Double or nothing: Patterns of equity fund holdings and transactions Stephen J. Brown NYU Stern School of Business David R. Gallagher University of NSW.

Double or nothing: Patterns of equity fund holdings and transactions Stephen J. Brown NYU Stern School of Business David R. Gallagher University of NSW Onno Steenbeek Erasmus University / ABP Investments Peter L. Swan University of NSW Frontiers of Finance 2005, Bonaire

Performance measurement Leeson Investment Managemen t Market (S&P 500) Benchmark Short-term Government Benchmark Average Return.0065.0050.0036 Std. Deviation.0106.0359.0015 Beta.06401.0.0 Alpha.0025 (1.92).0 Sharpe Ratio.2484.0318.0 Style: Index Arbitrage, 100% in cash at close of trading

Frequency distribution of monthly returns

Percentage in cash (monthly)

Examples of riskless index arbitrage …

Percentage in cash (daily)

Sharpe ratio of doublers

Informationless investing

Apologia of Nick Leeson “I felt no elation at this success. I was determined to win back the losses. And as the spring wore on, I traded harder and harder, risking more and more. I was well down, but increasingly sure that my doubling up and doubling up would pay off... I redoubled my exposure. The risk was that the market could crumble down, but on this occasion it carried on upwards... As the market soared in July [1993] my position translated from a £6 million loss back into glorious profit. I was so happy that night I didn’t think I’d ever go through that kind of tension again. I’d pulled back a large position simply by holding my nerve... but first thing on Monday morning I found that I had to use the 88888 account again... it became an addiction” Nick Leeson Rogue Trader pp.63-64

Infinitely many ways to lose money!  Manager trades S&P contracts  μ = 12.5%, σ = 20%, r = 5% per annum  Fired on a string of 12 losses (a drawdown of 13.5 times initial capital)  Probability of 12 losses =.024%  Trading 8 times a day for a year  Only 70% probability of surviving year!

Infinitely many ways to lose money!

Informationless investing  Zero net investment overlay strategy (Weisman 2002)  Uses only public information  Designed to yield Sharpe ratio greater than benchmark

Informationless investing  Zero net investment overlay strategy (Weisman 2002)  Uses only public information  Designed to yield Sharpe ratio greater than benchmark  Why should we care?  Sharpe ratio obviously inappropriate here

Informationless investing  Zero net investment overlay strategy (Weisman 2002)  Uses only public information  Designed to yield Sharpe ratio greater than benchmark  Why should we care?  Sharpe ratio obviously inappropriate here  But is metric of choice of hedge funds and derivatives traders

We should care!  Behavioral issues  Prospect theory: lock in gains, gamble on loss  Are there incentives to control this behavior?  Delegated fund management  Fund flow, compensation based on historical performance  Limited incentive to monitor high Sharpe ratios

Examples of Informationless investing  Doubling  a.k.a. “Convergence trading”  Covered call writing  Unhedged short volatility  Writing out of the money calls and puts

Forensic Finance  Implications of Informationless investing  Patterns of returns  Patterns of security holdings  Patterns of trading

Sharpe Ratio of Benchmark Sharpe ratio =.631

Maximum Sharpe Ratio Sharpe ratio =.748

Concave trading strategies

Hedge funds follow concave strategies R-r f = α + β (R S&P - r f ) + γ (R S&P - r f ) 2

Hedge funds follow concave strategies R-r f = α + β (R S&P - r f ) + γ (R S&P - r f ) 2 Concave strategies: t β > 1.96 & t γ < - 1.96

Hedge funds follow concave strategies ConcaveNeutralConvexN Convertible Arbitrage Dedicated Short Bias Emerging Markets Equity Market Neutral Event Driven Fixed Income Arbitrage Fund of Funds Global Macro Long/Short Equity Hedge Managed Futures Other 5.38% 0.00% 21.89% 1.18% 27.03% 2.38% 16.38% 4.60% 11.19% 2.80% 5.00% 94.62% 100.00% 77.25% 97.06% 72.64% 95.24% 82.06% 91.38% 86.62% 94.17% 91.67% 0.00% 0.86% 1.76% 0.34% 2.38% 1.57% 4.02% 2.18% 3.03% 3.33% 130 27 233 170 296 126 574 174 1099 429 60 Grand Total11.54%86.53%1.93%3318 R-r f = α + β (R S&P - r f ) + γ (R S&P - r f ) 2 Source: TASS/Tremont

Portfolio Analytics Database  36 Australian institutional equity funds managers  Data on  Portfolio holdings  Daily returns  Aggregate returns  Fund size  59 funds (no more than 4 per manager)  51 active  3 enhanced index funds  4 passive  1 international

Some successful Australian funds Fund Sharpe RatioAlpha FF AlphaBeta Skewnes s Kurtosi s Annual turnover 10.10170.08%0.10%0.90-0.52094.687820.69 (2.21)(2.58) 20.15000.16%0.17%1.110.08344.27770.79 (6.44)(5.88) 30.15590.19%0.20%1.080.73827.65401.18 (4.09)(4.36) 150.10790.09% 0.96-0.25584.17490.34 (2.66)(2.61) 260.09770.12%0.11%1.03-0.26673.43161.27 (2.42)(2.25) 350.18140.29%0.31%0.90-0.62485.12780.62 (3.02)(3.06)

Style and return patterns CategoryBeta Treynor Mazuy measure Modified Henriksson Merton measure Number of observations GARP 0.9608-0.0111 (-2.25) -0.0895 (-2.47) 2372 Growth 1.0367-0.0071 (-1.53) -0.0376 (-1.15) 1899 Neutral 1.0284-0.0011 (-0.29) -0.0210 (-0.72) 1313 Other 1.0067-0.0020 (-0.53) 0.0068 (0.21) 640 Value 0.7690-0.0126 (-2.01) -0.1082 (-2.36) 2250 Passive/ Enhanced 1.01460.0069 (1.50) 0.0457 (1.46) 859

Size and return patterns CategoryBeta Treynor Mazuy measure Modified Henriksson Merton measure Number of observations Largest 10 Institutional Manager No 0.9644 -0.0058 (-2.12) -0.04580 (-2.25) 6467 Yes 0.9059 -0.0100 (-2.25) -0.0779 (-2.56) 2866 Boutique firm No 0.9430 -0.0082 (-2.78) -0.0613 (-2.91) 6567 Yes 0.9543 -0.0045 (-1.23) -0.0428 (-1.53) 2766

Patterns of derivative holdings Fund Investmen t Style CallsPutsMonth end option positions FundNumberStrikeNumberStrike Concavity decreasing Concavity increasingTotal GARP 1 2 3 4 5 6 11 13 0.726 -0.061 0.099 0.041 -0.650 0.222 0.811 0.054 1.017 1.050 1.017 1.023 1.062 1.076 0.002 1.076 0.395 -0.122 0.021 0.008 -1.346 0.950 - 0.957 0.904 0.952 0.944 0.985 0.674 - 100% 29% 59% 77% 100% 71% 41% 23% 100% 80 246 79 898 18 11 8 11 Growth 15 16 17 18 -0.033 -0.039 -0.367 -0.059 1.056 1.060 1.067 1.023 - 0.107 0.108 - 0.951 0.913 27% 35% 13% 73% 100% 65% 87% 11 8 83 344 Neutral 21 22 24 -0.093 0.567 0.405 1.038 0.984 0.854 -0.093 - 0.947 - 10% 100% 90% 208 10 1 Other250.0791.1470.1470.96594%6%35 Value330.0500.91457%43%23 Passive/ Enhanced 38 39 -0.013 -0.026 0.948 1.036 -0.017 -0.041 0.955 0.959 9% 10% 91% 90% 340 613 Total38%62%3027

Doubling trades h 0 = S 0 – C 0 h 0 : Initial highwater mark S 0 : Initial stock position C 0 : Cost basis of initial position

Doubling trades h 0 = S 0 – C 0 S 1 = d S 0 C 1 = (1+r f ) C 0 Bad news!

Doubling trades h 0 = S 0 – C 0 S 1 = d S 0 + Δ 1 C 1 = (1+r f ) C 0 + Δ 1 Increase the equity position to cover the loss!

Doubling trades h 0 = S 0 – C 0 h 1 = u S 1 – (1+r f ) C 1 S 1 = d S 0 + Δ 1 C 1 = (1+r f ) C 0 + Δ 1 Good news! Δ 1 is set to make up for past losses and re-establish security position

Doubling trades h 0 = S 0 – C 0 h 1 = u S 1 – (1+r f ) C 1 S 1 = d S 0 + Δ 1 C 1 = (1+r f ) C 0 + Δ 1 Good news! Δ 1 is set to make up for past losses and re-establish security position Δ 1 = + S 0 h 0 - u d S 0 + (1+r f ) 2 C 0 u – (1+r f )

Doubling trades h 0 = S 0 – C 0 S 1 = d S 0 + Δ 1 C 1 = (1+r f ) C 0 + Δ 1 S 2 = d S 1 C 2 = (1+r f ) C 1 Bad news again!

Doubling trades h 0 = S 0 – C 0 h 2 = u S 2 – (1+r f ) C 2 S 1 = d S 0 + Δ 1 C 1 = (1+r f ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+r f ) C 1 + Δ 2 Good news finally!

Doubling trades h 0 = S 0 – C 0 h 2 = u S 2 – (1+r f ) C 2 S 1 = d S 0 + Δ 1 C 1 = (1+r f ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+r f ) C 1 + Δ 2 Good news finally! Δ 2 is set to make up for past losses and re-establish security position Δ 2 = + S 0 h 1 - u d S 1 + (1+r f ) 2 C 1 u – (1+r f )

Doubling trades h 0 = S 0 – C 0 S 1 = d S 0 + Δ 1 C 1 = (1+r f ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+r f ) C 1 + Δ 2 S 3 = d S 2 C 3 = (1+r f ) C 2 Bad news again!

Doubling trades h 0 = S 0 – C 0 S 1 = d S 0 + 1 C 1 = (1+r f ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+r f ) C 1 + Δ 2 Bad news again!

Observable implication of doubling Δ i = a + b 1 (1 - δ i ) h i-1 + b 2 V i + b 3 B i + b 4 δ i + b 5 G i On a loss, trader will increase position size by otherwise, position is liquidated on a gain, for all trades Δ i = + S 0 h i-1 - u d S i-1 + (1+r f ) 2 C i-1 u – (1+r f )

Observable implication of doubling Δ i = + S 0 h i-1 - u d S i-1 + (1+r f ) 2 C i-1 u – (1+r f ) Δ i = a + b 1 (1 - δ i ) h i-1 + b 2 V i + b 3 B i + b 4 δ i + b 5 G i < 0 > 0 ? On a loss, trader will increase position size by otherwise, position is liquidated on a gain,

Some successful Australian funds Fund Sharpe RatioAlpha FF AlphaBeta Skewnes s Kurtosi s Annual turnover 10.10170.08%0.10%0.90-0.52094.687820.69 (2.21)(2.58) 20.15000.16%0.17%1.110.08344.27770.79 (6.44)(5.88) 30.15590.19%0.20%1.080.73827.65401.18 (4.09)(4.36) 150.10790.09% 0.96-0.25584.17490.34 (2.66)(2.61) 260.09770.12%0.11%1.03-0.26673.43161.27 (2.42)(2.25) 350.18140.29%0.31%0.90-0.62485.12780.62 (3.02)(3.06)

Some successful Australian funds Fund Highwater mark on a loss Value of holdings on a loss Cost basis on a loss Value above highwater markRsq Gain from long buy short sell (one month) 10.0004-0.03730.056-0.0180.067-0.58% (0.24)(-2.82)(3.74)(-1.04) 20.0167-0.16730.014-0.8810.4210.50% (1.56)(-7.69)(1.19)(-11.55) 3-0.0023-0.1704-0.005-0.9820.642-0.27% (-0.19)(-8.22)(-0.39)(-39.16) 151.1659-0.91630.080-0.1700.185-1.30% (1.17)(-2.16)(0.57)(-0.22) 26-0.3633-0.1626-0.253-1.1330.4484.49% (-3.57)(-1.83)(-3.79)(-2.00) 35-0.0184-0.1297-0.081-1.0100.4202.63% (-0.45)(-3.30)(-1.80)(-2.48)

Sharpe ratio and doubling

Sector Patterns High Water Mark on a loss Mining and mineralsIndustrial Services Health and Biotechnology Gain above high water mark Value of Holdings on Loss Cost Basis on Loss Value of Holdings on Loss Cost Basis on Loss Value of Holdings on Loss Cost Basis on Loss Value of Holdings on Loss Cost Basis on Loss GARP 0.010-0.0290.009-0.0440.003-0.021-0.004-0.0640.031-0.791 (2.97)(-2.56)(1.71)(-3.62)(0.55)(-1.88)(-0.41)(-4.50)(2.22)(-5.14) Largest 0.012-0.0270.013-0.0390.006-0.015-0.004-0.0650.039-0.764 (3.20)(-3.07)(2.19)(-4.48)(1.14)(-2.19)(-0.62)(-5.66)(3.37)(-4.76) Domesti c 0.022-0.0440.022-0.0560.016-0.0270.000-0.0780.047-0.898 (3.18)(-4.23)(2.58)(-5.45)(2.07)(-3.18)(0.00)(-6.40)(3.74)(-11.99)

Seasonal patterns High Water Mark on a loss February - AprilMay - July August – October November – January Gain above high water mark Value of Holding s on Loss Cost Basis on Lo ss Value of Holding s on Loss Cost Basis on Lo ss Value of Holding s on Loss Cost Basis on Lo ss Value of Holding s on Loss Cost Basis on Lo ss GARP 0.009-0.0210.008-0.0400.010-0.018-0.003-0.0250.005-0.791 (2.77)(-1.34)(0.57)(-3.53)(1.19)(-1.64)(-0.51)(-2.43)(1.04)(-5.14) Largest 0.012-0.0230.002-0.0300.014-0.0180.000-0.0120.001-0.764 (2.98)(-2.38)(0.24)(-3.68)(1.65)(-2.63)(0.07)(-1.43)(0.21)(-4.76) Domest ic 0.021-0.0370.008-0.0440.022-0.0310.007-0.0260.009-0.897 (3.05)(-3.39)(0.74)(-4.22)(2.01)(-3.62)(0.96)(-2.46)(1.05) (- 11. 94 )

Return to long buy/short sell (monthly) CategoryRaw return Market Adjusted GARP0.28%0.33% (0.81)(0.91) Growth-0.07%-0.05% (-0.11)(-0.07) Neutral1.46%0.83% (1.84)(1.53) Other2.40%2.48% (1.99)(2.11) Value1.11%0.92% (2.06)(1.64) Passive/ Enhanced Passive -1.31%-0.80% (-2.40)(-3.07)

Return to long buy/short sell (monthly) Category Raw return Market Adjuste d Largest 10 Institutional Manager No0.86%0.65% (2.49)(1.99) Yes0.11%0.40% (0.33)(1.41) Boutique firm No0.42%0.40% (1.27)(1.35) Yes1.07%0.88% (2.24)(1.95)

National Australia Bank

A clear and present danger?  No evidence of informationless investing at fund level  Behavioral theories  Prospect theory  Lock in gains, gambling on losses  Narrow Framing  Consider only one gamble at a time  Window dressing  Doubling at end of fiscal year

Conclusion  Behavioral patterns of trading are common  Concave trading patterns create adverse incentives  Narrow framing limits negative consequences

Double or nothing: Patterns of equity fund holdings and transactions Stephen J. Brown NYU Stern School of Business David R. Gallagher University of NSW.

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Double or nothing: Patterns of equity fund holdings and transactions Stephen J. Brown NYU Stern School of Business David R. Gallagher University of NSW.

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