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

Michael Jacobs, Jr.1

Office of the Comptroller of the Currency

Ahmet K. Karagozoglu2

Hofstra University

Draft: December 2007

J.E.L. Classification Codes:

Keywords: Correlations, Forecasting, GARCH, DCC, Risk Management

1 Senior Financial Economist, Credit Risk Modelling, Risk Analysis Division, Office of the Comptroller of the Currency, 250 E Street SW, 2nd Floor, Washington, DC 20024, 202-874-4728, [email protected]. 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.2 Associate Professor, Hofstra University, Zarb School of Business, Department of Accounting and Finance.

mailto:[email protected]

Abstract

In this study we compare the time correlation modeling techniques, and document the effectiveness of various correlation forecasting models for different asset types, using a broad database from Commodity Research Bureau (CRB) and Bloomberg. First, examine time varying correlations are computed from different moving windows and pairs of assets, and build time series models to forecast correlation at different horizons. We then compare the properties of the simple correlation estimates and there forecasts to the Engle (2002) dynamic conditional correlation (DCC) model.

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1. Introduction and Summary Measurement and estimation of volatility and correlation are essential for pricing complex financial instruments as well as successful risk management practices. In the field of finance, we have known about the curial nature of correlations between assets starting with the fundamentals of portfolio theory, and that of volatility starting with the early option pricing models. Advances in econometrics have provided us with models that more accurately describe the time varying dynamics of volatility, such as ARCH / GARCH, as well as stochastic volatility models. These advances allowed financial economists to develop precise methods of pricing complex derivative securities, as well as facilitated more advanced and reliable risk management modeling techniques. Once practitioners started utilizing econometric methods that more accurately model the time varying dynamics of volatility, they could forecast future realized volatility, which paved the way for derivative instruments (such as variance swaps. volatility futures as well as options) that allow trading of volatility for risk management purposes.

In recent years, these 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. The next step in utilizing these enhanced correlation modeling techniques is forecasting correlations and the development of derivative securities on correlation between assets. These instruments would allow trading correlation risk and help with the management of risk that arises from changes in correlation between such assets. The explosive growth in credit default derivatives owes this success in part to the development of correlation modeling techniques.

Engle et al (2001) and Engle (2002) develops a new class of multivariate dynamic conditional correlation (DCC) models. He shows that these models have the flexibility of univariate GARCH models, coupled with parsimonious parametric models for the correlations, stating that “they are not linear but can often be estimated very simply with univariate or two step methods based on the likelihood function”. His paper presents evidence that they perform well in a variety of situations and give sensible empirical results. 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.

The purpose of this research project is first to compare the time varying volatility and correlation modeling techniques, and second to document the effectiveness of various correlation forecasting models for different asset types, using a broad database from Commodity Research Bureau and Bloomberg. These data will be representative of broad asset classes of

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relevance to portfolio and risk management as well as trading applications: various equity, commodity, currency, real estate and credit related indices; as well as individual assets such as pairs of different currencies, debt instruments, futures contracts etc. The empirical exercise will start with employing rolling historical correlations for different windows, describing the properties of these estimators for different pairs of assets, and building time series models to forecast correlation at different horizons. This will then be compared with exponentially smoothed correlation forecasts (used by RiskMetrics), as well as constant and dynamic correlation GARCH models, including Engle’s (2002) dynamic conditional correlation (DCC) model. It is expected that such empirical results would provide evidence for the applicability of different correlation forecasting models for various asset classes, highlighting the differences in univariate and multivariate modeling of correlations.

This paper will proceed as follows. In Section 2 we review the relevant literature in this area. In Section 3, we describe our empirical methodology, the moving window correlation estimation, exponentially smoothed and the DCC model. In Section 4 we describe the data and summary statistics. In Section 5 we report our estimation results and the comparison of the different correlation models. In Section 6 we conclude and provide directions for future research.

2. Review of the Literature

Gibson and Boyer (1998) develop a forecast evaluation methodology based on option pricing by forecasting the variance-covariance matrix of joint asset returns and using it in turn to generate a trading strategy for a package of simulated options. They state that the most accurate forecast will produce the most profitable trading strategy and that the package of simulated options can be chosen to be sensitive to correlation, to volatility, or to any arbitrary combination of the two. In their empirical application, they focus on the ability to forecast the correlation between two stock market indices. However, Engle (2002) indicates that for most asset classes implied correlations are not available, therefore applications of Gibson and Boyer methodology had been limited to pairs of underlying assets with active options markets.

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. Citing a voluminous literature (Bollerslev et al (1988), Bollerslev (1990), Engle and Kroner (1995), Engle and Mezrich (1996)), the author notes that traditionally time varying correlations have been estimated in multivariate GARCH frameworks, which are linear in the squares and cross-products of the data. He proposes that various implementations of the DCC model, which can capture the dynamics of the correlations structure independently of the volatility structure, yielding

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a model non-linear in the 2nd moments of the data yet retaining the tractability of univariate GARCH models. This is accomplished by a 2 step procedure, in which the volatility structure is first modeled by a series of univariate GARCH estimations, and then the covariance structure is modeled in a second step. According to various metrics, the bivariate version of the DCC performs favorably with other estimators, such as multivariate GARCH.

Guo (2003) examines the currency risk hedge when volatilities and correlations of forward currency contracts and underlying assets returns are all time-varying. He utilizes a multivariate GARCH model with time-varying correlations to fit the dynamic structure of the conditional volatilities and correlations, and estimates the conditional risk-minimizing hedge strategies, for an international portfolio of the US, UK and Switzerland stocks.

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, across all series, and dynamic realized correlations. They back-test the models on a six-dimensional exchange-rate time series using different goodness-of-fit criteria and provide empirical evidence of their strong predictive power. There is some evidence that constant correlation models (i.e. sample averages of correlations) outperform various more sophisticated models in forecasting the correlation matrix, an important input component for portfolio analysis. 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.

Hamerle, Liebig and Scheule (2006) tackle one of the main challenges of forecasting credit default risk in loan portfolios, i.e. forecasting the default probabilities and the default correlations. They derive a Merton-style threshold value model for the default probability that treats the asset value of a firm as unknown and use a factor model instead to demonstrate how default correlations can be easily modeled within this framework.

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. They apply this model to the Italian stock market and compare alternative correlation models for portfolio risk evaluation.

Wang and Nguyen (2007), using forward forecasting tests on dynamic conditional correlation (DCC), test contagion between Taiwanese and US stocks under asymmetry. Their empirical methodology first uses the iterated cumulative sums of squares (ICSS) algorithm to detect the structural breaks

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of market returns, creates dummy variables for breaks, estimates an EGARCH model of the conditional generalized error distribution, and finally computes dynamic conditional correlation coefficients of the DCC multivariate GARCH model.

3. Empirical Methodology and Econometric Models

We may define the conditional correlation between two zero-mean random variables ri and rj at time t as:

(1)

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. Note that we may define an h-step ahead forecast, denoted

similarly and it will posses the same properties:

(2)

Note that we may express (1) in terms of the standardized residuals , if we define the conditional standard deviation in terms

of :

(3)

(4)

Then it follows that the conditional correlation of the returns is just the conditional correlation of the standardized residuals:

(5)

A simple method of the conditional correlation (1) or (5) is given by the rolling window moving average estimator for length k (RWMA-k):

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(6)

While (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 (6) consistently estimates (1), and there is little guidance in how to choose the window l in practical applications. It may be undesirable that equal weight is given to returns from t-1 to t-k-1, and then zero weight to returns t-k-2 and earlier. An alternative in the same spirit as (6) is the exponentially weighted moving average, with smoothing parameter λ (EWMA- λ):

(7)

This leaves us the task of choosing the parameter λ, which RiskMetrics has chosen to be 0.94 for all assets. In lieu of these, we choose to treat λ as an unknown model parameter, and (under a normality assumption for εl,t) estimate this by maximum-likelihood (ML). We can do this by defining the conditional covariance of returns matrix in terms of λ:

(8)

Then under the model (7), the covariance matrix is seen to be an exponentially weighted average of the sample return cross-product matrix:

(9)

The parameter λ can be estimated by maximizing the log-likelihood function:

(10)

This is generalized in the multivariate GARCH framework, which has been typically formulated such that the elements of variance-covariance matrix are linear functions of the squares and cross-products of the returns, as in the model of Engle and Kroner (1995):

(11)Where vec(.) is the vectorization operator that stacks the elements of an

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matrix into an vector and . A problem with this specification is that the positive-definiteness of cannot be guaranteed. A useful restriction of (11) suggested by Engle and Kroner (1995) is:

(12)

The model that we consider here, the dynamic conditional correlation (DCC) of Engle et al (2001) and Engle (2002), is a generalization of the constant conditional correlation (CCC) model of Bollerslev (1990), which can be written as:

(13)

(14)

Where the are a series of univariate GARCH models. While there are various ways in which to parameterize the conditional correlation matrix , the one advocated by Engle (2002) is:

(15)

(16)

(17)

(18)

Where is an n-vector of ones, is the unconditional correlation matrix of and denotes Hadamard element-by-element multiplication. This follows the follows the MARCH specification for of Ding and Engle (2001), which guarantees its positive-semidefiniteness conditional on being positive-semidefinite. Engle (2002) shows that this is equivalent to expressing its elements as univariate GARCH(1,1) processes:

(19)

Where is the unconditional correlation between and , which gives rise to the correlation estimator:

(20)

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It is shown in Engle et al (2001) and Engle (2002) that the likelihood function in the DCC model is the sum of 2 components, the individual GARCH volatilities and a correlation term:

(21)

(22)

(23)

Where and denote the parameters of the volatility and correlations processes, respectively, and denotes the univariate GARCH volatilities. This suggests a 2-stage procedure, and indeed Engle et al (2001) demonstrate the consistency of such.

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4. Data and Summary Statistics

Daily data from Commodity Research Bureau database and Bloomberg will be used for the analysis. The 11 series under consideration fall into 4 groups: equities (S&P500 Index - SP500, S&P400 Midcap Index – SP400, S&P600 Smallcap Index – SP600, Russell 2000 Index – RUS2000 and NASDAQ Index - NASDAQ), interest rates (10 Year Treasury Yield-10YTY and 1 Year Treasury Yield-1YTY), commodities (Goldman Sachs Commodity Index – GSCI, CRB Precious Metals Index – CRBPMI, CRB Energy Index – CRBEI and PLX Precious Metals Index – PLXPMI) and credit ().

Tables 1 and 2 present basic summary statistics for the series, for there levels and returns, respectively. Jarra-Barque normality and Box-Pierce white noise tests are presented as well. In all cases, for all series in both levels and returns form, we can very strongly reject normality as well as the lack of autocorrelation. Furthermore, in Tables 2 these tests are performed for the squared returns, which is an indirect way of testing for GARCH effects. We can also strongly reject normality and white noise in the second moments of the return series.

The unconditional sample Pearson correlation matrix of the return series is exhibited in Table 3. We show the p-values for the null hypothesis of zero correlation on the lower diagonal for each pair. We can observe wide variation in the range of the correlations as well as some obvious patterns. First, as expected, all the index returns in each category are highly and positively correlated. In the case of equities, the unconditional correlations range from 0.77 (SP500 and RUS2000) to 0.97 (SP600 and RUSS2000) amongst the 5 index returns. Long and short term interest rates exhibit a rather strong positive correlation of 0.58, reflecting the dominance of the “level of interest rates” factor in explaining movements in the yield curve over the entire period. The correlations between the commodity indices are also positive, but highest between the GSCI and CRBEI (0.86), as compared to lower – albeit statistically significant - correlations between the commodity and precious metals indices (0.25 and 0.18 for GSCI/CRBPM! and GSCI /PLXPMI, respectively, and 0.15 between CRBEI and CRBPMI). Understandably, the two precious metals indices, PLXPMI and CRBPMI, have a high correlation of 0.60.

Turning to correlations between different groups, we first observe that the signs of the correlation interest rates and equities are mixed depending upon whether we are looking at long vs. short term rates, as well as the broader market vs. a small cap index. The 1-year Treasury rate has negligible correlation with the SP500 (-0.0007 with a p-value of 0.94), but mildly positive and significant correlation with the other indices (0.08 for SP400 and RUSS2000, 0.13 and 0.19 for NASDAQ and SP600, respectively). All the broader indices are negatively correlated with the 10 Year Treasury (-0.15, -0.07 and -0.05 for SP500, SP400 and RUSS200, respectively), the smaller cap

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indices exhibit mild positive correlation (0.03 and 0.11 for NASDAY and SP600, respectively). The correlations between the equity and commodity indices are mixed in sign, but generally rather low, and in some cases insignificant. CRBPMI ranges in correlation from negligible with SP500 and RUSS2000 (0.01 and 0.06, with p-values of 0.62 and 0.050, respectively), to slightly but significantly positively correlated to SP400 and RUSS2000 (0.04 and 0.06, respectively), to slightly but significantly negatively correlated to NASDAQ (-0.03). On the other hand, the PLXPMI is consistent in being slightly positively correlated to all equity indices, ranging from 0.05 with NASDAQ to 0.12 with SP400, in all cases highly statistically significant. While the GSCI is only very slightly (-0.02 and -0.04) inversely correlated with SP500 and NASDAQ, in the former case of only marginal significance (p-value = 0.05), it is not significantly correlated with the other indices: correlations (p-values) of 0.01, 0.02 and 0.03 (0.44, 0.12 and 0.12) with SP400, RUSS2000 and SP600, respectively. On the other hand, with the exception of SP600 (correlation = 0.01 and p-value = 0.42) the CRBEI is exhibits consistent small yet statistically significant correlation with equities: -0.06,-0.03, -0.05 and -0.04 with SP500, SP400, NASDAQ and RUSS2000, respectively. We observe mixed results for interest rates and commodities. GSCI exhibits small yet significant (insignificant) positive correlation with long (short) term interest rates, estimates of 0.03 for both, but with p-values of 0.04 (0.42). The CRBEI shows a similar pattern, significant (insignificant) positive correlation with long (short) term interest rates, estimates of 0.06 (0.01), but with p-values of 2.7E-06 (0.28). In the case of the CRBPMI, we see significant negative correlation with 10 Year Treasuries (estimate of -0.04 and p-value 8.4E-5), but positive and marginally significant correlation with 1 Year Treasuries (estimate of 0.02 and p-value 0.04). However, for the PLXPMI we observe negligible correlation to short term rates (estimate of 0.01 and p-value of 0.52), yet mild positive (and highly significant) correlation to long term rates (estimate of 0.09 and p-value of 6.5E-11).

5. Estimation ResultsIn this section we discuss the main empirical results of this study, the estimation of the correlation structures amongst the 11 asset return series under consideration. First, we analyze the rolling window moving average (RWMA) estimators for monthly, quarterly, 6 month and 1 through 3 years. In the section following that we compare this to the DCC model estimates, both in terms of the similarity of the correlation estimates, as well as the performance of the estimators in hedging.

5.1 Analysis of Moving Correlations

When comparing the average daily correlations based on six different windows for all fifty five pairs of assets, there were some patterns that appeared similar across different pairs. As the length of the rolling window of correlation increased (from a one month to the three years), we found two

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trends reappearing amongst all the asset pairs, while four characteristics were common amongst on a few.

Chart 1 shows average daily correlation increasing in the rolling window length for short term interest rates and an equity index, the case of the 1YTY and SP400 Index. Average daily correlation increases monotonically from 0.07 to 0.21 going from a 1 month to the 3 year window. This relationship held for the pairing of 1YTBY and all other equity index (SP500, RUSS2000, NASQAQ and SP600) as well, as well as for the correlation between long term interest rates (30TBY) and equity index returns with the exception of the SP600 small-cap index. A potential story here is that over a shorter term horizon, transient technical factors may be dominant (e.g., a tendency to move into stocks when interest rates – especially short term - are low), whereas over a longer time frame a factor representing the broader macroeconomic state may be at play (e.g., higher rates –especially long-term – signaling underlying economic strength), thereby augmenting the strength of the relationship. However, the levels of the unconditional correlation between interest rates and equity returns are at low absolute levels across windows. Further, holding periods are still only a day. While the 30YTY and CRBEI displayed a similar trend of an increasing correlation with the window length, the 1YTBY and CRBEI did not. However, the 1YTBY exhibited such a trend when compared to the GSCI Index, while the 30YTY did not. This could be attributed to the fact that the GSCI represents a diversified position in commodity futures, while the CRBEI most closely tracks crude oil, heating oil, and natural gas.

Chart 2 shows that the correlation between the 1YTY and 30YTY changes are decreasing in window length, from 0.63 for 1-month, down to 0.54 for 3 years. However, the level of the correlation remains high for all windows, and the gradient of change from short to long window is not dramatic. As noted in the discussion of the unconditional Pearson correlation matrix, this reflects the dominance of the “level of interest rates” factor in explaining variation in yields across the risk-free term structure. Other asset pairs that display this inverse trend between correlation and window length are the short term interest rate and precious metals pair, 1YTY and CRBPMI, as well as a commodity / precious metal s index pair, CRPMI and PHLXPMI. Chart 3 shows a hump shaped relationship between average daily correlation of PLXPMI precious metals and SP400 mid-cap equity indices and length of the rolling window. The average daily correlation increases from 0.11 at a 1-month window to 0.13 at a 6-month window, falling thereafter to 0.08 for a 3 year window. Pairs also exhibiting this pattern include PLXPMI and equity indices (NASDAQ, SP500, SP600 and RUSS2000), and the energy index CRBEI and the long term interest rate 30YTY.

In contrast, Chart 4 shows the inverse pattern, a reverse U-shape for the GSCI and SP500 equity index. In this case, the average daily correlation decreases from -0.01 at a 1-month window to -0.13 at a 1-year window,

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rising thereafter to -0.02 for a 3 year window. In both of these cases, the absolute values of the average daily correlations are small. Pairs also exhibiting this pattern include the commodity indices versus equity indices (SP400 and SP500), long term interest rates (30YTY) and the precious metals index CRBPMI.

Chart 5 illustrates another canonical pattern, a u-shape reflected across the y-axis, for the CRBPMI and NASDAQ. Whilst all average correlations are negative, they increase from -0.04 at the 1 month to -0.02 at the 2 year window, and then revert downward to -0.03 at the 3-year window. A similar pattern is observed between two other commodity and equity index pairs, CRNPMI / SP400 and CRBEI / RUSS2000. However, note that in all these cases the correlations are rather small, and for some windows not statistically distinguishable from zero.

Finally, Chart 6 illustrates a case of an uncorrelated series across windows, CRBEI and RUSS2000, where all correlations are less than 0.01. About half of our pairs, 23 out of 55, exhibit average daily correlations that are statistically indistinguishable from zero. Perhaps surprisingly, this include a fair number of equity index pairs (RUSS2000 vs. SP600, SP400 and NASQAQ; SP400 vs. NASQAQ, SP500 & SP600; and NASQAQ vs, SP600), a large number of commodity / equity index pairs (CRBEI vs. all of the equity indictors, GSVI vs. all but SP500, CRBPMI vs. SP500 and SP600), as well as a few commodity index / bond yield pairs (PHLX vs. IYTBY and 10YTBY). Having a correlation of approximately zero across each of the 6 rolling windows used, these pairs of assets are good candidates to put into a portfolio, as the movement of each asset return is almost completely independent of one another, thereby enabling portfolio diversification by reducing the risk for a given level of return.

Looking at the standard deviations of the rolling correlations calculated from the six different windows (Chart 7), one can see that the window used to measure correlation has a large impact on the time series properties of correlation. It is clear that there is a similar trend across all 55 asset pairs when comparing the rolling correlation’s standard deviation to the rolling window used. While standard deviations vary across asset pairs, they all have their highest coefficients when looking at the shortest 1 month rolling window, from the standard deviation of rolling correlations for the 55 asset pairs all decrease, respectively. The fact that moving averages based on shorter time spans fluctuate more than such with longer holds across asset pairs. Whether one is interested in the management of risk (VaR), pricing derivatives, or portfolio diversification, looking at average daily correlation between assets and being able to forecast their trend with precision is extremely significant. The irregular trends noted herein make it extremely hard to forecast correlation because of the volatility amongst the 6 different rolling windows. Time horizon is an important factor that must be considered when trying to forecast correlation. The volatility across the 6

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different rolling windows makes it more difficult to accurately forecast average daily correlation between any asset pairs. Charts 8A through 13A (8B through 13B) graphically present standard descriptive statistics, the standard deviations (averages with 95% confidence intervals) of the six rolling windows, which give further insight into how different observation windows effect the estimates. This is also presented in tabular form in Table 1. For example, when studying the SP500 vs. 30YTBY at a 1 month rolling window, the average daily correlation is 0.4312. When one uses the 3 year rolling, average daily correlation significantly drops to 0.1275, a 30% difference. Looking at the PHLXGSI as compared to RUSS2000 at a 1 month rolling window, the assets have an average daily correlation of 0.034, but when using the 3 year rolling window this significantly drops to 0.0088 (practically zero). These are just two examples of the substantial differences in distributional properties amongst moving averages correlation estimates of different rolling windows. Therefore, in any application (derivatives pricing, risk or portfolio management), time horizon and period used to forecast correlation (whether using rolling windows, or any other type of forecasting method) are two factors that are of the utmost importance.

A desirable property of a correlation estimator is confidence Intervals with low standard errors, and we see that this is difficult to achieve at high averaging frequencies. This inflation in a pair’s confidence interval may be attributed to a large (positive or negative) unexpected movement in the market, which may carry unduly large weight for shorter window lengths. Indeed, as we look at the confidence Intervals going from Graph 8B to graph 13B (i.e., moving from a 3 year to a 1 month rolling window), it is clear the confidence intervals increase as the rolling window length decreases. Therefore, some type of adjustment should be made to account for the degree of uncertainty, as the room for error on these rolling windows correlation estimates becomes very large when a few days are used (such as a 5 day or 22 day rolling window). For this reason it is advisable to use a longer averaging period for the correlation estimates. However, the longer the rolling window, the greater the potential mismatch with ones risk (or time) horizon is, as a greater number of uncertainties one obtains. In general, our results show more similarities between the 1 month (22 day) rolling windows compared to the 2 year and 3 year rolling windows. For example, partially as a result of this risk horizon, the standard deviation of the forecasted correlation differs most in the asset pairs of the interest rates (both long-Term and Short-Term) and the S&P500 (Graphs 1 through 3). In the next section, we focus on these, and the issue of varying holding periods.

5.2 Comparison of Moving Average to DCC Correlations for S&P 500 Index (SP500) Returns and 10-Year Treasury Bond Yield (10YTBY) Changes In this section, focusing on the broad equity (SP500) and long-term risk-free fixed income markets (10TBY), we compare the properties of moving average correlation estimates (for window lengths 2-, 22-, 66-, 126-, and

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252-day) with DCC estimated correlation. We do this for different holding periods: daily, weekly, quarterly and annually. Table 5.2.1 presents summary and distributional statistics of these estimators, as well as the (Spearman rank order) correlations between the moving (MC) and DCC correlations. The density plots Figures 5.2.1 through 5.2.5 depict the distributions of these graphically for each holding period, DCC as compared to the 5 rolling windows. Figures 5.2.6 through 5.2.8 graphically present some key distributional characteristics of the correlation estimates from Table 5.2.1 (means and standard deviations) and the correlations of the MCs to DCC. The panel plots Figures 5.2.9 through 5.2.13 depict the time series of the correlations graphically graphically, for each holding period, DCC as compared to the 5 rolling windows. Table 5.2.2 presents distributional statistics of hedge portfolios, as discussed in Section 3, based upon the various types of correlation estimators. In addition to MC and DCC, this includes the constant correlation (or naïve model), in which we base the portfolio delta upon simple univariate (unconditional) correlation coefficients.

The first and perhaps principle conclusion that we can draw form Table 5.2.1 is that DCC is most similar to the MCs for daily holding period as opposed to any horizon longer than this. Correlations between DCC and MC range from 0.71 for 5-day MC (MC5), rising to a peak of 0.88 for 22-day MC (MC22), and falling thereafter to 0.57 for 252-day MC (MC252). In the case of a weekly holding period, we obtain small albeit statistically significant negative correlations between DCC and the Mc ranging in -0.25 to -0.13, with the MC66 having the most negative correlation. The correlations between DCC and the MCs are also negative, but very small, at the monthly holding period, ranging from -0.02 to -0.01. At a quarterly holding period, these correlations are an order of magnitude lower and statistically indistinguishable from zero, ranging from 0.003 to 0.008. Finally, for an annual holding period, the correlations between DCC and MC are small but positive, ranging from 0.01 to 0.05 and peaking at MC66. This can be seen in Figure 5.2.1, which graphs the correlation of the DCC to MC correlation estimators for each holding period, were we can see that it plunges past the daily holding period below zero at one week and subsequently increases slightly to about zero for the remainder of the longer holding periods.

Comparing the distributional statistics for the MC and DCC estimators in Table 5.2.1, we see that at a daily holding period, DCC is similar to the MCs. The former has mean correlation of -0.27 and a standard deviation of 0.19, while the latter has means (standard deviations) ranging in -0.23 to -0.21 (0.29-0.53). Yet, for all other holding periods we see large differences. Average DCC correlations tend to be small for holding periods of a week and greater, ranging from -0.002 to -0.02 (with little variation across holding periods), while MC tend to be of greater magnitude and increase monotonically with holding period from the range of -0.24 to -0.22 (-0.10 to -0.07) for weekly (annually). This is also shown in Figure 5.2.12, were the mean DCC correlation is negative and close to the MCs for a daily holding period, then shooting small negative values away from the MC at weekly and

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greater, while the average MC correlations all increase gradually with holding period. At the same time, DCC has much less variation around the mean across holding periods, and this variation drops off and remains about constant at standard deviation ranging in 0.02-0.03 at a weekly holding period and after; whereas the variation in the MCs is much greater, increases monotonically with holing period, ranging in 0.30-0.52 (0.63-0.92) for daily (annual). This is shown in Figure 5.2.13, the sharp drop in standard deviation for DCC past a daily holding period, while all the other MC increase steadily with holding period (and with the lower window lengths uniformly more volatile).

The plots of the densities and time series of the correlation estimates, in Figures 5.2.1-5.2.5 and 5.6.6-5.2.10, respectively, sheds further light on radical difference between MC and DCC for holding periods of greater than a day. Focusing on the smoothed densities of the correlation estimators, while DCC is more concentrated and peaked than any of the MCs, it appears to be most similar to MC252. This as opposed to our conclusion from the correlation coefficient analysis in Table 5.2.1, in which we noted that MC22 seemed to be most like DCC. However, note the multi-modality of all of the MCs as compared to DCC. As we go to longer holding periods in the subsequent 4 graphs, note how concentrated DCC becomes about zero, while the MCs become increasingly diffuse. In Figure 5.2.2 for weekly holding periods, the MCs all appear almost uniformly distributed in the unit interval. As we move toward the longest holding period, this becomes increasingly bi-model with concentrations near -1 and 1 for the MCs, with this shape the most pronounced at the annual holding period in Figure 5.2.5. Note that the degree of the bi-modality is accentuated for the shorter moving windows, being greatest for MC5 and least for MC252, although it is evident for all of them. This suggests that while smoothing may be going wrong with the MC estimators, as discussed below, one is better off using a longer window if the holding period tend to be longer as well.

We can see this progression in the MCs toward instability in the time series plots of Figures 5.2.6 through 5.2.10 as well. In Figure 5.2.6, for a daily holding period, we see the typical pattern of the MCs becoming smoother as we increase the rolling window. Clearly MC5 stands out as rather unstable, but the others seem reasonably well behaved, and in fact mimic the shape of the DCC in that correlations are seen as going from generally negative through the late 1990’s to positive early in this decade, and recently dipping back to the other side of the x-axis. Of course, the scale of all the MCs is exaggerated as compared to DCC, a distortion introduced by not controlling for autocorrelation and GARCH effects in the series. However, as we move toward longer holding periods, we start to see a pronounced alternation in signs across all the MCs, with this cycling just becoming more gradual as we increase the window length. On the other hand, the DCC estimates are in general qualitatively similar to each other as we increase holding period, only moving less on average around and subject to more extreme blips as compared to a daily holding period.

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Figure 5.2.14 compares the DCC estimators across holding periods. As noted previously, we do not see the same radical qualitative change in the series as we did in the MCs with increasing holding period. In all cases, DCC correlations tend to be small (for many days, within the 95 percent confidence bounds for zero!), but subject to some sudden spikes. However, there are notable patterns or differences across the holding period. Daily DCC is the “jumpiest” and most variable of these. The weekly holding period DCC is for some reason distinct from the others, in that it exhibits the most cyclicality around zero (albeit this appears to be mostly noise), and the huge blips seen in the holding periods longer than this do not appear. Quarterly, monthly and annual holding period seem the most similar to each other in the time series of the estimates, but there are some perplexing differences: the annual shows a tremendous upward blip around the crash of 1987; monthly shares this, but has a huge negative shock shortly before this; and the quarterly DCC has only the latter negative shock at around that time. The distributions of the DCC for different holding period are shown in Figure 5.2.15, where the segmentation of daily versus all longer holding periods is clear, but where some of the anomalous difference across the latter are obscured.

The reason why we see such a divergence for the holding periods longer than is a direct result of the radically different estimation methodologies used between DCC and MC, and the fact that the holding periods are overlapping. In DCC, we first remove any dependence in the first and second moments of the returns processes through univariate ARMA and GARCH modeling. This is not so for the MCs, where the effects of dependence by virtue of the holding periods being overlapping is not controlled for, which is reflected in spuriously amplified correlation estimates. The implications of this misspecification in the MCs for long holding periods relative to DCC, if we are willing to accept DCC as a reasonable representation of the true process, has profound implications for applications of this estimation methodology. The results for the DCC estimation are telling us that equities and interest rates, at least as measured by the S&P 500 index and the 10-year U.S. Treasury bond, exhibit very low correlation for holding periods greater than a year; but there are a few periods in which correlation increases dramatically (e.g., for annual holding periods, while on average essentially nil, the maximum DCC correlation is 0.47, and excess skewness is 1.8). On the other hand, the MCs are giving us a severely downward biased and noisy estimate of the correlation (e.g., for monthly holding periods, average MC22 is -0.22, but ranges in -0.94 to 0.94!), an artifact of being constructed from overlapping returns.

7. Summary and Conclusions

In conclusion forecasting correlation is important for three main reasons; portfolio diversification, derivative pricing, and VaR. Accurate

17

pricing of derivatives on volatility and correlation depends significantly on how the underlying correlation and volatility fluctuate throughout different time periods. The assessment of the conditional return distribution is a significant factor that contributes to any type of financial risk supervision or management. According to Anderson, Bollerslev, Diebold, and Labys (2005), correlation is itself highly correlated with realized volatility, which they call the “volatility effect in correlation.” They point out that return correlations tend to rise on high-volatility days, which can be seen throughout viewing different charts and graphs in the appendix. I show how time varying information, through the use of different rolling windows, can have significant impacts on one’s financial decisions with respect to forecasting correlation.

In VaR models, managers must consider the risk factors of the portfolio their running, derived from the variations in value for a given pair of assets (for example the fluctuations of exchange rates). Other important risks that must be considered when, pricing a derivative or simply just trying to optimally diversify ones portfolio are the following; price risk, settlement risk, default risk, systematic risk, operational risk, and liquidity risk.

In today’s financial society, moving averages are used to reduce daily volatility or noise that interfere with identifying trends across time and across different rolling windows with respect to correlation and standard deviation. With the exponential amount of leverage being used today, there is an unknown level of risk that must be accounted for, calculated and attributed to the exponential usage of leverage.

“Globalization” is a word that is constantly heard throughout all areas of the public, not just the financial society. In regards to all asset groups, globalization has caused for an increase in correlation for a majority of securities in the financial society, making portfolio diversification, risk management, and derivative pricing harder and harder to achieve. The correlation dynamics documented in this paper raise significant issues for investors and others involved in the financial society, proving the significance of correlation.

According to Alexander (2001), the short rolling windows have trivial coefficients due to the high degree of multicollinearity between the assets. Therefore, this common problem makes it more difficult to effectively interpret the true strength of the effect each asset contributes to the total portfolio. Therefore if one decides to use a short rolling window they must be aware of multicollinearity and not use indicators that expose the same sort of information. When observing the shorter rolling window correlation coefficients of the asset pairs, there is a greater chance that the standard errors will be depressed creating inaccuracy. This inaccuracy can be applied to models used to forecast and manage risk, price derivatives, and optimize one’s portfolio.

18

It will be of great interest to observe the nature of the relationship between commodities (especially energy) and US Treasury’s rates (both Long Term and Short Term) over the next decade or two. While in a “normal” economic environment one would expect a positive term premium, the U.S.’s yield curve has experienced near inversion as of late, which has been exacerbated by technical supply factors (i.e., no issuance were of 30-year T-Bonds between October 2001 and August of 2005). On the other hand, commodities are affected by particular supply and demand factors, expectations of inflation, the weather, and other events. While bursts of expected inflation tends to depress the price of financial instruments, and in crease the prices of commodities, this relationship may not be stable over time, as events (such as a changes in supply-demand conditions or the activity of speculators) could differently alter commodity price expectations. A key factor affecting the direction of this relationship involves the status of the US dollar as today’s vehicle currency, with the country as a whole spending $7 for every $1 earned. This benefit of seignorage may only last for so long, with both the Euro and the Yuan waiting in the wings two potential candidates to share in this status. This will have implications for US Treasuries will not be considered the world’s safest investment, in which case the US government may find itself in a black hole of debt. A majority of today’s oil is denominated in US dollars along with many other assets traded throughout the world; if the vehicular currency changes so will the type of money an underlying security is denominated it. Therefore betting for this trend to remain constant over time for one’s portfolio, or for pricing a derivative may not be that accurate.

19

Appendix 1 – Tables and Figures

S&P 500 Equity Index

Goldman Sachs Commodity Index

10 Year Treasury Yield

CRB Precious Metals Index

CRB Energy Index



NASDQQ Equity Index

Russel 2000 Equity Index

S&P 600 Small Cap Equity Index

PLX Precious Metals Index

Minimum 4.40 98.92 1.9800 133.500 100.700 0.8200 31.46 238.10 40.52 103.75 41.851st Quartile 18.56 171.49 5.5920 238.290 164.800 4.2465 78.42 445.30 121.35 171.59 75.16Median 86.35 192.72 7.4450 268.940 187.995 5.7180 169.87 1028.19 211.54 202.93 90.893rd Quartile 251.79 215.64 8.9800 316.305 250.585 7.6890 424.12 1928.67 433.23 266.06 110.71Maximum 1527.46 509.63 15.2100 760.900 704.540 17.3100 817.95 5048.62 781.83 404.89 168.62Count 19,796 9,395 8,015 8,007 5,820 11,295 6,519 5,563 7,027 2,874 5,772 Mean: 240.86 202.34 7.6155 284.83 229.79 6.2111 255.0727 1264.21 278.72 220.31 92.99LCL Mean 235.67 201.03 7.5570 282.87 226.87 6.1571 249.8958 1239.76 274.32 217.62 92.36UCL Mean 246.05 203.66 7.6741 286.79 232.71 6.2652 260.2497 1288.66 283.11 223.01 93.63Standard Deviation 372.34 65.14 2.6733 89.55 113.71 2.9291 213.2241 930.25 187.79 73.68 24.61Skewness: 1.9405 2.0124 0.5200 1.5319 2.2552 0.9826 0.8543 1.1114 0.6895 0.7012 0.2944Kurtosis: 2.4991 5.7551 -0.1453 4.1222 4.8210 1.3780 -0.4877 1.1040 -0.6355 -0.3942 -0.5594JB Normality Test 17,568.9 19,288.2 368.2 8,790.3 10,555.0 2,709.2 857.4 1,426.1 674.9 254.0 158.8BP White Noise Test 197,270.8 92,262.3 79,563.6 78,055.4 57,283.6 111,978.0 64,692.5 55,158.3 69,640.3 28,114.9 55,350.8

Table 1: Summary Statistics (Index Levels)





CRB Energy Index



NASDQQ Equity Index




Minimum -22.8868% -18.4540% -7.7039% -8.6310% -26.0563% -20.6653% -14.4075% -12.0432% -13.3887% -6.2489% -26.3413%1st Quartile -0.4528% -0.5483% -0.4596% -0.5866% -0.8296% -0.5317% -0.4122% -0.4962% -0.3790% -0.6102% -1.2722%Median 0.0444% 0.0188% 0.0000% 0.0304% 0.0267% 0.0000% 0.0838% 0.1162% 0.1094% 0.1040% -0.0888%3rd Quartile 0.5257% 0.5926% 0.4321% 0.6907% 0.8856% 0.5451% 0.5553% 0.6554% 0.5344% 0.6949% 1.2039%Maximum 11.81% 7.53% 3.92% 8.81% 9.60% 25.12% 8.07% 13.25% 7.35% 5.45% 19.12%Count 19,043 9,182 7,103 7,735 5,613 10,711 6,304 5,376 6,795 2,774 5,578Mean: 0.0247% 0.0141% -0.0102% 0.0153% 0.0138% -0.0023% 0.0514% 0.0447% 0.0454% 0.0472% 0.0054%LCL Mean 0.0089% -0.0081% -0.0294% -0.0130% -0.0307% -0.0296% 0.0277% 0.0080% 0.0220% 0.0045% -0.0542%UCL Mean 0.0404% 0.0363% 0.0089% 0.0435% 0.0582% 0.0251% 0.0752% 0.0814% 0.0688% 0.0900% 0.0649%Standard Deviation 1.1071% 1.0844% 0.8243% 1.2688% 1.6983% 1.4455% 0.9603% 1.3729% 0.9844% 1.1483% 2.2692%Skewness: -0.6955 -0.6377 -0.0791 -0.2734 -0.8004 0.3782 -0.7901 -0.2454 -0.9894 -0.1225 0.0338Kurtosis: 22.2134 11.6998 3.1080 3.4858 12.5076 22.7955 13.1762 8.8146 11.8868 1.7030 6.7281

JB Normality Test (Returns) 392,839.2 52,929.6 2,860.6 4,005.6 37,114.9 231,936.8 46,178.9 17,421.5 41,047.8 340.1 10,499.6

BP White Noise Test (Returns) 137.2 16.4 23.9 25.9 26.1 74.9 125.0 37.4 201.6 31.1 18.7JB Normality Test (Squared Returns) 9.05E+09 6.34E+09 4.27E+07 2.27E+06 1.55E+09 1.14E+09 8.80E+08 1.20E+07 3.15E+08 3.61E+05 1.73E+08

BP White Noise Test (Squared Returns) 4,457.9 519.9 581.2 1,856.0 258.1 2,107.5 1,883.6 3,146.8 3,609.8 952.9 971.8

Table 2: Summary Statistics (Index Returns)





CRB Energy Index



NASDAQ Equity Index




S&P 500 Equity Index - -0.0211 -0.1504 0.0056 -0.0602 -7.2E-04 0.8395 0.7852 0.7723 0.8071 0.0801

Golman Sachs Commodity Index 0.0456 - 0.0256 0.2520 0.8600 0.0257 0.0096 -0.0413 0.0188 0.0299 0.1849

10 Year Treasury Yield 3.39E-37 0.0382 - 0.0241 0.0632 0.5791 -0.0727 0.0302 -0.0509 0.1053 0.0881

CRB Precious Metals Index 0.6237 2.38E-112 0.0419 - 0.1528 -0.0414 0.0374 -0.0324 0.0649 0.0152 0.5978

CRB Energy Index 6.43E-06 0.00E+00 2.73E-06 1.12E-30 - 0.0145 -0.0255 -0.0467 -0.0356 0.0129 0.1538

1 Year Treasury Yield 0.9407 0.4185 0.00E+00 8.39E-05 0.2800 - 0.0785 0.1340 0.0757 0.1871 0.0086

S&P 400 Equity Index 0.00E+00 0.4478 1.27E-14 3.04E-03 0.0558 6.12E-10 - 0.8675 0.9224 0.9263 0.1232

NASDAQ Equity Index 0.00E+00 0.0025 0.0283 1.76E-02 6.43E-04 1.23E-22 0.00E+00 - 0.8701 0.8315 0.0512

Russsel 2000 Equity Index 0.00E+00 0.1211 1.27E-14 8.86E-08 7.63E-03 5.98E-10 0.00E+00 0.00E+00 - 0.9748 0.1353

S&P 600 Small Cap Equity Index 0.00E+00 0.1154 3.45E-08 0.4232 0.4972 4.93E-23 0.00E+00 0.00E+00 0.00E+00 - 0.1086

PLX Precious Metals Index 2.11E-09 4.26E-44 6.45E-11 0.00E+00 1.17E-30 0.5233 2.67E-20 1.73E-04 3.39E-24 9.66E-09 -

Table 3: Correlation Matrix of Index Returns (P-Values on Below Diagonal)

Estim

ates

P-Values

20

Chart 1

Asset Pairs exhibiting a similar pattern: IR vs. EI IR vs. CI IR vs. EI IR vs CI EI vs. CI CI VS. CI IY vs. QQ IY vs. GI UY vs. MD UY vs. 1M SP vs. MD 1I vs. GS IY vs. R2 UY vs. QQ SP vs. X5 IY vs. SP UY vs. R2 IY vs. X5 UY vs. SP

UY vs. X5

Chart 2

Asset Pairs exhibiting a similar pattern: IR vs. CI CI vs. CI 1I vs. 1Y 1I vs. XA

Average of daily correlations based on different rolling windows for1-Yr T-Bill vs. S&P midcap Index

0.07

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0.21

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1 month 3 month 6 month 1 year 2 year 3 year

Length of rolling window of correlation

Correlation

IYUY

1I1MGIXA

MDQQR2SPX5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bill yield30yr T-bond yield

CRB prescious Metals Index CRB Energy IndexGoldman Sachs Commodity IndexPHLX Gold/Silver Index

S&P midcap 400 Index

S&P smallcap 600 IndexS&P 500 IndexRussell 2000 Index Nasdaq Composite Index

Average of daily correlations based on different rolling windows for1yr T-bill yield vs. 30yr T-bond yield

0.63

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0.58

0.55

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0.62



IYUY

1I1MGIXA

MDQQR2SPX5

Interest Rates

Commodity Indices

Equity Indices





Correlation

21

Chart 3

Asset Pairs exhibiting a similar pattern: CI vs. EI CI vs. IR XA vs. QQ 1I vs. UY XA vs. R2

XA vs. SP XA vs. X5

Chart 4

Asset Pairs exhibiting a similar pattern: CI vs. EI CI vs. IR CI vs. CI 1M vs. MD 1M vs. UY 1I vs. GI 1M vs. SP

Average of daily correlations based on different rolling windows forS&P midcap 400 Index vs. PHLX Gold/Silver Index

0.11

0.130.13 0.13

0.11

0.08

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Correlation

IYUY

1I1MGIXA

MDQQR2SPX5

Interest Rates

Commodity Indices

Equity Indices





Average of daily correlations based on different rolling windows for Goldman Sachs Commodity Index vs. S&P 500 Index

-0.02

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-0.036

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Correlation

IYUY

1I1MGIXA

MDQQR2SPX5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bi ll yield30yr T-bond yield

CRB prescious Metals Index CRB Energy IndexGoldman Sachs Commodity IndexPHLX Gold/Si lver Index



22

Chart 5

Asset Pairs exhibiting a similar pattern: CI vs. EI 1I vs. MD 1M vs. R2

Chart 6

Asset Pairs exhibiting a similar pattern: EI vs. EI IR vs. EI CI vs. EI CI vs. CI R2 vs. X5 GI vs. UY 1M vs. R2 XA vs. GI R2 vs. MD GI vs. 1Y 1M vs. X5 XA vs. UY

R2 vs. QQ GI vs. R2 1M vs. MD XA vs. 1Y MD vs. QQ 1M vs. 1I

MD vs. SP 1M vs. XA MD vs. SPX 1M vs. QQ

QQ vs X5 1I vs. SP QQ vs X5 1I vs. XP GI vs. R2

GI vs. X5 GI vs. QQ GI vs. MD

Average of daily correlations based on different rolling windows forCRB Precious Metals Index vs. NASDAQ Composite Index

-0.04

-0.03-0.03

-0.02

-0.02

-0.03

-0.04

-0.0375

-0.035

-0.0325

-0.03

-0.0275

-0.025

-0.022

-0.02

-0.0175

-0.015

-0.0125

-0.01



Correlation

IYUY

1I1MGIXA

MDQQR2SPX5

Interest Rates

Commodity Indices

Equity Indices





CRB Energy Index vs. Russell 2000 Index

0.010.00 0.00 0.00 0.00

0.01

0

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0.1

1 month 3 month 6 month 1 year 2 year 3 yearLength of rolling window of correlation

Correlation

Average of daily correlations based on different rolling windows for

IYUY

1I1MGIXA

MDQQR2SPX5

Interest Rates

Commodity Indices

Equity Indices





23

Chart 7

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STANDARD DEVIATION OF ROLLING CORRELATION WITH 6 DIFFERENT WINDOWS


Chart 8A

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Standard Deviation of Moving Correlations (based on 756-day window)12/28/1998 to 10/27/2006

Graph 8B

95% Confidence Interval for Average of Moving Correlations (based on 756-day window) 12/28/1998 to 10/27/2006

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Chart 10A


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Chart 11A


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Chart 12A


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Chart 13A


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Table 1 Stdev 1I1M 1I1Y 1IGI 1IMD 1IQQ 1IR2 1ISP 1IUY 1IX5 1IXA 1M1Y 1MGI 1MMD 1MQQ 1MR2 1MSP 1MUY 1MX5 1MXA 1YGI 1YMD 1YQQ 1YR2 1YSP 1YUY 1YX5 1YXA GIMD GIQQ GIR2 GISP GIUY GIX5 GIXA MDQQ MDR2 MDSP MDUY MDX5 MDXA QQR2 QQSP QQUY QQX5 QQXA R2SP R2UY R2X5 R2XA SPUY SPX5 SPXA UYX5 UYXA X5XA

1 month 0.25 0.24 0.26 0.24 0.24 0.24 0.25 0.26 0.25 0.17 0.25 0.10 0.24 0.24 0.24 0.25 0.24 0.24 0.25 0.26 0.38 0.33 0.36 0.40 0.22 0.35 0.25 0.25 0.25 0.25 0.25 0.25 0.24 0.26 0.08 0.06 0.08 0.40 0.07 0.30 0.07 0.09 0.37 0.10 0.29 0.11 0.38 0.03 0.30 0.43 0.12 0.30 0.37 0.25 0.303 month 0.18 0.16 0.19 0.17 0.16 0.16 0.16 0.18 0.16 0.10 0.15 0.08 0.16 0.16 0.15 0.16 0.15 0.15 0.16 0.16 0.31 0.27 0.29 0.34 0.18 0.28 0.17 0.16 0.17 0.16 0.17 0.15 0.15 0.18 0.06 0.05 0.06 0.32 0.05 0.25 0.05 0.07 0.30 0.08 0.23 0.09 0.31 0.03 0.25 0.37 0.09 0.23 0.30 0.17 0.246 month 0.15 0.13 0.16 0.13 0.12 0.13 0.13 0.15 0.13 0.09 0.12 0.07 0.12 0.12 0.11 0.13 0.12 0.12 0.14 0.12 0.27 0.23 0.25 0.30 0.16 0.25 0.14 0.13 0.14 0.12 0.13 0.12 0.12 0.16 0.04 0.04 0.06 0.29 0.05 0.22 0.04 0.06 0.26 0.07 0.20 0.08 0.27 0.02 0.22 0.33 0.08 0.20 0.26 0.15 0.211 year 0.11 0.11 0.12 0.11 0.10 0.10 0.11 0.13 0.10 0.08 0.08 0.07 0.08 0.09 0.08 0.09 0.10 0.08 0.11 0.08 0.23 0.19 0.20 0.25 0.13 0.20 0.11 0.09 0.10 0.09 0.09 0.09 0.08 0.13 0.04 0.03 0.05 0.24 0.04 0.19 0.03 0.05 0.21 0.05 0.17 0.08 0.22 0.02 0.19 0.27 0.08 0.17 0.22 0.13 0.182 year 0.06 0.09 0.07 0.08 0.08 0.08 0.08 0.12 0.08 0.07 0.05 0.05 0.06 0.07 0.06 0.06 0.09 0.06 0.07 0.06 0.14 0.11 0.12 0.15 0.10 0.12 0.08 0.06 0.07 0.06 0.06 0.07 0.06 0.08 0.03 0.02 0.04 0.15 0.03 0.16 0.02 0.04 0.13 0.03 0.15 0.06 0.13 0.01 0.16 0.17 0.06 0.14 0.13 0.12 0.153 year 0.04 0.07 0.05 0.06 0.05 0.06 0.06 0.11 0.06 0.06 0.04 0.04 0.04 0.06 0.05 0.05 0.08 0.04 0.04 0.04 0.10 0.08 0.08 0.12 0.08 0.09 0.05 0.04 0.05 0.05 0.04 0.07 0.04 0.05 0.02 0.02 0.04 0.11 0.02 0.13 0.02 0.03 0.09 0.03 0.11 0.05 0.10 0.01 0.13 0.13 0.05 0.11 0.09 0.10 0.12

Av erage 1I1M 1I1Y 1IGI 1IMD 1IQQ 1IR2 1ISP 1IUY 1IX5 1IXA 1M1Y 1MGI 1MMD 1MQQ 1MR2 1MSP 1MUY 1MX5 1MXA 1YGI 1YMD 1YQQ 1YR2 1YSP 1YUY 1YX5 1YXA GIMD GIQQ GIR2 GISP GIUY GIX5 GIXA MDQQ MDR2 MDSP MDUY MDX5 MDXA QQR2 QQSP QQUY QQX5 QQXA R2SP R2UY R2X5 R2XA SPUY SPX5 SPXA UYX5 UYXA X5XA1 month 0.11 -0.03 0.18 -0.02 -0.04 -0.01 -0.06 0.00 0.00 0.58 -0.01 0.88 0.03 -0.02 0.01 -0.02 0.01 0.02 0.13 0.02 0.07 0.09 0.08 0.05 0.63 0.09 -0.03 0.05 -0.01 0.03 -0.01 0.03 0.04 0.17 0.88 0.93 0.87 0.01 0.92 0.11 0.88 0.87 0.07 0.85 0.07 0.82 0.04 0.97 0.11 -0.01 0.81 0.08 0.04 -0.01 0.113 month 0.12 -0.03 0.18 0.00 -0.03 0.00 -0.05 0.01 0.01 0.58 -0.01 0.88 0.03 -0.03 0.00 -0.03 0.01 0.02 0.14 0.02 0.09 0.11 0.10 0.07 0.61 0.10 -0.03 0.04 -0.01 0.03 -0.02 0.03 0.04 0.18 0.88 0.93 0.88 0.02 0.92 0.13 0.89 0.87 0.08 0.86 0.08 0.83 0.05 0.97 0.12 0.01 0.82 0.09 0.05 0.00 0.136 month 0.11 -0.03 0.17 0.00 -0.03 0.00 -0.05 0.02 0.00 0.57 -0.01 0.88 0.02 -0.03 0.00 -0.04 0.01 0.02 0.14 0.02 0.11 0.12 0.12 0.09 0.60 0.12 -0.03 0.04 -0.02 0.02 -0.03 0.03 0.04 0.17 0.89 0.93 0.88 0.04 0.93 0.13 0.89 0.87 0.09 0.86 0.08 0.83 0.07 0.98 0.12 0.03 0.82 0.09 0.07 0.00 0.131 year 0.11 -0.03 0.16 0.00 -0.02 0.00 -0.05 0.01 0.00 0.57 -0.01 0.88 0.02 -0.03 0.00 -0.04 0.01 0.01 0.14 0.02 0.13 0.15 0.14 0.12 0.58 0.15 -0.03 0.04 -0.02 0.02 -0.04 0.04 0.03 0.17 0.89 0.93 0.88 0.07 0.93 0.13 0.89 0.87 0.12 0.86 0.07 0.83 0.09 0.98 0.12 0.06 0.82 0.09 0.09 0.00 0.122 year 0.12 -0.04 0.16 0.00 -0.02 0.01 -0.05 0.00 0.01 0.55 0.00 0.89 0.02 -0.02 0.00 -0.04 0.02 0.01 0.15 0.04 0.18 0.18 0.18 0.17 0.55 0.19 -0.02 0.04 -0.01 0.03 -0.03 0.04 0.04 0.18 0.88 0.94 0.88 0.12 0.93 0.11 0.89 0.87 0.16 0.86 0.05 0.83 0.14 0.98 0.11 0.11 0.83 0.07 0.14 0.01 0.113 year 0.12 -0.05 0.16 -0.01 -0.03 0.00 -0.06 -0.01 0.00 0.54 0.01 0.89 0.03 -0.01 0.01 -0.03 0.03 0.02 0.15 0.05 0.21 0.20 0.20 0.20 0.54 0.21 -0.02 0.05 0.00 0.03 -0.02 0.04 0.04 0.18 0.87 0.93 0.88 0.14 0.93 0.08 0.88 0.87 0.17 0.85 0.02 0.82 0.16 0.98 0.07 0.15 0.82 0.04 0.16 0.00 0.08STDEV 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.02 0.01 0.00 0.00 0.01 0.00 0.01 0.01 0.00 0.01 0.01 0.05 0.04 0.05 0.06 0.03 0.05 0.00 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.05 0.01 0.02 0.00 0.00 0.04 0.01 0.02 0.00 0.05 0.00 0.02 0.06 0.01 0.02 0.05 0.00 0.02Range 0.01 0.02 0.02 0.02 0.02 0.02 0.01 0.02 0.01 0.05 0.03 0.01 0.01 0.02 0.01 0.02 0.02 0.01 0.02 0.03 0.14 0.10 0.12 0.16 0.09 0.13 0.01 0.01 0.02 0.01 0.03 0.01 0.01 0.01 0.01 0.01 0.01 0.13 0.02 0.05 0.01 0.01 0.11 0.01 0.06 0.01 0.12 0.01 0.05 0.15 0.02 0.05 0.12 0.01 0.05

1I1M 1I1Y 1IGI 1IMD 1IQQ 1IR2 1ISP 1IUY 1IX5 1IXA 1M1Y 1MGI 1MMD 1MQQ 1MR2 1MSP 1MUY 1MX5 1MXA 1YGI 1YMD 1YQQ 1YR2 1YSP 1YUY 1YX5 1YXA GIMD GIQQ GIR2 GISP GIUY GIX5 GIXA MDQQ MDR2 MDSP MDUY MDX5 MDXA QQR2 QQSP QQUY QQX5 QQXA R2SP R2UY R2X5 R2XA SPUY SPX5 SPXA UYX5 UYXA X5XALB-95-1mo -0.40 -0.51 -0.34 -0.50 -0.52 -0.49 -0.56 -0.52 -0.50 0.24 -0.51 0.69 -0.45 -0.51 -0.48 -0.52 -0.48 -0.46 -0.37 -0.49 -0.68 -0.57 -0.64 -0.75 0.19 -0.62 -0.53 -0.45 -0.51 -0.46 -0.50 -0.46 -0.43 -0.35 0.72 0.80 0.70 -0.78 0.77 -0.49 0.74 0.68 -0.67 0.64 -0.50 0.60 -0.72 0.90 -0.49 -0.87 0.56 -0.52 -0.70 -0.51 -0.49UB-95-1 mo 0.62 0.44 0.69 0.47 0.44 0.48 0.44 0.53 0.49 0.92 0.49 1.08 0.51 0.47 0.49 0.47 0.50 0.50 0.64 0.53 0.82 0.76 0.80 0.84 1.06 0.79 0.48 0.55 0.50 0.52 0.48 0.53 0.52 0.68 1.04 1.05 1.04 0.81 1.07 0.72 1.03 1.05 0.80 1.06 0.64 1.05 0.79 1.04 0.71 0.86 1.06 0.67 0.78 0.49 0.72LB-95-3 mo -0.23 -0.34 -0.20 -0.34 -0.34 -0.32 -0.38 -0.35 -0.32 0.37 -0.32 0.72 -0.29 -0.34 -0.30 -0.36 -0.29 -0.29 -0.19 -0.30 -0.53 -0.42 -0.48 -0.60 0.26 -0.47 -0.37 -0.28 -0.35 -0.29 -0.36 -0.27 -0.27 -0.19 0.77 0.84 0.75 -0.63 0.82 -0.37 0.79 0.73 -0.52 0.70 -0.37 0.65 -0.56 0.92 -0.37 -0.73 0.63 -0.38 -0.55 -0.35 -0.36UB-95- 3mo 0.47 0.29 0.55 0.33 0.29 0.33 0.28 0.37 0.33 0.79 0.30 1.04 0.34 0.29 0.31 0.29 0.32 0.32 0.47 0.33 0.70 0.64 0.67 0.74 0.96 0.67 0.32 0.37 0.32 0.34 0.31 0.34 0.35 0.54 0.99 1.02 1.01 0.67 1.03 0.62 0.99 1.01 0.68 1.02 0.53 1.01 0.66 1.03 0.62 0.74 1.01 0.55 0.65 0.35 0.61LB-95-6 mo -0.18 -0.29 -0.15 -0.27 -0.27 -0.25 -0.31 -0.29 -0.25 0.40 -0.25 0.74 -0.22 -0.27 -0.22 -0.29 -0.23 -0.22 -0.14 -0.22 -0.44 -0.34 -0.38 -0.51 0.28 -0.38 -0.30 -0.22 -0.29 -0.22 -0.30 -0.20 -0.20 -0.14 0.80 0.86 0.77 -0.53 0.84 -0.31 0.81 0.75 -0.43 0.72 -0.32 0.67 -0.46 0.93 -0.31 -0.63 0.66 -0.32 -0.46 -0.30 -0.29UB-95-6 mo 0.41 0.23 0.48 0.26 0.22 0.25 0.22 0.32 0.26 0.75 0.22 1.03 0.27 0.22 0.23 0.21 0.26 0.25 0.42 0.26 0.65 0.59 0.61 0.68 0.91 0.61 0.25 0.30 0.25 0.27 0.24 0.27 0.28 0.49 0.98 1.01 0.99 0.62 1.02 0.57 0.98 0.99 0.62 1.00 0.47 0.99 0.60 1.02 0.56 0.68 0.99 0.49 0.59 0.30 0.55LB-95-1 yr -0.11 -0.25 -0.09 -0.22 -0.23 -0.20 -0.26 -0.25 -0.21 0.41 -0.17 0.75 -0.15 -0.20 -0.15 -0.21 -0.19 -0.14 -0.09 -0.14 -0.32 -0.23 -0.26 -0.38 0.32 -0.26 -0.24 -0.14 -0.21 -0.15 -0.22 -0.14 -0.13 -0.09 0.81 0.87 0.78 -0.41 0.86 -0.25 0.83 0.77 -0.31 0.76 -0.27 0.67 -0.34 0.95 -0.25 -0.48 0.67 -0.26 -0.34 -0.26 -0.24UB-95- 1 yr 0.33 0.18 0.40 0.22 0.18 0.21 0.17 0.27 0.21 0.72 0.15 1.02 0.19 0.15 0.16 0.13 0.22 0.17 0.36 0.19 0.59 0.52 0.55 0.61 0.84 0.55 0.18 0.22 0.18 0.19 0.15 0.21 0.20 0.43 0.96 1.00 0.99 0.55 1.00 0.51 0.95 0.98 0.54 0.97 0.42 0.98 0.53 1.01 0.50 0.60 0.97 0.43 0.52 0.26 0.48LB-95-2 yr 0.00 -0.21 0.02 -0.17 -0.17 -0.15 -0.21 -0.24 -0.15 0.41 -0.10 0.78 -0.09 -0.16 -0.11 -0.16 -0.15 -0.10 0.00 -0.07 -0.10 -0.05 -0.06 -0.14 0.35 -0.06 -0.18 -0.07 -0.15 -0.09 -0.15 -0.11 -0.08 0.01 0.83 0.89 0.79 -0.18 0.88 -0.21 0.85 0.80 -0.11 0.80 -0.24 0.71 -0.13 0.96 -0.21 -0.22 0.71 -0.22 -0.13 -0.23 -0.20UB-95-2 yr 0.24 0.13 0.30 0.17 0.13 0.17 0.11 0.24 0.17 0.70 0.11 1.00 0.14 0.12 0.12 0.08 0.20 0.13 0.29 0.15 0.46 0.41 0.42 0.48 0.75 0.43 0.13 0.16 0.13 0.15 0.09 0.19 0.15 0.34 0.94 0.98 0.97 0.42 0.99 0.43 0.93 0.94 0.42 0.93 0.35 0.94 0.40 1.00 0.43 0.45 0.94 0.35 0.40 0.24 0.41LB-95-3 yr 0.04 -0.18 0.06 -0.13 -0.13 -0.12 -0.17 -0.22 -0.12 0.42 -0.06 0.81 -0.06 -0.13 -0.09 -0.13 -0.12 -0.07 0.06 -0.03 0.00 0.03 0.03 -0.04 0.38 0.04 -0.12 -0.03 -0.11 -0.06 -0.10 -0.09 -0.04 0.07 0.83 0.89 0.80 -0.07 0.88 -0.18 0.85 0.80 -0.02 0.80 -0.21 0.72 -0.04 0.96 -0.18 -0.11 0.72 -0.18 -0.03 -0.19 -0.17UB-95-3 yr 0.20 0.08 0.26 0.11 0.08 0.12 0.05 0.21 0.11 0.65 0.08 0.97 0.11 0.10 0.10 0.06 0.18 0.10 0.23 0.13 0.41 0.36 0.37 0.45 0.69 0.38 0.07 0.13 0.11 0.13 0.06 0.18 0.13 0.28 0.92 0.97 0.97 0.36 0.98 0.34 0.92 0.93 0.36 0.90 0.24 0.92 0.35 0.99 0.33 0.40 0.93 0.26 0.35 0.20 0.32

Chart 14

Average Range of the average daily correlations across 6 rolling window's for each asset pair

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Asset pairs

Ran

ge

Table 2

Asst Description # of Days Start DateCRB PRECIOUS METALS INDEX 8007 1/17/1975CRB ENERGY INDEX (1977) 5820 9/1/1983T-BILL YIELD, 1-YEAR 11295 4/30/1953GOLDMAN SACHS COMMODITY INDEX 9395 12/31/1981S&P MIDCAP 400 INDEX 6519 1/2/1981NASDAQ COMPOSITE INDEX 5563 10/11/1984RUSSELL 2000 INDEX 7027 12/29/1978S&P 500 INDEX 19796 1/3/1928T-BOND YIELD, 30-YEAR 8123 1/31/1919S&P SMALLCAP 600 INDEX 2874 6/5/1995PHLX GOLD / SILVER INDEX 5772 12/19/1983

Asst Description # of Days Start DateCRB PRECIOUS METALS INDEX 8007 1/17/1975CRB ENERGY INDEX (1977) 5820 9/1/1983T-BILL YIELD, 1-YEAR 11295 4/30/1953GOLDMAN SACHS COMMODITY INDEX 9395 12/31/1981S&P MIDCAP 400 INDEX 6519 1/2/1981NASDAQ COMPOSITE INDEX 5563 10/11/1984RUSSELL 2000 INDEX 7027 12/29/1978S&P 500 INDEX 19796 1/3/1928T-BOND YIELD, 30-YEAR 8123 1/31/1919S&P SMALLCAP 600 INDEX 2874 6/5/1995PHLX GOLD / SILVER INDEX 5772 12/19/1983

31

Graph 1

Daily Correlations Across 2 Different Rolling Windows Across Time comparing the daily correlation of the 1-yr T-Bill Yield & the S&P500 compared to the 30-yr T-Bond & S&P500

-0.85

-0.65

-0.45

-0.25

-0.05

0.15

0.35

0.55

0.75

Date (YYYY,MM,DD)

Cor

rela

tion

1yr T-bill for1mo rolling window

30yr T-bond for 1mo rolling window

1yr T-bill for2yr rolling window

30yr T-bond for 2yr rolling window

32

Graph 2

Daily Correlations Across 6 Different Rolling Windows Across Time for the 1-yr T-Bill Yield & the S&P500

-0.82

-0.62

-0.42

-0.22

-0.02

0.18

0.38

0.58

0.78

Time (YYYY,MM,DD)

Cor

rela

tion







Graph 3

Daily Correlations Across 6 Different Rolling Windows Acrosss Time for the 30-yr T-Bond Yield vs. the S&P500

-0.82

-0.62

-0.42

-0.22

-0.02

0.18

0.38

0.58

0.78

Date (YYYY,MM,DD)

Corr

elat

ion







33

Holding Period Disributional Statistics

5 Day Moving Correlation





Dynamic Conditional Correlation

Stdev 0.5265 0.3845 0.3342 0.3116 0.2928 0.1888Skewness 0.5053 0.7398 0.9165 0.8637 0.8456 0.8362Kurtosis -0.8214 0.0166 0.3961 0.2795 0.0343 3.2906Minimum -0.9986 -0.9265 -0.8140 -0.7758 -0.6887 -0.98855th Percentile -0.9108 -0.7394 -0.6445 -0.6003 -0.5847 -0.5393Median -0.3229 -0.3057 -0.3050 -0.2903 -0.2898 -0.2839Mean -0.2197 -0.2347 -0.2319 -0.2269 -0.2205 -0.269395th Percentile 0.7574 0.5333 0.4452 0.4347 0.348 0.0557Maximum 0.9983 0.9404 0.8379 0.7781 0.6201 0.9991Corelation to DCC 0.7118 0.8626 0.7324 0.6436 0.5725Stdev 0.6481 0.4612 0.3728 0.3389 0.3110 0.0228Skewness 0.4709 0.5926 0.8547 0.8012 0.7787 0.0094Kurtosis -1.2249 -0.4723 0.3730 0.2404 -0.0093 -0.3088Minimum -0.2209 -0.9764 -0.8893 -0.8697 -0.2254 -0.07115th Percentile -0.9675 -0.8489 -0.7238 -0.6727 0.3966 -0.0442Median -0.4021 -0.3164 -0.3023 -0.2987 -0.2949 -5.30E-03Mean -0.2210 -0.2400 -0.2401 -0.2342 -0.2254 -6.61E-0395th Percentile 0.8975 0.6683 0.5127 0.4611 0.3966 0.0284Maximum 0.9997 0.9764 0.9059 0.8593 0.6460 0.0707Corelation to DCC -0.1288 -0.1921 -0.2464 -0.2435 -0.2111Stdev 0.7809 0.6077 0.4863 0.4109 0.3509 0.0219Skewness 0.3376 0.4101 0.5675 0.5759 0.6138 2.4945Kurtosis -1.6058 -1.1812 -0.6932 -0.4845 -0.2609 38.69Minimum -0.9925 -0.9953 -0.9469 -0.9227 -0.8941 -0.18855th Percentile -0.9925 -0.9485 -0.8571 -0.7862 -0.6848 -0.0516Median -0.4785 -0.3218 -0.3212 -0.3287 -0.2893 -0.0213Mean -0.1730 -0.1976 -0.2243 -0.2265 -0.2185 -0.020495th Percentile 0.9748 0.8629 0.7276 0.5861 0.5067 0.0119Maximum 0.9999 0.9828 0.9380 0.9861 0.7337 0.4167Corelation to DCC -0.0191 -0.0102 -0.0094 -0.0109 -0.0157Stdev 0.8427 0.6970 0.5720 0.4980 0.4257 0.0273Skewness 0.2716 0.3437 0.3265 0.2820 0.2167 -3.3287Kurtosis -1.7397 -1.4218 -1.1968 -1.0050 -0.9071 116.47Minimum -0.9999 -0.9976 -0.9922 -0.9654 -0.9194 -0.83855th Percentile -0.9965 -0.9767 -0.9178 -0.8468 -0.8059 -0.0556Median -0.5178 -0.8784 -0.2777 -0.2059 -0.1817 -0.0183Mean -0.1380 -0.1626 -0.1683 -0.1673 -0.1622 -0.018495th Percentile 0.9906 0.9525 0.7978 0.7142 0.5889 0.0202Maximum 0.9999 0.9929 0.9841 0.8948 0.7213 0.2679Corelation to DCC 0.0053 0.0072 0.0077 0.0034 0.0080Stdev 0.9229 0.8354 0.7467 0.6943 0.6131 0.0272Skewness 0.1230 0.1148 0.1173 0.1145 0.1099 1.8285Kurtosis -1.9046 -1.7916 -1.6607 -1.6248 -1.4514 23.7325Minimum -0.9999 -0.9994 -0.9967 -0.9936 -0.9758 -0.14205th Percentile -0.9991 -0.9945 -0.9779 -0.9513 -0.8995 -0.0462Median -0.5369 -0.2402 -0.1511 -0.1856 -0.0658 -0.0070Mean -0.0651 -0.0728 -0.0845 -0.0893 -0.0968 -0.006195th Percentile 0.9976 0.9923 0.9393 0.9070 0.8376 0.0350Maximum 0.9999 0.9980 0.9888 0.9631 0.9095 0.4700Corelation to DCC 0.0320 0.0471 0.0533 0.0489 0.0125

Table 5.2.1: Distributional Statistics for Rolling vs. Dynamic Conditional Correlations (S&P 500 Equity Index Returns vs.10 Year Treasury Yield Changes 1977-2006)

Daily

Weekly

Monthly

Quarterly

Annually

34

-1.0 -0.5 0.0 0.5 1.0Correlation

0.0

0.5

1.0

1.5

2.0

Pro

babi

lity

Den

sity

Figure 5.2.1: Densities of Correlation EstimatesS&P500 Index Daily Returns and 10 Year Treasury Yield Daily Changes 1977-2006

Dynamic Conditional CorrelationMoving 5-Day CorrelationMoving 22-Day CorrelationMoving 66-Day CorrelationMoving 126-Day CorrelationMoving 252-Day Correlation

-1.0 -0.5 0.0 0.5 1.0Correlation

0

5

10

15

Pro

babi

lity

Den

sity

Figure 5.2.2: Densities of Correlation EstimatesS&P500 Index Weekly Returns and 10 Year Treasury Yield Weekly Changes 1977-2006


35

-1.0 -0.5 0.0 0.5 1.0Correlation

0

4

8

12

Pro

babi

lity

Den

sity

Figure 5.2.3: Densities of Correlation EstimatesS&P500 Index Weekly Returns and 10 Year Treasury Yield Monthly Changes 1977-2006


-1.0 -0.5 0.0 0.5 1.0Correlation

0

2

4

6

8

Pro

babi

lity

Den

sity

Figure 5.2.4: Densities of Correlation EstimatesS&P500 Index Quarterly Returns and 10 Year Treasury Yield Quarterly Changes 1977-2006


36

-1.0 -0.5 0.0 0.5 1.0Correlation

0

4

8

12

Pro

babi

lity

Den

sity

Figure 5.2.5: Densities of Correlation EstimatesS&P500 Index Annual Returns and 10 Year Treasury Yield Annual Changes 1977-2006


0

1980 1985 1990 1995 2000 2005

DCC Correlation

0


0


-0.5

0.0

0.5


-0.5

0.0

0.5


-0.5

0.0

0.5

1980 1985 1990 1995 2000 2005


Figure 5.2.6: Moving Average vs. DCC Correlations (Daily S&P500 Index Returns & 10 Year T-Bill Yield Changes)

37

-0.0

50.

000.

05

1980 1985 1990 1995 2000 2005

DCC Correlation

0


0


-0.5

0.0

0.5


-0.5

0.0

0.5


-0.5

0.0

0.5

1980 1985 1990 1995 2000 2005


Fig. 5.2.7:Moving Average vs. DCC Correlations(Weekly S&P500 Index Returns & 10 Year T-Bill Yield Changes)

0.0

0.2

0.4

1980 1985 1990 1995 2000 2005

DCC Correlation

0


0


0


-0.5

0.0

0.5


-0.5

0.0

0.5

1980 1985 1990 1995 2000 2005


Fig. 5.2.8:Moving Average vs. DCC Correlations(Monthly S&P500 Index Returns & 10 Year T-Bill Yield Changes)

38

-0.5

0.0

1980 1985 1990 1995 2000 2005

DCC Correlation

0


0


0


0


-0.5

0.0

0.5

1980 1985 1990 1995 2000 2005


Fig. 5.2.9:Moving Average vs. DCC Correlations(Quarterly S&P500 Index Returns & 10 Year T-Bill Yield Changes)

0.0

0.2

0.4

1980 1985 1990 1995 2000 2005

DCC Correlation

0


0


0


0


0

1980 1985 1990 1995 2000 2005


Fig. 5.2.10:Moving Average vs. DCC Correlations(Annual S&P500 Index Returns & 10 Year T-Bill Yield Changes)

39

Figure 5.2.11: Correlations to DCC by Holding Period and Window Length: S&P 500 Returns and 10 Year Treasury Bond Yield Changes (1977-2006)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Daily Weekly Monthly Quarterly Annually

5 Day Moving Correlation 22 Day Moving Correlation 66 Day Moving Correlation126 Day Moving Correlation 252 Day Moving Correlation

Figure 5.2.12: Average Correlations by Holding Period and Estimation Method: S&P 500 Returns and 10 Year Treasury Bond Yield Changes (1977-2006)

-0.3000

-0.2500

-0.2000

-0.1500

-0.1000

-0.0500

0.0000Daily Weekly Monthly Quarterly Annually

5 Day Moving Correlation 22 Day Moving Correlation 66 Day Moving Correlation126 Day Moving Correlation 252 Day Moving Correlation Dynamic Conditional Correlation

40

Figure 5.2.13: Standard Deviation of Correlations by Holding Period and Estimation Method: S&P 500 Returns and 10 Year Treasury Bond Yield Changes (1977-2006)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Daily Weekly Monthly Quarterly Annually

5 Day Moving Correlation 22 Day Moving Correlation 66 Day Moving Correlation126 Day Moving Correlation 252 Day Moving Correlation Dynamic Conditional Correlation

0

1980 1985 1990 1995 2000 2005

Daily Holding Period

-0.0

50.

000.

05

Weekly Holding Period

0.0

0.2

0.4 Monthly Holding Period

-0.5

0.0

Quarterly Holding Period

0.0

0.2

0.4

Annual Holding Period

Fig. 5.2.14: Dynamic Conditional Correlations (S&P500 Index Returns & 10 Year T-Bill Yield Changes)

41

-1.0 -0.5 0.0 0.5 1.0Correlation

0

5

10

15

Pro

babi

lity

Den

sity

Figure 5.2.15: Densities of Dynamic Conditional Correlation EstimatesS&P500 Index Returns and 10 Year Treasury Yield Changes 1977-2006

Daily Holding PeriodWeekly Holding PeriodMonthly Holding PeriodQuarterly Holding PeriodAnnual Holding Period

42

Holding Period

Disributional Statistics






Dynamic Conditional Correlation

Constant Correlation

Stdev 0.0118 0.0100 0.0100 0.0101 0.0103 0.0069 0.0158Skewness 1.2425 -1.6764 -1.6631 -1.628 -1.5771 -0.9934 -1.9343Kurtosis 74.44 40.08 40.20 39.46 38.36 24.6421 52.3470Minimum -0.2289 -0.2288 -0.2288 -0.2288 -0.2287 -0.1525 -0.34325th Percentile -0.0155 -0.0147 -0.0146 -0.0147 -0.0150 -0.0101 -0.0230Median 6.60E-04 5.31E-04 5.78E-04 5.64E-04 6.08E-04 3.61E-04 7.50E-04Mean 5.40E-04 4.32E-04 4.67E-04 4.56E-04 4.64E-04 3.05E-04 7.50E-0495th Percentile 0.0162 0.0154 0.0154 0.0154 0.0156 0.0105 0.0239Maximum 0.2891 0.0866 0.0865 0.0865 0.0862 0.0578 0.1299Stdev 0.0415 0.0251 0.0240 0.0242 0.0246 0.0171 0.0381Skewness -2.4362 -0.6353 -0.7689 -0.7470 -0.7258 -0.3129 -0.71895Kurtosis 129.70 8.4132 9.9533 10.0036 10.0352 6.2536 15.375Minimum -0.9981 -0.3027 -0.3005 0.0024 -0.3001 -0.2085 -0.46925th Percentile -0.0426 -0.0356 -0.0316 -0.0351 -0.0358 -0.0237 -0.05325Median 3.05E-03 2.62E-03 3.16E-03 3.25E-03 3.10E-03 2.03E-03 4.56E-03Mean 2.03E-03 2.25E-03 2.44E-03 2.36E-03 2.30E-03 1.59E-03 3.57E-0395th Percentile 0.0445 0.0398 0.0375 0.037 0.0388 0.0259 0.0582Maximum 0.6913 0.1315 0.1318 0.1456 0.1626 0.1301 0.2928Stdev 0.1241 0.0687 0.0526 0.0521 0.0519 0.0350 0.07845Skewness -0.9333 -0.4137 -0.5498 -0.5276 -0.5755 -0.3545 -0.8196Kurtosis 40.74 21.1065 2.789 2.7077 2.6648 1.9913 4.8384Minimum 0.0109 -0.7139 -0.3450 -0.3460 -0.3436 -0.2268 -0.513755th Percentile -0.1268 -0.0849 -0.0752 -0.7359 -0.0745 -0.0499 -0.11055Median 0.0107 0.0107 0.0120 0.0112 0.0110 0.0079 0.0171Mean 0.011 0.0100 0.0099 0.0095 0.0094 0.0071 0.0157595th Percentile 0.1419 0.0969 0.0892 0.0894 0.0885 0.0604 0.1344Maximum 1.3339 0.7024 0.2119 0.2044 0.1993 0.1449 0.3396Stdev 0.2482 0.1374 0.1052 0.1042 0.1038 0.0578 0.12945Skewness -1.8666 -0.8274 -1.0996 -1.0552 -1.1510 -0.5037 -1.2201Kurtosis 81.48 42.213 5.578 5.4154 5.3296 2.3061 5.397Minimum 0.0218 -1.4278 -0.69 -0.692 -0.6872 -0.3083 -0.695855th Percentile -0.2536 -0.1698 -0.1504 -1.4718 -0.1490 -0.0720 -0.16125Median 0.0214 0.0214 0.024 0.0224 0.0220 0.0221 0.0498Mean 0.022 0.02 0.0198 0.019 0.0188 0.0214 0.046895th Percentile 0.2838 0.1938 0.1784 0.1788 0.1770 0.1123 0.2505Maximum 2.6678 1.4048 0.4238 0.4088 0.3986 0.2386 0.5064Stdev 0.7446 0.4122 0.3156 0.3126 0.3114 0.1734 0.38835Skewness -5.5998 -2.4822 -3.2988 -3.1656 -3.453 -1.5110 -3.6603Kurtosis 244.44 126.639 16.734 16.2462 15.9888 6.9182 16.191Minimum 0.0654 -4.2834 -2.07 -2.076 -2.0616 -0.9248 -2.087555th Percentile -0.7608 -0.5094 -0.4512 -4.4154 -0.4470 -0.2160 -0.48375Median 0.0642 0.0642 0.072 0.0672 0.066 0.0664 0.1494Mean 0.066 0.06 0.0594 0.057 0.0564 0.0642 0.140495th Percentile 0.8514 0.5814 0.5352 0.5364 0.5310 0.3370 0.7515Maximum 8.0034 4.2144 1.2714 1.2264 1.1958 0.7158 1.5192

Annually

Table 5.2.2: Hedge Portfolio Statistics (S&P 500 Equity Index vs.10 Year Treasury Bonds 1977-2006)

Daily

Weekly

Monthly

Quarterly

43

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