Are Hedge Funds Skewing their Investors?

Matthew C. Pollard∗

BNP Paribas Hedge Fund Center

London Business School

November 14, 2009

∗mpollard@london.edu, Slides and Paper available at matthewcpollard.com

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Figure 1: High Alpha? Basis Capital, cumulative returns.2

Figure 2: High Alpha? Basis Capital, cumulative returns.3

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: Pr(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?

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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”, http://hf-implode.com

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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

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The “kink” in histogram of returns across 8000 funds. Suggests misreporting.

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Problem

Hard for investors to identify ex-ante good managers.

Hedge Fund 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

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The Problem (Cont.)

Unskilled managers can engineer εt so that, in-sample

α̂ > 0 almost certainly

Just make εt very negatively skewed distribution with zero mean.

Foster and Young (2008) Binary option strategy

ε ∼

p (1− p)

−(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.

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Foster & Young (2008) show that extreme binary option strat-

egy is optimal, two ways:

ε ∼

p (1− p)

−(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.

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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.

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Figure 3: High Alpha? Basis Capital, cumulative returns.12

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Skewed Risk Measures

With payoff, true α = 0 , Pr(loss)=p

returns observed T years, sample alpha will be skewed

α̂ = p− Binom(T, p)

T

Mode(α̂) = p

Median(α̂) > 0

Mean(α̂) = 0

Other measures (Sharpe Ratio, Treynor Ratio,...) also inheritskewed distribution giving larger modes and medians.

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T-tests for α stop working

t =α̂− 0

std.err(α̂)

With probability (1− p)T

α̂ = p

s.e.(α̂) ≈ 0

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.

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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.

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Replication by Ruin Theorem

Observe cumulative returns of hedge fund.

Calculate alpha and cumulative alpha,

αt = RHFt − βRBENCH

t

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

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Proof: Construct a “ruining” martingale strategy with

Mt+∆t =

Mt × 11−p (1− p)

0 (p) ”ruin”

and independent draws. Probability that m ≥ m∗ at time T is

Pr(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.

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How do we construct this martingale?

1. Write binary options. Sell $ 11−p × AUM worth of options.

2. Martingale Doubling: pick any risky asset, Pr(rt > 0) = 12 .

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 ≤ −1Sell risky, declare loss

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The Skewness/Skill Performance Measure

The probability of α = 0, no skill given observed returns (m0, ...,mT ):

Pr(no skill|m) =1

m∗

Does not depend explicitly on T or α̂ or σM .

Always bounded between (0, 1) since m0 := 1.

Direct Colorado of replication theorem.

Intuitively: probability that a given cumulative return could beachieved 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.

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Alternative Derivation, Doob’s Maximal Inequality:

for all martingale sequences, max value bounded by

P

[max1≤t≤T

(Mt) ≥ x

]≤ E[MT ]

x

Set x∗ = max1≤t≤T

(mt). Assume Mt is a martingale.

Probability x∗ came from no-skill martingale, E[MT ] = 1 is:

Pr(no skill|x∗) = P

[max1≤t≤T

(Mt) ≥ x∗]≤ 1

x∗

Previous argument shows that this bound is perfectly tight.

Previous argument also doesn’t need to assume E[Mt+1|Ft] = Mt

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Why is there no α̂, p or T in the test?

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:α̂ = 1

1−p prior to ruin. So α̂ vanishes.Invariant to p is same as invariant to T .

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Number of Periods to Distinguish Skill and Skewers

With π confidence, a skilled fund earning average α per period

needs

T ≥log( 1π

)log(1+α)

trading periods before can confidently say fund is skilled.

Example

Fund achieved steady α = 4% per month against benchmark.

π = 10%,

T ≥ log(0.1)

log(1.04)= 59 months ∼ 5 years

If α halved to 2%, need double, ∼ 10 years. Very long time!

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Empirical Work – Predicting Likelihood of Blow-up

Factor model. excess return / “alpha” return:

mt := RFUNDt − β̂1F1,t + ...+ β̂8F8,t

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%.

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Identifying Skill Using Measure

Each fund in TASS, calculate their p(Skill) value

p(skill)i,t =1

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

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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.

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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-valuesUse logistic regression to predict blow-up fund, regression signifi-cant but p-values insignificant.

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