Modeling the Time Varying Dynamics of Correlations: Applications for Forecasting and Risk Management

Modeling the Time Varying Dynamics of Correlations:Applications forForecasting and Risk Management Michael Jacobs, Jr. Senior Financial Economist Credit Risk Modeling, Risk Analysis Division Office of the Comptroller of the Currency Ahmet K. Karagozoglu* Associate Professor, Department of Finance Frank G. Zarb School of Business Hofstra University * Corresponding author. Karagozoglu acknowledges the Summer Research Grant from the Frank G. Zarb School of Business which partially supported this research.  The views expressed herein are those of the authors and do not necessarily represent a position taken by of the Office of the Comptroller of the Currency or the U.S. Department of the Treasury. XLII meeting of the Euro Working Group in Financial Modeling

Time Varying Dynamics of Correlations • We evaluate the time series correlation modeling techniques and document the effectiveness of various correlation forecasting models for different asset types • Time varying correlations are computed from different moving windows (rolling window moving average - RWMA) • We compare the properties of the RWMA correlations to Engle (2002) dynamic conditional correlation (DCC) model estimates • Applications to portfolio modeling XLII meeting of the Euro Working Group in Financial Modeling

Volatility and Correlation • Accurate measurement and estimation of volatility and correlation are essential to many facets of finance • Portfolio management • Asset pricing-including complex derivative instruments • Risk management • Modeling time varying dynamics of volatility and correlations XLII meeting of the Euro Working Group in Financial Modeling

Volatility and Correlation • In recent years, advanced models of volatility have been augmented to simultaneously take into account the time varying dynamics of correlations between assets, in order to improve the pricing of derivatives securities, and to enhance risk management methods • Explosive growth in credit default derivatives owes this success in part to the development of correlation modeling techniques XLII meeting of the Euro Working Group in Financial Modeling

Dynamic Conditional Correlations • In portfolio management and optimization, term structure models (either of fixed income or futures), or large-scale vector autoregressions require the estimation of large time-varying covariance matrices • Engle et al (2001) and Engle (2002) develop a new class of multivariate dynamic conditional correlation (DCC) models • Engle shows that these models have the flexibility of univariate GARCH models, coupled with parsimonious parametric models for the correlations XLII meeting of the Euro Working Group in Financial Modeling

Dynamic Conditional Correlations • Prior to Engle’s work, time varying correlations have been estimated using simple univariate methods, such as rolling historical correlations and exponential smoothing, or with multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models that are linear in squares and cross products of the data • Dynamic conditional correlation (DCC) models provide more precise forecasts of future realized correlations, therefore aiding in the development of derivatives on correlations XLII meeting of the Euro Working Group in Financial Modeling

Literature • Engle (2002) discusses and analyzes the performance of the dynamic conditional correlation (DCC) model, introduced in a multivariate setting by Engle and Sheppard (2001), in a bivariate context and show that bivariate version of the DCC performs favorably with other estimators, such as multivariate GARCH • Audrino and Barone-Adesi (2006) utilize a multivariate GARCH model that allows for time-variations in the second moments, to estimate the time-varying conditional correlations by means of a convex combination of averaged correlations and find some evidence that constant correlation models (i.e. sample averages of correlations) outperform various more sophisticated models in forecasting the correlation matrix XLII meeting of the Euro Working Group in Financial Modeling

Literature • Kwan (2006) identifies some additional analytical properties of the constant correlation model and relates them to familiar portfolio concepts. By comparing computational times for portfolio construction, with and without simplifying the correlation matrix in a simulation study, he presents evidence for the model's computational advantage. • Billio and Caporin (2006) develop a generalization of the Dynamic Conditional Correlation multivariate GARCH model of Engle (2002), which introduces a block structure in parameter matrices, allowing for interdependence with a reduced number of parameters and apply their model to the Italian stock market and compare alternative correlation models for portfolio risk evaluation. XLII meeting of the Euro Working Group in Financial Modeling

Literature • Engle and Colacito (2006) evaluate alternative models of variances and correlations with an economic loss function, in the process constructing portfolios to minimize predicted variance subject to a required return and find that on average, dynamically correct correlations are worth around 60 basis points in annualized terms. • Wang and Nguyen (2007), using forward forecasting tests on dynamic conditional correlation (DCC), test contagion between Taiwanese and US stocks under asymmetry. XLII meeting of the Euro Working Group in Financial Modeling

Models for Correlation Estimation • We may define the conditional correlation between two zero-mean random variables ri and rj at time t as: • Therefore, the conditional correlation at time t will rely on information known at time t-1. This quantity is guaranteed to lie in the interval [-1,1] for possible realizations of these random variables as well as their linear combinations. 3.1.1 XLII meeting of the Euro Working Group in Financial Modeling

Models for Correlation Estimation • A simple method of the conditional correlation is given by the rolling window moving average estimator for length k (RWMA-k): • While (3.1.6) has the property that it is a well-defined correlation lying in [-1,1] for all t and k, it is not establish if (3.1.6) consistently estimates (3.1.1), and there is little guidance in how to choose the window k in practical applications. 3.1.6 XLII meeting of the Euro Working Group in Financial Modeling

Models for Correlation Estimation XLII meeting of the Euro Working Group in Financial Modeling

Evaluating Correlation Forecasts • We compare the RWMA estimators to the DCC estimates by various means: • Examine the distributional properties of the estimators. • Analyze measures of association, such as linear correlation and concordance between the RWMA and DCC estimators • We evaluate how effectively the correlation estimators perform in portfolio construction. • We consider various measures of predictive accuracy, ranging from the very standard point estimates of mean-squared error (MSE) type metrics commonly employed, to more general tests that can assess the statistical significance of the divergence between models, e.g. Diebold and Mariano (1995). XLII meeting of the Euro Working Group in Financial Modeling

Data • Sources: • Commodity Research Bureau (CRB) • Bloomberg • Daily data January 1995 to November 2007 • 9 asset groups, 33 variables, 528 pairs of correlations • 6 different RWMA correlation estimates: • 22, 66, 126, 252, 504 and 756 day rolling windows 1, 3 and 6 months; 1, 2 and 3 years • DCC estimates with optimal GARCH(p,q)’s XLII meeting of the Euro Working Group in Financial Modeling

Data XLII meeting of the Euro Working Group in Financial Modeling

Correlation Estimates: RWMA vs DCCCorrelation between S&P 500 and GS-Commodity Index XLII meeting of the Euro Working Group in Financial Modeling

Dynamics of RWMA Correlations XLII meeting of the Euro Working Group in Financial Modeling

Concordance between RWMA & DCCCorrelation between Commodity Variables and others XLII meeting of the Euro Working Group in Financial Modeling

Correlation between RWMA & DCCCorrelation between Commodity Variables and others XLII meeting of the Euro Working Group in Financial Modeling

Concordance between RWMA & DCCCorrelation between Currency Variables and others XLII meeting of the Euro Working Group in Financial Modeling

Correlation between RWMA & DCCCorrelation between Currency Variables and others XLII meeting of the Euro Working Group in Financial Modeling

Concordance between RWMA & DCCCorrelation between Real Estate Variables and others XLII meeting of the Euro Working Group in Financial Modeling

Correlation between RWMA & DCCCorrelation between Real Estate Variables and others XLII meeting of the Euro Working Group in Financial Modeling

Impact of Holding Period on CorrelationsCorrelation between S&P500 and 10yrT-Bond XLII meeting of the Euro Working Group in Financial Modeling

Portfolio ConstructionUsing correlation between S&P500 and 10yrT-Bond XLII meeting of the Euro Working Group in Financial Modeling

Conclusions • RWMA Correlations exhibit different patterns based on rolling windows • Relationship between RWMA and DCC estimates differ across assets groups • Only in daily holding period RWMA Correlations appear to be closer to DCC • DCC portfolios outperform all of the RWMA estimations in terms of volatility of portfolio returns (for S&P500 and 10yr Tbond) XLII meeting of the Euro Working Group in Financial Modeling

Modeling the Time Varying Dynamics of Correlations: Applications for Forecasting and Risk Management