Variance Covariance (Delta Normal) Approach of VaR Models: An Example From Istanbul Stock Exchange

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1 ISSN (Paper) ISSN (Ole) Vol.7, No.3, 206 Varace Covarace (Delta Normal) Approach of VaR Models: A Example From Istabul Stock Exchage Dr. Ihsa Kulal Iformato ad Commucato Techologes Authorty, Turkey Abstract May vestors desre to kow how much moey they ca lose for example a day or a te days. I ths study, varace-covarace approach of the VaR models s troduced to the reader. It estmates maxmum potetal loss for a gve probablty ad tme horzo. It shows moey type oe loss value. I a calculato process, frstly, portfolos are created. The, returs dstrbuto s detfed. Ad lastly, VaR values of portfolos are measured. Daly loss s calculated wth usg 252 days hstorcal data belogg to the year 205. Stocks are chose from Istabul Stock Exchage (BIST 00 Idex). Calculato s made for both 95 % ad 99 % cofdece level ad oe day ad te days holdg perods. Keywords: Rsk Measuremet, VaR, Varace-Covarace approach, correlato, portfolo rsk. Itroducto Rsk ad retur are the ma parameters of all vestmet process. Face theory accepts that vestors are rsk averse ad utlty maxmzer. I that sese, rsk maagemet appears as the most crtcal feld of vestmet evaluatos. May facal sttutos ad regulatory bodes also gve more mportace to rsk measuremet after global facal crss. There are may ways of calculatg market rsk cosstg of terest rate, exchage rate ad etc. Value at Rsk models (VaR) have bee appled sce 994. VaR models oly estmate quatfable rsk ad they are ot approprate to measure of poltcal ad regulatory rsks. Models measure probable maxmum loss of portfolo for gve cofdece level ad holdg perods. I other ways, vestors have a chace to kow ther potetal loss a reasoable boud to use VaR models. Dfferet VaR models show dfferet results. They have may pros ad cos both. They ca be classfed as parametrc ad o parametrc models. As a parametrc oe, varace covarace approach (also kow as delta ormal) s wdely used facal world. It s very practcal ad easy to use. It depeds o correlato ad covarace matrces to estmate varace ad stadard devaato of rsky asset portfolo. However, returs usg the aalyss should have a characterstcs of ormal dstrbuto. The am of ths paper s brefly to descrbe Varace-Covarace approach of VaR methods ad to measure maxmum loss portfolos cosstg of dfferet stocks. The rest of the paper s orgased as follows. I secto 2, VaR models are brefly troduced. I secto 3, varace-covarace approach s used emprcal studes to measure loss value. May statstcal techques are used ths secto. Fally, secto 4 cocludes the paper. 2. Varace-Covarace Approach of VaR Methods All vestors desre to mmze ther rsks o vestmets whle maxmzg returs. Thus, predcto of rsk s a key put all vestmet decsos. The facer wll vest ay kd of asset oly f the expected retur wll be hgher tha the perceved cost. Geerally, vestors face a trade off stuato whch a large but bad vestmet may result huge loss whle a good but small vestmet may result opportuuty cost. I that sese, rsk maagemet s a ecessary effort to maxmze the portfolos retur ad to mmze losses. Value at Rsk (VaR) s a wdely used as a rsk measuremet method calculatg the worst case losses over a predetermed tme perod ad at a predefed cofdece level (Johasso, 203: ). I 994, VaR s frstly appled by J. P. Morga creatg CredtMetrcs methodology, RskMetrcs ad RAROC models. The model was to be approprated ad appled by may compaes (Auas et al, 2009: 9). Whle regulatory groups have bee wdely promotg t as a bass for settg regulatory mmum captal stadards, may facal sttutos have bee developed ts dervates terally as a way of motorg ad maagg market rsk (Darbha, 200: 2). Basel Commttee of Bakg Supervso, USA Federal Reserve System ad USA Stock Commttee 995, Europea Uo Captal Requremets Drectve 996 proposed to use value at rsk method as oe for market rsk maagemet (Auas, 2009: 9). There are three ma assumptos of VaR models. Oe of them s statoary requremet meag that daly fluctuatos of returs are depedet from yesterday s or tomorrow s retur. It s related wth the radom walk theory of face. The secod assumpto s kow as o-egatvty requremet meag that facal assets do ot have egatve values. The thrd assumpto s related wth dstrbuto of facal data. VaR model assumes that facal hstorcal data are dstrbuted ormally (Alle et all, 2004: 8 9). VaR methods are geerally classfed two ma groups as parametrc whch s also called varace-covarace (or delta ormal) approach ad o-parametrc method cosstg of two smulato methods whch are called hstorcal smulato ad Mote 65

2 ISSN (Paper) ISSN (Ole) Vol.7, No.3, 206 Carlo smulato. There are both pros ad cos of these methodologes (Bozkaya, 203: 22). I ths study, varace covarace approach s preferred to calculate portfolo loss. Speed ad smplcty are the ma two advatages of ths method. Moreover, dstrbuto of returs eed ot be assumed to be statoary through tme, sce volatlty updatg s corporated to the parameter estmato (Bohdalova, 2007: 2 3). However, I ths approach, oly lear rsk s measured ad correlatos are assumed as stable. As a other drawbacks, t heavly reles o ormal dstrbuto ad ad returs the market are wdely beleved to have fatter tals tha a true to ormal dstrbuto. 3. Emprcal Aalyss 3.. Data ad Formulas Hstorcal data s usually used by VaR models to calculate maxmum (worst case) losses over a certa holdg perod at a gve cofdece terval. I that sese, (holdg) tme perod ad a cofdece level are the ma parameters of measuremet. Models express losses as oe term ad dollar values. The result shows us that losses wll ot be exceeded by the ed of the tme perod wth the specfed cofdece level (Darbha, 200: 2). I ths study, two hypothetcal portfolos are created at frst. They have same three compaes stocks but dfferet weghts. These three compaes have traded Istabul Stock Exchage (BIST). They are operatg a petroleum dustry. Whle TUPRS s operatg refery feld ad PETKIM s petrochemstry feld, AYGAZ s a LPG compay. Moreover, TUPRS ad AYGAZ deped o same ower but they are maaged separately. Table. Name ad Actvtes of Compaes Stocks Compay Name Busess Actvty AYGAZ Aygaz A.Ş. LPG 2 PETKIM Petkm PetroKmya Holdg A.Ş. Petrochemstry 3 TUPRS Türkye Petrol Rafeler A.Ş. Refery I ths study, market retur ad beta of compaes are estmated by usg daly returs (adjusted prce for US dollar). They are acheved from the Isyatrm database 2. Oe year perod (252 workg days) data s used. It belogs to a year of 205. Morever, BIST 00 dex s used as a market dex. To calculate for stocks daly retur; the formula s appled as follows:! = " #$%& '()* () " #$)* where R s a daly retur of share, Rt s a closg prce of share t date ad Rt- s a closg prce of share t - date To calculate the Idex (BIST 00) daly retur; the formula s appled as follows: +!,-.//= 023.// $4%023.// $)* 023.// $)* Where R Bst00 s a average retur for market, Bst00 t s a market retur t date, Bst00 t- s a market retur t- date. To calculate varace of stocks daly retur ad dex retur, I used the followg hstorcal volatlty formula: - å 2 ( R- Raverage) s 2 = = (3) Where s 2 s a varace of daly share retur, R s a daly retur of share, R average s average daly retur, s a sample sze (252 days) To measure how stocks vary together, stadard formula for covarace ca be used: å[( X - X ).( Y -Y )] Cov (X,Y) = - = (4) where the sum of the dstace of each value X ad Y from the mea s dvded by the umber of observatos mus oe. The covarace calculato eables us to calculate the correlato coeffcet, show as: Cov( X, Y) s.s Correlato Coeffcet = X Y (5) where s the stadard devato of each asset. However, f there are more tha two facal assets the portfolo, the correlato ad covarace matrces are eeded to solve equatos. To calculate stadard devato of portfolo (posto), the followg formula s appled: ( ) 66

3 ISSN (Paper) ISSN (Ole) Vol.7, No.3, 206 s p = å = 2 2 ( w. s ) + 2( åå = j= ( w. s. w. s. rj) (6) Where σ p s a stadard devato of portfolo, σ s a stadard devato of stocks, w s a weght of stocks a portfolo ad ρ j s a correlato coeffcet betwee stocks ad j Emprcal Results I ths study, excel fuctos ad data solver are used for all calculato. The calculato of varace- covarace model volves the followg steps: Step Determg Holdg Perod ad Cofdece Level (table 2) Step 2 Determg Portfolo (table 3 ad table 4) Step 3 Creatg a Probablty Dstrbuto (table 5) Step 4 Determg Correlatos betwee Assets (table 6 ad table 7) Step 5 - Calculatg the Volatlty of the Portfolo (table 8) Step 6 - Calculatg the VaR Estmate (table 9) Table 2. Ma Parameters of Calculatos Parameter Value Cofdece level %95 ad 99 % Tme Horzo day ad 0 days Sze of hstorcal data 252 days Testg perod A umber ad closg prce of stocks Porfolo ad Portfolo 2 are gve below. They have same stocks but dfferet weghts. ( ): Stocks Number of Stocks () Table 3. Dstrbuto of stocks Portfolo Closg Prce Market Value of Stocks (USD) (USD) (3) (3) = () x j J Weghts (%) AYGAZ , ,3 PETKIM , ,3 TUPRS , ,3 Market Value of Portfolo = Stocks Number of Stocks () Table 4. Dstrbuto of stocks Portfolo 2 Closg Prce Market Value of Stocks (USD) (USD) (3) (3) = () x Weghts (%) AYGAZ , PETKIM , TUPRS , Market Value of Portfolo = As see from Table 2 ad Table 3, total market value of portfolo equals to the sum of stocks values. Statstcal features of stocks ad market dex (BIST 00) are gve below. Table 5. Statstcal Features of Returs Stocks BIST 00 AYGAZ PETKIM TUPRS Stadard Devato 0, , , ,02 Varace 0, , , ,00044 Skewess 0, , , ,02924 Kurtoss, , , ,48746 Average 0, , , ,00028 Mmum -0,052-0, , ,065 Maxmum 0, , , , Asymmetry s statstcally measured by skewess evaluatg how ad whch way returs are dstrbuted aroud mea. It s used to determe whether data s ormally dstrbuted or ot. For ormal dstrbuto, skewess takes zero value ad returs are dstrbuted aroud mea equally. It meas that 50% percetage les below ad above the mea. Kurtoss quatfes how peaked s the dstrbuto. Tals of dstrbuto ca be expressed by 67

4 ISSN (Paper) ISSN (Ole) Vol.7, No.3, 206 kurtoss results. I ormal dstrbuto, kurtoss must be three therefore f data s ot ormally dstrbuted the a kurtoss values must be more tha three (Bozkaya, 203: 8). VaR models assume probablty dstrbuto s ormal dstrbuto however facal returs are ot ormally dstrbuted but very close to ormal dstrbuto. As see from Table 5, skewess ad kurtoss results are reasoable to accept that returs are dstrbuted ormally. I that sese, VaR model s ma assumpto s provded by hstorcal data. Varace-covarace approach uses matrces gvg chace to measure VaR value for a portfolo cosstg of hudreds of assets. As see from the formula (6), stadard devato of portfolo measuremet requres correlatos of each asset ad also covarace betwee them. Usg of varace covarace matrce s practcal way of calculatg stadard devato of portfolo. I ths study, t s demostrated how the parametrc methodology uses varace ad correlato matrces to calculate the varace, ad hece stadard devato, of a portfolo. Table 6: Corelato matrx of share s retur AYGAZ PETKIM TUPRS AYGAZ 0,6906 0, PETKIM 0,6906 0,56769 TUPRS 0, ,56769 The degree of depedece betwee two varables s measured by correlato detfyg what percetage ad drecto two varables move together. Portfolo rsk (volatlty) s smaller tha ts dvdual assets' rsks. I that sese, t s ecessary to kow the relato betwee assets to a portfolo varace. Correlato takes value betwee - ad + (Bozkaya, 203: 6). As see from Table 6, correlato betwee AYGAZ ad PETKIM s hgher tha AYGAZ TUPRS ad PETKIM TUPRS. However, both correlatos are postve ad hgh eough. Ths s ot good for dversfed portfolos. The correlato coeffcet ca be calculated as usg the covarace betwee the assets measurg how average value of two facal assets move together, how they vary together. Covarace helps facal maager to decde whch assets are smlar ad move together ad whch move verse (Bozkaya, 203:6). Table 7: Covarato matrx of share s retur BIST 00 AYGAZ PETKIM TUPRS BIST 00 0, ,0008 0, , AYGAZ 0,0008 0, , , PETKIM 0, , , ,00023 TUPRS 0, , , , Covarace matrce helps us to calculate volatlty of portfolos. Covarace values betwee stocks are multpled by each shares weghts ad the collected to fd portfolo volatlty. Table 8. Stadard Devato ad Varace of Portfolo ad Portfolo 2 Portfolo Portfolo 2 Stadard Devato 0, , Varace 0, , VaR s calculated as usg the followg formula: VaR = P* a * s * t (7) Where P s value of portfolo (or posto), α s cofdece level, σ s a volatlty of portfolo ad t s a holdg perod. For 95 % cofdece level, α s,65 ad for 99 % cofdece level, α s 2,33. VaR maybe calculated for dfferet tme legth. I ths study, both oe day ad te days holdg perods are take accout. If ayoe tres to fd VaR values more tha oe day such as 0 days, t eeds to multply daly volatlty results by 5678 Table 9. VaR Values of Portfolo ad Portfolo 2 Portfolo Portfolo 2 95 % 99% 95 % 99% VaR (oe day-usd ) VaR (te days-usd) Market rsk of ay portfolo ca be measured by VaR models. As see from Table 9, cofdece level 95 % ad oe day holdg perod, maxmum loss of Portfolo wll ot exceeded 2978 USD ad maxmum loss of Portfolo 2 wll ot exceeded 2990 USD. It meas that there s oly 5 % chace that the loss of ext day wll be greater tha 2978 USD for portfolo ad 2990 USD for portfolo 2. Wth cofdece level 95 % ad te days holdg perod, maxmum loss of Portfolo wll ot exceeded 940 USD ad maxmum loss of Portfolo 2 wll ot exceeded 9447 USD. Wth cofdece level 99 % ad oe day holdg perod, maxmum loss of Portfolo 68

5 ISSN (Paper) ISSN (Ole) Vol.7, No.3, 206 wll ot exceeded 423 USD ad maxmum loss of Portfolo 2 wll ot exceeded 4247 USD. It meas that there s oly % chace that the loss of ext day wll be greater tha 423 USD for Portfolo ad 4247 USD for Portfolo 2.Wth cofdece level 99 % ad te days holdg perod, maxmum loss of Portfolo wll ot exceeded 3370 USD ad maxmum loss of Portfolo 2 wll ot exceeded 342 USD. 4. Cocluso It s possble for vestors to estmate probable loss value of ther portfolos for dfferet holdg perods ad cofdece level. Varace covarace approach helps us to measure portfolo rsk f returs are dstrbuted ormally. I ths study, two hypothetcal portfolo to calculate potetal loss wth both 95% ad 99% cofdece level as well oe day ad te days holdg perods are created. As a ma cocluso, there s o huge dfferece betwee Portfolo ad Portfolo 2 results. It s thought that the portfolo was ot dversfed well. There were oly three stocks the portfolo but mportatly ther correlatos were ot low eough to decrease rsk adequately. Stocks have equal weghts Portfolo as 33,3 %. Stocks have dfferet weghts Portfolo 2, cosstg of 50% AYGAZ, 30 % PETKIM ad 20 % TUPRS. AYGAZ ad PETKIM have lower stadart devato tha TUPRS. Although Portfolo 2 has a lower stadart devato tha Portfolo, t gets greater VaR values for all categores. Because, AYGAZ has greater correlato coeffcets comparg wth other two stocks. Thus, whle rsk evaluato of oe stock s related wth especally volatlty characterstcs, rsk evaluato of portfolo s related wth correlato betwee rsky assets sde the portfolo. Refereces Alle, Lda, Jacob Boudoukh ad Athoy Sauders (2004). Uderstadg Market, Credt, ad Operatoal Rsk The Value At Rsk Approach, Blackwell Publshg. Auas, Povlas, Joas Nedzveckas, Ryts Krušskas (2009). Varace Covarace Rsk Value Model for Currecy Market, Egeerg Ecoomcs, No (6) Bozkaya, Muhammet (203). Comparso of Value At Rsk Models ad Forecastg Realzed Volatlty By Usg Itraday Data A Emprcal Study O Amerca Stock Exchages, Neoma Busess School, Master Of Scece I Face, December, Bohdalová, Mára (2007). A comparso of Value at Rsk methods for measuremet of the facal rsk, E- Leader, Prague. /Bohdalova. Darbha, Gagadhar (200). Value-at-Rsk for Fxed Icome portfolos A comparso of alteratve models, Natoal Stock Exchage, Mumba, Ida December. Johasso, Adreas ad Vctor Sowa (203). A comparso of GARCH models for VaR estmato three dfferet markets., Uppsala Uversty Departmet of Statstcs. 69