Transcript Document 7619307

Empirical Financial Economics
6. Ex post conditioning issues
Stephen Brown NYU Stern School of
Business
UNSW PhD Seminar, June 19-21 2006
Overview
A simple example
Brief review of ex post conditioning
issues
Implications for tests of Efficient
Markets Hypothesis
Performance measurement
Leeson
Market
Short-term
Investment
(S&P 500)
Government
Managemen Benchmark Benchmark
t
Average .0065
Return
.0050
.0036
Std. .0106
Deviation
.0359
.0015
1.0
.0
Beta .0640
Alpha .0025
.0
.0
(1.92)
100% in cash.0
at close of
Sharpe Style:
Ratio Index
.2484 Arbitrage,.0318
trading
-1
.0
-0 0 %
.5
0
0. %
00
0. %
50
%
1.0
0%
1.5
0%
2.
00
2. %
50
3. %
00
3. %
50
4. %
00
4. %
50
5. %
00
5. %
50
6. %
0
0
6. %
50
%
Frequency distribution of
monthly returns
35
30
25
20
15
10
5
0
Percentage in cash (monthly)
120 %
10 0 %
80%
60%
40 %
20 %
0%
31-Dec-198 9
15-May-1991
26 -Sep-1992
8 -Feb-1994
Examples of riskless index
arbitrage …
Percentage in cash (daily)
20 0 %
10 0 %
0%
-10 0 %
-20 0 %
-30 0 %
-40 0 %
-50 0 %
-6 0 0 %
31-Dec-198 9
15-May-1991
26 -Sep-1992
8 -Feb-1994
Is doubling low risk?
$1
$0
$-1
p=
1
2
Is doubling low risk?
$1
$0
$-3
1
p=
4
Is doubling low risk?
$1
$0
$-7
1
p=
8
Is doubling low risk?
$1
$0
$-15
1
p=
16
Is doubling low risk?
$1
$0
$-31
1
p=
32
Is doubling low risk?
$1
$0
$-63
1
p=
64
Is doubling low risk?
$1
$0
$-127
1
p=
128
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 win
Must win if play long enough
Repeated doubler
Reestablish position on a win
Must lose if play long enough
Infinitely many ways to lose
money!
Manager trades S&P contracts
  12.5%,   20%, rf  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!
30
1
24
0 .8
0 .7
18
0 .6
0 .5
12
0 .4
0 .3
6
0 .2
0 .1
0
1-Jan
0
21-Mar
9-Jun
28 -Aug
16 -Nov
Tenure of t rader
Sharpe rat io of repeat ed doubler
Probabilit y of surviving
Probabilit y of surviving
Sharpe rat io of survivors
0 .9
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:
dp   dt   dz

is expected return of manager
Lower
bound[0,
onT ]
is
Value at Risk (VaR)

with probability
The challenge of risk management
Performance and risk inferred from
logarithm of fund value:
dp   dt   dz
p* isp | A
But what the manager observes
A = {set of price paths where doubler has not
embezzled}
The challenge of risk management
Performance and risk inferred from
logarithm of fund value:
dp   dt   dz
p* isp | A
But what the manager observes
yet
A = {set of price paths where doubler has not
embezzled}
National Australia Bank
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
Effect of conditioning on observed
value paths
The logarithm of value follows a
[0, T ]
simple absolute diffusion
on
dp   dt   dz
Log price in unit s of annual st andard
deviat ion
Unconditional price paths
9
7
5
3
1
-1
-3
-5
0
2
4
6
Years
8
10
Effect of conditioning on observed
value paths
The logarithm of value follows a
[0, T ]
simple absolute diffusion
on
dp   dt   dz
What can we say about values we
observe?
[0,on
T]
A = {set of price paths observed
}
Log price in unit s of annual st andard
deviat ion
Absorbing barrier at zero
9
7
5
3
1
-1
-3
-5
0
2
4
6
Years
8
10
Log price in unit s of annual st andard
deviat ion
Conditional price paths
9
7
5
3
1
-1
-3
-5
0
2
4
6
Years
8
10
Effect of conditioning on observed
value paths
Define (t )  Pr[ A | p, t ]
Observed values follow an absolute
[0, T ] on
diffusion
dp*   * dt   dz
*    
2
p

Example: Absorbing barrier at
zero
*    
2
p

2 [ w]
p

, w
T  t (2 [ w]  1)
 T t
As T goes to infinity, conditional diffusion is
dp* 
2
p p
dt   dz
Expected return is positive, increasing in volatility and
decreasing in ex ante probability of failure
Log price in unit s of annual st andard
deviat ion
Expected value path
9
7
5
3
1
-1
-3
-5
0
2
4
6
Years
8
10
Emerging market price paths
2
p0  p  2
Value
1.5
1
1
p0  p  
2
0 .5
0
0
10
20
Years
30
40
Important result
*    
2
p

 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
Important result
*    
2
p

 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!
Important result
*    
2
p

 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!
Equity premium puzzle
With nonzero drift, as T goes to infinity
2 (1   ( p)
*   
 ( p)
  rf  4%
If true equity premium is zero,
10%(
an observed equity premium*of 6%
) implies 2/3 ex ante probability that the
market will survive in the very long
 ( p ) of
 .66
term given the current level
prices (
)
Log price in unit s of annual st andard
deviat ion
Unconditional price path
p0
9
7
5
3
pT
1
-1
-3
-5
0
2
4
6
Years
8
10
Log price in unit s of annual st andard
deviat ion
Conditional price paths
p0
9
7
p*T
5
3
1
-1
-3
-5
0
2
4
6
Years
8
10
Properties of survivors
High return
Low risk
Apparent mean reversion:
1
4  2
lim Var pT* 

T  T
2
4 
 .429204....
Variance ratio =
2
Variance of long holding period
returns
0 .0 45
Annualized variance
0 .0 4
0 .0 35
0 .0 3
0 .0 25
0 .0 2
0.0172
0 .0 15
0 .0 1
0 .0 0 5
0
0 .0 1
1
10 0
Holding period (years)
2 σ cut off
σ/2 cut off
σ² (4-Π) / 2
10 0 0 0
‘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
181
Chi-square 13.26 (0.00%)
Cross Product ratio 3.04(0.02%)
‘Hot Hands’ in mutual funds
Cross section regression of sequential
performance
 2  .034  0.30751
(3.37) (5.73)
R  0.155; N  181
2
‘Cold Hands’ in mutual funds
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%)
Persistence of Mutual Fund
Performance
400
350
300
250
200
150
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
100
50
0
Loser-Loser
Winner-Loser
Loser-Winner
Winner-Winner
Survivorship, returns and volatility
Index distributions by a spread
parameter

Pr[ x   y | x  y; x, y  0]

Pr[ x  y |  x   y ; x, y  0]Pr[ x   y | x  y; x, y  0]
Pr[ x  y | x, y  0]
1
1 2 1


2
21
2
Selection by performance selects by
Managers differ in volatility
Probabilit y
Manager y
Manager x
0%
a
Annual ret urn on fund asset s
Performance persists among
survivors
Conditional on x, y surviving both
periods:
Pr[ x2  y2 |  x   y ]  1  p p  0
2
Pr[ x   y | x1  y1 ]  1  q, q  0
2
 Pr[ x2  y2 | x1  y1 ]  1  2 pq  1
2
2
Summary of simulations with
different percent cutoffs
Panel 1: No Cutoff (N = 600)
2nd time
winner
2nd time
loser
1st time
winner
150.09
149.91
1st time
loser
149.91
150.09
Panel 2: 5% Cutoff (N = 494)
2nd time
winner
2nd
time
loser
1st time
winner
127.49
119.51
1st time
loser
119.51
127.49
Average Cross Product Ratio
1.014
Average Cross Product Ratio
1.164
Average Cross Section t -.004
Average Cross Section t 2.046
Risk adjusted return 0.00%
Risk adjusted return 0.44%
“Anomalies”
Persistence of mutual fund returns
Post-earnings announcement drift
Glamour vs. Value
“Anomalies”
Persistence of mutual fund returns
Post-earnings announcement drift
Glamour vs. Value
These effects are economically and statistically
significant
“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
“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
Post earnings drift
Earnings
surprise
decile
Using SUE as surprise
Using event period CAR
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
Glamour vs. Value
Book to Market
Year 1
Year 2
Year 3
Year 4
Year 5
Glamour
Q2
Q3
Q4
Value
0.000
0.000
0.000
0.000
0.037
(0.08)
(0.01)
(0.02)
(0.01)
(13.42)
0.000
0.000
0.000
0.001
0.035
-(0.01)
(0.05)
(0.00)
(0.31)
(11.62)
0.000
0.000
0.000
0.002
0.035
-(0.09)
(0.03)
-(0.06)
(1.06)
(10.81)
0.000
0.000
0.000
0.004
0.036
-(0.03)
-(0.02)
(0.08)
(1.82)
(10.22)
0.000
0.000
0.000
0.005
0.035
(0.05)
(0.03)
(0.03)
(2.68)
(9.26)
Stock splits
Rarely does a stock split come on a
decrease in security value:
T


pt 



t 0
A   price path such that pT 

T 



Approximate summation by integral
w(T ) 

A   p (t )  p, p (T ) 
 ; w(T )   p (t ) dt
T 

0
T
FFJR Redux
Cumulat ive residuals around st ock split
Cumulat ive average residual - Um
0 .4
0 .35
0 .3
0 .25
0 .2
0 .15
0 .1
0 .0 5
0
-30
-20
-10
0
10
20
Mont h relat ive t o split - m
30
Original FFJR results
Cumulat ive residuals around st ock split
Cumulat ive average residual - Um
0 .4
0 .35
0 .3
0 .25
0 .2
0 .15
0 .1
0 .0 5
0
-30
-20
-10
0
10
20
Mont h relat ive t o split - m
30
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!