On the correlation between commodity and equity returns: Implications for portfolio allocation

Author’s Accepted Manuscript On the correlation between commodity and equity returns: implications for portfolio allocation Marco J. Lombardi, Francesco Ravazzolo

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To appear in: Journal of Commodity Markets Received date: 28 January 2016 Revised date: 20 June 2016 Accepted date: 21 July 2016 Cite this article as: Marco J. Lombardi and Francesco Ravazzolo, On the correlation between commodity and equity returns: implications for portfolio a l l o c a t i o n , Journal of Commodity Markets, http://dx.doi.org/10.1016/j.jcomm.2016.07.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the correlation between commodity and equity returns: implications for portfolio allocation ∗ Marco J. Lombardi†

Francesco Ravazzolo‡

June 2016

Abstract In recent years several commentators hinted at an increase of the correlation between equity and commodity returns, blaming for that surging investment in commodity-related products. This paper investigates such claims by looking at various measures of correlation, and assesses what are the implications of this for asset allocation. We develop a time-varying Bayesian Dynamic Conditional Correlation model and find that joint modelling commodity and equity prices produces accurate forecasts, which lead to benefits in portfolio allocation. This, however, comes at the price of higher volatility. Therefore, the popular view that commodities are to be included in investment portfolio as hedging device is not grounded.

Keywords: Commodity prices, equity prices, correlation, Bayesian DCC, multi asset class investment. JEL Classification : C11, C15, C53, E17, G11.

∗

We thank seminar participants at the BIS, the Bank of Canada, the Norges Bank Investment Management, as well as at the ECB-NB “Modeling and forecasting oil prices”, MMF “Understanding Oil and Commodity Prices”, NMBU “Commodity Market Workshop” and CAMP “Forecasting and Analysing Oil Prices” workshops for helpful comments. We also gratefully acknowledge comments by Lutz Kilian, Dubravko Mihaljek and Sjur Westgaard. This paper is part of the research activities at the Centre for Applied Macro and Petroleum economics (CAMP) at BI Norwegian Business School. CAMP is supported by Statoil’s research program in petroleum economics. The manuscript was circulated initially with the title “Oil Price Density Forecasts: Exploring the Linkages with Stock Markets”. The views expressed in this paper are our own and do not necessarily reflect those of Bank for International Settlements. † Bank for International Settlements. Email: [email protected] ‡ Free University of Bozen-Bolzano and BI Norwegian Business School. Email: [email protected]

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1

Introduction

The past decade has witnessed wide swings in commodity prices, with oil under the spotlight. The upward trend in prices of the 2000s has been ascribed to booming demand at the global level, but fluctuations around it have been substantial. After the onset of the Great Recession, commodity prices fell sharply, but then swiftly rebounded as the recovery took hold. Commodity prices subsequently hovered around pre-recession levels, but since 2013 they entered again a phase of high volatility, triggered by the slowdown of emerging economies and the boom of shale oil production in the United States. Against this background, investing in commodities has generated hefty returns and has become increasingly popular, in spite of the high risks associated with this type of investment, which are due to the inherent volatility of commodity prices. Indeed, several fund managers started advising their customers to devote a share of their portfolios to commodity-related products as part of long-term diversification strategy. This is often motivated by the fact that, over the long run, commodities are believed to display low correlation with other asset classes, most notably with equities (Gorton and Rouwenhorst (2006), Bhardwaj et al. (2015)). Yet, over the last years, long-only commodity investment strategies have proven disappointing (Erb and Harvey (2016)). At the same time, substantial inflows into commodity-related investment products (Croft and Norrish (2013)) have led many commentators to speculate on whether commodities are increasingly behaving as an asset class. This was especially the case for oil, due to the high and growing liquidity of derivatives markets amid surging prices, as well as to the specific role of energy in firms’ and households’ consumption baskets. Researchers have therefore devoted efforts to examine whether financial inflows may affect the price formation mechanism, both in futures and physical markets. The empirical evidence on a lasting impact of financial investment on oil prices is, at best, scant (see Fattouh et al. (2013) for a recent survey). In the segment of commodities, however, financial investors 2

may have less commodity-specific knowledge and a different attitude compared to commercial players, and hence enter or exit trades based on their overall perceptions of the macroeconomic situation rather than market-specific factors. If this is the case, it would suggest that commodity prices are increasingly influenced by shocks coming from the demand side. This is a well-established fact in the literature on oil, following the seminal work by Kilian (2009). Moreover, for non-oil commodities, rising global demand may also play an indirect role, through the spillover of rising energy costs (see Lombardi et al. (2012) for a discussion of the correlation patterns of commodity prices). The fact that equity prices are also likely to have been largely driven by shocks related to the global economic activity, especially in the aftermath of the financial crisis, could then explain the increase in correlations. For example, Tang and Xiong (2012) report evidence of increased correlations between the returns of commodities that are included in indexes tracked by investment funds. In this paper, we try to shed some light on these issues. If one looks at the most recent years, commodity and equity prices appeared to be increasingly correlated, both being apparently more sensitive to news concerning the global macroeconomic environment rather than to idiosyncratic and market-specific shocks. We first provide a complete characterization of this phenomenon, by computing correlations using different methodologies and trying to identify relevant turning points. Our results suggest that correlations, after having hovered around zero for more than a decade, have indeed increased markedly since mid-2008. The result is robust to the use of both different commodity indices distinguished by a dominant or a non-dominant role of oil, and oil prices alone. Therefore, not only oil prices have been a driving force in the process, but the change in the correlation pattern since 2008 has been shared by many commodities. We then use constant parameter univariate and bivariate models for commodity and equity prices, and derive a time-varying Bayesian Dynamic Conditional Correlation (DCC) model (Della Corte et al.

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(2012)), which can account for the changes in the relationship between commodity and equity prices observed after 2008. This being established, we move to our main research question, i.e. whether joint modeling commodity and equity returns has an economic value in asset-allocation framework. To this end, we first conduct a density forecasting exercise and find that the DCC provides statistically superior density forecasts for commodities compared to a plain random walk model. However, we do not find statistically significant gains in forecasting equity prices. Then, we use a Bayesian framework which allows us to explicitly account for an active role of the investor by choosing portfolio weights that are optimal with respect to his or her subjective belief about the true return distribution and the associated fact that higher order moments and the full predictive densities of commodity and equity prices are uncertain and vary over time. Our results indicate that the use of a time-varying joint model for commodity and equity prices provides economic gains relative to passive strategies of buying and holding a fixed share of the portfolio in the given assets, especially at times of large price swings. At the same time, an investment strategy which also includes commodities in a portfolio produces substantially higher volatility compared to portfolios which do not include them. For short investment horizons this is evident for the whole sample and not only from September 2008 onwards, when correlations between equities and commodities increase. This is at odds with the common notion that commodities can serve as a hedge. The remainder of the paper is organized as follows. Section 2 provides a review of the existing literature on commodity and equity returns. Section 3 describes the data and documents changes in correlation over our sample. Section 4 presents bivariate models for commodity and equity prices and investigates their forecast accuracy. We then move to apply these findings to active asset allocation strategies in Section 5. Section 6 presents two further exercises for robustness analysis. Finally, Section 7 concludes.

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2

Literature review

The co-movements between commodity and equity prices have already been examined in the literature: Gorton and Rouwenhorst (2006) report that commodity returns display negative correlation with equity returns over a long sample, running from 1959 to 2004. B¨ uy¨ u¸sahin et al. (2010) explore the correlations in the pre-crisis period, failing to find evidence of significant changes. Using post-2008 data, B¨ uy¨ u¸sahin and Robe (2012) find that correlations between equities and commodities increased amid greater participation by speculators. Other papers focused more specifically on oil: Kilian and Park (2009) report that the response of equity prices to oil price shocks depends on the nature of the shocks; Cassassus and Higuera (2011) show that oil price changes are good predictors of equity returns; Chang et al. (2011) report evidence of volatility spillovers between oil and equity prices. The implications of such co-movements for asset allocation have been examined in some recent papers in the finance literature, see for example Bessler and Wolff (2015) and references therein. Some authors have also considered the different benefits provided by various commodity categories, see for example Belousova and Dorfleitner (2012). Our work builds on computing predictive densities based on the data and the prior, and therefore accounts for estimation risk in the portfolio allocation, see Zellner and Chetty (1965) and Brown (1978). Kandel and Stambaugh (1996) and Barberis (2000), show that accounting for parameter uncertainty in the investor’s portfolio decision problem can lead to quite different results; see also Avramov (2002), Cremers (2002), Kan and Zhou (2007) and Jacquier and Polson (2012). Ravazzolo et al. (2008), Guidolin and Na (2008) and Pettenuzzo and Timmermann (2011) extend their analysis to account for instability uncertainty. Della Corte et al. (2011) and Della Corte et al. (2012) assess the economic value of volatility and correlation timing, and, more generally, Jondeau and Rockinger (2012) of time-varying higher moments. We extend this literature by including in the 5

asset allocation problem a new asset class, i.e. commodities. The novelty of our approach lies in the more sophisticated econometric and forecasting framework, that allows us to incorporate predictions on time-varying correlations into the asset allocation problem. The closest paper to ours is Daskalaki and Skiadopoulos (2011) who also study the benefits of including commodities in a portfolio of traditional asset classes. They show that commodities are beneficial to non mean-variance investors in insample setting, but these benefits are not preserved out-of-sample. The major difference between our framework and Daskalaki and Skiadopoulos (2011) is how the problem of the portfolio choice is treated econometrically. Daskalaki and Skiadopoulos (2011), as most of the academics and practitioners do, estimate the parameters of the data generating process ignoring subjective beliefs and then plug these parameter values into a numerical solution algorithm for the investor’s optimization problem. Our work instead confirms the usefulness of Bayesian methods in the context of portfolio decision theory.

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Data and stylized facts

In this section, we will present some stylized facts about commodity and equity returns, most notably about their correlations. To start with, we collected weekly returns generated by the Morgan Stanley Capital International global equity index (MSCI) and the Standard & Poor’s Goldman Sachs commodity index (SPGSCI), starting from the first week of January 1980 until the last week of August 2015.1 The MSCI is an index of global equity, and we use it as a benchmark for an internationally-diversified equity portfolio.2 The SPGSCI is a tradable index that is readily available to market participants of the Chicago Mercantile Exchange and for this it often serves as a benchmark for investment in 1

Data was obtained from Bloomberg. In the robustness checks presented in section 6, however, we also consider the S&P500: although it is a US-centric equity index, it could help isolting the effects of exchange rate fluctuations, given that the MSCI index is quoted in US Dollars. 2

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the commodity markets and as a measure of commodity performance over time. The index currently comprises commodities from all commodity sectors - energy products, industrial metals, agricultural products, livestock products and precious metals, but it contains a much higher exposure to energy than other commodity price indices. In particular, WTI and Brent crude oil account for more than 50% of the composition weights, and the energy sector accrues for almost 80%.3 Many commodity-related investment products are actually based on the SPGSCI index, or even on oil prices themselves. Therefore, we see the fact that the index is strongly influenced by oil price developments as a plus.4 The weekly returns generated by these two indexes are approximately zero-mean, although the MSCI index has smaller variance but higher kurtosis. While it is not possible to discern any pattern from weekly returns, applying a 52-weeks moving average reveals comovements in the most recent years (Figure 1). Yet, it is also evident that returns only start moving in parallel since the 2000s. Looking at time-variation in the standard deviations also highlights that commodity prices are substantially more volatile than equities (Figure 2), especially since the 2000s. The idea that commodities should be a good hedge against equity price fluctuations can be deducted from expressing equity prices as the discounted value of future dividends. If prices of inputs in the production process (energy, metals, raw materials) increase, firms will see their profits shrink, other things equal, and will therefore have less dividend to distribute.5 Therefore, one would expect to see a negative correlation between commodity and equity retruns. In a sense, however, this suggests that commodity prices are driven exogenously. It is now widely acknowledged that this is not the case. Commodity price increases often come 3

At the beginning of the sample, the weight on energy products, and above oil, was even larger. We refer however to section 6 for a robustness analysis with a less energy-intensive commodity index, as well as with Brent and WTI crude oil prices alone. 5 While this is likely to be the case for each individual firm, at the aggregate level things may be less clear cut, since increases in input prices can be passed through customers. 4

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on the back of buoyant demand due to booming economic activity (Kilian (2009)). If this is so, the sign of the correlation becomes less clear-cut: both equity and commodity prices could increase on positive news about the global macroeconomic outlook materialize. Over the past years, market commentators have indeed often pointed at commodity and equity prices moving in the same direction as a consequence of more optimistic or more pessimistic expectations about the global economy. Kilian and Vega (2011) investigate formally the impact of the flow of macroeconomic news on the oil price over a sample running from 1983 to 2008, but do not find a systematic relationship. Figure 3 reports sample correlations computed on moving windows of different length; this is indeed what most commentators refer to when discussing co-movements of commodity and equity prices. It emerges clearly that correlations have been sometimes positive and sometimes negative, but on average have hovered around zero. In the recent episode, however, positive correlations appear to have been stronger and more persistent compared to the past: it started with the bursting of the financial crisis in 2008, and although correlations have declined since then, they are still positive and seem to have stabilized around a level which is somehow higher compared to the pre-recession.6 Using different windows alters somewhat the size of correlation, but the pattern remains. Instead of using rolling windows of different length, which spread the influence of extreme episodes or periods of market turbulence over time, one can use model-based approaches to estimating correlations. A popular approach is the Dynamic Conditional Correlation (DCC) model proposed by Engle (2002): 6

To be sure, this is just eyeball econometrics, and we are aware that different modelling choices – e.g. for the sample size, or for the way correlations are computed – may yield different conclusions. For example, Bhardwaj et al. (2015) acknowledge the spike in correlations during the phase of turbulence that followed the onset of the Great Recession, but maintain that correlations returned to historical levels thereafter.

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yt

= Φ(L)yt−1 + vt

vt

= Ht0.5 εt , εt ∼ N (0, IN ), Ht = Dt Rt Dt 0

2 , i = 1, ..., N Dt2 = diag{ωi } + diag{κi }vt−1 vt−1 + diag{λi }Dt−1 0

(1)

0

Qt = S(ıı − A − B) + Aεt−1 εt−1 + BQt−1 Rt = diag{Qt }−1 Qt diag{Qt }−1 where yt is a (N × 1) vector of dependent variables, S is the unconditional correlation 0

matrix of εt , A, B and S(ıı −A−B) are positive semidefinite matrices, L is the lag autoregressive order; see Appendix for model details. In a nutshell, the DCC is a multivariate GARCH model in which correlations are time-varying according to an autoregressive specification. As such, the DCC accounts for both the time-varying features of volatilities and correlations. The shocks εt are assumed normally distributed, but the residuals vt are not restricted to be so due to the time-varying matrix Ht and therefore allow to cope with typical features of daily returns, such as volatility clustering and kurtosis. The dynamic conditional correlations are reported in Figure 4. Although correlations appear to be smaller compared to those estimated using rolling windows, they still look persistently positive towards the final years of the sample. While this is in line with the findings of B¨ uy¨ u¸sahin and Robe (2012), it is nevertheless remarkable, as oil prices were very volatile throughout 2015. It should be no surprise that the correlation shoots up at times of heightened volatility, especially if one takes into account that, in the downturn and the subsequent economic recovery, both commodity and equity prices are likely to have been driven by the same common shocks, i.e. news on the shape of the global macroeconomic outlook. Nor it is surprising that correlations declined after such shocks faded away and the global economy stabilized. But it is nevertheless remarkable that seven years after the onset of the Great Recession, an instantaneous measure of correlation such as DCC still points to a

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substantially higher number compared to the pre-recession times. Since visual inspection reveals a persistently higher correlation towards the end of the sample, we formally tested for a breakpoint in the correlation pattern using the methodology proposed by Andrews (1993) and Andrews and Ploberger (1994). The results of the test point to a significant breakpoint on 5 September 2008, just before the bankruptcy of Lehmann Brothers7 Interestingly, the same breakdate is found on window-based and model-based correlations.

4

Joint forecasts of commodity and equity prices

Having established an increase in the correlation between commodity and equity returns, we move to check whether this can be exploited to achieve forecasting gains. The debate on the predictability of equity prices is still an open issue in empirical research, see for example Welch and Goyal (2008). Market efficiency theories imply not predictability, whether market friction theories imply predictability. Evidence of predictability of oil prices has, on contrary, been subject to a break in recent years: from the mid-90’s, research evidence suggests not predictability where future prices contain all the relevant information and alternative models cannot improve forecast accuracy. However, studies such as Baumeister and Kilian (2012) find that several (Bayesian) reduced-form Vector Autoregressive models outperform forecasts based on future prices in a real-time setting. Baumeister and Kilian’s models apply macroeconomic data to forecast oil prices, but they do not explore the linkage with equity prices. Furthermore, their analysis refers mainly to point forecasting. Kandel and Stambaugh (1996) and Barberis (2000), among others, discuss the role of parameter uncertainty and Bayesian analysis as tool to cope with for 7

The test was conducted using the average, sup and LM statistics, with the p-values tabulated by Hansen (1997). All test statistics are significant at the 1% level. As recommended, we cut off the first and last 10% of the observations.

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return predictability and for asset allocation. We produce weekly point and density forecasts for commodity and equity returns over the sample from 2005W1 to 2015W35 for a total of 557 weeks. We compute h = 1, 2, ..., 24 steps ahead forecast for each vintage using a bivariate L = 12 lags Bayesian Vector Autoregressive model with Minnesota type prior (VAR), see Clark and Ravazzolo (2015) for details,8 and a bivariate L = 12 lags Bayesian DCC model.9 The main advantage of density forecasting over a plain point forecast is that it can take into account higher moments, which are key ingredients for the asset allocation exercise and can lead to substantial economic gains. We compare these forecasts to a random walk (RW), i.e. the point forecast is taken to be today’s price and the predicted volatility is the sample standard deviation, as well as a Bayesian autoregressive model (AR). Bayesian inference on the listed models allows to derive complete predictive densities, whose statistical accuracy is evaluated in the forecasting exercise. More specifically, we consider several evaluation statistics for point and density forecasts previously proposed in the literature, see Billio et al. (2013) for a recent application. We evaluate commodity and equity forecasts separately in this section, and use marginal densities from bivariate models. We compare point forecasts in terms of Mean Square Prediction Errors (MSPE) t 1X 2 e , M SP Ek = ∗ t t=t k,t+1

where t∗ = t − t + 1 and e2k,t+1 is the square prediction error of model k. We evaluate the predictive densities using the Kullback-Leibler Information Criterion (KLIC) based 8

Our priors are based on statistical consideration; see Tu and Zhou (2010) for a discussion on economic based priors. 9 Further details on the model and the estimation algorithm are reported in the appendix. Della Corte et al. (2012) present an application to assess the economic value of time-varying correlation timing. We also investigate different autoregressive lag orders, L = 1 and L = 4, and performance are marginal inferior.

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measure; see for example Kitamura (2002), Mitchell and Hall (2005), Hall and Mitchell (2007), Amisano and Giacomini (2007), Kascha and Ravazzolo (2010). The KLIC distance between the true density p(yt+1 |y1:t ) of a random variable yt+1 and some candidate density p(˜ yk,t+1 |y1:t ) obtained from model k is defined as

Z KLICk,t+1 =

p(yt+1 |y1:t ) ln

p(yt+1 |y1:t ) dyt+1 , p(˜ yk,t+1 |y1:t )

= Et [ln p(yt+1 |y1:t ) − ln p(˜ yk,t+1 |y1:t ))].

(2)

where Et (·) = E(·|Ft ) is the conditional expectation given information set Ft at time t. An estimate can be obtained from the average of the sample information, yt+1 , . . . , yt+1 , on p(yt+1 |y1:t ) and p(˜ yk,t+1 |y1:t ):

KLIC k+1

t 1X = ∗ [ln p(yt+1 |y1:t ) − ln p(˜ yk,t+1 |y1:t )]. t t=t

(3)

The KLIC chooses the model which on average gives higher probability to events that have actually occurred. In reality we do not know the true density, but for the comparison of two competing models, it is sufficient to consider the Logarithmic Score (LS), which corresponds to the latter term in the above sum,

LSk = −

t 1X ln p(˜ yk,t+1 |y1:t ), t∗ t=t

(4)

for all k and to choose the model for which the expression in (4) is minimal, or as we report in our tables, the opposite of the expression in (4) is maximal. Since the distribution properties of a statistical test to compare density accuracy 12

performances, measured in terms of LS, are not derived when working with nested models and expanding estimation windows, as is our case, similarly to Clark and McCracken (2012) for point forecasting and to Groen et al. (2013) for density forecasting, we apply an encompassing t-test to compare the null of equal finite sample forecast accuracy, against the alternative that a model outperforms the RW benchmark.10 We also apply the Clark and West (2007) test in point forecast comparison and results are qualitative similar. Table 1 reports point and density forecast results. Absolute predictability for commodity returns is lower than absolute predictability for equity returns, meaning that MSPEs for commodity returns are higher than for equity returns and LS for commodity returns lower than for equity returns for all horizons. Data characteristics, such as volatilities, discussed in section 3 can explain the result. However, the evidence is somewhat different for relative predictability. The models produce very similar point forecasts; but the DCC model gives the highest LS for all the horizons and statistically significant up to 12-weeks ahead in forecasting commodities, but not in forecasting equity returns, see Brandt (2007) for a discussion on issues related to stock market (illusive) predictability.11 Therefore, equities in our DCC framework seem to contain relevant information to forecast commodities, whether the opposite is not supported by our analysis. Ferraro et al. (2015) find opposite evidence in a linear regression framework when investigating exchange rate and oil price predictability: oil prices forecast exchange rates, but exchange rates do not forecast oil prices. To sum up, the DCC model gives more accurate density forecasts relative to the RW benchmark for all the horizons up to 24 weeks with improvements in density forecasting statistically significant up to 12-weeks ahead. Therefore, a time-varying covariance matrix which can model instability in volatility and correlations between the two variables, as 10

Given that we maximise the LS, we use right-tail p-values for the LS test. The bivariate VAR could be extended with other financial and macroeconomic variables, for example as in Kilian and Park (2009) and Baumeister and Kilian (2012). We leave it for further research. 11

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discussed in Section 3 is an important ingredient to predict higher moments of the joint commodity and equity predictive density. It is useful for testing structural instability in an in-sample (ex-post) analysis and, above all for the topic of this paper, it provides gains in an out-of-sample (real-time) exercise. Figure 5 indicates that large part of the gains are in September 2008; but they also persist in the aftermath of the Lehman bankruptcy, as well as amid the oil price collapse of 2015.

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Asset allocation exercise

How should an investor read the results reported in the last two sections? On one hand, the finding that correlation has become positive in the aftermath of the crisis may suggest that commodities can no longer work as a hedge in one’s portfolio. On the other hand, the fact that commodity returns are predicted by equity returns may bring benefits if one follows a dynamic asset allocation strategy. To answer these questions, we need to investigate the economic value of jointly modelling commodities and equities in the setting of an investment strategy which allows both assets to be included in one’s portfolio. To this end, we develop an active short-term investment exercise. The investor’s portfolio consists of the equity index, the commodity index and risk free bonds only, see Kan and Zhou (2007) for an discussion on three-fund portfolios where one of the asset is the risk free.12 At the end of each week t, the investor decides upon the fraction αs,t+h of her portfolio to be held in stocks, αc,t+h in the commodity index and the remaining part in the risk free asset for the period t + h, based on the forecast of the commodity and stock index returns. We constrain αs,t+h , αc,t+h not allowing for short-sales or leveraging (see Barberis (2000)), to have at least 50% of the invested portfolio in risky assets at each period t + h, 12

The risk free asset is approximated by using the weekly federal funds rate; data was obtained from the Fred database at the Federal Reserve Bank of St Louis.

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and αc,t+h to be maximum 40% of the invested portfolio. We assume that the investor maximizes a power utility function: 1−γ Wt+h , u(Wt+h ) = 1−γ

γ > 1,

(5)

where γ is the coefficient of relative risk aversion and Wt+h is the wealth at time t + h, which is equal to

Wt+h = Wt ((1 − αs,t+h − αc,t+h ) exp(yf,t+h ) + αt+h exp(yf,t+h + y˜t+h )),

(6)

where Wt denotes initial wealth, αt+h = (αs,t+h , αc,t+h ) is (1, 2) vector of weights, yf,t+h the h-step ahead risk free rate and y˜t+h the (2, 1) h-step ahead bivariate forecast of the commodity and stock index returns in excess of the risk free made at time t. Our investor does not rebalance the portfolios in the period from t to t + h, but keeps positions on the three assets constant.13 When the initial wealth is set equal to one, i.e. W0 = 1, the investor solves the following problem:

 max P

αt+h ∈[0,1]2

αt+h ≤1

Et

((1 − αs,t+h − αc,t+h ) exp(yf,t+h ) + αt+h exp(yf,t+h + y˜t+h ))1−γ 1−γ

 .

The expectation Et (·) depends on the predictive density for the commodity and stock log excess returns, p(˜ yt+h |y1:t ) and the problem can be rewritten as: 13

In the case of dynamic asset allocation the long-run investor is allowed to rebalance her portfolio during the investment period, adjusting the portfolio weights to reflect new information that arrives. Solving the resulting dynamic programming problem is complicated due to the large number of state variables that enter the problem in a highly nonlinear way, see Barberis (2000) and Guidolin and Timmermann (2007).

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Z max P

αt+h ∈[0,1]2

αt+h ≤1

u(Wt+h )p(˜ yt+h |y1:t )d˜ yt+h .

(7)

We approximate the integral in (7) by generating G independent draws from the g , and then use a numerical optimization method to predictive density p(˜ yt+h |y1:t ), y˜t+h

find: G

max

αt+h ∈[0,1]2

1 X G g=1



g ))1−γ ((1 − αt+h ) exp(yf,t+h ) + αt+h exp(yf,t+h + y˜t+h 1−γ

 .

(8)

We consider an investor who can choose between different forecast densities of the (excess) commodity and equity returns yt+h to solve the optimal allocation problem described above. We include three cases in the empirical analysis below and assume the investor uses alternatively the density from the RW and AR univariate models for each series, the bivariate BVAR and the bivariate BVAR-DCC.14 Moreover, since the portfolio weights in the active investment strategies change every period, we include transaction costs of c = 0.5%, i.e. 50 basis points, see Daskalaki and Skiadopoulos (2011) and DeMiguel et al. (2009) for similar choices. We evaluate the different investment strategies for an investor with a risk aversion parameter γ = 5, as in Barberis (2000), by computing the ex post annualized mean portfolio return, the annualized standard deviation and the Sharpe ratio.15 We compare the wealth provided at time t + h by two resulting portfolios by determining the value of multiplication factor of wealth ∆ which equates their average utilities. For example, suppose we compare two strategies A and B. 14

In our Bayesian framework, the strategy based on the RW is also considered an active strategy. The RW implies that the investor subjective view is that current prices incorporate all future predictable information and those should be used to actively allocated his or her wealth. 15 Results are qualitatively similar for γ = 2, . . . , 6.

16

t X

u(WA,t+h ) =

t=t

t X

u(WB,t+h / exp(∆)),

(9)

t=t

where u(WA,t+h ) and u(WB,t+h ) are the wealth provided at time T + h by the two resulting portfolios A and B, respectively. Following West et al. (1993), we interpret ∆ as the maximum performance fee the investor would be willing to pay to switch from strategy A to strategy B.16 We infer the value added of strategies based on individual models and the combination scheme by computing ∆ with respect to three static benchmark strategies: holding only stock (∆s ), holding only commodities (∆c ), and holding 70% in equities and 30% in commodities (∆70−30 ).17 Finally we compute the certainty equivalent return (CER) for each strategy, in formula:

CERt+h = u−1 (Et (u(Wt+h ))),

(10)

where u−1 is the inverse of the power utility function defined in (5). Strategy with maximum CER is preferred. We do not report the final value as for other alternative measures, but plot how cumulative difference between any strategy A and investing 100% of the portfolio in the stock market:

CERDk,t+1 =

t X

(CERA,t+h − CERs,t+h ),

(11)

s=t

where k = RW, AR, ..., 70 − 30. If CERDk,t+h increases at observation t + h, this indicates that the strategy k gives higher CER than the benchmark strategy. Results in Table 2 strengthen the evidence reported in the section on point and density forecasting. The DCC models gives higher SR than the RW for horizons from two16

See, for example, Fleming et al. (2001) for an application with stock returns. Holding constantly 30% of the portfolio in the commodities provides the highest gains among passive strategies with positive weights on the commodities, in particular for short-term horizons. Moreover, a risk-parity portfolio applied to our full sample data will also suggest a weight to commodities around 30%. 17

17

weekes to 8-weeks and have positive ‘entrance fees’ relative to passive strategies for all the horizons. Gains are robust to reasonable transaction costs and are substantial compared to alternative models for horizons longer than two and up to eight weeks.. So, a joint modelling of commodity and equity returns with time-varying volatility can produce statistically and economically significant gains, above all to passive strategies of buying and holding. To shed light how such gains are made, Figure 6 plots the CER differential relative to a passive strategy of investing 100% of the portfolio in stock prices over the sample period for a two weeks horizons. For all the six investment horizons we consider, the gains arise mainly in the second part of 2008, i.e. during the most turbulent time of the recent financial crisis, when equity and commodity prices are likely to have been driven by the same common shocks. However, the whole period from the beginning of the Great Financial Crisis in August 2007 to the end of the sample shows that the active strategies lead to economic gains. Those yielded by the DCC are substantially higher than AR and VAR models. We now turn to the question of what are the relative advantages of a portfolio including oil and commodities in addition to equities and a risk-free asset. To answer this, we compare the results of Table 2 with those coming from a similar portfolio allocation exercise, in which the weight associated to commodities is set to zero. Results, reported in Table 3, suggest that the inclusion of commodities in one’s portfolio has boosted returns when using a DCC model, in particular for horizons of 2 to 8 weeks. However, this comes at the price of a substantially higher volatility, irrespective of the model used. Therefore, the common lore that commodities should serve as a hedge does not seem to be solidly grounded. We compare the realized 1-year moving window standard deviation of the portfolio based on DCC forecasts which actively invests in both stock and commodity indexes and the risk free to an active portfolio that invests only on stock and risk free using a forecast from the RW model (since it produced more accurate predictions that the

18

AR model). Figure 7 shows that the large increase in the volatility of the stock/commodity portfolio is attained in September 2008. Nevertheless, for horizons shorter than 8 weeks this portfolio has higher volatility than the portfolio without commodities for each week of the whole sample.18

6

Robustness analysis

To assess the robustness of our results, we implement three additional exercises. More specifically, the aim is to investigate whether our findings depend on the particular indices we choose, and whether our results can be generalized. First, the dynamics of the USD exchange rate could alter the correlation pattern, since non-US equities in the MSCI index are priced in US dollars. To control for this, we repeated our analysis with the S&P500 index in place of the MSCI index. Figure 8 shows that the pattern of the correlation between the two series is similar to the MSCI and SPGSCI case. The break date is estimated again to be September 5, 2008. Statistical and economic gains are similar to those discussed in the previous section. Second, the SPGSCI index has high weight on oil, in particular at the beginning of our sample. We substitute it with the Bloomberg Commodity Index (BCOM) and repeat the analysis. The BCOM index is composed of futures contracts on physical commodities and it is designed to minimize concentration in the different commodities. No commodity can compose less than 2% or more than 15% of the index, and no sector can represent more than 33% of the index (as of the annual weighting of the components). The weighs for each commodity included in BCOM are calculated to ensure that the relative proportion of 18

The finding is qualitatively similar if we use other measures of risk, such as the 1% Value at Risk (VaR). For example, the weekly 1% VaRs computed on the full sample associated to the strategy based on the model RW at horizons 2, 4 and 8 are -4.77%, -4.77%, -4.84%, respectively; the VaRs associated to the strategy based on the model DCC at horizons 2, 4 and 8 are -4.94%, -5.14%, -5.43%, respectively. Therefore, the strategy including commodities can generate higher negative returns and higher risk.

19

each of the underlying individual commodities reflects its global economic significance and market liquidity. Annual rebalancing and reweighting ensure that diversity is maintained over time. The index actually covers seven sectors and is computed as of January 1991. Figure 9 shows the DCC estimates between the BCOM and MSCI, and it is again similar to the MSCI and SPGSCI case, with a break date estimated on September 5, 2008. Statistical and economic gains are also of similar magnitude. Finally, we also tried using directly Brent and West Texas Index (WTI) crude oil prices. Figure 10 shows the estimated conditional correlations between, respectively, Brent (top panel) and WTI (bottom panel) and MSCI. Again, results are nearly identical to the MSCI and SPGSCI case, with the break date estimated on September 5, 2008. As we would expect, oil-prices have played a dominant role across commodities, but our evidence also shows that the change in correlation in 2008 has been generalized to many other commodities and is still evident in 2015 despite the large volatility in the final year of our sample.

7

Concluding remarks

We showed that the correlation between commodity and equity returns has substantially increased after the onset of the recent financial crisis. We have then investigated the joint predictability of commodity and equity returns, and its benefits in a dynamic asset allocation exercise. A bivariate Bayesian DCC model, which can account for time variation in the correlation pattern, produces accurate density forecasts of commodity prices and gives large economic gains in an asset allocation exercise. The value of an active strategy based on DCC forecasts is large compared to passive strategies during turbulent times. At the same time, an investment strategy which also includes commodities in a portfolio produces substantially higher volatility and not always produces higher Sharpe ratios.

20

This is at odds with the common notion that commodities serve as a hedge. Private and institutional investors have displayed an increasing appetite for oil and commodities over the past years. Our findings have far-fetching policy implications in this respect. On one side, our results provide empirical support for the inclusion of commodities in a portfolio. At the same time, however, we have also found that this comes at the cost of an increase in volatility and risk. Therefore, the growing appetite for commodities is likely to produce more volatile portfolios. Digging further into the financial stability implications of the increasing correlation of commodity and equity returns is a relevant subject for future research.

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Appendix A DCC: a Bayesian estimation algorithm The DCC model for the (N × 1) vector yt is formulated as: yt

= Φ(L)yt−1 + vt

vt

= Ht0.5 εt , εt ∼ N (0, IN ), Ht = Dt Rt Dt 0

2 , i = 1, ..., N Dt2 = diag{ωi } + diag{κi }vt−1 vt−1 + diag{λi }Dt−1 0

(12)

0

Qt = S(ıı − A − B) + Aεt−1 εt−1 + BQt−1 Rt = diag{Qt }−1 Qt diag{Qt }−1 0

where S is the unconditional correlation matrix of εt , A, B and S(ıı − A − B) are positive semidefinite matrices and L is the autoregressive order. In our exercise N = 2, therefore the parameters A and B reduce to scalar and above conditions to A > 0, B > 0, A+B < 1. Following Engle (2002), the log likelihood can be expressed as: T

1X 0 0 0 lnL = − (N ln(2π) + 2 ln |Dt | + vt Dt−1 Dt−1 vt − εt εt + ln |Rt | + εt−1 Rt−1 εt−1 ) 2 t=1

(13)

We estimate the DCC model using a Metropolis-Hastings algorithm. Define the vector 0

αi = (Φ0 , ..., ΦL , ωi , κi , λi , A, B) , with Φ(L) = (Φ0 , ..., ΦL ), i = 1, .., N , and αj the j-th element of it. The sampling scheme consists of the following iterative steps. Step 1: At iteration s, generate a point αj∗ from the random walk kernel αj∗ = αji−1 + j ,  ∼ N (0, Q),

(14)

where Q is a diagonal matrix and σj2 is its j-th diagonal element, and αjs−1 is the (s − 1)th iterate of αj . Therefore, we draw row elements of Φ0 , ..., ΦL and ωi , κi , λi , A, B

28

  independently. Then accept αj∗ as αjs with probability p = min 1, f (αj∗ )/f (αjs−1 ) , where f () is the likelihood of model (12) times priors. Otherwise, set αj∗ = αjs−1 . The elements of Q are tuned by monitoring the acceptance rate to lie between 25% and 50%. Step 2: After M iterations, we apply the following independent kernel MH algorithm. Generate αj∗ from αj∗ = µi−1 αj + j ,  ∼ N (0, Qαj ),

(15)

where µαj and Qαj are, respectively, the sample mean and the sample covariance of the first M iterates for αj . Then accept αj∗ as αji with probability " p = min 1,

f (αj∗ )g(αjs−1 ) f (αjs−1 )g(αj∗ )

# ,

(16)

where g() is a Gaussian proposal density (15). Priors We set normal priors for Φ(L) with mean and variance equal to OLS estimates. The priors for ωi , κi , λi , A, B are uniform distributed and satisfy the restrictions ωi > 0, κi > 0, λi >, κi + λi < 1, A > 0, B > 0, A + B < 1. We note that different priors for the coefficients A and B of the correlation matrix should be considered if the dimension of the model is larger than two, see discussion in Tokuda et al. (2012).

29

Table 1: Forecast accuracy for commodity and equity returns

h=1 RW AR VAR MSPE 10.810 10.912 10.917 LS -3.004 -3.063 -2.977 h=8 RW AR VAR MSPE 10.928 10.916 10.842 LS -3.075 -3.096 -3.105

RW MSPE 6.647 LS -3.441 RW MSPE 6.718 LS -3.120

h=1 AR VAR 6.778 7.003 -3.480 -3.581 h=8 AR VAR 6.705 6.791 -3.464 -3.718

DCC 10.940 -2.558∗ DCC 10.859 -2.562∗

DCC 6.987 -3.461 DCC 6.825 -3.677

Commodity index h=2 RW AR VAR 10.875 10.898 10.804 -3.035 -3.182 -3.018 h=12 RW AR VAR 10.874 10.943 11.053 -3.054 -3.168 -3.252 Equity index h=2 RW AR VAR 6.643 6.675 6.861 -3.099 -3.617 -3.642 h=12 RW AR VAR 6.743 6.736 6.754 -3.165 -3.716 -3.700

DCC 10.937 -2.629∗

RW 10.847 -3.064

DCC 11.018 -2.720∗

RW 10.765 -3.129

DCC 6.871 -3.375

RW 6.680 -3.088

DCC 6.758 -3.703

RW 6.877 -3.198

h=4 AR VAR DCC 10.945 11.067 11.068 -3.237 -3.046 -2.543∗∗ h=24 AR VAR DCC 10.816 10.813 10.830 -3.322 -3.347 -2.993 h=4 AR VAR 6.728 6.870 -3.421 -3.705 h=24 AR VAR 6.850 6.847 -3.724 -3.759

DCC 6.872 -3.581 DCC 6.880 -3.737

Notes: RW , AR, V AR, DCC denote models as defined in Section 4; MSPE is the Mean Square Prediction Error; LS is the average Logarithmic Score. Lower MSPE and higher LS imply more accurate forecasts. One ∗ and two ∗∗ stars represent rejections of the null hypothesis of equal predictability at 10% and 5%, respectively.

30

Table 2: Economic value of portfolios with commodities

Mean Ret St dev SR ∆s ∆c ∆ 70 − 30

RW 2.54 14.86 0.07 40.67 124.69 36.24

Mean Ret St dev SR ∆s ∆c ∆ 70 − 30

RW 2.58 14.96 0.07 123.23 633.52 179.60

RW Mean Ret 2.68 Mean Ret 1.21 St dev 14.86 SR -0.02 21.46 ∆s 105.48 ∆c 17.04 ∆ 70 − 30

Mean Ret St dev SR ∆s ∆c ∆ 70 − 30

RW 2.50 14.97 0.07 120.05 630.34 176.42

h=1 AR VAR DCC 2.54 1.79 2.22 14.42 14.92 15.18 0.08 0.02 0.05 45.01 27.54 32.48 129.02 111.56 116.50 40.58 23.11 28.05 h=8 AR VAR DCC 1.49 2.32 3.38 16.91 16.17 16.81 0.00 0.05 0.12 -48.63 15.65 150.82 461.66 525.94 661.11 7.73 72.02 207.18 h=1 AR VAR DCC 1.19 2.41 2.87 0.50 -0.18 0.32 14.41 14.92 15.18 -0.07 -0.12 -0.08 15.68 -0.75 5.19 129.02 83.27 89.21 11.25 -5.18 0.76 h=8 AR VAR DCC 1.39 2.22 3.29 16.91 16.17 16.81 -0.01 0.05 0.11 -52.58 11.86 127.13 461.66 522.15 657.42 3.79 68.23 183.50

No transaction costs h=2 RW AR VAR DCC 2.66 2.24 1.79 3.34 14.74 15.08 15.42 15.30 0.08 0.05 0.02 0.13 49.08 36.14 21.02 57.94 171.35 158.41 143.29 180.21 45.42 32.49 17.37 54.28 h=12 RW AR VAR DCC 3.11 0.90 1.95 2.82 15.07 18.13 16.57 17.10 0.11 -0.04 0.03 0.08 204.14 -164.30 -15.66 32.83 1120.76 752.32 900.95 949.45 341.58 -26.86 121.78 170.27 50 bp transaction costs h=2 RW AR VAR DCC 2.39 2.19 1.93 3.36 2.11 1.47 1.03 2.67 14.75 15.09 15.43 15.30 0.04 0.00 -0.03 0.08 37.81 20.50 5.41 44.34 160.08 158.41 127.68 166.61 34.15 16.84 1.76 40.68 h=12 RW AR VAR DCC 3.06 0.84 1.89 2.76 15.07 18.12 16.57 17.10 0.11 -0.04 0.02 0.08 201.88 -167.15 -18.48 129.96 1118.49 752.32 898.13 946.57 339.32 -29.71 118.96 167.39

RW 2.73 14.72 0.09 70.83 324.70 84.04 RW 3.77 15.69 0.15 537.04 2485.52 853.88

RW 2.27 2.52 14.72 0.07 74.85 328.72 88.06 RW 3.75 15.68 0.15 535.93 2484.41 852.77

h=4 AR VAR DCC 1.95 2.11 3.80 15.58 15.76 15.91 0.03 0.04 0.15 27.92 22.64 75.18 281.79 276.51 329.05 41.14 35.85 88.40 h=24 AR VAR DCC 1.69 2.60 3.51 18.17 15.72 17.00 0.01 0.07 0.12 32.43 304.89 364.38 1980.91 2253.38 2312.86 349.27 621.73 681.22 h=4 AR VAR DCC 2.35 2.74 3.04 1.69 1.88 3.57 15.58 15.76 15.91 0.01 0.03 0.13 20.47 15.96 78.63 281.79 269.82 332.50 33.68 29.17 91.85 h=24 AR VAR DCC 1.66 2.58 3.49 18.16 15.71 17.00 0.01 0.07 0.12 30.94 303.34 362.92 1980.91 2251.83 2311.40 347.78 620.18 679.76

Note: RW , AR, V AR, DCC denote individual models defined in Section 4; Mean Ret is the annualized mean portfolio return; St dev is the annualized standard deviation; SR is the Sharpe ratio. ∆s , ∆o , ∆f are the performance fees for switching from an active strategy to passive strategies holding only stock (rs ), only commodity (rc ), and the 70-30 passive strategy (rm ).

31

Table 3: Economic value of portfolios without commodities h=1 2.93 1.80 9.34 12.93 0.16 0.02 h=8 RW AR Mean Ret 3.38 2.41 St dev 9.30 12.93 SR 0.21 0.07

Mean Ret St dev SR

h=2 3.10 1.87 9.10 13.11 0.18 0.03 h=12 RW AR 3.41 2.51 9.41 11.99 0.21 0.09

h=4 3.25 2.31 9.21 13.00 0.20 0.06 h=24 RW AR 3.60 3.51 10.24 10.52 0.21 0.19

Note: RW , AR: denote individual models defined in Section 4; Mean Ret is the annualized mean portfolio return; St dev is the annualized standard deviation; SR is the Sharpe ratio.

32

Figure 1: Commodity and equity returns 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 1985

1990

1995

2000

M SC I

2005

2010

2015

SPG SC I

Note: Average returns over a 1-year moving window of commodity (SPGSCI) and equity (MSCI) indexes returns.

33

Figure 2: Commodity and equity volatilities 2.8 2.4 2.0 1.6 1.2 0.8 0.4 1985

1990

1995

2000

M SC I

2005

2010

2015

SPG SC I

Note: Standard deviations over a 1-year moving window of commodity (SPGSCI) and equity (MSCI) indexes.

34

Figure 3: Commodity and equity correlations 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 1985

1990

1995

2000 3M

2005

2010

2015

12M

Note: Sample correlations between commodity and equity returns computed on moving windows of different length: 3-months (3M); and 12-months (12M).

35

Figure 4: Dynamic Conditional Correlations between the MSCI and the SPGSCI .8 .6 .4 .2 .0 -.2 -.4 -.6 1985

1990

1995

2000

2005

2010

2015

Note: Correlations between commodity and equity returns estimated using a Dynamic Conditional Correlation model. The vertical dashed line marks the estimated breakpoint (5 September 2008).

36

Figure 5: Log-Score Differentials AR

VAR

DCC

AR

300

250

250

200

VAR

DCC

150

200

100 150 50 100 0 50

−50

0 −50 2005W1

−100

2007W1

2009W1

2010W52

2012W52

−150 2005W1

2014W52

2007W1

h=1 AR

VAR

DCC

AR 300

250

250

200

200

150

150

100

100

50

50

0

0

−50

−50

2007W1

2009W1

2010W52

2012W52

−100 2005W1

2014W52

2007W1

h=4 AR

VAR

2010W52

2012W52

2014W52

h=2

300

−100 2005W1

2009W1

2009W1

VAR

2010W52

DCC

2012W52

2014W52

h=8 DCC

AR

200

VAR

DCC

100

150 50 100 0

50 0

−50

−50 −100 −100 −150 2005W1

2007W1

2009W1

2010W52

2012W52

−150 2005W1

2014W52

h = 12

2007W1

2009W1

2010W52

h = 24

Note: Log-Score (LS) Differentials versus the Log-Score of the RW benchmark.

37

2012W52

2014W52

Figure 6: CER Differentials

RW

AR

VAR

DCC

COMMODITY

70−30

100

80

60

40

20

0

−20 2005W1

2007W1

2009W1

2010W52

2012W52

2014W52

Note: Certainty Equivalent Return (CER) Differential at an horizon of two weeks, with respect to the CER of the passive strategy holding only equities, with transaction costs set at 5 basis points.

38

Figure 7: Volatility for portfolios with and without commodities DCC

RW

DCC

18

18

16

16

14

14

12

12

10

10

8

8

6

6

4 2005W1

2007W1

2009W1

2010W52

4 2005W1

2012W52

2007W1

h=1 DCC

2009W1

RW

2010W52

2012W52

h=2

RW

DCC

20

RW

22 20

18

18 16 16 14

14

12

12 10

10

8 8 6 6 4 2005W1

4 2007W1

2009W1

2010W52

2 2005W1

2012W52

2007W1

h=4 DCC

RW

DCC 25

20

20

15

15

10

10

5

5

2007W1

2009W1

2010W52

2010W52

2012W52

h=8

25

0 2005W1

2009W1

0 2005W1

2012W52

h = 12

2007W1

2009W1

RW

2010W52

2012W52

h = 24

Note: Standard deviation of portfolio returns including commodities (using a DCC model for the dynamic asset allocation) and excluding commodities (using a RW model for the dynamic asset allocation).

39

Figure 8: Dynamic Conditional Correlations between the S&P500 and the SPGSCI .8 .6 .4 .2 .0 -.2 -.4 -.6 80

82

84

86

88

90

92

94

96

98

00

02

04

06

08

10

12

Note: Correlations between commodity and equity returns estimated using a Dynamic Conditional Correlation model. The vertical dashed line marks the estimated breakpoint (5 September 2008).

40

Figure 9: Dynamic Conditional Correlations between the BCOM and the MSCI

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 92

94

96

98

00

02

04

06

08

10

12

Note: Correlations between commodity and equity returns estimated using a Dynamic Conditional Correlation model. The vertical dashed line marks the estimated breakpoint (5 September 2008).

41

Figure 10: Dynamic Conditional Correlations between oil prices and the MSCI

Brent .8 .6 .4 .2 .0 -.2 -.4 -.6 90

92

94

96

98

00

02

04

06

08

10

12

14

04

06

08

10

12

14

WTI .8 .6 .4 .2 .0 -.2 -.4 -.6 90

92

94

96

98

00

02

Note: Correlations between Brent (top panel) and WTI (bottom panel) oil prices and equity returns estimated using a Dynamic Conditional Correlation model. The vertical dashed line marks the estimated breakpoint (5 September 2008 for both series).

42