Solving Market Index Biases Using Minimum Risk Indices

Transcription

1 Solvig Market Idex Biases Usig Miimum Risk Idices JORDI ANDREU Departamet Gestió Empreses Uiversitat Rovira i Virgili Av. Uiversitat Reus SPAIN jordi.adreuc@urv.cat SALVADOR TORRA Departameto de Ecoometría, Estadística y Ecoomía Española Uiversidad de Barceloa C/ Tiet Valezuela, 1-11; T4, P1, D106; Barceloa SPAIN storra@ub.edu Abstract: - The market, i cotrary to what is defeded i traditioal fiace, is ot a efficiet zero-sum game where hypotheses of the CAPM are fullfilled. I that situatio, the market portfolio is ot located i the efficiet frotier, ad passive ivestmets are ot optimal but biased. I this paper, the sample, costructio, efficiecy ad active biases are defied, ad trackig error is also aalysed. We propose Miimum Risk Idices (MRI) to solve market idex biases, ad to provide ivestors with ivestmets closer to the efficiet frotier. MRI (usig a Value-at-Risk Miimizatio approach) are calculated for three stock markets achievig iterestig results. Our idices are less risky ad more profitable tha curret Market Idices i the Argetiia ad Spaish markets. Two iovatios must be outlied: First, a error dimesio has bee icluded i the backtestig ad, secod, the Sharpe Ratio has bee used to select the best model from all models preseted Key-Words: -Idex Biases, Passive ivestig, Market Idices, VaR, Portfolio Optimizatio. 1 Itroductio Fiace has grow to iclude ideas such as the market, zero-sum games, efficiecy or the CAPM. Fiacial theory, as based o these four pillars, cocludes that a ivestor ca ot cosistetly beat the total market, thus meaig that a passive ivestmet strategy is the optimal strategy to follow if we operate i a efficiet zero-sum game market where CAPM hypotheses are fulfilled. To simplify this strategy, market idexes were created as refereces for passive ivestors, as proxies of a cocrete aalytical market. I this framework, total market idices are appropriate bechmarks for passive ivestmet. I [5] we proposed a exteded revisio of these four fudametal items achievig coclusios that are summarized here: a) markets are mius-sum games (moey is substracted from the system through commissios, fees ad other costs); b) markets are margially efficiet (because of ivestors irratioalities ad psychological biases, limited arbitrage, ad reported empirical aomalies); c) whe CAPM hypotheses are ot fulfilled, ivestors hold differet portfolios, therefore, the market portfolio is ot optimal ad ot located i the efficiet frotier. I that revised framework, passive ivestmet (ad idexig) is ot optimal but biased. These biases ca be see i Fig. 1, i a mea-variace aalysis. I a cocrete momet of time t, the Market Portfolio retur R is located mt ISSN: ISBN:

2 below the Efficiet Poit retur R et ad the Maximum Retur available at the market R ct. The market Idex retur R (proxy of the Market Portfolio), ad the Proxy Portfolio retur bt R pt (proxy of the idex), are usually located eve below i terms of retur ad probably sufferig from more risk. Payig attetio to retur 1 differeces, market idex biases are defied as follows: i) The sample bias (SB) is the differece betwee R bt ad R mt due to the stock sample selectio that make up a idex. ii) The costructio bias (CB) is the differece betwee R bt ad R mt due to the use of differet weightig criteria (price, equal or GDP) or methodologies (Laspeyres, Paasche or Geometric mea amog others) i the costructio of a idex, iii) Trackig error (TE) is the differece betwee R ad R due to commissios ad turover bt pt costs; iv) Efficiecy bias (EB) is the retur differece betwee R mt ad R et ; v) Active bias (AB) is defied as the differece betwee the Efficiet Poit ad the best ivestmet opportuity, that is, a opportuity cost. Mathematically, these biases ca be calculated usig expressios i Table 1. I that table, it is possible to foud biases defied i a momet of time (t ), ad also alog a time period -Average Absolute Bias (AA) ad Stadard Deviatio of a Bias (SD)-. Cocered with this situatio, the aim of this paper is to propose alterative Market Idices that solve, at least i part, some of the detected biases. There are three importat reasos for creatig ew Market Idices. First, there is a huge iterest i market risk maagemet after the last bearish market cotext ad fiacial disasters. Because of their special characteristics, such iterest is perhaps greater whe we speak about emerget markets. Secod, because market yield is still used as a essecial parameter ad trillios of dollars are ivested by followig the market. Third, market idex biases shows us that a capitalizatio-weighted Idex (ad eve the market portfolio) is ot always located i the efficiet frotier, therefore, there are other portfolios able to beat the market with lower 1 Market Idex Biases could be defied usig retur, risk, or risk-adjusted returs. We defied them usig retur i [5] assumed risk. I that paper, we propose a methodological approach to deal with this questio usig a parametric Value-at-Risk Miimizatio to build Miimum Risk Idices (MRI), bechmark portfolios with better risk-adjusted characteristics tha owadays market idices. The authors apply the method to the Spaish, Argetiia ad America stock markets durig the period, ad estimate Covariace matrices by differet legth movig averages. After that, the best alterative market idex is selected usig traditioal backtestig, error dimesio backtestig ad the Sharpe Ratio. Although the estimatio methods used here are very simple, results are iterestig. All idices are less risky tha the Spaish IBEX 35 ad the Argetiia Merval (curret Market Idex) ad, surprisigly, more profitable. This highlights oe idea: similar ivestmet strategies could beat the market, thus questioig the Efficiet Market Hypothesis, ad reiforcig the market bias aalysis. Possible applicatios of Miimum Risk Idices are clear: they could reduce the risk assumed by istitutioal ad mutual fuds that owadays follow curret Market Idices. They could also be used as a bechmark for risky assets or as a basis for developig derivatives. The structure of the paper is as follows. I Sectio 2 we discuss the theoretical framework of MRS. This aalysis is completed i Sectio 3 where the theoretical framework is applied to the Spaish, America ad Argetiia Stock Markets durig the period. Performace ad risk of MRS are recostructed ad the best idex is selected usig backtestig ad performace parameters. Sectio 4 gives the coclusios, Sectio 5 establishes future lies of research, ad Sectios 6 ad 7 cotai the appedix ad refereces. 2 Miimum Risk Idices Fiacial risk has historically bee aalyzed by multiple measures [16]. However, the icreasig volatility i fiacial markets, derivatives ad techological advaces force academics to ow treat market risk from other perspectives. Oe widely accepted measure is Value-at-Risk [18] with its evolutios [17, 12, 1]. All the VaR-based methods have several problems regardig leptokurtosis or skewess, so complemetary techiques as Stress testig, Coditioal VaR or Extreme Value Theory with the Expected Shortfall have bee aalysed [14 ISSN: ISBN:

3 or 15]. VaR has several problems, but if we ca determie a cotrolled sceario with some iterestig coditios, traditioal VaR methods are reliable eough [9] ad easier to calculate tha extreme value methods. I this paper we pay attetio to VaR as a tool for market risk maagemet ad portfolio optimizatio through the creatio of ew Market Idices by VaR miimizatio. If we wat to use VaR as a risk maagemet tool, we have to fid a method that istitutios ad ivestors fid easy to follow. These characteristics mea that parametric VaR is the most suitable method, however, it is ecessary to take ito accout the weakesses of the parametric approach: if returs are ot ormal, the VaR measure will ot be coheret [6]. I this study, the Cetral Limit Theorem should make Market Idex returs similar to a Gaussia distributio if the umber of shares formig the Idex is high eough. Ideed, as i the portfolio, we do ot iclude o-liear positios, ad we use weekly data at a 5% sigificace level to calculate the VaR, the parametric Gaussia approach is cosidered reasoably good [9]. The problem the is reduced to miimize Parametric VaR subject to o-egativity of stock weights ad o-leveraged possibilities, selectig the desired sigificace level. Oe of the most importat steps i this procedure is to estimate retur covariace matrix, ad this could be doe usig: i) the Historical Volatility Method; ii) the Movig Average Method; iii) the Expoetial Weighted Movig Average Method; or iv) The GARCH ad E-GARCH Methods. After the Covariace matrix has bee estimated usig oe of these methods, the miimizatio method ca be used to obtai the optimal weights each share must have withi the Idex to miimize the Idex s market risk. With the historical data available, we ca recostruct the performace ad evolutio of Miimum Risk Idices to compare returs ad risks betwee Miimum Risk Idices ad curret Market Idices. Oce the recostructio of Miimum Risk Idices is available for a certai market, the validity of each approximatio must be checked by a backtestig process. This process will establish how well the model applied to the data fits the real market. Here we ca see the importace of selectig the best Idex from all the approximatios. I our opiio the best model should be selected i accordace with two key ideas: The model s capacity to be accepted by a periodic backtestig process, ad the relatioship betwee retur ad risk i each idex. 3 Miimum Risk Idices i Real Markets Usig the theoretical framework developed i part 2, we geerate Miimum Risk Idices for each Stock Market we chose. As a example of how our methodology reacts to differet Market Idices, i this sectio we apply it to the Spaish Stock Market, to the America Stock Market, ad to the Argetiia Stock Market. These examples have two objectives. First, it is iterestig to test how Miimum Risk Idices work i Stock Markets with differet volatility ad efficiecy. Secod, each of the Idices represets a differet way to build a Market Idex, ad usig them i my approximatio is a first step to determiig the importace of differet samplig strategies, weightigs ad costructio rules i the calculatio of a market idex. The sample ad costructio ad efficiecy bias must be take ito accout whe aalysig the performace of a idex. I the Argetiia ad America markets, the MERVAL ad the DowJoes show importat costructio ad sample biases; first because they are weighted usig egotiatio ad price, ad secod because they are ot calculated usig a Laspeyres capitalizatio approach. The aim is to create Miimum Risk Idices based o the historical compositio of the IBEX35, the Dow Joes Idustrial Average SM ad the MERVAL for the period. To put it more simply: Miimum Risk Idices would be developed by takig ito accout oly the shares cotaied i each Idex i each period. I this way it is possible to determie whether a differet weightig i the compoets of the actual Idices usig a VaR Miimizatio criterio ca reduce risk ad to aalyse how this affects the profitability of Market Idices. We call our Idices IdexVaR35 (IVaR35 for the Spaish Market, IdexVaR30 (IVaR30) for the America Market, ad IdexVaRM (IVaRM) for the Argetiia Market. As we have metioed, there is ot just oe Miimum Risk Idex for each market, because with each estimatio criterio we ca create a Miimum Risk Idex. The Covariace matrix was estimated i all the markets by the simplest estimatio methods (the Historical method ad the Movig Average Method usig legths from 4, to 100 weeks) i order to explai the method s potetial beefits, although ISSN: ISBN:

4 the authors kow these estimates ca be improved by more complex methods. I the ed we decided to preset IVaR35, IVaR30 ad IVaRM Idices calculated oly by some of these Movig Averages as beig represetative of the short, medium ad log terms. Covariace matrices estimated with a few data (4-30 weeks) are problematical because the miimizatio process is difficult or rather ustable i some cases. Short-legth Movig Averages chage quickly i respose to fiacial data but they cosistetly uderestimate the VaR value ad cause problems iside the miimizatio process because the positive ad semi-defied Variace-Covariace Matrix coditio is sometimes ot fullfilled. Medium-legth Movig Averages (30-52 weeks) are more stable ad VaR measures closer to real values. Fially, log-legth Movig Averages (e.g weeks) are the most stable but are less able to adapt to volatile short-term chages. Despite the limited predictio capacity of Movig Averages, results with these approximatios are quite iterestig. 3.1 Volatility Aalysis The basic objective of the study is, by VaR miimizatio, to create Miimum Risk Idices that are less risky tha curret oes. Table 2 shows how our Idices are less risky tha the curret oes. From the data, it is easy to see how the reductio i volatitily is greater i the Spaish Market tha i the America or Argetiia Markets. It also shows that, i geeral, the loger the movig average, the less volatile, which meas that risk is reduced. This seems ot to be true i all the cases with the logest movig averages (52 ad 78 i the IVaR35, i the IVaR30 ad i the IVaRM) for which volatility is more or less the same or icreases slightly. As with loger legths, it is more difficult to estimate short chages i volatility, which could mea that there is a optimal movig average legth beyod which it is impossible to reduce risk usig the movig average method. Improved Movig Averages (i the IVaR35 ad i IVaRM) are a little more risky tha those with o improvemets. This result is ratioal because, firstly, multiple-step estimatio is applied to avoid uderestimatig the risk ad, secodly, the 0.01% weightig restricts oe asset so that the portfolio ca be less diversified ad risk rises. Volatility reductio is clearer i Fig. 2, which shows cumulative volatilities. The first picture shows the Spaish Market. The first lie represets the riskiest Idex, which i our case is the curret Market Idex (IBEX 35 ). The secod group of lies is made up of the MA10, MA10a ad MA10b approximatios. The third group, with half the IBEX 35 risk, is made up of the MA25, MA52, MA78 approximatios ad all their modificatios. The most stable approximatios are the modificatios a, especially MA52a, which is less risky tha MA25a ad MA78a. This agai idicates the existece of a optimal legth for movig averages beyod which it is impossible to better estimate the covariace matrix ad reduce risk with movig average methods. The secod picture shows the America Market. The first lie agai represets the riskiest Idex, which is ow the MA10 approximatio due to problems with the positive ad semi-defied covariace matrix coditio. The, ad before observatio 75, the secod riskiest Idex is the curret Dow Joes Idustrial AverageSM. Below the curret Market Idex, ad with less risk, we fid all the other MA approximatios. The least risky approximatios are MA25 ad MA30 ad the more data is used to build the MA, the riskier the Movig Average approximatio seems to be, which supports the idea of the optimal legth for movig averages. Fially, the third picture represets the Argetiia Market. The first lie (the riskiest Idex) is agai the curret Market Idex (MERVAL). Below we ca see the MA10,a,b as the secod group of riskiest approximatios, ad below this group, with less risk, the other MAs. Agai we ca establish the idea of optimal legth for movig averages. 3.2 Aalysis of extreme losses i VaR miimizatio [7, 13] show that ot allowig agets to assume more risk tha a certai VaR value or to develop VaR miimizatios ca icrease extreme losses, especially whe retur distributios are very differet from Normal distributios. These results appear basically whe distributios are heavily skewed or have log fat tails. I our case, the problems oticed by the above authors are ot excessively importat (see Table 3). For the shortest movig average, extreme losses are similar to those of IBEX 35 ad lower i the America ad Argetiia market ad decrease whe we icreased the movig average legth. There is a certai movig average legth whe extreme losses start to rise agai (Ma78 i IVaR35, MA70 i IVaR30 ad more difficult to defie i IVaRM), which agai ISSN: ISBN:

5 supports the existece of a optimal legth movig average. 3.3 VaR Aalysis Each approximatio has a differet VaR measure that evolves over time. Great chages i volatility that are commo i movig average approximatios are greater i short legth movig averages tha i log legth oes. O the other had, as these types of averages do ot attach differet weights to more recet data tha to older data, movig averages are idicators of past volatility, regardig iappropriate Covariace estimatios whe price treds chage. This problem decreases whe the legths are loger. Fially, we should poit out that short movig averages usually uderestimate VaR, so the losses beyod the VaR will be more frequet i those cases. The loger the movig average legth the lesser the uderestimatio of the VaR measure. 3.4 Retur Aalysis Fig. 3 shows all the returs of Movig Average approximatios. I the Spaish market, all our Idices have higher returs tha the IBEX 35. These results are surprisig but ot uique. Other authors have costructed portfolios able to beat the market [e.g. 3, 4, 8]. There are two reasos for these data. First, the Spaish Stock Market is sufferig efficiecy bias ad Miimum Risk Idices are harvestig part of it. Secod, the way the IBEX35 is costructed geerates sample ad costructio bias, eve they should be relatively small comparig with these biases i other markets. I cotrast, i the America Stock Market, o Miimum Risk Idex beats the market. I the IVaR30, the approximatios with the worst returs are MA10. The other approximatios performed quite well durig the bearish market, beig ear the actual idex or beatig it i some periods, but they performed worse tha the curret idex durig ad after the Iraq war i the bullish market. I the ed, the best approximatio i terms of profitability is the MA60, with a 20% lower retur tha the Dow Joes Idustrial AverageSM. The same two reasos could be put forward i this case. The efficiecy bias i the America market is clearly lower tha i the Argetiia or Spaish case, but the sample ad costructio biases i the Dow Joes are theoretically importat. How these biases seem to compesate amog them avoidig the outperformace of our idices is a iterestig idea to be aalysed i the future. Fially, i the Argetiia Market, all approximatios (except 78b) are able to beat the market. The same reasos put forward for the IBEX 35 are valid here. The efficiecy biases seem to be especially importat kowig a little about the Argetiia market. The costructio ad sample biases are very importat takig ito accout the sample is ot totally represetative of the market ad the idex is costructed usig a egotiatio weightig. It is ecessary to take ito accout some cosideratios. Firstly, it is essetial to discover how efficiecy affects these coclusios. This would mea calculatig efficiecy tests for each market ad comparig results, but I must leave this for further research. Secodly, I must deeply aalyse how other idex biases affect risk ad retur. Fially, it is essetial to aalyse how results could improve usig more powerful techiques to estimate variaces ad covariaces. 3.5 Normality Aalysis ad Backtestig Normality aalysis of logarithmic returs is ot very positive, as it ca be see i Tables 4, 5 ad 6. I all cases, the Normality Hypothesis has bee rejected except i the case of Merval. I the case of IVaR35, the distributios have leptokurtosis ad are slightly egatively skewed. I the case of IVaR30, the distributios are more leptokurtical, ad egative skewess is especially importat, affectig, as I have said, the profitability results. Fially, i the case of IVaRM, distributios are extremelly leptokutical ad skewess is positive because of the evolutio of Merval durig the period aalysed. If we look at the backtestig results, though real errors are more frequet tha the 5% sigificace level expected, they are ot very large (aroud 2% higher tha the VaR value i the IVaR35, 2.80% higher i the IVaR30, ad 2% higher i the IVaRM). Errors are more cotrolled i terms of frequecy i the case of IVaR30 tha for the IVaR35 or IVaRM, but they are less cotrolled i terms of magitude (the mea error i the America ad Argetiia cases is higher tha i the Spaish case). [7] observed that settig VaR limits o istitutios could lead to higher extreme losses tha whe these limits are ot set. We ca see from results, however, that this theoretical result is ot clear here. 3.6 Model Selectio We have see how parametric VaR miimizatio could create Miimum Risk Idices with less risk ad, i the Spaish ad Argetiia case, with ISSN: ISBN:

6 greater profitability tha curret market idices. I this paper we costruct 12 approximatios usig Movig Averages of differet legths for the Spaish ad Argetiia market ad 9 approximatios for the America market. It is ecessary ow to decide which is the best approximatio to use i each market. We this selectio should be doe o the basis of two ideas: (1) The model s capacity to explai reality or, i other words, its capacity to be accepted by the backtestig process. After determiig the umber of returs lower tha the VaR value (classic backtestig), it is importat to also measure the error magitude. This type of backtestig has ot yet bee developed ad here we oly propose a very simple method that deals with error magitude usig the Excess Total Loss (ETL) measure, which is defied as the total sum of all returs lower tha the VaR value over the studied period. We will choose those approximatios with the lowest ETL i order to take ito accout the risk out of the model. It is the ecessary to select those approximatios with less mea VaR or with less risk withi the model. As we ca see i Table 7, i the Spaish market, usig ETL the best approximatios are MA52a,b ad MA78a,b. Moreover, studyig the cotrolled risk withi the model, we coclude that the MA52a,b approximatios are the least risky. I the America Market, (see Table 8), the best approximatios usig the ETL are MA60, MA70, MA78, MA85 ad MA100. Usig the cotrolled risk, we coclude that the best approximatios are MA60, MA78 ad MA85. Fially, i the Argetiia market, the best approximatios by ETL are MA52a,b ad MA78a,b, ad, after usig the cotrolled risk measured by the VaR, we ca coclude that the best approximatios are MA52a ad MA52b. (2) The relatioship betwee retur ad risk, sice [10] criticize ot attachig importace to that poit i VaR calculatios. Here the authors use Sharpe s ratio to aalyse this relatioship. I the Spaish market, Sharpe s ratio i the MA52b approximatio is bigger tha i the MA52a approximatio so we ca coclude that MA52b is the best approximatio with which to costruct the Spaish Miimum Risk Idex. I the America market, Sharpe s ratio i the MA60 approximatio is the lowest of the selected MAs, so MA60 is the best approximatio with which to costruct the America Miimum Risk Idex. Fially, i the Argetiia market, Sharpe s ratio i MA52a is higher tha i MA52b, so it is reasoable to coclude that MA52a is the best approximatio to costruct the Argetiia Miimum Risk Idex. 4 Coclusios I this article we propose usig the VaR as a active risk measure to costruct Miimum Risk Idices to solve market idex biases. We have used the parametric VaR approach to costruct a very simple miimizatio problem i which the Covariace matrix amog asset returs has to be estimated. Covariace matrix estimatio ca be doe usig may methods. There are, therefore, may ways of costructig a Miimum Risk Idex oe for each way of estimatig the Covariace matrix so a method of selectig the best model is eeded. This selectio method must be based o the model s capacity to be accepted by the backtestig process, (takig ito accout error frequecy ad error magitude) ad the retur-risk relatioship. We apply this method to the Spaish, America ad Argetiia markets to create differet Miimum Risk Idices for the period. Usig the simplest Covariace matrix estimatio methods, we achieve iterestig results: our idices are less risky tha the curret oes (half the risk i the Spaish Market). Also, thaks to their optimal portfolio characteristics, the Spaish ad Argetiia cases achieved bigger returs tha those of the curret market Idices, cotrary to what is expected from the Efficiet Market Hypothesis. These results show that both markets suffer from efficiecy biases, ad that Miimum Risk Idices could partially solve this. Part of the results, i the Argetiia case, ca be due to the existece of a importat sample ad costructio bias created by how the Argetiia idex is built. This highlights a iterestig discussio that eeds to be dealt with care i future research ad which must be based o the followig ideas: (i) the ability to movig averages to estimate future covariace matrices ad the possibility of obtaiig better results with more complex estimatio methods; (ii) the ifluece of the weightig process ad other costructio rules o market idices (the sample ad costructio bias); (iii) the ifluece of market idex biases i the performace ad risk of idexes ad how biases are additive or ca be compesated amog them; (iv) the Miimum Risk Idex aproximatio i order to prove the efficiecy of a market ad to solve the efficiecy bias; ad (v) whether it is possible to obtai better results by ot limitig our Miimum Risk Idex shares to the curret Market Idex ISSN: ISBN:

7 compoets ad to the particular ad legal timig of chages i compoets. The potetial uses of Miimum Risk Idices are clear. Firstly, they are less risky ad i some cases more profitable tha curret oes, which makes them a suitable bechmark of risky assets for mutual fuds that curretly follow market idices or a suitable base for derivatives. Secodly, Miimum Risk Idices may geerate more stable Betas i the CAPM model, which is a possibility that must be developed i the future. 5 Future Lies of Research The results achieved by very simple methods i the examples preseted are iterestig but it must also be said that there is still a lot to do. First it is ecessary to determie whether better Covariace estimatios usig EWMA or GARCH or HAC methods ca achieve better results i terms of risk ad profitability. We also eed to determie how the sample ad costructio biases affect eficiecy tests, market idex performace ad the possibility of beatig it. Fially, methods for selectig the best model must be further developed sice here we have oly provided some geeral guidelies. I would like to thak Dr. Maxim Borrell, Dr. Salvador Torra, Dr. Daiel Liviao ad Professor Sebastia Cao for their commets/opiios regardig this paper. 6 Appedix I this sectio we will briefly explai how the Spaish IBEX35, the America Dow Joes Idustrial Average SM ad the Argetiia MERVAL are built. It is also importat to explai certai characteristics ad problems we foud ad solved by creatig the Miimum Risk Idex for each Market. The Spaish IBEX35 : The IBEX35 is built usig the 35 largest compaies i the Spaish Stock Market i terms of market capitalizatio ad liquidity. Every six moths the compoets of the Idex are checked, some shares are icluded or excluded but the total umber of assets is maitaied. The Idex is calculated usig a market capitalizatio weightig criterio. The Miimum Risk Idices we created for this market were amed IvaR35, ad comprise the 35 shares of the IBEX 35 at each momet with the optimal weight established by the VaR miimizatio process. We must poit out oe problem with the IBEX 35 Spaish Market Idex. I the six-moth revisio of the compositio of the IBEX 35, it is ormal to iclude shares ad compaies with very little history o the Stock Exchage because it is relatively easy to be both ew ad oe of the biggest 35 compaies i the Spaish Market. Durig the period of our aalysis we sometimes ecoutered this problem especially i because of the Iteret ad.com compaies that grew quickly at that time. This makes it difficult to obtai complete data for all the IBEX 35 compoets i some periods ad has importat cosequeces i Covariace matrix estimatio. After April 2000 we solved this problem with the followig techiques: a) Covariace matrix estimatio usig a multiplestep method: whe we did ot have complete data o the 35 shares, Covariace matrix estimatio was doe usig a multiple-step method, estimatig each idividual value i the covariace matrix with all the available data. b) 0.01% Weightig: the above solutio improved the results, but shares with short historical data teded to uderestimate risk ad therefore received high weights because of their artificial low risk. With this approximatio we forced these shares to have the miimum weight accepted for our study. The approximatios I fially developed are show i Table 10: The America Dow Joes Idustrial Average SM : The DJIA SM is built usig the 30 biggest compaies i the America Stock Market ad, for the sake of cotiuity, compositio chages are rare. Iclusios ad exclusios of shares are therefore rare ad basically related to corporate acquisitios or dramatic busiess evets. The Idex is calculated usig a price-weightig criterio. Usig available data for the DJIA SM we did ot eed to apply improvemets to the Covariace matrix estimatio. The good quality of these data meas that we used the methodology with a greater umber of movig average legths. The Miimum Risk Idex we created to this market was amed IVaR30. ISSN: ISBN:

8 The approximatios we developed are show i Table 11 The Argetiia MERVAL : The MERVAL is built usig the most traded compaies i the Argetiia Stock Market. The weights of each share i the idex are calculated usig the umber of trasactios of these shares i the Stock Market ad the Volume of these trasactios, so the Idex is calculated usig a egotiatio weightig criterio. The Miimum Risk Idices wereated for this market were amed IVaRM ad comprise the shares of the Merval at each momet with the optimal weight established by the VaR miimizatio process. Every three moths, the Merval compositio is chaged, ad it is possible, as i the IBEX 35, to fid compaies with very little historical data. Calculatios must the be improved usig the same techiques as for the the IBEX 35. Table 12shows the approximatios we used i this paper. It is importat to poit out that approximatio b is especially iflueced i the Merval by the fact that there are a lot of stocks with a short or o history whe they eter the Idex, this affects the performace ad backtestig of the approximatio. Refereces: [1] Abke, P. A. (2000), A empirical evaluatio of value at risk by sceario simulatio, Joural of Derivatives, pp [2] Alexader, J.; Baptista, M. (2003), Portfolio Performace Evaluatio Usig Value at Risk: The Reward-to-VaR Ratio, Joural of Portfolio Maagemet, v. 29, iss. 4, pp [3] Alexader, C.; Dimitriu, A. (2004), Idexig ad Statistical Arbitrage, The Joural of Portfolio Maagemet, v.31, iss.2, pp [4] Alexader, C.; Dimitriu, A. (2005), Idexig, coitegratio ad equity market regimes, Iteratioal Joural of Fiace ad Ecoomics, v.10, pp [5] Adreu, J. (2009), Market Idices: Bases, Biases ad Beyod. PhD Thesis. [6] Artzer, P.; Delbae, F.; Eber, J.M.; Heath, D. (1999), Coheret Measures of Risk, Mathematical Fiace, v.9,.3, pp [7] Basak, S.; Shapiro, A. (2001), Value-at-risk based risk maagemet: optimal policies ad asset prices, The Review of Fiacial Studies, v.14 (2), pp [8] Carosa, C. (2005), Passive ivestig: The Emperor Exposed?, Joural of Fiacial Plaig, v.18. iss.10, pp [9] Daielsso, J.; De Vries, C. G.; y Jorgese, B. N. (1998), The value of Value at Risk: Statistical, Fiacial, ad Regulatory Cosideratios. FRBNY Ecoomic Policy Review, Oct. [10] Dembo, R.; Freema, A. (1998), Seeig Tomorrow: rewritig the Rules of Risk. Wiley. [11] Hopper, G. P. (1996), Value at risk: a ew methodology for measurig portfolio risk, Federal Reserve Bak of Philadelphia Busiess Review, v.0, iss.0, pp [12] Hull, J. C.; White, A. W. (1998), Icorporatig volatility up-datig ito the Historical Simulatio method for value at risk, Joural of Risk, v.1 (1), pp.5-19, Fall. [13] Larse, N.; Mausser, H.; Uryasev, S. (2002), Algorithms for Optimizatio of Value-at-Risk, Fiacial Egieerig, e-commerce ad SupplyChai. Kluwer Academic Publishers, pp [14] Neftci, S. N. (2000), Value at Risk calculatios, extreme evets, ad tail estimatio, Joural of Derivatives, v.7, iss.3, pp [15] Pearso, N. D.; Smithso, C. (2002), VaR, the state of play, Review of Fiacial Ecoomics, v.11, pp [16] Pederse, C. S.; Satchell, S. E. (1998), A Exteded Family of Fiacial-Risk Measures, Geeva Papers o Risk ad Isurace Theory, v.23, iss.2, pp [17] Pritsker, M. (1997), Evaluatig value at risk methodologies: accuracy versus computatioal time, Joural of Fiacial Services Research, v.12 (2/3), pp [18] Riskmetrics (1995), Techical Documet. New York: Morga Guaratee Trust Compay, Global Research. ISSN: ISBN:

9 Table 1 Biases i Passive Ivestmet Strategies Bias Measure Defiitio Average Absolute (AA) Stadard Deviatio (SD) TE TEt = Rpt R bt SCB EB AB SCB EB AB t t t = R = R = R mt et bt R R R et ct mt AA AA AA AA TE SCB EB AB TE 1 t SD t= = 1 TE = ( TE 1 t= 1 t TE) SCB 1 t SD = t= = 1 SCB ( SCBt SCB) 1 t= 1 EB 1 t SD = t= = 1 EB ( EB 1 t= 1 AB 1 t SD = t= = 1 AB ( AB 1 t= 1 t EB) t AB) Nota: (TE)=Trackig Error; (SCB)=Sample ad Costructio Bias; (EB)= Efficiecy Bias; (AB)=Active Bias. Source: Author s ow. 2 ISSN: ISBN:

10 Table 2 Market Idices Stadard Deviatio Approximatio Stadard Deviatio Curret IBEX35 Market Idex Miimum Risk Idex Stadard Stadard Deviatio Deviatio DJIA SM MERVAL IVaR35 IVaR30 IVaRM MA10 MA MA10a MA10b MA25 MA MA25a MA25b MA MA52 MA MA52a MA52b MA MA MA78 MA MA78a MA78b MA MA ISSN: ISBN:

11 Table 3 Extreme Losses IBEX35 IVaR35 MA10 MA10a MA10b MA25 MA25a MA25b MA30 MA52 MA52a MA52b MA60 MA70 MA78 MA78a MA78b MA85 MA100 Highest Extreme Loss (%) DJIA SM 15.4 MERVAL IVaR30 IVaRM ISSN: ISBN:

12 Table 4 Backtestig process i the IVaR35 Normality Jarque- Probability Bera IBEX MA10 Mea VaR(%) Backtestig Errors % Errors Mea Error(%) MA10a MA10b MA MA25a MA25b MA MA52a MA52b MA MA78a MA78b Note: VaR value calculated at 5% sigificace level usig data available from 242 weeks. Errors: losses worse tha the VaR value. % Errors: real sigificace level. Mea Error (%) shows the mea loss exceedig the VaR value. ISSN: ISBN:

13 Table 5 Backtestig process i the IVaR30 Normality Jarque- Probability Bera DJIASM MA10 Mea VaR(%) Backtestig Errors % Errors Mea Error(%) MA MA MA MA MA MA MA MA Note: VaR value calculated at 5% sigificace level usig data available from 262 weeks. Errors: losses worse tha the VaR value. % Errors: real sigificace level. Mea Error (%) shows the mea loss exceedig the VaR value. ISSN: ISBN:

14 Table 6 Backtestig process i the IVaRM Normality Jarque- Probability Bera MERVAL MA10 Mea VaR(%) Backtestig Errors % Errors Mea Error(%) MA10a MA10b MA , MA25a MA25b MA MA52a MA52b MA MA78a MA78b Note: VaR value calculated at 5% sigificace level usig data available from 261 weeks. Errors: losses worse tha the VaR value. % Errors: real sigificace level. Mea Error (%) shows the mea loss exceedig the VaR value. ISSN: ISBN:

15 Table 7 Model Selectio i the Spaish Stock Market MA10 MA10a MA10b MA25 MA25a MA25b MA52 MA52a MA52b MA78 MA78a MA78b Errors Mea Error (%) Backtestig Mea VaR (%) Sharpe s Ratio ETL i Note: i Sharpe s ratio o-risk retur has bee cosidered equal to zero. Table 8 Model Selectio i the America Stock Market MA10 MA25 MA30 MA52 MA60 MA70 MA78 MA85 MA100 Errors Backtestig Mea VaR (%) Sharpe s Ratio ETL i Mea Error(%) Note: i Sharpe s ratio o-risk retur has bee cosidered equal to zero. ISSN: ISBN:

16 Table 9 Model Selectio i the Argetiia Stock Market MA10 MA10a MA10b MA25 MA25a MA25b MA52 MA52a MA52b MA78 MA78a MA78b Errors Backtestig Mea VaR (%) Sharpe s Ratio ETL i Mea Error(%) Note: i Sharpe s ratio o-risk retur has bee cosidered equal to zero. ISSN: ISBN:

17 Table 10 Approximatios used for Covariace matrix estimatio i the IVaR35 Approximatio Method Legth Improvemets Applied. Movig Average MA10 10 weeks Noe MA10a Movig Average 10 weeks Multiple-step Method MA10b Movig Average 10 weeks Multiple-step Method ad 0.01% Weightig MA25 Movig Average 25 weeks Noe MA25a Movig Average 25 weeks Multiple-step Method MA25b Movig Average 25 weeks Multiple-step Method ad 0.01% Weightig MA52 Movig Average 52 weeks Noe MA52a Movig Average 52 weeks Multiple-step Method MA52b Movig Average 52 weeks Multiple-step Method ad 0.01% Weightig MA78 Movig Average 78 weeks Noe MA78a Movig Average 78 weeks Multiple-step Method MA78b Movig Average 78 weeks Multiple-step Method ad 0.01% Weightig ISSN: ISBN:

18 Table 11 Approximatios used for Covariace matrix estimatio i the IVaR30 Approximatio Method Legth Improvemets Applied. Movig Average MA10 10 weeks Noe MA25 Movig Average 25 weeks Noe MA30 Movig Average 30 weeks Noe MA52 Movig Average 52 weeks Noe MA60 Movig Average 60 weeks Noe MA70 Movig Average 70 weeks Noe MA78 Movig Average 78 weeks Noe MA85 Movig Average 85 weeks Noe MA100 Movig Average 100 weeks Noe ISSN: ISBN:

19 Table 12 Approximatios used for Covariace matrix estimatio i the IVaRM Approximatio Method Legth Improvemets Applied. Movig Average MA10 10 weeks Noe MA10a Movig Average 10 weeks Multiple-step Method MA10b Movig Average 10 weeks Multiple-step Method ad 0.01% Weightig MA25 Movig Average 25 weeks Noe MA25a Movig Average 25 weeks Multiple-step Method MA25b Movig Average 25 weeks Multiple-step Method ad 0.01% Weightig MA52 Movig Average 52 weeks Noe MA52a Movig Average 52 weeks Multiple-step Method MA52b Movig Average 52 weeks Multiple-step Method ad 0.01% Weightig MA78 Movig Average 78 weeks Noe MA78a Movig Average 78 weeks Multiple-step Method MA78b Movig Average 78 weeks Multiple-step Method ad 0.01% Weightig ISSN: ISBN:

20 Figure 1 Market Idex Biases Note: Market Portfolio=Total Market; Proxy Portfolio= passive ivestor s portfolio used to proxy a Idex. Source: Authors ow ISSN: ISBN:

21 Figure 2 Market Idices cumulative volatility (IVaR35, IVaR30, IVaRM) VIBEX35 VMA10 VMA10A VMA10B VMA25 VMA25A VMA25B VMA52 VMA52A VMA52B VDOW30 VMA10 VMA100 VMA25 VMA30 VMA52 VMA60 VMA70 VMA78 VMA VMERVAL VMA10 VMA10A VMA10B VMA25 VMA25A VMA25B VMA52 VMA52A VMA52B Note: here, volatility is variace. I the first, secod ad third graphs, VIBEX35, VDOW30 ad VMERVAL are the cumulative volatilities of the IBEX35, the DowJoes Idustrial AverageSM ad the MERVAL, respectively, ad VMA are the cumulative volatilities of each movig average approximatio used for each market. ISSN: ISBN:

22 Figure 3 Evolutio of the idices (IVaR35, IVaR30, IVaRM) RIBEX35 RMA10 RMA10A RMA10B RMA25 RMA25A RMA25B RMA52 RMA52A RMA52B RDOW30 RMA10 RMA100 RMA25 RMA30 RMA52 RMA60 RMA70 RMA78 RMA RMERVAL RMA10 RMA10A RMA10B RMA25 RMA25A RMA25B RMA52 RMA52A RMA52B Note: 100 based. I the first, secod ad third graph RIBEX35, RDOW30 ad RMERVAL are the evolutio of IBEX35, DJIASM ad MERVAL, ad RMA are the evolutios of each movig average approximatio i each market. ISSN: ISBN: