Empirical Financial Economics

Ex post conditioning issues

Fama Fisher Jensen and Roll

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FFJR Redux

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FFJR Redux

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Overview

A simple example

Brief review of ex post conditioning issues

Implications for tests of Efficient Markets Hypothesis

Performance measurementLeeson InvestmentManagement

Market (S&P 500) Benchmark

Short-term Government Benchmark

Average Return

.0065 .0050 .0036

Std. Deviation

.0106 .0359 .0015

Beta .0640 1.0 .0Alpha .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

05

101520253035

-1.00%-0.

50%0.0

0%0.5

0%1.0

0%1.5

0%2.0

0%2.5

0%3.0

0%3.5

0%4.0

0%4.5

0%5.0

0%5.5

0%6.0

0%6.5

0%

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

$0$1

$-1 p = 12

Is doubling low risk?

$0$1

$-3 p = 14

Is doubling low risk?

$0$1

$-7 p = 18

Is doubling low risk?

$0$1

$-15 p = 116

Is doubling low risk?

$0$1

$-31 p = 132

Is doubling low risk?

$0$1

$-63 p = 164

Is doubling low risk?

$0$1

$-127 p = 1128

Is doubling low risk?

Is doubling low risk?

Only two possible outcomes

Will win game if play “long enough”

Bad outcome event extremely unlikely

Sharpe ratio infinite for managers who survive periodic audit

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

The case of the Repeated Doubler

Bernoulli game:Leave game on a winMust win if play long enough

Repeated doublerReestablish position on a winMust lose if play long enough

The challenge of risk management

Performance and risk inferred from logarithm of fund value:

dp dt dz

The challenge of risk management

Performance and risk inferred from logarithm of fund value:

is expected return of manager

Lower bound on with probability is

Value at Risk (VaR)

dp dt dz

[0, ]T

The challenge of risk management

Performance and risk inferred from logarithm of fund value:

But what the manager observes is

A = {set of price paths where doubler has not embezzled}

dp dt dz

* |p p A

The challenge of risk management

Performance and risk inferred from logarithm of fund value:

But what the manager observes is

A = {set of price paths where doubler has not embezzled}

dp dt dz

* |p p A

yet

National Australia Bank

Ex post conditioning

Ex post conditioning leads to problemsWhen inclusion in sample

depends on price pathExamples

Equity premium puzzleVariance ratio analysisPerformance measurementPost earnings driftEvent studies“Anomalies”

Effect of conditioning on observed value paths

The logarithm of value follows a simple absolute diffusion on

dp dt dz [0, ]T

Unconditional price paths

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-1

1

3

5

7

9

0 2 4 6 8 10

Years

Log

price

in u

nits

of a

nnua

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ndar

d de

viatio

n

Effect of conditioning on observed value paths

The logarithm of value follows a simple absolute diffusion on

What can we say about values we observe?

A = {set of price paths observed on }

dp dt dz

[0, ]T

[0, ]T

Absorbing barrier at zero

-5

-3

-1

1

3

5

7

9

0 2 4 6 8 10

Years

Log

price

in u

nits

of a

nnua

l sta

ndar

d de

viatio

n

Conditional price paths

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-1

1

3

5

7

9

0 2 4 6 8 10

Years

Log

price

in u

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of a

nnua

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Effect of conditioning on observed value paths

Define

Observed values follow an absolute diffusion on

( ) Pr[ | , ]t A p t

[0, ]T

* *dp dt dz

2* p

Stephen Brown, William Goetzmann and Stephen Ross “Survival” Journal of Finance 50 1995 853-873.

Example: Absorbing barrier at zero

2*

2 [ ] ,(2 [ ] 1)

p

w pwT t w T t

As T goes to infinity, conditional diffusion is2

*dp dt dzp p

Expected return is positive, increasing in volatility and decreasing in ex ante probability of failure

Expected value path

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-1

1

3

5

7

9

0 2 4 6 8 10

Years

Log

price

in u

nits

of a

nnua

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Emerging market price paths

0

0.5

1

1.5

2

0 10 20 30 40Years

Value

0 2p p 0

12

p p

Important result

Ex post conditioning a problem whenever inclusion in the sample depends on value path

Effect exacerbated by volatility

Induces a spurious correlation between return and correlates of volatility

2* p

Important result

Ex post conditioning a problem whenever inclusion in the sample depends on value path

Effect exacerbated by volatility

Induces a spurious correlation between return and correlates of volatility

A much misunderstood issue in empirical Finance!

2* p

Important result

Ex post conditioning a problem whenever inclusion in the sample depends on value path

Effect exacerbated by volatility

Induces a spurious correlation between return and correlates of volatility

A much misunderstood issue in empirical Finance!

2* p

Equity premium puzzle

With nonzero drift, as T goes to infinity

If true equity premium is zero, an observed equity premium of 6% ( ) implies 2/3 ex ante probability that the market will survive in the very long term given the current level of prices ( )

2 (1 ( )*( )

pp

4%fr

* 10%

( ) .66p

Unconditional price path

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0 2 4 6 8 10

Years

Log

price

in u

nits

of a

nnua

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pTp0

Conditional price paths

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-1

1

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Years

Log

price

in u

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of a

nnua

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d de

viatio

n pTp0

*

Properties of survivors

High returnLow riskApparent mean reversion:

Variance ratio =

21 4lim Var *2TT

pT

4 .429204....2

Variance of long holding period returns

00.005

0.010.0150.02

0.0250.03

0.0350.04

0.045

0.01 1 100 10000Holding period (years)

Annu

alize

d va

rianc

e

2 σ cutoff σ/2 cutoff σ² (4-Π) / 2

0.0172

‘Hot Hands’ in mutual funds

Growth fund performance relative to alpha of median manager 1984-1987

1986-87 winners

1986-87 losers Totals

1984-85 winners 58 33 91

1986-87 losers 33 57 90

Totals 91 90 181Chi-square 13.26 (0.00%) Cross Product ratio

3.04(0.02%)

‘Hot Hands’ in mutual funds

Cross section regression of sequential performance

2 1

2

.034 0.3075( 3.37) (5.73)

0.155; 181R N

Survivorship, returns and volatility

Index distributions by a spread parameter

Selection by performance selects by volatility

Pr[ | ; , 0]

Pr[ | ; , 0]Pr[ | ; , 0]Pr[ | , 0]

11 2 1212 2

x y

x y x y

x y x y

x y x y x y x yx y x y

Stephen Brown, William Goetzmann, Roger Ibbotson, Stephen Ross “Survivorship bias in performance studies” Review of Financial Studies, December 1992 553-580.

Managers differ in volatility

0

-2 5 % 0% 2 5 % 5 0% 7 5 % 1 00% 1 2 5 % 1 5 0% 1 7 5 % 2 00%

Annual return on fund assets

Prob

abilit

y

0% a

Manager x

Manager y

Performance persists among survivors

Conditional on x, y surviving both periods:

2 2

1 1

2 2 1 1

1Pr[ | ] 021Pr[ | ] , 02

1 1Pr[ | ] 22 2

x y

x y

x y p p

x y q q

x y x y pq

Stephen Brown, William Goetzmann, Roger Ibbotson, Stephen Ross “Survivorship bias in performance studies” Review of Financial Studies, December 1992 553-580.

Summary of simulations with different percent cutoffs

Panel 1: No Cutoff (N = 600) Panel 2: 5% Cutoff (N = 494)2nd time

winner

2nd time loser

2nd time

winner

2nd time loser

1st time winner 150.09 149.91 1st time

winner 127.49 119.51

1st time loser 149.91 150.09 1st time

loser 119.51 127.49Average Cross Product Ratio

1.014Average Cross Product Ratio

1.164Average Cross Section t

-.004Average Cross Section t

2.046Risk adjusted return 0.00% Risk adjusted return 0.44%

Prices of ten art works

0 1 2 3 4 5 6 7 8 9 100.25

2.5

25

Prices

Korteweg, Arthur G. and Kräussl, Roman and Verwijmeren, Patrick, Does it Pay to Invest in Art? (October 15, 2013). Available at : http://ssrn.com/abstract=2280099

Values of ten art works

0 1 2 3 4 5 6 7 8 9 100.25

2.5

25

Values

Why does price depart from value?

0 1 2 3 4 5 6 7 8 9 100.25

2.5

25

ValuesPrices

Selection equation

20 1 2

0 !

w r

w s l

t

e

t

a

Conclusion

Can only examine trading records of survivors

High risk associated with return ex post

Biased inferences about performance and risk

Be careful about what you can infer!