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Risk Containmentfor Hedge FundsAnish R. Shah, CFANorthfield Information ServicesAnish@northinfo.comApr 19, 2007

Raging Asset Class 30 fold growth in assets under management since1990 estimate > 2000 new funds launched last year in US equities: 5% of assets, but 30% of tradingvolume(source: sec.gov) Premium for top funds, e.g. Caxton 3/30,Renaissance 4/44, SAC 50% of profits

Extensive Literature Weisman, A. “InformationlessInvesting And Hedge Fund PerformanceMeasurement Bias,” Journal of Portfolio Management, , 2002, v28(4,Summer),80-91. Lo, A. “Risk Management for Hedge Funds: Introduction and Overview,”Financial Analysts Journal, 2001, v57(6,Nov/Dec), 16–33. Chow, G. & Kritzman, , M. “Value at Risk for Portfolios with Short Positions,”Journal of Portfolio Management, , 2002 v28(3,Spring), 73-81. Bondarenko, , O. “Market Price of Variance Risk and Performance of HedgeFunds,” SSRN working paper, , Mar 2004. Getmansky, , M. et al. “An econometric model of serial correlation and illiquidityin hedge fund returns,” Journal of Financial Economics, , 2004, v74(3,Dec), 529-609. Winston, K. “Long/short portfolio behavior with barriers,” Northfield ResearchConference, , 2006. Jorion, , P. “Risk management lessons from Long-Term Capital Management,”European Financial Management, , 2000, v6(3), 277-300. Carmona, R. & Durrleman, , V. “Pricing and Hedging Spread Options,” SIAMReview, , 2003, v45(4), 627-685685

Fundamental Idea For both investors and managers, hedge funds(though they may be benchmarked to long-only only orcash) are a totally different animal Non-Gaussian return distributions Liquidity and leverage/credit considerations Dynamic investment strategies Traditional measures of performance and risk –std dev, tracking error, β, α, Sharpe ratio – arenon-descriptive

Part I: Complications for the Investor Lo 2001, “Risk Management for HedgeFunds: Introduction and Overview”Weisman 2002, “InformationlessInvestingAnd Hedge Fund PerformanceMeasurement Bias” “How to manufacture performance with noskill”

From Lo 2001:1. No Skill α

The Secret: Short Volatility(selling insurance -risk is invisible until it happens)Writing optionsLo’s s example sells out of the money putsWriting synthetic options by ∆ hedging (dynamically(altering the mix of stock and cash)Executed without owning derivativesIssuing credit default swapsBetting that spreads return to typical levelse.g. LTCM, see Jorion 2000

Frequent Small Gains Exchangedfor Infrequent Large LossesS(T)Probability Option Writer Gain Option Writer Loss

Performance of Short Vol StrategyFrom Weisman 2002:

2. Estimated Prices forIlliquid SecuritiesValue of infrequently traded securities is estimatedEven operating in earnest, one is likely to undershoot both losses andgainsUnderestimate volatilityOverestimate value after a series of lossesi.e. exactly when positions must be liquidatedBehavior evidenced by serial correlation in returns A separate phenomenon: Up returns are, in general, shrunk byperformance fees. So, the return of the underlying investments (inparticular, downside) is more volatile than indicated by reported d returns

The Effect on Sharpe Ratio Suppose the estimate is a combination of present and pasttrue returns:r estimated tσ 2 estimatedestimated = (1-w)rt + wr t-1estimated = [(1[(1-w)2 + w 2 ] σ 2SR = (r-r(f ) / σw = 50% → Estimated SR ↑ 41%25% → ↑ 26%10% → ↑ 10%

3. Increasing Bets After Loss Weisman 2002 – St. Petersburg Investing If you lose $1 on the first bet, wager $2 on the next. If youlose that bet, wager $4 on the next, etc. Low probability of losing, but loss is extreme Can happen inadvertently – $10 long, $10 short, $10 cash.Lose on the shorts: $10 long, $12 short, $10 cash. Size ofbets jumps from 200% to 275%($20 on net $10 → $22 on net $8)

Performance ofSt. Petersburg StrategyFrom Weisman 2002:

Theory Meets RealityLTCM90% of return explained by monthly changes in credit spread1/98 → 8/98, lost 52% of its value. Leverage jumped from 28:1 ↑ 55:1Nick Maounis, , founder of Amaranth Advisors:"In September, 2006, a series of unusual and unpredictable marketevents caused the fund's natural-gas positions, including spreads, toincur dramatic losses”“We had not expected that we would be faced with a market that wouldmove so aggressively against our positions without the market offeringferingany ability to liquidate positions economically.”"We viewed the probability of market movements such as those thattook place in September as highly remote … But sometimes, even thehighly improbable happens.”

Addressing Short Volatility Bondarenko 2004 From set of options on an underlying, price a variancecontract. (Different than option implied vol. Contract isaverage realized variance over the interval) Over the interval, sample the underlying to measurerealized variance (Sampled – Priced) / Priced = the return to variance. Theaverage of this over time is the return premium to variance

Empirical Value of Short VolatilityThe premium is negative. i.e. the marketconsistently overestimates (overprices) varianceAdding the time series of variance returns as afactor in style analysis1) reveals a fund’s s exposure2) corrects alpha estimate to account for this source ofreturnBondarenko finds hedge funds as a group earn6.5% annually from shorting volatility

Addressing Serial Correlation Fit model that explicitly incorporates thestructure of serial correlation Getmansky 2004r reported t = Σ k θ k r t-k Σ k=1..K θ k = 1r t = µ + βm t + ε t ε t , m t ~ IID, mean 0var(m t ) = σ 2

Nonlinearities: Different Up andDown Market SensitivitiesFrom Lo 2001:r t = α + β - m - t + β + m + t + ε tLo also provides amodel to account forphase-locking behaviore.g. correlations acrossassets rising duringcatastrophic markets

Part II: Complications for theManager Chow 2002, “Value at Risk for Portfolioswith Short Positions” Winston 2006, “Long/short portfolio behaviorwith barriers”

Recall Usual Brownian Motion Modelfor Stock Price Movement dS/S = µ dt + σ dWdLogS = (µ(- ½ σ 2 ) dt + σ dW Although instantaneous return is normal,(1 + return) over time is lognormal:S T /S 0 = e [(µ - ½ σ 2)T + σW T ] Sum of lognormal ≠ lognormal

Lognormal has positive skew,limited downsidePositive skew in returns ...... becomes a long left tail for short positions. A separate issue:Long and wrong → exposure decreasesShort and wrong → exposure increases

Lognormal portfolio approximation,ok for long only,breaks down with long/shortFrom Van Royen, Kritzman, , Chow 2001:

A Better Frameworkfor Long/Short RiskModel each side of a long/short portfolio as a geometric Brownian motiondL/L = µ Ldt + σ LdW LdS/S = µ Sdt + σ SdW SdW LdW S= ρdtDynamics of L - S describe behavior of long/short portfolio Answer quantitative and qualitative questions (Winston 2006)“What is the expected time to hit drawdown?”“What is the probability the portfolio is > $110 in 1 year without t falling below adrawdown of $80 in the interim?”“How does increasing short-side side volatility affect the probability of ruin?”L - S is not a geometric Brownian motionSee mathematical literature for options on spreads

Ways to tame thenon-GBM, L – SApproximate L-S L S by a Brownian motion with the same mean and variance attime TLook at ratio, f = L / Sdf = dL/S– L dS/S 2 + L/S 3 d – 1/S 2 ddf/f = [µ[L - µ S + σ S2 - ρσ L σ S ] dt + σ L dW L - σ S dW S→ f is GBM Kirk approximation (used in Winston 2006)Interested in P(L – S < critical k) = P(L/[S+kS+k] ] < 1)let g(L,S) ) = L/[S+kS+k]will be approximating S/(S+kS+k) ) by S 0 /(S 0 +k)dg = dL/(S+k) – L dS/(S+k) 2 + L/(S+k) 3 d – 1/(S+k) 2 ddg/g = dL/L– dS/S [S/(S+kS+k)] + σ S2 [S/(S+k)] 2 dt – ρσ L σ S [S/(S+kS+k)]dt≈ dL/L– dS/S [S 0 /(S 0 +k)] + σ S2 [S 0 /(S 0 +k)] 2 dt – ρσ L σ S [S 0 /(S 0 +k)] dtwhich is BM

Applications Success and failure surfacesfrom Winston 2006:

Summary Hedge funds offer investment strategies poorlydescribed by traditional tools and measures. If investors aren’t t aware of the hidden risks, surelythey will select for them.e.g. 4:00 mile is fast, 3:30 mile = a goat? Managers of long/short portfolios are exposed tophenomena not present in long-only. only. Avoiding ablow-up requires extra vigilance.