2. 5/23/2015 Tactical Asset Allocation 1 Tactical Asset Allocation 2 session 6 Andrei Simonov

3. 5/23/2015 Tactical Asset Allocation 2 Agenda Statistical properties of volatility. –Persistence –Clustering –Fat tails Is covariance matrix constant? Predictive methodologies –Macroecon variables –Modelling volatility process: GARCH process and related methodologies –Volume –Chaos Skewness

4. 5/23/2015 Tactical Asset Allocation 3 Volatility is persistent Returns 2 are MORE autocorrelated than returns themselves. Volatility is indeed persistent. Akgiray, JB89

5. 5/23/2015 Tactical Asset Allocation 4 It is persistent for different holding periods and asset classes Sources: Hsien JBES(1989), Taylor&Poon, JFB92

6. 5/23/2015 Tactical Asset Allocation 5 Volatility Clustering, r t =ln(S t /S t-1 ).

7. 5/23/2015 Tactical Asset Allocation 6 Volatility clustering

8. 5/23/2015 Tactical Asset Allocation 7 Kurtosis & Normal distribution Kurtosis= 0 for normal dist. If it is positive, there are so-called FAT TAILS

9. 5/23/2015 Tactical Asset Allocation 8 Higher Moments & Expected Returns Data through June 2002

10. 5/23/2015 Tactical Asset Allocation 9 Higher Moments & Expected Returns Data through June 2002

11. 5/23/2015 Tactical Asset Allocation 10 Extreme events

12. 5/23/2015 Tactical Asset Allocation 11

13. 5/23/2015 Tactical Asset Allocation 12

14. 5/23/2015 Tactical Asset Allocation 13 Normal distribution: Only 1 observation in 15800 should be outside of 4 standard deviations band from the mean. Historicaly observed: –1 in 293 for stock returns (S&P) –1 in 138 for metals –1 in 156 for agricultural futures

15. 5/23/2015 Tactical Asset Allocation 14 What do we know about returns? Returns are NOT predictable (martingale property) Absolute value of returns and squared returns are strongly serially correlated and not iid. Kurtosis>0, thus,returns are not normally distributed and have fat tails -’ve skewness is observed for asset returns

16. 5/23/2015 Tactical Asset Allocation 15 ARCH(1) volatility at time t is a function of volatility at time t-1 and the square of of the unexpected change of security price at t-1. Ret(t)=  Ret(t-1)+  t  (t)=  (t) z(t)  2 (t)=  0 +  1  2 (t-1), z  ) If volatility at t is high(low), volatility at t+1 will be high(low) as well Greater  1 corresponds to more persistency

17. 5/23/2015 Tactical Asset Allocation 16 Simulating ARCH vs Normal Normal ARCH(1) ARCH(4)

18. 5/23/2015 Tactical Asset Allocation 17 GARCH= Generalized Autoregressive Heteroskedasticity volatility at time t is a function of volatility at time t-1 and the square of of the unexpected change of security price at t-1.  2 (t)=  0 +  1  2 (t-1)+  1  2 (t-1),  2 t ) If volatility at t is high(low), volatility at t+1 will be high(low) as well The greater , the more gradual the fluctuations of volatility are over time Greater  1 corresponds to more rapid changes in volatility

19. 5/23/2015 Tactical Asset Allocation 18

20. 5/23/2015 Tactical Asset Allocation 19

21. 5/23/2015 Tactical Asset Allocation 20 Persistence If (  1 +  )>1, then the shock is persistent (i.e., they accumulate). If (  1 +  )<1, then the shock is transitory and will decay over time For S&P500 (  1 +  )=0.841, then in 1 month only 0.841 4 =0.5 of volatility shock will remain, in 6 month only 0.01 will remain Those estimates went down from 1980-es (in 1988 Chow estimated (  1 +  )=0.986

22. 5/23/2015 Tactical Asset Allocation 21 Forecasting power GARCH forecast is far better then other forecasts Difference is larger over high volatility periods Still, all forecasts are not very precise (MAPE>30%) xGARCH industry

23. 5/23/2015 Tactical Asset Allocation 22 Options’ implied volatilities Are option implicit volatilities informative on future realized volatilities? YES If so, are they an unbiased estimate of future volatilities? NO Can they be beaten by statistical models of volatility behavior (such as GARCH)? I.e. does one provide information on top of the information provided by the other? –Lamoureux and Lastrapes: h t =  +  2 t-1 +  h t-1 +  implied –They find  significant.

24. 5/23/2015 Tactical Asset Allocation 23 Which method is better? (credit due: Poon & Granger, JEL 2003)

25. 5/23/2015 Tactical Asset Allocation 24 Straddles: a way to trade on volatility forecast Straddles delivers profit if stock price is moving outside the normal range If model predicts higher volatility, buy straddle. If model predicts lower volatility, sell straddle STST X Profit

26. 5/23/2015 Tactical Asset Allocation 25 Volatility and Trade Lamoureux and Lastrapes: Putting volume in the GARCH equiation, makes ARCH effects disappear. h t =  +  2 t-1 +  h t-1 +  Volume Heteroscedastisity is (at least, partially) due to the information arrival and incorporation of this information into prices. Processing of information matters!

27. 5/23/2015 Tactical Asset Allocation 26 What else matters? Macroeconomy

28. 5/23/2015 Tactical Asset Allocation 27 Macroeconomic variables (2)

29. 5/23/2015 Tactical Asset Allocation 28 Stock returns and the business cycle: Volatility NBER Expansions and Contractions January 1970-March 1997

30. 5/23/2015 Tactical Asset Allocation 29 Predicting Correlations (1) Crucial for VaR Crucial for Portfolio Management –Stock markets crash together in 87 (Roll) and again in 98... –Correlations varies widely with time, thus, opportunities for diversification (Harvey et al., FAJ 94)

31. 5/23/2015 Tactical Asset Allocation 30 Predicting Correlations (2) Use “usual suspects” to predict correlations Simple approach “up- up” vs. “down- down”

32. 5/23/2015 Tactical Asset Allocation 31 Predicting Correlations (3)

33. 5/23/2015 Tactical Asset Allocation 32 Chaos as alternative to stochastic modeling Chaos in deterministic non-linear dynamic system that can produce random-looking results Feedback systems, x(t)=f(x(t-1), x(t-2)...) Critical levels: if x(t) exceeds x 0, the system can start behaving differently (line 1929, 1987, 1989, etc.) The attractiveness of chaotic dynamics is in its ability to generate large movements which appear to be random with greater frequency than linear models (Noah effect) Long memory of the process (Joseph effect)

34. 5/23/2015 Tactical Asset Allocation 33 Example: logistic eq. X(t+1)=4ax(t)(1-x(t))

35. 5/23/2015 Tactical Asset Allocation 34 A=0.9 A=0.95

36. 5/23/2015 Tactical Asset Allocation 35 Hurst Exponent Var(X(t)-X(0))  t 2H H=1/2 corresponds to “normal” Brownian motion H )1/2 – indicates negative (positive) correlations of increments For financial markets (Jan 63-Dec89, monthly returns): IBM0.72 Coca-Cola0.70 Texas State Utility0.54 S&P5000.78 MSCI UK0.68 Japanese Yen0.64 UK £0.50

37. 5/23/2015 Tactical Asset Allocation 36 Long Memory Memory cannot last forever. Length of memory is finite. For financial markets (Jan 63-Dec89, monthly returns): IBM18 month Coca-Cola42 Texas State Utility90 S&P50048 MSCI UK30 Industries with high level of innovation have short cycle (but high H) “Boring” industries have long cycle (but H close to 0.5) Cycle length matches the one for US industrial production Most of predictions of chaos models can be generated by stochastic models. It is econometrically impossible to distinguish between the two.

38. 5/23/2015 Tactical Asset Allocation 37 Correlations and Volatility: Predictable. Important in asset management Can be used in building dynamic trading strategy (“vol trading”) Correlation forecasting is of somewhat limited importance in “classical TAA”, difference with static returns is rather small. Pecking order: expected returns, volatility, everything else… Good model: EGARCH with a lot of dummies

39. 5/23/2015 Tactical Asset Allocation 38 Smile please! Black- Scholes implied volatilities (01.04.92)

40. 5/23/2015 Tactical Asset Allocation 39 Skewness & Expected Returns Data through June 2002

41. 5/23/2015 Tactical Asset Allocation 40 Skewness & Expected Returns Data through June 2002

42. 5/23/2015 Tactical Asset Allocation 41 Skewness or ”crash” premia (1)  Skewness premium =Price of calls at strike 4% above forward price/ price of puts at strike 4% below forward price  1 The two diagrams following show:  That fears of crash exist mostly since the 1987 crash  This shows also in the volume of transactions on puts compared to calls

43. 5/23/2015 Tactical Asset Allocation 42 Skewness or ”crash” premia (2)

44. 5/23/2015 Tactical Asset Allocation 43

45. 5/23/2015 Tactical Asset Allocation 44Skewness See also movie from Cam Harvey web site.

46. 5/23/2015 Tactical Asset Allocation 45 Where skewness is coming from? Log-normal distribution Behavioral preferences (non-equivalence between gains and losses) Experiments: People like +’ve skewness and hate negative skewness.

47. 5/23/2015 Tactical Asset Allocation 46 Conditional Skewness, Bakshi, Harvey and Siddique (2002)

48. 5/23/2015 Tactical Asset Allocation 47 What can explain skewness? Stein-Hong-Chen: imperfections of the market cause delays in incorporation of the information into prices. Measure of info flows – turnover or volume.

49. 5/23/2015 Tactical Asset Allocation 48 Co-skewness Describe the probability of the assets to run-up or crash together. Examples: ”Asian flu” of 98,” crashes in Eastern Europe after Russian Default. Can be partially explained by the flows. Important: Try to avoid assets with +’ve co-skewness. Especially important for hedge funds Difficult to measure.

50. 5/23/2015 Tactical Asset Allocation 49 Three-Dimensional Analysis

51. 5/23/2015 Tactical Asset Allocation 50 Alternative Vehicles Alternate Asset Classes Often Involve Implicit or Explicit Options Source: Naik (2002)

52. 5/23/2015 Tactical Asset Allocation 51 Alternative Vehicles Alternate Asset Classes Often Involve Implicit or Explicit Options Source: Naik (2002)

53. 5/23/2015 Tactical Asset Allocation 52 Alternative Vehicles Alternate Asset Classes Often Involve Implicit or Explicit Options Source: Naik (2002)

54. 5/23/2015 Tactical Asset Allocation 53 Alternative Vehicles Alternate Asset Classes Often Involve Implicit or Explicit Options Source: Naik (2002)

55. 5/23/2015 Tactical Asset Allocation 54 Alternative Vehicles Alternate Asset Classes Often Involve Implicit or Explicit Options Source: Figure 5 from Mitchell & Pulvino (2000)

56. 5/23/2015 Tactical Asset Allocation 55 Alternative Vehicles Alternate Asset Classes Often Involve Implicit or Explicit Options -8 -6 -4 -2 0 2 4 6 -15-10-50510 Russell 3000 Index Returns Event Driven Index Returns LOWESS fit Source: Naik (2002)

57. 5/23/2015 Tactical Asset Allocation 56 Co-skewness for hedge funds Source: Lu and Mulvey (2001)