Empirical Financial Economics
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Transcript of Empirical Financial Economics
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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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
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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
( ) 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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Emerging market price paths
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Value
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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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pTp0
Conditional price paths
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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
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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
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-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
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Values
Why does price depart from value?
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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!