1. Empirical Financial Economics 6. Ex post conditioning issues Stephen Brown NYU Stern School of Business UNSW PhD Seminar, June 19-21 2006

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

3. Performance measurement Leeson Investment Managemen t Market (S&P 500) Benchmark Short-term Government Benchmark Average Return.0065.0050.0036 Std. Deviation.0106.0359.0015 Beta.06401.0.0 Alpha.0025 (1.92).0 Sharpe Ratio.2484.0318.0 Style: Index Arbitrage, 100% in cash at close of trading

4. Frequency distribution of monthly returns

5. Percentage in cash (monthly)

6. Examples of riskless index arbitrage …

7. Percentage in cash (daily)

8. $0 $1 $-1 p = 1 2 Is doubling low risk?

9. $0 $1 $-3 p = 1 4 Is doubling low risk?

10. $0 $1 $-7 p = 1 8 Is doubling low risk?

11. $0 $1 $-15 p = 1 16 Is doubling low risk?

12. $0 $1 $-31 p = 1 32 Is doubling low risk?

13. $0 $1 $-63 p = 1 64 Is doubling low risk?

14. $0 $1 $-127 p = 1 128 Is doubling low risk?

15.  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

16. 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

17. 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

18. 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!

19. Infinitely many ways to lose money!

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

21. 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)

22. 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}

23. 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

24. National Australia Bank

25. 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”

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

27. Unconditional price paths

28. 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 }

29. Absorbing barrier at zero

30. Conditional price paths

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

32. 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

33. Expected value path

34. Emerging market price paths

35. 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

36. 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!

37. 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!

38. 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 ( )

39. Unconditional price path pTpT p0p0

40. Conditional price paths pTpT p0p0 *

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

42. Variance of long holding period returns 0.0172

43. ‘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 583391 1986-87 losers 335790 Totals9190181 Chi-square 13.26 (0.00%)Cross Product ratio 3.04(0.02%)

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

45. ‘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 92029 1986-87 losers 27125152 Totals36145181 Chi-square 2.69 (10.10%)

46. Persistence of Mutual Fund Performance

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

48. Managers differ in volatility 0% a Manager x Manager y

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

50. 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 150.09149.91 1st time winner 127.49 119.51 1st time loser 149.91150.09 1st time loser 119.51 127.49 Average Cross Product Ratio 1.014 Average Cross Product Ratio 1.164 Average Cross Section t -.004Average Cross Section t 2.046 Risk adjusted return 0.00%Risk adjusted return 0.44%

51. “Anomalies”  Persistence of mutual fund returns  Post-earnings announcement drift  Glamour vs. Value

52. “Anomalies”  Persistence of mutual fund returns  Post-earnings announcement drift  Glamour vs. Value These effects are economically and statistically significant

53. “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

54. “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

55. Post earnings drift Earnings surprise decile Using SUE as surpriseUsing event period CAR Post event CARt-valuePost event CARt-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 50.0010.77-0.004-2.03 60.0084.29-0.003-1.62 70.0105.640.0000.28 80.0126.960.0010.45 90.02212.780.0074.12 100.02414.280.0179.26

56. Glamour vs. Value Book to Market GlamourQ2Q3Q4Value Year 10.000 0.037 (0.08)(0.01)(0.02)(0.01)(13.42) Year 20.000 0.0010.035 -(0.01)(0.05)(0.00)(0.31)(11.62) Year 30.000 0.0020.035 -(0.09)(0.03)-(0.06)(1.06)(10.81) Year 40.000 0.0040.036 -(0.03)-(0.02)(0.08)(1.82)(10.22) Year 50.000 0.0050.035 (0.05)(0.03) (2.68)(9.26)

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

58. FFJR Redux

59. Original FFJR results

60. 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!