Econometric Game 2006

Transcription

1 Econometric Game 2006 ABN-Amro, Amsterdam, April 27 28, 2006 Time Variation in Asset Return Correlations Introduction Correlation, or more generally dependence in returns on different financial assets or asset classes lies at the heart of modern portfolio theory and risk management. The possibilies for diversification crically depend on the sign and magnude of those correlations. Similarly, an adequate measure of risk of a portfolio requires a reliable estimate of the correlation between the assets in the portfolio; and more importantly, the validy of such estimates in times when really matters should be confirmed. One could choose to estimate population correlation coefficients by their sample analog based on a long sample of historical data. However, growing evidence has indicated that correlations are subject to time variation. This case centers around testing for this time variation, estimating models for time-varying correlations, and analyzing the implications for financial applications. The case The case consists of two parts, to be analyzed on the two consecutive days of the Econometric Game 2006, Thursday April 27 and Friday April 28. The first part of the case (to be analyzed on Thursday) centers around the specification, estimation and testing of econometric models for time-varying asset correlations. The time-variation in correlations is in line wh the well-known fact that the volatily of asset returns changes over time, a phenomenon that has lead to the populary of so-called GARCH models, and of the exponentially-weighted movingaverage approach advocated by RiskMetrics. Similar approaches can be devised for covariances and correlations, and this is the topic for the first day. The main reference here is the article Dynamic Condional Correlation: A Simple Class of Multivariate GARCH Models, by the Nobel laureate Robert F. Engle. The second part of the case (to be analyzed on Friday) focuses on the financial implications of asset correlations and their time-variation. A number of different topics are to be addressed here. First, the implications and relevance of correlation variation for portfolio selection and asset pricing are to be considered. Next, issues involving (portfolio) risk management are analyzed. Finally, the effect of correlation on a particular financial derivative will be addressed. 1

2 Data Two data-sets are provided for analysis, both in Excel and in EViews format. The first data-set (indices.xls and indices.wf1) consists of historical daily closing prices of a number of international stock market indices (in addion to a US bond index and a US Tbill rate), over the period January 1981 through December 2005 (not all series are available over the full period). The next data-set (stocks.xls and stocks.wf1) contains daily prices, over the same period, of the 30 stocks that are currently listed in the DowJones index. Both data-sets have been obtained from Datastream. All variables (except the TBill rate) are prices or indices, and should be converted to daily returns by taking log-differences for the estimation and analysis of correlations. No special adjustments have been made for bank holidays (on these days, the most recent closing prices are given, such that the return is zero; a dummy variable called notrade has been included to indicate these days). 2

3 Part 1 (Thursday) Inially focus on the correlation between two of the main US stock market indices: the S&P500 index and the Nasdaq (compose) index. The objective is to come up wh a model and corresponding estimate of the (time-varying) correlation between the returns on these two indices. The following ingredients may provide some guidance: First estimate a constant correlation over the full sample, and then over suably chosen subsamples. For example, you may investigate if the correlation differs over subsequent n-year periods, or whether differs between periods of bull and bear markets or periods of high and low market volatily. An idea of how to choose different periods may also be obtained from rolling window correlations (i.e., based on samples t {1,, n}, t {2,, n+ 1}, and so on, for some suable choice of n). Check whether the variation is statistically significant by constructing and carrying out an appropriate test for the hypothesis that the correlation is constant over the 25 years of the sample. A popular econometric model for time-varying correlations is Engle s dynamic condional correlation (DCC) model. This model starts from a suably chosen and estimated GARCH model for the two separate returns, and then continues wh an analysis of the time-varying correlation between the standardized residuals of these GARCH models. Thus, if the univariate GARCH models for the 2 two returns r, i= 1,2, are given by r ~ N( μ, σ ), then the second step involves the analysis of the correlation between zˆ ( ˆ )/ ˆ = r μ σ. Estimation of univariate GARCH models is straightforward in EViews or Stata; estimation of the second-step correlation model, as described in the article by Engle, requires programming and maximizing a likelihood function. Simpler solutions, which do not require such a likelihood analysis, are also possible, such as an exponentiallyweighted moving-average correlation estimator. Different models (constant correlation, different correlation over subsamples, DCC) can be compared in the degree to which they pass various misspecification tests. Some suggestions for such tests are given in Engle s article. Apply these tests or develop your own tests, discuss their outcome, and against which alternative they provide evidence. If none of the models pass the tests, try to come up wh extensions of your model to fix this. It is often thought that correlations increase in times of financial distress, in particular during stock market crashes. If this is true, then that would imply that portfolio diversification fails at times when is most needed. Examine whether the models considered earlier are able to describe such an increasing correlation in bad times ; if not, think of extensions of the models that can capture this, and try to test for this particular type of asymmetric response in correlation to posive and negative shocks. If time perms, would be useful to extend the analysis to other pairs of stock market indices. Reference Engle, R.F. (2002), Dynamic condional correlation: A simple class of multivariate GARCH models, Journal of Business & Economic Statistics, 20,

4 Part 2 (Friday) In this second part of the case, some implications of (time-variation in) correlations in asset returns are to be analyzed, based on the models and correlations estimated in the first part. The attention should be focused on three aspects. Because analyzing each topic may be too time-consuming, you may choose to analyze two out of these three issues. 1. Porfolio selection and asset pricing. Covariances play a central role in modern portfolio theory. Incorrectly imposing constant variances and covariances could lead to incorrect portfolio weights, and possibly incorrect conclusions about the possibily of abnormal returns after correcting for market risk (via a capal asset pricing model [CAPM] regression). Try to assess how important correlation variation really is for this purpose. For example, try to construct a dynamic portfolio of the S&P500 index and one or more other (European) indices which minimizes the portfolio variance of the combination at each point in time (hence wh time-varying weights), and compare this wh the variance of a portfolio based on a constant covariance matrix (hence constant portfolio weights). Alternatively, compare the results of a constant beta CAPM regression of one or more individual stock returns on the S&P500 (market) return wh the results when you allow for a time-varying beta (defined by the time-varing covariance of stock and market return divided by the time-varying variance of the market return). This may be based on a DCC-type model analyzed yesterday, but also on another time-varying correlation model. 2. Portfolio risk measurement. A portfolio consisting of two assets (such as the S&P500 index and the Nasdaq index), wh fixed portfolio weights, has a (timevarying) volatily (standard deviation) depending on the volatilies of the two assets and the correlation between them. The portfolio s volatily may be used as a measure of s risk, or used to construct other risk measures such as Value at Risk. Investigate the contribution of time-varying correlations to the portfolio risk, by comparing the risk measure for each day under constant correlation (but timevarying volatilies) wh the corresponding measure wh time-varying correlations. Engle (2002) also discusses the use of correlation models in constructing Value at Risk, and corresponding testing procedures. 3. Derivative pricing. Consider the effect of (variation in) correlation in the following financial product. At time t = 0, we invest an amount of 100 euro. After a month, we receive the inial amount plus the return on, which is defined as the maximum of the return on the S&P500 index, the Nasdaq index, and 4%. Thus, we receive the best payoff of the two stock indices and a bond paying an interest rate of 4%. The value of such a product is the discounted value (using the risk-free interest rate) of the expected payoff of the product (i.e., the maximum of the 3 returns), using the risk-neutral distribution. A recent article that investigates the value of such products, but using different (copula-) methods to characterize the dependence, is van den Goorbergh et al. (2005); see also Duan (1995) for the use of GARCH models in univariate option pricing. You are asked to investigate the value of such a financial product, by simulating a month (20 days) of daily returns using your volatily and correlation model, calculating the payoff (from the sum of daily returns), repeating this a large number of times, and calculating the average (discounted) payoff over all replications. This can be done for (i) constant volatily and correlation, (ii) GARCH volatily and constant correlation, and (iii) 4

5 GARCH volatily and DCC correlation. Again the question is: what is the contribution of time-varying correlations to the value? References Duan, J.-C. (1995), The GARCH option pricing model, Mathematical Finance, 5, Goorbergh, R.W.J. van den, C. Genest and B.J.M. Werker (2005), Bivariate option pricing using dynamic copula models, Insurance: Mathematics and Economics, 37,