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,
michael.jacobs@occ.treas.gov. 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.
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.
2
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,
3
using a broad database from Commodity Research Bureau and Bloomberg.
These data will be representative of broad asset classes of 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
4
correlations structure independently of the volatility structure, yielding 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
5
sums of squares (ICSS) algorithm to detect the structural breaks 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 ij ,t between two zero-mean random
variables ri and rj at time t as:
ij ,t 
Et 1  ri ,t rj ,t 
(1)
   
Et 1 ri 2,t Et 1 r j2,t
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 12,t h|t similarly and it will
posses the same properties:
ij ,t  h|t 
Et  ri ,t  h rj ,t  h 
(2)
   
2
2
Et ri ,t h Et 1 rj ,t h
Note that we may express (1) in terms of the standardized residuals  lt l  i, j , if
we define the conditional standard deviation hl ,t l  i, j in terms of r ,lt :
 
hl ,t  Et 1 rl 2,t
(3)
rlt  hlt  lt
l  i, j
(4)
Then it follows that the conditional correlation of the returns is just the conditional
correlation of the standardized residuals:
ij ,t 
Et 1   i ,t  j ,t 
   
Et 1  i ,t Et 1  j ,t
2
(5)
2
A simple method of the conditional correlation (1) or (5) is given by the rolling
window moving average estimator for length k (RWMA-k):
6
t 1
ˆ
(k )
ij ,t


s t  k 1
t 1
ri , s rj , s
k  0,.., t  1
t 1

2
s t  k 1
ri ,s

r
s t  k 1
(6)
2
j ,s
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 (EWMAλ):
t 1
ˆ
( )
ij ,t


t  s 1
s 1
t 1

(7)
t 1
ri ,s   t  s 1rj2,s
t  s 1 2
s 1
ri , s rj ,s
s 1
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:
Et 1  rt rtT   H t
(8)
Then under the model (7), the covariance matrix is seen to be an exponentially
weighted average of the sample return cross-product matrix:
Ht 
t 1
1 n
k
T
T
r
r


r
r

1


H




1     rt k 1rtTk 1 




t k t k 
t 1 t 1 
t 1
n k 1
k 1
(9)
The parameter λ can be estimated by maximizing the log-likelihood function:

 1 T
n log  2   log Ht   rtT Ht rt 


 2 t 1

ˆML  arg max L   | r1 ,.., rn   arg max 


(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):
vec  H t   vec  Ωt   Avec  H t 1   Bvec  rt rtT 
(11)
7
Where vec(.) is the vectorization operator that stacks the elements of an n  n
2
2
matrix into an n 2 1 vector and A, B  Rn n . A problem with this specification is
that the positive-definiteness of Ht cannot be guaranteed. A useful restriction
of (11) suggested by Engle and Kroner (1995) is:
H t  Ωt  AH t 1AT  Brt rtT BT
(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)
H t  Dt R t DTt
Dt  diag

hi ,t

(14)
Where the hi ,t are a series of univariate GARCH models. While there are various
ways in which to parameterize the conditional correlation matrix R t , the one
advocated by Engle (2002) is:
D2t  diag i   diag i  rt 1rtT1  diag  i  D2t 1
(15)
ε t  Dt1rt
(16)
R t  diag  Qt  Qt diag  Qt 
1
Qt  S
i i
T
n n
1
 A  B   A εt 1εTt 1  B Qt 1
(17)
(18)
Where i n is an n-vector of ones, S is the unconditional correlation matrix of
ε and denotes Hadamard element-by-element multiplication. This follows the
follows the MARCH specification for Qt of Ding and Engle (2001), which
guarantees its positive-semidefiniteness conditional on  i n iTn  A  B  , A, B being
positive-semidefinite. Engle (2002) shows that this is equivalent to expressing its
elements as univariate GARCH(1,1) processes:
qij ,t  ij     i ,t  j ,t  ij     qij ,t 1  ij 
(19)
Where  ij is the unconditional correlation between  i ,t and  j ,t , which gives rise
to the correlation estimator:
8
ij ,t 
qij ,t
(20)
qii ,t qij ,t
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:
L  θ, φ | r1 ,.., rn , ε1,.., εn   LV θ | r1,.., rn   LC φ | ε1,.., ε n 
(21)
ri 2,t
1 T  n 
LV  θ | r1 ,.., rn       log  2   log  hi ,t  
2 t 1  i 1 
hi ,t
(22)
LC  φ | ε1 ,.., ε n   

 
 
1 T
  log R t  εTt R t1εt  εTt εt 
2 t 1
(23)
Where θ  i , i , i  and φ  sij , aij , bij  denote the parameters of the volatility
and correlations processes, respectively, and hi ,t denotes the univariate GARCH
volatilities. This suggests a 2-stage procedure, and indeed Engle et al (2001)
demonstrate the consistency of such.
9
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 indices exhibit mild positive correlation
(0.03 and 0.11 for NASDAY and SP600, respectively).
10
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 (pvalues) 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 Results
In 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 trends
reappearing amongst all the asset pairs, while four characteristics were
common amongst on a few.
11
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, rising thereafter to 0.02 for a 3 year window. In both of these cases, the absolute values of the
12
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 yaxis, 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 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
13
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 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
14
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 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,
15
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.
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
16
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 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
17
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.
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
18
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
Table 1: Summary Statistics (Index Levels)
Minimum
1st Quartile
Median
3rd Quartile
Maximum
Count
Mean:
LCL Mean
UCL Mean
Standard Deviation
Skewness:
Kurtosis:
JB Normality Test
BP White Noise Test
S&P 500
Equity Index
4.40
18.56
86.35
251.79
1527.46
19,796
240.86
235.67
246.05
372.34
1.9405
2.4991
17,568.9
197,270.8
Goldman
Sachs
S&P 600 Small
Commodity
10 Year
CRB Precious CRB Energy
1 Year
S&P 400
NASDQQ
Russel 2000
Cap Equity
PLX Precious
Index
Treasury Yield Metals Index Index
Treasury Yield Equity Index
Equity Index
Equity Index
Index
Metals Index
98.92
1.9800
133.500
100.700
0.8200
31.46
238.10
40.52
103.75
41.85
171.49
5.5920
238.290
164.800
4.2465
78.42
445.30
121.35
171.59
75.16
192.72
7.4450
268.940
187.995
5.7180
169.87
1028.19
211.54
202.93
90.89
215.64
8.9800
316.305
250.585
7.6890
424.12
1928.67
433.23
266.06
110.71
509.63
15.2100
760.900
704.540
17.3100
817.95
5048.62
781.83
404.89
168.62
9,395
8,015
8,007
5,820
11,295
6,519
5,563
7,027
2,874
5,772
202.34
7.6155
284.83
229.79
6.2111
255.0727
1264.21
278.72
220.31
92.99
201.03
7.5570
282.87
226.87
6.1571
249.8958
1239.76
274.32
217.62
92.36
203.66
7.6741
286.79
232.71
6.2652
260.2497
1288.66
283.11
223.01
93.63
65.14
2.6733
89.55
113.71
2.9291
213.2241
930.25
187.79
73.68
24.61
2.0124
0.5200
1.5319
2.2552
0.9826
0.8543
1.1114
0.6895
0.7012
0.2944
5.7551
-0.1453
4.1222
4.8210
1.3780
-0.4877
1.1040
-0.6355
-0.3942
-0.5594
19,288.2
368.2
8,790.3
10,555.0
2,709.2
857.4
1,426.1
674.9
254.0
158.8
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
S&P 500
Equity Index
-22.8868%
-0.4528%
0.0444%
0.5257%
11.81%
19,043
0.0247%
0.0089%
0.0404%
1.1071%
-0.6955
22.2134
Goldman
Sachs
10 Year
CRB
1 Year
S&P 600
Commodity Treasury
Precious
CRB Energy Treasury
S&P 400
NASDQQ
Russel 2000 Small Cap
PLX Precious
Index
Yield
Metals Index Index
Yield
Equity Index Equity Index Equity Index Equity Index Metals Index
-18.4540%
-7.7039%
-8.6310%
-26.0563%
-20.6653%
-14.4075%
-12.0432%
-13.3887%
-6.2489%
-26.3413%
-0.5483%
-0.4596%
-0.5866%
-0.8296%
-0.5317%
-0.4122%
-0.4962%
-0.3790%
-0.6102%
-1.2722%
0.0188%
0.0000%
0.0304%
0.0267%
0.0000%
0.0838%
0.1162%
0.1094%
0.1040%
-0.0888%
0.5926%
0.4321%
0.6907%
0.8856%
0.5451%
0.5553%
0.6554%
0.5344%
0.6949%
1.2039%
7.53%
3.92%
8.81%
9.60%
25.12%
8.07%
13.25%
7.35%
5.45%
19.12%
9,182
7,103
7,735
5,613
10,711
6,304
5,376
6,795
2,774
5,578
0.0141%
-0.0102%
0.0153%
0.0138%
-0.0023%
0.0514%
0.0447%
0.0454%
0.0472%
0.0054%
-0.0081%
-0.0294%
-0.0130%
-0.0307%
-0.0296%
0.0277%
0.0080%
0.0220%
0.0045%
-0.0542%
0.0363%
0.0089%
0.0435%
0.0582%
0.0251%
0.0752%
0.0814%
0.0688%
0.0900%
0.0649%
1.0844%
0.8243%
1.2688%
1.6983%
1.4455%
0.9603%
1.3729%
0.9844%
1.1483%
2.2692%
-0.6377
-0.0791
-0.2734
-0.8004
0.3782
-0.7901
-0.2454
-0.9894
-0.1225
0.0338
11.6998
3.1080
3.4858
12.5076
22.7955
13.1762
8.8146
11.8868
1.7030
6.7281
Table 2: Summary Statistics (Index Returns)
Minimum
1st Quartile
Median
3rd Quartile
Maximum
Count
Mean:
LCL Mean
UCL Mean
Standard Deviation
Skewness:
Kurtosis:
JB Normality Test
(Returns)
BP White Noise Test
(Returns)
JB Normality Test
(Squared Returns)
BP White Noise Test
(Squared 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
137.2
16.4
23.9
25.9
26.1
74.9
125.0
37.4
201.6
31.1
18.7
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
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
Russel
1 Year
S&P 400 NASDAQ 2000
Treasury Equity
Equity
Equity
Yield
Index
Index
Index
S&P 600
Small
Cap
Equity
Index
Table 3: Correlation Matrix of Index Returns (P-Values on Below Diagonal)
Goldman
S&P 500 Sachs
10 Year
Equity
Commodity Treasury
Index
Index
Yield
S&P 500 Equity Index
Golman Sachs Commodity Index
10 Year Treasury Yield
CRB Precious Metals Index
0.0456
-0.0211
-
3.39E-37
0.0382
0.6237
2.38E-112
-0.1504
0.0056
0.0256
0.2520
0.8600
0.0241
0.0632
0.1528
0.0419
-
-0.0602 -7.2E-04
-
0.8395
0.7852
0.0257
0.0096
0.5791
-0.0727
-0.0414
0.0145
0.7723
0.8071
0.0801
-0.0413
0.0188
0.0299
0.1849
0.0302
-0.0509
0.1053
0.0881
0.0374
-0.0324
0.0649
0.0152
0.5978
-0.0255
-0.0467
-0.0356
0.0129
0.1538
0.0785
0.1340
0.0757
0.1871
0.0086
0.8675
0.9224
0.9263
0.1232
0.8701
0.8315
0.0512
0.9748
0.1353
6.43E-06
0.00E+00
2.73E-06 1.12E-30
1 Year Treasury Yield
0.9407
0.4185
0.00E+00 8.39E-05
0.2800
S&P 400 Equity Index
0.00E+00
0.4478
1.27E-14 3.04E-03
0.0558 6.12E-10
NASDAQ Equity Index
0.00E+00
0.0025
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
S&P 600 Small Cap Equity Index
0.00E+00
0.1154
3.45E-08
PLX Precious Metals Index
2.11E-09
4.26E-44
-
-
0.0283 1.76E-02 6.43E-04 1.23E-22 0.00E+00
0.4232
-
-
0.4972 4.93E-23 0.00E+00 0.00E+00 0.00E+00
6.45E-11 0.00E+00 1.17E-30
PLX
Precious
Metals
Index
0.5233 2.67E-20
-
0.1086
1.73E-04 3.39E-24 9.66E-09
P-Values
20
-
Estimates
CRB Energy Index
-
CRB
Precious CRB
Metals
Energy
Index
Index
Chart 1
Average of daily correlations based on different rolling windows for
1-Yr T-Bill vs. S&P midcap Index
0.21
0.19
0.17
IY
UY
1I
1M
GI
XA
0.15
Correlation
0.13
MD
QQ
R2
SP
X5
Interest Rates
1yr T -bi l l yi el d
30yr T -bond yi el d
Commodity Indices
CRB presci ous M etal s Index
CRB Energy Index
Gol dm an Sachs Com m odi ty Index
PHLX Gol d/Si l ver Index
Equity Indices
S&P m i dcap 400 Index
Nasdaq Com posi te Index
Russel l 2000 Index
S&P 500 Index
S&P sm al l cap 600 Index
0.21
0.18
0.13
0.11
0.11
0.09
0.09
0.07
0.07
0.05
1 month
3 month
6 month
1 year
2 year
3 year
Length of rolling window of correlation
Asset Pairs exhibiting a similar pattern: IR vs. EI IR vs. CI
IY vs. QQ
IY vs. R2
IY vs. SP
IY vs. X5
IR vs. EI
IY vs. GI
IR vs CI
UY vs. MD
UY vs. QQ
UY vs. R2
UY vs. SP
UY vs. X5
EI vs. CI
UY vs. 1M
CI VS. CI
SP vs. MD
SP vs. X5
1I vs. GS
Chart 2
Average of daily correlations based on different rolling windows for
1yr T-bill yield vs. 30yr T-bond yield
0.62
0.63
IY
UY
0.61
Correlation
0.6
1I
1M
GI
XA
0.60
0.58
0.58
0.56
MD
QQ
R2
SP
X5
Interest Rates
1yr T-bill yield
30yr T-bond yield
Commodity Indices
CRB prescious Metals Index
CRB Energy Index
Goldman Sachs Commodity Index
PHLX Gold/Silver Index
Equity Indices
S&P midcap 400 Index
Nasdaq Composite Index
Russell 2000 Index
S&P 500 Index
S&P smallcap 600 Index
0.55
0.54
0.54
0.52
1 month
3 month
6 month
1 year
2 year
3 year
Length of rolling window of correlation
Asset Pairs exhibiting a similar pattern:
IR vs. CI
CI vs. CI
1I vs. 1Y
1I vs. XA
21
Chart 3
Average of daily correlations based on different rolling windows for
S&P midcap 400 Index vs. PHLX Gold/Silver Index
0.14
IY
UY
1I
1M
GI
XA
0.13
0.13
0.13
0.13
MD
QQ
R2
SP
X5
0.12
Interest Rates
1yr T -bi l l yi el d
30yr T -bond yi el d
Commodity Indices
CRB presci ous M etal s Index
CRB Energy Index
Gol dm an Sachs Com m odi ty Index
PHLX Gol d/Si l ver Index
Equity Indices
S&P m i dcap 400 Index
Nasdaq Com posi te Index
Russel l 2000 Index
S&P 500 Index
S&P sm al l cap 600 Index
Correlation
0.11
0.11
0.11
0.1
0.09
0.08
0.08
0.07
1 month
3 month
6 month
1 year
2 year
3 year
Length of rolling window of correlation
Asset Pairs exhibiting a similar pattern:
CI vs. EI
CI vs. IR
XA vs. QQ
XA vs. R2
XA vs. SP
XA vs. X5
1I vs. UY
Chart 4
Average of daily correlations based on different rolling windows for
Goldman Sachs Commodity Index vs. S&P 500 Index
1 month
3 month
6 month
1 year
2 year
3 year
-0.006
-0.01
-0.011
-0.016
Correlation
-0.02
-0.021
-0.02
IY
UY
-0.026
-0.031
1I
1M
GI
XA
MD
QQ
R2
SP
X5
Interest Rates
1yr T -bi l l yi el d
30yr T -bond yi el d
Commodity Indices
CRB presci ous M etal s Index
CRB Energy Index
Gol dm an Sachs Com m odi ty Index
PHLX Gol d/Si l ver Index
Equity Indices
S&P m i dcap 400 Index
Nasdaq Com posi te Index
Russel l 2000 Index
S&P 500 Index
S&P sm al l cap 600 Index
-0.036
Asset Pairs exhibiting a similar pattern:
-0.03
-0.03
-0.04
Length of rolling window of correlation
CI vs. EI
CI vs. IR
1M vs. MD
1M vs. SP
CI vs. CI
1M vs. UY
1I vs. GI
22
Chart 5
Average of daily correlations based on different rolling windows for
CRB Precious Metals Index vs. NASDAQ Composite Index
1 month
3 month
6 month
1 year
2 year
3 year
-0.01
-0.0125
-0.015
-0.0175
-0.02
Correlation
-0.02
-0.02
-0.022
-0.03
-0.025
Interest Rates
1yr T -bi l l yi el d
30yr T -bond yi el d
Commodity Indices
CRB presci ous Metal s Index
CRB Energy Index
Gol dman Sachs Commodi ty Index
PHLX Gol d/Si l ver Index
Equity Indices
S&P mi dcap 400 Index
Nasdaq Composi te Index
Russel l 2000 Index
S&P 500 Index
S&P smal l cap 600 Index
IY
UY
-0.03
-0.0275
1I
1M
GI
XA
-0.03
-0.0325
MD
QQ
R2
SP
X5
-0.035
-0.0375
-0.04
-0.03
-0.04
Length of rolling window of correlation
Asset Pairs exhibiting a similar pattern:
CI vs. EI
1I vs. MD
1M vs. R2
Chart 6
Average of daily correlations based on different rolling windows for
CRB Energy Index vs. Russell 2000 Index
0.1
0.09
0.08
1I
1M
GI
XA
0.07
Correlation
Interest Rates
1yr T -bi l l yi el d
30yr T -bond yi el d
Commodity Indices
CRB presci ous Metal s Index
CRB Energy Index
Gol dman Sachs Commodi ty Index
PHLX Gol d/Si l ver Index
Equity Indices
S&P mi dcap 400 Index
Nasdaq Composi te Index
Russel l 2000 Index
S&P 500 Index
S&P smal l cap 600 Index
IY
UY
0.06
0.05
MD
QQ
R2
SP
X5
0.04
0.03
0.02
0.01
0.01
0.00
0.00
0.00
0.00
6 month
1 year
2 year
0.01
0
1 month
3 month
Length of rolling window of correlation
Asset Pairs exhibiting a similar pattern: EI vs. EI
IR vs. EI
R2 vs. X5
R2 vs. MD
R2 vs. QQ
MD vs. QQ
MD vs. SP
MD vs. SPX
QQ vs X5
QQ vs X5
CI vs. EI
GI vs. UY
GI vs. 1Y
GI vs. R2
3 year
CI vs. CI
1M vs. R2
1M vs. X5
1M vs. MD
1M vs. 1I
1M vs. XA
1M vs. QQ
1I vs. SP
1I vs. XP
GI vs. R2
GI vs. X5
GI vs. QQ
GI vs. MD
XA vs. GI
XA vs. UY
XA vs. 1Y
23
Asset Pair
1 month
3 month
6 month
1 year
2 year
3 year
1 month
3 month
6 month
1 year
2 year
3 year
1IGI
1I1Y
1I1M
1ISP
1IR2
1IQQ
1IMD
0.45
X5X
UYX A
A
UY X
SPX 5
A
SPX
SPU 5
Y
R2X
A
R2X
5
R2U
Y
R2S
QQX P
A
QQX
QQU 5
Y
QQS
P
QQR
2
MDX
A
MDX
MDU 5
Y
MDS
P
MDR
MDQ 2
Q
GIX
A
GIX
5
GIU
Y
GIS
P
GIR
2
GIQ
Q
GIM
D
1YX
A
1YX
5
1YU
Y
1YS
P
1YR
2
1YQ
Q
1YM
D
1YG
I
1MX
A
1MX
5
1MU
Y
1MS
P
1MR
2
1MQ
Q
1MM
D
1MG
I
1M1
Y
1IXA
1IX5
1IUY
Chart 7
STANDARD DEVIATION OF ROLLING CORRELATION WITH 6 DIFFERENT WINDOWS
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
average
LB-95
GIX5
GIXA
MDQ
Q
MDR
2
MDS
P
MDU
Y
MDX
5
MDX
A
QQR
2
QQS
P
QQU
Y
QQX
5
QQX
A
R2SP
GIUY
GIMD
GIQQ
GIR2
GISP
1YX5
1YXA
1YSP
1YUY
1MG
I
1MM
D
1MQ
Q
1MR
2
1MS
P
1MU
Y
1MX
5
1MX
A
1YGI
1YM
D
1YQQ
1YR2
1IX5
1IXA
1M1Y
1IUY
1IR2
1ISP
1IQQ
1IGI
1IMD
1I1Y
1I1M
Standard Deviation
GIQQ
GIR2
GISP
GIUY
GIX5
GIXA
MDQ
Q
MDR
2
MDS
P
MDU
Y
MDX
5
MDX
A
QQR
2
QQS
P
QQU
Y
QQX
5
QQX
A
R2SP
R2UY
R2X5
1YX5
1YXA
GIMD
1MX5
1MXA
1YGI
1YMD
1YQQ
1YR2
1YSP
1YUY
1MR2
1MSP
1MUY
1MGI
1MM
D
1MQQ
1IX5
1IXA
1M1Y
1IR2
1ISP
1IUY
1IGI
1IMD
1IQQ
1I1M
1I1Y
Chart 8A
Standard Deviation of Moving Correlations (based on 756-day window)
12/28/1998 to 10/27/2006
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
Asset Pairs
Graph 8B
95% Confidence Interval for Average of Moving Correlations
(based on 756-day window) 12/28/1998 to 10/27/2006
1.1
0.9
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.1
0
-0.2
-0.3
Asset Pairs
UB-95
25
1I
1M
1I
1Y
1I
G
1I I
M
1I D
Q
Q
1I
R
1I 2
S
1I P
U
Y
1I
X
1I 5
X
1M A
1
1M Y
1M G
I
1MMD
Q
1M Q
1M R2
1MSP
U
1M Y
1M X5
X
1Y A
1Y GI
M
1Y D
Q
1Y Q
R
1Y 2
S
1Y P
U
1Y Y
X
1Y 5
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
G P
IU
GY
IX
G 5
M IXA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
U
R Y
2
X
R 5
2
Correlation
1I
1M
1I
1Y
1I
G
1I I
M
1I D
Q
Q
1I
R
1I 2
S
1I P
U
Y
1I
X
1I 5
X
1M A
1
1M Y
1M G
I
M
1M D
Q
1M Q
1M R2
1MSP
U
1M Y
1M X5
X
1Y A
1Y GI
M
1Y D
Q
1Y Q
R
1Y 2
S
1Y P
U
1Y Y
X
1Y 5
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
G P
IU
GY
IX
G 5
M IXA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
U
R Y
2
X
Standard Deviation
Chart 9A
Standard Deviation of Moving Correlations (based on 504-day window)
12/26/1997 to 10/27/2006
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
Asset Pairs
Graph 9B
95% Confidence Interval for Average of Moving Correlations (based on 504-day window
12/26/1997 to 10/27/2006
1.1
0.9
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.1
0
-0.2
-0.3
average
Asset Pairs
LB-95
UB-95
26
1I
1M
1I
1Y
1I
G
1I I
M
1I D
Q
Q
1I
R
1I 2
S
1I P
U
Y
1I
X
1I 5
X
1M A
1
1M Y
1M G
I
M
1M D
Q
1M Q
1M R2
1MSP
U
1M Y
1M X5
X
1Y A
1Y GI
M
1Y D
Q
1Y Q
R
1Y 2
S
1Y P
U
1Y Y
X
1Y 5
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
G P
IU
GY
IX
G 5
I
M XA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
Standard Deviation
M
1I
1
Y
1I
G
1I I
M
D
1I
Q
Q
1I
R
2
1I
S
P
1I
U
Y
1I
X
5
1I
X
1M A
1
1M Y
1M GI
M
1M D
Q
1M Q
R
1M 2
S
1M P
U
1M Y
X
1M 5
X
A
1Y
G
1Y I
M
1Y D
Q
1Y Q
R
1Y 2
S
1Y P
U
1Y Y
X
1Y 5
X
G A
IM
G D
IQ
Q
G
IR
G 2
IS
G P
IU
Y
G
IX
G 5
I
M XA
D
Q
M Q
D
R
M 2
D
S
M P
D
U
M Y
D
X
M 5
D
X
Q A
Q
R
Q 2
Q
S
Q P
Q
U
Q Y
Q
X
Q 5
Q
X
R A
2S
R P
2U
R Y
2X
R 5
2X
1I
1
Standard Deviation
Chart 10A
Standard Deviation of Moving Correlations (based on 252-day window)
12/27/1996 to 10/27/2006
0.3
0.25
0.2
0.15
0.1
0.05
0
Asset Pairs
Graph 10B
95% Confidence Interval for Average of Moving Correlations
(based on 252-day window) 12/27/1996 to 10/27/2006
1.2
1.1
0.9
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.1
0
-0.2
-0.3
-0.4
-0.5
average
Asset Pairs
LB-95
UB-95
27
1I
1M
1I
1Y
1I
G
1I I
M
1I D
Q
Q
1I
R
1I 2
S
1I P
U
Y
1I
X
1I 5
X
1M A
1
1M Y
1M G
I
M
1M D
Q
1M Q
1M R2
S
1M P
U
1M Y
1M X5
X
1Y A
1Y GI
M
1Y D
Q
1Y Q
R
1Y 2
S
1Y P
U
1Y Y
X
1Y 5
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
G P
IU
GY
IX
G 5
M IXA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
U
R Y
2
X
Standard Deviation
1I
1M
1I
1Y
1I
G
1I I
M
1I D
Q
Q
1I
R
1I 2
S
1I P
U
Y
1I
X
1I 5
X
1M A
1
1M Y
1M G
I
1MMD
Q
1M Q
1M R2
1MSP
U
1M Y
1M X5
X
1Y A
1Y GI
M
1Y D
Q
1Y Q
R
1Y 2
S
1Y P
U
1Y Y
X
1Y 5
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
G P
IU
GY
IX
G 5
M IXA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
U
R Y
2
X
Standard Deviation
Chart 11A
Standard Deviation of Moving Correlations (based on 126-day window)
6/28/1996 to 10/27/2006
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Asset Pairs
Graph 11B
95% Confidence Interval for Average of Moving Correlations
(based on 126-day window) 6/28/1996 to 10/27/2006
1.1
0.9
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.1
0
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
average
Asset Pairs
LB-95
UB-95
28
1I
1M
1I
1Y
1I
G
1I I
M
1I D
Q
Q
1I
R
1I 2
S
1I P
U
Y
1I
X
1I 5
X
1M A
1
1M Y
1M G
I
1MMD
Q
1M Q
1M R2
1MSP
U
1M Y
1M X5
X
1Y A
1Y GI
M
1Y D
Q
1Y Q
R
1Y 2
S
1Y P
U
1Y Y
X
1Y 5
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
G P
IU
GY
IX
G 5
M IXA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
U
R Y
2
X
R 5
Standard Deviation
1I
1M
1I
1Y
1I
G
1I I
M
1I D
Q
Q
1I
R
1I 2
S
1I P
U
Y
1I
X
1I 5
X
1M A
1
1M Y
1M G
I
1MMD
Q
1M Q
1M R2
1MSP
U
1M Y
1M X5
X
1Y A
1Y GI
M
1Y D
Q
1Y Q
R
1Y 2
S
1Y P
U
1Y Y
X
1Y 5
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
G P
IU
GY
IX
G 5
M IXA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
U
Standard Deviation
Chart 12A
Standard Deviation of Moving Correlations (based on 66-day window)
4/3/1996 to 10/27/2006
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Asset Pairs
Graph 12B
95% Confidence Interval for Average of Moving Correlations
(based on 66-day window) 4/3/1996 to 10/27/2006
1.2
1.1
0.9
0.8
1
0.7
0.6
0.5
0.4
0.3
0.2
-0.1
0.1
-0.2
0
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
average
Asset Pairs
LB-95
UB-95
29
1
I1
M
1
I1
Y
1
IG
1 I
IM
1 D
IQ
Q
1
IR
1 2
IS
1 P
IU
Y
1
IX
1 5
IX
1 A
M
1
1 Y
M
1 G
M I
M
1 D
M
Q
1 Q
M
1 R2
M
S
1 P
M
U
1 Y
M
1 X5
M
X
1 A
Y
1 GI
Y
M
1 D
Y
Q
1 Q
Y
R
1 2
Y
S
1 P
Y
U
1 Y
Y
X
1 5
Y
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
G P
IU
GY
IX
G 5
M IXA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
U
R Y
2
Standard Deviation
1
I1
M
1
I1
Y
1
IG
1 I
IM
1 D
IQ
Q
1
IR
1 2
IS
1P
IU
Y
1
IX
1 5
IX
1 A
M
1
1 Y
M
1 G
M I
1 MD
M
Q
1 Q
M
1 R2
M
1 SP
M
U
1 Y
M
1 X5
M
X
1 A
Y
1 GI
Y
M
1 D
Y
Q
1 Q
Y
R
1 2
Y
S
1 P
Y
U
1 Y
Y
X
1 5
Y
X
G A
IM
G D
IQ
GQ
IR
G 2
IS
GP
IU
GY
IX
G 5
M IXA
D
Q
M Q
D
M R2
D
M SP
D
U
M Y
D
M X5
D
Q XA
Q
Q R2
Q
Q SP
Q
U
Q Y
Q
Q X5
Q
X
R A
2
S
R P
2
U
R Y
2
X
R 5
Standard Deviation
Chart 13A
Standard Deviation of Moving Correlations (based on 22-day window)
1/31/1996 to 10/27/2006
0.425
0.45
0.375
0.4
0.325
0.35
0.275
0.25
0.3
0.225
0.175
0.15
0.2
0.125
0.075
0.05
0.1
0.025
0
Asset Pairs
Graph13B
95% Confidence Interval for Average of Moving Correlations
(based on 22-day window) 1/31/1996 to 10/27/2006
1.2
1.1
0.9
0.8
1
0.7
0.6
0.5
0.4
0.3
0.2
-0.1
0.1
-0.2
0
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
average
LB-95
Asset Pairs
UB-95
30
Table 1
Stdev
1 month
3 month
6 month
1 year
2 year
3 year
1I1M
0.25
0.18
0.15
0.11
0.06
0.04
1I1Y
0.24
0.16
0.13
0.11
0.09
0.07
1IGI
0.26
0.19
0.16
0.12
0.07
0.05
1IMD
0.24
0.17
0.13
0.11
0.08
0.06
1IQQ
0.24
0.16
0.12
0.10
0.08
0.05
1IR2
0.24
0.16
0.13
0.10
0.08
0.06
1ISP
0.25
0.16
0.13
0.11
0.08
0.06
1IUY
0.26
0.18
0.15
0.13
0.12
0.11
1IX5
0.25
0.16
0.13
0.10
0.08
0.06
1IXA
0.17
0.10
0.09
0.08
0.07
0.06
1M1Y
0.25
0.15
0.12
0.08
0.05
0.04
1MGI
0.10
0.08
0.07
0.07
0.05
0.04
1MMD
0.24
0.16
0.12
0.08
0.06
0.04
1MQQ
0.24
0.16
0.12
0.09
0.07
0.06
1MR2
0.24
0.15
0.11
0.08
0.06
0.05
1MSP
0.25
0.16
0.13
0.09
0.06
0.05
1MUY
0.24
0.15
0.12
0.10
0.09
0.08
1MX5
0.24
0.15
0.12
0.08
0.06
0.04
1MXA
0.25
0.16
0.14
0.11
0.07
0.04
1YGI
0.26
0.16
0.12
0.08
0.06
0.04
1YMD
0.38
0.31
0.27
0.23
0.14
0.10
1YQQ
0.33
0.27
0.23
0.19
0.11
0.08
1YR2
0.36
0.29
0.25
0.20
0.12
0.08
1YSP
0.40
0.34
0.30
0.25
0.15
0.12
1YUY
0.22
0.18
0.16
0.13
0.10
0.08
1YX5
0.35
0.28
0.25
0.20
0.12
0.09
1YXA
0.25
0.17
0.14
0.11
0.08
0.05
GIMD
0.25
0.16
0.13
0.09
0.06
0.04
GIQQ
0.25
0.17
0.14
0.10
0.07
0.05
GIR2
0.25
0.16
0.12
0.09
0.06
0.05
GISP
0.25
0.17
0.13
0.09
0.06
0.04
GIUY
0.25
0.15
0.12
0.09
0.07
0.07
GIX5
0.24
0.15
0.12
0.08
0.06
0.04
GIXA
0.26
0.18
0.16
0.13
0.08
0.05
MDQQ
0.08
0.06
0.04
0.04
0.03
0.02
MDR2
0.06
0.05
0.04
0.03
0.02
0.02
MDSP
0.08
0.06
0.06
0.05
0.04
0.04
MDUY
0.40
0.32
0.29
0.24
0.15
0.11
MDX5
0.07
0.05
0.05
0.04
0.03
0.02
MDXA
0.30
0.25
0.22
0.19
0.16
0.13
QQR2
0.07
0.05
0.04
0.03
0.02
0.02
QQSP
0.09
0.07
0.06
0.05
0.04
0.03
QQUY
0.37
0.30
0.26
0.21
0.13
0.09
QQX5
0.10
0.08
0.07
0.05
0.03
0.03
QQXA
0.29
0.23
0.20
0.17
0.15
0.11
R2SP
0.11
0.09
0.08
0.08
0.06
0.05
R2UY
0.38
0.31
0.27
0.22
0.13
0.10
R2X5
0.03
0.03
0.02
0.02
0.01
0.01
R2XA
0.30
0.25
0.22
0.19
0.16
0.13
SPUY
0.43
0.37
0.33
0.27
0.17
0.13
SPX5
0.12
0.09
0.08
0.08
0.06
0.05
SPXA
0.30
0.23
0.20
0.17
0.14
0.11
Average
1 month
3 month
6 month
1 year
2 year
3 year
STDEV
Range
1I1M
0.11
0.12
0.11
0.11
0.12
0.12
0.00
0.01
1I1Y
-0.03
-0.03
-0.03
-0.03
-0.04
-0.05
0.01
0.02
1IGI
0.18
0.18
0.17
0.16
0.16
0.16
0.01
0.02
1IMD
-0.02
0.00
0.00
0.00
0.00
-0.01
0.01
0.02
1IQQ
-0.04
-0.03
-0.03
-0.02
-0.02
-0.03
0.01
0.02
1IR2
-0.01
0.00
0.00
0.00
0.01
0.00
0.01
0.02
1ISP
-0.06
-0.05
-0.05
-0.05
-0.05
-0.06
0.01
0.01
1IUY
0.00
0.01
0.02
0.01
0.00
-0.01
0.01
0.02
1IX5
0.00
0.01
0.00
0.00
0.01
0.00
0.00
0.01
1IXA
0.58
0.58
0.57
0.57
0.55
0.54
0.02
0.05
1M1Y
-0.01
-0.01
-0.01
-0.01
0.00
0.01
0.01
0.03
1MGI
0.88
0.88
0.88
0.88
0.89
0.89
0.00
0.01
1MMD
0.03
0.03
0.02
0.02
0.02
0.03
0.00
0.01
1MQQ
-0.02
-0.03
-0.03
-0.03
-0.02
-0.01
0.01
0.02
1MR2
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.01
1MSP
-0.02
-0.03
-0.04
-0.04
-0.04
-0.03
0.01
0.02
1MUY
0.01
0.01
0.01
0.01
0.02
0.03
0.01
0.02
1MX5
0.02
0.02
0.02
0.01
0.01
0.02
0.00
0.01
1MXA
0.13
0.14
0.14
0.14
0.15
0.15
0.01
0.02
1YGI
0.02
0.02
0.02
0.02
0.04
0.05
0.01
0.03
1YMD
0.07
0.09
0.11
0.13
0.18
0.21
0.05
0.14
1YQQ
0.09
0.11
0.12
0.15
0.18
0.20
0.04
0.10
1YR2
0.08
0.10
0.12
0.14
0.18
0.20
0.05
0.12
1YSP
0.05
0.07
0.09
0.12
0.17
0.20
0.06
0.16
1YUY
0.63
0.61
0.60
0.58
0.55
0.54
0.03
0.09
1YX5
0.09
0.10
0.12
0.15
0.19
0.21
0.05
0.13
1YXA
-0.03
-0.03
-0.03
-0.03
-0.02
-0.02
0.00
0.01
GIMD
0.05
0.04
0.04
0.04
0.04
0.05
0.00
0.01
GIQQ
-0.01
-0.01
-0.02
-0.02
-0.01
0.00
0.01
0.02
GIR2
0.03
0.03
0.02
0.02
0.03
0.03
0.00
0.01
GISP
-0.01
-0.02
-0.03
-0.04
-0.03
-0.02
0.01
0.03
GIUY
0.03
0.03
0.03
0.04
0.04
0.04
0.00
0.01
GIX5
0.04
0.04
0.04
0.03
0.04
0.04
0.00
0.01
GIXA
0.17
0.18
0.17
0.17
0.18
0.18
0.00
0.01
MDQQ
0.88
0.88
0.89
0.89
0.88
0.87
0.00
0.01
MDR2
0.93
0.93
0.93
0.93
0.94
0.93
0.00
0.01
MDSP
0.87
0.88
0.88
0.88
0.88
0.88
0.01
0.01
MDUY
0.01
0.02
0.04
0.07
0.12
0.14
0.05
0.13
MDX5
0.92
0.92
0.93
0.93
0.93
0.93
0.01
0.02
MDXA
0.11
0.13
0.13
0.13
0.11
0.08
0.02
0.05
QQR2
0.88
0.89
0.89
0.89
0.89
0.88
0.00
0.01
QQSP
0.87
0.87
0.87
0.87
0.87
0.87
0.00
0.01
QQUY
0.07
0.08
0.09
0.12
0.16
0.17
0.04
0.11
QQX5
0.85
0.86
0.86
0.86
0.86
0.85
0.01
0.01
QQXA
0.07
0.08
0.08
0.07
0.05
0.02
0.02
0.06
R2SP
0.82
0.83
0.83
0.83
0.83
0.82
0.00
0.01
R2UY
0.04
0.05
0.07
0.09
0.14
0.16
0.05
0.12
R2X5
0.97
0.97
0.98
0.98
0.98
0.98
0.00
0.01
R2XA
0.11
0.12
0.12
0.12
0.11
0.07
0.02
0.05
SPUY
-0.01
0.01
0.03
0.06
0.11
0.15
0.06
0.15
SPX5
0.81
0.82
0.82
0.82
0.83
0.82
0.01
0.02
SPXA
0.08
0.09
0.09
0.09
0.07
0.04
0.02
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
LB-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
UB-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
LB-95-3 mo -0.23 -0.34 -0.20 -0.34
UB-95- 3mo 0.47 0.29 0.55 0.33
LB-95-6 mo -0.18 -0.29 -0.15 -0.27
UB-95-6 mo 0.41 0.23 0.48 0.26
-0.34
0.29
-0.27
0.22
-0.32
0.33
-0.25
0.25
-0.38
0.28
-0.31
0.22
-0.35
0.37
-0.29
0.32
-0.32
0.33
-0.25
0.26
0.37
0.79
0.40
0.75
-0.32
0.30
-0.25
0.22
0.72
1.04
0.74
1.03
-0.29
0.34
-0.22
0.27
-0.34
0.29
-0.27
0.22
-0.30
0.31
-0.22
0.23
-0.36
0.29
-0.29
0.21
-0.29
0.32
-0.23
0.26
-0.29
0.32
-0.22
0.25
-0.19
0.47
-0.14
0.42
-0.30
0.33
-0.22
0.26
-0.53
0.70
-0.44
0.65
-0.42
0.64
-0.34
0.59
-0.48
0.67
-0.38
0.61
-0.60
0.74
-0.51
0.68
0.26
0.96
0.28
0.91
-0.47
0.67
-0.38
0.61
-0.37
0.32
-0.30
0.25
-0.28
0.37
-0.22
0.30
-0.35
0.32
-0.29
0.25
-0.29
0.34
-0.22
0.27
-0.36
0.31
-0.30
0.24
-0.27
0.34
-0.20
0.27
-0.27
0.35
-0.20
0.28
-0.19
0.54
-0.14
0.49
0.77
0.99
0.80
0.98
0.84
1.02
0.86
1.01
0.75
1.01
0.77
0.99
-0.63
0.67
-0.53
0.62
0.82
1.03
0.84
1.02
-0.37
0.62
-0.31
0.57
0.79
0.99
0.81
0.98
0.73
1.01
0.75
0.99
-0.52
0.68
-0.43
0.62
0.70
1.02
0.72
1.00
-0.37
0.53
-0.32
0.47
0.65
1.01
0.67
0.99
-0.56
0.66
-0.46
0.60
0.92
1.03
0.93
1.02
-0.37
0.62
-0.31
0.56
-0.73
0.74
-0.63
0.68
0.63
1.01
0.66
0.99
-0.38
0.55
-0.32
0.49
LB-95-1 yr
UB-95- 1 yr
LB-95-2 yr
UB-95-2 yr
LB-95-3 yr
UB-95-3 yr
-0.23
0.18
-0.17
0.13
-0.13
0.08
-0.20
0.21
-0.15
0.17
-0.12
0.12
-0.26
0.17
-0.21
0.11
-0.17
0.05
-0.25
0.27
-0.24
0.24
-0.22
0.21
-0.21
0.21
-0.15
0.17
-0.12
0.11
0.41
0.72
0.41
0.70
0.42
0.65
-0.17
0.15
-0.10
0.11
-0.06
0.08
0.75
1.02
0.78
1.00
0.81
0.97
-0.15
0.19
-0.09
0.14
-0.06
0.11
-0.20
0.15
-0.16
0.12
-0.13
0.10
-0.15
0.16
-0.11
0.12
-0.09
0.10
-0.21
0.13
-0.16
0.08
-0.13
0.06
-0.19
0.22
-0.15
0.20
-0.12
0.18
-0.14
0.17
-0.10
0.13
-0.07
0.10
-0.09
0.36
0.00
0.29
0.06
0.23
-0.14
0.19
-0.07
0.15
-0.03
0.13
-0.32
0.59
-0.10
0.46
0.00
0.41
-0.23
0.52
-0.05
0.41
0.03
0.36
-0.26
0.55
-0.06
0.42
0.03
0.37
-0.38
0.61
-0.14
0.48
-0.04
0.45
0.32
0.84
0.35
0.75
0.38
0.69
-0.26
0.55
-0.06
0.43
0.04
0.38
-0.24
0.18
-0.18
0.13
-0.12
0.07
-0.14
0.22
-0.07
0.16
-0.03
0.13
-0.21
0.18
-0.15
0.13
-0.11
0.11
-0.15
0.19
-0.09
0.15
-0.06
0.13
-0.22
0.15
-0.15
0.09
-0.10
0.06
-0.14
0.21
-0.11
0.19
-0.09
0.18
-0.13
0.20
-0.08
0.15
-0.04
0.13
-0.09
0.43
0.01
0.34
0.07
0.28
0.81
0.96
0.83
0.94
0.83
0.92
0.87
1.00
0.89
0.98
0.89
0.97
0.78
0.99
0.79
0.97
0.80
0.97
-0.41
0.55
-0.18
0.42
-0.07
0.36
0.86
1.00
0.88
0.99
0.88
0.98
-0.25
0.51
-0.21
0.43
-0.18
0.34
0.83
0.95
0.85
0.93
0.85
0.92
0.77
0.98
0.80
0.94
0.80
0.93
-0.31
0.54
-0.11
0.42
-0.02
0.36
0.76
0.97
0.80
0.93
0.80
0.90
-0.27
0.42
-0.24
0.35
-0.21
0.24
0.67
0.98
0.71
0.94
0.72
0.92
-0.34
0.53
-0.13
0.40
-0.04
0.35
0.95
1.01
0.96
1.00
0.96
0.99
-0.25
0.50
-0.21
0.43
-0.18
0.33
-0.48
0.60
-0.22
0.45
-0.11
0.40
0.67
0.97
0.71
0.94
0.72
0.93
-0.26
0.43
-0.22
0.35
-0.18
0.26
-0.11
0.33
0.00
0.24
0.04
0.20
-0.25
0.18
-0.21
0.13
-0.18
0.08
-0.09
0.40
0.02
0.30
0.06
0.26
-0.22
0.22
-0.17
0.17
-0.13
0.11
Chart 14
Average Range of the average daily correlations
across 6 rolling window's for each asset pair
0.18
0.16
0.14
Range
0.12
0.10
0.08
0.06
0.04
0.02
GIX5
GIXA
MDQ
Q
MDR
2
MDS
P
MDU
Y
MDX
5
MDX
A
QQR
2
QQS
P
QQU
Y
QQX
5
QQX
A
R2S
P
R2U
Y
R2X
5
R2X
A
SPU
Y
SPX
5
SPX
A
UYX
5
UYX
A
X5X
A
1I1M
1I1Y
1IGI
1IMD
1IQQ
1IR2
1ISP
1IUY
1IX5
1IXA
1M1
Y
1MG
I
1MM
D
1MQ
Q
1MR
2
1MS
P
1MU
Y
1MX
5
1MX
A
1YG
I
1YM
D
1YQ
Q
1YR
2
1YS
P
1YU
Y
1YX
5
1YX
A
GIM
D
GIQQ
GIR2
GISP
GIUY
0.00
Asset pairs
Table 2
Asst Description
CRB PRECIOUS METALS INDEX
CRB ENERGY INDEX (1977)
T-BILL YIELD, 1-YEAR
GOLDMAN SACHS COMMODITY INDEX
S&P MIDCAP 400 INDEX
NASDAQ COMPOSITE INDEX
RUSSELL 2000 INDEX
S&P 500 INDEX
T-BOND YIELD, 30-YEAR
S&P SMALLCAP 600 INDEX
PHLX GOLD / SILVER INDEX
# of Days Start Date
8007 1/17/1975
5820 9/1/1983
11295 4/30/1953
9395 12/31/1981
6519 1/2/1981
5563 10/11/1984
7027 12/29/1978
19796 1/3/1928
8123 1/31/1919
2874 6/5/1995
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
0.75
0.55
0.35
Correlation
0.15
-0.05
-0.25
-0.45
-0.65
19
96
06
24
19
96
12
16
19
97
06
11
19
97
12
03
19
98
06
01
19
98
11
20
19
99
05
19
19
99
11
10
20
00
05
05
20
00
10
27
20
01
04
25
20
01
10
18
20
02
04
16
20
02
10
08
20
03
04
03
20
03
09
26
20
04
03
23
20
04
09
16
20
05
03
11
20
05
09
02
td
at
e
-0.85
Date (YYYY,MM,DD)
1yr T-bill for
1mo rolling window
30yr T-bond for
1mo rolling window
1yr T-bill for
2yr rolling window
30yr T-bond for
2yr rolling wind
32
Graph 2
D ai l y C o r r el ati o n s Acr o ss 6 D i ffer en t R o l l i n g W i n d o ws Acr o ss Ti me fo r
th e 1-yr T-B i l l Yi el d & th e S&P500
0.78
0.58
0.38
Correlation
0.18
-0.02
-0.22
-0.42
-0.62
-0.82
tda
te
606
199
24
612
199
16
706
199
11
712
199
03
806
199
01
811
199
20
905
199
19
911
199
10
005
200
05
010
200
27
104
200
Time
1yr T-bill for
1mo rolling window
1yr T-bill for
3mo rolling window
25
110
200
18
204
200
16
210
200
08
304
200
03
309
200
26
403
200
23
409
200
16
503
200
11
5
200
(YYYY,MM,DD)
1yr T-bill for
6mo rolling window
1yr T-bill for
1yr rolling window
1yr T-bill for
2yr rolling window
Graph 3
D ail y C o r r elatio n s Acro ss 6 D i ffer ent R o l l i n g W i n d o ws Acro sss Ti me fo r th e
30-yr T-B o n d Yi eld vs. th e S&P500
0.78
0.58
0.38
Correlation
0.18
-0.02
-0.22
-0.42
-0.62
-0.82
6 01
19 9
02
25
17
12
04
02
23
20
11
08
30
26
19
17
09
04
29
24
17
6 06
6 12
7 06
7 12
8 06
8 11
9 05
9 11
0 05
0 10
1 04
1 10
2 04
2 10
3 04
3 09
4 03
4 09
5 03
19 9
19 9
19 9
19 9
19 9
19 9
19 9
19 9
20 0
20 0
20 0
20 0
20 0
20 0
20 0
20 0
20 0
20 0
20 0
Date (YYYY,MM,DD)
30yr T-bond for
1mo rolling window
30yr T-bond for
3mo rolling window
30yr T-bond for
6mo rolling window
30yr T-bond for
1yr rolling window
30yr T-bond for
2yr rolling window
33
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
5 Day
22 Day
66 Day
126 Day
252 Day
Dynamic
Holding
Moving
Moving
Moving
Moving
Moving
Conditional
Period Disributional Statistics Correlation Correlation Correlation Correlation Correlation Correlation
Stdev
0.5265
0.3845
0.3342
0.3116
0.2928
0.1888
Skewness
0.5053
0.7398
0.9165
0.8637
0.8456
0.8362
Kurtosis
-0.8214
0.0166
0.3961
0.2795
0.0343
3.2906
Minimum
-0.9986
-0.9265
-0.8140
-0.7758
-0.6887
-0.9885
5th Percentile
-0.9108
-0.7394
-0.6445
-0.6003
-0.5847
-0.5393
Median
-0.3229
-0.3057
-0.3050
-0.2903
-0.2898
-0.2839
Mean
-0.2197
-0.2347
-0.2319
-0.2269
-0.2205
-0.2693
95th Percentile
0.7574
0.5333
0.4452
0.4347
0.348
0.0557
Maximum
0.9983
0.9404
0.8379
0.7781
0.6201
0.9991
Corelation to DCC
0.7118
0.8626
0.7324
0.6436
0.5725
Stdev
0.6481
0.4612
0.3728
0.3389
0.3110
0.0228
Skewness
0.4709
0.5926
0.8547
0.8012
0.7787
0.0094
Kurtosis
-1.2249
-0.4723
0.3730
0.2404
-0.0093
-0.3088
Minimum
-0.2209
-0.9764
-0.8893
-0.8697
-0.2254
-0.0711
5th Percentile
-0.9675
-0.8489
-0.7238
-0.6727
0.3966
-0.0442
Median
-0.4021
-0.3164
-0.3023
-0.2987
-0.2949
-5.30E-03
Mean
-0.2210
-0.2400
-0.2401
-0.2342
-0.2254
-6.61E-03
95th Percentile
0.8975
0.6683
0.5127
0.4611
0.3966
0.0284
Maximum
0.9997
0.9764
0.9059
0.8593
0.6460
0.0707
Corelation to DCC
-0.1288
-0.1921
-0.2464
-0.2435
-0.2111
Stdev
0.7809
0.6077
0.4863
0.4109
0.3509
0.0219
Skewness
0.3376
0.4101
0.5675
0.5759
0.6138
2.4945
Kurtosis
-1.6058
-1.1812
-0.6932
-0.4845
-0.2609
38.69
Minimum
-0.9925
-0.9953
-0.9469
-0.9227
-0.8941
-0.1885
5th Percentile
-0.9925
-0.9485
-0.8571
-0.7862
-0.6848
-0.0516
Median
-0.4785
-0.3218
-0.3212
-0.3287
-0.2893
-0.0213
Mean
-0.1730
-0.1976
-0.2243
-0.2265
-0.2185
-0.0204
95th Percentile
0.9748
0.8629
0.7276
0.5861
0.5067
0.0119
Maximum
0.9999
0.9828
0.9380
0.9861
0.7337
0.4167
Corelation to DCC
-0.0191
-0.0102
-0.0094
-0.0109
-0.0157
Stdev
0.8427
0.6970
0.5720
0.4980
0.4257
0.0273
Skewness
0.2716
0.3437
0.3265
0.2820
0.2167
-3.3287
Kurtosis
-1.7397
-1.4218
-1.1968
-1.0050
-0.9071
116.47
Minimum
-0.9999
-0.9976
-0.9922
-0.9654
-0.9194
-0.8385
5th Percentile
-0.9965
-0.9767
-0.9178
-0.8468
-0.8059
-0.0556
Median
-0.5178
-0.8784
-0.2777
-0.2059
-0.1817
-0.0183
Mean
-0.1380
-0.1626
-0.1683
-0.1673
-0.1622
-0.0184
95th Percentile
0.9906
0.9525
0.7978
0.7142
0.5889
0.0202
Maximum
0.9999
0.9929
0.9841
0.8948
0.7213
0.2679
Corelation to DCC
0.0053
0.0072
0.0077
0.0034
0.0080
Stdev
0.9229
0.8354
0.7467
0.6943
0.6131
0.0272
Skewness
0.1230
0.1148
0.1173
0.1145
0.1099
1.8285
Kurtosis
-1.9046
-1.7916
-1.6607
-1.6248
-1.4514
23.7325
Minimum
-0.9999
-0.9994
-0.9967
-0.9936
-0.9758
-0.1420
5th Percentile
-0.9991
-0.9945
-0.9779
-0.9513
-0.8995
-0.0462
Median
-0.5369
-0.2402
-0.1511
-0.1856
-0.0658
-0.0070
Mean
-0.0651
-0.0728
-0.0845
-0.0893
-0.0968
-0.0061
95th Percentile
0.9976
0.9923
0.9393
0.9070
0.8376
0.0350
Maximum
0.9999
0.9980
0.9888
0.9631
0.9095
0.4700
Corelation to DCC
0.0320
0.0471
0.0533
0.0489
0.0125
Weekly
Monthly
Quarterly
Annually
34
Figure 5.2.1: Densities of Correlation Estimates
S&P500 Index Daily Returns and 10 Year Treasury Yield Daily Changes 1977-2006
Dynamic Conditional Correlation
Moving 5-Day Correlation
Moving 22-Day Correlation
Moving 66-Day Correlation
Moving 126-Day Correlation
Moving 252-Day Correlation
Probability Density
2.0
1.5
1.0
0.5
0.0
-1.0
-0.5
0.0
0.5
1.0
Correlation
Figure 5.2.2: Densities of Correlation Estimates
S&P500 Index Weekly Returns and 10 Year Treasury Yield Weekly Changes 1977-2006
Dynamic Conditional Correlation
Moving 5-Day Correlation
Moving 22-Day Correlation
Moving 66-Day Correlation
Moving 126-Day Correlation
Moving 252-Day Correlation
Probability Density
15
10
5
0
-1.0
-0.5
0.0
0.5
1.0
Correlation
35
Figure 5.2.3: Densities of Correlation Estimates
S&P500 Index Weekly Returns and 10 Year Treasury Yield Monthly Changes 1977-2006
Dynamic Conditional Correlation
Moving 5-Day Correlation
Moving 22-Day Correlation
Moving 66-Day Correlation
Moving 126-Day Correlation
Moving 252-Day Correlation
Probability Density
12
8
4
0
-1.0
-0.5
0.0
0.5
1.0
Correlation
Figure 5.2.4: Densities of Correlation Estimates
S&P500 Index Quarterly Returns and 10 Year Treasury Yield Quarterly Changes 1977-2006
Dynamic Conditional Correlation
Moving 5-Day Correlation
Moving 22-Day Correlation
Moving 66-Day Correlation
Moving 126-Day Correlation
Moving 252-Day Correlation
Probability Density
8
6
4
2
0
-1.0
-0.5
0.0
0.5
1.0
Correlation
36
Figure 5.2.5: Densities of Correlation Estimates
S&P500 Index Annual Returns and 10 Year Treasury Yield Annual Changes 1977-2006
Dynamic Conditional Correlation
Moving 5-Day Correlation
Moving 22-Day Correlation
Moving 66-Day Correlation
Moving 126-Day Correlation
Moving 252-Day Correlation
Probability Density
12
8
4
0
-1.0
-0.5
0.0
0.5
1.0
Correlation
Figure 5.2.6: Moving Average vs. DCC Correlations (Daily S&P500 Index Returns & 10 Year T-Bill Yield Changes)
1980
1985
1990
1995
2000
2005
252 Day Moving Correlation
-0.5
-0.5
0.0
0.0
0.5
0.5
126 Day Moving Correlation
66 Day Moving Correlation
-0.5
0
0.0
0.5
22 Day Moving Correlation
5 Day Moving Correlation
0
0
DCC Correlation
1980
1985
1990
1995
2000
2005
37
Fig. 5.2.7:Moving Average vs. DCC Correlations(Weekly S&P500 Index Returns & 10 Year T-Bill Yield Changes)
1980
1985
1990
1995
2000
2005
252 Day Moving Correlation
-0.5
-0.5
0.0
0.0
0.5
0.5
126 Day Moving Correlation
66 Day Moving Correlation
-0.5
0
0.0
0.5
22 Day Moving Correlation
5 Day Moving Correlation
-0.05
0
0.00
0.05
DCC Correlation
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)
1980
1985
1990
1995
2000
2005
252 Day Moving Correlation
-0.5
-0.5
0.0
0.0
0.5
0.5
126 Day Moving Correlation
66 Day Moving Correlation
0
0
22 Day Moving Correlation
5 Day Moving Correlation
0.0
0
0.2
0.4
DCC Correlation
1980
1985
1990
1995
2000
2005
38
Fig. 5.2.9:Moving Average vs. DCC Correlations(Quarterly S&P500 Index Returns & 10 Year T-Bill Yield Changes)
1980
1985
1990
1995
2000
2005
252 Day Moving Correlation
-0.5
0
0.0
0.5
126 Day Moving Correlation
66 Day Moving Correlation
0
0
22 Day Moving Correlation
5 Day Moving Correlation
-0.5
0
0.0
DCC Correlation
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)
1980
1985
1990
1995
2000
2005
252 Day Moving Correlation
0
0
126 Day Moving Correlation
66 Day Moving Correlation
0
0
22 Day Moving Correlation
5 Day Moving Correlation
0.0
0
0.2
0.4
DCC Correlation
1980
1985
1990
1995
2000
2005
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)
1
0.8
0.6
0.4
0.2
0
Daily
Weekly
Monthly
Quarterly
Annually
-0.2
5 Day Moving Correlation
126 Day Moving Correlation
-0.4
22 Day Moving Correlation
252 Day Moving Correlation
66 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)
0.0000
Daily
Weekly
Monthly
Quarterly
Annually
-0.0500
-0.1000
-0.1500
-0.2000
-0.2500
-0.3000
5 Day Moving Correlation
126 Day Moving Correlation
22 Day Moving Correlation
252 Day Moving Correlation
66 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)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Daily
Weekly
5 Day Moving Correlation
126 Day Moving Correlation
Monthly
22 Day Moving Correlation
252 Day Moving Correlation
Quarterly
Annually
66 Day Moving Correlation
Dynamic Conditional Correlation
Fig. 5.2.14: Dynamic Conditional Correlations (S&P500 Index Returns & 10 Year T-Bill Yield Changes)
0.0
0.2
0.4
Annual Holding Period
Quarterly Holding Period
0.0
-0.5
0.2
0.0
0.4
Monthly Holding Period
Weekly Holding Period
-0.05
0
0.00
0.05
Daily Holding Period
1980
1985
1990
1995
2000
2005
41
Figure 5.2.15: Densities of Dynamic Conditional Correlation Estimates
S&P500 Index Returns and 10 Year Treasury Yield Changes 1977-2006
Daily Holding Period
Weekly Holding Period
Monthly Holding Period
Quarterly Holding Period
Annual Holding Period
Probability Density
15
10
5
0
-1.0
-0.5
0.0
0.5
1.0
Correlation
42
Table 5.2.2: Hedge Portfolio Statistics (S&P 500 Equity Index vs.10 Year Treasury Bonds 19772006)
Daily
5 Day
22 Day
66 Day
126 Day
252 Day
Dynamic
Holding Disributional
Moving
Moving
Moving
Moving
Moving
Conditional Constant
Period Statistics
Correlation Correlation Correlation Correlation Correlation Correlation Correlation
Stdev
0.0118
0.0100
0.0100
0.0101
0.0103
0.0069
0.0158
Skewness
1.2425
-1.6764
-1.6631
-1.628
-1.5771
-0.9934
-1.9343
Kurtosis
74.44
40.08
40.20
39.46
38.36
24.6421
52.3470
Minimum
-0.2289
-0.2288
-0.2288
-0.2288
-0.2287
-0.1525
-0.3432
5th Percentile
-0.0155
-0.0147
-0.0146
-0.0147
-0.0150
-0.0101
-0.0230
Median
6.60E-04 5.31E-04 5.78E-04 5.64E-04 6.08E-04 3.61E-04 7.50E-04
Mean
5.40E-04 4.32E-04 4.67E-04 4.56E-04 4.64E-04 3.05E-04 7.50E-04
95th Percentile
0.0162
0.0154
0.0154
0.0154
0.0156
0.0105
0.0239
Maximum
0.2891
0.0866
0.0865
0.0865
0.0862
0.0578
0.1299
Stdev
0.0415
0.0251
0.0240
0.0242
0.0246
0.0171
0.0381
Skewness
-2.4362
-0.6353
-0.7689
-0.7470
-0.7258
-0.3129 -0.71895
Kurtosis
129.70
8.4132
9.9533
10.0036
10.0352
6.2536
15.375
Minimum
-0.9981
-0.3027
-0.3005
0.0024
-0.3001
-0.2085
-0.4692
5th Percentile
-0.0426
-0.0356
-0.0316
-0.0351
-0.0358
-0.0237 -0.05325
Median
3.05E-03 2.62E-03 3.16E-03 3.25E-03 3.10E-03 2.03E-03 4.56E-03
Mean
2.03E-03 2.25E-03 2.44E-03 2.36E-03 2.30E-03 1.59E-03 3.57E-03
95th Percentile
0.0445
0.0398
0.0375
0.037
0.0388
0.0259
0.0582
Maximum
0.6913
0.1315
0.1318
0.1456
0.1626
0.1301
0.2928
Stdev
0.1241
0.0687
0.0526
0.0521
0.0519
0.0350
0.07845
Skewness
-0.9333
-0.4137
-0.5498
-0.5276
-0.5755
-0.3545
-0.8196
Kurtosis
40.74
21.1065
2.789
2.7077
2.6648
1.9913
4.8384
Minimum
0.0109
-0.7139
-0.3450
-0.3460
-0.3436
-0.2268 -0.51375
5th Percentile
-0.1268
-0.0849
-0.0752
-0.7359
-0.0745
-0.0499 -0.11055
Median
0.0107
0.0107
0.0120
0.0112
0.0110
0.0079
0.0171
Mean
0.011
0.0100
0.0099
0.0095
0.0094
0.0071
0.01575
95th Percentile
0.1419
0.0969
0.0892
0.0894
0.0885
0.0604
0.1344
Maximum
1.3339
0.7024
0.2119
0.2044
0.1993
0.1449
0.3396
Stdev
0.2482
0.1374
0.1052
0.1042
0.1038
0.0578
0.12945
Skewness
-1.8666
-0.8274
-1.0996
-1.0552
-1.1510
-0.5037
-1.2201
Kurtosis
81.48
42.213
5.578
5.4154
5.3296
2.3061
5.397
Minimum
0.0218
-1.4278
-0.69
-0.692
-0.6872
-0.3083 -0.69585
5th Percentile
-0.2536
-0.1698
-0.1504
-1.4718
-0.1490
-0.0720 -0.16125
Median
0.0214
0.0214
0.024
0.0224
0.0220
0.0221
0.0498
Mean
0.022
0.02
0.0198
0.019
0.0188
0.0214
0.0468
95th Percentile
0.2838
0.1938
0.1784
0.1788
0.1770
0.1123
0.2505
Maximum
2.6678
1.4048
0.4238
0.4088
0.3986
0.2386
0.5064
Stdev
0.7446
0.4122
0.3156
0.3126
0.3114
0.1734
0.38835
Skewness
-5.5998
-2.4822
-3.2988
-3.1656
-3.453
-1.5110
-3.6603
Kurtosis
244.44
126.639
16.734
16.2462
15.9888
6.9182
16.191
Minimum
0.0654
-4.2834
-2.07
-2.076
-2.0616
-0.9248 -2.08755
5th Percentile
-0.7608
-0.5094
-0.4512
-4.4154
-0.4470
-0.2160 -0.48375
Median
0.0642
0.0642
0.072
0.0672
0.066
0.0664
0.1494
Mean
0.066
0.06
0.0594
0.057
0.0564
0.0642
0.1404
95th Percentile
0.8514
0.5814
0.5352
0.5364
0.5310
0.3370
0.7515
Maximum
8.0034
4.2144
1.2714
1.2264
1.1958
0.7158
1.5192
Weekly
Monthly
Quarterly
Annually
43
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