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ForecastingVolatility of Portfolios of Indexes

ForecastingVolatility of Portfolios of Indexes. A Portfolio Management’s Perspective Silverio Foresi Quantitative Strategies Group Goldman Sachs Asset Management. Key Points. Smoother Models, Multivariate Volatility decoupling volatilities from correlations Portfolios Diagnostics

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ForecastingVolatility of Portfolios of Indexes

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  1. ForecastingVolatility of Portfolios of Indexes A Portfolio Management’s Perspective Silverio Foresi Quantitative Strategies Group Goldman Sachs Asset Management

  2. Key Points • Smoother Models, Multivariate Volatility • decoupling volatilities from correlations • Portfolios Diagnostics • optimal portfolios and their problems • scatterplot diagnostics work with Giorgio De Santis and Adrien Vesval at GSAM

  3. Covariance Matrix Returns in excess of expectations

  4. Portfolio Variance • Variance • Variance of portfolio w

  5. Estimating the Covariance Matrix

  6. Forecasting with Smoothers • Moving averages of n obs • Exponential smoothing of n obs • P controls “persistence” of vol estimates

  7. Generalized Smoothers • Vec model (Engle) • Very highly parameterized • Special cases • Baba, Engle, Kroner, and Kraft • C = 0, AB appropriate scalars: exp. smoother

  8. Volatilities and Correlations • Variance decomposition • Constant correlation model (Bollerslev) • univariate GARCH for volatilities • constant correlations • Dynamic correlation (Engle , Engle & Sheppard) • univariate GARCH for D • time-varying R estimated by exponential smoothing on the residuals of GARCH

  9. Multi-Decay Model • Different persistence for volatilities, Pi • Common persistence for correlations, Pcor • Assembly

  10. Multi-Decay Model (2) • Q-MLE: estimates for Pcor, PEQ, PFI, PFX • …Pcor = PEQ = PFI = PFX? • No • Volatilities differ by asset class PEQ = 0.65, PFI = 0.66, and PFX= 0.58 • Correlations move more slowly than volatilities Pcor = 0.97 • Details: De Santis-Vesval (2001)

  11. Diagnostics on Portfolio Vol

  12. Interesting Portfolios • Mean/Variance • Global Minimum Variance • Minimum Tracking Error ... optimal portfolios depend on estimated covariance matrix

  13. Problem with Optimal Portfolios • Example: true PEQ is low • if one uses low PEQ =Pcor : optimal portfolios change too fast (chasing correlations) ... underestimate vol • use high PEQ may look better on average • Implications • it may help decouple correlations from volatilities • it may help to have diagnostics based on portfolios independent of estimated covariance

  14. Other Interesting Portfolios • Equilibrium • Equally weighted • Random ... do not depend on estimated cov

  15. Experiments Pcor = PEQ • Simulated Data • Random Portfolios, no benchmanrk, sum to 1 • Real Data: • Equity Returns for 18 Country Indexes from 5/5/1983 to 3/29/2002 (4931 daily obs) • Random Portfolios, no benchmanrk, sum to 1 • Optimal Portfolio • cash benchmark (zero beta) • target TE = 7.5%

  16. Diagnostics: Smoothing Forecasts Consider many portfolios wp(p = 1, 2, ... ,P) • Calculate Vart-1(wp’rt) for all t,p • Assign portfolios to bins,by forecastedVar • average forecasts of all portfolios in every bin • average realizations in every bin Average out forecast error • Diagnostics based on scatterplot • realizations vs forecasts • vs other characteristics (past e2, corr?) • nonlinearities

  17. Simulated Data Random Portfolios

  18. More Diagnostics Standardized portfolio residuals Distribution • Unconditional • is E(e) = 0 and E(e2) = 1? • Conditional: can we explain e or e2 -1 with • lags of past e and e2? (autocorrelation) • sign of past e or e2 -1?

  19. Simulated Data Random Portfolios

  20. Simulated Data Random Portfolios vol sinusoidal

  21. Real Data Random Portfolios Interesting region

  22. Real Data Min TE Portfolio

  23. Real Data Min TE Portfolio

  24. Back to Key Points • Decoupling volatilities from correlations • Correlations move more slowly than vols • Portfolios Diagnostics • Random portfolios allow us to average out forecast errors • Nonlinearities (scatterplot diagnostics)

  25. Diagnostics for Covariance Matrix Use estimated Covariance to standardize residuals Distribution • Unconditional • is E(e) = 0 and E(e * e’) = I? • Conditional: can we explain e or e2 -1 with • lags of past e and e2? (autocorrelation) • sign of past e or e2 -1?

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