Betas calculated with GARCH models provide new · PDF filefor a Portfolio selection with...

XXXIII ASTIN Colloquium

March 2002

Betas calculated with GARCHmodels provide new parameters

for a Portfolio selection withEfficient Frontier

Betas calculated with GARCHBetas calculated with GARCHmodels provide new parametersmodels provide new parameters

for a Portfolio selection withfor a Portfolio selection withEfficient FrontierEfficient Frontier

Dr. Ricardo A. TagliafichiDr. Ricardo A. Tagliafichi

Assumptions of CAPM

All investors choose mean-variance efficient portfolioswith one period horizon, although they need not haveidentical utility functions


All investors have the same subjective expectations onthe means, variances and co variances of returns

The market is fully efficient so far there arn´t anytransactions costs, indivisibilities, taxes or constraintson borrowing or lending at risk-free rate

The Efficient Frontier


∑=

=k

iiiP RXR

1

ij

k

i

k

ijj

ji

k

iiiP XXX σσσ ∑∑∑

=≠==

+=1 11

22

The mean ofa Portfolio is:

The volatilityof Portfolio is:

∑=

=k

iiX

11The condition is:

The Efficient Frontier


The covariance of apair of returns is:

The correlationcoefficient is:

n

RRRRn

iii∑

=

−−= 1

2211

2,1

)()(α

21

2,12,1 σσ

σρ =

The Effect of DiversificationXXXIII ASTIN Colloquium

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Number of Stocks

Varia

nce o

f Por

tfolio Xi = 1/N

ijiP NN

Nσσσ 11 22 −+=

Delineating Efficient PortfoliosXXXIII ASTIN Colloquium

Short sales are allowed but ....

Delineating Efficient PortfoliosXXXIII ASTIN Colloquium

Short sales are disallowed but ....

Riskless lending and borrowing exists...

p

FP RRσ

θ−

=

Subject to: 11

=∑=

n

iiX and 0≥iX for all i

Riskless lending and borrowing are not allowed

Maximize

Maximize: ij

k

i

k

ijj

ji

k

iiiP XXX σσσ ∑ ∑∑

=≠==

+=1 11

22

Subject to: ∑ ∑= =

=≥=n

i

n

iPiiii RRXiallforXX

1 1;0;1

The assets returns and theBetas


If we consider the relationship between themarket and assets returns, the asset return can bewritten as follows: imiii RaR εβ ++=

The variance of an asset return is: 2222imii εσσβσ +=

The covariance of returns between stocks i and j is thefollowing: 2

mjiij σββσ =

Considering Betas ….


The expected return of a portfolio is:

∑ ∑= =

+=n

i

n

imiiiiP RXXR

1 1βα

The variance of a portfolio is:

∑ ∑∑ ∑= =

≠= =

++=n

i

n

i

n

ijj

n

iimjijimiiP i

XXXX1 1 1 1

222222εσσββσβσ

Rearranging the formulas


The variance of a portfolio is:

∑=

+=n

iimPP i

X1

22222εσσβσ

To estimate Beta and alpha we usethe regression analysis technique

tt miiim

imi RR βα

σσ

β −== ;2

Delineating a Portfolio withBetas


Ratio of excess on return to beta is:i

Fi RRβ−

The method to select which stocks areincluded in the optimum portfolio are:

• Estimate the ratio of each stock andrank them from highest to lowest

• Invest in all stocks for which the ratio isgreater than a particular C*

The cut-off rate C*


( )

∑

∑

=

=

+

−

=i

j

jm

i

j

jFjm

i

j

j

RR

C

12

22

12

2

1ε

ε

σβ

σ

σβ

σ

To estimate C* is necessary to rank the assetsby the ratio of excess of return to Beta andestimate the value of each Ci that depends onthis ranking

The economic significance of Ci


i

FPiPi

RRC

ββ )( −

=Change the previousexpression by

Where ββββiP is the expected change in the rate ofreturn of asset i associated with 1% of change ofoptimal portfolioIf we consider the inclusion of an asset in theoptimal portfolio the excess of return on Betamay be greater than Ci as:

ii

Fi CRR

>−β

The economic significance of Ci


The previous equation may be rearranged as:

)()( FPiPFi RRRR −>− βThe left hand side is the expected excessof return of an individual assetThe right hand side is the expected excessof return on a particular stock based onthe performance of the optimal portfolio

The percentage invested ineach asset Xi


∑=

includedi

ii Z

ZX

−−= *

2 CRRZi

Fiii

iβσ

βε

If short sales are notallowed the values of Zimust be all positive Thecut-off is the value thatallows the difference to bepositive between the ratioof excess of return on Betaand C*

Some results .....XXXIII ASTIN Colloquium

0.0115432.19321.02600.0118Telefonica

0.0118074.23880.47810.0056Metrogas

0.0301724.19441.15060.0347BancoGalicia

0.0387081.39441.21230.0469PerezCompanc

0.0602402.60741.13210.0682Siderca

Excess ofReturn on

Beta

EDBeta

CExcess of

return

AStocks 2

iεσ

Portfolio selectionXXXIII ASTIN Colloquium

000.033506Telefonica

000.038551Metrogas

000.039259BancoGalicia

000.040926PerezCompanc

10.009420.044436Siderca

XiZiCiStocksC* cut-offrate is

0.044436

to obtainvalues ofZi > 0

The presence of ARCHXXXIII ASTIN Colloquium

The step to evaluate the presence ofheteroscedasticity in the model and in consequenceaccept the presence of ARCH in the estimation ofBeta is the following:

1) Analyze the ACF and PACF of the squaredresiduals of the traditional regression model. If theQ Statistic concludes that the squared residuals area black noise there are presence of Arch and isnecessary to do a most formal test like theLagrange Multiplier test proposed by Engle (1982)

The Arch LM testXXXIII ASTIN Colloquium

Once the presence of Arch using the Q. Statistic isaccepted, it may regress the squared residualsfitted in the Beta model on a constant and on the qlagged values as:

2233

222

211

2qtqtttot −−−− ++++= εαεαεαεααε L

If there are no Arch o Garch effect the coefficientsααααi should be 0. Also is used the coefficient TR2

which converges to a χχχχ2 with q degrees of freedom.If TR2 is large it can be accept the presence of Archwith q lags of incidence

The new Betas calculatedwith Arch models


Assuming that the errors of themodel were generated by ttt xy βε −=and applying an Arch(1) nowεεεεt is given by ( )2

110 −+= ttt v εααε

Then the conditional variance is 2110 −+= tth εαα

The appropriate log likelihood function is:( )

( )21110

1

2

21ln

2)2(ln

2

−−

=

−+=

−

−

−

−= ∑

ttt

T

ttttt

xyh

xyhhTT

βαα

βπl

The symmetric model


imiii RaR εβ ++=Now we can model the squares residuals with

a Garch (p,q) ∑ ∑= =

−− +++=p

it

q

iitiitit vh

1 1

222 βεαωε

Estimate the traditional regression model as:

Using the appropriate log likelihood functionwe obtain the new Betas with minimumvariance

The asymmetric models


We can use, to model the squared residuals,the following asymmetric models:

tt

t

t

ttt v

hh ++++=−

−

−

−−

1

1

1

121

2 )log()log( αε

γεβωε

caseotherindandifdwhere

vdh

t

tt

tttttt

001

1

11

211

21

21

2

=<=

++++=

−

−−

−−−−

εεγεαβϖε

Egarch

Tarch

Results applying symmetric andasymmetric models


GarchRMGarchRM

0.01120.0116Garch (1,1)1.054

(0.013)1.026

(0.018)Telefonica

0.06120.0602Garch (2,2)1.1142

(0.0145)1.1321

(0.0194)Siderca

0.03280.0302Tarch (1,1)1.059

(0.020)1.151

(0.0245)Banco Galicia

0.01260.0118Garch (2,1)0.4467(0.016)

0.478(0.025)

Metrogas

0.05070.0387Tarch (1,1)0.925

(0.008)1.2123(0.014)

PerezCompanc

RankingGarchModel

BetaStock

The new coefficientsXXXIII ASTIN Colloquium

0.01122.19901.50400.0118Telefonica

0.01264.24730.44670.0056Metrogas

0.03284.24401.05950.0347BancoGalicia

0.05071.44900.92500.0469PerezCompanc

0.06122.60921.11420.0682Siderca

Excess ofReturn on

Beta

EDBeta

CExcess of

return

AStocks

Portfolio selectionXXXIII ASTIN Colloquium

000.035968Telefonica

000.044015Metrogas

000.044994BancoGalicia

0.255850.001990.047601PerezCompanc

0.744150.005810.044761Siderca

XiZiCiStocksC* cut-offrate is

0.047601

to obtainvalues ofZi > 0

Conclusions


The presence of Garch effects in the model used tocalculate the values of Beta, permit to enforce theidea of obtaining best coefficients, with minimumvariance. In all observed cases, this premises havebeen confirmed in a real situation. These resultsmay be reproduced in several assets in differentmarkets.

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