Time Variation in Asset Return Correlations: Econometric Game solutions submitted by Oxford University

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1 Time Variation in Asset Return Correlations: Econometric Game solutions submitted by Oxford University June 21, 2006 Abstract Oxford University was invited to participate in the Econometric Game organised by VSAE for the first time in April We were delighted to enter a team of six, consisting of four M.Phil students and two D.Phil students. We had a thoroughly enjoyable 2 days in Amsterdam and found the organisers, other teams and locals to be very welcoming and friendly. To our astonishment and delight we won the Game, and this article outlines our solutions to the cases on Time Variation in Asset Return Correlations. 1 Day 1 The Game consisted of two cases over the two days, based on analysing Time Variation in Asset Return Correlations. The first part of the case asked us to assess the evidence for time varying asset return correlations. As preliminary data analysis, we calculated the correlations between returns for the NASDAQ and S&P500 over rolling windows. Fitting a cubic spline to the correlation data gave us an occular impression of the changing correlations. There was strong evidence of time variation in asset return correlations, and so we estimated dynamic conditional correlation (DCC) models based on Engle (2002). Two different specifications were estimated. For the first set of estimations a GARCH model was estimated, while in the second specification an EGARCH model was estimated instead. Using the residuals from the first stage estimation, a second stage estimation was conducted to obtain estimates of the parameters of the time-varying correlation matrix. The estimation method for the second stage was maximum likelihood. All stages of these estimations were implemented using the MATLAB m. files written by Kevin Sheppard (see: 2 Day 2 The second part of the case focused on the financial implications of asset correlations and their time-variation. We were asked to consider the implications and relevance of correlation variation for portfolio selection and risk management. We first examined the implications of time-varying correlations for the CAPM model using data on a number of stocks listed in the DowJones index, and secondly we computed the Value at Risk measure to investigate the contribution of time-varying correlations to the portfolio risk. The report that we submitted for the second day of the game is printed below. 3 Report submitted for Case Portfolio selection and asset pricing Analysis for the CAPM model was based on a sub-set of the stocks data. We obtained a balanced panel by excluding six of the stocks in which the sample does not commence on 01/01/1981. This results in a panel of 24 stocks with 6312 observations. We did not use the no trading days data given 1

2 2.00 Time varying beta (ALCOA) Constant CAPM beta (ALCOA) Figure 1: Time-varying DCC and constant parameter CAPM βs for ALCOA in the sample as this was inconsistent with the indices data, and used the indices data to determine the number of no trading days. 1 It should be noted that there are a number of outliers in the data for both individual stocks and indices, which is evident by a simple plot of the distribution of the data. Furthermore, all returns series fail normality. These outliers can be identified by the outlier detection option in PcGets, although observe that as there are regime changes throughout the sample period, primarily due to the burst of the dot-com bubble, such that sub-samples are used to detect the outliers based on sub-sample equation standard errors. Also observe that indicator saturation techniques developed by Hendry, Johansen and Santos (2005), which detect both outliers and regime-changes, could be implemented, although its performance on GARCH data has not been established. Notable outliers include Black Monday on 19/10/1987, the impact of 9/11, the start of the Gulf War on oil and energy prices, and November and December 2000, in which the US elections had a substantial impact. Obviously, robustness of the results could be assessed by removing these outliers. The importance of the varying correlations between individual stock returns and the market is best highlighted by constructing a constant β CAPM model and comparing it to an optimal portfolio resulting from Engle s (2002) 2-step DCC approach. In table 3.1, we report the α and β coefficients of all 24 stocks over the sample period ( ). 2 Unsurprisingly, the α intercept is very close to 0 and statistically insignificant. We utilize the positive and highly significant β coefficients to construct our constant CAPM investment portfolio and report the average annualised return in table 2. To illustrate the difference between the time-varying DCC CAPM β and the constant CAPM β, we plot both over the full sample period for ALCOA inc. in Figure 1 and find that the constant β corresponds to the average of the time-varying βs. This is also the case for the remaining 23 stocks. As a proof that Engle (2002) also made a contribution to the financial industry as well as the academic literature, we form an investment portfolio on the basis of the time-varying and the constant β. As illustrated in table 2, the mean annualized return of the time-varying DCC β portfolio equals to 11.21% and is a substantially higher return compared to the constant β portfolio return of 10.55%. Furthermore, the time-varying β portfolio return exhibits a lower annualized standard deviation. The market indices are highly correlated during times of crisis. Our DCC model allows us to examine whether this is true of the individual stocks that form the indices. Figure 2 shows correlation between ALCOA and American Express and ALCOA and ALTRA Group. During the recent time 1 The definition of a no trading day is given as one in which there is no change in stock price for 2 consecutive days for 3 given stocks. Hence, there is a difference between the number of defined no trading days between individual stocks and indices. 2 We construct our CAPM using the S&P500 as a proxy for the market return. Undoubtedly, the S&P500 does not capture the whole market portfolio but this seams a good approximation for our purposes. 2

3 α σ α α t-stat β σ β β t-stat 3M ALCOA ALTRIA GROUP INCO AMERICAN EXPRESS AMERICAN INTL.GP BOEING CATERPILLAR COCA COLA DU PONT E I DE NEMOURS EXXON MOBIL GENERAL ELECTRIC GENERAL MOTORS HEWLETT-PACKARD INTEL INTERNATIONAL BUS.MACH JP MORGAN CHASE & CO JOHNSON & JOHNSON MCDONALDS MERCK & CO PFIZER PROCTER & GAMBLE UNITED TECHNOLOGIES WAL MART STORES WALT DISNEY Table 1: Time invariant CAPM parameters obtained from OLS regressions of stock returns on the S&P500 constant β varying β mean annualised return 10.55% 11.21% annualised st. Dev 18.99% 18.35% Table 2: Optimal dynamic portfolio with fixed and time-varying βs 3

4 0.50 COV(ALCOA, American Express) COV(ALCOA, ALTRA Group) Figure 2: Time-varying correlations between ALCOA and American Express and ALCOA and ALTRA Group. of crisis, the correlation between ALCOA and American Express stocks was relatively low despite the correlation between market indices being high during this time. In contrast the correlation between ALCOA and ALTRA group stocks followed the same pattern as that between market indices. Implications for investment strategies are reported below. 3.2 Portfolio risk measurement We investigated the impact of allowing for time-varying correlation on the measurement of portfolio risk. To do so, we looked at a portfolio composed of equal shares of NASDAQ and S&P500, and computed the Value at Risk (VaR) for each day, first assuming constant correlation, then permitting the correlation to vary over time. This was done first using a rolling correlation estimator, and then the more satisfactory DCC approach. On the basis of the first part of the case, we would expect that since correlations tend to increase in times of financial distress, the model with a constant correlation would underestimate the VaR at such times, which could have important implications for investment strategies. Our results are displayed in Figure 3. In the first graph we see the ratio of the 95% VaR computed using constant correlations over the 95% VaR computed using the rolling correlation estimator. Until the burst of the dot-com bubble (approximately observation 5000) no systematic bias is discernible, but the two measures often differ by up to 2%. However throughout the period post 2000 the ratio is below one, which implies that the constant correlation VaR consistently underestimates the true value. In Figure 4 we see the ratio of the 95% VaR computed using constant correlations over the 95% VaR computed using the DCC estimator. We again can observe the same phenomenon as in the first graph. However the bias appears to be less severe. We explain this by the fact that the rolling estimator has only a very short memory whereas the correlations calculated under DCC approach are informed by the entire history of the process. As a result, the period of high correlations immediately following the bursting of the bubble raises the rolling window estimates of the VaR above those found in the DCC model. Thus, our results indeed confirm that allowing correlations between components of their portfolios to vary over time may improve investors assessment of risk. 3.3 Conclusion We find strong evidence of time-varying correlations between the returns on both indices and individual stocks. The correlations between returns on assets increase during periods of high volatility, evident by the increases from 2000 onwards, around the time of the dot-com bubble. Establishing 4

5 3M ALCOA ALTRIA AMERICAN AMERICAN BOEING GROUP EXPRESS INTL.GP. ω κ λ σ ω σ σ σ λ CATER COCA DU PONT E EXXON GENERAL GENERAL PILLAR COLA I DE NEMOURS MOBIL ELECTRIC MOTORS ω κ λ σ ω σ σ σ λ HEWLETT INTEL INTERNATIONAL JP MORGAN JOHNSON & MCDONALDS PACKARD INTEL BUS.MACH. CHASE & CO. JOHNSON ω κ λ σ ω σ σ σ λ MERCK PFIZER PROCTER & UNITED WAL MART WALT & CO. GAMBLE TECHNOLOGIES STORES DISNEY ω κ λ σ ω σ σ σ λ S&P500 ω α κ β λ σ ω σ α σ σ σ β σ λ Table 3: Parameter estimates for the DCC model with 24 stocks and S&P500 index. 5

6 Ratio VaR constant corr / rolling window corr Figure 3: VaR ratio of constant correlations to rolling correlation estimator Ratio VaR constant corr / DCC Figure 4: VaR ratio of constant correlations to DCC estimator 6

7 causal links is more difficult; periods of high volatility may drive increases in correlations due to common shocks, or a structural change may have occured which has resulted in a permanent shift in the level of correlations. The implications for investment decisions are substantive. If we accept the hypothesis that higher volatility drives an increase in correlations, risk diversification is most difficult precisely during those periods in which it is most required. During periods of high volatility it is difficult to diversify risk, even between different type of assets (e.g. high-tech versus low-tech stocks) although there is evidence that correlations do not increase to the same extent between US and European assets. Hence, risk diversification may be achieved through international markets. 4 References Engle, R. F. (2002). Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models. Journal of Business & Economic Statistics, 20(3), pp