double or nothing: patterns of equity fund holdings and transactions

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

www.stern.nyu.edu/~sbrown

Performance measurement

Leeson Investmen

tManagem

ent

Market (S&P 500)

Benchmar

k

Short-term

Government

Benchmark

Average Return

.0065 .0050 .0036

Std. Deviation

.0106 .0359 .0015

Beta .0640 1.0 .0

Alpha .0025(1.92)

.0 .0

Sharpe Ratio

.2484 .0318 .0

Style: Index Arbitrage, 100% in cash at close of trading

Frequency distribution of monthly returns

0

5

10

15

20

25

30

35

Percentage in cash (monthly)

0%

20%

40%

60%

80%

100%

120%

31-Dec-1989 15-May-1991 26-Sep-1992 8-Feb-1994

Examples of riskless index arbitrage …

Percentage in cash (daily)

-600%

-500%

-400%

-300%

-200%

-100%

0%

100%

200%

31-Dec-1989 15-May-1991 26-Sep-1992 8-Feb-1994

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

Sharpe ratio of doublers

-0.2

-0.1

0

0.1

0.2

0.3

0.4

All Doublers

Doublers who have notyet embezzled

Sharpe Ratio ofMarket

Informationless investing


Zero net investment overlay strategy (Weisman 2002)

Uses only public informationDesigned to yield Sharpe ratio greater than

benchmark

Why should we care?

Sharpe ratio obviously inappropriate here


Zero net investment overlay strategy (Weisman 2002)

Uses only public informationDesigned to yield Sharpe ratio greater than benchmark

Why should we care?

Sharpe ratio obviously inappropriate hereBut is metric of choice of hedge funds and derivatives

traders

We should care!

Agency issuesFund flow, compensation based on

historical performanceGruber (1996), Sirri and Tufano

(1998), Del Guercio and Tkac (2002)

Behavioral issuesStrategy leads to certain ruin in the

long term

Examples of Informationless investing

Doubling a.k.a. “Convergence trading”

Covered call writing

Unhedged short volatilityWriting 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

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

Sharpe ratio = .631

Maximum Sharpe Ratio

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

MaximumSharpe RatioStrategy

Sharpe ratio = .748

Short Volatility Strategy

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

Shortvolatility

Sharpe ratio = .743

Doubling

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

Doubling(upper 5%)

Doubling(median)

Doubling(lower 5%)

Sharpe ratio = .046

Doubling (no embezzlement)

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

Doubling(upper 5%)

Doubling(median)

Doubling(lower 5%)

Sharpe ratio = 1.962

Concave trading strategies

-200%

-150%

-100%

-50%

0%

50%

100%

-50% 0% 50% 100%

Benchmark

Doubling(median)

MaximumSharpe RatioStrategy

Hedge funds follow concave strategies

R-rf = α + β (RS&P- rf) + γ (RS&P- rf)2

Concave strategies: tβ > 1.96 & tγ < -1.96

Hedge funds follow concave strategies

ConcaveNeutra

lConve

x N

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%0.00%0.86%1.76%0.34%2.38%1.57%4.02%2.18%3.03%3.33%

13027233170296126574174

109942960

Grand Total 11.54% 86.53% 1.93% 3318

R-rf = α + β (RS&P- rf) + γ (RS&P- rf)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

Ratio AlphaFF

AlphaBet

aSkewne

ssKurtos

is

Annual turnov

er

1 0.1017 0.08% 0.10% 0.90 -0.5209 4.6878 20.69

(2.21) (2.58)

2 0.1500 0.16% 0.17% 1.11 0.0834 4.2777 0.79

(6.44) (5.88)

3 0.1559 0.19% 0.20% 1.08 0.7382 7.6540 1.18

(4.09) (4.36)

16 0.1079 0.09% 0.09% 0.96 -0.2558 4.1749 0.34

(2.66) (2.61)

27 0.0977 0.12% 0.11% 1.03 -0.2667 3.4316 1.27

(2.42) (2.25)

36 0.1814 0.29% 0.31% 0.90 -0.6248 5.1278 0.62

(3.02) (3.06)

Style and return patterns

Category Beta

Treynor Mazuy

measure

Modified Henriksson

Merton measure

Number of observatio

ns

GARP

0.96347

-0.01105(-2.30)

-0.08989(-2.52)

2395

Growth

1.03670

-0.00708(-1.53)

-0.03762(-1.15)

1899

Neutral

1.02830

-0.00110(-0.29)

-0.02092(-0.71)

1313

Other

1.00670

-0.00196(-0.53)

0.00676(0.21)

640

Value

0.76691

-0.01215(-1.93)

-0.10350(-2.24)

2250

Passive/Enhanced

1.01440

0.00692(1.51)

0.04593(1.47)

859

Size and return patterns

Category Beta

Treynor Mazuy

measure

Modified Henriksson Merton measure

Number of observation

s

Largest 10 Institution

al Manager

No

0.9627

-0.00645(-2.25)

-0.05037(-2.34)

6100

Yes

0.8819

-0.01306(-2.60)

-0.10095(-2.92)

2397

Boutique firm

No

0.9322

-0.01029(-3.12)

-0.07616(-3.23)

5709

Yes

0.9556

-0.00452(-1.25)

-0.04184(-1.49)

2788

Incentives and return patterns

Category Beta

Treynor Mazuy

measure

Modified Henriksson

Merton measure

Number of

observations

Annual Bonus

No

0.9819

0.00013(0.03)

0.01233(0.35)

308

Yes

0.9386

-0.00857(-3.32)

-0.06720(-3.56)

8189

Domestic owned

No

0.9739

-0.00990(-2.80)

-0.07282(-2.79)

4262

Yes

0.9053

-0.00652(-1.86)

-0.05557(-2.18)

4235

Equity Ownershi

p by senior staff

No

0.9322

-0.01029(-3.12)

-0.07616(-3.23)

5709

Yes

0.9556

-0.00452(-1.25)

-0.04184(-1.49)

2788

Patterns of derivative holdings

Fund Investm

ent Style

Calls Puts Month end option positions

Fund

Number Strike Number Strike

Concavity decreasing

Concavity increasing Total

GARP 123456

1113

0.726-0.0610.0990.041-0.6500.2220.8110.054

1.0171.0501.0171.0231.0621.0760.0021.076

0.395-0.1220.0210.008-1.346

0.950

0.9570.9040.9520.9440.985

0.674

100%29%59%77%

100%100%100%

71%41%23%

100%

8024679

89818118

11

Growth 15161718

-0.033-0.039-0.367-0.059

1.0561.0601.0671.023

0.1070.108

0.9510.913

27%

35%13%

73%100%65%87%

118

83344

Neutral 212224

-0.0930.5670.405

1.0380.9840.854

-0.093 0.947 10%100%100%

90%

208101

Other 25 0.079 1.147 0.147 0.965 94% 6% 35

Value 33 0.050 0.914 57% 43% 23

Passive/ Enhanced

3839

-0.013-0.026

0.9481.036

-0.017-0.041

0.9550.959

9%10%

91%90%

340613

Total 38% 62% 3027

Doubling trades

h0 = S0 – C0

h0 : Initial highwater mark

S0 : Initial stock position

C0 : Cost basis of initial position

Doubling trades

h0 = S0 – C0

S1 = d S0

C1 = (1+rf ) C0

Bad news!

Doubling trades

h0 = S0 – C0

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

Increase the equity position to cover the loss!

Doubling trades

h0 = S0 – C0 h1 = u S1 – (1+rf) C1

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

Good news!

1 is set to make up for past losses and re-establish security position

Doubling trades

h0 = S0 – C0 h1 = u S1 – (1+rf) C1

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

Good news!


1 = + S0 h0 - u d S0 + (1+rf)2 C0

u – (1+rf)

Doubling trades

h0 = S0 – C0

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1

C2 = (1+rf ) C1

Bad news again!

Doubling trades

h0 = S0 – C0 h2 = u S2 – (1+rf) C2

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1 + 2

C2 = (1+rf ) C1 + 2

Good news finally!

Doubling trades

h0 = S0 – C0 h2 = u S2 – (1+rf) C2

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1 + 2

C2 = (1+rf ) C1 + 2

Good news finally!


2 = + S0 h1 - u d S1+ (1+rf)2 C1

u – (1+rf)

Doubling trades

h0 = S0 – C0

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1 + 2

C2 = (1+rf ) C1 + 2

S3 = d S2

C3 = (1+rf ) C2

Bad news again!

Doubling trades

h0 = S0 – C0

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1 + 2

C2 = (1+rf ) C1 + 2

S3 = d S

2C

3 = (1+rf ) C

2

Bad news again!

Doubling trades

h0 = S0 – C0

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1 + 2

C2 = (1+rf ) C1 + 2S

3 = d S2

C3 = (1+r

f ) C2

Bad news again!

Doubling trades

h0 = S0 – C0

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1 + 2

C2 = (1+rf ) C1 + 2S3 =

d S

2

C3 =

(1+r

f ) C2

Bad news again!

Doubling trades

h0 = S0 – C0

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1 + 2

C2 = (1+rf ) C1 + 2

S3 =

d S

2

C3 =

(1+r

f ) C2

Bad news again!

Doubling trades

h0 = S0 – C0

S1 = d S0 + 1

C1 = (1+rf ) C0 + 1

S2 = d S1 + 2

C2 = (1+rf ) C1 + 2

Bad news again!

Observable implication of doubling

i = a + b1 (1 - i) hi-1 + b2 Vi + b3 Bi + b4 i + b5 Gi

On a loss, trader will increase position size by

otherwise, position is liquidated on a gain,

for all trades

i = + S0 hi-1 - u d Si-1+ (1+rf)2 Ci-1

u – (1+rf)


i = + S0 hi-1 - u d Si-1+ (1+rf)2 Ci-1

u – (1+rf)


Vi = (1 - i) d Si-1 , the value of security on a loss




i = + S0 hi-1 - u d Si-1+ (1+rf)2 Ci-1

u – (1+rf)


Bi = (1 - i) (1 + rf ) Ci-1 , the cost basis of the security




i = + S0 hi-1 - u d Si-1+ (1+rf)2 Ci-1

u – (1+rf)


Gi = I (Si – Ci – hi) , the measure of gain once highwatermark is reached




i = + S0 hi-1 - u d Si-1+ (1+rf)2 Ci-1

u – (1+rf)


< 0< 0 > 0> 0> 0 ?




Fund

Sharpe

Ratio AlphaFF

AlphaBet

aSkewne

ssKurtos

is

Annual turnov

er

1 0.1017 0.08% 0.10% 0.90 -0.5209 4.6878 20.69

(2.21) (2.58)

2 0.1500 0.16% 0.17% 1.11 0.0834 4.2777 0.79

(6.44) (5.88)

3 0.1559 0.19% 0.20% 1.08 0.7382 7.6540 1.18

(4.09) (4.36)

16 0.1079 0.09% 0.09% 0.96 -0.2558 4.1749 0.34

(2.66) (2.61)

27 0.0977 0.12% 0.11% 1.03 -0.2667 3.4316 1.27

(2.42) (2.25)

36 0.1814 0.29% 0.31% 0.90 -0.6248 5.1278 0.62

(3.02) (3.06)


Fund

Highwater mark on a

loss

Value of holdings on a loss

Cost basis on

a loss

Value above

highwater mark Rsq

1 0.0004 -0.0266 0.0327 -0.0119 0.0442

(0.21) (-2.39) (2.19) (-0.86)

2 0.0346 -0.1301 0.0300 -0.8614 0.3924

(2.97) (-6.45) (2.29) (-9.52)

3 0.0366 -0.1125 0.0216 -0.9771 0.6098

(2.58) (-6.02) (1.57) (-33.69)

16 0.6981 -0.9135 0.0167 -0.6133 0.1406

(0.69) (-2.06) (0.13) (-0.91)

27 -0.0712 -0.3305 -0.1205 -1.3277 0.3930

(-0.71) (-4.18) (-2.02) (-2.32)

36 -0.0226 -0.0973 -0.0935 -1.0166 0.3947

(-0.55) (-2.38) (-2.08) (-2.52)

Sharpe ratio and doubling

0

0.05

0.1

0.15

0.2

0.25

-8 -6 -4 -2 0 2 4 6

t-value of value on a loss

Sha

rpe

ratio

of w

eekly

retu

rns

Do managers lack an equity stake?

0

0.05

0.1

0.15

0.2

0.25

-8 -6 -4 -2 0 2 4 6


Sha

rpe

ratio

of w

eekly

retu

rns

Is fund owned by a bank or life insurance company?

0

0.05

0.1

0.15

0.2

0.25

-8 -6 -4 -2 0 2 4 6


Sha

rpe

ratio

of w

eekly

retu

rns

Is fund one of 10 largest in Australia?

0

0.05

0.1

0.15

0.2

0.25

-8 -6 -4 -2 0 2 4 6


Sha

rpe

ratio

of w

eekly

retu

rns

Is fund large (not a boutique manager)?

0

0.05

0.1

0.15

0.2

0.25

-8 -6 -4 -2 0 2 4 6


Sha

rpe

ratio

of w

eekly

retu

rns

Style and return patterns

Category Beta

Treynor Mazuy

measure

Modified Henriksson

Merton measure

Number of observatio

ns

GARP

0.96347

-0.01105(-2.30)

-0.08989(-2.52)

2395

Growth

1.03670

-0.00708(-1.53)

-0.03762(-1.15)

1899

Neutral

1.02830

-0.00110(-0.29)

-0.02092(-0.71)

1313

Other

1.00670

-0.00196(-0.53)

0.00676(0.21)

640

Value

0.76691

-0.01215(-1.93)

-0.10350(-2.24)

2250

Passive/Enhanced

1.01440

0.00692(1.51)

0.04593(1.47)

859

Style and trading patterns

Category

Highwater

mark on a loss


Cost basis on

a loss

Value above

highwater mark Rsq

GARP 0.0086 -0.0584 0.0028 -0.7957 0.4281

(2.45) (-7.93) (0.58) (-5.30)

Growth 0.0352 0.0291 -0.0498 -0.3429 0.1339

(1.04) (0.99) (-1.66) (-0.92)

Neutral 0.0005 -0.0208 0.0035 -0.2161 0.0341

(0.07) (-1.89) (0.35) (-3.69)

Other 0.0277 -0.0242 -0.0074 -0.0712 0.0586

(1.84) (-1.75) (-0.60) (-0.60)

Value -0.0006 0.0081 -0.0104 -0.1172 0.0113

(-0.07) (0.88) (-1.28) (-1.85)

Passive/ Enhanced

0.0901 -0.0769 0.0535 -0.2307 0.0089

(2.06) (-1.54) (1.61) (-0.98)

Size and return patterns

Category Beta

Treynor Mazuy

measure

Modified Henriksson Merton measure

Number of observation

s


al Manager

No

0.9627

-0.00645(-2.25)

-0.05037(-2.34)

6100

Yes

0.8819

-0.01306(-2.60)

-0.10095(-2.92)

2397

Boutique firm

No

0.9322

-0.01029(-3.12)

-0.07616(-3.23)

5709

Yes

0.9556

-0.00452(-1.25)

-0.04184(-1.49)

2788

Size and trading patterns

Category

Highwater

mark on a loss


Cost Basis

Value above

highwatermark Rsq


al Manager

No 0.0384 0.0250 -0.0443 -0.4393 0.0630

(1.36) (0.92) (-1.62) (-1.26)

Yes 0.0077 -0.0159 0.0011 -0.7627 0.3017

(2.05) (-3.01) (0.24) (-4.82)

Boutique firm

No 0.0015 -0.0040 -0.0093 -0.7502 0.1607

(0.24) (-0.44) (-1.03) (-4.75)

Yes 0.0097 -0.0270 -0.0184 -0.2847 0.0751

(0.66) (-1.42) (-1.07) (-4.23)


Category Beta

Treynor Mazuy

measure

Modified Henriksson

Merton measure

Number of

observations

Annual Bonus

No

0.9819

0.00013(0.03)

0.01233(0.35)

308

Yes

0.9386

-0.00857(-3.32)

-0.06720(-3.56)

8189

Domestic owned

No

0.9739

-0.00990(-2.80)

-0.07282(-2.79)

4262

Yes

0.9053

-0.00652(-1.86)

-0.05557(-2.18)

4235

Equity Ownershi

p by senior staff

No

0.9322

-0.01029(-3.12)

-0.07616(-3.23)

5709

Yes

0.9556

-0.00452(-1.25)

-0.04184(-1.49)

2788


Category

Highwater

mark on a loss

Value of

holdings on a loss

Cost Basis

Value above

highwater Rsq

Annual Bonus

No 0.0259 -0.0233 -0.0026 0.0388 0.0420

(1.52) (-1.55) (-0.20) (0.25)

Yes 0.0016 -0.0040 -0.0093 -0.7493 0.1601

(0.25) (-0.45) (-1.04) (-4.74)

Domestic owned

No 0.0025 0.0265 -0.0395 -0.0756 0.1229

(0.48) (1.24) (-1.57) (-0.95)

Yes 0.0148 -0.0228 0.0069 -0.9023 0.2063

(2.21) (-2.79) (0.99) (-12.68)

Equity Ownershi

p by senior staff

No 0.0015 -0.0040 -0.0093 -0.7502 0.1607

(0.24) (-0.44) (-1.03) (-4.75)

Yes 0.0097 -0.0270 -0.0184 -0.2847 0.0751

(0.66) (-1.42) (-1.07) (-4.23)

National Australia Bank

Incentives are not everything!

No evidence of doubling in asset allocation

Large institutional funds are organized and compensated on a specialist team basis

Behavioral explanations:Prospect theoryNarrow framing

Conclusion

Informationless investing can be dangerous to your financial health

Funds as a whole do not seem to use these techniques

However, some of most successful funds have interesting trading patterns … associated with

Large, decentralized controlShort term incentive compensation

double or nothing: patterns of equity fund holdings and transactions

Documents

sharpe ratio greater

doublingsharpe ratio

embezzlementsharpe ratio

concave strategiesrrf

rsp rf rsp rf2 source

swan university of

public informationdesigned

inappropriate herebut