Empirical Financial Economics

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Empirical Financial Economics

• 6. Ex post conditioning issues

Stephen Brown NYU Stern School of Business UNSW
PhD Seminar, June 19-21 2006
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Overview

• A simple example
• Brief review of ex post conditioning issues
• Implications for tests of Efficient Markets
  Hypothesis

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Performance measurement
Leeson Investment Management Market (SP 500) Benchmark 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
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Frequency distribution of monthly returns
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Percentage in cash (monthly)
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Examples of riskless index arbitrage
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Percentage in cash (daily)
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Is doubling low risk?
1
0
-1
1
p
2
9
Is doubling low risk?
1
0
-3
1
p
4
10
Is doubling low risk?
1
0
-7
1
p
8
11
Is doubling low risk?
1
0
-15
1
p
16
12
Is doubling low risk?
1
0
-31
1
p
32
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Is doubling low risk?
1
0
-63
1
p
64
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Is doubling low risk?
1
0
-127
1
p
128
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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

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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 didnt think Id ever go through that
kind of tension again. Id 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
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The case of the Repeated Doubler

• Bernoulli game
• Leave game on a win
• Must win if play long enough
• Repeated doubler
• Reestablish position on a win
• Must lose if play long enough

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Infinitely many ways to lose money!

• Manager trades SP contracts
• 
  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!

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Infinitely many ways to lose money!
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The challenge of risk management

• Performance and risk inferred from logarithm of
  fund value

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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)

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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
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The challenge of risk management

• Performance and risk inferred from logarithm of
  fund value
• But what the manager observes is

yet
A set of price paths where doubler has not
embezzled
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National Australia Bank
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Ex post conditioning

• Ex post conditioning leads to problems
• When inclusion in sample depends on price path
• Examples
• Equity premium puzzle
• Variance ratio analysis
• Performance measurement
• Post earnings drift
• Event studies
• Anomalies

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Effect of conditioning on observed value paths

• The logarithm of value follows a simple absolute
  diffusion on

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Unconditional price paths
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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
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Absorbing barrier at zero
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Conditional price paths
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Effect of conditioning on observed value paths

• Define
• Observed values follow an absolute diffusion on

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Example Absorbing barrier at zero
As T goes to infinity, conditional diffusion is
Expected return is positive, increasing in
volatility and decreasing in ex ante probability
of failure
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Expected value path
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Emerging market price paths
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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

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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 well understood peril of empirical finance!

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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 well understood peril of empirical finance!

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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 ( )

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Unconditional price path
p0
pT
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Conditional price paths
pT

p0
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Properties of survivors

• High return
• Low risk
• Apparent mean reversion
• Variance ratio

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Variance of long holding period returns
0.0172
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Hot Hands in mutual funds
Growth fund performance relative to alpha of median manager 1984-1987 Growth fund performance relative to alpha of median manager 1984-1987 Growth fund performance relative to alpha of median manager 1984-1987 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 90 181
Chi-square 13.26 (0.00) Chi-square 13.26 (0.00) Cross Product ratio 3.04(0.02) Cross Product ratio 3.04(0.02)
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Hot Hands in mutual funds
Cross section regression of sequential performance
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Cold Hands in mutual funds
Growth fund performance relative to alpha of zero 1984-1987 Growth fund performance relative to alpha of zero 1984-1987 Growth fund performance relative to alpha of zero 1984-1987 Growth fund performance relative to alpha of zero 1984-1987
1986-87 winners 1986-87 losers Totals
1984-85 winners 9 20 29
1986-87 losers 27 125 152
Totals 36 145 181
Chi-square 2.69 (10.10) Chi-square 2.69 (10.10)
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Persistence of Mutual Fund Performance
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Survivorship, returns and volatility

• Index distributions by a spread parameter
• Selection by performance selects by volatility

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Managers differ in volatility
Manager y
Manager x
a
0
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Performance persists among survivors

• Conditional on x, y surviving both periods

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Summary of simulations with different percent
cutoffs
Panel 1 No Cutoff (N 600) Panel 1 No Cutoff (N 600) Panel 1 No Cutoff (N 600) Panel 2 5 Cutoff (N 494) Panel 2 5 Cutoff (N 494) 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 119.51 127.49
Average Cross Product Ratio 1.014 Average Cross Product Ratio 1.014 Average Cross Product Ratio 1.014 Average Cross Product Ratio 1.164 Average Cross Product Ratio 1.164 Average Cross Product Ratio 1.164
Average Cross Section t -.004 Average Cross Section t -.004 Average Cross Section t -.004 Average Cross Section t 2.046 Average Cross Section t 2.046 Average Cross Section t 2.046
Risk adjusted return 0.00 Risk adjusted return 0.00 Risk adjusted return 0.00 Risk adjusted return 0.44 Risk adjusted return 0.44 Risk adjusted return 0.44
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Anomalies

• Persistence of mutual fund returns
• Post-earnings announcement drift
• Glamour vs. Value

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Anomalies

• Persistence of mutual fund returns
• Post-earnings announcement drift
• Glamour vs. Value

These effects are economically and statistically
significant
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Anomalies

• Persistence of mutual fund returns
• Post-earnings announcement drift
• Glamour vs. Value

These effects are economically and statistically
significant
We cannot rule out market inefficiency as an
explanation
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Anomalies

• Persistence of mutual fund returns
• Post-earnings announcement drift
• Glamour vs. Value

These effects are economically and statistically
significant
We cannot rule out market inefficiency as an
explanation
Magnitude affected by survival and volatility
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Post earnings drift
Earnings surprise decile Using SUE as surprise Using SUE as surprise Using event period CAR Using event period CAR
Earnings surprise decile Post event CAR t-value Post event CAR t-value
1 -0.030 -16.10 -0.011 -5.79
2 -0.026 -14.93 -0.009 -4.95
3 -0.021 -12.14 -0.005 -2.57
4 -0.012 -6.77 -0.006 -3.59
5 0.001 0.77 -0.004 -2.03
6 0.008 4.29 -0.003 -1.62
7 0.010 5.64 0.000 0.28
8 0.012 6.96 0.001 0.45
9 0.022 12.78 0.007 4.12
10 0.024 14.28 0.017 9.26
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Glamour vs. Value
Book to Market Book to Market Book to Market Book to Market Book to Market
Glamour Q2 Q3 Q4 Value
Year 1 0.000 0.000 0.000 0.000 0.037
(0.08) (0.01) (0.02) (0.01) (13.42)
Year 2 0.000 0.000 0.000 0.001 0.035
-(0.01) (0.05) (0.00) (0.31) (11.62)
Year 3 0.000 0.000 0.000 0.002 0.035
-(0.09) (0.03) -(0.06) (1.06) (10.81)
Year 4 0.000 0.000 0.000 0.004 0.036
-(0.03) -(0.02) (0.08) (1.82) (10.22)
Year 5 0.000 0.000 0.000 0.005 0.035
(0.05) (0.03) (0.03) (2.68) (9.26)
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Stock splits

• Rarely does a stock split come on a decrease in
  security value
• Approximate summation by integral

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FFJR Redux
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Original FFJR results
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Conclusion

• Ex post conditioning a well known peril of
  empirical finance
• High risk associated with return ex post
• The Efficient Markets Hypothesis is a statement
  about conditional expectations
• Be careful about what you can infer!