Empirical Financial Economics 6. Ex post conditioning issues Stephen Brown NYU Stern School of Business UNSW PhD Seminar, June

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

Performance measurement Leeson Investment Managemen t Market (S&P 500) Benchmark Short-term Government Benchmark Average Return Std. Deviation Beta Alpha.0025 (1.92).0 Sharpe Ratio Style: Index Arbitrage, 100% in cash at close of trading

Frequency distribution of monthly returns

Percentage in cash (monthly)

Examples of riskless index arbitrage …

Percentage in cash (daily)

$0 $1 $-1 p = 1 2 Is doubling low risk?

$0 $1 $-3 p = 1 4 Is doubling low risk?

$0 $1 $-7 p = 1 8 Is doubling low risk?

$0 $1 $-15 p = 1 16 Is doubling low risk?

$0 $1 $-31 p = 1 32 Is doubling low risk?

$0 $1 $-63 p = 1 64 Is doubling low risk?

$0 $1 $-127 p = 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 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  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!

The challenge of risk management  Performance and risk inferred from logarithm of fund value:

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)

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}

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} yet

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  “Anomalies”

Effect of conditioning on observed value paths  The logarithm of value follows a simple absolute diffusion on

Unconditional price paths

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 }

Absorbing barrier at zero

Conditional price paths

Effect of conditioning on observed value paths  Define  Observed values follow an absolute diffusion on

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

Expected value path

Emerging market price paths

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

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!

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!

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 ( )

Unconditional price path pTpT p0p0

Conditional price paths pTpT p0p0 *

Properties of survivors  High return  Low risk  Apparent mean reversion:  Variance ratio =

Variance of long holding period returns

‘Hot Hands’ in mutual funds Growth fund performance relative to alpha of median manager winners losers Totals winners losers Totals Chi-square (0.00%)Cross Product ratio 3.04(0.02%)

‘Hot Hands’ in mutual funds Cross section regression of sequential performance

‘Cold Hands’ in mutual funds Growth fund performance relative to alpha of zero winners losers Totals winners losers Totals Chi-square 2.69 (10.10%)

Persistence of Mutual Fund Performance

Survivorship, returns and volatility  Index distributions by a spread parameter  Selection by performance selects by volatility

Managers differ in volatility 0% a Manager x Manager y

Performance persists among survivors  Conditional on x, y surviving both periods:

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 st time winner st time loser st time loser Average Cross Product Ratio Average Cross Product Ratio Average Cross Section t -.004Average Cross Section t 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 surpriseUsing event period CAR Post event CARt-valuePost event CARt-value

Glamour vs. Value Book to Market GlamourQ2Q3Q4Value Year (0.08)(0.01)(0.02)(0.01)(13.42) Year (0.01)(0.05)(0.00)(0.31)(11.62) Year (0.09)(0.03)-(0.06)(1.06)(10.81) Year (0.03)-(0.02)(0.08)(1.82)(10.22) Year (0.05)(0.03) (2.68)(9.26)

Stock splits  Rarely does a stock split come on a decrease in security value:  Approximate summation by integral

FFJR Redux

Original FFJR results

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!