Skew Berlin2009


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Presented at Humboldt University zu Berlin

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Skew Berlin2009

  1. 1. Are Hedge Funds Skewing their Investors? Matthew C. Pollard∗ BNP Paribas Hedge Fund Center London Business School November 14, 2009 ∗, Slides and Paper available at
  2. 2. 1 Skill or Skew? Hard to distinguish between skill and negative skew . Simple strategies induce negative skewness, zero expected returns All performance measures over sample returns are inflated. Are managers “skewing their clients”? I produce robust measure: P r(skill|skew), skill p-values The probability a fund achieves returns by simple skew strategy Intuitive, simple, based on Doob’s Maximal inequality. Construct a “probability of skill” index for 8000 funds. Out-of-sample test: does p−value predict blow-up funds?
  3. 3. Malpractice in Hedge Funds Theft: Boston Provident Partners (13-11-2009), among many Ponzi Funds: Bernard Madoff, Allen Stanford, Bayou Group. Blown-Up: LTCM (-86%), JWM Partners (-52%), Basis Capi- tal LLP (-89%), Bear Sterns Structured Credit (-79%), Ellington Partners (-75%), ING Diverisified yield (-53%), Absolute Capital (-67%), Bear Sterns High Grade Credit Funds (-85%). At least 117 funds have imploded since 2006. “Hedge Fund Implode-O-Meter”,
  4. 4. Literature on Manipulating Performance Theory Manipulating Sharpe Ratios – Goetzmann et al., 2002 Manipulation-Proof Performance Measures – Goetzmann Martingale Doubling Strategies – Brown et al., 2005 Incentives, Skewed Strategies – Foster and Young, 2008 Empirical “Capital Decimation Partners” – Lo, 2001 “Do Hedge Fund Managers Misreport Returns?” – Bollen, 2009
  5. 5. The “kink” in histogram of returns across 8000 funds. Suggests misreporting.
  6. 6. Problem Hard for investors to identify ex-ante good managers. Hedge F und Returni,t = α + βi,t Benchmarkst + εt “Skilled”: α > 0, “Unskilled” α ≤ 0. Before fees, industry has positive alpha Net of fees, industry has insignificant zero alpha. Hard to do inference on alpha, omitted variables, low persistence
  7. 7. The Problem (Cont.) Unskilled managers can engineer εt so that, in-sample α > 0 almost certainly ˆ Just make εt very negatively skewed distribution with zero mean. Foster and Young (2008) Binary option strategy  p (1 − p) ε∼ −(1 − p) (p) where p < 0.5. Smaller p gives more negative skew.. If p = 0.1, big loss (-90%) experienced every 10 periods on average.
  8. 8. Foster & Young (2008) show that extreme binary option strat- egy is optimal, two ways:  p (1 − p) ε∼ −(1 − p) (p) 1. Maximizes total value of performance fee (high water mark) 2. Hardest to discern true and sample alpha – only after big loss.
  9. 9. Case Study: Basis Capital LLP “Asia-Pacific focused Relative Value and Absolute Return fund.” “Equity type returns with bond market volatility,” “Highly diversified holdings.” “The Fund offers investors a high level of transparency.” $1 billion assets under management Australian Hedge Fund of the Year Award in 2006 S&P award “five-star” managers rating, four consecutive years.
  10. 10. Skewed Risk Measures With payoff, true α = 0 , P r(loss)=p returns observed T years, sample alpha will be skewed Binom(T, p) α=p− ˆ T Mode(ˆ ) = p α Median(ˆ ) > 0 α Mean(ˆ ) = 0 α Other measures (Sharpe Ratio, Treynor Ratio,...) also inherit skewed distribution giving larger modes and medians.
  11. 11. T-tests for α stop working α−0 ˆ t= std.err(ˆ ) α With probability (1 − p)T α = p ˆ s.e.(ˆ ) ≈ 0 α t → ∞ T = 5 and p = 0.1, 60% probability of getting unbounded t-value. T = 10 and p = 0.1, 39% probability of getting unbounded t-value.
  12. 12. Strategies for Engineering Skew? Static: short OTM options, long distressed securities, long Catastrophe bonds, securitized insurance, junk bonds, long CDOs, CLOs, short CDSs, VIX. Dynamic: Merger arbitrage (long target, short acquirer) Convergence arbitrage (long/short two assets of equal value) Carry Trade (short low yield currencies, long high yield ) The Martingale doubling system (double position upon x% loss) Easy to implement (e.g. Martingale doubling with any asset) All earn at least a zero expected return.
  13. 13. Replication by Ruin Theorem Observe cumulative returns of hedge fund. Calculate alpha and cumulative alpha, HF BEN CH αt = Rt − βRt Mt = (1 + α1 )(1 + α2 )...(1 + αt ) Then, any maximum cumulative return m∗ = max(M0 , ..., MT ) can be replicated with no skill with at-least probability 1 m∗ Only assumption: Managers can trade, or dynamically replicate, options
  14. 14. Proof: Construct a “ruining” martingale strategy with  Mt × 1 (1 − p) 1−p Mt+∆t = 0 (p) ”ruin” and independent draws. Probability that m ≥ m∗ at time T is P r(MT ≥ m∗ ) = (1 − p)-log(m )/log(1-p) ∗ 1 = m∗ This assumes zero-risk premia. If positive, 1/m∗ is lower bound. May also be proved by Doob’s Maximal Inequality.
  15. 15. How do we construct this martingale? 1 1. Write binary options. Sell $ 1−p × AUM worth of options. 2. Martingale Doubling: pick any risky asset, Pr(rt > 0) = 1 . 2 For each trading time (∆t, 2∆t, ..., T ) { if YTD cumulative return < 1/(1 − p) Increase risky position by eσ∆t if YTD cumulative return ≥ p Sell risky, go 100% cash. if YTD cumulative return ≤ −1 Sell risky, declare loss
  16. 16. The Skewness/Skill Performance Measure The probability of α = 0, no skill given observed returns (m0 , ..., mT ): 1 Pr(no skill|m) = ∗ m Does not depend explicitly on T or α or σM . ˆ Always bounded between (0, 1) since m0 := 1. Direct Colorado of replication theorem. Intuitively: probability that a given cumulative return could be achieved by negatively skewed, zero alpha strategy. For Pr(no skill|m)< 0.05, need fund to 20-fold outperform. Hard to reject no-skill hypothesis.
  17. 17. Alternative Derivation, Doob’s Maximal Inequality: for all martingale sequences, max value bounded by E[MT ] P max (Mt ) ≥ x ≤ 1≤t≤T x Set x∗ = max (mt ). Assume Mt is a martingale. 1≤t≤T ∗ Probability x came from no-skill martingale, E[MT ] = 1 is: 1 Pr(no skill|x∗ ) = P max (Mt ) ≥ x∗ ≤ 1≤t≤T x∗ Previous argument shows that this bound is perfectly tight. Previous argument also doesn’t need to assume E[Mt+1 |Ft ] = Mt
  18. 18. Why is there no α, p or T in the test? ˆ All these terms cancel out. Geometry: alpha is the average gradient in cum. return series. p trades off higher growth with higher probability of ruin Trade off is exact: Distribution of the maximum attained is invariant to p: P (maximum replicator cum return < x) = 1 − 1/x. In skewed strategy: 1 α = 1−p prior to ruin. So α vanishes. ˆ ˆ Invariant to p is same as invariant to T .
  19. 19. Number of Periods to Distinguish Skill and Skewers With π confidence, a skilled fund earning average α per period needs 1 log π T ≥ log(1+α) trading periods before can confidently say fund is skilled. Example Fund achieved steady α = 4% per month against benchmark. π = 10%, log(0.1) T ≥ = 59 months ∼ 5 years log(1.04) If α halved to 2%, need double, ∼ 10 years. Very long time!
  20. 20. Empirical Work – Predicting Likelihood of Blow-up Factor model. excess return / “alpha” return: ˆ ˆ mt := Rt U N D − β1 F1,t + ... + β8 F8,t F Eight Factors are from Hsieh & Fung, 2004: S&P, Russell 2000, Treasury, Credit Spread, Emerging Mkt, MSCI EAFE, VIX index, US Currency Basket. Fitted out-of-sample, rolling window. Betas unique to fund. Linear Replication of individual funds, R2 ∼ 60% to 70%.
  21. 21. Identifying Skill Using Measure Each fund in TASS, calculate their p(Skill) value 1 p(skill)i,t = maxt (Mt,i ) Question: does low p-skill predict negatively skewed crash funds? Should by theory. Does it in practice? Test: create balance sample of crashed and non-crashed hedge funds, n = 86. Use p(skill)i,t in period prior to crash. Regress logistic regression: logit(1crash,i ) = α + βpi + γcontrolsi
  22. 22. Controls are size, age, fund net flows, fund strategy dummy Result: Pseudo R2 = 0.21%, coefficient for p correct sign, insignificant. Strategy dummy and fund inflows singificant. Convertable arbitrage strategies much higher blow up likelihood Higher Age of fund reduces tendency to blow up. Fund net flow insignificant.
  23. 23. Conclusion Theory: Developed a robust test of hedge fund alpha Test is probability performance can be replicated with no skill. Simple statistic, Doobs Maximal Inequality Power and coverage of test is good, t-test fails with skewed returns. Empirical: TASS hedge fund database, evidence of highly skewed funds. Fund replication by factors to get p-values Use logistic regression to predict blow-up fund, regression signifi- cant but p-values insignificant.