Extreme Risk Analysis July 2009

Transcription

1 Extree Risk Analysis Lisa R. Goldberg Michael Y. Hayes Jose Menchero Indrajit Mitra Quantitative risk anageent allows for qualitative notions such as optiality and expected returns to be put on a quantitative footing. It copleents subjective risk considerations with an objective, statistical perspective. Broadly, quantitative risk anageent consists of two distinct eleents. The first is easureent, which involves quantifying the overall risk of a portfolio. The second step is analysis, which involves gaining insight into the sources of risk, and deterining whether or not they accurately reflect the views of the portfolio anager. Risk analysis is ost coonly considered in the context of a particular risk easure, leading to concepts such as ean-variance optiization, and a definition of beta in ters of variances and covariances. Many volatility (or variance) based analytics can be extended to cover a broad class of risk easures, allowing for new perspectives of risk to be understood using standard analytics. The 2008 arket turoil provides fresh otivation to take as broad a view of risk as possible. Financial risk is ulti-faceted and ust be easured and analyzed fro ultiple perspectives. More generally, the lackluster perforance of conventional quantitative risk anageent practices highlights the need for an expansive, dynaic fraework that is effective in all econoic cliates. To this end, we unite the acadeic literature on financial risk with a practitioner s perspective. First, we elucidate shortfall, which is the expected loss in a bad period and is a natural copleent to volatility. Shortfall is essential to a risk anageent paradig that provides eaningful analysis in both cal and turbulent arkets. Second, we apply industry-standard (volatility-based) analysis to shortfall using several exaples. By analyzing volatility and shortfall in conjunction we arrive at insights that cannot be obtained by considering either easure on its own. Finally, in a technical Appendix, we set down the atheatical details of a fraework for generalized risk analysis. This fraework applies to volatility and shortfall, and it extends without odification to a diverse class of risk easures. 1 Risk Measures An influential paper by Markowitz (1952) arked the beginning of quantitative risk anageent by proposing volatility (standard deviation) as a risk easure. Volatility continues to play a central role in finance for any reasons. First, volatility always favors diversification over concentration, and siple ean-variance optiization probles adit analytic solutions. Furtherore, robust econoetric odels (e.g., ulti-factor risk odels) have been developed to forecast volatility. Finally, volatility can be traded directly on the open arket (through the VIX index in the US and related indices in other countries) and indirectly using derivatives. If portfolio returns were norally distributed, then the ean and volatility would fully characterize risk. However, in any cases, portfolio returns are aterially non-noral. Extree equity returns occur far ore frequently than would be predicted by a noral distribution, resulting in a heavy loss tail populated by headline disasters such as the collapse of Lehan Brothers, the Long Ter Capital debacle, Black Monday, and other arket disruptions. If a portfolio contains significant weight in bonds or derivatives, the risk profile can take on arbitrary shapes, so 1 The aterial in the appendix extends in perfect analogy to the class of coherent risk easures, and with soe odification, to the class of convex risk easures. Further details are in Föller and Scheid (2004) MSCI Barra. All rights reserved. 1 of 20 lease refer to the disclaier at the end of this docuent. Electronic copy available at:

2 Extree Risk Analysis additional easures are indispensible. For a non-noral portfolio, no single easure copletely describes portfolio risk. In particular, volatility is insensitive to the difference between loss and gain (a distinction that is irrelevant for the syetrical noral distribution). The ost proinent risk easure after volatility is value at risk (VaR), which was developed at J Morgan in the early 1990s. VaR is coonly expressed in ters of loss ( L ) relative to the ean: ( E[ ]) L= R R. (1) Value at risk is specified by a given confidence level and investent horizon, and it easures the axiu expected loss under ordinary arket conditions. For instance, a portfolio loss is expected to exceed its 95% one-day VaR on average once every 20 days. Matheatically, value at risk is defined as VaR = QL ( ), (2) where QL ( ) is the specified percentile of the centered loss distribution. In principle, value at risk copleents volatility because it is a downside easure: it takes account of a portfolio s losses and ignores its gains. This contributed to the inclusion of VaR in the Basel II regulatory fraework. However, certain drawbacks of VaR are well-appreciated (The Econoist, 2004). Soe of these drawbacks relate to the way in which VaR is estiated or interpreted. Even if those are addressed, probles reain in the very definition of VaR. VaR, unlike volatility, ay encourage concentration over diversification: lowering your value at risk can lead to a ore concentrated portfolio. A scheatic exaple shows how this can happen. 2 Exaple 1: Value at Risk and Diversification Consider investing in two corporate bonds that are contractually identical; the only difference between the bonds is that they are issued by different firs. Is it better to invest in one of the bonds or to diversify with soe of each? Suppose that the annual default probability of both issuers is 0.7%. This eans that each fir has just under a one-in-a-hundred chance of defaulting over the next twelve onths. Defaults are iprobable events and value at risk does not raise any concern: hold only one of the two bonds and your 99% VaR (considering only default risk) is zero. This is because VaR is a worst-case scenario for an ordinary year. In any one of the 99 taer years in a typical hundred, the issuer does not default and investent in either bond pays dividends. How risky is it to put half the oney in each of the two bonds? The probability that soething 3 goes wrong approxiately doubles to 1.4% and as a result, the 99% value at risk increases to little less than half the principal. In other words, diversification raises value at risk fro zero to alost half the portfolio value. What went wrong? Value at risk punishes the two-bond portfolio for a greater probability of an adverse event. However, it is relatively insensitive to the event's agnitude. The agnitudes of the adverse events differ (aterially) by a factor of two. A single- 2 Exaples of this appear in the acadeic literature; this is adapted fro Föller and Scheid (2004), Exaple This calculation is valid if the default events are independent, or even if they have a odest positive correlation MSCI Barra. All rights reserved. 2 of 20 lease refer to the disclaier at the end of this docuent. Electronic copy available at:

3 Extree Risk Analysis bond portfolio loses its entire principal, while the two-bond portfolio loses at least half, but usually no ore than half, of its principal. Figure 1 One-day shortfall and Value at Risk, at 95% confidence level, plotted versus nuber of options in the portfolio. The portfolio is long an ETF, and short a variable nuber of out-of-the-oney call options Risk (ercent) Shortfall VaR Nuber of Options As an upper bound for loss on an ordinary day, VaR is also a lower bound for loss on an extraordinary day. It gives no indication of what to expect on an extraordinary day. While it is a easure of downside risk, VaR is not a easure of extree risk. Are typical breaches of a VaR liit ild infractions or egregious violations? This question is beyond the scope of VaR; it can be answered only by a true easure of extree risk. Shortfall (S) is a natural extension of value at risk, and is a true extree risk easure. It is the expected portfolio loss given that VaR has been breached: [ L L ] S = E > VaR. (3) In other words, shortfall is an estiate of what to expect in a bad period. It is ipossible to hide tail risk fro shortfall the way it can be hidden fro VaR. In Exaple 1, shortfall favors the twobond portfolio over a single-bond portfolio. More generally shortfall, like volatility but unlike VaR, will always encourage diversification. We investigate the difference between VaR and shortfall using a set of portfolios that are long an ETF and short call options on the ETF MSCI Barra. All rights reserved. 3 of 20 lease refer to the disclaier at the end of this docuent. Electronic copy available at:

4 Extree Risk Analysis Exaple 2: The Risk of a Short osition in Call Options Out-of-the-oney call options are coonly sold to enhance portfolio returns. The option issuer receives a preiu when the option is written. Under ordinary arket conditions, the underlying security experiences, at best, a odest gain, and the option expires out of the oney. However, an extree gain to the underlying can trigger a devastating loss to the issuer of the option. Using an epirically realistic (non-noral) distribution for the portfolio, we show that shortfall is a better gauge of the risk of the short position than VaR. Consider a faily of portfolios, each coposed of a single share of an ETF and a short position in a variable nuber of identical European call options. The underlying chosen is the MSCI US Broad Market index, 4 and the analysis date is October 23, The spot price of the index is taken as $60, and the strike price of each option is $64, with one day to expiration. If the short position consists of ore than one option, the portfolio has an infinite down side the potential to lose an arbitrarily large aount of oney since there is no upper liit to the value of the ETF. What is the one-day risk of holding such a portfolio? The risk of this faily of portfolios as viewed through the lens of VaR and shortfall is illustrated in Figure 1. As the plot clearly shows, VaR does not significantly increase with the nuber of call options sold, while shortfall increases roughly linearly with the nuber of short positions. The portfolio loses value fro two types of scenarios: (a) the ETF experiences a oderate loss, in excess of the call preius, and (b) the ETF experiences a large gain, so the call options are exercised. Losses of the first kind are largely insensitive to the nuber of calls sold. By contrast, losses resulting fro the calls being exercised increase in direct proportion to the nuber of calls written. If the call option is sufficiently deep out of the oney, i.e., the strike price is sufficiently higher than the spot price, ost of the losses are fro the ETF losing value. In particular, the sallest of the large losses are of this kind. This iplies that for such a portfolio, VaR would be quite insensitive to the nuber of short positions. More iportantly, it disregards the ipact of large losses resulting fro scenarios (b). Shortfall, on the other hand, averages over all large losses including scenarios of the second kind, for which a portfolio short 10 call options will lose 10 ties as uch as a portfolio short a single call option. 5 Risk Analysis Once risk has been easured, the next step is to understand the sources of risk and how they interact. Different investors ay be interested in different types of sources, such as individual securities, asset classes, sectors, industries, currencies, or style factors fro a particular risk odel. ortfolio risk can be analyzed in ters of any type of source, so we ake our discussion generic by considering sources without reference to their type. 4 We copute the distribution of one-day portfolio returns using the flexible sei-paraetric odel described by Goldberg et al. (2008) and Barbieri et al. (2008). The odel uses ore than 36 years of historical daily returns to the MSCI US to siulate price changes of the index and option. We standardize these returns by their historical volatility and then scale by the current volatility to give a consistent view of arket history over the course of both cal and turbulent arkets. We use the distribution of ETF prices at aturity of the option to directly copute the distribution of option payoffs and portfolio values. The Black-Scholes-Merton odel is used only to copute the call preiu given the ters and conditions of the contract and the iplied volatility forecast of the odel. 5 Deep out-of-the-oney call options are cheap and constitute a sall portion of the initial portfolio value MSCI Barra. All rights reserved. 4 of 20 lease refer to the disclaier at the end of this docuent.

5 Extree Risk Analysis Risk Sources Fix an investent period and let r denote the return to source over this period. Then the portfolio return over the period is given by a su, R= x r, (4) where x is the portfolio exposure to the source. It is the job of the portfolio anager to deterine the source exposures at the start of a period. The source returns are rando variables, whose realized values are known only at the end of the investent period. Risk Contributions ortfolio risk is not a weighted su of source risks, so there is no direct analog to Equation 4 for risk. However, there is a parallel to Equation 4 in ters of arginal contributions to risk (MCR). The arginal contribution to risk of a source is the approxiate change in portfolio risk when increasing the source exposure by a sall aount x, while keeping all other exposures fixed, σ σ MCR x. (5) The contribution of a return source to portfolio risk is given precisely by the product of the source exposure and the arginal contribution to risk, as shown by Litteran (1996) in the context of volatility σ, and these risk contributions su to the portfolio risk: σ = x MCR σ. (6) In the Appendix, we show that Equation 6 holds for a large class of risk easures including shortfall, and it facilitates eaningful, even-handed decopositions of different risk easures. X-Siga-Rho Risk Attribution Additional insight can be provided by a refineent of the Litteran (1996) decoposition for volatility given in Equation 6. As discussed in Menchero and oduri (2008), the arginal contribution to volatility can be expressed as the product of the stand-alone volatility of the source and the correlation of the source return with the portfolio, ( ) MCR = r R. (7) σ σ, ρ This gives rise to the x-siga-rho risk attribution forula, σ = x σ ρ( r, R). (8) This decoposition provides an intuitive fraing of the concept of arginal contribution, as well as a deeper risk analysis. For instance, two assets with the sae arginal contribution to risk ay have very different volatility characteristics depending on their stand-alone volatilities σ ρ r R. and correlations ( ), Equation 8 ay be applied to shortfall using the shortfall-iplied correlation, as shown in the Appendix. Shortfall correlation easures the likelihood of coincident extree losses. By contrast, linear correlation easures the overall tendency of two sources to ove together. 6 6 There is a conceptual parallel between shortfall correlation and exceedence correlation, as defined in Longin and Solnik (2001). Cherny and Madan (2007) provide a discussion of correlations iplied by the faily of coherent risk easures MSCI Barra. All rights reserved. 5 of 20 lease refer to the disclaier at the end of this docuent.

6 Extree Risk Analysis Shortfall correlation shares useful properties with linear correlation, such as scale independence, and an upper bound of 1. It also has a lower bound, which need not be equal to -1 for an asyetric return distribution. Exaining Equations 6, 7, and 8, we see that the shared eleent is the arginal contribution to risk. For the special class of linear, convex risk easures (which includes volatility and shortfall), any additional analytics are shared through the central arginal contribution (MCR). Any one of these easures will encourage diversification, and ay be decoposed using the x-siga-rho forula. The relationship between arginal contribution to risk and various analytic tools is shown in Figure 2. Further details are provided in the Appendix. Figure 2 Scheatic diagra showing relationships between arginal contribution to risk and other widely used easures. Here, denotes a generalized risk easure. Er [ ] MCR MCR Coponent Inforation Ratio E[ R] MCR Generalized Beta Iplied Returns Marginal Contribution To Risk Risk Contribution Generalized Correlation x MCR MCR 2009 MSCI Barra. All rights reserved. 6 of 20 lease refer to the disclaier at the end of this docuent.

7 Extree Risk Analysis We illustrate the insights provided by parallel x-siga-rho decopositions of volatility and shortfall in two scheatic exaples. In both cases, the added perspective relies on estiates that reflect the non-norality of portfolio return distributions. Exaple 3: ortfolio Insurance Out-of-the-oney put options are coonly used to insure a portfolio against large losses. The cost of insurance is the price of the option preiu. When the portfolio does not suffer a severe loss, the option expires out of the oney and the preiu is lost. In contrast, a severe loss to the underlying portfolio leads to a large, positive option payoff. Therefore, the value of portfolio insurance depends on the likelihood of a severe loss to the underlying portfolio. This qualitative stateent can be ade quantitative with a parallel analysis of volatility and shortfall. An investor seeks to easure the reduction of risk when insuring an exchange traded fund (ETF) on the MSCI US Broad Market Index with an out-of-the-oney put option on the sae index. The spot price of the index is $60 and the option strike is $50. The analysis date is October 23, 2008, with the option expiration at 20 days. 7 We analyze portfolio risk in ters of both volatility and shortfall. Volatility paints an incoplete picture of this risk due to the non-norality of the return distribution of the portfolio. 8 Since a larger loss fro the index generates greater option profits, the diversification benefit of holding put options increases as the risk easure becoes ore concentrated in the tail of the portfolio distribution. We consider the risk of the portfolio as the option weight is varied. In Figure 3, we plot the contributions to 99% shortfall fro the option and the index. Initially, the option strongly reduces the risk of the portfolio. The portfolio risk is iniized for an option weight of about 7%. Eventually, however, increasing the option weight becoes ore of a gable on a arket crash, thus increasing the portfolio risk. In Figure 4, we show the contributions of the option and the index to portfolio volatility as the option weight is varied. Qualitatively, the results are siilar to Figure 3, with the portfolio risk first declining and then increasing as the option weight is varied. Note the relative aount by which the risk is reduced: the put option is uch ore effective at reducing 99% shortfall risk than at reducing portfolio volatility. 7 As in the previous exaple, we use the flexible sei-paraetric odel described by Goldberg et al. (2008) and Barbieri et al. (2008). 8 This portfolio non-norality originates fro the non-linear dependence of the option on the underlying security, as well as the non-norality of returns to the underlying security MSCI Barra. All rights reserved. 7 of 20 lease refer to the disclaier at the end of this docuent.

8 Extree Risk Analysis Figure 3 Contribution to 99% shortfall fro index and option, as option weight is varied. The fully hedged option weight is indicated by the vertical dashed line. The iniu risk portfolio occurs for an option weight of about 7% % Shortfall Contribution Index Option Total Option Weight (ercent) Figure 4 Contribution to volatility fro index and option, plotted as option weights are varied. The fully hedged option weight is indicated by the vertical dashed line. 10 Volatility Contribution (percent) Option Index Total Option Weight (ercent) 2009 MSCI Barra. All rights reserved. 8 of 20 lease refer to the disclaier at the end of this docuent.

9 Extree Risk Analysis In Table 1, we use the x-siga-rho ethodology to decopose three risk easures: volatility, 95% shortfall, and 99% shortfall. The horizon in all cases is taken to be one day. The portfolio is fully hedged, eaning that there are equal nuber of shares in the ETF and the option. This results in an index weight of 98.86%, and an option weight of 1.14%, as indicated in Colun 1. In the x-siga-rho fraework, the risk contribution is given by the product of the exposure (x), the stand-alone risk (siga), and the generalized correlation (rho). Note that the exposures are the sae for all three risk easures. Table 1 Risk decoposition for volatility, 95% shortfall, and 99% shortfall, using the x-siga-rho forulation. The portfolio consists of an index and an out-of-the-oney put option on the sae index. 95% 95% 95% 99% 99% 99% Stand-alone Volatility Volatility Stand-alone Shortfall Shortfall Stand-alone Shortfall Shortfall Asset Exposure Volatility Correlation Contrib Shortfall Correlation Contrib Shortfall Correlation Contrib Index ut Option Total The option contributes -169 basis points to 95% shortfall. Note that the shortfall correlation is This value is noteworthy since linear (volatility-iplied) correlation can never fall below -1. Shortfall correlation, on the other hand, can be less than -1 due to the asyetry of the risk easure and return distribution. As shown in the Appendix, the shortfall correlation between the option loss ( ) l and the portfolio loss L is given by the ratio of two ters: (1) the expected option loss on days when the portfolio VaR is breached, and (2) the shortfall of the option. The distribution of centered option losses is shown in Figure 5. The right side of the distribution represents the worst 5% of option scenarios, when the 95% option VaR is breached. The left side of the distribution represents option losses on the worst 5% of portfolio scenarios, showing that the option perfors well when the portfolio suffers losses. The nuerator of Equation A17 is the ean of the left-side of the distribution, or %. The denoinator of Equation A17 is the ean of the right side of the distribution, or 70.6%. The correlation of is the ratio of these two nubers MSCI Barra. All rights reserved. 9 of 20 lease refer to the disclaier at the end of this docuent.

10 Extree Risk Analysis Figure 5 artial histogra of centered option losses, defined as the option loss relative to the ean option loss ( 14.74%). The theoretical axiu centered option loss is therefore %. The right side of the distribution is for bad option days, defined as days when the 95% option VaR (60.2%) is breached. The expected centered option loss on bad option days is 70.6%. The left side of the distribution is for days when the 95% portfolio VaR is breached. The ean centered option loss on bad portfolio days is % (i.e., the option perfors well on bad portfolio days). The 95% shortfall correlation is the ratio of the two nubers, i.e., Note that the left side of the distribution has a long tail, and all losses beyond -280% are tried for plotting purposes Bad Option Days Count Bad ortfolio Days Excess Option Loss (percent) In the final exaple, we exaine a portfolio coposed of two assets that are uncorrelated by the standard volatility easure, but that nevertheless tend to experience large coincident losses. This ay be the result of financial contagion, where an extree ove in one asset triggers an extree ove in another. Standard volatility and correlation easures are insensitive to the risk of extree coincident losses. However this risk can be easured using shortfall and shortfalliplied correlation. Exaple 4: Coincident Extree Losses We consider two portfolios, each coposed of two equally-weighted assets whose returns follow standard noral distributions. The portfolios are distinguished only by the joint distributions of the two assets. First, we use a noral copula, which ensures that the joint distribution is also norally distributed. Copulas provide a ethod of forulating ultivariate return distributions with given statistical properties (Nelson, 1999). The second way in which we for the joint distribution is to use the t-copula, which exhibits a greater likelihood of joint extree losses. We stress that although the asset returns generated by the t-copula are uncorrelated using the conventional definition, they are not independent. Therefore, even though the asset returns are 2009 MSCI Barra. All rights reserved. 10 of 20 lease refer to the disclaier at the end of this docuent.

11 Extree Risk Analysis norally distributed, the portfolio returns are fat tailed due to the increased likelihood of both assets experiencing extree siultaneous losses. To copute the risk of these portfolios, we take one illion rando draws fro the two bivariate distributions. In Table 2, we perfor an x-siga-rho attribution of volatility and shortfall for the case of a noral copula. Since the portfolio is equally weighted, the exposure to each asset is 50%. Following the standard coputations for volatility, the correlation of each asset with the portfolio is 1 2, or The portfolio volatility is 0.71%, copared to a stand-alone volatility of 1% for each asset. Table 2 X-siga-rho risk decoposition for two identical assets assuing a noral copula. 95% 95% 95% 99% 99% 99% Stand-alone Volatility Volatility Stand-alone Shortfall Shortfall Stand-alone Shortfall Shortfall Asset Exposure Volatility Correlation Contrib Shortfall Correlation Contrib Shortfall Correlation Contrib A B Total For 95% shortfall, the stand-alone risk of each asset is 2.06%. For noral distributions, the ratio of 95% shortfall to volatility is The shortfall correlation is the sae as the volatility correlation (i.e., 0.71). The 95% shortfall for the portfolio is 1.46, which is exactly 2.06 ties the portfolio volatility. This is as expected, since the portfolio returns are norally distributed. More interesting is the risk decoposition for the t-copula, given in Table 3. Note that the volatility of the portfolio is unchanged. The correlation of each asset with the portfolio is also unchanged copared to Table 2. In other words, as easured by volatility, the t-copula portfolio is no riskier than the joint-noral portfolio. Table 3 X-siga-rho risk decoposition for two identical assets assuing a t-copula. 95% 95% 95% 99% 99% 99% Stand-alone Volatility Volatility Stand-alone Shortfall Shortfall Stand-alone Shortfall Shortfall Asset Exposure Volatility Correlation Contrib Shortfall Correlation Contrib Shortfall Correlation Contrib A B Total The shortfall easure, however, reflects the increased likelihood of extree loss. The standalone shortfall of each asset is still 2.06%, since the asset returns individually are norally distributed. However, now the portfolio 95% shortfall is increased to 1.59, which is 2.24 ties the portfolio volatility. That is, the portfolio returns are now fat tailed. The x-siga-rho ethodology cleanly captures the increased risk and attributes it to increased correlation. The 95% shortfall correlation is 0.77, and increases to 0.85 as we ove deeper within the tail (i.e, 99% shortfall). Figure 6 illustrates the effects of the joint distribution on portfolio risk. In Figure 6(a), we plot portfolio shortfall versus confidence level. As we ove further into the tail, the portfolio with 2009 MSCI Barra. All rights reserved. 11 of 20 lease refer to the disclaier at the end of this docuent.

12 Extree Risk Analysis returns following the t-copula appears riskier, due to the likelihood of coincident extree losses. In Figure 6(b), we plot the shortfall correlation versus confidence level. For the joint noral distribution, shortfall correlation is 0.71 at all confidence levels. For the t-copula portfolio, however, the shortfall correlation increases with confidence level, thus indicating the increased likelihood of large coincident losses. Figure 6 anel (a) shows the shortfall for a portfolio of two identical assets, assuing a joint noral distribution (dashed line) and a t-copula (solid line). anel (b) shows the shortfall correlation under the sae two distributional assuptions. Shortfall (ercent) Shortfall Correlation (a) t-copula Noral Confidence Level (ercent) (b) t-copula Noral Confidence Level (ercent) 2009 MSCI Barra. All rights reserved. 12 of 20 lease refer to the disclaier at the end of this docuent.

13 Extree Risk Analysis Conclusion The unrelenting turbulence that has plagued financial arkets since 2007 highlights the iportance of taking a broad, dynaic view of risk. This eans that investors need to extend standard risk anageent practices to include easures of extree risk. Shortfall, which is the expected loss given that the VaR threshold has been breached, is the ost iportant easure of extree risk. It probes the tails of portfolio return distributions, prootes diversification, and is easily aenable to the tools of standard risk analysis. arallel decopositions of portfolio volatility and shortfall provide insights that cannot be obtained through the lens of a single risk easure. We illustrate two such insights in this article. The first is a deep understanding of the value of downside protection: the diversification benefits of an out-of-the-oney put are uch greater for shortfall than for volatility. The second is that shortfall correlation easures the likelihood of extree events occurring in tande, which is a risk that cannot be detected by linear correlation. The key to extending industry-standard (volatility-based) risk analysis to shortfall is the arginal contribution to risk. Its central and unifying role in risk analysis is illustrated scheatically in Figure 2, and is docuented atheatically in the Appendix. The risk analysis paradig set down there can be used as a blueprint for the evolution of risk anageent that will accopany the growth in our understanding of arket dynaics. Acknowledgents We thank Angelo Barbieri, Aaron Brown, atrick Burke-d'Orey, Vladislav Dubikovsky, Alexei Gladkevich, Alec Kercheval, Julie Lefler, D.J. Orr, Jo Robbins, eter Shepard, Jerey Stau, and Fred Winik for their insights and support. We are grateful to Rodney Sullivan and two anonyous referees for thorough and thoughtful reviews of an earlier version of this article MSCI Barra. All rights reserved. 13 of 20 lease refer to the disclaier at the end of this docuent.

14 Extree Risk Analysis Appendix We define the shortfall of portfolio to be where L denotes the centered portfolio loss, S [ L L ] = E > VaR, (A1) ( E[ ]) L= R R, (A2) and VaR is the portfolio value at risk. It should be stressed that both value at risk and shortfall depend on investent horizon and the confidence level. For notational siplicity, however, these additional subscripts will be suppressed. Generalized Risk Attribution We write the portfolio return R as a su of return contributions fro various sources (e.g., assets, sectors, or factors), R= xr, (A3) where x denotes the exposure or weight and r denotes the return of source. Since ost of the discussion in this article is about loss, it is convenient to express portfolio loss as a su of contributions fro various sources (e.g., assets, sectors, or factors), where can be decoposed using Euler s theore L= xl, (A4) l denotes the centered loss of source. Any risk easure = that is scale invariant x. (A5) x Most failiar risk easures, including volatility, VaR, and shortfall are scale invariant. This eans that ultiplying all exposures by soe constant λ also ultiplies the risk by the sae constant. The risk contribution of a particular source is therefore identified as where RC = x MCR, (A6) MCR is the arginal contribution to risk of source with respect to risk easure. Generalized Beta The conventional definition of beta is β cov( r, R) σ ( r, R) 2 where written in ters of arginal contributions, =, (A7) 2 σ is the variance of R. A few lines of algebra reveal that conventional beta can also be β 1 r R = MCR σ. (A8) σ (, ) 2009 MSCI Barra. All rights reserved. 14 of 20 lease refer to the disclaier at the end of this docuent.

15 Extree Risk Analysis The generalized beta, for risk easure (, ), is a generalization of Equation A8 1 l L = MCR β Note that in Equation A9, we have switched fro returns to losses. Generalized Correlation Correlation is custoarily defined as ρ ( r R) cov( r, R) σ σ,. (A9) =, (A10) where σ is the volatility of r, and obvious how to generalize correlation for other risk easures. However, as with beta, correlation can also be expressed in ters of arginal contributions, ρ (, ) σ is the volatility of R. Written in this for, it is not 1 r R = MCR σ. (A11) σ The generalized correlation, for risk easure 1 l L = MCR (, ), is given by ρ Using Equation A6 and A12, the risk contribution of a particular source is and portfolio risk. (A12) RC = x ρ, (A13) is attributed to sources according to the x-siga-rho fraework,. (A14) = x ρ That is, the portfolio risk is decoposed into the product of three ters: (1) the size of the position, (2) the stand-alone risk of the source, and (3) the generalized correlation of the portfolio and the source. roperties of Shortfall Correlation The shortfall correlation between portfolio loss L and coponent l is given by Equation A12. Substituting Equation A4 into Equation A1, and taking partial derivatives, we obtain S = > xke [ lk L VaR ]. (A15) x x k Using the fact that the partial derivative with respect to the value at risk is zero, as discussed by Bertsias et al. (2004), this expression siplifies to S x [ l L ] = E > VaR. (A16) 2009 MSCI Barra. All rights reserved. 15 of 20 lease refer to the disclaier at the end of this docuent.

16 Extree Risk Analysis Substituting Equation A16 and the definition for stand-alone shortfall into Equation A12, we obtain [ l L> ] [ l l > ] S E VaR ρ = E VaR. (A17) Like linear correlation, shortfall correlation is scale independent. This eans that scaling one of the returns/losses by a constant leaves the correlation unchanged. In other words, [ λ ] [ λ λ λ ] [ ] [ ] E l L> VaR E l L> VaR = E l l > VaR E l l > VaR where we have used the linearity property of expectations., (A18) Another iportant property of shortfall correlation pertains to the range of possible values. Standard correlation, of course, is bounded between [ 1,1]. By contrast, the bounds of shortfall correlation are given by: S ρ. (A19) S S 1 * * Here, S is the shortfall of the gain of the stand-alone distribution. If the return distribution for * source is syetric, then S = S and the shortfall correlation is bounded fro below by 1, just as with standard correlation. If, however, the loss tail is different fro the gain tail, then the stand-alone shortfall of the losses can exceed the stand-alone shortfall for the gains. In this case, the shortfall correlation can be less than 1, as with Exaple 3 in the ain body. Reverse Optiization Suppose that the objective is to axiize the ratio of expected excess return 9 to risk, ER [ ]. This quantity represents a generalized inforation ratio, which reduces to the conventional ratio when is volatility. For an unconstrained optial portfolio, the derivative with respect to x ust equal zero, [ ] 1 [ ] [ ] E R E R E R = = 0 2 x x x. (A20) Using Equation A3 and the definition of arginal contribution, Equation A20 can be rewritten as [ ] E r [ ] E R = MCR. (A21) This says that the expected excess return of the source is proportional to the arginal contribution to risk, with the constant of proportionality being the generalized inforation ratio. 9 When evaluating portfolios using the Sharpe ratio, it is iportant to use the return in excess of the risk-free rate of cash. For anagers evaluated relative to a benchark, it is iportant to consider the return in excess of the benchark return (see Sharpe (2001) ) MSCI Barra. All rights reserved. 16 of 20 lease refer to the disclaier at the end of this docuent.

17 Extree Risk Analysis Generalized Coponent Inforation Ratios The generalized inforation ratio can be rewritten in the following for, ER [ ] x E[ r] x MCR =. (A22) x MCR Grouping ters and siplifying, we find We identify ( [ ] ) ER [ ] Er [ ] = RB MCR. (A23) E r MCR as the generalized coponent inforation ratio. This says that the generalized inforation ratio of the portfolio is given by the risk-weighted generalized coponent inforation ratios MSCI Barra. All rights reserved. 17 of 20 lease refer to the disclaier at the end of this docuent.

18 Extree Risk Analysis REFERENCES Barbieri, A., V. Dubikovsky, A. Gladkevich, L. R. Goldberg, and M. Y. Hayes, Evaluating Risk Forecasts with Central Liits. Working aper, MSCI Barra. Bertsias, D., G. J. Lauprete, and A. Saarov, Shortfall as a Risk Measure: roperties, Optiization and Applications. Journal of Econoic Dynaics and Control, 28: Brinson, G.., L. Hood, and G. Beebower, Deterinants of ortfolio erforance. Financial Analysts Journal, vol.42, no. 4: Cherney, A. S. and D. B, Madan, 2007, Coherent Measureent of Factor Risks, working paper, Moscow State University and University of Maryland. The Econoist, Too Clever By Half, Jan 22, Föller, H. and A. Scheid, Stochastic Finance: An Introduction in Discrete Tie, Second Edition, Walter degruyter, Berlin. Goldberg L. R., G. Miller and J. Weinstein, Beyond Value at Risk: Forecasting ortfolio Loss at Multiple Horizons. Journal of Investent Manageent, vol. 6, no. 2: Jorion,., Value at Risk, McGraw-Hill. Litteran, R., Hot Spots and Hedges. Journal of ortfolio Manageent: Longin, F. and B. Solnik, 2001, Extree Correlations of International Equity Markets, Journal of Finance, vol. LVI, no. 2: Markowitz, H.M., ortfolio Selection, Journal of Finance, vol. 7, no. 1: Menchero J., and V. oduri, Custo Factor Attribution. Financial Analysts Journal, vol. 64, no. 2: Nelson, R., An Introduction to Copulas, Springer-Verlag. Sharpe W. F., Budgeting and Monitoring the Risk of Defined Benefit ension Funds. Working aper, Stanford University MSCI Barra. All rights reserved. 18 of 20 lease refer to the disclaier at the end of this docuent.

19 Extree Risk Analysis Contact Inforation Aericas Aericas Atlanta Boston Chicago Montreal New York San Francisco Sao aulo Staford Toronto (toll free) Europe, Middle East & Africa Asterda Cape Town Frankfurt Geneva London Madrid Milan aris Zurich (toll free) Asia acific China Netco China Teleco Hong Kong Singapore Sydney Tokyo (toll free) (toll free) MSCI Barra. All rights reserved. 19 of 20 lease refer to the disclaier at the end of this docuent.

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