompose :

• Shiller (1981) documents excess volatility in the

equity market. We can use VD to prove it:

• The theoretical stock price:

based on actual future dividends

• The actual stock price:

[ ] yVar

[ ] ( )[ ] ( )[ ] x yVar E x y E Var yVar x x || +=

t t t I P E P *=

( )LL

2

21*

11 r

D

r

D P t t

t +

++

= ++




Variance Decomposition

What do we observe in the data? The opposite!!

t t t t t I P Var E I P E Var P Var *** +=

[ ] positive P Var P Var t t +=* positive

[ ]t t P Var P Var >*




Lecture Notes in Financial

Econometrics

Taylor Approximations inFinancial Economics


Financial Econometrics 71

TA in Finance



• Several major applications in finance require the use of

Taylor series approximation.

• See three applications in the following table.

• We will also derive a test statistic for the Sharpe ratio –

the price of risk – which is based on the delta method,

which in turn is using the first order Taylor

approximation.



TA in Finance: Major



Applications



Expected Utility Bond Pricing Option Pricing

First Oder Mean Duration Delta

Second Order Volatility Convexity Gamma

Third Order Skewness

Fourth Order Kurtosis



Taylor Approximation• Taylor series is a representation of a function as

an infinite sum of terms that are calculated from

the values of the function's derivatives at a single

point.

• Taylor approximation is written as:

• It can also be written with notation:

....))((!3

1))((

!2

1))((

!1

1)()( 3

00

'''2

00

''

00

'

0 +−+−+−+= x x x f x x x f x x x f x f x f

( )∑∞

=−=

0 00

!

)()(

n

nn

x xn

x f x f

∑



Maximizing Expected Utility of



Terminal Wealth

).1)......(1)(1(1

)1(

21 K T T T

T K T

R R R Rwhere

RW W

+++

+

+++=+

+=

The invested wealth is

The investment horizon is K periods.

The terminal wealth, the wealth in the end of the investment

horizon, is

T W



Transforming to Log Returns



)1ln(

.....

)1ln(

)1ln(

22

11

K T K T

T T

T T

Rr

Rr

Rr

++

++

++

+=

+=

+=



Power utility as a Function of



• Assume that the investor has the power utility function

of the from

• The cumulative log return (CLR) over the investment

horizon is:

• The terminal wealth as a function of CLR is



K T K T W W u ++ =γ

γ

1)(

K T T T r r r r +++ +++= ...21

)exp()1( r W RW W T T K T =+=+

Log Return

Power utility as a Function of



• Then the utility function of terminal wealth can be

expressed as a function of CLR

• We can use the TA to express the utility as a function

of the moments of CLR.

• That is, we will approximate around



)exp(11

)( r W W W u T k T K T γ γ γ

γ γ == ++

)( K T W u + )(r E =µ

Log Return

The Utility Function





])(24

1)(

6

1

)(2

11)[exp()]([

....))(exp(24

1

))(exp(6

1))(exp(

2

1

))(exp()exp()(

43

2

44

3322

γ γ

γ µγ γ

γ µ µγ

γ µ µγ γ µ µγ

γ µ µγ γ

µγ γ

γ

γ γ

r Kur r Ske

r Var W

W u E

r

r r

r W W

W u

T K T

T T K T

+

++≈

+−+

−+−+

−+=

+

+

Approximation

Expected Utility



Let us assume that log return is normally distributed:

The exact solution under normality is

[ ]

++≈⋅

== →

4422

42

821)exp(:

3,0),(~

σ γ σ γ γµ γ

σ σ µ

γ T W E Then

Kur Ske N r



[ ]

+=⋅ 2

2

2

exp σ γ

γµ

γ

γ

T W E

Taylor Approximation – Bond Pricing



• Taylor approximation is also used in bond pricing.

• The bond price is:

where y0 is the yield to maturity.

• Assume that y0 changes to y1

• The delta (the change) of the yield to maturity is written

as:

( )∑ =

+

=n

i i

i

y

CF y P

1

0

0

1

)(

01 y y y −=∆



Changes in Yields and Bond



Pricing• Using Taylor approximation we get

2

010

''

010

'

01

))((!2

1))((

!1

1)()( y y y P y y y P y P y P −+−+≈

2

010

''

010

'

01 ))((!2

1))((!11)()( y y y P y y y P y P y P −+−≈−⇒



Duration and Convexity



• Dividing by P(y0 ) yields

•Instead of: we can write “-MD”

(Modified Duration).

•Instead of: we can write “Con”(Convexity).

)(2

)(

)(

)(

0

2

0

''

0

0

'

y P

y y P

y P

y y P

P

P ∆⋅+

∆⋅≈

∆

)(

)(

0

0

'

y P

y P

)()(

0

0

''

y P y P



The Approximated Bond Price



Change• It is given by

• What if the yield to maturity falls?

2

2 yCon y MD

P

P ∆⋅+∆⋅−≈

∆



The Bond Price Change when



• The change of the bond price is:

• According to duration - bond price should increase.

• According to convexity - bond price should also

increase.• Indeed, the bond price rises.

2

2 yCon y MD

P

P ∆⋅+∆⋅−=∆



Yields Fall




• And what if the yield to maturity increases?

• Again, the change of the bond price is:

• According to duration – the bond price should

decrease.

2

2 yCon y MD

P

P ∆⋅+∆⋅−=

∆



Yields Increase


Yi ld I



• According to convexity - bond price should increase.

• So what is the overall effect?

• Notice: the influence of duration is always stronger

than that of convexity as the duration is a first order effect while the convexity is second order.



Yields Increase

Option Pricing



• A call price is a function of the underlying asset price

• What is the change in the call price when the

underlying asset pricing changes?

• Focusing on the first order term – this establishes the

delta neutral trading strategy.

( ) ( )

( ) ( )2

2

2

2

2

1)(

2

1

)()(

t st st

t st st s

P P P P P C

P P P

C

P P P

C

P C P C

−Γ+−∆+≈

−∂

∂

+−∂

∂

+≈



Delta Neutral Strategy



Suppose that the underlying asset volatility increases.

Suppose further the implied volatility lags behind.

The call option is then underpriced – buy the call.

However, you take the risk of fluctuations in the price

of the underlying asset.

To hedge that risk you sell Delta units of theunderlying asset.






Econometrics

OLS, MLE, MOM



Ordinary Least Squares (OLS)

h l i i h i



• The goal is to estimate the regression parameters.

• Using optimization can help us reach the goal.

• Consider the regression:

, ,

∑=

=

−=

+=

T

T

t

t t t

t t t

f

x y

x y

1

2

)( ε

β ε

ε β

β

=

T y

y

Y .

1

=

KT T

K

x x

x x

X

,,,1

,,,1

1

111

K

KKKKK

K

=

T

E

ε

ε

.

1



• First Order Conditions

L t d i th f ti ith t t b t d



• Let us derive the function with respect to beta and

make the derivative equal to zero

• Recall: X and Y are observations based on real data.

• Could we choose other to obtain smaller

Y X X X

Y X X X f

'1'

'')(

)(ˆ

02)(2

−=

=−=∂

∂

β

β β β

β ∑=

=T

T

t E E 1

2' ε



• o! Because the optimization minimizes the quantity

The OLS Estimator



.

• We know:

• Is the estimator unbiased for ?

, So – it is indeed unbiased.

E X X X

E X X X X X X X E X X X X E X Y

'1'

'1''1'

'1'

)(ˆ

)()(ˆ)()(

ˆ

−

−−

−

+=

+=+=→+=

β β

β β β β β

β β

[ ][ ] 0)(

)(

ˆ

'1'

'1'

==

=

−

−

−

E E X X X

E X X X E

E β β

E '



• What about the Standard Error of ?β

The Standard Errors of the OLS Estimates



• What about the Standard Error of ?

• Reminder:

β

( ) ( )

−⋅−=

'ˆˆ)ˆ( β β β β β E VAR

( ) 1'''

'1'

)(ˆ

)(ˆ

−

−

=−

=−

X X X E

E X X X

β β

β β



Continuing:

The Standard Errors of the OLS Estimates



where K is the number of explanatory variables in

the regression specification.

[ ]

E E K T

X X X X X X X EE E X X X

X X X EE X X X E

ˆˆ1

1ˆ

ˆ)(ˆ

ˆ)()()(

)()(

'2

21'

21'1'''1'

1'''1'

−−=

=∑=

=

−

−−−

−−

ε

ε β

ε

σ

σ σ



MLE

• We next turn to resorting to Maximum Likelihood as a



• We next turn to resorting to Maximum Likelihood as a

powerful tool for both estimating regression parameters

as well as testing models.

• The MLE is an asymptotic procedure and it a

parametric approach in that the distribution of the

regression residual must be specified explicitly.



Implementing MLE

A h ( )2N

iid



• Assume that

• Let us estimate and using MLE; then derive

the joint distribution of these estimates.

• Under normality, the probability distribution

function ( pdf ) of the rate of return takes the form

( )2,~ σ µ N r t

µ 2σ

( )

−−=

2

2

2 2

1exp

2

1)(

σ

µ

πσ

t t

r r pdf



The Joint Likelihood

( ) ( ) ( ) ( )rpdfrrrpdfrrrpdfrrrpdfL ×××== 322121



• Since stock returns are assumed iid - it follows that

• Now take the natural log of the joint likelihood:

( ) ( ) ( )

( )

−

−=

×××=

∑=

−T

T

t T

T

r L

r pdf r pdf r pdf L

1

2

22

21

2

1exp2

σ

µ πσ

K

( ) ( ) ∑=

−−−−=

T

t

t r T T L1

2

2

21ln

22ln

2ln

σ µ σ π

( ) ( ) ( ) ( )T T T T r pdf r r r pdf r r r pdf r r r pdf L ×××== KKKK 322121 ,,,



MLE: Sample Estimates

• Derive the first order conditions



• Derive the first order conditions

Since - the variance estimator is not

unbiased.

( )( )∑ ∑

∑∑

= =

==

−=→=−

+−=

∂

∂

=→=

−

=∂

∂

T

t

T

t

t t

T

t t

T

t

t

r

T

r T L

r T

r L

1 1

22

4

2

22

112

ˆ1

ˆ0

2

ln

1

ˆ0

ln

µ σ

σ

µ

σ σ

µ σ

µ

µ

22ˆ σ σ ≠ E



MLE: The Information Matrix

• Take second derivatives:



• Take second derivatives:

( )

( )

( )

∑

∑

∑

=

=

=

−=

∂

∂→

−−=

∂

∂

=

∂∂

∂→

−−=

∂∂

∂

−=

∂

∂

→−=−=∂

∂

T

t

t

T

t

t

T

t

T L E

r T L

L E

r L

T L

E

T L

1422

2

4

2

422

2

1

2

2

42

2

21

2

2

222

2

2

ln

2

ln

0lnln

ln1ln

σ σ

σ

µ

σ σ

σ µ σ

µ

σ µ

σ µ σ σ µ



MLE: The Covariance Matrix

• Set the information matrix ∂= ln)(2

θ LEI



• Set the information matrix

• Then the asy. distribution of the parameters is

In our context, the information matrix is derived as

∂∂∂−=

'ln)(θ θ

θ L E I

→

→

−

−−

T

T

T

T

T

T INVERSE MULTIPLY

4

2

4

2"1"

4

2

2,0

0,

2,0

0,

2,0

0,

σ

σ

σ

σ

σ

σ

( )1)(,0~

−∧

− θ θ θ I N T



To Summarize

•The asy distribution of the parameters is



The asy. distribution of the parameters is

In our context, the joint distribution of the sample

mean and variance assuming IID normal is

( )1)(,0~

−∧

− θ θ θ I N T

−−

−

∑∑

∑

==

=

4

2

2

11

2

1

2,0

0,,

0

0~

)1(1

1

σ

σ

σ

µ

N

r T

r T

r T

T T

t

t

T

t

t

T

t

t



The Sample Mean and Variance

• Notice that when returns are normally distributed –



Notice that when returns are normally distributed

the sample mean and variance are independent.

• That would not be the case otherwise.

• Departing from normality, the covariance between the

sample mean and variance is the skewness (see below).• Notice also that the ratio obtained by dividing the

variance of the variance by the variance of the mean is

smaller than one as long as volatility is below 70%.

• Clearly, the mean return estimate is more noisy.'Prof. Doron Avramov


Departing from Normality:

Setting Moment Conditions



Setting Moment Conditions• We know:

• If:

•Then:

• That is, we set two momentum conditions.

( )

( )22

σ µ

µ

=−

=

t

t

r E

r E

( )( )

−−

−=

22σ µ

µ θ

t

t

t r

r g

( )

=

0

0t g E



Method of Moments (MOM)

• There are two parameters: 2σµ



There are two parameters:

• Stage 1: Moment Conditions

• Stage 2: estimation

,σ µ

( )

−−−=

22σ µ

µ

t

t

t r

r g

01

1

^

=∑=

T

t

t g T



Method of Moments

• Continue estimation:



Continue estimation:

( )

( )[ ]

( )∑

∑

∑

∑

=

=

=

−=

=−−

=

=−

T

t

t

T

t

t

T

t

t

t

r

T

r T

r T

r

T

1

22

1

22

1

ˆ1

ˆ

0ˆˆ1

1ˆ

0ˆ1

µ σ

σ µ

µ

µ



T t ,,1 K=

MOM: Stage 3



)'

t t g g E S =

− 4

43

3

2

,

,

σ µ µ

µ σ

( ) ( ) ( )( ) ( ) ( ) ( )

+−−−−−−

−−−−=

422423

232

2,

,

σ µ σ µ µ σ µ

µ σ µ µ

t t t t

t t t

r r r r

r r r E S



MOM: stage 4

M



• Memo:

• Stage 4: derive wrt and take the expected

value

( )( )

−−

−=

22σ µ

µ θ

t

t

t r

r g

( )θ t g θ

( )( )

−

−=

−−−

−=

∂∂=

1,0

0,1

1,2

0,1

µ θ

θ

t

t

r E

g E D



MOM: The Covariance Matrix

Stage 5:



g

( )S

DS D

=∑

=∑−−

θ

θ

11'



The Covariance Matrices

=∑21 0,σ

θ =∑ 322 ,µ σ θ



∑42,0 σ

θ

−

∑4

43, σ µ µ θ

• Sample estimates of the Skewness and Kurtosis are

3

3 σ µ ×== sk SK

( )∑=

−=T

t

t r r T 1

3

3

1µ ( )∑

=

−=T

t

t r r T 1

4

4

1µ

sk is the skewness of the standardized

return

kr is the kurtosis of the standardizedreturn

4

4 σ µ ×== kr KR

110

Under Normality the MOM Covariance

Matrix Boils Down to



1

44

43

32

2

2,0

0,θ θ

σ σ µ µ

µ σ Σ=

=−===∑



MOM: Estimating Regression Parameters

• Let us run the time series regression



• There are two moment conditions:

t mt t r r ε α +⋅+=

( )

( ) 0

0

=

=

t mt

t

r E

E

ε

ε

( )( )[ ]

[ ][ ]',',1

'

β α θ

θ

β α

β α

==

−=

⋅−−

⋅−−=

mt t

t t t

mt mt t

mt t

t

r x

xr x

r r r

r r g



MOM: Estimating Regression Parameters

• Estimation:



( )

=

=

=

=−

×

−

=

−

=

==

∑∑∑∑

mT

m

T

T

t

t t

T

t

t t

T

t

t t

T

t

t t

r

r

X

R X X X r x x x

x xT

r xT

,1

,1

011

1

2

'1'

1

1

1

'^

1

^'

1

KK

θ

θ

=

T r

r

R K

1



MOM: Estimating Standard Errors

• Estimation of the covariance matrix



T

X X

x xT

g

T D

x xT g g T S

t

T

t t

T

t

t

T

t t

T

t

t t

T

t

t T

'

)'(

1)(1

ˆ)'(

1

)'()(

1

11^

^

2

1

^

1

^

−=−=∂

∂

=

==

∑∑

∑∑

==

==

θ

θ

ε θ θ



MOM: Estimating Standard Errors

• The covariance matrix estimate is thus



•Then, asymptotically we get

1

1

21

11

)'(ˆ)'()'(

)'(

−

=

−

−−

=Σ

=Σ

∑ X X x x X X T

DS DT

t

t t t

T T T

ε θ

θ



),0(~)(^

θ θ θ Σ− N T





Econometrics

Hypothesis Testing



Overview

A short brief of the major contents for today’s class:



• Hypothesis testing

• TESTS: Skewness, Kurtosis, Bera-Jarque

• A first step to testing asset pricing models

• Deriving test statistic for the Sharpe ratio



Hypothesis Testing

• Let us assume that a mutual fund invests in value

t k ( t k ith hi h ti f b k t k t)



stocks (e.g., stocks with high ratios of book-to-market).

• Performance evaluation is mostly about running theregression of excess fund returns on the market premium

• The hypothesis testing for examining performance is

H0

: means no performanceH1: Otherwise

it

e

mt ii

e

it R R ε β α ++=

=iα



Hypothesis Testing

• Errors emerge if we reject H 0 while it is true, or whend t j t H h it i



we do not reject H 0 when it is wrong:

Reject H0Don’t

reject H0

Type 1

error

Good

decisionH0

Good

decision

Type 2 error H1

α True state

of world



Hypothesis Testing - Errors

• is the first type error (size), while is the secondtype error (related to power)

α β



type error (related to power).

• The power of the test is equal to 1- .

• We would prefer both and to be as small as

possible, but there is always a trade-off.

• When decreases increases and vice versa.

•The implementation of hypothesis testing requires the

knowledge of distribution theory.

β

α

α β



Skewness, Kurtosis & Bera

Jarque Test Statistics



q

• We aim to test normality of stock returns.

• We use three distinct tests to examine normality.



• TEST 1 - Skewness (third moment)

Test I – Skewness



• The setup for testing normality of stock return:

H0:

H1: otherwise

• Sample Skewness is

( 2,~ σ µ N Rt

∑=

−=

T

t

H t

T N

R

T S

1

36

,0~ˆ

ˆ1 0

σ

µ



122

• Multiplying S by we get ( )10~ NST T

Test I – Skewness



• Multiplying S by , we get

• If the statistic value is higher (absolute value) than thecritical value e.g., the statistic is equal to -2.31, then

reject H 0, otherwise do not reject the null of nomrality.

( )1,0~6

N S 6



123

TEST 2 - Kurtosis



• Kurtosis estimate is:

• After transformation:

∑=

−=

T

t

H t

T N R

T K 1

424

,3~ˆ

ˆ1 0

σ

µ

( ) ( )1,0~324

0

N K T

H

−



124

• The statistic is: ( ) ( )2~3 222 χ−+= KT

ST

BJ

TEST 3 - Bera-Jarqua Test



The statistic is:

• Why ?

( ) ( )23246

χ + K S BJ

)2(2 χ



125

• If X1~N(0,1) , X2~N (0,1) & then:21 XX ⊥

TEST 3 - Bera-Jarqua Test



If X 1 N(0,1) , X 2 N (0,1) & then:

• and are both standard normal and

they are independent random varaibles.

( )2~22

2

2

1 χ X X +

21 X X ⊥

S T 6 ( )324 − K

T



126

Chi Squared Test

• In financial economics, the Chi square test is

implemented quite frequently in hypothesis testing



implemented quite frequently in hypothesis testing.

• Let us derive it.

• Suppose that:

Then:

( ) ( )

( ) ( ) ( ) N R R y y

I N R y

21''

2

1

~

,0~

χ µ µ

µ

−∑−=

−∑=−

−



127

Testing the CAPM

• For one the chi squared is used to test the CAPM



• For one, the chi squared is used to test the CAPM

• There are a few tests based on time series and crosssection specifications. Let us start with the time series.

• The CAPM says that:• Thus, we first run the time-series market regressions:

)()(

e

mi

e

i R E R E β =

N

e

m N N

e

N

e

m

e

R R

R R

ε β α

ε β α

++=

++=

KKKKKKKK

1111



128

• The expected excess return for asset i is given by:

)()( emii

ei R E R E β α +=

Testing the CAPM



• According to the CAPM, the intercept i is restricted

to be zero for every test asset.

• So, the joint hypothesis test is:

• While doing the estimation, we will never get .

• Rather, we examine whether the estimate is equal to

zero statistically. For example, the sample estimate could be -- nevertheless this estimate could be

insignificant, or it is statistically equal to zero.

)(α

otherwise H

H N

:

0:

1

210 === α α α K

005.0ˆ =α



129

0ˆ =α

The Test Statistic

• Estimate :

=α

α .ˆ

ˆ1

α



• If:

and

• We get:

• Later we will develop the exact analytics of this test.

N α ˆ

( ) ( )

( )α

α

α

α α µ

∑

∑→∑

,0~ˆ

,~ˆ,~

0

N

N N R

H

( ) N

H 21'

0

~ˆˆ χ α α α

−

∑



130

Joint Hypothesis Test

• Y th ti i i



• You run the time series regression:

,22110 t t t t

x x y ε β β β +++= T t ,,2,1 K=



131


• We have three orthogonal conditions:( ) 0=t E ε



• We have three orthogonal conditions:

• Using matrix notation:

( )

( ) 0

0

2

1

=

=

t t

t t

x E

x E

ε

ε

=×

T

T

y

y

Y ..

1

1

=×

T T

T

x x

x x

x x

X

21

2212

2111

3

,,1

,,1

,,1

KKKK[ ]'32113 ,, β β β β = x

11331 ×××× += T T T E X Y

=×

T

T E

ε

ε

.

.

1

1



132

• Let us assume that: , are iid (2

,0~ ε σ ε N t T ε ε ε ε K

321 ,,




• Joint hypothesis testing:

( )( ) Y X X X

X X N '1'

21'

ˆ

,~ˆ−

−

= β

σ β β ε

otherwise H

H

:

0,1:

1

200 == β β



133

• Define: ,

= 1,0,0

0,0,1 R

= 0

1

q




• The joint test becomes:

otherwise H

q R H

H

:

0

1

1,0,0

0,0,1

:

1

2

0

2

1

0

0

0

=

=

×

=

β

β

β

β β

β



134

q

• Returning to the testing:

otherwise H

q R H

:

0:

1

0 =− β




• Under H0:

• Chi squared:

test statistic

( )',~ˆ R Rq R N q R

β β β ∑−−

( )',0~ˆ0

R R N q R H

β β ∑−

( ) ( ) ( ) ( )2~ˆˆ 21''

χ β β β q R R Rq R −∑−−



135

•Yet, another example:

t t t t t t t x x x x x y ε β β ++++++= 55443322110

Tt 321=




, ,

=×

T

T

y

y

Y .

1

1

=×

T T

T

x x

x x

X

51

5111

6

,,,1

.

.

,,,1

K

K [ ]',, 61016 β β β β K=×

=×

T

T

ε

ε

ε .

1

1

T t ,,3,2,1 K=

E X Y += β



136

• Joint hypothesis test:

H 7,4

1,

3

1,

2

1,1: 532100 ===== β β β β β




• Here is a receipt:1.

2.

3.

4.

otherwise H :

432

1

( )

( )( ) β ε

ε

σ β β

σ

β

β

∑=

⋅

−

=

−=

=

−

−

21'

'2

'1'

,~ˆ

ˆˆ

6

1ˆ

ˆˆ

ˆ

X X N

E E

T

X Y E

Y X X X



137

5.

=

=1

2

1

1

000100

0,0,0,0,1,00,0,0,0,0,1

qR




5.

6.

7.

8.

9.

=

=

7

4

1

3,

1,0,0,0,0,00,0,1,0,0,0

0,0,0,1,0,0 q R

( )( )

( )( ) ( ) ( )5~ˆˆ

,0~ˆ

,~ˆ

0:

21''

'

'

0

0

0

χ β β

β

β β

β

β

β

H

H

q R R Rq R

R R N q R

R Rq R N q R

q R H

−∑−

∑−

∑−−

=−

−



138

•There are five degrees of freedom implied by the five

restrictions on53210 ,,,, β β β β β




• The chi-squared distribution is always positive.

53210 ,,,, βββββ



139

Shrinkage Methods

• One of the problems with the tests we have just



displayed is that the decision is binary: Reject or donot reject (Classical econometrics) the null.

• The possibility of partly rejecting/accepting the null,

for example, does not exist.



140

Shrinkage Methods and the CAPM

• One can advocate a Bayesian approach in which

the proposed model (CAPM) is recognized to be non



the proposed model (CAPM) is recognized to be non-

perfect, but at the same time it is worth something.

• For example, let us assign a 50% weight to theCAPM and 50% to data.



141

Shrinkage Methods and the CAPM

• 50% CAPM :

• 50% Data:

)()( emiei R E R E β =

) )e

m

ee

m

e R E R E R R 1111111 β α ε β α +=→++=



• Data based estimate is the sample mean. (Check it!)

• Let us consider three Mutual funds, with each of the

fund managers has one of the following beliefs:

• M 1 – The CAPM is true

• M 2 – The CAPM is wrong

• M 3 - Mix (equally weights) between CAPM and the Data

) )mm 1111111 ββ

0=→α

0≠→ α



142

Shrinkage and Performance

• Over the last several decades, the third Mutual fundwould have had the best performance.



• Using the mixing method improves the estimation of expected return.

• The Black Litterman method (coming up later) is also

a type of a shrinkage approach - it is more rigorous than

the one presented here.

• The weights are on the model versus views onexpected returns, either absolute or relative.



143

Estimating the Sample Sharpe Ratio

• You observe time series of returns on a stock, or a bond, or any investment vehicle (e.g., a mutual fund or

h d f d) (



a hedge fund):

• You attempt to estimate the mean and the variance of

those returns, derive their distribution, and test whether

the Sharpe Ratio of that investment is equal to zero.

• Let us denote the set of parameters by

• The Sharpe ratio is equal to σ

µ θ

f r SR

−=)(

]',[ 2σ µ θ =

( T r r r ,,, 21 K



MLE vs. MOM

• To develop a test statistic for the SR, we canimplement the MLE or MOM, depending upon our

ti b t th t di t ib ti



assumption about the return distribution.

•Let us denote the sample estimates by

MLE MOM

( )2,~ σ µ N r iid

t

( ) ( )1,0~ˆθ θ θ ∑− N T

a

( ) ( )2,0~ˆθ θ θ ∑− N T

a

∧

θ

returns depart from

iid normal



MLE vs. MOM

• As shown earlier, the asymptotic distribution usingeither MLE or MOM is normal with a zero mean but

distinct variance covariance matrices



distinct variance covariance matrices.

( ) ( )1,0~ˆθ θ θ ∑− N T

MLE ( ) ( )2,0~ˆθ θ θ ∑− N T

MOM



Distribution of the SR Estimate:The Delta Method

• We will show that ( ) ( )2,0~ˆSR

a

N SR RS T σ −



• We use the Delta method to derive

• The delta method is based upon the first order Taylor

approximation.

2

SRσ



Distribution of the SR Estimate:The Delta Method

• The first-order TA is∂SR



•The derivative is estimated at

( ) ( ) ( )( ) ( ) ( ) 12

2111

1221

11

ˆ'

ˆ

ˆ'

ˆ

××

×

××

×

−⋅∂∂

=−

⇒−⋅∂

∂

+=

θ θ θ

θ θ

θ θ θ θ θ

SRSRSR

SR

SRSR

θ



Distribution of the Sample SR

( ) ( )[ ] ( )( )'

0ˆˆ θ θ θ

θ θ E SR

E SRSR E =

−

∂∂

=−





( ) ( )( )

( ) ( )[ ] ( ) ( )[ ]

2

'

ˆˆ

ˆˆˆ

θ θ θ θ

θ θ θ θ θ

θ

SRSR E SRSRVAR

E VAR

MOREOVER

−=−

−−=

∂

The Variance of the SR

• Continue: ( )( )θ

θ θ θ θ θ

=

∂∂−−

∂∂ SRSR E

'

'ˆˆ



( )( )

θ θ

θ θ θ θ θ

θ

θ ∂∂∑

∂∂=

=∂∂

−−

∂∂=

SRSR

SR E SR

'

'

'ˆˆ



First Derivatives of the SR

• The SR is formulated as

•Let us derive:( ) 5.02σ

µ f

r SR

−=



( ) ( ) ( )32

5.02

2 22

1

1

σ

µ

σ

µ σ

σ

σ µ

f f r r

SR

SR

−−=

−−

=∂∂

=

∂

∂

−



The Distribution of SR under MLE

• Continue:

−∂∂ 2' 01 rSRSR σµ

1



,

( ) +−⇒

−=∂

∂∑∂

∂

2

43

1

'

ˆ211,0~ˆ

2,0

0,

2,1

RS N RS SRT

r SRSR f

σ

σ

σ

µ

σ θ θ θ

−−

32σ

µ

σ

f r



The Delta Method in General

• Here is the general application of the delta method

• If: ( )θ θ θ ∑− ,0~ˆ N T



• Then let be some function of :

• where is the vector of derivatives of

with respect to

( )

( )θ d θ

( ) ( ) ( )'

,0~ˆ

θ θ θ θ θ D D N d d T ×∑×−( )θ D ( )θ d

θ



Hypothesis Testing

• Does the S&P index outperform the Rf ?



H 0: SR=0

H 1: Otherwise

• Under the null there is no outperformance.

• Thus, under the null

)1,0(~ˆ1

ˆ

)ˆ1,0(~ˆ

2

2

N RS

RS T

RS N RS T

+

+



Lecture Notes in




The Efficient Frontier and theTangency Portfolio



Testing Asset Pricing Models

• Central to this course is the introduction of test

statistics to examine the validity of asset pricing



models.

• There are time series as well as cross sectional

asset pricing tests.



Testing Asset Pricing Models

• Time series tests correspond to only those caseswhere the factors are portfolio spreads, such as

excess return on the market portfolio as well as the



excess return on the market portfolio as well as the

SMB (small minus big), the HML (high minus low),

and the WML (winner minus loser) portfolios.

• The cross sectional tests apply to both portfolio and

non-portfolio based factors.

• Consumption growth in the CCAPM is a goodexample of a factor which is not a return spread.



Time Series Tests and the TangencyPortfolio

• Interestingly, time series tests are directly linked tothe notion of the tangency portfolio and the efficient

frontier.



• Here is the efficient frontier, in which the tangency

portfolio is denoted by T.

T



Economic Interpretation of the Time

Series Tests• Testing the validity of the CAPM entails the time

series regressions:t

e

mt

e

t r r ε β α ++= 1111



• The CAPM says: H0:

H1: Otherwise

Nt

e

mt N N

e

Nt r r ε β α ++=

KKKKKKK

N α α α === K21



Economic Interpretation of the Time

Series Tests• The null is equivalent to the hypothesis that the

market portfolio is the tangency portfolio.



• Of course, even if the model is valid – the market portfolio WILL NEVER lie on the estimated

frontier.

• This is due to sampling errors.

• The question is whether the market portfolio isclose enough, statistically, to the tangency portfolio.



What about Multi-Factor Models?

• The CAPM is a one-factor model.



• There are several extensions to the CAPM.

• The multivariate version is given by the K -factor

model:



Testing Multifactor Models

Nt K NK N N N

e

Nt

t K K

e

t

f f f r

f f f r

ε β β β α

ε β β β α

+++++=

+++++=

K

KKKKKKKKKKKKKKK

K

2211

1121211111



• The null is again: H0:

H1: Otherwise

• In the multi-factor context, the hypothesis is that

some optimal combination of the factors is thetangency portfolio.

N α α α === K21



The Efficient Frontier: Investable Assets

Consider N risky assets whose returns at time t are:

=t

t

R

R K

1



The expected value of return is denoted by:

×

Nt

N t

R1

( )

( )

( )

µ

µ

µ

=

=

=

N Nt

t

t

R E

R E

R E KK

11



The Covariance Matrix

The variance covariance matrix is denoted by:

1

2

1 ,,, N σ σ KKK



( )( )[ ]

=−−=

2

222

,

,,,'

N

N

t t R R E V

σ

σ σ µ µ

KKKKK

KKKKKK

KKK



Creating a Portfolio

• A portfolio is investing is the N assets.

=×

N

N

w

w

w K

1

1



• The return of the portfolio is:

• The expected return of the portfolio is:

N N p Rw Rw Rw R +++= K2211

( ) ( ) ( )

∑= ==+++=

=+++= N

iii N N

N N p

wwwww

R E w R E w R E w R E

12211

2211

'µ µ µ µ µ K

K




• The variance of the portfolio is:

N N p p wwwww RVAR 111221

2

1

2

1

2 σ σ σ σ K++==





22

2211

2222

221221

N N N N N N

N N

wwwww

wwwww

σ σ σ

σ σ σ

++++

+

+++++

K

M

K


• Thus ∑∑= =

= N

i

N

i

ij ji ji p ww1 1

2 ρ σ σ σ



where is the coefficient of correlation.• Using matrix notation:

1111

2 '××××

= N N N N

P wV wσ

ij ρ



The Case of Two Risky Assets

• To illustrate, let us consider two risky assets:2211 Rw Rw R p +=

22222



• We know:

• Let us check:

• So it works!

22122111 2 σ σ σ σ wwww RVAR pt p ++==

( ) 2

2

2

21221

2

1

2

1

2

1

2

212

12

2

1

21

2 2,,, σ σ σ σ σ σ σ σ wwww

wwww p ++=

=



Dominance of the Covariance

• When the number of assets is large, the

covariances define the portfolio’s rate of return.



• To illustrate, assume that all assets have the samevolatility and pairwise correlations.

• Then an equal weight portfolio’s variation is

cov)1( 2

2

22 =→

−+=

∞→

ρ σ ρ

σ σ N

p

N

N N N



The Efficient Frontier: ExcludingRiskfree Asset

The optimization program:

min Vww'



s.t

where is the Greek letter iota ,

and where is the expected

return target set by the investor.

1' =ι w

pw µ µ ='

ι

=

×

1

1

1K

N ι

pµ



The Efficient Frontier: Excluding

Riskfree AssetUsing the Lagrange setup:

1



( ) ( )

µ λ ι λ

µ λ ι λ

µ µ λ ι λ

1

2

1

1

21

21

0

''1'2

1

−− +=

=−−=∂∂

−+−+=

V V w

Vww L

wwVww L p



The Efficient Frontier: WithoutRiskfree Asset

• Let

ι ι

µ µ µ ι

1

1

1

'

''

−

−

−

=

==

V c

V bV a



• The optimal portfolio is:

( )

( )ι µ

µ ι

11

11

2

1

1

−−

−−

−=

−=

−=

aV cV d

h

aV bV

d

g

abcd

p N N N

h g w µ 111

*

×

+=××



Examine the Optimal Solution

• That is, once you specify the expected return target,

the optimal portfolio follows immediately.



• Let us check whether the sum of weights is equal

to 1.




( ) ( ) 1

1

''

1

'

'''

211

+=

−−

d

d

bdVVbd

h g w pµ ι ι ι



Indeed, .

( ) ( )( ) ( ) 0

1''

1'

1'''

11 =−=−=

==−=−=−− acac

d

V aV c

d

h

d abcd V aV bd g

ι ι µ ι ι

µ ι ι ι ι

1' =ι w




• Let us now check whether the expected return on

the portfolio is equal to .

R ll

pµ

h



• Recall:

• So we need to show

( )µ µ µ µ

µ

''' h g w

h g w

p

p

+=+=

1'

0'

=

=

µ

µ

h

g




( ) ( ) 01''1' 11 =−=−= −− ababd

V aV bd

g µ µ µ ι µ



Try it yourself: prove that .1' =µ h



Efficient Frontier The optimization program also delivers the shape of

the frontier.



inefficient part

A



Efficient Frontier

• Where point A stands for the Global Minimum

Variance Portfolio (GMVP).



• The efficient frontier reflects the investment

opportunities; this is the supply side.

• Points below A are inefficient since they are being

dominated by other more attractive portfolios.



The Notion of Dominance

If

AB

A B

µ µ

σ σ

≥

≤



and there is at least one strong inequality, then portfolio B dominates portfolio A.

A B



The Efficient Frontier with Riskfree

AssetIn practice, there is no really a riskfree asset. Why?

C dit i k



• Credit risk.

• Inflation risk.

• Interest rate risk and the horizon effect.



Setting the Optimization in the Presenceof Riskfree Asset

• The optimal solution is given by:

min

s t or

Vww'

( ) pfRww µιµ + '1' pef wR µµ+ '



s.t or,

and the tangency portfolio takes the form:

• The tangency portfolio is investing all the funds in

risky assets.

( ) p f Rww µ ι µ =−+ 1 p f w R µ µ =+

( )( ) e

e

f

f

V V

RV RV w

µ ι µ

ι µ ι ι µ

1

1

1

1

*

'' −

−

−

−

=⋅−

⋅−=



The Investment Opportunities• However, the investor could select any point in the

line emerging from the riskfree rate and touching the

efficient frontier in point T.



• The location depends on the attitude toward risk.

T



Fund Separation• Interestingly, all investors in the economy will mix

the tangency portfolio with a riskfree asset.• The mix depends on preferences.

• But the proportion of risky assets will be equal



• But the proportion of risky assets will be equalacross the board.

• One way to test the CAPM is indeed to examinewhether all investors hold the same proportions of

risky assets.

• Obviously they don’t!



General Equilibrium• The efficient frontier reflects the supply side.

• What about the demand?

• The demand side can be represented by a set of

indifference curves



indifference curves.



General Equilibrium• Why is the slope of indifference curve positive?

• The equilibrium obtains when the indifference

curve tangents the efficient frontier

• No riskfree asset:



No riskfree asset:

A



General EquilibriumWith riskfree asset:

A



A



Maximize Utility / Certainty

Equivalent ReturnIn the presence of a riskfree security, the tangency

point can be found by maximizing a utility functionf h f



point can be found by maximizing a utility functionof the form

where is the relative risk aversion.

2

2

1 p pU γσ µ −=

γ



Maximize Utility / Certainty

Equivalent Return• Notice that utility is equal to expected return minus

a penalty factor.



a penalty factor.

• The penalty factor positively depends on the risk aversion (demand) and variance (supply).



Utility Maximization( )

e

e

e

f

V w

Vww

U

Vwww RwU

µ γ

γ µ

γ µ

1* 1

0

'2

1'

−

=

=−=∂∂

⋅−+=



Notice that here we get the same tangency portfolio

where reflects the fraction of weights invested in

risky assets. The rest is invested in the riskfree asset.

e

e

V V w

µ ι µ 1

1

*

' −

−

=*w



Mixing the Risky and RiskfreeAssets

• So if then all funds (100%) areinvested in risky assets

eV w µ γ

1* 1 −=

eV µ ι γ 1' −=



( )invested in risky assets.

• If then some fraction is invested in

riskfree asset.

• If then the investor borrows money

to leverage his/her equity position.

eV µ ι γ 1' −>

eV µ ι γ 1' −<



The Exponential Utility Function

• The exponential utility function is of the form

where

• Notice that

( ) ( )W W U λ −−= exp 0>λ

( ) ( ) 0exp' >−= WWU λλ



• That is, the marginal utility is positive but

diminishes with an increasing wealth.

( ) ( )

( ) ( ) 0exp''

0exp

2 <−−=

>=

W W U

W W U

λ λ

λ λ



The Exponential Utility Function

• Indeed, the exponential preferences belong to the

λ =−='

''

U

U ARA



, p p g

class of constant absolute risk aversion (CARA).

• For comparison, power preferences belong to the

class of constant relative risk aversion (CRRA).



Exponential: The Optimization

Mechanism• The investor maximizes the expected value of the

exponential utility where the decision variable is theset of weights w and subject to the wealth evolution



p yset of weights w and subject to the wealth evolution.

• That is

( )[ ]t t W W U E 1+w

max

t s. et f t t Rw RW W 11 '1 ++ ++='Prof. Doron Avramov



Mechanism• Let us assume that V N R ee

t ,~1 µ +



• Then

mean variance

( )[ ]VwwW w RW N W t

e

f t t ','1~ 2

1 µ +++




MechanismIt is known that for

2,~ x x N x σ µ



( )[ ]

+=

22

2

1expexp x x aaax E σ µ




MechanismThus,

( )[ ] ( )( ) ( )

⋅+−−=−− +++ W VARW E W E t t t

21expexp 1

211 λ λ λ



( )[ ] ( )( ) ( )

( )

( )( )

−−+−−=

+++−−=

VwwW wW RW

VwwW w RW

t

e

t f t

t

e

f t

'2

1

'exp1exp

'2

1'1exp

2

22

λ µ λ λ

λ µ λ




Mechanism Notice that

t W λ γ =



where is the relative risk aversion coefficient.γ



Exponential: The OptimizationMechanism

• So the investor ultimately maximizes

w

max

− Vwww e '2

1' γ µ



• The optimal solution is .

• The tangency portfolio is the same as before

eV w µ γ

1* 1 −=

e

e

V

V

w µ ι

µ 1

1*

' −

−

=




MechanismConclusion:

The joint assumption of exponential utility and



j p p y

normally distributed stock return leads to the well-

known mean variance solution.



Quadratic Preferences

• The quadratic utility function is of the formwhere( ) 2

2W

bW aW U −+= 0>b

1−=∂∂ bW WU



• Notice that the first derivative is positive for

02

2

<−=∂∂

∂

bW U

W

W

b1

<




The utility function looks like( )W U



W

b

1




• It has a diminishing part – which makes no sense – because we always prefer higher than lower wealth

• Utility is thus restricted to the positive slope part

• Notice thatbW

WU

RRA ===''

γ



• Notice that

• The optimization formulation is given by

bW W

U RRA

−=−==

1'γ

wmax

t s. et ft t t Rw RW W 11 '1 ++ ++=

( )[ ]t t W W U E 1+




• Avramov and Chordia (2006 JFE ) show that theoptimization could be formulated as

( )wV w

R

w ee

ft

e 1''1

2

1'

−+

−

− µ µ µ w

max



• The solution takes the form

( ) ft

γ

ee

e

ft

V

V Rw

µ µ

µ

γ

1'

1*

1

1−

−

+

−=




• The tangency portfolio is

e

e

V

V

w µ ι

µ 1

1*

' −

−

=



• The only difference from the previously presented

competing specifications is the composition of risky

and riskfree assets.



The Sharpe Ratio of the TangencyPortfolio

• Notice that is actually the squaredSharpe Ratio of the tangency portfolio.

• Let us prove it

eeV µ µ 1' −

1*

eV µ−



( )

( )21

1'

**'2

1

1'*'*'*'

1

*

'

'1

'

e

ee

TP

e

eee

f

e

f TP

e

V V wV w

V

V w Rww R

V

V w

µ ι µ µ σ

µ ι

µ µ µ ι µ µ

µ ι

µ

−

−

−

−

−

==

==−+=−

=



The Sharpe Ratio of the Tangency

Portfolio

• Thus,

N ti th t TP i b i t f th t

( ) ee

TP

f TP

TP V R

SR µ µ σ

µ 1'

2

2

2 −=−

=



• Notice that TP is a subscript for the tangency

portfolio.



Lecture Notes in




Testing Asset Pricing Models:

Time Series Perspective



Why Caring about Asset

Pricing Models?• An essential question that arises is why would both

academics and practitioners invest huge resources in

developing and testing asset pricing models.



• It turns out that pricing models have crucial roles in

various applications in financial economics – bothasset pricing as well as corporate finance.

•In the following, I list five major applications.



1 – Common Risk Factors

• Pricing models characterize the risk profile of a firm.

• In particular, systematic risk is no longer stock returnvolatility – rather it is the loadings on risk factors.



y g

• For instance, in the single factor CAPM the market

beta – or the co-variation with the market –

characterizes the systematic risk of the firm.




• Likewise, in the single factor (C)CAPM theconsumption growth beta – or the co-variation with

consumption growth – characterizes the systematic risk

of the firm.

• In the m lti factor Fama French (FF) model there are



• In the multi-factor Fama-French (FF) model there are

three sources of risk – the market beta, the SMB beta,and the HML beta.




• Under FF, other things being equal (ceteris paribus), afirm is riskier if its loading on SMB beta is higher.

• Under FF, other things being equal (ceteris paribus), a

firm is riskier if its loading on HML beta is higher.





2 – Moments for Asset

Allocation• Pricing models deliver moments for asset allocation.

• For instance, the tangency portfolio takes on the form



e

e

TP V

V wµ ι µ 1

1

' −

−

=



2 – Asset Allocation

Under the CAPM, the vector of expected returns and thecovariance matrix are given by:

e

m

e βµ µ =

∑+= 2' mV σ ββ



where is the covariance matrix of the residuals in the

time-series asset pricing regression.

We denoted by the residual covariance matrix in the

case wherein the off diagonal elements are zeroed out.

∑

Ψ



2 –Asset Allocation

The corresponding quantities under the FF model are

++= µ β µ β µ β µ HML HMLSMLSML

e

m MKT

e



where is the covariance matrix of the factors.

∑+∑= ' β β F V

F ∑



3 – Discount Factors

• Expected return is the discount factor, commonlydenoted by k , in present value formulas in general

and firm evaluation in particular:

( )∑=T

t

t

k

CF PV

1



• In practical applications, expected returns are

typically assumed to be constant over time, an

unrealistic assumption.

( )∑= +t

t k 1 1



3 – Discount Factors• Indeed, thus far we have examined models with

constant beta and constant risk premiums

where is a K -vector of risk premiums.

Wh f t t d th i k i i

λ β µ '=e

λ



• When factors are return spreads the risk premium is

the mean of the factor.

• Later we will consider models with time varying

factor loadings.



4 – Benchmarks

• Factors in asset pricing models serve as benchmarksfor evaluating performance of active investments.

• In particular, performance is the intercept (alpha) in

the time series regression of excess fund returns on a

set of benchmarks (typically four benchmarks in



set of benchmarks (typically four benchmarks in

mutual funds and more so in hedge funds):

t t WMLt HML

t SMB

e

t MKT MKT

e

t

WML HML

SMBr r

ε β β

β β α

+×+×+

×+×+= ,



5 – Corporate Finance

• There is a plethora of studies in corporate finance thatuse asset pricing models to risk adjust asset returns.

• Here are several examples:

o Examining the long run performance of IPO firm.



o Examining the long run performance of SEO firms.

o Analyzing abnormal performance of stocks going

through splits and reverse splits.



5 – Corporate Finance

o Analyzing mergers and acquisitions

o Analyzing the impact of change in board of

directors.

o Studying the impact of corporate governance on

h i f



the cross section of average returns.

o Studying the long run impact of stock/bond

repurchase.



Time Series Tests

• Time series tests are designated to examine the validityof models in which factors are portfolio based, or

factors that are return spreads.

• Example: the market factor is the return difference

between the market portfolio and the riskfree asset.



p

• Consumption growth is not a return spread.

• Thus, the consumption CAPM cannot be tested using

time series regressions, unless you form a factor

mimicking portfolio (FMP) for consumption growth.



Time Series Tests• FMP is a convex combination of asset returns

having the maximal correlation with consumptiongrowth.

• The statistical time series tests have an appealingeconomic interpretation. In particular:



• Testing the CAPM amounts to testing whether the

market portfolio is the tangency portfolio.

• Testing multi-factor models amounts to testing

whether some optimal combination of the factors isthe tangency portfolio.



Testing the CAPM

• Run the time series regression:

ee

t

e

mt

e

t

rr

r r

εβα

ε β α

++=

++=

M

1111



• The null hypothesis is:

Nt mt N N Nt r r ε β α ++=

0: 210 ==== N H α α α K



Testing the CAPM

In the following, I will introduce four times series teststatistics:

• WALD.

• Likelihood Ratio.



• GRS (Gibbons, Ross, and Shanken (1989)).

• GMM.



The Distribution of .

• Recall, is asset mispricing.• The time series regressions can be rewritten using a

vector form as:

α

α

111111 NX t

X

e

mt NX NX NX

e

t r r ε β α +⋅+=



• Let us assume that

for t=1,2,3,…,T

• Let be the set of all

parameters.

∑

NxN

iid

Nxt N ,0~1

ε

( )( )'',',' ε α θ vech=




• Under normality, the likelihood function for is

α

( ) ( ) ( )

−−∑−−−∑=−−

e

mt

e

t

e

mt

e

t t r r r r c L β α β α θ ε 1'

2

1

2

1

exp

t ε



where c is the constant of integration (recall theintegral of a probability distribution function is

unity).




• Moreover, the IID assumption suggests that

α

( )

( ) ( )

−−∑−−−×

∑=

∑=

−

−

T

t

e

mt

e

t

e

mt

e

t

T

T

N

r r r r

c L

1

1'

221

2

1exp

,,,

β α β α

θ ε ε ε K



• Taking the natural log from both sides yields

t

( ) ( ) ( ) ( )∑=

− −−∑−−−∑−∝T

t

e

mt

e

t

e

mt

e

t r r r r T

L

1

1'

2

1ln

2

ln β α β α




Asymptotically, we have

where

α

( ))(,0~ˆ

θ θ θ Σ− N

( )1

2

'

ln−

∂∂−=Σ

θθ

α θ E



' ∂∂ θ θ




Let us estimate the parameters

α

( ) ( )

( ) ( )1

1

1

ln

ln

−

=

−

×−−∑=∂

−−∑=

∂

∂

∑∑

eT

ee

T

t

e

mt

e

t

rrr L

r r L

βα

β α

α



( )

( ) 1

1

'11

1

2

1

2

ln −

=

−−

=

∑

∑+∑−=

∑∂∂

×∑=

∂

∑

∑T

t

t t

mt

t

mt t

T L

r r r

ε ε

β α

β




Solving for the first order conditions yields

α

( )( )∑ −−

⋅−=T

e

m

e

mt

ee

t

e

m

e

r r ˆˆ

ˆˆˆˆ

µ µ

µ β µ α



( )( )

( )∑

∑

=

=

−= T

t

e

m

e

mt

t

r 1

2

1

ˆˆ

µ β




Moreover,

α

∑

∑=

=

=∑

T e

t

e

T

t

t t

r T

T 1

'

1ˆ

ˆˆ1ˆ

µ

ε ε



∑

∑

=

=

=T

t

e

mt

e

m

t

t

r T

T

1

1

1µ

µ




• Recall our objective is to find the variance-covariance matrix of .

• Standard errors could be found using the information

matrix.

α

α






• The information matrix is constructed as follows

α

( )

( ) ( ) ( )

( ) ( ) ( )

∂∂∂∑∂∂

∂

∂∂

∂

∂∂

∂

−=ln

,ln

,ln

'

ln,

'

ln,

'

ln

222

222

L L L

L L L

E I

α β α α α

θ



( )

( ) ( ) ( )

∑∂∑∂∂

∂∑∂∂

∂∑∂∂ ∑∂∂∂∂∂∂

'

ln,

'

ln,

'

ln'

,

'

,

'222 L L L

β α

β β β α β



The Distribution of the Parameters

• Try to establish yourself the information matrix.

• Notice that and are independent of -

thus, your can ignore the second derivatives withrespect to in the information matrix if your

objective is to find the distribution of and .

α

β

∑

∑α

β



j

• If you aim to derive the distribution of then

focus on the bottom right block of the information

matrix.

β



∑


• We get:

• Moreover,

α

∑

+2

ˆ

ˆ1

1,~ˆ

m

e

m

T N

σ

µ α α



∑⋅ 2ˆ

11,~ˆmT

N σ

β β

( )∑−∑ ,2~ˆ T W T




• Notice that W (x,y) stands for the Wishartdistribution with x=T-2 degrees of freedom and a

parameter matrix .

α

∑= y





The Wald Test

• Recall, if then

• Here we testwhere

( )∑,~ µ N X ( ) ( ) ( ) N X X 21 ~ˆ χ µ µ −∑− −

0ˆ:

0ˆ:

1

0

≠

=

α

α

H

H ( )α α ∑,0~ˆ

0

N H



• The Wald statistic is

0:1 ≠α H

( N 21 ~ˆˆ'ˆ χ α α α

−∑



The Wald Test

which becomes:

2

11

12

1

ˆ1

ˆˆ'ˆˆˆ'ˆ

ˆ

ˆ1

mm

e

m

RS

T T J

+

∑=∑

+=

−−

−

α α α α

σ

µ

ˆ



where is the Sharpe ratio of the market factor.m RS



Algorithm for Implementation

• The algorithm for implementing the statistic is as

follows:

• Run separate regressions for the test assets on the

common factor:

11

121

21 TxxTxTx

e

t X r ε θ += ,1 1

e

mr

M



where

11221

1121

Tx N

x N

TxTx

e

N

Tx xTx

X r ε θ +=

M

[ ]',

,12

iii

e

mT

Tx

r X

β α θ =

=M




• Retain the estimated regression intercepts

and

• Compute the residual covariance matrix

[ ]'ˆ,,ˆ,ˆˆ21 N α α α α K=

=

∧∧∧

N TxN

ε ε ε ,,1 K

∧∧

=∑ εε '1ˆ



=∑ ε ε T




•Compute the sample mean and the sample variance

of the factor.•Compute J 1.





The Likelihood Ratio Test

• We run the unrestricted and restricted specifications:

un:

res:

t

e

mt

e

t r r ε β α ++=

**

t

e

mt

e

t r r ε β +=

( )∑,0~ N t ε

** ,0~ ∑ N t ε



• Using MLE, we get:




( )

=∑

=

∑

∑

∑

=

=

=

'ˆˆ1ˆ

ˆ

1

***

1

2

1*

T

r

r r

T

t

t t

T

t

e

mt

T

t

e

mt

e

t

ε ε

β



( )∑−∑

∑

+

,1~ˆ

ˆˆ

11,~ˆ

*

22

*

T W T

T N

mm σ µ β β




where again W is the Wishart distribution, this timewith T-1 degrees of freedom.





The LR Test

• Using some algebra, one can show that

( ) ( ) [ ][ ] ( ) N T LR J

T

L L LR

2*

2

**

~ˆlnˆln2

ˆln

ˆln2lnln

χ ∑−∑=−=

∑−∑−=−=

2 J



• Thus, − = 1exp1 T T J

+⋅= 1ln

12

T

J T J



GRS (1989)

Theorem: let

let where

and let A and X be independent then

( )∑,0~1

N X Nx

( )∑,~1

τ W A Nx

N ≥τ



1,

1 ~'1

+−−+−

N N F X A X N

N τ

τ



GRS (1989)

In our context:

( )

( )

2

,~

ˆ

,0~ˆˆˆ1

02

12

−=

∑∑=

∑

+=

−

T

whereW T A

N T X H

m

m

τ

τ

α σ µ



Then:

This is a finite-sample test.

( )1,~ˆˆ'ˆˆ

ˆ1

1 1

12

3 −−∑

+

−−= −

−

N T N F N

N T J

m

m α α σ

µ



GMM

I will directly give the statistic without derivation:

where

( )( ) )(~ˆ''ˆ 2111'

4

0

N R DS D RT J H

T T T χ α α ⋅=−−−

NxN NxN N

N Nx I R

=

20,



( ) N

m

e

m

e

m

T I D ⊗

+

−=22

ˆˆ,ˆ

,1

σ µ µ

emµ ˆ



GMM

• Assume no serial correlation but heteroskedasticity:

( )

[ ]'

1

''

,1

ˆˆ1

e

mt t

T

t

t t t t T

r x

where

x xT

S

=

⊗= ∑=

ε ε



• Under homoskedasticity and serially uncorrelatedmoment conditions: J 4=J 1.

• That is, the GMM statistic boils down to the WALD.



The Multi-Factor Version of Asset Pricing Tests

J 2 follows as described earlier.

( ) ( ) N T J

F r

F F F

Nxt

Kxt

NxK Nx Nx

et

χ α α µ µ

ε β α

~ˆˆ'ˆˆˆˆ1 11

1'

1

1111

−−

− ∑∑+=

+⋅+=



where is the mean vector of the factor basedreturn spreads.

( ) ( ) K N T N F F F F N

K N T J −−−−− ∑∑+−−= ,

111'

3 ~ˆˆ'ˆˆˆˆ1 α α µ µ

F µ ˆ



The Multi-Factor Version of Asset Pricing Tests

• is the variance covariance matrix of the factors.

•For instance, considering the Fama-French model:

eµ ˆ

F ∑

,,

2 ˆ,ˆ,ˆ HMLmSMBmm σ σ σ



=

HML

SMB

m

F

µ

µ

µ

µ

ˆ

ˆˆ

=∑2

,,,

,

2

,

ˆˆ,ˆ

ˆ,ˆ,ˆˆ

HMLSMB HMLm HML

HMLSMBSMBmSMB F

σ σ σ

σ σ σ



The Current State of Asset PricingModels

o The CAPM has been rejected in asset pricing

tests.

o The Fama-French model is not a big success.

o Conditional versions of the CAPM and CCAPM



display some improvement.

o Should decision-makers abandon a rejected

CAPM?



Should a Rejected CAPM beAbandoned?

• Not necessarily!

• Assume that expected stock return is given by

where

• You estimate using the sample mean and CAPM:

f mi f ii R R −++= µ β α µ 0≠iα

iµ



g p



( ) ∑=

=T

t

it i RT 1

1 1µ

( ) ( ) f mi f i R R −+= µ β µ ˆˆˆ 2

Mean Squared Error (MSE)

The quality of estimates is evaluated based on

the Mean Squared Error (MSE)

( ) ( )( )2

211 ˆii E MSE µ µ −=





( ) ( )( )22 ˆ ii E MSE µ µ −=

MSE, Bias, and Noise of Estimates

• Notice that

• Of course, the sample mean is unbiased thus

• However, the CAPM is rejected, thus

(

estimateVar bias MSE += 2

( ) ( )11 ˆiVar MSE µ =





( ) ( ))222 îi Var MSE µ α +=

The Bias-Variance Tradeoff

• It might be the case that is significantly

lower than - thus even when the CAPM is

rejected, still zeroing out could produce a smaller mean square error.

( )2îVar µ

( )1îVar µ

iα





When is the Rejected CAPM Superior?

( )( ) ( ) ( ) ( )[ ]

( )( ) ( )emii

imiii

Var Var

RT

RT

Var

µ β µ

ε σ σ β σ µ

ˆˆˆ

11ˆ

2

22221

=

+==





When is the Rejected CAPM Superior?Using variance decomposition

( )( )

( ) ( ) ( ) ( )

( )[ ]+=

+

=+

⋅=

+==

222

22

2

2

2

2

22

2

ˆ1

ˆ

ˆ

ˆ1ˆ

1

ˆ

ˆ

ˆ|ˆˆˆ|ˆˆ)ˆˆ(ˆ

miim

mi

m

e

mi

e

mi

m

ie

m

em

emi

em

emi

emii

h

SRT

T

E

T

Var

T

E

E

Var Var E Var Var

σ β ε σ

σ β

σ

µ ε σ µ β

σ

ε σ µ

µ µ β µ µ β µ β µ





=

2

2

ˆ

ˆ

m

e

mm E SR

where

σ

µ

When is the Rejected CAPM Superior?• Then

where is the R squared in the market regression.

Si i ll th ti f th i

( )( )( )( )

( )( )

( ) 22

2

222

1

2

1ˆ

ˆ R RSR

R

SR

Var

Var m

i

miim

i

i +−=+=σ

σ β ε σ

µ

µ

2 R

SR



• Since is small -- the ratio of the varianceestimates is smaller than 1.



mSR

Example• Let

• For what values of it is sill preferred to use

th CAPM?

( )

3.0

05.0ˆ

ˆ

01.0

2

2

2

=

=

=

R

E

R

m

em

i

σ

µ

σ

0≠iα



the CAPM?• Find such that the MSE of the CAPM is smaller.



iα

Example

( )

( ) ( ) ( )

665001060

1

1

01.0601

3.07.005.0

11

2

2

1

222

1

2

<

⋅

++×

<++−=

i

i

im

MSE R RSR MSE

MSE

α

α

α





%528.101528.0

665.001.060

=<×⋅<

i

i

α

α

Economic versus StatisticalFactors

• Factors such as the market portfolio, SMB, HML,

WML, liquidity, credit risk, as well as bond based

factors are pre-specified.

• Such factors are considered to be economically

based



based.

• For instance, Fama and French argue that SMB and

HML factors are proxying for underlying state

variables in the economy.'Prof. Doron Avramov


Economic versus StatisticalFactors

• Statistical factors are derived using econometric procedures on the covariance matrix of stock return.

• Two prominent methods are the factor analysis andthe principal component analysis (PCA).

• Such methods are used to extract common factors.



• The first factor typically has a strong (about 96%)

correlation with the market portfolio.

• Later, I will explain the PCA.'Prof. Doron Avramov


The Economics of Time SeriesTest Statistics

Let us summarize the first three test statistics:

+⋅= 1ln 12

T

J T J

2

1

1

ˆ1

ˆˆ'ˆ

m RS

T J

+

∑=

− α α



2

1

3

ˆ1

ˆˆ'ˆ1

m RS N

N T J

+

∑−−=

− α α



The Economics of the TimeSeries Tests

• The J 4 statistic, the GMM based asset pricing test, is

actually a Wald test, just like J 1, except that the

covariance matrix of asset mispricing takes accountof heteroskedasticity and often even potential serial

correlation.

• Notice that all test statistics depend on the quantity



• Notice that all test statistics depend on the quantity

α α ˆˆ'ˆ 1−∑



The Economics of the TimeSeries Tests

• GRS show that this quantity has a very insightful

representation.

• Let us provide the steps.





• Consider an investment universe that consists of N+1

assets - the N test assets as well as the market portfolio.

• The expected return vector of the N+1 assets is given

by

h i h i d d

( )''ˆ,ˆˆ

11111

=

+ xN

e

x

e

m x N

µ µ λ

eˆ

Understanding the Quantity .α α ˆˆ'ˆ1−

∑



where is the estimated expected excess return on

the market portfolio and is the estimated expected

excess return on the N test assets.

e

mµ ˆeµ ˆ



• The variance covariance matrix of the N+1 assets is

given by

where

is the estimated variance of the market factor

( ) ( )

=Φ

++

V m

mm

N x N

ˆ,ˆˆ

ˆ'ˆ,ˆˆ

2

22

11

σ β

σ β σ

2

ˆ mσ


∑



is the estimated variance of the market factor.

is the N -vector of market loadings and is the

covariance matrix of the N test assets.

mσ

β V



• Notice that the covariance matrix of the N test assets

is

• The squared tangency portfolio of the N+1 assets is

∑+= ˆˆ'ˆˆˆ 2

mV σ β β

ˆˆˆˆ 12


∑



λ λ ˆ'ˆ 12 −Φ=TP RS



• Notice also that the inverse of the covariance matrix

is

• Thus, the squared Sharpe ratio of the tangency

( )

∑−

∑−∑+=Φ

−

−−−

−

β

β β β σ

ˆˆ

ˆ'ˆ,ˆˆ'ˆˆˆ

1

1112

1 m

1ˆ, −∑


∑



Thus, the squared Sharpe ratio of the tangency portfolio could be represented as



( ) ( )

221

122

1'

2

2

ˆˆˆˆ'ˆ

ˆˆ'ˆˆˆ

ˆˆˆˆˆˆˆˆˆˆ

mTP

e

m

ee

m

e

m

e

mTP

RSRS

or

RS RS

RS

∑

∑+=

−∑−+

=

−

−

−

αα

α α

µ β µ µ β µ σ µ


∑



mTP RS RS −=∑ α α



• In words, the quantity is the difference

between the squared Sharpe ratio based on the N+1assets and the squared Sharpe ratio of the market

portfolio.

• If the CAPM is correct then these two Sharpe ratios

are identical in population, but not identical in sample

due to estimation errors.

α α ˆˆ'ˆ 1−∑


∑



due to estimation errors.



• The test statistic examines how close the two sample

Sharpe ratios are.

• Under the CAPM, the extra N test assets do not add

anything to improving the risk return tradeoff.

• The geometric description of is given in

the next slide.α α ˆˆ'ˆ 1−∑


∑





1Φ

TP

2Φ Rf

r

σ


∑



2

2

2

1

1 ˆˆ'ˆ Φ−Φ=∑ − α α

σ



• So we can rewrite the previously derived test

statistics as


∑

( )

( )1,~ˆ

ˆˆ1

~

ˆ1

ˆˆ

2

22

3

2

2

22

1

−−−×−−=

+

−=

N T N F RS RS N T J

N

RS

RS RS T J

mTP

m

mTP χ



( ),ˆ1 23

+×=

RS N m



Asset Pricing Models withTime Varying Beta

• We consider for simplicity only the one factor

CAPM – extensions follow the same vein.

• Let us model beta variation with the lagged dividend

yield or any other macro variable – again for



y y gsimplicity we consider only one information,

predictive, macro, or lagged variable.



Asset Pricing Models withTime Varying Beta

• Typically, the set of predictive variables contains the

dividend yield, the term spread, the default spread,

the yield on a T-bill, inflation, lagged market return,

market volatility market illiquidity etc



market volatility, market illiquidity, etc.



Conditional Models

Here is a conditional asset pricing specification:

)|( 1

1

110

=

++=+=

++=

−

−

emt

emt

t t t

t iiit

it

e

mt it i

e

it

z r E

bz a z

z

r r

µ

η

β β β

ε β α



0),cov(

)|( 1

=−

t it

mt mt

η ε

µ



Conditional Asset PricingModels

• Substituting beta back into the asset pricing equation

yields.

• Interestingly, the one factor conditional CAPM

becomes a two factor unconditional model – the first

it t

e

mt i

e

mt ii

e

it z r r r ε β β α +++= −110



factor is the market portfolio, while the second is the

interaction of the market with the lagged variable.'Prof. Doron Avramov


Conditional Asset PricingModels

• You can use the statistics J 1 through J 4 to test such

models.

• If we have K factors and M predictive variables then

the K -conditional factor model becomes a K+KM-

unconditional factor model.

If l l th k t b t i t i ll th



• If you only scale the market beta, as is typically the

case, we have an M+K unconditional factor model.



Conditional Moments

Suppose you are at time t – what is the discount factor

for time t+2?

2121202 +++++ +++= it t

e

mt i

e

mt ii

e

it z r r r ε β β α

( ) 212120 ++++ +++×++= it t t

e

mt i

e

mt ii bz ar r ε η β β α

+++++= eeee rzbrrar εηββββα



2121212120 ++++++ +++++= it t mt it mt imt imt ii r z br r ar ε η β β β β α



Conditional Moments

t

e

mi

e

mi

e

miit

e

it z ba z r E µ β µ β µ β α 1102 +++=+

( ) t

e

mi

e

miii z ba µ β µ β β α 110 +++=

• Notice that here the discount factor, or the conditional

expected return, is no longer constant through time.



•Rather, it varies with the macro variable.



Conditional Moments

Could you derive a general formula – in particular –

you are at time t what is the expected return for time

t+T as a function of the model parameters as well as

?

t z





Conditional Moments

• Next, the conditional covariance matrix – the

covariance at time t+1 given – is given by

where and are the N -asset versions of

[ ] ( )( ) Σ+++=+2'

10101 mt t t e

t z z z r V σ β β β β

0

β 1

β 0i

β

t z



and respectively.1i β



Conditional Moments

• Could you derive a general formula – in particular –

you are at time t what is the conditional covariance

matrix for time t+T as a function of the model

parameters as well as ?

• Could you derive general expressions for the

t z



conditional moments of cumulative return?



Conditional versusUnconditional Models

• There are different ways to model beta variation.

Here we used lagged predictive variables; other

applications include using firm level variables such

as size and book market to scale beta as well as



modeling beta as an autoregressive process.



Conditional versusUnconditional Models

• You can also model time variation in the risk

premiums in addition to or instead of beta variations.

• Asset pricing tests show that conditional models

typically outperform their unconditionalcounterparts



counterparts.



Different Ways to Model BetaVariation

• The base case: beta is constant, or time invariant.

• Case II: beta varies with macro conditions

t t t

t iiit

bz a z

z

ε

β β β

++=

+=

−

−

1

110






• Case III: beta varies with firm-level size and the

book-to-market ratio

• Case IV: beta is some function of both macro and

firm-level variables as well as their interactions:

1,21,10 −− ++= t iit iiiit bm size β β β β

= bmsizezfβ



1,1,1 ,, −−−= t it it it bm size z f β




Case V: beta follows an auto-regressive AR(1) process

it t iit vba ++= −1, β β





Single vs. Multiple Factors• Notice that we describe the case of a single factor

single macro variable.

• We can expand the specification to include more

factors and more macro and firm-level variables.

• Even if we expand the number of factors it is

common to model variation only in the market beta,

while the other risk loadings are constant.



• Some scholars model time variations in all factor

loadings.'Prof. Doron Avramov


Testing Conditional Models• You can implement the J 1-J 4 test statistics only to

those cases where beta is either constant or it varieswith macro variables.

• Those specifications involving firm-levelcharacteristics require cross sectional tests.

• The last specification (AR(1)) requires filtering

methods involving state space representations.





Lecture Notes in


GMVP and Tracking ErrorVolatility





GMVP

• Of particular interest to academics and practitioners isthe Global Minimum Volatility Portfolio.

• For two distinct reasons:

1. No need to estimate the notoriously difficult to

estimate .

2. Low volatility stocks have been found to

t f hi h l tilit t k

µ



outperform high volatility stocks.



GMVP Optimization

min

s.t

• Solution:

• No analytical solution in the presence of portfolio

Vww'

1' =ι w

ι ι

ι 1

1

' −

−

=V

V wGMVP



• No analytical solution in the presence of portfolio

constraints –such as no short selling.'Prof. Doron Avramov


GMVP Optimization

• Ex ante, the GMVP is the lowest volatility portfolioamong all efficient portfolios.

• Ex ante, it is also the lowest mean portfolio, but ex

post it performs reasonably well in delivering high

payoffs.





The Tracking Error Volatility(TEV) Portfolio

• Actively managed funds are often evaluated based on

their ability to achieve high return subject to some

constraint on their Tracking Error Volatility (TEV).

• In that context, a managed portfolio can be

decomposed into both passive and active components.

• TEV is the volatility of the active component.





The Tracking Error Volatility(TEV) Portfolio

• The passive component is the benchmark portfolio.

• The benchmark portfolio changes with the

investment objective.

• For instance, if you invest in TA 100 stocks the

proper benchmark would be the TA100 index.• If you invest in corporate bonds traded in TASE the



If you invest in corporate bonds traded in TASE the

benchmark could be TelBond 60.



Tracking Error Volatility: TheBenchmark and Active Portfolios

• Let q be the vector of weights of the benchmark

portfolio.

• Then the expected return and variance of the benchmark portfolio are given by

where, as usual, µ and V are the vector of expected

Vqq

q

B

B

'

'

2

=

=

σ

µ µ



, , µ p

return and the covariance matrix of stock returns.



Tracking Error Volatility: TheBenchmark and Active Portfolios

• The matrix can be estimated in different methods –

most prominent of which will be discussed here.

• The active fund manager attempts to outperform this benchmark.

• Let x be the vector of deviations from the benchmark,or the active part of the managed portfolio.

V



• Of course, the sum of all the components of x, by

construction, must be equal to zero.'Prof. Doron Avramov


The Mathematics of TEV• So the fund manager invests w=q+x in stocks, q is the

passive part of the portfolio and x is the active part.

• Notice that is the tracking error variance.

• Also notice that the expected return and volatility of

the chosen portfolio are

Vx x'2 =ξ σ

222

'2

'

ξ σ σ σ

µ µ µ

++=

+=

Vxw

x

B p

B p





The Mathematics of TEV• The optimization problem is formulated as

ϑ

ι

µ

=

=

Vx

xt s

x x

'

0'..

'max





The Mathematics of TEVThe resulting active part of the portfolio, x, is given by

where

−±= − ι µ ϑ

c

aV

e x 1

cV

aV

bV

=

==

−

−

−

'

'

'

2

1

1

1

ι ι

ι µ

µ µ



ecba =− /2



Questions• Is the sum of the x components equal to zero?

• Prove that if the benchmark is the GMVP then all x

components are equal to zero.

• What if the benchmark is another efficient portfolio – does this result still hold?

• Does the active part of the portfolio depend on the benchmark composition?





Caveats about the Minimum TEVPortfolio

• Richard Roll (distinguished UCLA professor) points

out that the solution is independent on the benchmark.

• Put differently, the active part of the portfolio x is

totally independent of the passive part q.

• Of course, the overall portfolio q+x is impacted by q.

• The unexpected result is that the active manager paysno attention to the assigned benchmark. So it does not



really matter if the benchmark is S&P or any other

index.'Prof. Doron Avramov


TEV with total VolatilityConstraint (based on Jorion – an Expert in

Risk Management)

• Given the drawbacks underlying the TEV portfolio we

add one more constraint on the total portfolio volatility.

• The derived active portfolio displays two advantages.

• First, its composition does depend on the benchmark.

• Second, the systematic volatility of the portfolio is

controlled by the investor



controlled by the investor.



TEV with total volatilityconstraint

• The optimization is formulated as

• Home assignment: deri e the optimal sol tion

2)()'(

'

0'..

'max

p

x

xqV xq

Vx x

xt s

x

σ

ϑ

ι

µ

=++

=

=



• Home assignment: derive the optimal solution.



Lecture Notes in


Estimating the Large ScaleCovariance Matrix





Estimating the CovarianceMatrix:

• There are various applications in financial economics

which use the covariance matrix as an essential input.

• The Global Minimum Variance Portfolio, theminimum tracking error volatility portfolio, the mean

variance efficient frontier, and asset pricing tests are

good examples.

• In what follows I will present the most prominent



w o ows w p ese e os p o e

estimation methods of the covariance matrix.'Prof. Doron Avramov


The Sample Covariance Matrix

(Denoted S )

•This method uses sample estimates.• Need to estimate N(N+1)/2 parameters which is a lot.

• You can use excel to estimate all variances and co-variances which is tedious and inefficient.

• Here is a much more efficient method.

• Consider T monthly returns on N risky assets.



• We can display those returns in a T by N matrix R.'Prof. Doron Avramov


• Estimate the mean return of the N assets – and denote

the N -vector of the mean estimates by .

• Next, compute the deviations of the return

observations from their sample means:

where is a T vector of ones. Then the samplecovariance matrix is estimated as

∧

µ

∧∧

−= 'µ ι T R E

T ι ∧∧

= E E S '1

The Sample Covariance Matrix

(Denoted S )



T



The Equal Correlation BasedCovariance Matrix (Denoted F ):

• Estimate all N(N-1)/2 pair-wise correlations betweenany two securities and take the average.

• Let be that average correlation, let be theestimated variance of asset i, and let be the

estimated variance of asset j, both estimates are the i-th

and j-th elements of the diagonal of S .• Then the matrix F follows as

ρ ii s jj s

ii sii F =),(



jjii s s ji F

−

= ρ ),('Prof. Doron Avramov


The Factor Based CovarianceMatrix:

• Consider the time series regression

where is an N vector of returns at time t and

is a set of K factors. Factor means are denoted by

• Notice that the mean return is given by

t t t e F r +×+= β α

t r

Fµβαµ ×+=

t F

F µ



F µ β α µ ×+



The Factor Based CovarianceMatrix

• Thus deviations from the means are given by

• The factor based covariance matrix is estimated by

Here, is an N by K matrix of factor loadings and

is a diagonal matrix with each element represents the

t F t t e F r +−×=− )( µ β µ

∧∧∧∧

Ψ+Σ= ' β β FF V

∧ β Ψ



idio syncratic variance of each of the assets.



The Factor Based CovarianceMatrix – Number of Parameters

• This procedure requires the estimation of NK betas aswell as K variances of the factors, K(K-1)/2

correlations of those factors, and N firm specific

variances.

• Overall, you need to estimate NK+K+K(K-1)/2+N

parameters, which is considerably less than N(N+1)/2

since K is much smaller than N .

• For instance using a single factor model – the number



For instance, using a single factor model the number

of parameters to be estimated is only 2N+1.'Prof. Doron Avramov


Steps for Estimating the Factor Based Covariance Matrix

1) Run the MULTIVARIATE regression of stock

returns on asset pricing factors

where

( ) ( ) N T N K K T N T N K K T N T N T E F E F R

××++×××××××+⋅=++=

1111''' θ β α ι (

[ ][ ] β α θ

ι

,

,

=

= F F T (



[ ]β,




2) Estimate

and retain only.

( )[ ] ( ) [ ] β α θ ˆ,ˆ''''ˆ

1'1

===

−−

F F F R R F F F

((((((

θ

β

)






3) Estimate the covariance matrix of E

4) Let be a diagonal matrix with the

component being equal to the component

of .

N T T N N N E E T ××× =∆ '

1ˆ

( ) thii −,

( ) thii −,

∆

∧

Ψ






5) Compute:

where is the mean return of the factors.

6) Estimate:

K

f T K T K T

F V ×

××× ⋅−=1

'

1µ ι

f µ ˆ

K T T K K K F V V

T ×××

⋅=∑ '1ˆ






7)

This is the estimated covariance matrix of stock

returns.

8) Notice – there is no need to run N individual

regressions! Use multivariate specifications

∧

×××××

∧Ψ+∑=

N N N K K K F

K N N N V 'ˆˆˆ β β



regressions! Use multivariate specifications.



A Shrinkage Approach - Based ona Paper by Ledoit and Wolf (LW)

• There is a well perceived paper (among Wall Street

quants) by LW demonstrating an alternative approach

to estimating the covariance matrix.• It had been claimed to deliver superior performance in

reducing tracking errors relative to benchmarks as well

as producing higher Sharpe ratios.

• Here are the formal details.





A Shrinkage Approach - Based ona Paper by Ledoit and Wolf (LW)

• Let S be the sample covariance matrix, let F be the

equal correlation based covariance matrix, and let δ be

the shrinkage intensity. S and F were derived earlier.

• The operational shrinkage estimator of the covariance

matrix is given by

• Notice that F is the shrinkage target.

S F V ×−+×=∧∧

*)1(* _

δ δ





The Shrinkage IntensityLW propose the following shrinkage intensity, based on

optimization:

where T is the sample size and k is given as

and where with being the (i,j)

component of S and is the (i,j)

component of F.

=

∧

1,min,0max*T

k δ

γ η π −=k

∑∑= =

−= N

i

N

j

ijij s f 1 1

2)(γ ij s

ij f



322

The Shrinkage Intensity

and with and being the time t return on asset i

and the time-series average of return on asset i,

∑∑= =

= N

i

N

j

ij

1 1

π π

∑= −−−=

T

t ij j jt

iit ij sr r r r T 1

2 _ _

))((

1

π

∑ ∑ ∑= = ≠=

++=

N

i

N

i

N

i j j

ij jj

jj

iiijii

ii

jj

ii s

s

s

s

1 1 ,1

,,

_

2ϑ ϑ

ρ π η

∑=

−−−

−−=

T

t

ij j jt iit iiiit ijii sr r r r sr r T 1

_ _

2

_

, ))(()(1ϑ

∑=

−−−

−−=

T

t

ij j jt iit jj j jt ij jj sr r r r sr r T 1

_ _ 2

_

, ))(()(1

ϑ

it r _

ir



respectively.'Prof. Doron Avramov


The Shrinkage Intensity – ANaïve Method

If you get overwhelmed by the derivation of theshrinkage intensity it would still be useful to use a

naïve shrinkage approach, which often even works

better. For instance, you can take equal weights:

S F V 2

1

2

1 _

+=





Backtesting• We have proposed several methods for estimating the

covariance matrix.

• Which one dominates?

• We can backtest all specifications.

• That is, we can run a “horse race” across the various

models searching for the best performer.• There are two primary methods for backtesting –

rolling versus recursive schemes.



o g ve sus ecu s ve sc e es.



The Rolling Scheme

• You define the first estimation window.

• It is well perceived to use the first 60 sampleobservations as the first estimation window.

• Based on those 60 observations derive the GMVP

under each of the following methods:





Competing Covariance Estimates• The sample based covariance matrix

• The equal correlation based covariance matrix

• Factor model using the market as the only factor

• Factor model using the Fama French three factors

• Factor model using the Fama French plus Momentum

factors

• The LW covariance matrix – either the full or the



naïve method.'Prof. Doron Avramov


Out of Sample Returns• Then given the GMVPs compute the actual returns on

each of the derived strategies.

• For instance, if the derived strategy at time t is

then the realized return at time t+1 would be

where is the realized return at time t+1 on all the N investable assets.

t w

11, ' ++ ×= t t t p Rw R

1+t R





Out of Sample Returns• Suppose you rebalance every six months – derive the

out of sample returns also for the following 5 months

• Then at time t+6 you re-derive the GMVPs.

22, ' ++ ×= t t t p Rw R

33, ' ++ ×= t t t p Rw R

44, ' ++ ×= t t t p Rw R

55, ' ++ ×= t t t p Rw R

66, ' ++ ×= t t t p Rw R



y



The Recursive Scheme• A recursive scheme is using an expanding window.

• That is, you first estimate the GMVPs based on thefirst 60 observations, then based on 66 observations,

and so on, while in the rolling scheme you always use

the last 60 observations.

• Pros: the recursive scheme uses more observations.

• Cons: since the covariance matrix may be time

varying perhaps you better drop initial observations.





Out of Sample Returns• So you generate out of sample returns on each of

the strategies starting from time t+1 till the end of sample, which we typically denote by T .

• Next, you can analyze the out of sample returns.

• For instance, you can form the table on the next

page and examine which specification has been able

to deliver the best performance.





Out of Sample ReturnsRolling Scheme Recursive Scheme

S F MKT FF FF+MOM LW S F MKT FF FF+MOM LW

Mean

STD

SR

SP (5%)

alpha

IR





Out of Sample Returns• In the above table:

• Mean is the simple mean of the out of samplereturns

• STD is the volatility of those returns• SR is the associated Sharpe ratio obtained by

dividing the difference between the mean return and

the mean risk free rate by STD.

• SP is the shortfall probability with a 5% threshold



applied to the monthly returns.'Prof. Doron Avramov


Out of Sample Returns• In the above table:

• alpha is the intercept in the regression of out of sample EXCESS returns on the contemporaneous

market factor (market return minus the riskfree rate).

• IR is the information ratio – obtained by dividing

alpha by the standard deviation of the regression

error, not the STD above.•Of course, higher SR, higher alpha, higher IR are

associated with better performance.





Lecture Notes in


Principal Component Analysis

(PCA)





PCA• The aim is to extract K common factors to

summarize the information of a panel of rank N .• In particular, we have a T×N panel of stock returns

where T is the time dimension and N (<T ) is the

number of firms – of course K<< N

• The PCA is an operation on the sample covariance

[ ] N

N T

R R R ,,1 K=×



matrix of stock returns.'Prof. Doron Avramov


Principal Component Analysis(PCA)

• To understand the PCA let us master the notion of

eigen-vector and eigen-value.

• An eigen-vector of a squared matrix is a non zerovector which, when multiplied by the matrix, yields

a vector parallel to the origin.





Principal Component Analysis (PCA)• To illustrate:

• Let

• Hence

• is the first eigen-value. is the

corresponding eigen vector

=

2,11,2 A

11 3

3

3 x Ax =

=

31 =λ

=11

1 x

1 x





Eigen Values and Eigen Vectors• How to find eigen values and eigen vectors?

• Set det(B)=0 and solve

−

−=

−=−=

λ

λ

λ

λ λ

2,1

1,2

,0

0, A I A B

( )1,3

034det

21

2

===+−=

λ λ λ λ B





Principal Component Analysis (PCA)

• Let ,

• hence

• is the second eigen-value. is the

corresponding eigen vector.

=

2,1

1,2 A

12 =λ

= 3

3

2 x

22 13

3 x Ax =

=

2 x





Eigen Values and Eigen Vectors• Finding the eigen vector

1,1

32,1

1,2

==

=

y x

y

x

y

x

3,1

12,1

1,2

−==

=

y x

y

x

y

x





PCA• You have returns on N stocks for T periods

[ ] R R

T

V

R R R R

let R R R R

R R

N

N N N

T N

T

~'

~1ˆ

~,,

~,

~~

~~

,,

21

111

111

=

=

−=−=

∧∧

××

K

K

K

µ µ



T 'Prof. Doron Avramov


PCA• Extract K -eigen vectors corresponding to the largest

K -eigen values.• Each of the eigen vector is an N by 1 vector.

• The extraction mechanism is as follows.• The first eigen vector is obtained as

1

'

1 ˆwV w

11

'

1 =ww

1maxw

t s.





PCA• is an eigen vector since

• is therefore the highest eigen value.

1w

1'11

111

ˆ

ˆ

wV w

Moreover wwV

=

=

λ

λ

1λ





PCA• Extracting the second eigen vector

t s.

2

maxw 2

'

2ˆwV w

0

1

2

'

1

2

'

2

=

=

ww

ww





PCA• The optimization yields:

• The second eigen value is smaller than the first due

to the presence of one extra constraint in theoptimization – the orthogonality constraint.

12

'

22

222

ˆ

ˆ

λ λ

λ

<=

=

wV w

wwV





PCA• The K -th eigen vector is derived as

t s.

max K K wV w ˆ'

0

0

0

1

1

'

2

'

1

'

'

=

=

=

KK

K

K

K K

ww

ww

ww

ww

M



01 =− K K ww 'Prof. Doron Avramov


PCA• The optimization yields:

121

' ˆ

ˆ

λ λ λ λ

λ

<<<<=

=

− K K K K K

K K K

wV w

wwV





PCA• Then - each of the K-eigen vectors delivers a unique

asset pricing factor.• Simply, multiply excess stock returns by the eigen

vectors:

11

11

11

×××

×××

⋅=

⋅=

N K

N T

e

T K

N N T

e

T

w R F

w R F

M





PCA• Recall, the basic idea here is to replace the original

set of N variables with a lower dimensional set of K -factors ( K<<N ).

• The contribution of the j-th eigen vector to explain

the covariance matrix of stock returns is

∑=

N

i

i

j

1

λ

λ





PCA• Typically the first three eigen vectors explain over

and above 95% of the covariance matrix.

•What does it mean to “explain the covariancematrix”? Coming up soon!





Understanding the PCA:Digging Deeper

• The covariance matrix can be decomposed as

[ ]

='

'

,,0

0,

,,ˆ

1,1

1

N

N

w

w

wwV M

K

M

K

K

λ

N λ O

P f D A






• If some of the are either zero or negative – the covariance matrix is not properly defined -- it is

not positive definite.

• In fixed income analysis – there are three prominent

eigen vectors, or three factors.

• The first factor stands for the term structure level,the second for the term structure slope, and the third

for the curvature of the term structure.

s−λ

P f D A






• In equity analysis, the first few (up to three)

principal components are prominent.

• Others are around zero.

• The attempt is to replace the sample covariance

matrix by the matrix which mostly summarizes

the information in the sample covariance matrix.

V ~

'Prof Doron Avramov






• The matrix is given by

• Of course, the dimension of is N by N .

• However, its rank is K , thus the matrix is not

invertible.

[ ]

=×

'

'

,,0

0,

,,

~1,1

1

k

k N N

w

w

wwV MK

M

K

K

λ

k

O

V ~

V ~

'Prof Doron Avramov



invertible.'Prof. Doron Avramov


• This is the same as asking: what does it mean to

explain the sample covariance matrix?

Is close to ?V

~

V ˆ

'Prof Doron Avramov





• Let us represent returns as

M

t Kt K t t t f f f r 1121211111 ε β β β α +++++= K

t r 1~

Nt Kt NK t N t N N Nt f f f r ε β β β α +++++= K2211

Nt r ~

T t ,,1K=∀

T t ,,1K=∀

Is close to ?V

~

V ˆ

'Prof Doron Avramov



Prof. Doron Avramov


• where are the principal component based

factors and are the exposures of firm i tothose factors.

Kt t f f K1

iK i β K1

Is close to ?V

~

V ˆ




Prof. Doron Avramov


• is closed enough to - if

1. The variances of the residuals cannot be

dramatically reduced by adding more factors.

2. The pairwise cross-section correlations of theresiduals cannot be considerably reduced by

adding more factors.

Is close to ?V

~

V ˆ

V ~

V

'Prof. Doron Avramov359




The Contribution of the PC-Sto Explain Portfolio Variation

• Let be the exposures of firm i to

the K common factors.

• Let be an N -vector of portfolio weights:

• Recall, R is a T×N matrix of stock returns.

[ ]'

11

,, iK i K

i β β β K=×

pw

[ ] Np p p p wwww ,,, 21 K= '






• The portfolio’s rate of return is

• Moreover, the portfolio time t return is given by

11 ×××

⋅= N

p

N T T

p w R R

11

'

2211××

⋅≡+++= N

t

N

p Nt Npt pt p pt r wr wr wr w R K






We can approximate the portfolio’s rate of return as:

( ) Kt K t t p pt f f f w R 12121111

~ β β β +++= K

( )

( ) Kt NK t N t N Np

Kt K t t p

f f f w

f f f w

β β β

β β β

++++

+

++++

K

KKKKKKKKKKKKK

K

2211

22221212





The Contribution of the PC-S

to Explain Portfolio VariationThus,

11

'

12

1

'

2

111

'

1

2211

~

××

××

××

⋅=

⋅=

⋅=

+++=

N

K

N

p K

N N

p

N N p

Kt K t t pt

w

w

w

where f f f R

β δ

β δ

β δ

δ δ δ

M

K





The Contribution of the PC-S

to Explain Portfolio Variation

Notice that

are the K loadings on the common factors,

or they are the risk exposures, while

are the K -realizations of the factors at time t .

K δ δ K1

Kt t f f K1





Explain the Portfolio Variance

• The actual variation is

• The approximated variation is

• Both quantities are quite similar.

p p pt wV w R ˆ)('2

=σ

)var()var()var()~( 2

2

2

21

2

1

2

Kt K t t pt f f f R δ δ δ σ +++= K


Fi i l E i365




Explaining the Portfolio

Variance

The contribution of the i-th PC to the overall portfoliosvariance is:

∑=

k

j j j

ii

f

f

1

2

2

)var(

)var(

δ

δ


Fi i l E t i366




Asy. PCA: What if N>T?• Then create a T×T matrix and extract K eigen

vectors – those eigen vectors are the factors

and is the T -vector of cross sectional (across-stocks) mean of returns.

'ˆ

ˆ'ˆ1ˆ

11 N N

T N T N T R E

where

E E N

V

××

∧

××⋅−=

=

ι µ

µ ˆ

V






Other Applications• PCA can be implemented in a host of other

applications.

• For instance, you want to predict economic growth

with many predictors, say M where M is large.

• You have a panel of T×M predictors, where T is the

time-series dimension.






Other Applications• In a matrix form

where is the m-the predictor realized at time t .

=×

TM T T

M

M T Z Z Z

Z Z Z

Z

,,,

,,,

21

11211

K

M

K

tm Z






Other Applications• If T>M compute the covariance matrix of Z – then

extract K principal components such that yousummarize the M -dimension of the predictors with a

smaller subspace of order K<<M.

• You extract eigen vectors.

11

1 ,,

×× M

K

M

ww K






Other Applications• Then you construct K predictors:

11

11

11

×××

×××

⋅=

⋅=

M K

M T T K

M M T T

w Z Z

w Z Z

M






What if M>T ?• If M>T then you extract K eigen vectors from the

T×T matrix.

• In this case, the K -predictors are the extracted K eigen vectors.

• Be careful of a look-ahead bias in real time

prediction.





The Number of Factors in PCA• An open question is: how many factors/eigen vectors

should be extracted?

• Here is a good mechanism: set - the highest

number of factors.

• Run the following multivariate regression for



max K

max,,2,1 K K K=

N T N K K T N T N T E F R

×××××× +++= ''11

β α ι



The Number of Factors in PCA• Estimate first the residual covariance matrix and then

the average of residual variances:

• Where is the sum of diagonal elements in



( ) ( ) N

V tr K

E E K T

V N N

ˆˆ

ˆ'ˆ1

1ˆ

2 =

−−=

×

σ

V tr ˆ V ˆ



The Number of Factors in PCA

Compute for each chosen K

and pick K which minimizes this criterion.



( ) ( ) ( )

+

⋅+

+=T N

NT

NT

T N K K K K PC lnˆˆ

max

22 σ σ



Lecture Notes inFinancial Econometrics

The Black-Litterman (BL) Approach

for Estimating Mean Returns:Basics, Extensions, and

Incorporating Market Anomalies





Estimating Mean Returns – The Bayesian Perspective

• Thus far, we have been dealing with classical, also termedfrequentist, econometrics.

• Indeed, GMM, MLE, etc. are all classical methods.

• The competing perspective is the Bayesian.• The BL approach is Bayesian.

• Theoretically, the Bayesian approach is the most appealing

for decision making, such as asset allocation, securityselection, and policy making in general.

• Major advantages: accounting for estimation risk, model

risk, informative priors, and it is numerically tractable.





Estimating Mean Returns –

The Bayesian Perspective

• To explain the difference between classical and Bayesian

methods, assume that you observe the market returns over T

periods:

• The classical approach computes the sample mean return,

which is a stochastic (random) variable, and then specifies a

distribution for the mean return.

mT mm R R R ,,, 21 K







• For instance, assume that

• Then the sample mean is distributed as

2,~ mmt N R σ µ



T N r m

2

,~σ

µ





•The Bayesian approach is very different – the unobserved

actual mean return is the stochastic variable.

•The econometrician specifies both prior as well as likelihood

(data based) distributions for the mean return:

Prior: Likelihood:2,~ p p N σ µ µ 2,~ L L N σ µ µ






The Posterior Distribution

• The inference is then based on the posterior distributionwhich combines the prior and the likelihood.

• In particular, given the above specified prior and likelihood based distributions, the posterior density is given by

2~

,

~

~ σ µ µ N






The Posterior Distribution

The mean and variance of the posterior are

( L p ww µ µ µ 11 1~ −+=

1

22

2 11~−

+=

L p σ σ σ





The Posterior Mean Return• The posterior mean return is a weighted average of the prior

mean and the sample mean with weights proportional to the

precision of the prior versus the sample means, where

precision defined as the reciprocal of the variance.

• In particular,

22

2

1 11

1

L p

pw

σ σ

σ

+=

22

2

12 11

1

1

L p

Lww

σ σ

σ

+=−=;





Estimating Mean Returns• It is notoriously difficult to propose good estimates for

mean returns.

• The sample means are quite noisy – thus asset pricing

models -even if misspecified -could give a good guidance.

• To illustrate, you consider a K -factor model (factors are portfolio spreads) and you run the time series regression

112

121

1111 ×××××× +++++= N

t Kt N

K t N

t N N N

e

t e f f f r β β β α K





Estimating Mean Returns• Then the estimated excess mean return is given by

where are the sample estimates of factor

loadings, and are the sample estimates of the

factor mean returns.

K f K f f

e µ β µ β µ β µ ˆˆˆˆˆˆˆ21 21 +++= K

K β β β ˆˆ,ˆ 21 K

K f f f µ µ µ ˆˆ,ˆ21K





The BL Mean Returns• The BL approach combines a model (CAPM) with some

views, either relative or absolute, about expected returns.

• The BL vector of mean returns is given by

Ω+

∑⋅

Ω+

∑=

××

−

××

−

××

−

××

−

×

−

××× 1

1

1

1

11

1

11

111

'' K

v

K K K N N

eq

N N N K K K K N N N N BL P P P µ µ τ τ µ





Understanding the BL Formulation

• We need to understand the essence of the following

parameters, which characterize the mean return vector

• Starting from the matrix – you can choose any of the

specifications derived in the previous meetings – either the

sample covariance matrix, or the equal correlation, or an

asset pricing based covariance, or you could rely on the LWshrinkage approach – either the complex or the naïve one.

veq P µ τ µ ,,,,, Ω∑

Σ





Constructing Equilibrium

Expected Returns• The , which is the equilibrium vector of expected

return, is constructed as follows.

• Generate , the N ×1 vector denoting the weights of

any of the N securities in the market portfolio based on

market capitalization. Of course, the sum of weights must be unity.

• Then, the price of risk is where and

are the expected return and variance of the market portfolio.

• Later, we will justify this choice for the price of risk.

eqµ

MKT ω

2

m

m Rf

σ

µ γ

−= 2

mσ m

µ





Constructing EquilibriumExpected Returns

• One could pick a range of values for going from 1.5 to2.5 and examine performance of each choice.

• If you work with monthly observations, then switching to the

annual frequency does not change as both the numerator

and denominator are multiplied by 12.

• Having at hand both and , the equilibrium return

vector is given by

γ

γ

MKT ω γ

MKT

eq ω γ µ Σ=





Constructing EquilibriumExpected Returns

• This vector is called neutral mean or equilibrium expectedreturn.

• To understand why, notice that if you have a utility function

that generates the tangency portfolio of the form

then using as the vector of excess returns on the N assets

would deliver as the tangency portfolio.

e

e

TP wµ ι

µ 1

1

' −

−

∑∑

=

eqµ

MKT ω





What if you Directly Applythe CAPM?

• The question being – would you get the same vector of

equilibrium mean return if you directly use the CAPM?

• Yes, if…





The CAPM based Expected

Returns• Under the CAPM the vector of excess returns is given by

( ) ( )( )

MKT

e

m

m

MKT

N

e

m

MKT

m

MKT

ee

m

e

m

e

e

m N N

e

ww

CAPM

wwr r r r

∑=∑

=

∑===

=

×

××

γ µ σ

µ

σ σ σ β

µ β µ

21

222

11

:

',cov,cov





What if you use Directly the

CAPM?

Since and

then

( ) MKT ee

m w'µ µ = ( ) MKT ee

m wr r '=

eq MKT

m

e

me w µ σ µ µ =∑= 2





What if you use Directly theCAPM?

• So indeed, if you use (i) the sample covariance matrix, rather than any other specification, as well as (ii)

then the BL equilibrium expected returns and expected

returns based on the CAPM are identical.

2

m

m Rf

σ

µ

γ

−=





The P Matrix: Absolute Views• In the BL approach the investor/econometrician forms some

views about expected returns as described below.

• P is defined as that matrix which identifies the assets

involved in the views.

• To illustrate, consider two "absolute" views only.

• The first view says that stock 3 has an expected return of 5%

while the second says that stock 5 will deliver 12%.





The P Matrix: Absolute Views• In general the number of views is K .

• In our case K=2.

• Then P is a 2 ×N matrix.

• The first row is all zero except for the fifth entry which is

one.

• Likewise, the second row is all zero except for the fifth entry

which is one.





The P Matrix: Relative Views• Let us consider now two "relative views".

• Here we could incorporate market anomalies into the BL paradigm.

• Market anomalies are cross sectional patterns in stock

returns unexplained by the CAPM.

• Example: price momentum, earnings momentum, value, size,

accruals, credit risk, dispersion, and volatility.





Black-Litterman: Momentum and

Value Effects

• Let us focus on price momentum and the value effects.

• Assume that both momentum and value investing outperform.

• The first row of P corresponds to momentum investing.• The second row corresponds to value investing.

• Both the first and second rows contain N elements.





Winner, Loser, Value, and

Growth Stocks

•Winner stocks are the top 10% performers during the past sixmonths.

•Loser stocks are the bottom 10% performers during the past six

months.

•Value stocks are 10% of the stocks having the highest book-to-

market ratio.

•Growth stocks are 10% of the stocks having the lowest book-

to-market ratios.





Momentum and Value Payoffs

• The momentum payoff is a return spread – return on anequal weighted portfolio of winner stocks minus return on

equal weighted portfolio of loser stocks.

• The value payoff is also a return spread – the returndifferential between equal weighted portfolios of value and

growth stocks.





Back to the P Matrix

• Suppose that the investment universe consists of 100 stocks

• The first row gets the value 0.1 if the corresponding stock is a

winner (there are 10 winners in a universe of 100 stocks).

• It gets the value -0.1 if the corresponding stock is a loser (there

are 10 losers).

• Otherwise it gets the value zero.

• The same idea applies to value investing.

• Of course, since we have relative views here (e.g., return on

winners minus return on losers) then the sums of the first row

and the sum of the second row are both zero.





Back to the P Matrix• More generally, if N stocks establish the investment universe

and moreover momentum and value are based on deciles (thereturn difference between the top and bottom deciles) then

the winner stock is getting 10/N

while the loser stock gets -10/N .

• The same applies to value versus growth stocks.

• Rule: the sum of the row corresponding to absolute views is

one, while the sum of the row corresponding to relative

views is zero.





Computing the Vector

• It is the K ×1 vector of K views on expected returns.

• Using the absolute views above

• Using the relative views above, the first element is the

payoff to momentum trading strategy (sample mean); the

second element is the payoff to value investing (samplemean).

vµ

[ ]'0.05,0.12=vµ





The Matrix

• is a K ×K covariance matrix expressing uncertainty

about views.

• It is typically assumed to be diagonal.

• In the absolute views case described above denotes

uncertainty about the first view while denotes

uncertainty about the second view – both are at thediscretion of the econometrician/investor.

ΩΩ

( )1,1Ω( )2,2Ω





The Matrix

• In the relative views described above: denotes

uncertainty about momentum. This could be the sample

variance of the momentum payoff.

• denotes uncertainty about the value payoff. This is thecould be the sample variance of the value payoff.

Ω

( )1,1Ω

( )2,2Ω





Deciding Upon

• There are many debates among professionals about the right

value of .• From a conceptual perspective it should be 1/T where T

denotes the sample size.

• You can pick

• You can also use other values and examine how they

perform in real-time investment decisions.

τ

τ

1.0=τ





Maximizing Sharpe Ratio

• The remaining task is to run the maximization program

• such that each of the w elements is bounded below by 0 and

subject to some agreed upon upper bound, as well as the sum

of the w elements is equal to one.

ww

w BLw

Σ''

maxµ





Extending the BL Model to Incorporate Sample

Moments

• Consider a sample of size T , e.g., T=60 monthly

observations.

• Let us estimate the mean and covariance (V ) of our N assets

based on the sample.

• Then the vector of expected return that serves as an input for

asset allocation is given by

where[ ] [ ] sample sample BL sample T V T V µ µ µ

11111

)/()/(−−−−−

+∆+∆=

[ ] 111 ')(−−− Ω+Σ=∆ P P τ





Class Notes in


Risk Management:Down Side Risk Measures





Downside Risk

•Downside risk is the financial risk associatedwith losses.

• Downside risk measures quantify the risk of

losses, whereas volatility measures are bothabout the upside and downside outcomes.

• That is, volatility treats symmetrically up and

down moves (relative to the mean).• Or volatility is about the entire distribution while

down side risk concentrates on the left tail.'Prof. Doron Avramov




Downside Risk

• Example of downside risk measures

• Value at Risk (VaR)

• Expected Shortfall

• Semi-variance

• Maximum drawdown• Downside Beta

• Shortfall probability

• We will discuss below all these measures.





Value at Risk (VaR)• The says that there is a 5% chance that the

realized return, denoted by R, will be less than .

• More generally,

%95VaR

%95VaR−

µ %95VaR−

( ) α α =−≤ −1Pr VaR R

%5





Value at Risk (VaR )( ) ( )( )α σ µ α

σ

µ α

α 1

1

11 −−

−− Φ+−=⇒Φ=−−

VaRVaR

where

, the critical value, is the inverse cumulative

distribution function of the standard normal evaluated at α.

• Let α=5% and assume that

• The critical value is

( )α 1−Φ

( )2

,~ σ µ N R

( ) 64.105.01 −=Φ −





Therefore

Value at Risk (VaR)

Check:

If

Then

( 2,~ σ µ N R

( )

( ) ( ) 05.064.164.1Pr

64.1Pr

Pr Pr 1

=−Φ=−<=

−−<−=

−−≤

−=−≤ −

z

R

VaR RVaR R

σ µ σ µ

σ µ

σ

µ

σ

µ α α

( )( ) σ µ σ α µ α 64.11

1 +−=Φ+−= −−VaR





Example: The US Equity Premium

• Suppose:

• That is to say that we are 95% sure that the future equity

premium won’t decline more than 25%.• If we would like to be 97.5% sure – the price is that the

threshold loss is higher.

• To illustrate,

( )( ) 25.020.064.108.0

20.0,08.0~

95.0

2

=⋅−−=⇒ VaR

N R

( ) 31.020.096.108.0975.0 =⋅−−=VaR





VaR of a Portfolio• Evidently, the VaR of a portfolio is not necessarily lower

than the combination of individual VaR-s – which is

apparently at odds with the notion of diversification.

• However, recall that VaR is a downside risk measure while

volatility which diminishes at the portfolio level is a

symmetric measure.





Backtesting the VaR • The VaR requires the specification of the exact distribution

and its parameters (e.g., mean and variance).

• Typically the normal distribution is chosen.

• Mean could be the sample average.

• Volatility estimates could follow ARCH, GARCH,

EGARCH, stochastic volatility, and realized volatility, all

of which are described later in this course.

• We can examine the validity of VaR using backtesting.'Prof. Doron Avramov




Backtesting the VaR • Assume that stock returns are normally distributed with

mean and variance that vary over time

• The sample is of length T .

• Receipt for backtesting is as follows.

• Use the first, say, sixty monthly observations to estimate

the mean and volatility and compute the VaR.• If the return in month 61 is below the VaR set an indicator

function I to be equal to one; otherwise, it is zero.

T t ,,2,1 K=∀2,~ t t t N r σ µ


418



Backtesting the VaR • Repeat this process using either a rolling or recursive

schemes and compute the fraction of time when the next

period return is below the VaR.

• If α=5% - only 5% of the returns should be below the

computed VaR.

• Suppose we get 5.5% of the time – is it a bad model or just

a bad luck?


419



Model Verification Based on

Failure Rates• To answer that question – let us discuss another example

which requires a similar test statistic.

• Suppose that Y analysts are making predictions about the

market direction for the upcoming year. The analysts

forecast whether market is going to be up or down.

• After the year passes you count the number of wrong

analysts. An analyst is wrong if he/she predicts up movewhen the market is down, or predict down move when the

market is up.


420



Model Verification Based onFailure Rates

• Suppose that the number of wrong analysts is x.

• Thus, the fraction of wrong analysts is P=x/Y – this is thefailure rate.


421



The Test Statistic

• The hypothesis to be tested is

• Under the null hypothesis it follows that

Otherwise H

P P H

:

:

1

00 =

( ) ( ) x y x P P x y x f

−−

= 00 1


422



The Test Statistic

• Notice that

( )

( ) ( )

( )( )1,0~

1

1

00

0

00

0

N y P P

y P x Z

Thus

y P P xVaR

y P x E

−

−=

−=

=


423



Back to Backtesting VaR: A

Real Life Example

• In its 1998 annual report, JP Morgan explains: In 1998,daily revenue fell short of the downside (95%VaR) band

on 20 trading days (out of 252) or more than 5% of the

time (252×5%=12.6).

• Is the difference just a bad luck or something more

systematic? We can test the hypothesis that it is a bad luck.

Otherwise H

x H

:

6.12:

1

0 =14.2

25295.005.0

6.1220=

⋅⋅

−= Z


424



Back to Backtesting VaR: A

Real Life Example

• Notice that you reject the null since 2.14 is higher than the

critical value of 1.96.

• That suggests that JPM should search for a better model.• They did find out that the problem was that the actual

revenue departed from the normal distribution.


425



Expected Shortfall (ES): Truncated

Distribution

• ES is the expected value of the loss conditional upon the

event that the actual return is below the VaR.

• The ES is formulated as

[ ]α α −− −≤−= 11 | VaR R R E ES


426



Expected Shortfall (ES) and the

Truncated Normal Distribution

• Assume that returns are normally distributed:

( 2,~ σ µ N R

( )( )( )

α

α φ σ µ

σ α µ

α

α

1

1

1

1 |−

−

−−

Φ+−=⇒

Φ+≤−=⇒

ES

R R E ES


427





where is the pdf of the standard normal density

e.g

This formula for ES is about the expected value of a

truncated normally distributed random variable.

( )⋅φ

( ) 0.10396164.1 =−φ


428





Proof:

since

( )

( )( )( )

( )( )α

α φ σ µ κ

κ σφ µ

σ µ

α

1

0

01

2

|

,~

−

−

Φ−=Φ−=−≤ VaR x x E

N x

( )α σ

µ

κ α 11

0

−−

Φ=−−

=VaR


429



Expected Shortfall: Example

Example

( )

( ) 40.025.0

06.020.008.0

025.0

96.120.008.0

32.005.0

10.020.008.0

05.0

64.120.008.0

%20

%8

%5.97

%95

=+−≈−+−=

=+−≈−

+−=

=

=

φ

φ

σ

µ

ES

ES


430





• Previously, we got that the VaRs corresponding to those parameters are 25% and 31%.

• Now the expected losses are higher, 32% and 40%.

• Why?• The first lower figures (VaR) are unconditional in nature

relying on the entire distribution.

• In contrast, the higher ES figures are conditional on the

existence of shortfall – realized return is below the VaR.


431



Expected Shortfall in

Decision Making

• The mean variance paradigm minimizes portfolio volatility

subject to an expected return target.

• Suppose you attempt to minimize ES instead subject toexpected return target.


432



Expected Shortfall with

Normal Returns

• If stock returns are normally distributed then the ES chosen

portfolio would be identical to that based on the mean

variance paradigm.• No need to go through optimization to prove that assertion.

• Just look at the expression for ES under normality to

quickly realize that you need to minimize the volatility of

the portfolio subject to an expected return target.


433



Target Semi Variance

• Variance treats equally downside risk and upside potential.

• The semi-variance, just like the VaR, looks at the downside.

• The target semi-variance is defined as:

where h is some target level.

• For instance,

• Unlike the variance,

f Rh =

( ) ( )[ ]20,min hr E h −=λ

( )22 µ σ −= r E


434




• The target semi-variance:

I. Picks a target level as a reference point instead of the

mean.

II. Gives weight only to negative deviations from a

reference point.


435

T t S i V i




• Notice that if

where

and are the PDF and CDF of a variable,

respectively

• Of course if

then

( 2,~ σ µ N r

( )

−Φ

+

−+

−−= σ

µ

σ

µ

σ σ

µ

φ σ

µ

σ λ

hhhh

h 1

2

22

Φφ ( )1,0 N

µ =h

( )2

2σ λ =h


436



Maximum Drawdown (MD)

• The MD (M) over a given investment horizon is the largest

M-month loss of all possible M-month continuous periods

over the entire horizon.

• Useful for an investor who does not know the entry/exit

point and is concerned about the worst outcome.

• It helps determine the investment risk.


437



Down Size Beta

• I will introduce three distinct measures of downsize beta –

each of which is valid and captures the down side of

investment payoffs.

• Displayed are the population betas.

• Taking the formulations into the sample – simply replace the

expected value by the sample mean.


438



Downside Beta

• The numerator in the equation is referred to as the co-semi-

variance of returns and is the covariance of returns below

on the market portfolio with return in excess of onsecurity i.

• It is argued that risk is often perceived as downside

deviations below a target level by market participants andthe risk-free rate is a replacement for average equity market

returns.

( ) ( )[ ][ ]( )

[ ]

2

)1(

)0,min(

)0,min(

f m

f m f i

im R R E

R R R R E

−

−−= β

f R

f R


439



Downside Beta

where and are security i and market average return

respectively.

•One can modify the down side beta as follows:

( ) ( )[ ][ ]( )[ ]2

)2(

0,min

0,min

mm

mmiiim

R E

R R E

µ

µ µ β

−

−−=

iµ mµ

( )[ ] ( )[ ][ ]

( )[ ]2

)3(

0,min

0,min0,min

mm

mmii

im R E

R R E

µ

µ µ

β −

−−

=


440



Shortfall Probability• We now turn to understand the notion of shortfall

probability.• While VaR specifies upfront the probability of

undesired outcome and then finds the threshold level,

shortfall probability gives a threshold level and seeks

for the probability that the outcome is below that

threshold.

• We will thoroughly study the implications of shortfall

probability for long horizon investment decisions.


441

Shortfall Probability in Long




Horizon Asset Management

• Let us denote by R the cumulative return on the

investment over several years (say T years).

• Rather than finding the distribution of R we analyze

the distribution of

which is the continuously compounded return over the

investment horizon.

( ) Rr += 1ln


442





Horizon Asset Management• The investment value after T years is

• Dividing both sides of the equation by we get

• Thus

( )( ) ( )T T R R RV V +++= 111 210 K

0V

( )( ) ( )T T R R R

V

V +++= 111 21

0

K

( )( ) ( )T R R R R +++=+ 1111 21 K


443





Horizon Asset Management

• Taking natural log from both sides we get

• Assuming that

• Then using properties of the normal distribution, we

get

T r r r r +++= K21

2,~ σ µ T T N r


444

( ) T t N r

IID

t ,...,1,~ 2 =∀σ µ

Sh tf ll P b bilit d L H i



Shortfall Probability and Long Horizon

• The normality assumption for log return implies the log

normal distribution for the cumulative return – more later.

• Let us understand the concept of shortfall probability.

• We ask: what is the probability that the investment yields a

return smaller than a threshold level (e.g., the riskfree rate)?

• To answer this question we need to compute the value of ariskfree investment over the T year period.


445





• The value of such a riskfree investment is

where is the continuously compounded risk free rate.

( )T

f r RV V f

+= 10

f Tr V exp0=

f r


446




• Essentially we ask: what is the probability that

• This is equivalent to asking what is the probability

that

• This, in turn, is equivalent to asking what is the

probability that

rf T

V V <

00 V

V

V V rf T <

<

00

lnlnV

V

V

V rf T


447





• So we need to work out

• Subtracting and dividing by both sides of

the inequality we get

• We can denote this probability by

f Tr r p <

µ T σ T

−< σ

µ f

r T z P

−Φ=

σ

µ f r T Shortfall probabilit y


448

Shortfall Probability and long horizon



Shortfall Probability and long horizon

• Typically which means the probability

diminishes the larger T .

• Notice that the shortfall probability can be written as a

function of the Sharpe ratio of log returns:

µ < f r

SRT SP −Φ=


449

Example



Example

• Take r=0.04, µ=0.08, and σ=0.2 per year. What is the

Shortfall Probability for investment horizons of 1, 2, 5, 10,

and 20 years?

• Use the excel normdist function.

• If T=1 SP=0.42• If T=2 SP=0.39

• If T=5 SP=0.33

• If T=10 SP=0.26 • If T=20 SP=0.19


450

Cost of Insuring against Shortfall




• Let us now understand the mathematics of insuring against

shortfall.

• Without loss of generality let us assume that

• The investment value at time T is a given by the random

variable

10 =V

T V


451





• Once we insure against shortfall the investment value

after T years becomes

– If you get

– If you get

T V f T Tr V exp>

f T Tr V exp< f Tr exp


452





• So you essentially buy an insurance policy that pays 0

if

Pays if

• You ultimately need to price a contract with terminal

payoff given by

f T Tr V exp>

f T Tr V exp<T f V Tr −exp

T f V Tr −exp,0max


453




Cos o su g g s S o

• This is a European put option expiring in T years with

a. S=1

b. .

c. Riskfree rate given by

d. Volatility given by

e. Dividend yield given by

f Tr K exp=

f r

σ

0=δ


454





• The B&S formula indicates that

• Which, given the parameter outlined above, becomes

( ) ( ) ( )12 expexp d N T S d N Tr K Put f −−−−−= δ

−−

= T N T N Put σ σ

2

1

2

1


455





To show the pricing formula of the put use the following:

and

while


456

d ln S / K r T

T 1

21

2=+ − +( ) ( )δ σ

σd d T 2 1= − σ

( ) ( )

( ) ( )22

11

1

1

d N d N

d N d N

−=−

−=−




g g

• The B&S option-pricing model gives the current put

price P as

where

and is

( ) ( )21 d N d N Put −=

12

12

d d

T d

−=

= σ

( )d N d z prob <


457




Cos o su g g s S o

• For (per year)2.0=σ PT (years)

0.081

0.185

0.2510

0.35200.4230

0.5250





g g

• We have found out that the cost of the insurance increases

in T , even when the probability of shortfall decreases in T

(as long as the Sharpe ratio is positive).

• To get some idea about this apparently surprising outcome

it would be essential to discuss the expected value of the

investment payoff given the shortfall event.

• It is a great opportunity to understand down side risk when

the underlying distribution is log normal rather than

normal.


The Expected Value of



p

Cumulative Return• Two proper questions emerge at this stage:

1. What is the expected value of cumulative return during

the investment horizon ?

2. What is the conditional expectation – conditional on

shortfall ?

• We assume, without loss of generality, that the initial

invested wealth is one.

)( T V E

)exp(| f T T Tr V V E <


The Expected Value of



p

Cumulative Return• Notice that, given the tools we have acquired thus far,

finding the conditional expectation is a nontrivial task

since is not normally distributed – rather it is log-

normally distributed since.

• Thus, let us first display some properties of the log normal

distribution.

),(~)ln( 2σ µ T T N V T

T V


The Log Normal Distribution



g

• Suppose that x has the log normal distribution. Then the

parameters µ and σ are, respectively, the mean and the

standard deviation of the variable’s natural logarithm,

which means

where z is a standard normal variable.

• The probability density function of a log-normal

distribution is

z

e xσ µ +

=


The Log Normal Distribution



g

• If x is a log-normally distributed variable, its expected

value and variance are given by

( )( )

0,2

1,,

2

2

2

ln

>Π

=−

− xe

x x f

x

xσ

µ

σ σ µ

[ ]

[ ] ( ) ( ) [ ]( )22

2

1

11222

2

x E eee xVar

e x E

−=−=

=+

+

σ σ µ σ

σ µ


The Mean and Variance of T V



• Using moments of the log normal distribution, the mean

and variance of are

• Next, we aim to find the conditional mean.

( )1)exp()2exp()(

)2

1exp()(

22

2

−+=

+=

σ σ µ

σ µ

T T T V Var

T T V E

T

T

T V


The Mean of a Variable that has the

Truncated Log Normal Distribution




where is a normalizing constant.

• Let us make change of variables:

( )( )

∫

∞

−

−

=c

ln

2

12

2x

1xc)>x|E(xcF dxe

x

σ

µ

π σ

( )cF1/

dte=dxand e=xt-ln(x) tt µ σ µ σ σ

σ

µ ++⇒=







then: ( ) ( )

( )

( ) ( )( )

( )( )

( )∫

∫

∫

∫

∞

−

−−

+

∞−

++−−

∞

−

++−

∞

−+−

Π=

Π=

Π=

=Π

=

σ

µ

σ

σ µ

σ

µ

σ µ σ

σ

µ

µ σ

σ

µ µ σ σ

σ

cln 2

1

5.0

cln

5.021

cln2

1

cln2

1

22

22

2

2

21

2

1

2

1

2x

1c)>x|E(xcF

dt ee

dt e

dt e

dt ee

t

t

t t

t t







• Let us make another change of variables:

• The integral is the complement CDF of the standard normal

random variable.

( )( )

( )

∫

∞

−−

−+

Π= σ σ

µ

σ µ

cln

2

1

5.02

2

2

1

c)>x|E(xcF dvee

v

dt=dv-t=v ⇒σ








• Thus, the formula is reduced to:

( ) ( ) ( )

( ) ( )

++−

Φ=

−−Φ−=

+

+

σ

σ µ

σ σ µ

σ µ

σ µ

25.0

2

5.0LN

ln

ln1c)>x|(xEcF

2

2

c

e

ce








• In the same way we can show that:

( )( ) ( )

( ) ( )( )

( )( )

( ) ( )

−−Φ⋅=

Π=Π=

=Π⋅

=≤⋅

+

−−

∞−

−+−

∞−

−−+

−

∞−

++−

−−

∫∫

∫∫

σ

σ µ

σ

σ µ

σ σ

µ

σ µ σ

µ σ

σ µ

σ

µ µ σ

σ µ

25.0

ln

2

1

5.0

ln

2

1

5.0

ln

2

1ln21

0LN

ln

2

1

2

1

2

1c)x|(xEcF

2

22

22

2

2

ce

dveedt ee

dt edxe x

x

cv

ct

ct t

xc



Punch Lines



( )

( )

+−Φ

++−Φ

⋅=>σ

µ

σ

σ µ

c

c

ln

ln

(x)Ec)x|(xE

2

LNLN

( )

( )

−Φ

−−Φ

⋅=≤

σ

µ

σ σ µ

c

c

ln

ln

(x)Ec)x|(xE

2

LNLN



The Expected Value given Shortfall



• The expected value given shortfall is

or

• Notice that the denominator is the shortfall probability.

[ ]

−Φ

−−Φ

⋅=

+

σ

µ σ

σ µ

σ µ

T

T Tr T

T T Tr

e shortfall V E f

f

T T

T

2

2

12

|

[ ]( )( )

( )SRT

SRT e shortfall V E

T T

T ⋅−Φ

+−Φ⋅=

+ σ σ µ 2

2

1

|



The Expected Value given Shortfall



• Thus,

• Which means that the shortfall probability times the expected

value given shortfall is equal to the unconditional expected

value times a factor smaller than one.

• That factor diminishes with higher Sharpe ratio and/or with

higher volatility.

[ ] ( )σ +−Φ=× SRT V E shortfall V E shortfall ob t T ][|)(Pr



The Horizon Effect



Numerical example

• Let’s take . For different values of the conditional expectation over horizon T looks like:

%10%,5 == σ f r f r >µ



The Horizon Effect



• Previously we have shown that even when the shortfall

probability diminishes with the investment horizon, the costof insuring against shortfall rises.

• Notice that the insured amount is

• The expected value of that insured amount given shortfall

sharply rises with the investment horizon, which explains the

increasing value of the put option.

T f V Tr −exp



Value at Risk with Log Normal



Distribution

• We have analyzed VaR when quantities of interest arenormally distributed.

• It is challenging to extend the analysis to the case wherein the

log normal distribution is considered.

• Analytics follow.



VaR with Log Normal Distribution



• We are looking for threshold, VaR, such that

• Then in order to find the threshold we need to calculate

quantile of lognormal distribution:

where is as defined earlier.

( ) ( )000Pr V VaRCDF V Var V V T =<=α

( )21

0 ,; σ µ α T T CDF V VaR −⋅=

( )α σ µ 1

0

−Φ⋅+

⋅=

T T

eV ( )α 1−Φ






• Specifically,

( ) ( ) ( )( )0000 lnlnPr Pr V VaRV V V Var V V T T <=<=α

( ) ( )

−<

−=

σ

µ

σ

µ

T

T V VaR

T

T V V T 00 lnlnPr

( )

−Φ=

σ

µ

T

T V VaR 0ln






and then

That is to say that there is a α% probability that the investment

value at time T will be below that VaR.

( )( )

( )α σ µ

α σ

µ

1

0

10ln

−Φ⋅+

−

=

Φ=

−

T T eV VaR

T

T V VaR



VaR with t Distribution



• Suppose now that stock returns have a t

distribution with ν degrees of freedom and

expected return and volatility given by and σ

• The pdf of stock return is formulated as



( ) ( )

2

12)(

121,2

1,,|

+−

−+⋅⋅=

v

v

x

v Bvv x f

µ

σ σ µ

Partial Expectation



• Let -standardized r.v distribution with .

Than,



( ) ( )( )

=

+⋅

⋅=⋅=≤

+−

∞−∞− ∫∫ dxv

x

v Bvdxv x f x z X X PE

v

z z 2

1

2

121,2

1,|

( )

( )( )

1,1

121,2

1

|1121,2

1

212

1

2

2

1

2

−+⋅−=

+

−−⋅

⋅

=

+

−−⋅

⋅=

+

+

−

∞−

−−

v

z vv z f

v

z

v

v

v Bv

v x

vv

v Bv

v

z

v

σ

µ −= xY ( )v x F ,1,0|

Partial Expectation



And thus

Where is x quantile of



( ) ( )1

, 21

1

1

−+⋅−= −

−

−

v

qv

q

vq f ES T α

α

α σ µ α

( ) xVaRq x −= 1 nt T ~

Long Run Return when Periodic



The sum of independent t-distributed random variables is not t-

distributed. So we have no nice formula for expected shortfall

in the long run. However, it can be approximated by normal

with zero mean variance:

Return has the t- Distribution



−σ µ

2,~

. v

v N r

approx for 2≥v


R h h Di ib i



• Approximation makes sense for large v’s when t coincides

with normal distribution.

• However, simulation studies show that for sufficient

number of periods this approximation works well enough.





R h h Di ib i



• However simulations shows that for sufficient number of

periods this approximation works well enough.

• Let . The next graphs show normal

curve fit to the sum of t r.v.s (over T periods); sampleestimates vs. predicted parameters are includes




05.0;01.0 == t t σ µ


R t h th t Di t ib ti






-3 -2 -1 0 1 2 30

100

200

300

400

500

600

700

800

900

10=T

274.01.0

==

N

N

σ µ

273.0ˆ

102.0ˆ

==

σ

µ

3=v





-4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400




100=T

866.0

1

=

=

N

N

σ

µ

856.0ˆ011.1ˆ

==

σ µ

3=v





-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80

50

100

150

200

250

300

350




10=T

177.01.0

==

N

N

σ µ

174.0ˆ

099.0ˆ

==

σ

µ

10=v





-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50

50

100

150

200

250

300

350




100=T

559.01

== N

N

σ µ

566.0ˆ994.0ˆ

==σ µ

10=v

VaR Expected Shortfall using Extreme

Val e Theorem (EVT) Witho t



• Estimating VaR and ES using EVT can be done in two

ways:

• The idea of sub-periods;

• The idea based on exceedances over a threshold.

• Both ideas are based on assumption that extreme events of

many distributions (normal, student-t etc.) are distributed

with one of the extreme value distributions family. There isa lot of material on an issue and jut summarized the most

important points.

Value Theorem (EVT) - WithoutDistribution Assumption



• Divide the sample into m non overlapping subsamples (to

Sub-Periods



• Divide the sample into m non-overlapping subsamples (to preserve iid assumption) of length n:

• Then using minimum of each sub-period estimate GEV

density of the following form



mnmnnnn r r r r r r KKKK 1211 ||| ++

( )

( )

wher e

e

z GEV z

z

e

z

,

0,1

1

≠

= −

+− −

ξ ξ

0, ≠ξ σ

µ −=

x z

• Using estimations of and we can compute VaR

Sub-Periods

nn µξ nσ



• Using estimations of and we can compute VaR

• Here we used the assumption that returns are iid and thensingle period return is related to return over sub-period n as

follows



( )

( )[ ]

( )[ ]

−⋅−+

≠−−⋅−+

=

−

α σ µ

ξ α

ξ

σ µ

α

ξ

1lnln

0,11ln

n

n

VaR

nn

n

n

nn

n

0, ≠nξ

( ) ( ) ( )[ ]nn

t

nnn cr P cr P cr P ∏=

<=<=<1

nn µ ξ , nσ

• The result of the EVT are also relevant for the task of

Long Term VaR



• The result of the EVT are also relevant for the task of converting short-term VaR into long-term VaR. Assume

applies to a single-period return for large R.

Then we have for a T-period return

• It follows that a multi-period VaR forecast of a fat tailed

return distribution under the iid assumption is given by



ξ 1

1

~ −

≤≤

⋅

≤∑ RT Rr P

T t

t

( ) ( )α α ξ 11 VaRT VaR T ⋅= −

( ) ξ 1~ −≤ R Rr P

Exceedances over a Threshold

F i h h ld h di i l di ib i





( ( hl yhl P y F h >≤−= | for hl y F −≤≤0

( )( (

( ) y F

y F yh F y F h

−

−+=

1

)1(

• For a given threshold h conditional excess distribution

function on losses is defined as

• In terms of F this can be written as

• For a large class of underlying distribution functions F the

Exceedances over a Threshold



For a large class of underlying distribution functions F theconditional excess distribution function , for h large,

is well approximated by generalized Pareto distribution:

• For

( ) y F h



( )

−

≠

+−

=

−

−

σ

ξ

ξ σ

ξ

ye

y

yGP

1

0,111

0, ≠ξ

0≥ξ [ ]hl z F −∈ ,0 if and [ ]ξ σ −,0 if 0<ξ

• Using this model VaR and ES can be expressedl ti ll

Long Horizon VaR



Using this model VaR and ES can be expressedanalytically:

First, using (1) we can isolate

• Then is replaced by GP CDF and F(h) by the sample

estimate , where n is the total number of

observations and is the number of exceedances above

the threshold h:

( ) y F h

( )n N n h−

h N

( )l F



( ([ ] ( (h F y F h F l F h +−= 1

( ) ( )( ) ξ ξ

111

−−+−= hl n

N l F h

• Next

Long Horizon VaR

ξ



Next

And

is mean excess function for GP distribution which

equals to



( ) ( )

−

+==

−

− 11

ξ

ξ

σ α α

h N

nh F VaR

( ( ( (( α α α α VaRl VaRl E VaR ES >−+= |

Long Horizon VaR





( )( )

ξ

ξ σ

ξ

α α

+

−+

+

=

11

hVaR ES

• Given a s sample and some threshold h, parameters , andcan be estimated using MLE. σ ξ



Class Notes in


Testing the Black&ScholesFormula



Option Pricing



d ln S / K r T

T

1

21

2=+ − +( ) ( )δ σ

σ

d d T 2 1= − σ

C S,K, ,r,T, Se N d Ke N d - T -rT ( ) = ( ) ( )σ δ δ1 2−

P S,K, ,r,T, Ke N d Se N d -rT - T ( ) = ( ) ( )σ δ δ− − −2 1

• The B&S call Option price is given by

• The put Option price is

Where and



Option Pricing



• There are six inputs required:

S- Current price of the underlying asset.K- Exercise/Strike price.

r- Continuously compounded riskfree rate.

T- Time to expiration.

- Volatility.

- Continuously compounded dividend yield.

σ

δ



The B&S Economy



• The B&S formula is derived under several assumptions:

• The stock price follows a geometric Brownian motion(continuous path and continuous time).

• The dividend is paid continuously and uniformly over

time.

• The interest rate is constant over time.



The B&S Economy



• The underlying asset volatility is constant over time

and it does not change with the option maturity or with

the strike price.

• You can short sell or long any amount of the stock.

• You can borrow and lend in the riskfree rate.

• There are no transactions costs.



Testing the B&S Formula

M k R bi t i l ll ti th t d t f



• Mark Rubinstein analyzes call options that are deep out of

the money.

• He considers matched pairs: options with the same striking

price, on the same underlying asset (stock), but with

different time to maturity (expiration date).

• He examines overall 373 pairs.

• If B&S is correct then the implied volatility (IV) of thematched pair is equal. Time to maturity plays no role.



Testing the B&S Formula

H R bi t i fi d th t f th 373 i d



• However, Rubinstein finds that our of the 373 examined

matched pairs – shorter maturity options had higher IV.

• Under the null – the expected value of such an outcome is

373/2=186.5.

• Is the difference statistically significant?



The Failure Rate based Test

Statistic



•Use the failure rate test developed earlier to show that time to

expiration does play a major role.

•That is to say that the constant volatility assumption is

strongly violated in the data.



The Volatility Smile for Foreign

Currency Options



Currency Options



Implied

Volatility

Strike

Price

Implied Distribution for

Foreign Currency Options



g y p

•Both tails are heavier than the lognormal distribution.

•It is also “more peaked” than the lognormal distribution.



The Volatility Smile for Equity

Options



p



Implied

Volatility

Strike

Price

Implied Distribution for Equity

Options



p

•The left tail is heavier and the right tail is less heavy than thelognormal distribution.



Ways of Characterizing the

Volatility Smiles



y•Plot implied volatility against K/S0 (The volatility smile is

then more stable).•Plot implied volatility against K/F0 (Traders usually define an

option as at-the-money when K equals the forward price, F0,

not when it equals the spot price S0).

•Plot implied volatility against delta of the option (This

approach allows the volatility smile to be applied to some non-

standard options. At-the money is defined as a call with a delta

of 0.5 or a put with a delta of −0.5. These are referred to as 50-delta options).



Possible Causes of Volatility

Smile



• Asset price exhibits jumps rather than continuous changes.

• Volatility for asset price is stochastic:

- In the case of an exchange rate volatility is not heavily

correlated with the exchange rate. The effect of a

stochastic volatility is to create a symmetrical smile.

- In the case of equities volatility is negatively related to s

stock prices because of the impact of leverage. This isconsistent with the skew that is observed in practice.



Volatility Term Structure



•In addition to calculating a volatility smile, traders also

calculate a volatility term structure.

•This shows the variation of implied volatility with the time to

maturity of the option.

•The volatility term structure tends to be downward sloping

when volatility is high and upward sloping when it is low



Example of a Volatility Surface

K /S 0

0 90 0 95 1 00 1 05 1 10





0.90 0.95 1.00 1.05 1.10

1 mnth 14.2 13.0 12.0 13.1 14.5

3 mnth 14.0 13.0 12.0 13.1 14.2

6 mnth 14.1 13.3 12.5 13.4 14.3

1 year 14.7 14.0 13.5 14.0 14.8

2 year 15.0 14.4 14.0 14.5 15.1

5 year 14.8 14.6 14.4 14.7 15.0

Class Notes in



Class Notes in


Time Varying Volatility Models



Volatility Models

• We describe several volatility models commonly applied in



y y pp

analyzing quantities of interest in finance and economics:

• ARCH

• GARCH

• EGARCH

• Stochastic Volatility

• Realized and implied Volatility



Volatility Models

• All such models attempt to capture the empirical evidence



that volatility is time varying (rather than constant) as well

as persistent.

• The EGARCH captures the asymmetric response of

volatility to advancing versus diminishing markets.

• In particular, volatility tends to be higher (lower) during

down (up) markets.



h

ARCH(1)

( )10N=+= t t

er

σεε µ



where ( )1,0~ N et

2221

21 t t t t t t e E E σ σ ε == −−

2

1

2

−+=

=

t t

t t t

w

e

αε σ

σ ε

so

is the conditional variance.2t σ



ARCH (1)22

1 σ ε =−t E is the unconditional variance.



2

1

2

−+= t t w E E αε σ

2

1−+= t E w ε α

2

1

2

1 −−+= t t e E E w σ α

( )21−+= t E w σ α

2

2

2

1

22

−− ===⇒ t t t E E E σ σ σ σ

( )α

σ −

=1

2 w E t



518

ARCH (1)

( ) ( )[ ]• Fat tail?



( )

( )

( )[ ]

( )[ ]2

221

4

1

22

4

4

t t t

t t

t

t

e E E

E E

E

E

σ

ε

ε

ε µ

+==

−

−

( )( )( )[ ]

( )( )( )

( )

( )( )33

3

3

22

4

22

4

222

1

44

1

≥=

=+= −

−

t

t

t

t

t t t

t t t

E

E

E

E

e E E

e E E

σ

σ

σ

σ

σ

σ



519

ARCH (1)

• The last step follows because:



( ) ( ) ( ) 02242 ≥−= t t t E E Var ε ε ε

so

• Yes – fat tail!

(

( )

122

4

≥t

t

E

E

ε

ε



520

h

GARCH(1,1)

( )10~ Ne+= t t

e

r

σε

ε µ



where ( )1,0~ N et

( ) ( ) ( )2

1

2

1

2

2

1

2

1

2

−−

−−

++=

++=

=

t t t

t t t

t t t

E E w E

w

e

σ β ε α σ

βσ αε σ

σ ε



521

GARCH(1,1)

222 ++= σ β σ α σ w



( )( )3

321

113

1

1

224

2

>−−−

−−++=

−−=

β α αβ

β α β α µ

β α σ



522

where

EGARCH

( )1,0~ N et

=+= t t

e

r

σε

ε µ



•The first component

is the absolute value of a normally distribution variable

minus its expectation.

( ) ( )2

1

1

1

1

12 ln2ln −−

−

−

− +−

−+=

=

t

t

t

t

t t

t t t

w

e

σ β σ ε γ

π σ ε α σ

σ ε

π π σ ε 22

1

1

1 −=

− −

−

−t

t

t e

1−t e



523

•The second component is

EGARCH

1

1

1−

−

− = t

t

t eγ σ ε γ



• Notice that the two normal shocks behave differently.

•The first produces a symmetric rise in the log conditional

variance.

•The second creates an asymmetric effect, in that, the log

conditional variance rises following a negative shock.

1−t



524

•More formally if then the log conditional variance

EGARCH

01 <−t e



rises by .

•If then the log conditional variance rises by

•This produces the asymmetric volatility effect – volatility is

higher during down market and lower during up market.

γ α +

01 >−t eγ α −



525

Stochastic Volatility (SV)

• There is a variety of SV models.

• A popular one follows the dynamics



• A popular one follows the dynamics

where

where

• Notice, that unlike ARCH, GARCH, and EGARCH, here

the volatility itself has a stochastic component.

t t t t r ε σ µ += ( )1,0~ N t ε

( ) ( ) t t t t vη σ γ γ σ ++= −110 lnln ( )

( )0,cov

1,0~

=t t

t

v

N v

ε



526

Realized Volatility

• The realized volatility (RV) is a very tractable way to



measure volatility.

• It essentially requires no parametric modeling approach.

• Suppose you observe daily observations within a trading

month on the market portfolio.

• RV is the average of the squared daily returns within that

month.



527

Realized Volatility

• Of course, volatility varies on the monthly frequency but it



is assumed to be constant within the days of that particular

month.

• If you observe intra-day returns (available for large US

firms) then daily RV is the sum of squared of five minute

returns.



528

Implied Volatility (IV)

• The B&S call Option price is given by



d ln S / K r T

T 1

21

2=+ − +( ) ( )δ σ

σd d T 2 1= − σ

C S,K, ,r,T, Se N d Ke N d - T -rT ( ) = ( ) ( )σ δ δ1 2−

P S,K, ,r,T, Ke N d Se N d -rT - T ( ) = ( ) ( )σ δ δ− − −2 1

• The put Option price is

Where and



529

Implied Volatility (IV)

• In the traditional option pricing practice one inserts into the

f l ll h i i h k i h ik



formula all the six parameters, i.e., the stock price, the strike

price, the time to expiration, the cc riskfree rate, the cc

dividend yield, and stock return volatility.

• IV is that volatility that if inserted into the B&S formula

would yield the market price of the call or put option.

• As noted earlier, IV in not constant across maturities or across strike prices.



530

Class Notes in




Stock Return Predictability,Model Selection, and Model

Combination



531

Return Predictability

• If log returns are IID – there is no way you can deliver better



prediction for stock return than the current mean return.

• That is, if

where is the continuously compounded return and is

the set of information available at time t .

t t r ε µ += where ( )2,0~ σ ε N iid

t

[ ]

[ ] 21

1

|

|

σ

µ

=

=

+

+

t t

t t

I r Var

I r E

t r t I



532

Return Predictability

• Also note that the variance of a two-period return

i l

1++ t t r r



is equal to

• That is, variance grows linearly with the investment horizon,

while volatility grows in the rate square root.

( ) ( ) ( ) 2

11 2,cov2 σ =++ ++ t t t t r r r Var r Var



533

Variance Ratio Tests

• However, is it really the case?

P h k l d



• Perhaps stock returns are auto-correlated, or

then:

( ) 0,cov 1 ≠+t t r r

( )( )

( ) ( ) ( )( )t

t t t t

t

t t

r Var r r r Var r Var

r Var r r Var VR

2,cov2

2

1112

+++ ++=+=

( )( ) ( ) ρ σ σ +=+=

+

+ 1,cov

11

1

t t

t t

r r

r r



534


• Test:

0:

0:0

≠

=

ρ

ρ

H

H



• The test statistic is

0:1 ≠ ρ H

( ) ( )1,012 N VRT d

→−



535


• More generally,



no auto correlation

Otherwise H

VR H g

:

1:

1

0 =

( ) s

g

st

g t t t

g g

s

r gVar

r r r Var VR ρ ∑=

++

−+=

+++=

1

1

121K



536

Variance Ratios

• Test statistic:

12

gd



e.g

( )

−→− ∑

−

+

1

114,01

g

s

d

g g

s

N VRT

2= g

( ) [ ]1,012 N VRT d

→−

3= g

( )

→−920,013 N VRT

d



537

Predictive Variables

• In the previous specification, we used lagged returns to



forecast future returns or future volatility.

• You can use a bunch of other predictive variables, such as:

• The term spread.

• The default spread.

• Inflation.

• The aggregate dividend yield



538

Predictive Variables

• The aggregate book-to-market ratio.



• The market volatility.

• The market illiquidity



539

Predictive Regressions

• To examine whether stock returns are predictable, we can

run a predictive regression.



p g

• This is the regression of future excess log or gross return on

predictive variables.

• It is formulated as:

122111 ++ +++++= t Mt M t t t z b z b z bar ε K



Predictive Regressions

• To examine whether either of the macro variables can predict

future returns test whether either of the slope coefficients is



p

different from zero.

• Use the t -statistic or F -statistic for the regression R-squared.

• There is a small sample bias if (i) the predictive variables arehighly persistent, (ii) the contemporaneous correlation

between the predictive regression residual and the innovation

of the predictor is high, or (iii) the sample is small.



Long Horizon Predictive

Regressions

zbar ++ ' ε



• The dependent variable is the sum of log excess return over

the investment horizon, which is K periods.

• Since the residuals are auto correlated compute the standard

errors for the slope coefficient accounting for serial

correlation and often for heteroskedasticity.

• For instance you can use the Newey-West correction.

K t t t K t t z bar ++++ ++= ,1,1 ' ε



Newey-West Correction

• Rewriting the long horizon regression



[ ][ ]','

,1 ''

,1'

,1

ba

z x

xr

t t

K t t t K t t

=

=+= ++++

β

ε β



γ

543

Newey-West Correction

• The estimation error of the regression coefficient is



represented by

• where is the Newey-West given by serially correlated

adjusted estimator

( ) S X X Var ˆ'ˆ 1−= β

S ˆ

∑∑ +=−

−= ⋅−

=

T

jt

jt t

K

K j T K

j K

S 1

1ˆ ε ε



γ

β Var

544


RegressionsTradeoff:



•Higher K – better coverage of dependence.

•But we loose degrees of freedom.

•Feasible solution:

3

1

T K ∞




RegressionsE.g K=1:



•Here we have no serial correlation.

1,0,1 ==−= j j j

∑=

=T

t

t

T

S 1

21ˆ ε




RegressionsE.g K=2:



2,1,0,1,2 ===−=−= j j j j j

∑∑∑ =−

−=

−

=+ ⋅++⋅=

T

t

t t

T

t

t

T

t

t t

T T T

S 2

1

1

21

1

1

1

2

111

2

1ˆ ε ε ε ε ε

+= ∑ ∑

=

−

=

+

T

t

T

t

t t t

T 1

1

1

1

21ε ε ε



In the Presence of

Heteroskedasticity−−

11

111 T T



( )

[ ]∑∑

∑∑

+−

=−−

−=

==

⋅⋅−=

=

1

1

'

1

'

1

'

1ˆ

1

ˆ

11

ˆ

K T

t

jt jt t t

K

K j

t

t t

t

t t

x xT K

j K S

x xT S x xT T Var

ε ε

β



Out of Sample Predictability

• There is ample evidence of in-sample predictability, but little

evidence of out-of-sample predictability.



• Consider the two specifications for the stock return evolution

• Which one dominates? If then there is predictability

otherwise, there is no.

:1 M t t t bz ar ε ++= −1

:2 M t t r ε µ +=

1 M



Out of Sample Predictability

• One way to test predictability is to compute the out of sample

:2

( )∑ −T

tt r r 2

1ˆ



• Where is the return forecast assuming the presence of

predictability, and is the sample mean (no predictability).

• Can compute the MSE (Mean Square Error) for both models.

( )

( )∑

∑

=

=

−−= T

t

t t

t

t t

OOS

r r

R

1

2

1

1,

2 1

1,t r

t r



Model Selection

• When M variable are potential candidates for predicting stock

returns there are linear combinations of predictive models. M 2



• In the extreme, the model that drops all predictors is the no-

predictability or IID model.

The one that retains all predictors is the all inclusive model.• Which model to use?

• One idea (bad) is to implement model selection criteria.



Model Selection Criteria

( ) Lm AIC ln22 −=



where L is the maximized value of the likelihood function.

• Bayesian posterior probability

( ) ( )

( ) ( ) 11

1111

ln2ln

2222

−−−−=−− −−−=

−=

mT m R R

mT T R R

LT m BIC



Model Selection

• Notice that all criteria are a combination of goodness of fit

and a penalty factor.



and a penalty factor.

• You choose only one model and disregard all others.

• Model selection criteria have been shown to exhibit very

poor out of sample predictive power.



Model Combination

• The other approach is to combine models.

• Bayesian model averaging (BMA) computes posterior



yes ode ve g g ( ) co pu es pos e o

probabilities for each model then it uses the posterior

probabilities as weights to compute the weighted model.

• There are more naive combinations.

• Such combination methods produce quite robust predictors

not only in sample but also out of sample.



Avramov Doron Financial Econometrics

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