Tail Risk Management_PIMCO Paper 2008

8/8/2019 Tail Risk Management_PIMCO Paper 2008

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T ail R isk M a n a g e m e n tViNEER B H A N S A L I

ViNEER B H A N S A Uis 3 portfolio ni;in.iji(;r andth e hejd of Analyrics atPIMCO in NewportBeach, CA.bhansali@pitnco,cnm The events unfolding as of the writingt this article show the importanceof controlling portfolio exposuresto fat-tail events. The obvious ben-efit of having the appropriate amount of dis-aster insurance is survival. When applied to

financial market participants, a secondary ben-efit is observedthe survivors are better ableto take advantage of the reduced liquidity thataccompanies tail events and position themselvesfor attractive prospective returns. Thus, tailinsurance is an offensive strategy for the longterm, even though it may appear to be a defen-sive strategy in the short term.

The desire to outperform peers andindices over the last decade forced herds ofportfolio managers to reduce or void insurancein their portfolios for improbable, yet possible,events. This trend culminated in a spiraiingasset debacle whose end has yet to be realized.The emergence of structured products thatwere inefficient packaged sales of insurance wasa natura] consequence of the demand for highreturns in a low-return world (Bhansali[2007a]). The very real quandary a portfoliomanager now faces is one that requires the esti-mation of the optimal amount of insurance fora portfolio to protect against tail events. Tailevents, by their very nature, are hard to quan-tify with traditional models, wliich pay littleheed to rare events. Most investment processesrely on harvesting risk premia from systematicfactors, whose variations are measured and

exposures scaled based on history; by design,this statistical averaging process underweightsthe impact of tail events.

The purpose of this article is threefold.First, I will provide a characterization of tailevents that is quantifiable; second, I will pro-vide algorithms to quantify the probability andseverity of these events; and third, 1 will proposea selection fbimework for tail hedges that mayconsist of securities, options, and strategies.

TAIL RISK AS MACRO RISK

Tail risk at the portfolio level is almostalways a systemic risk. A systemic risk is onthat puts pressure on the ability to fimd lev-ered holdings. In episodes of systemic risk,everyone desires liquidity, but no one is willingto provide it. According to Bookstaber [2007],the tight-coupling of financial markets requiresthat liquidity be available to the system at alltimes at some price, or the financial systemwill start to break down. Financing and liq-uidity are macro risks and their proper valuation requires macro models. Proper hedgeconstruction requires a macro view and usesmacro tools and markets.

Khandani and Lo [2007] have observedthat new strategies in the market, especially thoseprovided by hedge funds, are characterized bynew betas (systematic selUng of liquidity), whichcan be withdrawn very quickly. Many of thesestrategies are highly correlated; hence, a shock

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to one type of strategy causes ripples in other strategies.Further, the nature of the relationships is not static, so thatafinancialmarket dislocation in a strategy that is runningmany uncorrelated strategies can rapidly influence otherstrategies. Therefore, a tail-risk hedge portfolio that pro-vides insurance against tail events should carry a "beta" tosystemic and macro risks.A statistical analysis of the returns of broad asset classesover the long term shows the presence of a handful oflatent factors, or principal components, that explain thevariation of most traditional investable assets. The strikingobservation x)m analyses such as these is that the presenceof regimes follows a similar systematic behavior and loadsmost heavily on monetary policy. In other words, withsuitable leads and lags, the quadrants of early and bt e reces-sions and early and late expansions in the developed mar-

kets (the U.S., in particular) can be mapped to early andlate periods of Fed easing and tightening. In a finance-driven economy such as the U.S.. it is not surprising thatthe tails which develop in these periods can be hedgedwith macro instruments which respond to central bankactivity. Deleveraging risk is. to a large degree, a mone-tary policy risk; thus, tail events become macro events.This observation has far-reaching consequences.The main consequence is that tail risk becomes a macrorisk. To forecast and control against tail risk, it is neces-sary to step outside the world of historical estimates and

calibration and to forecast structural changes (not easy!)and imagine improbable, high-severity scenarios. Theimm ense benefit of thinking abou t tail risk in terms of amacro risk is that the construction of hedges is siinpli-fied. At any p oint in time , macro markets are the deepestmarkets. A macro market is a broad bond, stock, foreignexchange (forex), credit, or commodity market, or a com-bination of them. Despite pockets of illiquidity in themacro markets (e.g., the credit market could dry u p withforex barely impacted), typically some sort of insuranceremains attractively priced for a sufficiently long perioddue to the sheer mass of capital reallocation needed toalign the markets.When systemic crises happen, correlations rise inabsolute value. This provides a fi-ee lu nch because a com-pletely disconnected macro asset class in norm al times canbecome a good hedge against the tails in distressed times.In other words, if the cost of credit hedging is too highin the midst of a systemic crisis, then a hedge from theforex or equity o ption markets might actually be a mo reefficient credit hedge over a long enou gh holding

horizondespite the apparent basis risk associated withit in normal times.VALUATION OF TAIL-RISK HEDGES

Proper evaluation of the cost of insuring against tailrisk is the second m ost important factor to evaluate. Creditmarket hedges were extremely cheap in 2007 because ofthe incessant selling of credit insurance through struc-tured products and the demand driven by excess liquidity.Today, these same hedges by some measures are. at best,fairly priced. Thus, hedging credit with credit marketinstruments might no longer be an attractive solution.The estimation of option prices and sensitivities in gen-eral is based on the academic foundations of continuoustrading and risk-neutral investors.These ideal conditions are rarely met in turbulentmarkets. As a result, what are now considered traditionalmodels of option pricing are flawed due to their over-simplifying assumptions (Bhansali [2007b]). A viable alter-native approach is to nm scenarios that investors are averseto and evaluate the expected value of portfolios underthose scenarios with and without insurance assets. Thedifference in these two p ortfolio values is indicative of afair price for the insurance from the investor's perspective,not the market price of the insurance. We all know thatcatastrophe insurance can trade too cheaply in the natural

insurance markets (i.e., the best time to buy insurance iswhen it is not needed). Obviously, it is possible for thesame to hold true in the world of finance, especially in aworld of innovative financial engineering that ports risksfi-om one type of market to another.More than one method can be used to implementtail hedges, but our framew ork adopts a natural unity thatwould seem obvious to someone w ho is involved in insur-ance. The simplest way to insure portfolios is to buy highquality "insurance" securities. For example, credit marketcrises almost always occur with deleveraging and a grab

for liquidity which usually results in Treasuries rallyingin price terms , especially those with the shortest ma turi -ties. Therefore, sho rt-term cash and Trea-suries are naturalasset-based hedges for tail risk, ut these securities canbecome overpriced due to technical reasons.Th e second way to insure portfolios is to buy co n-tingent claims, or "option-like" securities. A year ago,out-of-the-money tranches on CDX and ITRAXXindices were literally being given away due to the dem andfor constant proportion debt obligations (C PDO s), levered

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super seniors, and other default-remote structured trans-actions. These option-like payofife were priced below theirtheoretical expected value under a systemic risk outcome.

The third alternative to insure portfolios is to investin strategies that are negatively correlated to tail risk.Among the traditional estabhshed strategies. Exhibit 1demonstrates that only the systematic, trend-followingmanaged futures strategy provides positive correlation totail-risk indicators, such as the CBOE Volatility Index(VIX), while also being largely uncorrelated to the stockmarket. Copious amounts of research have been done todemonstrate that trend-foil owing strategies empirically

behave like a long position in lookback straddles, andhence are naturally long tail risk (Fung and Hsieh [2001]).

The fourth approach is to move the portfolio offthe optimal frontier (i.e.. to accept less return for the sameamount of risk), which explicitly recognizes that the sim-plest meanvariance optimalfit>ntierfalsely assumes thatrisk can be measured by volatility (second moment) aloneand that the investor has perfect forecasting ability, whichis almost never true. Oneexample of this is to reduce theexposure to spread products, such as corporate bonds, orlow-quality mortgages, which have higher yield becauseof embedded default and illiquidity options. When it

E X H I B I T 1Correlation of Typical Hedge Fund Strategies and VIX Index for Different Periods

Positive Correlation (Lower vdatlity, Higher Return)

Composas Convenible Dedicaled EmergingHedge Fund Arbitrage Short Siae Markets Equity Even lOnven Dlstr3Sd Risk Fixed income Global Macro Lorig/Shori MsnaQBd Multi.Market Artltrage AMa^ Equity Futtir w SlratagNeutral

Source: PIMCO.

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becomes evident that techjiicals have caused risk premiato compress to rich-enough levels, these assets promise ahigh probabihty of low-to-negative future returns. Asshorts, these assets might actually become candidates fortail risk control.To reiterate, I am not advocating portfolio insur-ance in which exposures are dynamically adjusted tochanging market conditions. The problem w ith this typeof zero-cost tail solution is that it assumes liquidity willbe present in crises, yet liquidity typically all but evapo-rates during systemic shocks. In the approach discussedhere, the portfolio is subjected to possible, but rare, super-shocks, and insurance may be purchased using one ot thefour techniques just described. Clearly, there is an upfront,or running, cost to the tail hedge, but the role of the tail-risk portfolio manager is to reduce the cost of these hedges

over the hedge horizon . This might not be aschallengingas it sounds.Frequently, when the horizon is long enough,hedges can almost be bought for "free." For instance,recently long-dated options on the forex, such as thedollaryen exchange rate, had zero carry cost over a one-year horizon due to the roll-up of forward rates from theinterest rate diiferential between dollar rates and yen rates.If, in periods of stress, one of the outcomes is deleveragingand aflight o low-yielding currencies such as the yen, thedirecrionality of the forex movement and the associated

increase in volatihty makes the real-world expected valueof the package much higher than the theoretical, risk-neutral value. Thus, the possibility of asymmetric payoffsin particular states-of-the-world makes the hedges worthmore to the investor than their stand-alone theoreticalvalue. For mo re discussion of how this strategy may resultin excessive risk-takin g and presence of cheap tail hedges,see Bhansali [2008].To ascertain the value of a particular tail hedge, it isessential that the scenario analysis is performed with manyvariations of the inputs for the parameters. Th e inputs are

more important than the level of refinement of the models.In the simple language of BlackScholes options, this boilsdown to running the performance of option positionsunder various m aturity, rate, volatility, and spot rate envi-ronments, using the option model simply as a crude non-linear transformation machine between inputs and outputs.Special attention must be paid to consistently taking thevolatilities and correlations to ex treme values because th eunderlying assumption of jo in t lognormality undervaluesthe tails of the distribution (see Bhansali and Wise [2001 ]).

It is also possible to approach the problem by spec-ifying other distributions with naturally fat tails, such asthe Levy distribution, b ut moving away from simple dis-tributions, in practice, creates more problems with intu-ition than is compensated for by more accurate empirics.In any case, today's computational power makes it easyto substitute empirical distributions that can be evaluatednumerically, especially on the tails, in lieu of theoretical,closed-form solvable models.

The extension of Black-Scholes to credit for thiscapability is the Gaussian copula model. While used andabused, this modela simple and an eficient way to valueoptions on portfolio lossesstill has the ability to providegood inmition about the performance of tranches in periodsof stress. Exhibit 2 shows the performance of a 7-100tranche on the 1G8 index for bo th instantaneous and on e-year-delayed shocks. The first scenario in Exhibit 2 showsthe percentage impact on a long protection tranche posi-tion for an instantaneous shock. The second scenarioshows the percentage impact on a tranche position for ashock at a one-year horizon.

Tranche returns can be different as the implied basecorrelation moves around. As correlations rise, the seniortranches pick up additional return due to a higher prob-ability of many names defaulting simultaneously, a hall-mark of systemic risk. The one-year-horizon shockaccurately captures the cost as the tranchewhich is anoption on lossesshortens in maturity and rolls downthe credit curve, which typically is upward sloping.

Using credit derivatives as a reference, there isanother way to look at systemic shocks, in Bhansali,Gingrich, and Longstaff [2008], index spreads for variousindices were broken down using a three-jump approach,which was first used by Longstaff and Rajan. In Exhibit 3.the idiosyncratic component is the darker solid line, thesector risk is the dotted line, and the eco nomy -wide riskis the light solid line. The economy-wide, or systemic, riskis currently at highly elevated levels. Because the econom y-wide risk is carried by senior and super senior tranches,it provides a microscope for the valuation of hquidity andsystemic risk in the market.If we proxy systemic risk by the fraction of spreadallocated to the systemic factor, or roughly speaking thefraction of expected losses accounted for by default-remotesuper senior tranches, we can construct a high-frequencymeasure for a systemic risk factor. Regression on thisfactor of other seemingly unrelated marketssuch asmunicipals, asset-backed securities, and other spread

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E X H I B I T 2Tranche-Shock Scenario Analysis

PriceDateindextenorfundedattPointdetPointCoupon(bp)

6/26/08IG810

07.00%

100.00%82.8

Delay for shocksspread Boost (bp)-25-100102550

10020 0

(years): 0(Attachment %) 64.69(Detachment %) 0.00

-1.3-0.53

00.521.322.665.32

10.38

97.030

-0.350.460.991.522.323.656.24

11.12

32.340

-2.36-1.58-1.05-0.510.311.714.499.83

Delay for shocksSpread Boost (bp)-25-10

0102550

10020 0

(years): 1(Attachment %) 64.69(Detachment %) 0.00

-2.07-1.4

-0,94-0.49

0.21.383.718.24

97.030

^1.17-0.47

00 4 61.162,334.63

9

32.340

-3.07-2 42-1.96

-1.5'0 .790.422,877,64

Tramhe-stmk .senario analysis for a 7-00 senior tranche on the O-ycar 1C8 nraitumt-gradi- index for mrious shocks to ihc imderlytii}; spread acl andhase correlations.Source:

product shows significant betas to systemic risk. In o therwords, many asset classes carry large am oun ts of systemicilliquidity risk. We found, with signifficant i-statistics, thatfrom June 2006 through March 2008, for every 10 bpswidening of the systemic component, without adjustingfor duration, Lehman Brothers indices moved as follows:

the ABS Floating Rate Index returned negative 53 bps,the U.S. High Yield Index returned negative 58 bps, theMunicipal Bond Index returned negative 51 bps, theCM BS Index returned negative 63 bps, and the Securi-tized Index re turne d negative 38 bps. Each of these assetclasses carries hq uidity risk (which is different from credit72 TAIL RISK MANAGEMENT SUMMER, 2(K)8


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E X H I B I T 3Allocation of CDX 5Y Index lo Idiosyncratic, Sector-wide, and Systemic Risk Factors

100

90

80

70

60

40

30

20

10

Idiosyncraticsectorwidesystemic

6/19/08

Sep-03 Feb-04 Jul-04 Dec-04 May-05 Ocl-05Source: Bhansali, G ingrich, and Longstajf [2008].

Mar-6 Aug-06 Dec-06 May-07 Oct-07 Mar-OS Aug-08

risk) and by stressing our liquidity risk measure we showhow che ;isset classes can be expected to perform in periodsof stress. It should be no su rprise, therefore, that as of thiswriting, m unicipal bonds which have one of the largestbetas to this risk fictor without much direct credit risk(especially general obligation bonds of natural A AA-ratedstates)are trading at higher yield levels than Treasuries,despite the fact that they are tax exempt!

Thus, the construction of a tail portfoho hedgedepends to a great extent on the following three fac-tors: first, the scenario behav ior of the portfolio; second ,the scenario behavior of the hedges, net of cost; andthird, the probability of the scenario occurring. Findingthe best combination requires knowledge of how theportfoHo to be hedged behaves under various shocks.

Th e best comb ination of hedges is the o ne that producesthe worst return to the portfolio under the stress scenarioand controls the risk, at horizon, to the dangerous fac-tors in the stress scenario.This second point is critical because it enables theportfolio to survive to play in the multiperiod game. Thegoal is for your risks to adjust down in periods of stress sothat you have liquidity while everyone else is cbmo ring forit. This might appear to s u r e s t that tail hedges always haveto be custom designed. Fortunately, because correlationsbetween risk factors increase in periods of stress, this needis mitigated for all but th e m ost exotic portfolios. For mostinvestment portfoHos, the underlying exposure to marketfactors, such as the Lehman Agrgate or S&P 5U() indices,is all that is needed to op timize tail hedges.

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The critical reader might wonder why cheap tail hedgescan be found in almost all market environments. There aremany potential explanations. For example, speculative demandof particular types and classes of assets may drive these assetprices very high. In low-return periods, as observed until themiddle of 2007, "yield hogs" increased options sales as asource of cany. The belief of mean-reversion participants isthat out-of-the-money options will never be exercised. Thisis generally true except when the leverage in the marketplaceincreases due to the majority of market participants simulta-neously executing levered option sales, which forces thenotional size to increase in order to generate the same carry.At some point, this type o f system becomes unstableto small noise, creating a domino effect of forced liquida-tion or of hedgers all trying to hedge at the same time. Th e

following cost calculation can illustrate this point: In June2007, to buy protection on the 7-100 tranche on the IG810-year index cost 15.63 bps per year. For a spread dura-tion of 7.45 on the underlying index, the cost per unit ofspread duration was 2 bps. Because the tranche rolls downto a shorter m aturity, the annual roll-down cost ot 2.67 bpsis added to the previous number (2 bps) tor a total 4.67 bpsof cost per unit of spread duration. To buy protection onone billion notional, the total lifetime cost is about $12milhon, which equals the annual cost times PVOl of 7.45.The same tranche in February 2008 was trading at91 bps per year. The new running cost per year of spreadduration is 10.1 bps and the annual roll-down cost is 2.2bps for a total of 12.3 bps per year. The new total lifetimecost is about $67 milHon. The delta of the tranche beforethe 2007 crisis was approximately 0.53. When multipliedby the spread duration of the 10-year index of 7.45, theresult is an approximate sensitivity of 3.95% per 100 bpsof index widening. In February 2008, the delta rose to0.77 as the index widened and moved closer to being atthe money, resulting in almost 5.66% of spread risk.Evaluating these changes in the market and in risk,it is immediately clear w^hy super senior levered notes,

which essentially sold protec tion using these structures inthe form of securities, are in severe distress today. First,the mark-to-market loss at five times is huge, equalingthe net present value of premium change; second, themark-to-market is more variable because the delta hasincreased; and third, the collateral that has to be postedto compensate for mark-to-market fluctuations is muchmore expensive due to lower T-bill rates and higherLIB OR rates. The fact that all of these events happ enedsimultaneously is typical of systemic tail events.

Hedges are also available for long-term investorswh o are willing to com mit capital to them because of thenatural habitat formation of option market participantsand the hmitations of models used by short-h orizo n par-ticipants, such as option market makers. Most traditionaloption trading models are only appropriate for short-termtrading, roughly for the time it takes a dealer desk tounload the package to another customer. Imphcit in thisframework is the assumption that deltas, gammas, vegasand other greeks can be computed and executed. \X/lienthe standard analysis is extended to longer-dated options,the unavailability of reference instrum ents, failure o f risk-neutral pricing, continuous trading, and limitationsimposed by counterparty risk controls all increase thelikelihood of mispricing.Neither is the probability question for tail eventssatisfactorily answered by reference to traded optio n prices.The reason is that the pricing of tail options in particularcarries a significant amount of risk premium compensa-tion to the seller, also known as lottery ticket risk, whichalters the probabihty distribution. Simulation based onpast observations is also no t a totally satisfactory approachas each crisis differs in severity and magnitude.The practical approach for estimating probabilitiescan follow a number of parallel paths. One approach is tosample from historical events with replacement and mag-nify the ra re-event likelihood by som e scale factor, whicheffectively reshapes the tails. Simultaneously, changes ina tail-risk indicator, such as the VIX, can be measured toindicate the likelihood of a crisis versus a normal environm ent. Correlations and volatilities fix)m a distressedregime are used for this purpose. Regardless of how theprobabilities are estimated, the important point is thatbecause tail hedges are usually cheap in the context oflong-lived portfolios, the probability calculation is relatively less critical than the know ledge that potential hedgeexist at the right price. Recognizing this distinction canbe the difference between survival and almost certain ruin

OTHER IMPLEMENTATION CONCERNSOn ce the m acroportfolio tail hedges are put in placemuch active management is still needed. As the cost oinsurance across markets and maturities changes, a "dash-board " that shows the cost of hedges for different ho ri-zons in different markets can provide direction onrebalancing the tail hedge portfolio. Frequent opportunities arise to add pseudo alpha (value tliat reduces th e c

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of the hedge), particularly during times of rollsfor indicesor due to short-term imbalances in pricing. The port-folio manager faces the choice of keeping the hedges static{i.e., letting them age) or keeping them current in themost on-the-run versions by evaluating the trade-ofFbetween basis risk and liquidity/transactions costs.Also key are the issues of monetization andfinancing. W hen the rare event does happen, can hedgesetciently be converted into cash? In our experience,because tail risks are accom panied by deleveraging, trans-action efTiciency is skewed to the part that is long thehedges; in other words, in a demand surge, the seller ofreinsurance is the price setter for hquidity Financing isalso a concern because hedges are usually implementedthrough derivatives. Payment for out-of-the-moneyequity o ptions and forex options is usually m ade upfi-ont,where as paym ent for credit derivatives is as-you-go. Th estructure of the current crisis has revealed that counter-party risk is very important. Many participants found thatinsurance written by mo noline insurers was not as solidas they had thought.CONCLUSION

In this article. I have discussed why tail risks in po rt-folio construction should be thought of as systemic risks.Because systemic risks are macro risks, a proper tail-riskhedging program should take into account the relativepricing of broad macro markets and strategies. A tail-riskhedging program should also evaluate combinations ofhedging alternatives to tnid the best combination forimmunizing the portfolio against improbable, but notimpossible, shocks. Lest it appear that tail-risk hedgingis only abou t crisis events, note that the current low yieldson long-term Treasuries may indicate that although creditinsurance is flilly priced, tail inflation risk is still cheap.That is, the tails to watch out for might arise from anuncontrollable rise in inflation that erodes the value of thedollar and the fixed coupons of long-term bonds. Indeed,stochastic simulations show that in a world where thecentral bank is simultaneously pursuing inflation-tar-geting and an asymmetric policy against deflation, yieldcurves will be flatter when the economy is robust andsteeper when the central bank is forced to ease aggres-sively (Bhansali, Dorsten, and Wise [2007]). While onlytime will show the truth or falseness of this prognostica-tion, o ne th ing is certain the hedge for this type of tailrisk will not be a micro hedge.

ENDNOTEI wish lo thank cl ients and PI M C O colleagues who paidattention to the need to hedge tail r isk before the subpriiuecrisis of 2007. I would also like to thank Bob Gingrich and

Francis Longstafffor their collaboration on recent research.REFERENCESl ihansali , Vineer . "Markowitz Bites Back: The Failure ofCAPM, Compression of Risky Asset Spreads, and the PathBack to Normalcy." PIMCO Viewpoints, 2007a.

. ' 'Putting Economics (Back) into Quantitative Models."Journal of PortfoUo Management (Spring 20()7b), pp. 637f).. "Correlation Risk: What the Market Is Telling Us andDoes It M ake Sense?" in Credit Risk: Miidds, Dtvehprncni, ana Ma

agement, ed. Niklas Wagner. Chapman and Hall/CH^C Press, 20()7c.. "Voladlity and the C arry Trae!' Journal of Fixed Income,Vol. 17 . No. 3 (Winter 2007) , pp . 72-84.

Bhansali , Vineer . Matthew Dorsten , and Mark Wise. "Asym-metr ic Monetary Policy Rules and the Yield Curve." Workingpaper, 2007. Available at www.ssrn.com.Bhansaii. Vineer, Robert Gingrich, and Francis LongstafF. "Sys-ceniic Risk: What Is the Market Tell ing Us?" For thcoming inFinancial Analysts Journal, 2 0 0 8 .Bhansali, Vineer, and Mark Wise. "Forecasting Portfolio Riskin Normal and Stressed MAvket$y Journal of Risk, Vol. 4, No. I(Fall 2001).Bookstaber , Richard . A Demon of Our Own Drsi'ijn: Markers,Hedge Funds, and the Perils of Financial Innouation. Hobokci i , N]Wiicy . 2007.Fung, Will iam, and David Hsieh . "The Risk of Hedge FundStrategies: Theory and Evidence from Trend Followers." 77ifRet'iew of Financial Studies, Vol 14, No, 2 (2(){)1), pp, 3 1 3 - 3 4 1 .Kliantlani, Amir, and Andrew Lo. "Wliat H^ippc-iifd to the Quantsin August 2007?" Working paper. MIT (Nt)vember 2007).LongstafF Francis A, and Arvind Rajan. "An Enipcricnl An a-lyzisofthe Pricing of Collateralized Debt t")bligations."_/i'Hni/of Finance, Vol. 63. No. 2 (April 2008) pp. 529.

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Tail Risk Management_PIMCO Paper 2008