Kernfachkombinaionen: Invesmenanalyse Porfoliomanagemen (Volailiy Predicion) O. Univ.-Prof. Dr. Engelber J. Dockner Insiu für Beriebswirschafslehre Universiä Wien A-0 Brünnersrasse 7 Tel.: [43] () 477-3805 Fax: [43] () 477-38054 Univariae Volailiy Models Sources of changing volaily (heroscedasiciy) Changes in he informaional flow Changes in he rading volume Changes in macroeconomic variables Financial leverage Financial crisis Volailiy is a measure of risk (usually calculaed on he basis of STD of reurns) Volailiy is no direcly observable
Alernaive Volailiy Models As a saring poin we assume ha reurns can be divided ino a predicable par and an unpredicable par R + = E ( R + ) + ε+ = µ + ε+ Predicable Unpredicable Under his assumpion he condiional variance of he reurns is given by σ+ = E ( R + ) µ E ( ε+ ) How can he condiional variance be esimaed? Esimaion of Condiional Variances The simples form o esimae he condiional variances is given by σ = T i = ω ( )( r µ ) i i where ω i () are he weighs of hisorical squared mean adjused reurns, which can vary over ime. Depending on he choice of he weighs ω i () we can disinguish differen ypes of condiional volailiy models Naive model, EWMA, ARCH, GARCH models
Alernaive Volailiy Models Naive model (changing window) σ k = ( r i k i = µ ) Window Reurn Mean Reurn Characerisics of he naive model Consan weighs Depending on he choice of he window No volailiy clusering Simple calculaion Alernaive Volailiy Models 0 and 40 day volailiies for he ATX ATX 60% 50% 40% 30% 0% 0% 0% 990 99 99 993 994 995 996 0 Tage 40 Tage 3
Alernaive Volailiy Models 0 and 40 day volailiies for AUA AUA 40% 35% 30% 5% 0% 5% 0% 5% 0% / 4/94 3/4/ 94 5/ 5/ 94 7/ 7/94 9/6/ 94 / 7/94 / 0/ 95 0 Tage 40 Tage Alernaive Volailiy Models Exponenially weighed moving average (EWMA) i ( λ) λ ( r i µ ) = ( λ)( r µ ) λσ i= σ = Characerisics of he EWMA model Dynamic adjusmen of weighs Non saionary ( Random Walk ) Good fi in empirical applicaions Used as volailiy model in RiskMerics Sable relaionship 4
Alernaive Volailiy Models ARCH Model (Auoregressive Condiional Heeroskedasiciy Model, ARCH(p)-Model) σ p = ω + i = α ( r µ ) i i Characerisics of he ARCH model Dynamic weighs ha can be esimaed Difference o he EWMA model because of he consan and no auoregressive erm for volailiy Choice of oder p needs o be deermined empirically Alernaive Volailiy Models GARCH Model (Generalized Auoregressive Condiional Heroscedasiciy Model) σ = ω + p i= i + i µ ) + q α ( r β σ i= ARCH-par GARCH-par Characerisics of he GARCH model Dynamic weighs Corresponds o EWMA wih more general srucure Volailiy clusering i i 5
GARCH(,) Reurn Model Sochasic model formulaion r = c + u wih u = ε h and ε N(0,) and h = ω + α u - + β h - Uncondiional variance is given by σ ε ϖ = α β Volailiy Models - Example We use German DAX daa Daily daa from 986 o 997 We sar ou wih prices Calculae he reurns Check reurns if hey are whie noise Check he squared reurns if here is srucure Esimae a GARCH (,) model Calculae he volailiies based on an empirically esimaed GARCH(,) model 6
DAX-Example 0.0 0.05 0.00-0.05-0.0-0.5 /06/86 /06/89 9/06/93 7/07/97 RETDAX 5000 4000 3000 000 000 0 /06/86 /06/89 9/06/93 7/07/97 DAXINDX DAX-Example 7
DAX-Example DAX-Example Variable Coefficien Sd. Error -Saisic Prob. C 0.000630 0.00093 3.64653 0.00 Variance Equaion C 8.8E-06 7.08E-07.5538 0.0000 ARCH() 0.36660 0.006688 0.4383 0.0000 GARCH() 0.8669 0.0674 69.9044 0.0000 R-squared -0.00059 Mean dependen var 0.000349 Adjused R-squared -0.00480 S.D. dependen var 0.0330 S.E. of regression 0.0340 Akaike info crierion -8.78860 Sum squared resid 0.47557 Schwarz crierion -8.780866 Log likelihood 9660.63 Durbin-Wason sa.00790 8
DAX-Example 0.06 0.05 0.04 0.03 0.0 0.0 0.00 /06/86 /06/89 9/06/93 7/07/97 GARCHVOL 9