predic6ng’capmresiduals’ volality’with’aggregate...

Vincent Blaclard & Younes Chajia

Predic6ng CAPM residuals vola6lity with aggregate public

sen6ment

Predicting CAPM abnormal returns volatility

!   Part 1 : Predic6ng macro-‐economic sta6s6cs with public sen6ment analysis

!   Part 2 : CAPM model : cri6cs and refinement !   Part 3 : Results

Summary


Part 1 : Predic6ng macro-‐economic sta6s6cs with public

sen6ment analysis


!   Research paper from Johan Bollen et al. (2010) !   Predicts with a 3 day lag the Dow Jones Industrial Average movement with a precision of 86.7%

!   10 millions tweets selected among 500 millions tweets from March to December 2008

TwiVer may predicts the stock market

Claims



Methodology !   Each tweet is analyzed with two mood assessment tools

!   Opinion Finder measures posi6ve versus nega6ve mood from text content

!   GPOMS measures 6 different mood dimensions from text content : Calm, Alert, Sure, Vital, Kind and Happy

!   Strong correla6on between Calm scores and DJIA



Observa4ons and cri4cs !   Why Calm score is relevant for the stock markets ? !   No all tweets from the collec6on were made in the United States

!   Raw approach : no use of common accepted model in the financial industry

!   Strong increase of volume of tweets (350 millions per month against 50 millions per month when the analysis was made)


Predic6ng macro-‐events with Google

Google Flu Trends !   Strong correla6on was found between an aggrega6on of search queries on Google and the flu ac6vity (reported by US centers for disease control)


Market anxiousness and vola6lity

Defini4ons !   Vola6lity is defined as the standard devia6on of the stock returns

!   Implied Vola6lity is derived from Black-‐Scholes op6ons pricing formula for example

C(S, t) = N(d1)S ! N(d2 )Ke!r(T!t )

d1,2 =ln( SK)+ (r ±

! implied2

2)(T ! t)

! implied T ! t


Market anxiousness and vola6lity

Signature !   Vola6lity skewness (equity op6ons market) is the signature of the rela6onship between market anxiousness and vola6lity


Part 2 : CAPM, cri6cs and refinement


Capital Asset Pricing Model

The equa4on !   Introduced by Sharpe (1964) and Linter (1965) !   Predicts the rela6onship between between risk and the return of a poriolio

Ri ! RF =! i + "i "(RM ! RF )+ #! Ri return of an asset or a specific poriolio ! Rf risk-‐free rate ! Rm return of the market !  ε error



Interpreta4on and resolu4on !   The return of an asset or a poriolio is propor6onal to the covariance between its return and the market returns

!   Resolu6on with the classical OLS es6mator

!i =cov(Ri,RM )var(RM )

Y = X! +"

!̂ = (X 'X)!1X 'Y



Cri4cs !   Widely used for simplicity but CAPM fails in general to predict returns with accuracy

!   βi only reflects the market varia6on and not the nature of the stocks returns

!   Fama and French observed that some stocks performed beVer than the market : small caps and stocks with a high-‐book-‐to-‐market ra6o



Fama French three factors model !   Introduced in 1993

Ri ! RF =! i + "i "(RM ! RF )+ bsSMB + bvHML + #!  SMB or Small Minus Big : measure of the historic excess returns

of small caps against big caps !  HML or High Minus Low : measure of the historic excess

returns of value poriolio against growth poriolio !  ε abnormal Fama French return



Cri4cs !   The classic OLS es6mator supposes that the residual ε has a constant variance (spherical variance assump6on)

!   In case of heteroskedas6city, this assump6ons will not hold and we will have to apply a generalized least square es6mator

E(! '! | X) =" 2Id


!   It is possible to use ARCH/GARCH model to predict the variance of the residual from the CAPM regression

!   In that case, abnormal returns are independent between each others but their variance follows a specific process

Es6ma6on of the error

ARCH/GARCH effect in abnormal returns

Ri ! RF =! i + "i "(RM ! RF )+# t zt!  σt is variance of the residual ! zt is a strong white noise (random gaussian independent with

mean zero and variance one)


!   The Generalized Least Squares es6mator is :

!   In case of ARCH/GARCH effects, Ω is diagonal and easy to inverse


Different es4ma4on of beta with GLS

!̂GLS = (X '!"1X)"1X '!"1Y

! = E(! '! | X)with

! =

!1.. 0..

0 ..! T

"

#

$$$$$$

%

&

''''''


!   An ARCH (Auto Regressive Condi6onal Heteroskedas6city) process models the effect of vola6lity clustering : resilience of vola6lity level over 6me

!   The variance of the innova6on is a linear func6on of the size of the squared previous innova6ons


ARCH (q) process

! t2 ="0 + "i#t!i

2

i=1

q

"


!   A GARCH process is an ARCH process combined with an ARMA process (Auto Regressive Moving Average)

!   More complex but allows more flexibility

!   Introduced because ARCH models required in general long lags in the condi6onal variance equa6on


GARCH (p,q) process

! t2 ="0 + "i#t!i

2

i=1

q

" + $ j! t! j2

j=1

p

"


!   Aims to capture the asymmetric effect : a posi6ve returns shock generate less vola6lity than a nega6ve return


EGARCH (p,q) process

ln(! t2 ) ="0 + "i

#t! t

!

"##

$

%&&

i=1

q

' + "*i#t! t

(µ)

*+

,

-.

i=1

q

' + $ j ln!2t( j

j=1

p

'


Excess vola6lity

CAPM equa4on for the excess vola4lity

!   The equa6on for CAPM is the following :

!   We can take the variance of the following equa6on :

!   It is possible to predict the short-‐term evolu6on of σt because it follows a specific process

Ri ! RF =! i + "i "(RM ! RF )+# t zt

var(Ri ) = var(RM )+! t2


Refinement of the es6ma6on of σt

Observa4ons

!   It is possible to track down the excess vola6lity of an asset based on vola6lity clustering

!   The innova6ons must be explained by other hidden variables

!   We want to see if addi6onal informa6on from aggrega6on of Google queries can provide a beVer es6mate of this excess vola6lity


Refinement of the es6ma6on of σt

Refinement proposed

!   We add a Google Trends component to the regression of the ARCH/GARCH es6ma6on.

!   The objec6ve is to try to explain more explicitly the variance of the innova6on

! t2 ="0 + "i#t!i

2

i=1

q

" + $ j! t! j2

j=1

p

" + % kGt!kk=1

r

"!  Gt is the component rela6ve to Google Trends or micro

blogging ac6vity such as TwiVer


Part 3 : (Very) Preliminary results


Data

Trades and Quotes data

!   AMD stock: TAQ data from 01/01/2011 to 12/31/2011 !   Compute a daily vola6lity for price !   Split the year into an es6ma6on period and a simula6on period of equal length


Data


Conclusion

predic6ng’capmresiduals’ volality’with’aggregate...

Documents