Empirical Financial Economics

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 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. 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

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

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

15. 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

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 yet A = {set of price paths where doubler has not embezzled}

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 AsTgoes 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 p0 pT

40. Conditional price paths pT * p0

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

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

45. ‘Cold Hands’ in mutual funds

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 Manager y Manager x a 0%

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

50. Summary of simulations with different percent cutoffs