Double or nothing Patterns of equity fund holdings
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 Market Short-term Investment (S&P 500) Government Managemen Benchmark t Average. 0065 Return . 0050 . 0036 Std. . 0106 Deviation . 0359 . 0015 1. 0 Beta. 0640 Alpha. 0025. 0. 0 (1. 92) 100% in cash. 0 at close of Sharpe. Style: Ratio Index. 2484 Arbitrage, . 0318 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-rf = α + β (RS&P- rf) + γ (RS&Pr f) 2
Hedge funds follow concave strategies R-rf = α + β (RS&P- rf) + γ (RS&Pr f) 2 Concave strategies: tβ > 1. 96 & tγ < 1. 96
Hedge funds follow concave strategies R-rf = α + β (RS&P- rf) + γ (RS&PConcave Neutral Convex N r f) 2 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 Source: TASS/Tremont 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 1099 429 60
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 Sharpe Fund Ratio Alpha FF Alpha Beta Skewnes s Kurtosi Annual s turnover 1 0. 1017 0. 08% (2. 21) 0. 10% (2. 58) 0. 90 -0. 5209 4. 6878 20. 69 2 0. 1500 0. 16% (6. 44) 0. 17% (5. 88) 1. 11 0. 0834 4. 2777 0. 79 3 0. 1559 0. 19% (4. 09) 0. 20% (4. 36) 1. 08 0. 7382 7. 6540 1. 18 15 0. 1079 0. 09% (2. 66) 0. 09% (2. 61) 0. 96 -0. 2558 4. 1749 0. 34 26 0. 0977 0. 12% (2. 42) 0. 11% (2. 25) 1. 03 -0. 2667 3. 4316 1. 27 35 0. 1814 0. 29% (3. 02) 0. 31% (3. 06) 0. 90 -0. 6248 5. 1278 0. 62
Style and return patterns Treynor Mazuy measure Modified Henriksson Merton measure Number of observations Category Beta 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. 0146 0. 0069 (1. 50) 0. 0457 (1. 46) 859
Size and return patterns Largest 10 Institutional Manager Boutique firm Treynor Mazuy measure Modified Henriksson Merton measure Number of observations Category Beta No 0. 9644 -0. 0058 (-2. 12) -0. 04580 (-2. 25) 6467 Yes 0. 9059 -0. 0100 (-2. 25) -0. 0779 (-2. 56) 2866 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 Calls Puts Month end option positions Concavity decreasing Fund Number Strike 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. 957 0. 904 0. 952 0. 944 0. 985 0. 950 - 0. 674 - 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% Neutral 21 22 24 -0. 093 0. 567 0. 405 1. 038 0. 984 0. 854 -0. 093 - 0. 947 - Other 25 0. 079 1. 147 0. 965 Value 33 0. 050 0. 914 Passive/ Enhanced 38 39 -0. 013 -0. 026 0. 948 1. 036 GARP Growth -0. 017 -0. 041 100% 29% 59% 77% Concavity increasing 71% 41% 23% 100% Total 80 246 79 898 18 11 73% 100% 65% 87% 11 8 83 344 10% 100% 90% 208 10 1 94% 6% 35 57% 43% 23 0. 955 0. 959 9% 10% 91% 90% 340 613 Total 38% 62% 3027 35% 13%
Patterns of derivative holdings Fund Investmen t Style Calls Puts Month end option positions Concavity decreasing Fund Number Strike 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. 957 0. 904 0. 952 0. 944 0. 985 0. 950 - 0. 674 - 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% Neutral 21 22 24 -0. 093 0. 567 0. 405 1. 038 0. 984 0. 854 -0. 093 - 0. 947 - Other 25 0. 079 1. 147 0. 965 Value 33 0. 050 0. 914 Passive/ Enhanced 38 39 -0. 013 -0. 026 0. 948 1. 036 GARP Growth -0. 017 -0. 041 100% 29% 59% 77% Concavity increasing 71% 41% 23% 100% Total 80 246 79 898 18 11 73% 100% 65% 87% 11 8 83 344 10% 100% 90% 208 10 1 94% 6% 35 57% 43% 23 0. 955 0. 959 9% 10% 91% 90% 340 613 Total 38% 62% 3027 35% 13%
Patterns of derivative holdings Fund Investmen t Style Calls Puts Month end option positions Concavity decreasing Fund Number Strike 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. 957 0. 904 0. 952 0. 944 0. 985 0. 950 - 0. 674 - 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% Neutral 21 22 24 -0. 093 0. 567 0. 405 1. 038 0. 984 0. 854 -0. 093 - 0. 947 - Other 25 0. 079 1. 147 0. 965 Value 33 0. 050 0. 914 Passive/ Enhanced 38 39 -0. 013 -0. 026 0. 948 1. 036 GARP Growth -0. 017 -0. 041 100% 29% 59% 77% Concavity increasing 71% 41% 23% 100% Total 80 246 79 898 18 11 73% 100% 65% 87% 11 8 83 344 10% 100% 90% 208 10 1 94% 6% 35 57% 43% 23 0. 955 0. 959 9% 10% 91% 90% 340 613 Total 38% 62% 3027 35% 13%
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+rf ) C 0 Bad news!
Doubling trades h 0 = S 0 – C 0 S 1 C 1 Increase the = d S 0 + Δ 1 equity position = (1+rf ) C 0 + Δ 1 to cover the loss!
Doubling trades h 0 = S 0 – C 0 h 1 = u S 1 – (1+rf) C 1 S 1 = d S 0 + Δ 1 C 1 = (1+rf ) 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+rf) C 1 Good news! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 is set to make up for past losses and re-establish security position Δ 1 = h 0 - u d S 0 + (1+rf)2 C 0 u – (1+rf) + S 0
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 C 2 = (1+rf ) C 1
Doubling trades h 0 = S 0 – C 0 h 2 = u S 2 – (1+rf) C S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 Good news finally!
Doubling trades h 0 = S 0 – C 0 h 2 = u S 2 – (1+rf) C S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 is set to make up for past losses Good news and re-establish security position finally! Δ 2 = h 1 - u d S 1+ (1+rf)2 C 1 u – (1+rf) + S 0
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 S 3 = d S 2 C 3 = (1+rf ) C 2
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 S 3 = d S 2 C 3 = (1+rf ) C 2
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 S 3 = C 3 = d S 2 (1+r f ) C 2
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 S C 3 =d (1 S 2 +r f )C 2
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 SC 3 =3 =d rf + (S 1 2 )C 2
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 SC 3 =3 =d rf + (S 1 2 )C
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 SC 3 =3 =d +r (S 1 2
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + Δ 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2 SC 3 =3 =d (
Doubling trades h 0 = S 0 – C 0 Bad news again! S 1 = d S 0 + 1 C 1 = (1+rf ) C 0 + Δ 1 S 2 = d S 1 + Δ 2 C 2 = (1+rf ) C 1 + Δ 2
Observable implication of doubling On a loss, trader will increase position size by hi-1 - u d Si-1+ (1+rf)2 Ci-1 Δi = u – (1+rf) + S 0 otherwise, position is liquidated on a gain, Δi = a + b 1 (1 - δi) hi-1 + b 2 Vi + b 3 Bi + b 4 δi + b 5 Gi for all trades
Observable implication of doubling On a loss, trader will increase position size by hi-1 - u d Si-1+ (1+rf)2 Ci-1 Δi = + S 0 u – (1+rf) otherwise, position is liquidated on a gain, Δi = a + b 1 (1 - δi) hi-1 + b 2 Vi + b 3 Bi + b 4 δi + b 5 Gi >0 >0 <0 >0 ? <0
Some successful Australian funds Sharpe Fund Ratio Alpha FF Alpha Beta Skewnes s Kurtosi Annual s turnover 1 0. 1017 0. 08% (2. 21) 0. 10% (2. 58) 0. 90 -0. 5209 4. 6878 20. 69 2 0. 1500 0. 16% (6. 44) 0. 17% (5. 88) 1. 11 0. 0834 4. 2777 0. 79 3 0. 1559 0. 19% (4. 09) 0. 20% (4. 36) 1. 08 0. 7382 7. 6540 1. 18 15 0. 1079 0. 09% (2. 66) 0. 09% (2. 61) 0. 96 -0. 2558 4. 1749 0. 34 26 0. 0977 0. 12% (2. 42) 0. 11% (2. 25) 1. 03 -0. 2667 3. 4316 1. 27 35 0. 1814 0. 29% (3. 02) 0. 31% (3. 06) 0. 90 -0. 6248 5. 1278 0. 62
Some successful Australian funds Highwater mark on a loss Value of holdings on a loss Cost basis on a loss Value above highwater mark 1 0. 0004 (0. 24) -0. 0373 (-2. 82) 0. 056 (3. 74) 2 0. 0167 (1. 56) -0. 1673 (-7. 69) 3 -0. 0023 (-0. 19) 15 Rsq Gain from long buy short sell (one month) -0. 018 (-1. 04) 0. 067 -0. 58% 0. 014 (1. 19) -0. 881 (-11. 55) 0. 421 0. 50% -0. 1704 (-8. 22) -0. 005 (-0. 39) -0. 982 (-39. 16) 0. 642 -0. 27% 1. 1659 (1. 17) -0. 9163 (-2. 16) 0. 080 (0. 57) -0. 170 (-0. 22) 0. 185 -1. 30% 26 -0. 3633 (-3. 57) -0. 1626 (-1. 83) -0. 253 (-3. 79) -1. 133 (-2. 00) 0. 448 4. 49% 35 -0. 0184 (-0. 45) -0. 1297 (-3. 30) -0. 081 (-1. 80) -1. 010 (-2. 48) 0. 420 2. 63% Fund
Sharpe ratio and doubling
Sector Patterns Mining and minerals Value of Holdings on Loss Cost Basis on Loss Gain above high water mark -0. 021 -0. 004 -0. 064 0. 031 -0. 791 (-3. 62) (0. 55) (-1. 88) (-0. 41) (-4. 50) (2. 22) (-5. 14) 0. 013 -0. 039 0. 006 -0. 015 -0. 004 -0. 065 0. 039 -0. 764 (-3. 07) (2. 19) (-4. 48) (1. 14) (-2. 19) (-0. 62) (-5. 66) (3. 37) (-4. 76) Domestic 0. 022 -0. 044 0. 022 -0. 056 0. 016 0. 000 -0. 078 0. 047 -0. 898 (3. 18) (-4. 23) (2. 58) (-5. 45) (2. 07) (-3. 18) (0. 00) (-6. 40) (3. 74) (-11. 99) GARP Largest High Water Mark on a loss Value of Holdings on Loss 0. 010 Health and Biotechnology Industrial Services Cost Value of Basis Holdings on Loss Cost Value of Cost Basis Holdings Basis on Loss -0. 029 0. 009 -0. 044 0. 003 (2. 97) (-2. 56) (1. 71) 0. 012 -0. 027 (3. 20) -0. 027
Seasonal patterns February April GARP Largest Domesti c High Water Mark on a loss Value of Holding s on Loss Cost Basis on Lo ss 0. 009 -0. 021 (2. 77) May - July Value of Holding s on Loss Gain above high water mark -0. 003 -0. 025 0. 005 -0. 791 (-1. 64) (-0. 51) (-2. 43) (1. 04) (-5. 14) 0. 014 -0. 018 0. 000 -0. 012 0. 001 -0. 764 (-3. 68) (1. 65) (-2. 63) (0. 07) (-1. 43) (0. 21) (-4. 76) -0. 044 0. 022 -0. 031 0. 007 -0. 026 0. 009 -0. 897 Value of Holding s on Loss Cost Basis on Lo ss Value of Holding s on Loss 0. 008 -0. 040 0. 010 -0. 018 (-1. 34) (0. 57) (-3. 53) (1. 19) 0. 012 -0. 023 0. 002 -0. 030 (2. 98) (-2. 38) (0. 24) 0. 021 -0. 037 0. 008 (-3. 39) (0. 74) November – January Cost Basis on Lo ss (3. 05) August – October (-4. 22) (2. 01) (-3. 62) (0. 96) (-2. 46) (1. 05) (11. 94)
Return to long buy/short sell (monthly) Category GARP Growth Neutral Other Value Passive/ Enhanced Passive Raw return 0. 28% (0. 81) -0. 07% (-0. 11) 1. 46% (1. 84) 2. 40% (1. 99) 1. 11% (2. 06) -1. 31% (-2. 40) Market Adjusted 0. 33% (0. 91) -0. 05% (-0. 07) 0. 83% (1. 53) 2. 48% (2. 11) 0. 92% (1. 64) -0. 80% (-3. 07)
Return to long buy/short sell (monthly) Raw return Market Adjuste d No 0. 86% (2. 49) 0. 65% (1. 99) Yes 0. 11% (0. 33) 0. 40% (1. 41) No 0. 42% (1. 27) 0. 40% (1. 35) Yes 1. 07% (2. 24) 0. 88% (1. 95) Category Largest 10 Institutional Manager Boutique firm
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
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