Downside Risk: Implications for Financial Management

1. Downside Risk: Implications for Financial Management Robert Engle NYU Stern School of Business Tilburg April 22, 2004

2. WHAT IS ARCH? • Autoregressive Conditional Heteroskedasticity • Predictive (conditional) • Uncertainty (heteroskedasticity) • That fluctuates over time (autoregressive)

3. THE SIMPLEST PROBLEM – WHAT IS VOLATILITY NOW? • One answer is the standard deviation over the last 5 years • But this will include lots of old information that may not be relevant for short term forecasting • Another answer is the standard deviation over the last 5 days • But this will be highly variable because there is so little information

4. THE ARCH ANSWER • Use a weighted average of the volatility over a long period with higher weights on the recent past and small but non-zero weights on the distant past. • Choose these weights by looking at the past data; what forecasting model would have been best historically? This is a statistical estimation problem.

5. FINANCIAL ECONOMETRICS • THIS MAY ALSO BE THE BIRTH OF FINANCIAL ECONOMETRICS • STATISTICAL MODELS DEVELOPED SPECIFICALLY FOR FINANCIAL APPLICATIONS • TODAY THIS IS A VERY POPULAR AND ACTIVE RESEARCH AREA WITH MANY APPLICATIONS

6. FROM THE SIMPLE ARCH GREW: • GENERALIZED ARCH (Bollerslev) a most important extension • Tomorrow’s variance is predicted to be a weighted average of the • Long run average variance • Today’s variance forecast • The news (today’s squared return)

7. AND • EGARCH (Nelson) very important as it introduced asymmetry • Weights are different for positive and negative returns

8. GJR-GARCH TARCH STARCH AARCH NARCH MARCH SWARCH SNPARCH APARCH TAYLOR-SCHWERT FIGARCH FIEGARCH Component Asymmetric Component SQGARCH CESGARCH Student t GED SPARCH NEW ARCH MODELS

9. ROLLING WINDOW VOLATILITIESNUMBER OF DAYS=5, 260, 1300

10. ARCH/GARCH VOLATILITIES

11. CONFIDENCE INTERVALS

12. FINANCIAL APPLICATIONS

13. VALUE AT RISK • Future losses are uncertain. Find a LOSS that you are 99% sure is worse than whatever will occur. This is the Value at Risk. • One day in advance • Many days in advance • This single number (a quantile) is used to represent a full distribution. It can be misleading.

14. CALCULATING VaR • Forecast the one day standard deviation– GARCH style models are widely used. Then: • Assuming normality, multiply by 2.33 • Without assuming normality, multiply by the quantile of the standardized residuals. • For the example, multiplier = 2.65

15. MULTI-DAY HORIZONS • If volatility were constant, then the multi-day volatility would simply require multiplying by the square root of the days. • Because volatility is dynamic and asymmetric, the lower tail is more extreme and the VaR should be greater.

16. TWO PERIOD RETURNS • Two period return is the sum of two one period continuously compounded returns • Look at binomial tree version • Asymmetry gives negative skewness Low variance High variance

17. MULTIPLIER FOR 10 DAYS • For a 10 day 99% value at risk, conventional practice multiplies the daily standard deviation by 7.36 • For the same multiplier with asymmetric GARCH it is simulated from the example to be 7.88 • Bootstrapping from the residuals the multiplier becomes 8.52

18. OPTIONS • Traded options always have multiple days to expiration. • Hence the distribution of future price levels is negatively skewed. • Thus the Black Scholes implied volatility should depend on strike if options are priced by GARCH. • A skew in implied volatility will result from Asymmetric GARCH, at least for short maturities.

19. IMPLIED VOLATILITY SKEW FOR 10 DAY OPTION • From simulated (risk neutral) final values, find average put option payoff for each strike. • Calculate Black Scholes implied volatilities and plot against strike. • Notice the clear downward slope. This would be zero for constant volatility.

20. PUT PRICES

21. PUT IMPLIED VOLATILITIES

22. PRICING KERNEL • The observed skew is even steeper than this. • Engle and Rosenberg(2002) explain the difference by a risk premium • Investors are especially willing to pay to avoid a big market drop. • Others describe this in terms of jumps and risk premia on the jumps

23. WHAT IS NEXT? MULTIVARIATE MODELS- DCC or Dynamic Conditional Correlation HIGH FREQUENCY MODELS- Market Microstructure

24. THE MULTIVARIATE PROBLEM • Asset Allocation and Risk Management problems require large covariance matrices • Credit Risk now also requires big correlation matrices to accurately model loss or default correlations • Multivariate GARCH has never been widely used – it is too difficult to specify and estimate

25. Dynamic Conditional Correlation • DCC is a new type of multivariate GARCH model that is particularly convenient for big systems. See Engle(2002) or Engle(2004).

26. DCC • Estimate volatilities for each asset and compute the standardized residuals or volatility adjusted returns. • Estimate the time varying covariances between these using a maximum likelihood criterion and one of several models for the correlations. • Form the correlation matrix and covariance matrix. They are guaranteed to be positive definite.

27. HOW IT WORKS • When two assets move in the same direction, the correlation is increased slightly. • When they move in the opposite direction it is decreased. • This effect may be stronger in down markets. • The correlations often are assumed to only temporarily deviate from a long run mean

28. Two period Joint Returns • If returns are both negative in the first period, then correlations are higher. • This leads to lower tail dependence Up Market Down Market

29. DCC and the Copula • A symmetric DCC model gives higher tail dependence for both upper and lower tails of the multi-period joint density. • An asymmetric DCC or ASY-DCC gives higher tail dependence in the lower tail of the multi-period density.

30. Testing and Valuing Dynamic Correlations for Asset Allocation Robert Engle and Riccardo Colacito NYU Stern

31. A Model for Stocks and Bonds • Daily returns on S&P500 Futures • Daily returns on 10-year Treasury Note Futures • Both from DataStream from Jan 1 1990 to Dec 18 2002

32. SUMMARY STATISTICS

34. Volatilities and Correlations

37. THE FORMULATION • Solve a series of portfolio problems with a riskless asset • Where r0 is the required excess return and µ is a vector of excess expected returns • With the true covariance matrix you can achieve lower volatility or higher required returns than with the incorrect one.

38. INTERPRETING RESULTS • A number such as 105 means required excess returns can be 5% greater with correct correlations without increasing volatility. • E.g. a 4% excess return with incorrect correlation would be a 4.2% return with correct correlations. • With 10% required return, the value of such correlations is 50 basis points.

40. AN EXPERIMENT • Simulate 10,000 days of the DCC model documented above. • One investor knows the volatilities and correlations every day, Ω. • The other only knows the unconditional volatilities and correlations, H • What is the gain to the informed investor?

41. VALUE GAINSStocks vs Bonds, Simulated Data, Full Covariance

42. Extreme Correlations(simulated data, full covariance)

43. Volatility ratiosStocks vs Bonds ( actual data with estimated DCC)

44. SP500 vs. DOW JONES • Correlation and return structure of equity indices is very different • Unconditional correlations are about .9 • Asymmetry is greater • Expected returns are probably nearly equal • RESULTS ARE ABOUT THE SAME

45. VALUE GAINSSP500 vs DOW, Simulated Data, Full Covariance

46. MONTHLY REBALANCING • Monthly rebalancing lies between rebalancing every day and never rebalancing. • The monthly joint distribution is asymmetric with important lower tail dependence. • Daily myopic rebalancing takes account of this asymmetry • Additional gains are possible with daily multi-period optimization

47. INTEGRATING RISK MANAGEMENT AND ASSET ALLOCATION • Asset Allocation is considered monthly because only at this frequency can expected returns be updated • Within the month, volatilities can be updated • Rebalancing can be done with futures – portfolio volatility can be reduced • Risk management can be done with futures or other derivatives • In this way, firms can integrate risk management and asset allocation

48. CONCLUSIONS • The value of accurate daily correlations is moderate – maybe 5% of the required return. Possibly why asset allocation is done monthly and ignores covariances. • On some days, the value is much greater. Possibly why risk management is done daily. • Additional value may flow from coordinating these decisions.