An Empirical Analysis on the

Significance and Implications of

Forecasting Correlations Amongst Assets

David Reed Spinowitz Honors Essay

Advisor: Dr. Karagozoglu Hofstra University

Frank G. Zarb School of Business Hempstead, NY Dspino9@aol.com

Abstract:

In this paper I conduct an empirical analysis through interpretations and descriptive

charts showing what is happening to correlations over time based on six different rolling

windows, and the crucial implications this has on three central areas in today’s widening and

deepening financial society. With a total of fifty-five asset pair correlations between eleven

different assets, I will show the effect of rolling windows on correlation and will also determine

if these asset pairs change over time, how they alter over time, and whether it is a good or bad

alteration. This paper also entails a description on the importance of correlation with respect to;

portfolio diversification, derivative pricing, and VaR

Acknowledgements:

My gratitude and appreciation goes out to Ahmet K. Karagozoglu, Ph.D. for his support,

time, useful discussions, comments, and valuable research guidance which helped a great deal to

improve the quality and structure of this paper. Most of all I thank him for introducing me to this,

and increasing my awareness to this important field.

First Draft: May 3, 2007

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I. Introduction

Is it possible for one to forecast the correlations purely by monitoring its historical price

movements? If so, what does the accuracy of the forecast depend on and what type of impact can

these forecasted correlations have in regards to making financial decisions? Benjamin Graham

once said, “In the short run, the market is the voting machine, but in the long run it is a weighting

machine.” This “weighting machine,” or moving average, is debatably the most commonly used

and perhaps the most effective and helpful way as a means to identify the trends of a price and to

forecast correlation amongst asset pairs.

Under Markowitz’s theory (1952), a portfolio made up of investments with different

levels of risk, including some with a substantial amount of volatility and others that are fairly

secure, can actually moderate the overall risk the portfolio poses without sacrificing its potential

return. Volatility is one of the most important factors that contribute to the future correlation

between two securities. Using a moving average, one will reduce this volatility or daily

fluctuations to calculate an indication of the trend of correlation over time. The specified time

window used to forecast correlation with a moving average should be based on one’s objectives

in regards to time, the volatility of the underlying securities being analyzed, and the specific

asset class they are categorized in to. As time continues and new data is generated, the average

“moves” along with the value and price of the underlying security. For example a 1 month (22

day) moving average represent the movement in prices (or yield) over the past 22 days with each

new day’s data added as the oldest day’s data is dropped from the average. Accurate forecasts of

correlations are essential components in today’s financial society that enable one to price, and

trade derivatives, form diversified optimal portfolios, and calculate risk management or VaR

(value at risk).

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One of the major skills a substantial amount of today’s society is trying to achieve is the

ability to build wealth and mitigating risks over time. Successively and continuously building

wealth is influenced by several important factors. A few fundamental investment principles that

contribute a significant amount of weight in regards to the success of this objective, is to

maintain a balanced portfolio, with respect to asset allocation and diversification. Each of these

principles depends a great deal on correlation and the accuracy of forecasted correlation over

time.

Asset diversification is significantly the most important factor that contributes to the

success in meeting an investor’s financial goals. This allocation involves assigning percentages

of the total investment portfolio to various asset classes, which is based on the correlations

between those assets. When ones portfolio is allocated across an assortment of asset classes,

strong returns from one asset class that’s thriving can offset small or negative returns from asset

classes that are performing poorly. Asset allocation takes into account the different ways that

stocks, bonds, cash, hybrids and alternative investments historically have performed and has

used those characteristics to improve the chances of achieving a desired total return over the long

term, while trying to mitigate risk per unit of return.

As a result of the considerably low correlation with conventional investments, alternative

investments such as futures on commodities may provide the opportunity to enhance the return

potential of an investor’s overall portfolio while maintaining or even lowering the portfolio’s

overall level of risk or volatility, exhibiting a higher Sharpe Ratio which is an indicator of the

excess return per unit of risk (measured by standard deviation).

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With the possibility of a significant enhancement of ones returns, there is also a high

degree of risk that might not parallel an investor’s risk profile. These risks are derived from

several factors including the speculative strategies carried out by some of the fund managers and

the illiquidity inherent in certain investment structures. For the reason that these investments are

highly illiquid, investors can’t touch their money for a few years (varies from investment to

investment) therefore one has to acknowledge they will be invested for a significant period of

time, increasing the uncertainties in regards to correlation and their time varying characteristics.

In today’s world where anything has the possibility of being securitized and

collateralized, the significance of default risk correlation (especially between an entity and its

counterparty in a CDO), has been greatly increasing. According to Patel, (2003) “there are

currently four major data sets used to estimate correlation using various types of models: equity

prices, cash market credit spreads, credit default swap credit spreads, and default - based data.

Besides correlation between reference entities and counterparties, there is also correlation

between loss given default and the probability of default.”

Based on Hull, Predescu, and White’s findings (2005) “The basic Gaussian copula

survival time model provides a good approximation to our base-case structural model when the

correlation parameter in the model is set equal to the asset correlation in the structural model.

Even for relatively large credit spreads the prices given by the two models are fairly close. The

advantage of the structural model is that it provides a way of simultaneously modeling credit

rating changes and defaults. Also it is possible to make extensions to the model while

maintaining its economic integrity as a structural model.” Credit derivatives are extremely

helpful instruments for the management of credit risk within the banking and investment society.

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When trying to forecast events in the future, there is an additional amount of correlation

risk added. As a result of the growth of securitization and the market uncertainties in regards to

volatility and correlation, the price paid or sold, in reality may not reflect the fair price of a

security. VaR is an extremely important element in the process of risk management and gives

another reason to accurately forecast correlation. If risk is measured based on correlation, the

default risks of certain underlying securities, volatility and other uncertainties in regards to time

are factors that must be forecasted with minimal error and precision.

As the complexities of the securities markets continue to deepen, volatility is becoming

harder and harder to forecast, even with the Black Scholes Model (knowing at least ten different

ways to derive it for more detail oriented purposes). In today’s society, correlation is relatively

more significant and substantially harder to forecast and determine a way to sufficiently assign a

number to its risk. Relatively speaking, efficient allocation of ones portfolio is extremely

important considering the scarcity of asset classes that offer diversification today, compared to

ten years ago.

II. Literature Review

II.i Articles covered in Literature:

Elton, Gruber and Spitzer (2006) analyzed different forecasting techniques with the hopes

of finding the best method of forecasting correlation coefficients with the lowest measured error.

There are different methods available to estimate expected value. Historical returns are often

used in the forecasting of expected return. Elton and Gruber (1973) and Elton, Gruber and

Ulrich (1978) developed the Constant Correlation Model (CCM) using historical average

correlation coefficients to predict future correlation. This appeared to produce better results over

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the Single Index Model (SIM) where security returns are believed to be related to each other

exclusively by their responses to one common factor. These men displayed that all pair wise

correlation coefficients were equal to the mean correlation coefficient. Chan Korceski and

Lakonishok (1999) studied the difference of correlation coefficient and covariance. They tested

different forecasting methods and concluded the CCM had lowest mean squared errors, the

lowest amount by which the estimator differs from the amount to be estimated. Jagannathan and

Ma (2003) looked at the estimated return distribution parameters to obtain the best correlation

with the smallest amount of variance in a portfolio by using Global minimum Variance Portfolio.

They limited the amount invested in each security and believed they would end up with the same

result as if they decreased the extreme covariance estimates toward the mean. Ledoit and Wolf

(2004) still believed the CCM to be the best forecasting tool. Engle and Colacito (2003) used

other forecasts to predict the covariance between columns of a data mix consisting of variances

of variables along a main diagonal and the co variances between each pair of variables in another

matrix position. By using a random set of forecasted expected returns for the securities they

attempted to forecast the variance with the goal of finding the least amount of variance for each

expected return. This article also addressed the question if using time series to improve

forecasting was better than just looking at prior period’s value. Auto Regressive Integrated

Moving Average (ARIMA) was looked at, which used a long time series of average correlation

coefficients. Another method was the Exponential smoothing Value Model to estimate average

future correlations. It was concluded that averaging the pairwise correlation overtime increases

the forecasting reliability.

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Portfolio theory is often used in the analysis of risk versus return in the marketplace.

Harry Markowitz (1996), a Nobel Laureate, explained how portfolio theory could measure risk

versus return of assets. Mathematical equations are used to show comparisons of risk and return

in an investor’s portfolio. When estimating expected returns an analyst would need to look at

expected volatility for that security, examining if stocks are correlated to each other and/or to

market events. Diversification will result by examining correlation in a portfolio, thus resulting

in a safer investment. If the belief is that the market is unstable, perhaps interest rates will

increase in the near future, one using portfolio theory would increase what they believe the

instability of the interest rate to be and the mathematical equations would decide on what all this

means for the portfolio as a whole. Via this method, it is shown that an asset’s return and risk is

being analyzed at the same time based on external factors, which are inputted into the equation.

An asset will display if it moves up or down with certain volatility relative to itself and any other

assets in the portfolio. By means of using this technique of analysis, a portfolio could be

constructed of assets that do not move together based on certain events. One asset may increase

in value and another asset my decrease, resulting in a better-balanced, diversified portfolio.

There have been recent developments in financial econometrics dealing with time-

varying volatility, stressing parsimonious models that are easily estimated according to

Andersen, Bollersley, Christoffersen and Diebold (2005). There are three themes discussed

throughout the paper; 1) the issue of portfolio level and asset level modeling, emphasizing the

distinction between risk measurement and risk management. 2) The issues concerning low-

frequency data vs. high-frequency data, and the associated issue of parametric vs. nonparametric

volatility measurement. 3) The issue of unconditional vs. conditional risk measurement. The

paper attempted to demonstrate the power and potential of dynamic financial econometric

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methods for financial risk management. Standard methods (i.e. historical simulation) rely on

false assumptions. GARCH volatility models offer a convenient framework for modeling

important dynamic features of returns, including volatility mean-reversion, long-memory, and

asymmetries. Risk management requires a specified multivariate density model and standard

multivariate GARCH models are too heavily parameterized to be useful when dealing with large

scale problems. However, recent advances in multivariate GARCH models have potential to be

of use for medium-scale problems

Ledoit and Wolf (2004) strongly emphasize that using the sample covariance matrix for

portfolio optimization is extremely inefficient and inept. In its place they propose, “the matrix

obtained from the sample covariance matrix through a transformation called shrinkage. This

tends to pull the most extreme coefficients towards more central values, thereby systematically

reducing estimation error where it matters most.” According to Jobson and Korbie, (1980) the

sample covariance matrix is estimated with a substantial amount of error when the number of

stocks under consideration is relatively large. According to Sharpe (1963) and others, there is a

compromise between a substantially high structured estimator and a sample covariance matrix in

order to arrive at an efficient risk model

Ahmed (2000) attempts to forecast correlations for mutual funds using classified models

that are broken down based on index, historical and mean. According to him, he is the first to

estimate the correlation structure of mutual funds via forecasting models. “Test periods are

repeated by moving one year forward while dropping the first year. This process creates eight,

M1-M8, data sets for estimating correlations.” “This study classifies a mutual fund into one of

four style categories on the basis of the sensitivity of a fund’s standardized returns to those of the

four Wilshire indexes” – LCG, SCG, LCV, and SCV. The forecasted correlation estimates for

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these mutual funds are derived from three different categories of models; “historical, mean, and

index.” “This study also compares both the forecasting ability of each model in intra- and inter-

style subs-samples.” The most attractive model will be the one with the lowest overall MSE or

“mean of the absolute errors.” Ahmed is trying to determine which method can be used to

forecast correlation with the most accuracy and the lowest errors. “This paper uses a return-based

classification method similar to Gallo and Lockwood (1997).” “Many studies have shown form

size, price-to-earnings ratios, and book-to-market ratios explain the cross-sectional variation in

stock returns.” Ahmed speaks of historical models, mean models and index models throughout

the beginning of his paper. “The index models, assumes securities move together because of their

response to a set of common factors.” “In its simplest form, the market model shows changes in

stock returns being correlated with changes in a broad market index.” He discusses two types of

index models; Style Index models and Multi-Style index models. The Style-Index model

“attempts to capture the influence of a fund’s commonality with a certain investment style. In

this model two factors, market index and style index are assumed to affect fund returns.” With

the Multi-Style index, “Eun and Resnick (1992) use a multi-index model in which stock returns

are dependant on three sized based indexes representing large, mid and small capitalization

firms” (LCV, SCG, SCV, LCV). Ahmed also discusses the Farma-French 3-Factor model

originally introduced by Korthari and Warner (1997) “Results indicate that estimates from

future correlations from the Multi-Style Index, Dynamic and Fama-French 3-Factor models have

the lowest prediction errors. Moreover, the relative ranks of Multi-Style Index and Fama-French

3-Factor models have lower dispersion across different forecasting time periods and in sub-

samples of funds belonging to similar or different ‘style’ categories” Besides being good for

explaining the historical structure of correlation among funds, the FF 3-Factor model can also be

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used to forecast correlations. “In contrast, Eun and Resnick (1992) find the aggregate mean

model analogous to the Style Mean model in this paper to be the best performer for stocks.”-

(Ahmed, 2000)

Nobel Prize winner Robert Engle (2002) says that two of the basic components factored

into to any pricing model are the forecasts of both correlation and volatility (Engle 2002).

“Simple methods such as rolling historical correlations and exponential smoothing are widely

used. More complex methods, such as varieties of multivariate generalized autoregressive

conditional heteroskedasticity (GARCH) or stochastic volatility, have been extensively

investigated in the econometric literature and are used by a few sophisticated practitioners.” “In

most cases, the number of parameters in large models is too big for easy optimization. In this

article, dynamic conditional correlation (DCC) estimators are proposed that have the flexibility

of univariate GARCH but not the complexity of conventional multivariate GARCH. These

models, which parameterize the conditional correlations directly, are naturally estimated in two

steps—a series of univariate GARCH estimates and the correlation estimate.” “In this article, the

accuracy of the correlations estimated by a variety of methods is compared in bivariate settings

where many methods are feasible.” “An alternative simple approach to estimating multivariate

models is the Orthogonal GARCH method or principle component GARCH method. This was

advocated by Alexander (1998, 2001). The procedure is simply to construct unconditionally

uncorrelated linear combinations of the series r. Then univariate GARCH models are estimated

for some or all of these, and the full covariance matrix is constructed by assuming the

conditional correlations are all zero.” “The goal for this proposal is to find specifications that

potentially can estimate large covariance matrices. In this article, only bivariate systems were

estimated to establish the accuracy of this model for simpler structures.” In this paper Engle

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proposes the Dynamic Conditional Correlation (DCC) model. “These have the flexibility of

univariate GARCH models coupled with parsimonious parametric models for the correlations.

They are not linear but can often be estimated very simply with univariate or two-step methods

based on the likelihood function.” “The main finding is that the bivariate version of this model

provides a very good approximation to a variety of time-varying correlation processes. This is

true whether the criterion is mean absolute error, diagnostic tests, or tests based on value at risk

calculations.” Engle’s DCC first adjusts for differences between the variances than updates

correlations as new information is received.

Robert Engel’s “auto recessive conditional heteroskedasticity” model analyzes random

variables with different variances form the mean and how it applies to economic data over time.

Point forecast is the average of all possible results. The variance measures the difference

between the outcomes. The variance is calculated by averaging the squared deviations of each

result form this point forecast. The longer in time one goes out, the higher the variance found,

the less certainty of the future. There is a volatility clustering when large changes in prices are

followed by other large changes and small changes are usually followed by small changes.

Volatility is calculated by taking the square root of the rate at which the variance of returns

grows over time. Carr (2007) found a correlation using the ARCH Model between business

cycles and the uncertainty of inflation (not inflation itself).

According to Picerno (2000), during the time frame of 1965 – 1985 a portfolio consisting

of 20 stocks lowered the measure of risk, the standard deviation rate to 10%. During the time

frame of 1965 – 1997 50 stocks were required to maintain a 10% standard deviation. This

demonstrates that individual stocks are becoming more volatile. With this increase in individual

stock volatility, an increase in market volatility was not found to exist. Market volatility has

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been flat and the volatility of individual stocks has increased. With a decrease in correlation,

stocks don’t move together in the same direction as much. This decreases the volatility of the

market as a whole, if one stock goes up, and there is less correlation, the other stock may go up.

Based on this, the more stocks one has in their portfolio, the greater the diversification. At some

point, the additional benefit of decreased risk will end as you keep adding more stocks to your

portfolio. It is important to try to find the middle of the road where one has the right amount of

stocks in their portfolio to decrease risk. Individual stock volatility has gone up over the years

for many possible reasons. Large companies are breaking up, creating smaller more specialized

companies with greater risk. Institutional investors have more effect on the market. If they all

follow one another such as one selling a specific stock and many institutional investors follow or

one buys a stock and they all follow, this will increase the volatility of that stock. Perhaps

Benjamin Graham in his well-known portfolio analysis theory of 1949 may be correct, where a

portfolio requires 10 to 30 stocks to provide just enough diversification.

Bernstein (2007) found that approximately 7 years ago, when the stock market took a

plunge, consumers invested in asset classes other then blue chip stocks to increase their

diversification and hopefully increase returns to their portfolios. The asset classes used in this

study all moved in opposite directions to the S&P 500. Consumers were looking for

uncorrelated assets to protect and diversify their portfolio. Based on the research performed in

this paper, it is concluded that as of today, the correlation to the S&P 500 index of blue chip

stocks has decreased for many of the asset classes used in this study. It is interesting to note that

the asset classes, high quality corporate bonds and Treasury bonds, had the strongest negative

correlation. Based on this, a consumer would need to hold these bonds in their portfolio along

with the blue chip stocks to maintain an efficiently diversified portfolio. The next group of

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assets, which would provide slight diversification along with blue chip stocks, would be

commodities, T Bills and gold. An investor attempting today to diversify their portfolio by

focusing on different market sectors would have a hard time doing this. Based on this paper,

Consumer Staples appears to be the only market sector that is not increasingly correlated to the

S&P 500 index.

Bernstein (2007) believes if an investor wanted to invest in asset classes over a time

frame of less then 1 year, they would be better off investing in assets which are cash equivalents

such as T-Bills. A longer time frame for investing would allow one to invest in riskier assets thus

potentially resulting in greater returns the longer the investment horizon is. If one invests for at

least 10 years in the asset classes analyzed in this study, the probability of them not earning a

positive return on their investment decreases to 0% for all asset classes studied in this paper.

This study also showed that if they used rolling windows over only 10 years in time, depending

on what decade you studied, a different asset would be the preferential asset, bearing a higher

return in relation to its risk. For example, in the 1970’s Gold was the winner, in the 1980s it was

Non-U.S. Stocks with Art as a close second. In the 1990s the S&P 500 took the prize. In our

current decade, Real Estate and small stock asset classes are ahead the most for the decade so far.

Diversification is an important part of asset management as well as investing in the long run.

In an analysis by Lien (2005) correlations of international currencies did not stay

consistent over time. This could be due to factors such as the country’s economic situation,

monetary policy, a change in commodity prices of that country and political events as well.

When analyzing the correlation between international currencies, one should at least examine a

6-month correlation to obtain a better understanding of the currencies relationships to each other.

By examining the relationships between currencies, an investor would see what makes sense to

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invest in. For example since EUR / USD has almost a 100% negative correlation to the USD /

CHF, a portfolio consisting of investments in each pair would result in no change in your

portfolio, one’s loss would wipe out the others gain.

A portion of today’s investors believes that the stock prices of Taiwan are influenced by

those of US, in which case a spillover result is created. Kuan and Thanh (2007) are testing this

belief and will determine if there is an interaction between US and Taiwan stock markets with an

asymmetric effect. To determine if an external shock could alter the relationship between stock

markets in a country or region, Kuan and Thanh considered the importance of contagion (The

probability of substantial economic changes in one country or area spreading to other countries

or areas). To test contagion between the U.S. and Taiwan stock markets, Kuan and Thanh used

forward forecasting (one-step tests and N-step tests). They concluded contagion does in fact exist

between U.S. and Taiwan stock markets based on the substantial increase of correlation

coefficients displaying the same trend among the markets. Their results displayed that if

contagion exists between the two countries, the average correlation coefficients exhibit a

significant decrease. This indicates that the stream of capital will flow into the country with the

higher return from the country with the lower return, relatively speaking.

There has been much research conducted to reinforce the belief of increased

diversification with the use of international stocks. Li and Rouwenhorst (2005) looked at the

correlation of intentional equity. When looking over the past 150 years, correlation was found to

vary greatly. This correlation was observed as being very high during times when the barriers of

trade were diminishing between countries. This research analyzes the past 150 years in order to

properly understand the effects of international diversification on a portfolio. Over 150 years of

data, from 1850 to present, were analyzed to determine the relationships between international

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equity markets. There was found to be a substantial amount of volatility overtime in this study.

The biggest changes in volatility were during the late 19th Century, The Great Depression and the

late 20th Century. This displays the inconsistency of benefits when investing in international

equities. Over time, with the increase in integration between the economies of different

countries, came positive and negative changes for an investor of international equities. There

was more opportunity to invest in different countries, however with the addition of emerging

markets came and increase in risk. The correlation of international equities over time may be

affected by many variables such as fiscal or monetary policy, legal conditions in that country,

and the specialization of the country’s industry or even a cultural difference. An investor should

also look at the effects of correlation as well as the number of international markets available to

invest in.

A common uncomplicated belief is to diversify your portfolio into domestic and

international investments. McReynolds (2002) believes this strategy would not have been

beneficial to many investors in the 1990s due to the domestic market greatly outperforming the

international market. Even in the year 2000 when stocks started to decrease in value,

international equities also decreased. In the 1990s the correlation between U.S. markets and

international markets has increased, almost doubling. Diversification still allows investors to

decrease volatility in their portfolio. Since we do not have a crystal ball and cannot precisely

predict the future, it is still a wise choice to diversify by investing in international equity.

As volatility in the U.S. economy increases, the returns of particular emerging markets

decrease according to Bernstein (2007). A negative correlation was found for most of the

emerging markets studied to the VIX Index. As U.S. Market volatility increased as measure by

the VIX Index, the returns on most of the emerging markets would decrease. Certain monetary

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conditions with higher debt make these emerging markets a riskier asset to invest in. The asset

class of emerging markets based on their negative correlation to the market volatility will under

perform as an asset class overall if U.S. market volatility increases. The U.S. market volatility

has increased according to the VIX Index, going from 11.6 at the end of 2006 to 14.6 after the

first quarter of 2007. Investing in a more conservative emerging market is suggested. If central

banks want to tighten the money supply and increase interest rates in a specific country, the

market there will have greater deficits and not be as good an investment. In addition to rising

interest rates, a recession domestically in the United States, and a downturn in commodities

could hurt the emerging markets. The emerging markets that are not as affected by these

elements are considered to be a more defensive emerging market and are believed to be a better

investment during volatile periods in the U.S. market.

De Chiara and Raab (2002) analyzed the commodity index fund (DJ – AIGCI), over a 10-

year period, from 1991 – 2001. The non-correlation between the commodities of this index is

from the microeconomic factors that affect the commodities. Some commodities are affected by

supply and demand factors while others are affected by the expectation of inflation, the weather,

and other events, which may move a specific commodities value. When inflation goes up, the

price of financial instruments goes up and interest rates go down. However, not all commodities

are affected in this manner. As previously stated, other events such as a change in supply and

demand could alter the price of commodities. “The principal benefits which historically have

been derived from diversified commodity exposure include: long term positive returns, robust

overall negative correlations with stocks and bonds, and generally reliable increases in the

Sharpe Ratio” (De Chiara and Raab, 2002) If there were expectations that a commodity’s supply

will decrease, an investor would expect prices to increase in the future. Expectations of inflation

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are up and they may pay a higher interest rate to finance because they expect a higher price (a

return) at a later date. This is an instance where there is a positive correlation of the DJ-AIGCE

and interest rates. There are some commodity groups that will be affected by inflation.

Commodities as a whole are negatively correlated to stocks and bonds. An investor would be

better protected by adding a broad index of commodities such as the DJ – AIGCI to their

portfolio as opposed to adding just an individual commodity.

Simons (2005) found that the JOC – ECRI index of industrial commodities follows bond

yields. Bond yields no not follow this index in the short term. The problem with analyzing and

comparing this index is that the items that make up the index don’t move together as a whole.

Even if we just look at the metal segment of this index, there is still no movement together.

There was found to be no perfect clues as to what will drive the world’s economy. It is best to

look at all the different pieces and try to put this all together as a whole in order to increase our

knowledge of the market over a long-term outlook.

Currencies are affected by many market condition variables such as interest rates, supply

and demand, the political environment, economic growth, etc. In an analysis, Lien (2006)

concluded that Commodity prices are highly correlated to certain currencies. There is a lot of

volatility in the price of oil as displayed in 2005. Canada, a net oil exporter, benefited when the

price of oil increased. Japan, an oil importer, was hurt from this price increase in oil. High oil

reserves, the country’s location close to the U.S., political and economic instability in the Middle

East and South America, puts Canada in a very good position. Japan is a very big importer of

oil. The Japan Yen is highly correlated to oil prices. When oil prices increase greatly, the

economy of Japan is hurt with a lower value in the Yen. When examining oil importing or

exporting, an investor may choose to trade the Canadian Dollar against the Japanese Yen. As

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expected, the price of oil is a main indicator as to the direction of the Canadian Dollar to the

Japanese Yen. Gold is another commodity, which was examined in this paper. The Australian

dollar is heavily correlated to the price of gold. Australia is the third largest producer of gold in

the world. When the price of gold increases in value, the value of the Australian dollar moves up

as well. Australia, with its close proximity to New Zealand, exports many New Zealand goods.

Their economic condition is connected to Australia. An investor who trades commodities may

also want to trade currencies based on their correlation. It would also just be wise when trading

commodities to at least watch the movements in the currency market.

There are a substantial amount of uncertainties when it comes to analyzing financial risk,

and forecasting the amount of risk associated to each unit of return. Lanza, Manera and McAleer

(2004) found that in today’s society there has been a relatively small amount of attention towards

the magnitude of correlations in the shocks to volatility. The volatility in the conditional

variance and the serial correlation to the mean make up the components of these shocks. This

paper attempts to estimate the dynamic conditional correlations in the returns on WTI oil, trying

to determine if the various future and forward returns are in fact substitutes or complements.

Lanza, Manera and McAleer found that the dynamic volatilities in the WTI oil forward and

future prices can be either interdependent or independent over different time periods. The only

cases which displayed low volatility were the returns of the three-month and six-month futures.

Drucker (2005) found that from the beginning of 2000, when the stock market began its

plunge, to the end of 2004, investments in REITs performed extremely well, earning an

annualized total return of 22.5%. The equity market at this time turned bearish resulting in an

annualized return of 2.3% for the same time period. REITs as an alternative investment allow the

investor to diversify their portfolios. It enables them to purchase an interest in real estate without

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having to put down a lot of cash. REITs have been correlated with inflation. As inflation

increases a tenant’s rent increases, which generates more income for the landlord. REITs are

usually not thought of as being highly correlated to other assets. An investor would assume that

if interest rates fall, REITs performance would improve. With the lowering of interest rates,

comes a lowering of mortgage rates, which results in a decreased cost of borrowing. In addition,

if investors are earning less on other investments such as bonds, they may look elsewhere for

alternative investments such as REITs. However, it is noted in this article that there has not been

a strong correlation shown between interest rates and REITs. Between the middle of 2003 and

the beginning of 2004, the yield on a 5-year treasury increased 110 basis points and REITs return

increased by 20%. Between February 1996 and August 1996, yields on 5-year treasuries

increased 99 basis points while REITs increased slightly by 8%. The performance of REITs is

affected by factors other than interest rates. In the two years preceding the stock market crash in

2000, there was an oversupply of REITs on the market and share price was depressed. In this

period, the main interest of many investors was technology stocks. At the beginning of 2000,

money started flowing back to other investments such as REITs. It is interesting to note that

recently real estate values don’t seem to be as strongly correlated to values of REITs. Real estate

valuations have been increasing greatly the last 5 years. Commercial properties, however, have

not been so lucky. Occupancy rates have been suffering and rentals have decreased. Profit

margins of commercial properties have suffered. REITs are also affected by external factors

such as the government’s tax ruling on qualified corporate dividends. The dividends of REITs

don’t qualify for the 15% rate and are taxed at an ordinary income rate up to 35%. Investors

have been advised to hold REITs in tax-deferred accounts due to this higher tax rate.

19

The media sector of the market has historically been negatively correlated to the

homebuilding sector. Bernstein (2007) found that since 1998 the S&P 500 homebuilders index

has been negatively correlated to the S&P 500 Media index. Looking at 8 years out from 1998, a

negative correlation was found of –0.9 between these 2 sectors. The real estate market is

weakening. The S&P 500 Homebuilders index was down 30% for the first half of 2006

compared to a gain of 2% for the S&P 500 index. As real estate values increase, homeowners

may have more money to spend by taking out loans against the increased values of their homes.

If real estate values continue to decline for 2007 this would continue to effect consumer spending

which would affect the discretionary sector of the stock market. The media sector would be a

wise investment due to its negative correlation to the housing market.

Hedge funds have been increasingly correlated over most of the years analyzed in this

study to stocks. Bernstein (2007) believes that even with this positive correlation, hedge funds

were a worthwhile investment due to the extra value added over the period studied. Certain

mathematical equations were used in this analysis: Alpha, a positive alpha is basically the

additional return earned by the investor for taking the additional risk and not just accepting the

market return. Alpha is a funds excess return relative to its benchmark return, which is zero. It

measures the skill of the manager, taking out the return from the market. Information Ratio (IR),

uses the idea of tracking error, the numerator is the return in excess of a benchmark (µ-r) and the

denominator is the standard deviation of the differences between the portfolio returns and the

benchmark returns, i.e. the tracking error. Basically it is a measure of the excess return compared

to the benchmark. This ratio measures consistency based on performance. IR measures the value

added by a manager relative to their benchmark. The Sharpe Ratio is one of the most important

non-trivial risk adjusted performance measures. Taking the, risk free rate over a specific period

20

of time (r), subtracted from the return strategy over that specific period of time (µ) all over the

standard deviation of returns (σσσσ). A higher ratio means a better investment, more return for each

unit of assumed risk. When using these mathematical equations, one first removes the portion of

returns coming from correlations to stocks and bonds. This is performed in order to state the

value added solely by hedge funds.

II.ii Models and Empirical Results from Literature:

The average forecasted pair wise correlation from each forecasting model analyzed by

Elton, Gruber and Spitzer (2006) was altered to make them equal to the best forecast of the

average. The forecasted accuracy was measured with the root mean squared error, the amount by

which the estimator differs from the quantity to be estimated. By using this measure of error,

each technique had the same mean so the difference in forecasting reliability was solely from he

forecasted differences from the common mean. Minimum variance based on future correlation

forecasts and historic returns and variances for each security were analyzed as well each year for

33 years. The rolling average of the last 5 years average correlations and an exponential smooth

of the 5 averages outperformed the other forecasting techniques by obtaining a lower root mean

error. It was also found that when the firms were divided into 30 different industries a lower

error resulted. This research also analyzed to see if it is better to use more than one forecasting

technique. It was found that by putting equal weight to each forecasting technique, using a

rolling average of individual pair wise correlations of 30 industry groupings and using historical

forecasts, the best forecasting resulted with the lowest error.

In Engle (2002) eight different methods are used to estimate the correlations—two

multivariate GARCH models, orthogonal GARCH, two integrated DCC models, and one mean

21

reverting DCC plus the exponential smoother from Risk Metrics and the familiar 100-day

moving average. The methods and their descriptions are as follows:

1. Scalar BEKK—scalar version with variance targeting 2. Diag BEKK—diagonal version with variance targeting 3. DCC IMA—DCC with integrated moving average estimation 4. DCC LL INT—DCC by log-likelihood for integrated process 5. DCC LL MR—DCC by log-likelihood with mean reverting model 6. MA100—moving average of 100 days 7. EX .06—exponential smoothing with parameter = .06 8. OGARCH—orthogonal GARCH or principle components GARCH

Engle uses three different performance measures for the above models. He first compares the

MSE or mean absolute error of the estimated correlations. An F test is than conducted from the

regression output derived from the second measure, testing of the autocorrelation of the squared

standardized residuals. The third measure evaluates the estimator or forecasting the VaR using an

F test. In four out of the six cases the DCC mean reverting model has the least MAE. Very close

second and third-place models are DCC integrated with log-likelihood estimation and scalar

BEKK. The second standardized residual is tested for remaining autocorrelation in its square.

This is the more revealing test because it depends on the correlations. From all these

performance measures, the DCC methods are either the best or very near the best method.

Choosing among these models, the mean reverting model is the general winner, although the

integrated versions are close behind and perform best by some criteria. Generally the log-

likelihood estimation method is superior to the IMA estimator for the integrated DCC models.

First the correlation between the Dow Jones Industrial Average and the NASDAQ composite is

examined for 10 years of daily data ending in March 2000. Then daily correlations between

stocks and bonds, a central feature of asset allocation models, are considered. Finally, the daily

correlation between returns on several currencies around major historical events including the

22

launch of the Euro is examined. The correlation between the Dow and NASDAQ was estimated

with the DCC integrated method.

The second empirical example demonstrated by Engle (2002) is the correlation between

domestic stocks and bonds. “Taking bond returns to be minus the change in the 30-year

benchmark yield to maturity, the correlation between bond yields and the Dow and NASDAQ.

Although it is widely reported in the press that the NASDAQ does not seem to be sensitive to

interest rates, the data suggests that this is true only for some limited periods, including the first

quarter of 2000, and that this is also true for the Dow. Throughout the decade it appears that the

Dow is slightly more correlated with bond prices than is the NASDAQ.” “Currency correlations

show dramatic evidence of non-stationary. It is seen that this is the only data set for which the

integrated DCC cannot be rejected against the mean reverting DCC.”

High quality corporate bonds and Treasury bonds show the most negative correlation to

blue chip stocks, therefore resulting in the most diversification to the S&P 500. Bernstein (2007)

stated that in one-year time, from the beginning of 2006 to 2007, there has been a drastic change

in the correlation of T-bills to the S&P 500 going from a high negative correlation to a very

slight negative correlation. At the beginning of 2006, T-bills were considered to be the asset

class with the highest negative correlation to the S&P 500 index. Hedge funds, Real Estate, and

the MSCI EAFE have all increased their correlation to the S&P 500, and now all three of these

asset classes are highly correlated to the S&P 500, resulting in little diversification within a

portfolio. After the stock market crash of 2000, small company stocks have moved with the

market. The Russell 2000 index is also currently highly correlated to the S&P 500, moving in the

same direction as the index consistently for the past year. Art, as an asset class, also moves with

significant positive correlation to the S&P 500. Commodity futures as measured by the Goldman

23

Sachs commodity Index and Gold have decreased its correlation to the S&P 500, resulting in

only a slight correlation at the beginning of this year.

Bernstein (2007) found the majority of assets studied did reflect that over the years, the

probability of negative absolute returns did decrease the longer the asset was held in a portfolio.

Long Term Treasuries and the S&P 500 decreased the most (7-8%) in probability of negative

absolute return from holding the asset for only 1 month to holding it for 1 quarter. If an investor

decided to hold their investment for 3 years instead of just 1 month, this probability of obtaining

a negative return decreased by the following amounts:

S&P 500 25% T Bills 0% Art 7% Small Caps 32% Commodities 4% Real Estate 22% Non-US Stocks 21% Gold 8% Long Term Treasuries 33% One could observe that Commodities, Gold, and Art did not decrease as much as the other assets

by changing the time frame from 1 month to 3 years. T Bills is also an exception, bearing no risk

over the years. In a 5-year time frame Commodities, Gold and Art decreased the least in

probability of negative return. At the 10 period time frame of investment, all asset classes

reviewed showed 0% probability of negative absolute return.

For a one month period, the EUR / USD and the AUD / USD were highly correlated so

when the Euro goes up compared to the U.S. Dollar the AUD will also increase above the U.S.

Dollar (Lien). As the time frame increased this high positive correlation decreased. When

looking at 6 month correlations from 3/29/04 to 3/31/05, one could see an average high negative

correlation for the EUR / USD compared to the USD/ JPY and to the USD / CHF. A high

positive correlation during this period was found for the correlations of the EUR / USD to the

AUD / USD, The GBP / USD and the NZD / USD. Correlation over a time frame as short as one

month to one year changed.

24

Kuan and Thanh (2007) took a three-step approach when faced with the circumstances.

First they applied the “iterated cumulative sums of squares algorithm (ICSS) of Inclan and Tiao”

in order to determine the breakpoints on the markets of U.S. and Taiwan. Then a univariate

GED-EDGARCH model is used and dummy variables for structural breaks are brought into

variance equation to account for the break and asymmetry. This step is repeated until all the sub-

samples are statically insignificant. The fact that this model allows for asymmetry while omitting

the need to artificially impose positive constraints is extremely beneficial. To test the existence

of the contagion effect the DCC-GARCH model is then used to calculate approximately the

dynamic conditional correlation coefficients with structural break dummy variables. Engle’s

DCC model is estimated with a two step approach; equity returns, which is a series of univariate

GARCH estimates and the standardized noise, which is the correlation estimate. Kuan and

Thanh then used a one-step and N-step forward forecast test to determine whether the likelihood

of contagion occurred is larger than the given significant level at each break. The One-step test

standardizes every point in the series to have the value fluctuation around 0 while the N-step

tests the difference of value between the forecasting and the regression period. The One-step test

noticeably demonstrates the time-points of contagion. The N-step test that the correlation values

of the regression and the forecasting period are the same.

Li and Rouwenhorst (2005) found the variance of stocks decrease as more stocks are

added on to a portfolio. The variance will stop improving at a certain point when the number of

equities added becomes too large. In 1950 a portfolio of country indexes had a 90% risk

reduction. Around the year of 2000, this risk reduction fell to 65%. The high correlation of this

last period under analysis of 1972 to 2000 was also observed during the Great Depression of the

1930s. Historical data on international equity returns have been made available basically only

25

during the last 30 years. Based on the low amount of historical data before that time period, an

analysis on diversification of asset classes in different countries could not be performed. The

variance of a portfolio invested internationally was 10-30% of the variance of a portfolio

invested in only one market. A portfolio that invests in different international equities portrays

more diversification than a portfolio, which only invests, in one country. The only countries

where total return data was found since 1872 were France, Germany, The United Kingdom and

the United States. Correlation ranges were –0.073 during WW I (1915-1918) to 0.475 in most

recent period under analysis (1972-2000). Correlation during this period between The U.S. and

the U.K. ranged from approximately 0 to 50%. The correlation between Germany and France

ranges from –0.175 to 0.62. Substantial variation was found between correlations of

international markets over time. The returns for these countries were highly correlated at the

beginning of the 20th Century (1890-1914), the Great Depression (1930s), the Bretton Woods Era

(1946-1971) and the present period (1972-2000).

McReynolds (2002) in a 5 year study between domestic and international markets shows

the correlation of domestic markets with international markets. In the 5 years, 1997 to 2001,

correlation was up to 0.8. In the years 1985-1990, this correlation was only 0.43. Correlation in

the latter time period increased to almost a perfect market correlation of 1.0. This correlation

also seems to be more correlated when the market is in a downward trend. When markets

increase in value, the correlation of U.S. equity to international equity decreases. When

analyzing S&P 500 index from approximately 1972-2002, a standard deviation of 17.14 was

found. Looking at international equity during the same time period resulted in a standard

deviation for the MSCI EAFE of 22.5. If a portfolio was equally divided with these assets

during this period, its standard deviation was 17.47. This would result in a portfolio that was just

26

slightly more volatile than the portfolio with U.S. equity exclusively. Returns during this 30-

year period for the S&P 500 index were 12.24%, MSCI EAFE –11.15%, and the split portfolio

resulted in a return of 12.05%.

Bernstein (2007) compared the emerging markets 12-month performance correlation

(local currency) to the 12-month change in the VIX form December 1995 to March 2007. He

found that many emerging markets including Brazil, India and China, displayed a negative

correlation. Russia displayed approximately a .0 correlation and Hungary, Morocco, Venezuela

and Turkey had a positive correlation of their returns to U.S. market volatility. When looking at

the U.S. market return compared to the VIX, by using the S&P 500 and the Russell 2000, a

negative correlation was found. Another comparison was performed by analyzing the emerging

market’s financial condition, surplus or deficit and the market valuation of its equity based on

forward PE ratios. When looking at each study, it was found that countries such as India and

South Africa may be risky investments based on their high negative correlation to U.S. market

volatility and a deficit in their economy. Countries such as Brazil and Taiwan may be better

investments due to their correlation not being as negatively correlated to the VIX and each

country has a surplus account.

De Chiara and Raab (2002) performed an analysis consisting of a series of 5-year

windows from 1991 to 2001 showing negative correlation between the different Dow Jones

commodity sector indexes compared to the S&P 500 index and a Lehman Bond index. Positive

correlation was found between the commodity indexes and unexpected inflation. As a result of

the prices of financial instruments decreasing as inflation tends to rise, all else held constant, the

relationship between an index such as the DJ-AIGCI, and the course of the 3 month T-bill, would

most likely be positively correlated. Cooper and Aluminum was found to have a correlation of

27

returns greater than 0.50 to each other. Over longer time frames, only metals such as zinc seem

to determine the movement of long-term bond yields. There was no correlation of nickel to

yields on these bonds. From 1995 to the middle of 2002, there was a high correlation of

aluminum prices and long-term bond yields. As the dollar depreciates, the price of cooper

increases substantially. As the dollar strengthens, price of cooper decreases but the effect is not

as great.

Lien (2006) found Australia, Canada and New Zealand to have the highest correlation of

their currencies with commodities. From 2003 – 2005, there was an 80% correlation between the

Canadian Dollar and the price of oil. Historically as the prices of oil increased, the value of the

U.S. Dollar to the Canadian dollar decreased. The New Zealand Dollar to the U.S. Dollar and

the Australian Dollar to the U.S. dollar had a 96% positive correlation to each other when

analyzed from 2003 – 2005. Australian and New Zealand currency are both very positively

correlated to the price of gold.

Lanza, Manera and McAleer(2004) used several variations of Engles’ GARCH(1,1)

model. The Dynamic Conditional Correlation model of Engle (2002) and the Constant

Conditional Correlation GARCH model of Bollerslev (1990) where the estimated multivariate

models used. Due to the fact that Bollerslevs’ CCC model doesn’t account for asymmetric

shocks, an asymmetric GARCH model was proposed by Jagannathan and Runnkle (1992). To

incorporate the dynamics of time Engle proposed the DCC model (2002) and Tse and Tsui

proposed the Variable Conditional Correlation model. Although the dynamic conditional

correlations vary substantially, only 10% of the variations are not an economically meaningful

range of variance, while exhibiting a strong negatively distorted distribution.

28

The media industry was shown to be the most negatively correlated to the homebuilding

sector (-0.90) from June 1998 to May 2006. Bernstein (2007) looked at this data from

September of 1989 to May 2006, and found that the auto industryalso had a high negative

correlation to the homebuilding sector (-0.66). All the sectors in the S&P 500 Consumer

Discretionary Industries Index from 1998 to 2006 which were all negatively correlated to the

homebuilding sector were: Media, Automobiles, Leisure equipment and Products and Auto

Components. The sectors with high positive correlation (0.75- 0.92) to the homebuilder’s index

were: Specialty Retail, Hotel Restaurants and Leisure, Textile and Apparel, and Multilane Retail.

A statistical method, Regression Analysis, was used by Bernstin (2007) to find

correlations between different variables in order to forecast future values. Alpha of hedge fund

indexes were compared against U.S. stocks (S&P 500 Total Return Index) and U.S. Bonds

(Merrill Lynch Domestic Master Fixed Income Index). When looking at the correlation of the

hedge fund indexes to the stock market index, the hedge fund indexes produced positive Alpha

and IR which have grown in the past year. The Sharpe Ratio was higher for the hedge fund

indexes for most of the time period studied when compared to U.S. stock. Hedge fund indexes

did better then bonds since the middle of 2005.

II.iii Data from Literature:

Elton, Gruber and Spitzer (2006), in a study of stocks, took the average correlation within

groups of firms and between groups were found over 5 years and used to forecast the next year.

This was repeated for each year from 1968 to 2001. The firms were grouped according to the

industry type and firm attributes and weekly returns from the prior years were measured.

29

In Honggang and Gao’s, (2006) study, they used the raw daily price index data on DJIA

taken from Yahoo’s financial database and defined daily return as the difference of the log of the

close price index. They used 19,397 returns from 19,398 transaction dates data covering the

period from October 1 of 1928 to December 30 of 2005.

Ledoit and Wolf, (2004) studied the out-of-sample performance of their shrinkage

estimator, using historical data from DataStream to provide them with monthly U.S. stock

information. They used this information to create several value-weighted indexes to serve as

their benchmarks. “Starting in February 1983, the methodology is as follows. At the beginning of

each month, we select the N largest stocks as measured by their market value. The market values

of the stocks define their index weight. At the end of the month, we observe the (real) returns of

the individual stocks and, given their weights, compute the return on the index. This prescription

is repeated every month until the end of December 2002. As far as the benchmark size N is

concerned, we employ N = 30; 50; 100; 225; 500. This range covers such important benchmarks

as DJIA, Xetra DAX, DJ STOXX 50, FTSE 100, NASDAQ-100, NIKKEI 225, and S&P 500.”

“The out-of-sample period ranges from 02/1983 until 12/2002, so a total of 239 monthly realized

excess returns are obtained. Since the results depend on the monthly forecasts, which are

random. They repeated this process fifty times and further calculated the mean summary

statistics.

Each asset class used in the report by Bernstein (2007) was correlated to the S&P 500

calculated from monthly correlations based on a 5 year rolling window. This was done in order

to smooth out any short-term fluctuations and to use a complete cycle of information. The asset

classes analyzed were: Commodities (Goldman Sachs Commodity Index), Long-term

Treasuries, High-Grade Corporate Bonds, Cash (T-bills), Small Stocks (Russell 2000), Non-U.S.

30

Stocks (MSCI EAFE), Real Estate (NAREIT Index), Hedge Funds (HFRI Hedge Fund Index),

Art (Art Market Research) and Gold. Bernstein’s (2007) analysis used 12-month rolling returns,

January-to-January, February-to-February, etc. By analyzing the returns in this manner, certain

events which might effect the returns in a certain month would be removed and not tamper with

the results. We began looking at these returns from December 1969 until June 2006 with the

exception of real estate (began January 1972) and Art (began January 1976).

Data was analyzed by Lien (2005) for 1 month, 3 months, 6 months, and 1 year to

correlate pairs of international currencies. The currencies used were: USD-U.S. Dollar: EUR-

euro: AUD- Australian Dollar: JPY-Japanese Yen; GBP-Pound: NZD-New Zealand Dollar:

CHF- Swiss Franc; CAD-Canadian Dollar.

The results of Kuan and Thanh (2007) are based on five years of historical data from

January 1, 1997 to October 31, 2001 and include 1,128 observations. They used three big

composite indexes from the US (NYSE, S&P 500, NASDAQ) and the Taiwan weighted stock

index. To compare the indexes on relative terms, they all started at 100 on January 1, 1997.

Cross sectional and time series data on historical returns on international markets were

used in the analysis of Li and Rouwenhorst (2005). Data was taken from the following: Global

Financial Data (GFD), Jorion and Goetzmann- a sample of equity markets, Ibbotson Associates-

database of international markets–(IA), and the International Finance Corporation –database of

emerging markets (IFC). Overall this study looked at 84 international equity markets. For some

countries data did not exist or was not available for certain years. Information only in the last

three decades was easily available.

In McReynold’s (2002) study, a rolling 5-year correlation was used for data collected

from the Center for Research in Security Prices to reflect U.S. equities and Morgan Stanley

31

Capital International Europe, Australia, and Far East index (MSCI EAFE) to represent

international equities.

The CBOE Volatility index (VIX) was used by Bernstein (2007) and was compared to

the returns of various emerging markets. The VIX is the ticker symbol for the Chicago Board

Options Exchange Volatility Index. The VIX is a measure of near term volatility of S&P 500

index options. This index shows the investors expectations of 30-day volatility using a wide

range of index options of the S&P 500 to show what an investor expects future volatility to be.

De Chiara and Raab, (2002) used the DJ-AIGCI and all commodity basket data prior to

the index launch on July 14, 1998, along with other return and correlation calculations, are

historical estimations using available data. DJ-AIGCI and commodity basket data prior to 1998

were calculated by based on the 1998-1999 percentage weights, with commodity basket weights

and calculations adjusted for data availability. The U.S. Treasury Bond return was calculated

using the Lehman Brothers Long Term Treasury Bond Index. Using a weekly rolling 3 month

Treasury Bill Yield rate from CRB they calculated the collateral yield.

In his study, Simons (2005) used The Journal of Commerce – Economic Cycle Research

Institute (JOC – ECRI) index of industrial commodities (this index includes the London Metal

Exchange’s –LME, which includes copper, aluminum, nickel, led, and zinc.

Lanza, Manera and McAleer, (2004) used the univariate and multivariate GARCH

models using daily data on WTI oil one month forward prices and one-, three-, six, and twelve-

month futures prices, along with their associated returns, for the period January 3, 1985 through

January 16, 2004.

As stated in Durcker’s (2005) paper, the NAREIT is actually the oldest family of indexes,

launched its first index in 1972 and is the industries most all-inclusive and offers uniformity for

32

long-term tracking. “Its total composite for all publicly traded equity REITs includes 190 names.

NAREIT also compiles a mortgage REIT index; a hybrid index that includes both equity and

mortgage REITs; and the Real Estate 50, an equity index that comprises the 50 largest companies

and is primarily targeted toward institutional investors that seek a large-cap benchmark.” “MSCI

and Wilshire are the other main indexes in this industry. MSCI, which took over management of

the Morgan Stanley RMS index last November, weeds out microcaps (those with a market cap of

less than about $300 million) using liquidity and minimum-market-cap screens. The 122 secu-

rities in it also exclude mortgage REITs, which more closely reflect interest rates than direct real

estate exposure. The Wilshire indexes, with 92 names, banish smaller caps, net lease companies,

and mortgage and health-care REITs”. “Unlike NAREIT, MSCI and Wilshire both adjust for

free float, which means they reduce the weight of rarely traded securities, such as those owned

by insiders. That adjustment enables many managers to replicate market performance more

closely. They can match their portfolios against an index that has eliminated shares that would be

unavailable for purchase.”

Bernstein (2007) examined correlation during the time period’s: June 1998 to May 2006

and from September 1989 to May 2006. Starting from the year 1998 was believed to be a better

choice due to the belief that speculation moved the market and not just the basic fundamentals at

the beginning of that period. Correlation was compared between the S&P 500 Consumer

Discretionary Industries Index and the S&P 500 Homebuilders Index. The Homebuilders Index

is made up of 5 companies and is a measure of confidence of U.S. homebuilders.

Rolling 5 year (60 month) windows were used in this study by Bernstein (2007), from

December 1998 to March 2006. The Hedge fund indexes used were the HFRI and the CSFB. To

33

analyze the U.S. Stock market the S&P 500 total return index was used and the Merrill Lynch

Domestic Master Fixed Income Index was used to analyze bonds.

III. Data

For this empirical analysis displaying the significance and implications of correlations

over time I used daily data from the Commodity Research Database (CRB). Using Stata, a

correlation matrix was created* of fifty-five asset pairs from ten different assets coming from

three asset classes: Commodities, Interest Rates, and Equities1. (See Table 2)

The Commodities asset class consisted of four assets: Precious Metals, Energy,

Gold/Silver and Goldman Sachs Commodity Index. The Interest Rate asset class consisted of

two assets: the one-year Treasury Bill and the thirty year Treasury Bond. The Equities asset class

consisted of five assets: S&P smallcap 600, S&P madcap 400, S&P 500, NASADAQ, and the

Russell 2000. The eleven asset sources, and the amount of daily data obtained from a specified

date are displayed in Table 1 of the Appendix. For each of the 55 asset pairs 6 different moving

windows were created: a 1 month (22 day), 3 month (66 day), 6 month 122 day), 1 year (252

day), 2 year (504 day), and a 3 year (756 day).

1 MVCORR function used in Stata to generate rolling window correlations is created by Christopher F Baum (2004) and can be downloaded from http://fmwww.bc.edu/repec/bocode/m/mvcorr.ado

34

IV. Empirical Results

When comparing the average daily correlations based on six different windows for all

fifty five pairs of assets there appeared to be some similar and unique trends amongst the

different asset pairs. Based on the graphs and data there seemed to be similar and unique trends

when comparing the three different asset classes; Commodities, Equities, and Interest Rates. As

the length of the rolling window of correlation increased (from a one month rolling window to

the three year rolling window), I found two trends that were constantly reappearing amongst the

asset pairs, and four trends that were common amongst a select few asset pairs.

Assuming normality and holding all else equal, the formula to calculate the sample

standard deviation (which is merely a measure of the dispersion of the return distribution for

each particular asset) of xi is:

( ) ( )11

2−−∑

=

= nxn

i

ix µσ

Assuming normality and holding all else equal, the formulas I used to calculate the

correlations for the 55 asset pairs for the n month (n day) moving average is:

( )( )[ ]

( ) ( )

n

xx

n

xx

nyyxx

ni

i

i

ni

i

i

ni

i

ii

AB

∑∑

∑+−

−=

+−

−=

+−

−

−−

×−−

=)1(

1

2)1(

1

2

)1(

1

1

ρ or

)(*)(

),(

BStdevAStdev

BACOVAB =ρ

This formula was repeated for each of the six rolling windows, where n=22 for the 1 month,

n=66 for the 3 month, n=122 for the 6 month, n=252 for the 1 year), n=504 for the 2 year, and

n=756 for the 3 year.

Evaluating the averages of the daily correlations based on different rolling windows

between the 1 year T-Bill and the S&P midcap 400 Index, the data and charts demonstrated that

there is a constant increase in the strength of their relationship as the length of the rolling

35

window increases. From the 1 month rolling window to the 3 year rolling window the data

indicated a constant increase in their correlation from .07 to.21, respectively (see chart 1). Based

on the data and empirical results, this trend, of a direct relationship between asset pair

correlation’s and the length of the rolling window used to calculate their average daily

correlation was a repetitive trend when comparing short and long term treasury yields to both

equity indexes. This repetitive trend may be attributed to the difference in trading volume

between the two assets, causing a positive relationship between their average daily correlation

and the length of the rolling window used. Another possible reason for the lower correlations

seen in the shorter rolling windows can be partially attributed to the low levels of volatility our

economy has been recently experiencing.

I observed similar trends across the Long-Term US Treasury Yields when compared to

the five equity indexes used, (which covered the different styles and sectors of the equity

industry). This trend was displayed when comparing both short term and long-term interest rates

to a majority of the equity indexes. While the 30 Year T-Bond displayed this trend of an

increasing correlation with the CRB Energy Index, the 1-year T-Bill did not. However the 1-year

T-Bill had a similar trend when compared to the Goldman Sachs Commodity Index and the 30

year T-Bond did not. This could be attributed to the fact that the Goldman Sachs Commodity

Index represents a diversified position in commodity futures, while the CRB Energy Index is

comprised of crude oil, heating oil, and natural gas. Knowledge of this relationship between the

differences in daily average rolling correlation across different windows in regards to the future

of energy and the direction its headed should and could be taken advantage of.

The relationship between commodities (especially energy) and US Treasury’s (both Long

Term and Short Term) should be carefully observed over the next decade or two. The differences

36

between the 1-month and 30 year Treasuries can be attributed to other factors besides their time

difference. In a normal economic environment one would receive a higher rate on the 30 year

compared to the 1-month and other T-bills. However, the U.S.’s yield curve has been flirting

with inversion for the past two years; in other words, there are times where one would receive a

lower rate on the 30 year T-Bond compared to the T-Bills. Also there were no 30-year T-Bonds

issued between October 2001 and August of 2005.

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, of

multicollinearity, makes it is 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 efficiently manage risk, price

derivatives, and optimize ones portfolio.

The standard deviation of the forecasted daily correlations may portray a constant flat

relationship across the different rolling windows, which can be attributed to no unusual

expectations in the markets. This chart displays the second major trend observed throughout my

analysis of the relationship between an asset pair’s correlation and the length of the rolling

window for their average daily correlations. When comparing the average of daily correlations

for the Russell 2000 Index and the CRB Energy Index to the length of the rolling window the

data clearly confirmed that a flat relationship existed. With a correlation of approximately zero

across each of the 6 rolling windows used, this pair of assets proves to be good assets to put into

37

a portfolio for two main reasons. First, with a correlation of almost zero, the movement of each

asset return is almost completely independent of one another. This independence enables

portfolio diversification by reducing the risk for a given level of return. When selecting assets for

ones portfolio, their time horizon is an important factor that has significant weight and must be

considered. While the relationship between other assets pairs varies from one rolling window to

the next, the relationship (correlation) remains constant for these two assets across all 6 rolling

windows. See chart 6 for similar asset pairs that exhibit this similar leveled trend across the six

rolling windows.

Chart 2 demonstrates an inverse (negative) relationship when comparing the average

daily correlation between the 1-year T-Bill and the 30-year T-Bond as the length of the rolling

window increases. Similar asset pairs that display this inverse relationship between correlation

and the length of the rolling window are seen when comparing precious metals to gold and

silver, along with short-term interest rate yields.

After analyzing chart 3 and the relationship of the average daily correlation between Gold

and Silver and other equity indexes as the length of the rolling window increases, an irregular

movement is displayed. The average daily correlation increases between the asset pair across

time, from the 2-month rolling window to the 6-month rolling window. The correlation between

the asset pairs after the 6 month rolling window begin to decline and display a trend of an

inverted relationship from the 1 year rolling window to the 3 year rolling window.

Similar to the previous movement in terms of irregularity, but different in terms of the

trend of the daily correlations is displayed in chart 4. This chart shows the relationship between

an asset pair’s correlation and the length of the rolling window for their average daily

38

correlations decreases from the 1 month rolling window to the 6 month rolling window. Then

from the 1 year rolling window to the 3 year rolling window the trend increases.

The average of daily correlations based on 6 different rolling windows for the CRB

Precious Metals Index and the NASDAQ Composite Index appear to exhibit a direct relationship

in regards to their correlation and the length of the rolling window used, (similar to the assets in

chart 1). However, the 3-year rolling window suddenly diverts off the trend and decreases by a

substantial amount. When comparing precious metals to midcap and smallcap equities and when

comparing long-term treasury yields to gold and silver a similar trend is observed.

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 relationship with precision is extremely significant. The above irregular trends

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.

Studying the standard deviation of rolling correlation with six different windows (see

chart 7), one can see that the window used to measure correlation has a big impact on what one

sees in regards to the time series properties of correlation. From this chart 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 1 month rolling window. From the 1-month rolling

window to the 3-year rolling window the standard deviation of rolling correlation for the 55 asset

39

pairs decrease respectively. The fact that moving averages based on shorter time spans fluctuate

more than moving averages with longer time spans holds true across all 55 asset pairs.

Looking at the “Descriptive Statistics” chart in the appendix one can witness how

different observation windows effect the correlation over time. Looking at the S&P 500 Index

compared to the 30 year T-Bond 1 month rolling window, the assets have an average daily

correlation of .4312. When one uses the 3 year rolling window to compare the two, the average

daily correlation significantly drops to .1275 (a 30% difference in value). Then one can look at

the PHLX Gold/Silver Index compared to the Russell 2000 Index 1 month rolling window, the

assets have an average daily correlation of .034. When one uses the 3 year rolling window to

compare the two, the average daily correlation significantly drops to .0088, practically zero.

These are just two examples of the substantial differences between different moving averages

(rolling windows). Therefore when one is trying to; price a derivative, manage risk, or diversify

their portfolio, their time horizon and the time 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 possible cause for a sudden increase in the spread of an asset pair’s confidence interval

may be attributed to a large, positive or negative, unexpected movement in the market. As a

result of a large standard deviation and standard error, confidence intervals can be relatively

wide. Comparing the Confidence Intervals for the one-month and 3 year rolling windows one

can see the variances of the value is proportional to the variance of the correlation forecast.

Therefore some type of adjustment should be made to account for the degree of uncertainty.

Looking at the irregularity of chart 3, one should adjust their significance downward for

uncertainties on correlation based on the concave shape across the 6-month and 1 year rolling

40

windows. Conversely the reciprocal action should be taken across the 6-month and 2 year rolling

windows in chart 4 based on the convexity of the trend. As you look at the confidence Intervals

from Graph 8B to graph 13B, moving from a 3 year rolling window to a 1 month rolling

window, it is clear the confidence interval increases as the rolling window used decreases.

There was a substantial amount of fluctuations across all 55 asset pairs when observing

the rolling window correlation estimates which decreased as the length of the rolling window

used increased. For this reason it is advisable to use a long averaging period on historical

volatility estimates. However, the longer the rolling window, the further ones risk (or time)

horizon is, the greater number of uncertainties one obtains. In general there are more similarities

between the 1 month (22 day) rolling windows compared to the 2 year and 3 year rolling

windows due to the fact that uncertainties increase with the risk horizon of ones timeline. 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.

V. 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 correlated with realized

volatility, which they call the “volatility effect in correlation.” They point out that return

41

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.

Based on my literature review correlations are important and should be estimated well

and forecasted with precision and caution. There have been recent developments in financial

econometrics dealing with time-varying volatility, stressing parsimonious models that are easily

estimated according to Andersen, Bollersley, Christoffersen and Diebold (2005). GARCH

volatility models offer a convenient framework for modeling important dynamic features of

returns, including volatility mean-reversion, long-memory, and asymmetries. Risk management

requires a specified multivariate density model and standard multivariate GARCH models are

too heavily parameterized to be useful when dealing with large scale problems. However, recent

advances in multivariate GARCH models have potential to be of use for medium-scale problems.

Nobel Prize winner Robert Engle (2002) says that two of the basic components factored

into to any pricing model are the forecasts of both correlation and volatility (Engle 2002).

“Simple methods such as rolling historical correlations and exponential smoothing are widely

used. More complex methods, such as varieties of multivariate generalized autoregressive

conditional heteroskedasticity (GARCH) or stochastic volatility, have been extensively

investigated in the econometric literature and are used by a few sophisticated practitioners.” “In

most cases, the number of parameters in large models is too big for easy optimization. In this

article, dynamic conditional correlation (DCC) estimators are proposed that have the flexibility

of univariate GARCH but not the complexity of conventional multivariate GARCH. These

models, which parameterize the conditional correlations directly, are naturally estimated in two

42

steps—a series of univariate GARCH estimates and the correlation estimate.” “In this article, the

accuracy of the correlations estimated by a variety of methods is compared in bivariate settings

where many methods are feasible.” “An alternative simple approach to estimating multivariate

models is the Orthogonal GARCH method or principle component GARCH method. This was

advocated by Alexander (1998, 2001). The procedure is simply to construct unconditionally

uncorrelated linear combinations of the series r. Then univariate GARCH models are estimated

for some or all of these, and the full covariance matrix is constructed by assuming the

conditional correlations are all zero.” “The goal for this proposal is to find specifications that

potentially can estimate large covariance matrices. In this article, only bivariate systems were

estimated to establish the accuracy of this model for simpler structures.” In this paper Engle

proposes the Dynamic Conditional Correlation (DCC) model. “These have the flexibility of

univariate GARCH models coupled with parsimonious parametric models for the correlations.

They are not linear but can often be estimated very simply with univariate or two-step methods

based on the likelihood function.” “The main finding is that the bivariate version of this model

provides a very good approximation to a variety of time-varying correlation processes. This is

true whether the criterion is mean absolute error, diagnostic tests, or tests based on value at risk

calculations.” Engle’s DCC first adjusts for differences between the variances than updates

correlations as new information is received.

Robert Engel’s “auto recessive conditional heteroskedasticity” model analyzes random

variables with different variances form the mean and how it applies to economic data over time.

Point forecast is the average of all possible results. The variance measures the difference

between the outcomes. The variance is calculated by averaging the squared deviations of each

result form this point forecast. The longer in time one goes out, the higher the variance found,

43

the less certainty of the future. There is a volatility clustering when large changes in prices are

followed by other large changes and small changes are usually followed by small changes.

Volatility is calculated by taking the square root of the rate at which the variance of returns

grows over time. Carr (2007) found a correlation using the ARCH Model between business

cycles and the uncertainty of inflation (not inflation itself).

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.

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

relationships 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

44

unknown level of risk that must be accounted for, calculated and attributed to the exponential

usage of leverage.

Looking at chart 6, the standard deviation of the forecasted daily correlations may portray

a constant flat relationship across the different rolling windows, which can be attributed to no

unusual expectations in the markets. With a correlation of approximately zero across each of the

6 rolling windows used, this 55 asset pairs proves to be good assets to put into a portfolio for two

main reasons. First, with a correlation of almost zero, the movement of each asset return is

almost completely independent of one another. This independence enables portfolio

diversification by reducing the risk for a given level of return. When selecting assets for ones

portfolio, their time horizon is an important factor that has significant weight and must be

considered. While the relationship between other assets pairs varies from one rolling window to

the next, the relationship (correlation) remains constant for these two assets across all 6 rolling

windows.

Based on my observations, the shorter the rolling window used, the more volatile and

unpredictable the correlations are. These fluctuations decreased as the length of the rolling

window used increased, across all 55 asset pairs. As a result of the fluctuations of the shorter

rolling windows, an asset pair’s correlation might go from positive to negative, negative to

positive, or just experience a significant change in their correlation. If this data based on the

short term rolling windows were to be used for further analysis, such as forming an optimal

diversified portfolio, the results will most likely significantly fluctuate and be misleading. As a

result caution should be taken when using short rolling windows as a means of forecasting.

45

References

Andersen, Torben G.,Time Bollerslev, Peter F. Christoffersen and Francis X. Diebold. “Practical

Volatility and Correlation Modeling for Financial Market Risk Management.” Risks of Financial Institutions, July 22, 2005.

Alexander, C. (2001). Market Models: A Guide to Financial Data Analysis. Wiley, Chichester. Ch.5

Forecasting Volatility and Correlation

Berger, Eric and David Klein, “Stochastic Interest Rates: A Crucial Correlation”, Bloomberg, July 1997.

Berger, Eric and Aaron Gross, “Value at Risk: Testing the Depths of Derivatives”, Bloomberg, March 1998.

Bernstein, Richard, “Updated: Uncorrelated Assets Are Now Correlated”, Merrill Lynch U.S. Strategy Update, March 5, 2007.

Bernstein, Richard, “All Emerging Markets Are Not The Same” The Merrill Lynch Ric Report, April 10,2007

Bernstein, Richard, “Individual Investors Have The Advantage, Will They Use It? Media: The Anti Housing Consumer Play”, Merrill Lynch Ric Report, 2007.

Bernstein, Richard, “A Simple Risk Reduction Tool: Time” Merrill Lynch U.S. Strategy Update, 2007.

Bernstein, Richard, “Hedge Fund Update: Alpha / Information Rates (IR) and Correlation”, Merrill Lynch Ric Report, 2007.

Broszkiewicz , Suwaj, Weron R. Makagon and A. Wylomansak. “On Detecting and Modeling Periodic Correlation in Financial Data” Physics A: Statistical and Theoretical Physics, Vol 336, Issues 1-2, May 1,2004:196-205

Bruce, Robert, “Moving – and Far From Average” Focus Technical analysis-Bloomberg, December 1998.

Carr, Peter, “The Value of Volatility” Bloomberg, January 2007:134-137 www.bloomberg.com/news/marketsmag/qcorner/qcorner_jan07.pdf>

Chaplin, Geoff, “Credit Derivatives: mine’s a big one”,Reech Capital, PLC, www.sungard.com/reech/menus/documents/whitepapers/big_one.pdf

Christopher F., “Rolling RegressionsWith Stata”, Boston College, North American Stata Users’ Group Meetings, August 11, 2004.

De Chiara, Adam and Daniel M. Raab, “Benefits of Real Asset Portfolio Diversification” Euromoney International Commodities Review, 2002:3-10

Drucker, Vanessa, “High Fliers – Real Estate Investment Trusts Have Soared During The Past Five Years, But Where Do REITs Go From Here?” Bloomberg Wealth Manager, May 2005:51-57

46

Elton, Edwin J., Martin J. Gruber and Jonathan Spitzer, “Improved Estimates of Correlation Coefficients And Their Impact on the Optimum Portfolios”, European Financial Magement, vol 12,issue3, June2006 pg 303.

Engle, Robert F., “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models” Journal of Business and Economics Statistics, vol 20,No3: 339-350.

Gluck, Jeremy, “Derivatives: After last fall’s market meltdown and Brazil’s currency problems this year, a lot of people are taking a fresh, hard look at all sorts of derivative-related issues.” Bloomberg, March 1999.

Kuan-Min, Wang and Nguyen Thi Thanh-Binh, “Testing for contagion Under Asymmetric Dynamics: Evidence From The Stock Markets between U.S. and Taiwan”, Physica A: Statistical and Theoretical Physics, vol 376, March 15, 2007: 422-432.

Lanza, Alessandro, Matteo Manera and Michael McAleer, “Modeling Dynamic Conditional Correlations in WTI Oil Forward and Futures Returns”,NOTA DI LAVORO, APRIL 2004.

Ledoit, Olivier and Michael Wolf, “Honey, I Shrunk The Sample Covariance Matrix” Journal of Portfolio Management, November 2003.

Li, Honggang and Yan Gao, “Statistical Distribution and Time Correlation of Stock Runs”, Physica A: Statistical Mechanics and its Applications, Vol 377, issue 1, April 1, 2007: 193-198.

Li, Ling Feng and K. Geert Rouwenhorst, “Long-Term Global Market Correlations” Journal of Business vol 78, No 1, 2005.

Lien, Kathy, “Commodity Prices and Currency Movements”, January 25,2006 www.investopedia.com/articles/forex/06/commoditycurrencies.asp

Lien, Kathy, “Using Currency Correlations To Your Advantage”, Forex Capital Markets, May 19,2005. www.investopedia.com/articles/forex/05/051905.asp

Markowitz, Harry, “Risk Management: Improving Your Odds in the Crapshoot”, Bloomberg Forum, July 1996: 75-77.

McReynolds, Rebecca, “Suit Yourself”, Bloomberg Wealth Manager, September 2002: 64-68.

Parvez, Ahmed, “Forecasting Correlation Among Equity Mutual Funds”, Journal of Banking and Finance, April 13, 2000: 1187- 1208.

Patel, Navroz, “Credit Risk Cracking Correlation”, Tavakoli Structured Finance, Inc, Vol 16, No 7, www.tavakolistructuredfinance.com/credit_risk.html

Picerno, James, “Quality Control”, Bloomberg Wealth Manager, July/Aug 2000: 37-40.

Simons, Howard L., “The Bond Between Yields and Metals: It May Not Be As Solid As Analysts Say” Stocks, Futures and Options Magazine, Vol 4, No 9, September 2005.

Stohl, Andrew, “Calculating Moving Averages”, Focus Technical Analysis-Bloomberg, Ausut 2002.

47

Simons, Howard L., “The Bond Between Yields and Metals: It May Not Be As Solid As Analysts Say”

Stocks, Futures and Options Magazine, Vol 4, No 9, September 2005.

Stohl, Andrew, “Calculating Moving Averages”, Focus Technical Analysis-Bloomberg, Ausut 2002.

48

APPE*DIX

Symbol Asset *umber of Days Start Date

1I CRB Precious Metals Index 8,007 1/17/1975

1M CRB Energy Index (1977) 5,820 9/1/1983

GI Goldman Sachs Commodity Index 9,395 12/31/1969

XA PHLX Gold/Silver Index 5,772 12/19/1983

Interest Rates

IY T-Bill Yield, 1-Year 11,295 4/30/1953

UY T-Bond Yield, 30-Year 8,123 1/31/1919

Equity Indicies

MD S&P midcap 400 Index 6,519 1/2/1981

QQ S&P Small cap 600 Index 2,874 6/5/1995

R2 S&P 500 Index 19,796 1/3/1928

SP *ASDAQ Composite Index 5,563 10/11/1984

X5 Russell 2000 Index 7,027 12/29/1978

Commodity Indicies

49

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 for 1-Yr T-Bill vs. S&P midcap Index

0.07

0.09

0.11

0.13

0.18

0.21

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

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

Length of rolling window of correlation

Corre

latio

n

IY

UY

1I

1M

GI

XA

MD

QQ

R2

SP

X5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bil l yield

30yr T-bond yield

CRB prescious Metals Index

CRB Energy Index

Goldman Sachs Commodity Index

PHLX Gold/Silver Index

S&P midcap 400 Index

S&P smallcap 600 Index

S&P 500 Index

Russell 2000 Index

Nasdaq Composite Index

Average of daily correlations based on different rolling windows for

1yr T-bill yield vs. 30yr T-bond yield

0.63

0.61

0.60

0.58

0.55

0.54

0.52

0.54

0.56

0.58

0.6

0.62

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

IY

UY

1I

1M

GI

XA

MD

QQ

R2

SP

X5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bil l yield

30yr T-bond yield

CRB prescious Metals Index

CRB Energy Index

Goldman Sachs Commodity Index

PHLX Gold/Si lver Index

S&P midcap 400 Index

S&P smallcap 600 Index

S&P 500 Index

Russell 2000 Index

Nasdaq Composite Index

Corre

latio

n

50

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 for S&P midcap 400 Index vs. PHLX Gold/Silver Index

0.11

0.13 0.13 0.13

0.11

0.08

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

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

Corre

latio

n

IY

UY

1I

1M

GI

XA

MD

QQ

R2

SP

X5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bill yield

30yr T-bond yield

CRB prescious Metals Index

CRB Energy Index

Goldman Sachs Commodity Index

PHLX Gold/Silver Index

S&P midcap 400 Index

S&P smallcap 600 Index

S&P 500 Index

Russell 2000 Index

Nasdaq Composite Index

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

-0.02

-0.03

-0.04

-0.03

-0.02

-0.01

-0.036

-0.031

-0.026

-0.021

-0.016

-0.011

-0.006

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

Length of rolling window of correlation

Corre

latio

n

IY

UY

1I

1M

GI

XA

MD

QQ

R2

SP

X5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bil l yield

30yr T-bond yield

CRB prescious Metals Index

CRB Energy Index

Goldman Sachs Commodity Index

PHLX Gold/Si lver Index

S&P midcap 400 Index

S&P smallcap 600 Index

S&P 500 Index

Russell 2000 Index

Nasdaq Composite Index

51

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 for

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

Length of rolling window of correlation

Corre

latio

n

IY

UY

1I

1M

GI

XA

MD

QQ

R2

SP

X5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bi ll yield

30yr T-bond yield

CRB prescious Metals Index

CRB Energy Index

Goldman Sachs Commodity Index

PHLX Gold/Silver Index

S&P midcap 400 Index

S&P smallcap 600 Index

S&P 500 Index

Russell 2000 Index

Nasdaq Composite Index

CRB Energy Index vs. Russell 2000 Index

0.01 0.00 0.00 0.00 0.00 0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

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

Corre

latio

n

Average of daily correlations based on different rolling windows for

IY

UY

1I

1M

GI

XA

MD

QQ

R2

SP

X5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bil l yield

30yr T-bond yield

CRB prescious Metals Index

CRB Energy Index

Goldman Sachs Commodity Index

PHLX Gold/Silver Index

S&P midcap 400 Index

S&P smallcap 600 Index

S&P 500 Index

Russell 2000 Index

Nasdaq Composite Index

52

Chart 7

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

3 month

6 month

1 year

2 year

3 year

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Asset Pair

STA*DARD DEVIATIO* OF ROLLI*G CORRELATIO* WITH 6 DIFFERE*T WI*DOWS

1 m

onth

3 m

onth

6 m

onth

1 year

2 year

3 year

53

1I1M1I1Y1IG

I1IM

D1IQQ1IR21ISP1IU

Y1IX51IXA1M1

Y1MG

I1MM

D 1MQQ

1MR2

1MSP 1MUY

1MX5

1MXA

1YGI 1YMD

1YQQ

1YR2

1YSP

1YUY

1YX5

1YXA

GIMD

GIQQ

GIR2

GISP

GIUY

GIX5

GIXAMDQ

Q MDR2 MDSP

MDUY MDX5 MDXA QQR2

QQSP QQUY

QQX5

QQXA R2SP

R2UY

R2X5 R2XA

SPUY SPX5

SPXA UYX5

UYXA X5XA

1 Month

3 Month

6 Month

1 Yea

r3 Yea

r5 Yea

r0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Ass

et P

air

s

Chart

7A

Cro

ss S

ect

ional S

tandard

Devi

ation A

cross

6 R

ollin

g W

indow

s fo

r all A

sset P

air

s

1 M

onth

3 M

onth

6 M

onth

1 Y

ear

3 Year

5 Y

ear

54

Chart 8A

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

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

Asset P

airs

Standard D

eviation of M

oving C

orrelations (based on 756-day w

indow

)12/28/1998 to 10/27/2006

Graph 8B

95%

C

onfidence Interval for A

verage of M

oving C

orrelations

(based on 756-day w

indow

) 12/28/1998 to 10/27/2006

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

1.1

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

Asset Pairs

Standard Deviation

average

LB-95

UB-95

55

Chart 9A

Standard D

eviation of M

oving C

orrelations (based on 504-day w

indow

)

12/26/1997 to 10/27/2006

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18 1

I1M

1I1Y

1IGI1IMD

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA1M1Y

1MGI 1MMD1MQQ

1MR21MSP1MUY

1MX51MXA

1YGI 1YMD1YQQ

1YR21YSP1YUY

1YX51YXAGIMDGIQQ

GIR2GISP

GIUY

GIX5GIXA MDQQMDR2

MDSPMDUYMDX5

MDXAQQR2

QQSPQQUYQQX5

QQXA

R2SPR2UY

R2X5R2XASPUY

SPX5SPXA

UYX5UYXA

X5XA

Asset Pairs

Standard Deviation

Graph 9B

95%

C

onfidence Interval for A

verage of M

oving C

orrelations (based on 504-day w

indow

)

12/26/1997 to 10/27/2006

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

1.1 1

I1M

1I1Y

1IGI1IMD

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA1M1Y

1MGI 1MMD1MQQ

1MR21MSP1MUY

1MX51MXA

1YGI 1YMD1YQQ

1YR21YSP1YUY

1YX51YXAGIMDGIQQ

GIR2GISP

GIUY

GIX5GIXA MDQQMDR2

MDSPMDUYMDX5

MDXAQQR2

QQSPQQUYQQX5

QQXA

R2SPR2UY

R2X5R2XASPUY

SPX5SPXAUYX5UYXA

X5XA

Asset Pairs

Correlation

average

LB-95

UB-95

56

Chart 10A

Standard D

eviation of M

oving C

orrelations (based on 252-day w

indow

)

12/27/1996 to 10/27/2006

0

0.05

0.1

0.15

0.2

0.25

0.3

1I1M

1I1Y

1IGI1IMD

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

GIXAMDQQ

MDR2

MDSP

MDUY

MDX5

MDXA

QQR2

QQSP

QQUY

QQX5

QQXA

R2SP

R2UY

R2X5

R2XA

SPUY

SPX5

SPXA

UYX5

UYXA

X5XA

Asset Pairs

Standard Deviation

Graph 10B

95%

C

onfidence Interval for A

verage of M

oving C

orrelations

(based on 252-day w

indow

) 12/27/1996 to 10/27/2006

-0.5

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

1.1

1.2 1

I1M

1I1Y

1IGI1IMD

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA1M1Y

1MGI 1MMD1MQQ

1MR21MSP

1MUY

1MX51MXA

1YGI1YMD

1YQQ

1YR2

1YSP

1YUY

1YX5

1YXA

GIMD

GIQQ

GIR2

GISP

GIUY

GIX5

GIXAMDQQ

MDR2MDSPMDUY

MDX5MDXAQQR2QQSPQQUY

QQX5QQXA

R2SP

R2UY

R2X5

R2XASPUY

SPX5SPXA

UYX5UYXA

X5XA

Asset Pairs

Standard Deviation

average

LB-95

UB-95

57

Chart 11A

Standard D

eviation of M

oving C

orrelations (based on 126-day w

indow

)

6/28/1996 to 10/27/2006

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 1

I1M

1I1Y

1IGI1IMD

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA1M1Y

1MGI 1MMD1MQQ

1MR21MSP1MUY

1MX51MXA

1YGI 1YMD1YQQ

1YR21YSP

1YUY

1YX51YXA

GIMD

GIQQ

GIR2

GISP

GIUY

GIX5

GIXA MDQQ

MDR2MDSPMDUY

MDX5MDXAQQR2QQSPQQUY

QQX5QQXA

R2SP

R2UY

R2X5R2XASPUY

SPX5SPXA

UYX5UYXA

X5XA

Asset P

airs

Standard Deviation

Graph 11B

95%

C

onfidence Interval for A

verage of M

oving C

orrelations

(based on 126-day w

indow

) 6/28/1996 to 10/27/2006

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

1.1

1I1M

1I1Y

1IGI1IMD

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA1M1Y

1MGI 1MMD1MQQ

1MR21MSP1MUY

1MX51MXA

1YGI 1YMD1YQQ

1YR21YSP1YUY

1YX51YXAGIMDGIQQ

GIR2GISP

GIUY

GIX5GIXA MDQQ

MDR2

MDSPMDUYMDX5

MDXAQQR2

QQSPQQUYQQX5

QQXA

R2SPR2UY

R2X5R2XASPUY

SPX5SPXA

UYX5UYXA

X5XA

Asset Pairs

Standard Deviation

average

LB-95

UB-95

58

Chart 12A

Standard D

eviation of M

oving C

orrelations (based on 66-day w

indow

)

4/3/1996 to 10/27/2006

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4 1

I1M

1I1Y

1IGI1IM

D

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA1M1Y

1MGI 1MMD

1MQQ

1MR21MSP1MUY1MX51MXA

1YGI 1YMD1YQQ

1YR21YSP1YUY

1YX51YXAGIM

DGIQQ

GIR2GISPGIUY

GIX5GIXA MDQQMDR2

MDSP

MDUYMDX5

MDXAQQR2

QQSP

QQUYQQX5

QQXAR2SPR2UY

R2X5R2XASPUY

SPX5SPXAUYX5UYXAX5XA

Asset Pairs

Standard Deviation

Graph 12 B

95%

C

onfidence Interval for A

verage of M

oving C

orrelations

(based on 66-day w

indow

) 4/3/1996 to 10/27/2006

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

1.1

1.2 1

I1M

1I1Y

1IGI1IMD

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA1M1Y

1MGI 1MMD1MQQ

1MR21MSP1MUY

1MX51MXA

1YGI 1YMD1YQQ

1YR21YSP1YUY

1YX51YXAGIM

DGIQQ

GIR2GISP

GIUY

GIX5GIXA MDQQMDR2

MDSP

MDUYMDX5

MDXAQQR2

QQSP

QQUYQQX5

QQXA

R2SPR2UY

R2X5R2XASPUY

SPX5SPXA

UYX5UYXA

X5XA

Asset P

airs

Standard Deviation

average

LB-95

UB-95

59

Chart 13A

Standard D

eviation of M

oving C

orrelations (based on 22-day w

indow

)

1/31/1996 to 10/27/2006

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0.25

0.275

0.3

0.325

0.35

0.375

0.4

0.425

0.45 1

I1M

1I1Y

1IGI1IMD

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA

1M1Y

1MGI1MMD

1MQQ

1MR2

1MSP

1MUY

1MX5

1MXA

1YGI1YMD

1YQQ

1YR2

1YSP

1YUY

1YX5

1YXA

GIMD

GIQQ

GIR2

GISP

GIUY

GIX5

GIXAMDQQ

MDR2MDSP

MDUY

MDX5MDXA

QQR2QQSP

QQUY

QQX5QQXA

R2SP

R2UY

R2X5

R2XA

SPUY

SPX5

SPXA

UYX5

UYXA

X5XA

Asset P

airs

Standard Deviation

Graph13B

95%

C

onfidence Interval for A

verage of M

oving C

orrelations

(based on 22-day w

indow

) 1/31/1996 to 10/27/2006

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

1.1

1.2

1I1M

1I1Y

1IGI1IMD

1IQQ

1IR2

1ISP

1IUY

1IX5

1IXA1M1Y

1MGI 1MMD1MQQ

1MR21MSP1MUY

1MX51MXA

1YGI1YMD1YQQ

1YR2

1YSP

1YUY

1YX5

1YXA

GIMD

GIQQ

GIR2

GISP

GIUY

GIX5

GIXA MDQQ

MDR2MDSPMDUY

MDX5MDXAQQR2QQSPQQUY

QQX5QQXA

R2SP

R2UY

R2X5R2XASPUY

SPX5SPXA

UYX5UYXA

X5XA

Asset Pairs

Standard Deviation

average

LB-95

UB-95

60

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 0.30

3 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 0.24

6 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 0.21

1 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 0.18

2 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 0.15

3 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

Average 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 0.11 -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 0.11

3 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 0.13

6 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 0.13

1 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 0.12

2 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 0.11

3 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 0.08

STDEV 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 0.02

Range 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 X5XA

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 -0.70 -0.51 -0.49

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 0.78 0.49 0.72

LB-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 -0.36

UB-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 0.61

LB-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 -0.29

UB-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 0.55

LB-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 -0.24

UB-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 0.48

LB-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 -0.20

UB-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 0.41

LB-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 -0.17

UB-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

61

Chart 14

Average R

ange o

f th

e a

verage d

aily c

orrela

tions

across 6

rollin

g w

indow

's for e

ach a

sset pair

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

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

GIXAMDQ

Q

MDR2

MDSP

MDUY

MDX5

MDXA

QQR2

QQSP

QQUY

QQX5

QQXA

R2SP

R2UY

R2X5

R2XA

SPUY

SPX5

SPXA

UYX5

UYXA

X5XA

Asset pair

s

Range

62

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Range

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

Asset pairs

Chart 7C

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

IY

UY

1I

1M

GI

XA

MD

QQ

R2

SP

X5

Interest Rates

Commodity Indices

Equity Indices

1yr T-bill yield

30yr T-bond yield

CRB prescious Metals Index

CRB Energy Index

Goldman Sachs Commodity Index

PHLX Gold/Silver Index

S&P midcap 400 Index

S&P smallcap 600 Index

S&P 500 Index

Russell 2000 Index

Nasdaq Composite Index

Pair

1MR2

R2X5

1MX5

R2SP

1YXA

MDR2

QQSP

GIR2

QQR2

GIX5

1MMD

GIXA

1IX5

1I1M

GIUY

MDQQ

GIMD

QQX5

1ISP

UYXA

MDSP

1MXA

1MQQ

MDX5

GIQQ

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.02

0.02

0.02

0.02

Range < .02

Average 0.04

Max 0.16

Min 0.01

RANGE

Range

0.05

0.14

0.10

0.12

0.16

0.09

0.13

0.13

0.05

0.11

0.06

0.12

0.05

0.15

0.05

0.12

0.05

1IXA

Pairs w/ range >.04

1YSP

1YR2

1YQQ

1YMD

MDXA

MDUY

1YX5

1YUY

R2XA

R2UY

QQXA

QQUY

X5XA

UYX5

SPXA

SPUY

63

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

tdat

e19

9606

2419

9612

1619

9706

1119

9712

0319

9806

0119

9811

2019

9905

1919

9911

1020

0005

0520

0010

2720

0104

2520

0110

1820

0204

1620

0210

0820

0304

0320

0309

2620

0403

2320

0409

1620

0503

1120

0509

0220

0603

0120

0608

23

Date (YYYY,MM,DD)

Corr

ela

tion

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 window

64

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

tdate

19960624

19961216

19970611

19971203

19980601

19981120

19990519

19991110

20000505

20001027

20010425

20011018

20020416

20021008

20030403

20030926

20040323

20040916

20050311

20050902

20060301

20060823

Time (YYYY,MM,DD)

Correlation

1yr T-bill for

1mo rolling window

1yr T-bill for

3mo rolling window

1yr T-bill for

6mo rolling window

1yr T-bill for

1yr rolling window

1yr T-bill for

2yr rolling window

1yr T-bill for

3yr rolling window 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

19960102

19960625

19961217

19970612

19971204

19980602

19981123

19990520

19991111

20000508

20001030

20010426

20011019

20020417

20021009

20030404

20030929

20040324

20040917

20050314

20050906

20060302

20060824

Date (YYYY,MM,DD)

Correlation

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

30yr T-bond for

3yr rolling window

65

Graph 4

Daily Correlations of the 1-yr T-Bill Across 6 Different Rolling Windows and its Reactions Across Time

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

TIME (1996-2006)

Co

rrela

tio

n

1yr T-bill for

1mo rolling window

1yr T-bil l for

3mo roll ing window

1yr T-bill for

6mo rolling window

1yr T-bi ll for

1yr roll ing window

1yr T-bi ll for

2yr roll ing window

1yr T-bi ll for

3yr roll ing window

66

Chart

5

S&

P 5

00 v

s Inte

rest R

ate

s

-0.85

-0.65

-0.45

-0.25

-0.05

0.15

0.35

0.55

0.75 td

ate 19

9604

29 1996

0826 19

9612

23 1997

0423 19

9708

20 1997

1217 19

9804

20 1998

0817 19

9812

14 1999

0415 19

9908

12 1999

1209 20

0004

07 2000

0807 20

0012

04 2001

0404 20

0108

02 2001

1130 20

0204

03 2002

0731 20

0211

26 2003

0328 20

0307

28 2003

1121 20

0403

24 2004

0723 20

0411

18 2005

0321 20

0507

19 2005

1114 20

0603

16 2006

0714

Tim

e

Correlation

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 w

indow

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82