Tactical Asset Allocation 2 session 6

1. Tactical Asset Allocation 2session 6 Andrei Simonov 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 Tactical Asset Allocation

3. Volatility is persistent • Returns2 are MORE autocorrelated than returns themselves. Volatility is indeed persistent. Akgiray, JB89 Tactical Asset Allocation

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

5. Volatility Clustering, rt=ln(St/St-1). Tactical Asset Allocation

6. Volatility clustering Tactical Asset Allocation

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

8. Higher Moments & Expected Returns Data through June 2002 Tactical Asset Allocation

9. Higher Moments & Expected Returns Data through June 2002 Tactical Asset Allocation

10. Extreme events Tactical Asset Allocation

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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 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 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)=f Ret(t-1)+et e(t)= s(t) z(t) s2(t)=a0+a1e2(t-1), z~N(0, 1) • If volatility at t is high(low), volatility at t+1 will be high(low) as well • Greater a1 corresponds to more persistency Tactical Asset Allocation

16. Simulating ARCH vs Normal ARCH(4) ARCH(1) Normal 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. s2(t)=a0+b1s2(t-1)+ a1e2(t-1), e~N(0, s2t) • If volatility at t is high(low), volatility at t+1 will be high(low) as well • The greater b, the more gradual the fluctuations of volatility are over time • Greater a1 corresponds to more rapid changes in volatility Tactical Asset Allocation

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20. Persistence • If (a1+b)>1, then the shock is persistent (i.e., they accumulate). • If (a1+b)<1, then the shock is transitory and will decay over time • For S&P500 (a1+b)=0.841, then in 1 month only 0.8414=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 (a1+b)=0.986 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 Tactical Asset Allocation

22. Options’ implied volatilities • Are option implicit volatilities informative onfuture realized volatilities? YES • If so, are they an unbiased estimate of futurevolatilities? NO • Can they be beaten by statistical models ofvolatility behavior (such as GARCH)? I.e. doesone provide information on top of theinformationprovided by the other? • Lamoureux and Lastrapes: ht= w + ae2t-1 + bh t-1+ gsimplied • They find gsignificant. Tactical Asset Allocation

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

24. Straddles: a way to trade on volatility forecast Profit • 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 X ST Tactical Asset Allocation

25. Volatility and Trade • Lamoureux and Lastrapes: Putting volume in the GARCH equiation, makes ARCH effects disappear. ht= w + ae2t-1 + bh t-1+ gVolume • Heteroscedastisity is (at least, partially) due to the information arrival and incorporation of this information into prices. • Processing of information matters! Tactical Asset Allocation

26. What else matters? Macroeconomy Tactical Asset Allocation

27. Macroeconomic variables (2) Tactical Asset Allocation

28. Stock returns and the business cycle:VolatilityNBER Expansions and ContractionsJanuary 1970-March 1997 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) Tactical Asset Allocation

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

31. Predicting Correlations (3) 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 x0, 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) Tactical Asset Allocation

33. Example: logistic eq. • X(t+1)=4ax(t)(1-x(t)) Tactical Asset Allocation

34. A=0.9 A=0.95 Tactical Asset Allocation

35. Hurst Exponent • Var(X(t)-X(0)) t2H • 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): IBM 0.72 Coca-Cola 0.70 Texas State Utility 0.54 S&P500 0.78 MSCI UK 0.68 Japanese Yen 0.64 UK £ 0.50 Tactical Asset Allocation

36. Long Memory • Memory cannot last forever. Length of memory is finite. • For financial markets (Jan 63-Dec89, monthly returns): IBM 18 month Coca-Cola 42 Texas State Utility 90 S&P500 48 MSCI UK 30 • 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. 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 Tactical Asset Allocation

38. Smile please!Black- Scholes implied volatilities (01.04.92) Tactical Asset Allocation

39. Skewness & Expected Returns Data through June 2002 Tactical Asset Allocation

40. Skewness & Expected Returns Data through June 2002 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 onputs compared to calls Tactical Asset Allocation

42. Skewness or ”crash” premia (2) Tactical Asset Allocation

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44. Skewness See also movie from Cam Harvey web site. 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. Tactical Asset Allocation

46. Conditional Skewness, Bakshi, Harvey and Siddique (2002) 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. 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. Tactical Asset Allocation

49. Three-Dimensional Analysis Tactical Asset Allocation

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