Post on 10-Mar-2018
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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
∑=
=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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
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 ….
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
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*
XXXIII ASTIN Colloquium
( )
∑
∑
=
=
+
−
=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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
∑=
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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
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
XXXIII ASTIN Colloquium
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.