between stock markets and crude oil
from a risk management perspective
Dragoù Iorga
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics Author: Dragoù Iorga Student nr: 11372125
Email: iorga dragos paul@yahoo.com Date: August 1, 2017
Supervisor: Andrei Lalu Second reader: . . .
responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other
than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for
the supervision of completion of the work, not for the contents.
Abstract
This paper investigates the conditional variances and correlations between three major European stock markets (Sweden, The Nether-lands, Norway) and the market for crude oil. Volatility spillover effects are also discussed. The analysis is done between the daily returns of the OSX, AEX and OMX stock indexes and the daily returns of crude oil spot price listed on Brent during the period 2000-2017. The BEKK model and the DCC model have been implemented. The BEKK has identified volatility spillover effects across markets after 2009. Corre-lations between the aforementioned indexes and crude oil have also drastically increased after 2009. Results are similar albeit the fact that Sweden has immensely reduced it’s dependency on the commod-ity. This sparks the conversation whether market participants might be irrational. Relative to the DCC, the BEKK model assumes more volatility in the daily covariances of the markets. When constructing optimal portfolio strategies, the BEKK model stands out as the best option in terms of hedging crude oil price risk.
Keywords: Crude oil, Stock markets, Volatility Spillovers, Risk Management, BEKK, DCC, OMX, AEX, OSX, Optimal portfolio design
Preface and Acknowledgments
Foremost, I would like to thank my thesis supervisor and coordinator Andrei Lalu for sharing his vast knowledge and for providing me with the best support and advice. With his guidance I’ve managed to successfully complete this thesis, and thus the master program in Actuarial Science and Mathematical Finance.
I would also like to express my gratitude towards the professors of the University of Amsterdam. Their quality of teaching and immense help have provided me with the most resourceful learning experience over the past year.
Of course, I would also like to thank my family for all the sacrifices they have made for me to enjoy my student years and for their continuous support to my personal development.
Contents
1 Introduction 1
2 Literature Review 2
3 Model Description 4
3.1 The BEKK-GARCH Model . . . 5
3.2 The DCC-GARCH Model . . . 6
4 Data description 8 5 Empirical Analysis and Results 11 5.1 Volatility Spillovers . . . 11
5.1.1 Swedish OMX and Crude Oil . . . 11
5.1.2 Dutch AEX and Crude Oil . . . 13
5.1.3 Norwegian OSX and Crude Oil . . . 14
5.2 Dynamic Conditional Correlations and Variances . . . 16
6 Portfolio Design and Hedging Strategy 19 6.1 Portfolio Investment Weights . . . 19
6.2 Portfolio Hedge Ratios . . . 20
6.3 Crude Oil Hedge Effectiveness . . . 20
7 Conclusion 23
1
Introduction
Oil is an important resource for consumption in many industries and it is considered the most vital raw material used as a key input in various production processes. Being the main driver of the Industrial Revolution, from the beginning of the 19th Century, it has ensured economic and technological development of the society that we live in. It can be considered one of the most controversial goods to place ownership on, establishing the worlds largest industry of the past two centuries. With large spillover effects embedded across the global economy, Hamilton (1983) [27] in his seminal paper analyzed the behavior of macroeconomic activity that is due to oil price changes. He concluded that oil shocks had a statistically significant contribution to the 1972 recession in the US. A lot of research has followed afterwards, proving that oil shocks have an important influence on a country's GDP. Rodriguez and Sanchez (2005) [34] studied the effect oil price shocks have on the real GDP of the main OECD industrialized countries during 1972-2001. They concluded that oil price shocks are one of the largest source of variation to the real GDP. However oil price changes affect not only a nations GDP, but also other essential economic indicators such as unemployment rates (Uri, 1996)[70], inflation rates (LeBlanc & Chinn, 2004), interest rates (Cologni & Manera, 2006) [15] and most importantly for this research the stock market.
According to Mussa (2000) [56], the linkage between the market for crude oil and the stock market may be established through the mass effect it has on the sentiment of mar-ket participants, macroeconomic indicators, corporate earnings and monetary policies. Jones (2004)[36] argued that since stock values should reflect the market’s expectations of the future profitability of companies, the transmission effect that oil price shocks have on the stock market cannot be negligible. The present value of an asset is determined as the future discounted value and any factors affecting its future valuation should already be mirrored in current stock prices and returns. Therefore oil price fluctuation can be considered a major source of risk that is transposed to the economy and from there on to the financial mar-kets. Recent financial instability due to the economic crisis of 2008 and elevated geopolitical conflicts have caused a lot of uncertainty among investors. Moreover, increasing integration between countries and economies has raised the risk of volatility transmission across markets. Measuring these various sources of risk and volatility behavior has long been of main impor-tance to academics and practitioners. Hedge funds, traders, asset management companies and pension funds are heavily dependent on developing proper risk management strategies for their investment portfolios. This requires accurate measurements of dynamic covariances and correlations between assets.
As a result, the aim of this paper is to examine the conditional correlations and volatility spillover between these two markets; how it has evolved across time. This will shed some light on how stocks react when exposed to market risk stemming from crude oil price fluctuations. Because the relationship between national stock markets and crude oil price movements may be determined by the country’s dependency on oil, three distinct markets have been chosen with different clusters of characteristics.
Despite the topical importance of the ”black gold”, universally it becomes apparent that the ”Age of Oil” is actually coming to an end and countries are transitioning to fully renewable energy systems. The Renewable Energy Directive 2009/28/EC released by the European Commission in 2009, establishes an overall policy for the production and usage of energy from renewable resources for all the countries within the European Union (EU). Member states are required to switch at least 20% of their total energy dependency to renewable resources by 2020. On top of the list stands Sweden who managed to reach an outstanding 56.2% of its renewable energy sources shares (RES) already surpassing its 2020 target1. Also an agreement has been signed by its government and adjacent political parties to set the goal of implementing an entirely renewable electricity system, with a target of
1
http://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:52017DC0057&qid= 1488449105433&from=EN
100% renewable electricity production by 2040, therefore eliminating any link whatsoever with the market of this traded commodity2. Swedish crude oil imports, as of 2016 stand at 3.46% of total EU crude oil imports. Surprisingly, at the other side of the spectrum stands The Netherlands reaching only 7.6% of its 2020 required targets in RES. The Netherlands is also the 4th main European oil importer, holding 11% of the total imported crude oil in the EU3. For reference, the main crude oil importer is Germany with 14% of the Union’s total. As part of The Netherlands’ total imported traded goods, crude oil lies second at 7.66% with a total value of $34.8B, the first one being refined petroleum with 7.68%4. The third country included in the analysis is Norway which is one of the world’s major oil exporters. Currently being the 8th largest oil exporter globally, in 2016 crude oil exports accounted for about 25% of its total external traded goods5. Nevertheless, these statistics are not meant to point a finger at Norway’s limited environmental awareness, since 98% of all electricity production in Norway comes from renewable sources, mostly from hydropower generating systems6. Therefore in the bucket of countries meant for the analysis, there are two countries which have their economies heavily dependent on crude oil, and oppositely Sweden which is successfully transitioning to a fully renewable energy system. This makes an interesting case for analyzing dynamic variances, covariances and correlations and to see whether volatility spillovers effects are present. Given the characteristics of the countries selected, this will reveal whether stock behavior rationally reflects the impact of crude oil price fluctuations on real cash flows.
The remainder of the paper is organized as follows. Section 2 provides a review on the al-ready existing research which treats the relationship between the stock market and the crude oil market. The analysis will be conducted with the use of two multivariate GARCH models, namely the Baba, Engle, Kraft and Kroner model (BEKK) and the Dynamic Conditional Correlation (DCC) model. Both will be discussed in Section 3. Section 4 and 5 describe the data and the empirical results. Since the findings of this paper are of great relevancy for fi-nancial actors, Section 6 presents the implications for portfolio design and hedging strategies. The hedging effectiveness of the two models will also be discussed. Section 7 concludes.
To the author’s best of knowledge, this paper is the first one to study the dynamic covari-ances and volatility spillovers between these three European stock indexes and crude oil. It is also the first paper to advice on optimal asset allocation and hedging strategies in relation to these particular stock markets and crude oil. Because two models are being implemented, it is also the first paper to compare their crude oil price risk hedging effectiveness.
2
Literature Review
Jones and Kaul (1996)[35] were one of the first to analyze the impact oil price shocks have on the stock markets. With the use of quarterly data and a standard cash-flow dividend valuation model, the researchers tried to explain movements in the stock prices due to oil price shocks, by analyzing the impact it has on real cash flows as a proxy. Successful results were found for US and Canada indicating a negative relationship between oil prices and stock returns, however for Japan and UK results were inconclusive. In the same year Huang et al. investigated the relationship between daily oil futures returns traded on the New York Mercantile Exchange (NYMEX) and daily US stock returns with the use of a multivariate vector autoregressive (VAR) model. The research concluded that oil futures are not in any sort of connection with stock market returns, except for oil and gas companies.
2http://www.government.se/49d8c1/contentassets/8239ed8e9517442580aac9bcb00197cc/ ek-ok-eng.pdf 3 https://ec.europa.eu/energy/en/data-analysis/eu-crude-oil-imports 4 http://atlas.media.mit.edu/en/profile/country/nld/ 5 http://www.norskpetroleum.no/en/production-and-exports/exports-of-oil-and-gas/ 6 https://www.regjeringen.no/en/topics/energy/renewable-energy/ renewable-energy-production-in-norway/id2343462/
Sadorsky (1999) [62] led the research forward in analyzing the effects of oil price shocks on the US stock market. With the use of a VAR-GARCH approach, and by taking into consideration the asymmetry of oil price shocks, Sadorsky [62] showed that there is a negative relationship between oil price movements and stock returns. In addition, he found that the dynamics of oil price shocks have changed over time. After 1986, oil price shocks had a stronger impact on stock returns. In accordance with these results, other papers on different countries followed. Papapetrou (2001)[59] showed that such a negative relationship holds for Greece as well. Impulse response functions from the VAR-GARCH model show that oil prices are important in explaining stock price fluctuations. The results suggest that a positive oil price shock depresses real stock returns. Hammoudeh and Aleisa (2004) [28] examined the links between GCC stock markets and NYMEX oil futures. In addition to finding significant predictive power of oil futures to stock returns, authors draw an interesting conclusion with regards to Saudi Arabia. Their findings suggest that there is a bi-directional relationship between the Saudi Arabia stock market and the NYMEX futures price at a 5% significance level. This means that the Saudi Arabian stock market has a predictive power for oil futures prices.
Given the fact that emerging countries are expected to experience rapid growth and thus would consume a fair share of the world's oil resources, Basher and Sadorsky (2006) [4] conduct research on 21 countries within these scopes. They employed a multi-factor model that allows for a series of risk factors to be included. Results indicated that oil price risk carries a significant weight in explaining emerging market stock returns. Oil price risk is positive and statistically significant at the 10% level in most of the models included in the study. On the other hand, Maghyereh (2004) [49] had reached contradicting conclusions, showing that there is a very weak relationship between the emerging stock markets and crude oil price shocks. Park and Ratti (2008) [60] with the use of a VAR model, study the effects of oil price shocks on the covariance of oil price volatility and returns for the U.S. and 13 European countries during 1986–2005. They found that oil price shocks have a statistically significant impact on the covariance in the same month or within one month. Moreover, they concluded that the effect of oil price shocks is better captured by considering the price listed on Brent than by the national oil price that reflects the offsetting movement in the exchange rate.
However, there is scarce research in analyzing volatility transmission mechanisms across the crude oil market and the stock market. Most of the existing research has been analyzing this phenomena for crude oil in relationship with other commodities or financial instruments (see Lien and Yang (2008) [48], Du et al. (2011) [18], Serra (2011 , Kumar et al. (2012) etc.). Ewing and Thompson (2007) [22] were one of the first to alter the direction of the research towards this subject matter. They investigated the cyclical co-movements of crude oil prices with industrial production, consumer prices, unemployment, and stock prices. By allowing for oil prices to lag, lead, or be contemporaneously correlated with movements in the macroeconomic variables they are able to better capture the dynamic behavior and cross-correlations among them. They concluded that crude oil prices are pro-cyclical and lag stock prices by 6 months. Malik and Hammoudeh (2009) [50], with the use of a BEKK-GARCH model, analyzed volatility spillovers between the oil market and the equity markets of the U.S. and of the three major OPEC oil exporters from the Gulf (Bahrain, Kuwait and Saudi Arabia). They found significant positive results in volatility transmission and emphasized again the bi-directional relationship that exists between the Saudi equity market and the crude oil market. This accentuates further the major role Saudi Arabia plays in the global oil market.
Aloui and Jammazi (2009)[1] applied a univariate two regime-switching MS-E-GARCH model to examine the relationship between crude oil shocks and UK, French and Japanese stock markets. Their researched showed that oil price increases play a significant role in de-termining the volatility of real returns within these markets. Bhar and Nikolovann (2010)[6] have examined the dynamic correlation between the stock market and oil prices for
Rus-sia. Russia's stock exchange is one of the most concentrated markets around oil producing companies with 19% of its total stock market. By employing a univariate E-GARCH model, they showed that oil price movements have a strong effect on the Russian equity returns and volatility behavior. Given the time-varying fluctuations of the correlation, the authors were able to link much better historical happenings with data indications. Choi and Hammoudeh (2010) [30] applied a DCC-GARCH model and indicated increasing correlations between oil prices listed on Brent, WTI and copper, gold and silver but decreasing correlations with the S&P500 index. Chang, McAleer, and Tansuchat (2013) [14] also investigated the condi-tional correlations and volatility spillovers between crude oil (WTI and Brent benchmarks) and FTSE100, NYSE, Dow Jones and S&P500 stock indices. The authors have incorporated 4 multivariate GARCH models. Results showed little evidence of volatility spillover effects which lead to the idea that there is a smaller than expected dependency between crude oil returns and stock index returns. In the end, given the asymmetric effects of positive and negative oil price shocks, the VARMA-AGARCH model was chosen to be superior.
A major point in all the aforementioned papers is that the explanatory parameters of the dynamic multivariate GARCH models have been found to be significant. Which indi-cates that the assumption of constant correlation or covariance between oil price returns and stock returns is false. This has major implications for risk management, hedging and asset allocation strategies. Previously in 2011, the same group of researchers (Chang, McAleer, and Tansuchat)[12] analyzed the time-dependent covariance between the crude oil spot and futures returns for the two major benchmarks, Brent and WTI. Five different multivariate GARCH models have been implemented for comparison. Based on their results, they man-aged to derive optimal portfolio hedge ratios (OHR) accordingly. The key in any hedging strategy is actually deriving these OHRs between assets which in the end do display signifi-cant variability over time. Ballie and Myers (1991) [3] evaluated the performance of constant hedge ratios derived using linear regression against dynamic OHRs based on a bivariate VEC-GARCH model across six different commodities. Based on their tests, they concluded that inarguably time-varying OHRs result in a much more effective hedging strategy than constant hedge ratios. Kroner and Sultan (1993) [44] also proved, using foreign currencies as an example of assets, that time-varying hedge ratios result in a greater risk-reduction effect for an investment portfolio.
3
Model Description
An important reasoning for discussing various modeling approaches, is that it provides important insights as to how big is the model risk and what is the impact to the hedging effectiveness of the investment portfolio. If the model used is insufficiently accurate, the risk of loss due to ineffective hedging also increases substantially. Therefore in this paper, two multivariate conditional volatility models will be implemented. More specifically the empirical analysis will be conducted with the use of the BEKK-GARCH model developed by Engle and Kroner (1995) [21], and the DCC-GARCH model developed by Engle (2002) [19].
The motivation behind using these two famous models stands by the fact that they allow for time-varying relationships in the conditional correlation and covariance matrices of the paired assets. This has been proven to be more in line with what actually happens in the financial markets, as correlation and covariances between assets tend to change as a response to an ever-evolving market environment. Tse (2000) [68] successfully showed that the correlations across national stock market returns, for example, are time varying.
The models therefore entail similar required properties, but are different in many aspects, each with its own advantages and limitations. It’s been argued by McAleer et. al (2010) [10] that the BEKK model has as its primary function the goal of deriving conditional covariances from which indirectly, conditional correlations can be computed. On the other hand, the DCC model puts emphasis on the construction of the conditional correlations, which in turn
permits for the derivation of the conditional covariances. Caporin and McAleer (2008) [9] perform a comparison between the scalar BEKK and the DCC and conclude that both share common structural and asymptotic characteristics. They managed to show that from a risk management perspective there shouldn’t be any bias in choosing one over the other, both generating close empirical results for portfolio allocation or risk evaluation.
Engle and Colacito (2006) [20] peform an interesting evaluation of several multivariate GARCH models, including the diagonal BEKK, the DCC model and the asymmetric DCC model. With the use of a Diebold-Mariano test, they asses their predictive power of returns for stock and bonds, and show that both of the DCC models outperform the diagonal BEKK at 5% significance level. However, in terms of accurately estimating volatility ratios for the S&P 500 and the Dow Jones, the diagonal BEKK was preferred. In terms of minimizing a portfolio variance comprising of stocks and bonds, all three models perform well, slightly favoring the asymmetric DCC. One drawback of the DCC however, is that when dealing with portfolios that have more than two assets, the DCC tends to lose from its explanatory power of the correlation structure. They also concluded that accurate estimation of the conditional correlation during times where it is expected to be at the extremes can be as important as explaining 30–40% of the required return. Gould and Bos (2007) [8] also tackle this issue, including both the general BEKK and the DCC model among their models for comparison. Their analysis showed that in terms of capturing the correlation structure, the DCC model has a lower MSE than the BEKK. As for risk minimization effectivenes, for a hedge portfolio composed of futures and shares of S&P 500, the BEKK is preferred on an overall level. Chang, Serrano and Martin (2013) [14] also compare the hedging effectiveness for currencies of 4 different multivariate models (CCC, VARMA-AGARCH,BEKK,DCC)and reach the conclusion that both BEKK and DCC perform the best out of the four, with very close results.
Based on this paper’s analysis, more clarification will be brought as to the behaviour of the two models in terms of their effectiveness and flexibility and in relation to crude oil and stock indexes specifically. Each of the three stock indexes will be paired with the price of crude oil.
3.1
The BEKK-GARCH Model
If a bivariate series of data is taken as yt = (yr,t, yo,t)0 consisting of real stock returns
and real crude oil price changes, the BEKK model typically employs the following equations for the conditional mean and conditional variances:
yt = E(yt|Ft−1) + t = α + m X l=1 φlyt−l+ t (1) t = H 1 2 t zt (2) Ht = A0A00+ p X i=1 Ai(t−i0t−i)A0i+ q X j=1 BjHt−jBj0 (3)
for t = 1, 2, ...n where φl is a 2 x 2 matrix, t = (er,t, eo,t)0 is the shock term in the
mean equation (1) and t|Ft−1 ∼ N (0, Ht) with Ft−1 representing the past information
available at time t − 1. zt is a 2 x 1 vector (zrt, zot)0 of standardized i.i.d. residuals. The
time-varying conditional variance-covariance of the residuals Ht is a k x k matrix with
k=2. In MGARCH models, since Ht is a variance matrix, its positive definiteness is key to
be able to properly describe it. As can be observed, the BEKK model ensures this by using a quadratic decomposition for the parameter matrices. A0 is a lower triangular matrix, and
B matrix encompasses GARCH effects. A more explicit representation of the model would be the following: 1,t 2,t =h11,t h12,t h21,t h22,t 1/2 z1,t z2,t (4) where Ht = h11,t h12,t h21,t h22,t =a011 0 a021 a022 a011 a012 0 a022 + p
X
i=1 a11,i a12,i a21,i a22,i 21,t−i 1,t−12,t−i 1,t−i2,t−1 22,t−i a11,i a21,i a12,i a22,i + qX
j=1 b11,j b12,j b21,j b22,j h11,t−j h12,t−j h21,t−j h22,t−j b11,j b21,j b12,j b22,jStationarity is ensured if a2xx+ b2xx< 1 with x = 1, 2. As can already be seen, the model suffers from the ”curse of dimensionality”. The number of parameters needed for estimation is k2(p + q) + k(k + 1)/2 which increases hastly with p, q and k. In the bivariate case of order 1, with p = q = 1 and k = 2, which will be the aspect of the model used in this paper, the numbers of parameters needed for estimation is 11, which is still acceptable. Generally speaking, in order to wane the number of estimated parameters, Bauwens (2006) showed that restrictions on the BEKK can be imposed so that a diagonal or scalar structure may be used for modelling. However, neither representation allows any more for volatility spillovers. In this paper due to the desire for analyzing precisely this phenomena between the three stock indexes and the real price of crude oil, the general form will be used with no restrictions imposed in the estimation of the BEKK.
In the equation for the conditional variance of real stock returns h11,t, it can easily be
observed that it depends on its own lagged conditional variances through the parameter a11
and lagged shocks through b11as well as the ones stemming from real oil price changes which
determine the size of volatility spillovers between the two 7. The parameter a12 in Eq.(4)
captures the effect a real oil price shock at time t − 1 has on the conditional variance of real stock returns at time t, and b12 measures the impact real oil price’s conditional variance at
time t − 1 has on the conditional variance of real stock returns at time t. Other cross-terms can be seen in its equation, however they lack any direct interpretation.
Estimation of the BEKK parameters is done through the maximum likelihood method (MLE). Under the assumption of Gaussianity, the following log-likelihood function is con-sidered l(θ) = −T 2ln(2π) − T X t=1
(
1 2ln(H 2 t) + 1 2tH −1 t 0 t)
(5)where T is the number of observations in the sample and θ is a vector of unknown parameters (a11,i, a12,i, ..., b11,j, ...)0.
3.2
The DCC-GARCH Model
Assuming again a vector containing a bivariate series yt = (yr,t, yo,t)0 with the same
information as previously mentioned, the DCC-GARCH model employs the following struc-ture: yt = E(yt|Ft−1) + t = α + m X l=1 φl· yt−l+ t (6)
under the assumption that yt|Ft−1 ∼ N (0, Ht) for t = 1, 2, ...n. To accommodate for
heteroskedasticity, the shock terms are written as:
t = Dtzt (7)
and the conditional covariance matrix of the return vector is given by:
Ht = DtRtDt (8)
with Dt being a k x k with k = 2 diagonal matrix of conditional variances, with the
diagonal elements following univariate GARCH processes:
hii,t = ωi+ p X l=1 ζi,l· i,t−l + q X l=1 ψi,l· hii,t−l (9)
with i = 1, .., k and ζi,l and ψi,l representing ARCH and GARCH effects respectively. In
this research p = q = 1 in which case stationarity is ensured if ζi+ ψi < 1. With these
being said, more specifically Dt looks like this:
ph11,t . . . 0 0 ph22,t ... .. . . . . ... 0 . . . phkk,t
If equation 8 is multiplied with the transpose of the residuals, the following arises: t0t = Dtztzt0Dt (10)
and
E(t0t|Ft−1) = DtE(ztz0t|Ft−1)Dt (11)
it can be noted that the conditional correlation matrix of the residuals is equal to the conditional covariance matrix of the standardized residuals which means that (11) = (8)
If Rt is assumed to be constant, the CCC model of Bollerslev (1990) can be observed.
Engle (2002) proposed the following structure for the conditional covariance matrix of zt:
Rt = {diag(Q −1/2 t )}Qt{(diag(Q −1/2 t )} (12) where Qt = (1 − θ1− θ2)Q + θ1zt−1zt−10 + θ2Qt−1 (13)
θ1 and θ2 represent the ARCH and GARCH effects respectively. Q is the unconditional
variance matrix of zt. If θ1 = θ2 = 0 then Qt = Q and Rt becomes constant.
Specific time-dependent correlations can be extracted through: ρ12,t = q12,t √ q11,tq22,t = (14) = q (1 − θ1 − θ2)q12+ θ1z1,t−1z2,t−1+ θ2q12,t−1 ((1 − θ1 − θ2)q11+ θ1z21,t−1+ θ2q11,t−1)((1 − θ1− θ2)q22+ θ1z2,t−12 + θ2q22,t−1)
where the denominators are elements from the diag(Q−1/2t ) matrix which has the following structure: √ q11,t . . . 0 0 √q22,t ... .. . . . . ... 0 . . . √qkk,t
If the DCC model is of the same order as the BEKK model, the numbers of paramateres needed for estimation is (k + 1)(k + 4)/2 which in the bivariate case leads to 9 parameters. As Engle (2002)[19] pointed out, estimation is done through a two step method. Assuming normality in the distributions, first, the series of the univariate GARCH equations are fitted based on the following log-likelihood function:
L(λ) = −T 2ln(2π) − T X t=1
(
1 2ln(D 2 t) + 1 2tD −1 t 0t)
(15)where λ contains the parameters of Dt. As for the Rt with η containing its respective
parameters, the log-likelihood function is:
L(λ, η) = −T 2ln(2π) − T X t=1
(
1 2ln(R 2 t) + 1 2ztR −1 t z 0 t)
(16)T is the number of observations in the sample. Optimization is achieved by fulfilling the following equations:
ˆ
λ = arg max{L(λ)} (17)
and then recursively:
max
η L(ˆλ, η) (18)
A drawback of the model is that it doesn’t allow for volatility spillovers. Also, as Huang (2010) [47] stressed out and as can be seen in equation 14, another limitation would be that the conditional correlations follow the same dynamic structure.
4
Data description
Daily prices for the 3 national stock indexes represented by the Stockholm Stock Ex-change (OMX) the Amsterdam Stock ExEx-change (AEX) and the Oslo Stock ExEx-change (OSX) have been extracted from Datastream. Daily prices for crude oil listed on the Brent bench-mark have been extracted from the The U.S. Energy Information Administration(EIA) web-site. The data has been corrected for country specific inflation as CP IPi,tj
i,t/100 with i = 1, .., 4
for each country, including the US particularly for the price of crude oil. t stands for the month and j stands for the day within that particular month. The Consumer Price In-dexes(CPI) have been extracted from the OECD Data library. All prices are being expressed in dollars, conversion being done with the use of the real exchange rate (RER): X · CP If
CP Id
where X is the nominal exchange rate in units of domestic (d) currency per unit of foreign (f ) currency. Daily nominal exchange rates have been extracted from Datastream. By cor-recting all prices and exchange rates for inflationary effects, any common trend that may have been layered on the data has been removed. Thus any sort of correlation is isolated within specific drivers of the real prices. After testing for cointegration between the real price of each stock index with the real price of crude oil through Johansen’s trace test, the null hypothesis of any long-term trend between the paired time-series can be rejected at 1% significance level8. The Augmented-Dickey-Fuller (ADF) test for unit root indicates that the series are I(1). The returns of the daily real price indexes and real price of crude oil are calculated at a continously compounded rate: ri = ln(Pi,t/Pi,t−1). The time span for all
individual 4401 data points is from 30th of December 1999 to 30th of December 2016. To analyze the presence of any time-variation in the volatility spillovers effects, two intervals will be considered: 30th of December 1999 to 30th of December 2008 and 30th of December 2008 to 30 of December 2016. Results of this analysis will indicate whether there could be any structural breaks present in the volatility transmission of the time-series. All analysis, modeling and tests are done in the R programming tool. Main packages used are Tsay’s ”MTS” and ”rmgarch”.
Figure 2
Daily real prices are plotted in Figure 1. We can already observe how the AEX and OMX move somewhat together, whilst OSX and the Brent price of oil follow a separate common pattern, especially in the first decade of the 21st century. However, during that period the series for the real price of crude oil is more volatile. Log-returns are displayed in Figure 2. The series clearly exhibit volatility clustering. The sample means are very close to 0. From Table 1 it can be concluded that real crude oil returns appear to be the most volatile with the highest standard deviation. The Jarque-Bera test rejects the null-hypothesis for all the return series9. Nevertheless, for parsimonious reasons normality will be assumed on the return distributions. High excess kurtosis emphasizes the natural movement of the stock indexes where returns are relatively stable and evolve more closely around the mean. It also implies heavy tails. A negative skewness for the AEX, OSX and Brent returns indicate that negative gains are more likely than positive gains. The slight opposite holds for OMX. The Multivariate Ljung-Box (LBQ) test has been performed on the real returns and on the squared real returns10. The statistics from Table 2 show evidence for serial correlation except for crude oil. Strong evidence for conditional heteroskedasticity for all the series indicate that GARCH models may be used for modeling.
Table 1: Descriptive Statistics
Returns Mean Max Min SD Skewness Kurtosis
Jarque-Bera OSX 0.00023 0.10993 -0.11864 0.01762 -0.39594 8.07389 4833.70* AEX -0.00015 0.12313 -0.11851 0.01589 -0.10413 9.80005 8483.50* OMX -0.00005 0.13703 -0.10314 0.01838 0.02541 7.40806 3562.00* Brent 0.00010 0.18137 -0.19884 0.02268 -0.13960 8.23141 5030.60*
An ADF test has been performed on the real returns in order to test for unit roots. The results of the test are included in Table 3. The null-hypothesis can be rejected for all series at the 1% level of significance.
Table 2: Ljung-Box Test Returns LBQ(10) LBQ2(10) OMX 51.372* 2092.5* AEX 72.004* 2958.7* OSX 28.314* 3668.0* Brent 9.2552a 450.68* *Significance at 1% a No serial correlation
Table 3: ADF Test
Returns None Constant Constant + Trend OMX -49.6703* -49.6651* -49.6653* AEX -47.9708* -47.9713* -47.9742* OSX -48.5953* -48.6026* -48.5997* Brent -46.0813* -46.0776* -46.0865* *Significance at 1%
9See Appendix Note C for the specification of the test 10See Appendix Note D for the specification of the test
5
Empirical Analysis and Results
5.1
Volatility Spillovers
This section presents an analysis on the volatility spillover effects between real crude oil returns and 3 stock index returns, namely the OMX, AEX and OSX. Only the BEKK model will be used as among the two it is the only one allowing for such cross-effect. Specifically for this discussion, the time span has been split in two intervals. This will help identifying whether recent trends of the countries in implementing renewable energy systems, have had any sort of impact on the apparent relationship between stock markets and the crude oil market. The time threshold chosen is 2009 which is the year the Renewable Energy Directive has entered into force. For both periods, the series have been tested for cointegration and unit roots11.
5.1.1 Swedish OMX and Crude Oil
Based on the Akaike Information Criterion(AIC), the lag number that should be used in the VAR mean equation for the period before 2009 is 7, and for the period afterwards 2. If we look at the partial autocorrelation function (PACF) for the period 2000-2009, there is significant serial correlation in the returns for both series up until t − 712. Afterwards it
cuts-off with a few random outliers. For the period 2009-2017 the memory of the processes is much weaker with no significant serial correlation at higher order lags. By recursively fitting a VAR model with the help of Ordinary Least Squares (OLS) and then testing whether the residuals behave like white noise, serial correlation has been successfully removed at the lags indicated by the AIC. The same methodology has been carried for all the upcoming analysis. Insignificant terms above 5% are dropped from the equations. Results are in Table 4 & 5. The LBQ test on the squared residuals indicates strong (G)ARCH effects. For the period before 2009, there are no mean spillover effects between the real returns of OMX and real crude oil returns. Surprisingly after 2009, we can see that real crude oil returns have a significant negative effect on the OMX real return.
Table 4: Mean Equations 2000-2009 Parameters Coefficients Standard
Error φ11,t−2 -0.06609** 0.02053 φ11,t−3 -0.05604** 0.02053 φ11,t−5 -0.06452** 0.02057 φ11,t−7 0.04874* 0.02056 φ22,t−6 -0.05062* 0.02059 φ22,t−7 0.05103* 0.02059 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 29.055 (0.899) Ljung-Box2(10) Q-Stat = 1550.935 (2.20E-16)
11See Appendix Note E for test results 12See Appendix Note F for the PACF plots
Table 5: Mean Equations 2009-2017 Parameters Coefficients Standard
Error φ11,t−1 -0.04498* 0.02445 φ21,t−1 -0.03530* 0.02396 φ11,t−2 -0.06592** 0.02439 **Significance at 1% * Significance at 5% Ljung-Box (10) Q-Stat = 44.965 (0.272) Ljung-Box2(10) Q-Stat = 1122.902 (2.20E-16)
A BEKK(1,1) model is then assumed on the structure of the residuals. If the model is a good fit, the LBQ test should indicate that the standardized residuals are stable over time. The standardized residuals zt have been obtained through Cholesky decomposition13.
Results from Tables 6& 7 confirm that the models have indeed captured all the serial corre-lation and the (G)ARCH effects up to lag 10. For the period between 2000 and 2009 there are no volatility spillover effects between the OMX and real crude oil, with the spillover parameters being insignificant. The volatility of the OMX real returns depends on own past shock and on own past volatilities. The same holds for the real returns on crude oil. However, after 2009, the data indicates strong volatility spillover between crude oil and the Swedish stock market. The conditional OMX volatility is positively affected by real crude oil price shocks from the previous day. This shows that even though the country is becoming highly dependent on renewable energy resources, the price of crude oil still has a significant impact on the fluctuations of its stock market. More so after 2009 even.
Table 6 OMX: BEKK(1,1) 2000-2009 Parameters Coefficients Standard
Error a011** 3.70E-03 5.85E-04 a021 5.63E-04 1.16E-03 a022** 0.004756 9.92E-04 a11** 0.359778 3.23E-02 a21 7.50E-03 4.04E-02 a12 1.50E-02 1.55E-02 a22** 0.225395 2.69E-02 b11** 0.909230 1.87E-02 b21 -3.69E-04 2.05E-02 b12 -5.52E-03 8.52E-03 b22** 0.953638 1.42E-02 **Significance at 1% Ljung-ox(10) Q-Stat = 37.158 (0.598) Ljung-Box(10)2Q-Stat = 39.904(0.159)
Table 7 OMX: BEKK(1,1) 2009-2017 Parameters Coefficients Standard
Error a011* 0.00373 6.57E-04 a021* 0.00159 7.98E-04 a022* 0.00330 1.25E-03 a11** 0.29238 3.13E-02 a21 0.01128 2.94E-02 a12** 0.10789 2.52E-02 a22** 0.31186 6.38E-02 b11** 0.92719 1.92E-02 b21 0.00386 1.35E-02 b12 0.02166 1.47E-02 b22** 0.92688 3.73E-02 **Significance at 1% Ljung-Box(10) Q-Stat = 47.866 (0.184) Ljung-Box(10)2 Q-Stat = 48.464(0.169)
5.1.2 Dutch AEX and Crude Oil
For the Dutch market, the mean equations have been fitted with the use of a VAR(7) for the first period and a VAR(1) for the subsequent one. Serial correlation has been removed from the residuals. The LBQ test indicates heteroskedasticity in the shock series. Results for the mean equations are in Table 8 & 9. There is no mean spillover between the two assets in either of the periods. Even though The Netherlands is a major oil importer, its real stock returns are not affected by real crude oil price movements.
Table 8: AEX Mean Equations 2000-2009 Parameters Coefficients Standard Error
φ11,t−3 -0.07329** 2.05E-02 φ11,t−4 0.08397** 2.04E-02 φ11,t−5 -0.08448** 2.04E-02 φ11,t−6 0.05109* 2.04E-02 φ11,t−8 0.08815** 2.05E-02 φ22,t−6 -0.04895* 2.06E-02 φ22,t−7 0.04671* 2.06E-02 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 28.606 (0.910) Ljung-Box2(10) Q-Stat = 1058.880 (2.20E-16)
Table 9: AEX Mean Equations 2009-2017 Parameters Coefficients Standard Error
φ22,t−1 0.04025* 0.01937
**Significance at 1%
* Significance at 5%
Ljung-Box(10) Q-Stat = 43.583 (0.321) Ljung-Box2(10) Q-Stat = 900.507 (2.20E-16)
As can be observed from Tables 10 & 11, volatility spillover effects are only present in the period of 2009-2017, although not quite as strong as in the case of Sweden, which is somewhat dubious. Nevertheless in both cases, it can be argued that after the financial crisis of 2008, investors became more sensitive to oil price information and to its major explanatory power as to the development of global economies. Therefore crude oil price shocks may have been transmitted more powerfully across markets, despite less and less usage.
Past crude oil volatility has no effect on the present volatility of the AEX with b12and
b21being insignificant. Lagged own past price shocks and volatilities play an important role
in explaining the development of the present volatility. The effect is also persistent with coefficients for both assets being above 0.89 in both periods of time.
Table 10:AEX BEKK(1,1) 2000-2009 Parameters Coefficients Standard
Error a011** 0.00329 3.56E-04 a021 0.00059 9.45E-04 a022** 0.00475 1.00E-03 a11** 0.38026 2.51E-02 a21 -0.03426 2.98E-02 a12 0.00935 1.18E-02 a22** 0.22298 2.56E-02 b11** 0.89618 1.45E-02 b21 0.01784 1.38E-02 b12 -0.00392 6.71E-03 b22** 0.95350 1.41E-02 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 44.809 (0.277) Ljung-Box(10)2Q-Stat = 38.558(0.197)
Table 11:AEX BEKK(1,1) 2009-2017 Parameters Coefficients Standard
Error a011** 0.00306 3.83E-07 a021 0.00166 2.72E-01 a022 0.00327 1.26E-01 a11** 0.29380 3.06E-14 a21 0.00763 8.42E-01 a12* 0.07733 1.51E-03 a22* 0.30898 6.69E-03 b11** 0.93033 2.22e-16 b21 -0.00067 9.72E-01 b12 -0.02284 2.90E-01 b22** 0.92783 2.22e-16 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 32.206 (0.805) Ljung-Box(10)2 Q-Stat = 36.933 (0.609)
5.1.3 Norwegian OSX and Crude Oil
A VAR(7) and a VAR(1) are employed for the two time intervals. The residuals are stable as they seem to behave like white noise. Interesting is to observe in both periods through φ12,t−1, the significant positive effect increases in the real returns of crude oil have on the
real returns of the Norwegian stock market. This shows the confidence that investors gain in its stock market once the real price of crude oil goes up. The effect is felt within a day.
During 2000-2009, all else held constant, a 1% increase in the real return of crude oil leads to a 6% increase in the real return of the OSX. In the second period chosen for the analysis, all else held constant, a 1% increase in the real return of crude oil leads to a 11% increase in the real return of the OSX. Mean spillover effects are much stronger during 2009-2017. Results are shown in Tables 12 & 13.
Table 13: OSX Mean Equations 2009-2017 Parameters Coefficients Standard Error
φ11,t−1 -0.09933** 0.02435
φ12,t−1 0.11233** 0.02435 **Significance at 1%
* Significance at 5%
Ljung-Box(10) Q-Stat = 47.055 (0.127) Ljung-Box2(10) Q-Stat = 1099.169 (2.20E-16)
Table 12: OSX Mean Equations 2000-2009 Parameters Coefficients Standard Error
φ12,t−1 0.06132** 1.43E-02 φ11,t−2 -0.05445* 1.97E-02 φ11,t−4 0.0611* 1.98E-02 φ11,t−5 -0.04008* 1.97E-02 φ11,t−7 0.03738* 1.98E-02 φ21,t−1 0.06083* 2.74E-02 φ21,t−3 0.0819* 2.73E-02 φ22,t−6 -0.04432* 1.98E-02 φ22,t−7 0.04708* 1.99E-02 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 33.931 (0.991) Ljung-Box2(10) Q-Stat = 1189.2590 (2.20E-16)
A BEKK(1,1) model is fitted to capture (G)ARCH effects. The model indicates that volatility spillover effects are only present in the period after 2009. Significant a12 and b12
show that the volatility of OMX strongly dependends on past crude oil price shocks and also on past crude oil volatility. Nevertheless, most of the present information with regards to present volatility of both assets, is given by own lagged price shocks and volatilities with the respective parameters being highly significant and with large coefficients.
Table 14:OSX BEKK(1,1) 2000-2009 Parameters Coefficients Standard
Error a011** 0.00368 0.00048 a021 0.00141 0.00087 a022** 0.00456 0.00100 a11** 0.36452 0.03080 a21 0.00208 0.03430 a12 -0.00131 0.01420 a22** 0.22877 0.02846 b11** 0.90318 0.01784 b21 0.01407 0.01790 b12 -0.00596 0.00876 b22** 0.95009 0.01625 **Significance at 1% Ljung-Box(10) Q-Stat = 40.233 (0.459) Ljung-Box(10)2 Q-Stat = 32.885 (0.239)
Table 15:OSX BEKK(1,1) 2009-2017 Parameters Coefficients Standard
Error a011** 0.00390 1.05E-06 a021** 0.00205 2.69E-04 a022** 0.00344 4.92E-07 a11** 0.22015 9.15E-08 a21 -0.04695 3.60E-01 a12** 0.166251 7.15E-04 a22** 0.31146 4.07E-13 b11** 0.94934 2.22e-16 b21 0.02999 1.68E-01 b12** -0.05908 2.02E-04 b22** 0.91568 2.22e-16 **Significance at 1% Ljung-Box(10) Q-Stat = 39.126 (0.509) Ljung-Box(10)2 Q-Stat =38.299 (0.547)
5.2
Dynamic Conditional Correlations and Variances
To further analyze the time-varying relationship between the stock market and the crude oil market the dynamic conditional variances and covariances have been computed by both models. The full period of 2000-2017 is being considered for the following section. This provides a rich data source for both models.14.
Firstly, interesting to see, is that the BEKK model fails to identify volatility spillover any longer. This shows how sensitive the model is to the time included in the data source. It indicates that structural breaks may need to be taken into account as they can cause quite different coefficients. This leaves room for further research. Nevertheless, the volatility of each asset is highly dependent on own past price shocks and own past volatilities with all the parameters being significant at 1% confidence level.Before implementing the DCC model, the test for constant correlation proposed by Engle & Sheppard (2002) [19] has been performed for all the paired assets. If the correlation between the two is constant, the covariance matrix of the standardized residuals should be the identity matrix I2 and the
series should behave like white noise15. As can be observed from the Appendix Note I, the test clearly rejects the null of constant correlation, with the significance going up hastily as the number of lags is increased. Therefore the DCC model proves to be quite useful.
Figure 3: BEKK vs DCC: OMX/Crude Oil Correlation
Figure 4: BEKK vs DCC: AEX/Crude Oil Correlation
Figure 5: BEKK vs DCC: OSX/Crude Oil Correlation
14See Appendix Note H for tables with coefficients and test statistics 15See Appendix Note I for the specification of the test
Moving forward, the correlations from the BEKK model have been obtained indirectly from the variance-covariance matrix Ht of the residuals. For the DCC model, its coefficients
are significant indicating again that the assumption of constant correlation is unrealistic16. According to the Results in Appendix Note I, significant and close to 1 values for the para-mater θ2 indicate that correlations exhibit strong persistence over time. The correlations
in the plots have been aggregated over the month 17. What can be observed, is that the BEKK consistently assumes that the correlations are much more noisy than what the DCC model shows, with the effect being stronger when considering daily data. The same holds for the conditional variances and covariances18. The BEKK model also implies stronger correla-tions, with averages being higher than the ones estimated by the DCC for all the countries. According to both models, Norway has the highest correlation, barely reaching negative grounds in the past 17 years. Sweden has the weakest links, and The Netherlands stands in the middle. Correlations peak close around the financial crisis of 2008 when the stock markets collapsed and the price of oil went down significantly as well. Mid 2008 the price of oil reached record high at $ 146.69 per barrel on the ICE Futures exchange and by December 2008 it would drop to $32. At that moment, having short positions in crude oil when the two markets exhibit strong correlations, would have greatly decreased the variance of any equity portfolio19. What can also be observed from Figures 3-5 is that the level of correlation for all stock markets has drastically increased after 200920. This may explain the volatility spillover effects identified by the BEKK in the period of 2009-2017, indicating the stronger link between stock markets and crude oil
Table 16: Average Conditional Correlations
Model Sweden The
Netherlands Norway BEKK 0.224 (0.215) 0.258 (0.217) 0.376 (0.185) DCC 0.205 (0.209) 0.223 (0.211) 0.342 (0.160)
*Elements in the brackets indicate the standard deviation of the cor-relations. It can be observed that the BEKK assumes the correlations are more volatile.
On a different note, what can explain the variation in the correlation between the stock market and crude oil? In recent years, successful research has pointed out that stock markets may respond differently to oil price shocks depending on the nature of the shock. Kilian and Park (2009)[40] showed that demand side oil price shocks influence stocks more than the supply side oil price shocks. Demand side shocks themselves can also be split in different sources. Once there is uncertainty about the future supply of oil, companies will start increas-ing their oil reserves, drivincreas-ing its price up. This has been defined as increases in precautionary demand for oil, which exhibits a negative influence on stock prices, due to an unexpected capital outflow. On the other hand, if there is a period of global economic expansion and the demand for oil increases due to this resourceful cycle, than this should cause a positive effect on stock prices. Given this seminal discovery, Filis, Degiannakis and Floros (2011) [25] tried to link the source of the shock with the development of the correlation for several oil export-ing and oil importexport-ing country’s stock markets (although makexport-ing this differentiation between
16See Appendix Note J
17See Appendix Note K for daily plots of the correlations 18See Appendix Note L
19See Appendix Note N to see the role correlation plays in minimizing the variance of a portfolio 20See Appendix Note O for comparative correlation averages
countries did not change their results). Their research confirmed the findings of Kilian and Park. Supply side oil price shocks (OPEC’s productions cuts, hurricanes, strikes etc.) do not seem to have a significant impact on the correlation between oil and stock markets. On the other hand, precautionary demand shocks lead to a negative correlation between crude oil and stock market prices, whereas aggregate demand-side shocks are causing a positive relationship.
Assessing the source of price shock for crude oil is very closely related to the theory of storage. As Weymar (1966) [71] explained in his paper, for consumption commodities, there may be a certain advantage to holding the physical asset rather than holding futures contract on it. For crude oil for example, holding oil in inventory allows for it to be used as an input in the production process, whilst holding a futures contract does not pertain the same benefit. This advantage gets stronger when there’s a risk of a future supply shortage of oil which forces companies to increase their current reserves and thus causing an increase in precautionary demand. The benefit associated to holding the physical commodity is captured by the convenience yield which is given by the cost-of-carry model (under no arbitrage):
F0eyT = S0(r+u)T (19)
where F0 represents the futures price, u represents the storage costs (as a constant
proportion of the spot price) and y is defined as the convenience yield. According to Hull (1988) [32], the convenience yield measures the extent to which the left-hand side of Eq. 19 is less than the right-hand side, that is by how much to futures prices differ from the spot prices. Therefore one could say that the convenience yield reflects the market’s expectations concerning the future availability of the commodity. If the risk of a supply shortage is high, companies will increase their oil reserves now, driving the spot price up, the futures price down and the convenience yield up. This state is known as backwardation. Alternatively, if inventories are high then the risk of crude oil shortage is small, and the convenience yield also tends to be small. Alquist, Bauer and Diez (2014) [2] compute a term structure for the convenience yield and and decompose it through the Principal Component Analysis (PCA) equivalent to the term structure of interest rates. Authors manage to show that increases in the level of the term structure of the convenience yield has a significant and positive explanatory power as to future global oil production. The slope of the convenience yield is found to be significant and positive for describing movements in the aggregate demand for commodities. A positive slope in the convenience yield’s term structure may indicate that companies assign a higher value to future inventories than to today’s inventories as they expect an increase in the aggregate demand in the future. They’ve also found significant predictive power of the level and the slope principals as to the future scarcity of oil inventories. The coefficient of the level component is significant for predicting the movements of oil reserves up to 3 months in the future. The negative coefficient is in line with the theory of storage as an increase in the level of the term structure tends to indicate scarce inventories. A negative sign for the slope coefficient of the convenience yield indicates that for a upward sloping curve, future crude oil stocks will have a higher value in the future than they would today, which in turn means that oil is expected to be more scarce in the future. Therefore, according to Alquist, Bauer and Diez, the term structure of the convenience yield has a predictive power in explaining the different sources of oil price shocks.
Does the convenience yield carry any sort of predictive power with regards to the corre-lation between stock markets and oil? This will not be addressed further as it is beyond the scope of the paper. Although this leaves room for future research as to possible implications for the correlation between the stock market and crude oil.
6
Portfolio Design and Hedging Strategy
From a risk management perspective, it is highly important for an investor or any fi-nancial institution to build an optimal investment portfolio where asset allocation decisions are made based on the most reliable available information. Eliminating risk drivers as one necessitates is also highly dependent on this. Therefore in order to properly reflect one’s attitude towards risk in investment decisions, an accurate estimation of time-varying condi-tional variances and covariances is essential to make such decisions. In the following section, to address and measure how big model risk is, the BEKK and the DCC will be compared and tested for their crude oil price risk hedging effectiveness ability.
6.1
Portfolio Investment Weights
In order to construct an optimal portfolio design, the method of Kroner and Ng (1998) [43] is followed. The minimum-variance portfolio weights will be derived, which provide for the lowest risk, within a mean-variance utility framework. Therefore optimal portfolio weights (OPW) are given by the following21:
wt = h1,t− h12,t h1,t+ h2,t− 2h12,t (20) with wt = 0 if wt < 0 wt if 0 < wt 6 1 1 if wt > 1 (21)
where wt represents the percentage of the portfolio invested in crude oil and 1 − wt
in the stock index. h12,t represents the covariance between real crude oil returns and real
stock index returns. h1,t is the variance of the stock index. Results are in Table 17-19. Most
diversification benefits from investing in crude oil are obtained for the Swedish OSX, as the correlation between the markets is the lowest. The DCC consistently assigns more weight to investing in crude oil than the BEKK. The Pearson correlation between the portfolio weights across different models stands at around 0.69 for all countries, which is quite low. This emphasizes the difference that the choice of the model makes. In the case of The Netherlands, the AEX assigns even 11% higher weight to crude oil than the BEKK.
Table 17: OMX/Crude Oil Model Optimal Portfolio Weights Optimal Hedge Ratios BEKK 0.309 0.201 DCC 0.322 0.199
Table 18: AEX/Crude Oil Model Optimal Portfolio Weights Optimal Hedge Ratios BEKK 0.214 0.187 DCC 0.238 0.177
Table 19: OSX/Crude Oil Model Optimal Portfolio Weights Optimal Hedge Ratios BEKK 0.236 0.312 DCC 0.253 0.289
6.2
Portfolio Hedge Ratios
Like in the paper of Kroner and Sultan (1993) [44], risk-minimizing hedge ratios will also be derived, only this time for a portfolio invested in stock index and crude oil. To minimize the risk of a portfolio that is 1$ long in Asset 1 (stock index), the investor should short $β of Asset 2 (crude oil). The risk minimizing hedge ratio is given as22:
βt−1 =
h12,t
h2,t
(22) This time, the hedge ratios derived from the BEKK model are higher than the ones from DCC. The DCC relatively understimates the covariance between the two assets and this can be seen in the calculations for the OPWs and OHRs. However, which model is most hedge effective and provides the biggest protection?
6.3
Crude Oil Hedge Effectiveness
As Ku et. al (2007) [45] put it, a model can be considered superior in terms of accuracy and effectiveness if it provides the largest variance decrease for any hedged portfolio compared to an unhedged portfolio. Ignoring transaction costs, this can be measured with the help of the Hedge Effectiveness Index given by:
HEI =
[
V arunhedged− V arhedged V arhedged]
(23)
In order to evaluate the performance of the two models the analysis is twofold. On one hand, in-sample hedging effectiveness has been calculated in order to have an idea as to what is the potential risk reduction that can be reached through crude oil hedging. On the other hand, risk managers and hedge fund managers unfortunately cannot foresee the future. Hence they have to rely on the best available forecasts. As a result, an out-of-sample analysis has also been implemented. Starting with the sample data from 2000 to end 2008, next day forecasts have been estimated throughout the whole year of 2009.
Table 20: Stock Index/Crude Oil HEI 2009-2010
In-sample HEI% Out-of-sample HEI%
Model OMX AEX OSX OMX AEX OSX
BEKK 0.369 0.378 0.488 0.326 0.345 0.424 DCC 0.294 0.326 0.387 0.237 0.276 0.356
From Table 20 it can be seen that based on OHRs derived from in-sample data, the BEKK stands out as the most hedge effective for all countries. For an equity portfolio invested in the Norwegian stock exchange, hedging for crude oil price risk could have reduced the portfolio variance without affecting returns by almost 50%. Crude oil hedging has the least impact on the Swedish OMX reducing it’s variance by only 37%. The Netherlands stands closely to Sweden in the middle. Figures 6-9 include a visual representation of the hedge effect on the equity portfolios.
Figure 6: BEKK vs DCC: OMX/Brent Spot Hedge Effectiveness
Figure 7: BEKK vs DCC: AEX/Brent Spot Hedge Effectiveness
Figure 8: BEKK vs DCC: OSX/Brent Spot Hedge Effectiveness
Based on both models, forecasts have been produced for the equity portfolios volatility23. From the graphs in Appendix Note P it can be seen that between the two, the DCC manages
to capture better it’s own fitted conditional variances. The normalized mean squared error (NMRSE) confirms that the DCC does indeed a better job at forecasting it’s own fitted volatilities in most of the cases. But be that as it may, for a more sensible comparison of their forecasting accuracy, a different proxy should be used for the actual volatility. This has been a highly debated topic among researchers as to what is the right volatility proxy (see Hansen (2005), Andersen and Bollerslev (1998), Hansen and Lunde(2003)). In this paper however, the performance of the models will be measured in terms of their crude oil risk hedging effectiveness. The results will then show which model is more realistic. On an out-of-sample basis, the BEKK still outperforms the DCC model. The differences between in-sample and out-of in-sample HEI% are comparable for both models. Both underestimate the volatilities which explain the under-hedging in the out-of-sample framework. Nonetheless, it seems that even though the BEKK relatively underperforms in terms of forecasting it’s in-sample variances, the fact that it assumes more volatility seems to be in line with what happens in reality. The Norwegian stock exchange still enjoys the most reduction in it’s volatility at 42% less.
Figure 9: BEKK vs DCC: OMX/Brent Spot Hedge Effectiveness
Figure 11: BEKK vs DCC: OSX/Brent Spot Hedge Effectiveness
It is obvious that the BEKK model is the most hedge effective in terms of eliminating oil price risk, consistently outperforming the DCC model. For Norway, the BEKK reduces the portfolio variance by 20% more when compared with the DCC. For Sweden the difference is even larger, with the gap encompassing 38%.
7
Conclusion
This paper investigates the dynamic variances and correlations together with the pres-ence of volatility spillover effects between crude oil and three major European stock markets, namely the OMX, AEX, and OSX. Results from the BEKK model indicate that for all three countries, there is a significant risk of volatility transmission across markets. However it is only after the end of 2008 that such phenomena can be observed. Even though countries become increasingly reliant on alternative sources of energy, their stock indexes are still af-fected by crude oil price shocks. This is regardless of the fact that the country is importing or exporting the commodity, or neither. So one could say that markets might be irrational. Such information is of great interest for portfolio managers, risk managers and traders. Therefore to actually see what exactly are the implications, optimal portfolio weights and hedge ra-tios have been calculated. This requires proper estimation of covariances and correlations between markets. Two models have been discussed, in particular the BEKK and the DCC. Results have shown that BEKK assumes more volatility in the way stock markets and crude oil move together over time. This has proven to place BEKK as a better option in terms of accuracy and hedging effectiveness in every case discussed. This is in contrast with the findings of Caporin and McAleer (2008) but are in line with the findings of Gould and Bos (2007). This shows that model risk is not negligible with different models leading to quite different results. Nevertheless, adding crude oil as part of an investment portfolio does miti-gate some of its risk degree, having reduced its variance without affecting returns by up to 49% over the period 2009-2010 based on the in-sample data. Depending on the strength of the correlation, the hedging benefits are also stronger.
Limitations of the paper include the scarce amount of models discussed. Hence the possi-ble presence of asymmetric behavior as a response to oil price shocks has not been addressed at all. Also important to note is that standardized residuals at lags higher than order 10 still exhibit GARCH effects. Therefore there is still room for improvement in the goodness of fit of the models. Given the limited resources R provides for the BEKK modeling, higher lags have not been implemented. The standard GARCH (1,1) for both models was then chosen for comparison reasons. Future research could also aim at how the correlation between crude oil and stock markets may evolve over time given different sources of oil price shocks.
8
Appendix
Note A
h11,t = a2011+a2111,t−12 +2a11a121,t−12,t−1+a21222,t−1+b211h11,t−1+2b11b12h12,t−1+b212h22,t−1 h21,t = a011a012+a21(a1121,t−1+a121,t−12,t−1)+a22(a111,t−12,t−1+a122,t−1)+ b21(b11h11,t−1+ (b12h21,t) + b22(b11h12,t−1+ b12h22,t−1) h21,t = h12,t h22,t = a021a012+ a0222 + a22121,t−1+ 2a22a211,t−12,t−1+ a22222,t−1+ b221h11,t−1+ 2b21b22h21,t−1+ b22h22,t−1Note B
Assume a vector autoregression (VAR) model in the levels of a vector xt:
xt = k
X
i=1
Aixt−i+ ut (24)
For k>1, the model in levels can be rewritten as:
∆xt = Πxt−1+ k−1
X
j=1
Πi∆xt−i+ ut (25)
The matrix Π can be written in terms of the vector or matrix of adjustment parameters α and the vector or matrix of cointegrating vectors β as
Π = αβ0 (26)
If the matrix Π equals a matrix of zeros, that is, Π = 0 then the variables are not cointe-grated and the relationship reduces to the vector autoregression in the first differences
∆xt = k−1
X
i=1
Πi∆xt−i+ ut (27)
The null hypothesis H0 for Johansen’s trace test is that the number of cointegration vectors
is r = r∗ < k, vs. the alternative hypothesis Ha that r = k.
OMX Johansen Cointegration Test 2000-2017
Test 10% 5% 1%
r 61 2.49 6.50 8.18 11.65
AEX Johansen Cointegration Test 2000-2017
Test 10% 5% 1%
r 61 3.39 6.50 8.18 11.65
r = 0 8.67 12.91 14.9 19.19
OSX Johansen Cointegration Test 2000-2017
Test 10% 5% 1%
r 61 2.95 6.50 8.18 11.65
r = 0 4.98 12.91 14.9 19.19
Note C
The Jarque - Bera test is a measure on the sample data to see whether it follows a normal distribution. The test statistic is defined as
J B = n − k + 1
6 (S
2+1
4(C − 3)
2), (28)
where n is the number of observations, S is the sample skewness, C is the sample kurtosis, and k is the number of regressors:
S = µˆ3 ˆ σ3 = 1 n n P i=1 (xi− ¯x)3 (n1 n P i=1 (xi− ¯x)2)3/2 (29) C = µˆ4 ˆ σ4 = 1 n n P i=1 (xi− ¯x)4 (n1 n P i=1 (xi− ¯x)2)2 , (30) ˆ
µ3and ˆµ4are the estimates for the third and fourth central moments, ¯x is the sample mean,
and ˆσ2 is the estimate for the variance.
Note D
The (Multivariate) Ljung-Box test has the following hypothesis: H0: The data is independently distributed (white noise)
H1: The data exhibits serial correlation.
The test statistic has the following form:
Q = n(n + 2) h X k=1 ˆ p2k n − k (31)
where n is the size of the sample, ˆp2k is the sample autocorrelation at lag k, and h is the number of lags. Under H0, the statistic Q follows a χ2h. At the significance level α, the null
Q > χ21−α,h
where χ21−α,h is the 1 − α - quantile of the chi-square distribution with h degrees of freedom. For multivariate distributions the equation looks like this:
M LBQ = T (T + 2) K X j=1 1 T − jtr{C0jC −1 00 C 0 0jC −1 00 } (32)
Note E
OMX Johansen Cointegration Test 2000-2009
Test 10% 5% 1%
r 61 1.74 6.50 8.18 11.65
r = 0 7.72 12.91 14.9 19.19
OMX Johansen Cointegration Test 2009-2017
Test 10% 5% 1%
r 61 0.56 6.50 8.18 11.65
r = 0 12.38 12.91 14.9 19.19
AEX Johansen Cointegration Test 2000-2009
Test 10% 5% 1%
r 61 1.96 6.50 8.18 11.65
r = 0 6.14 12.91 14.9 19.19
AEX Johansen Cointegration Test 2009-2017
Test 10% 5% 1%
r 61 0.73 6.50 8.18 11.65
r = 0 12.4 12.91 14.9 19.19
OSX Johansen Cointegration Test 2000-2009
Test 10% 5% 1%
r 61 0.93 6.50 8.18 11.65
OSX Johansen Cointegration Test 2009-2017
Test 10% 5% 1%
r 61 0.05 6.50 8.18 11.65
r = 0 12.70 12.91 14.90 19.19
Note F
Figure 12: PACF OMX
Figure 14: PACF OSX
Figure 15: PACF Brent Spot
Note G
For a positive definite matrix A, the Cholesky decomposition is of the form:
A=LLT
where L is a lower triangular matrix, also called the Cholesky factor of A, with real and positive diagonal entries and LT is the transpose of L. L may also be considered the square root matrix of A. Given the fact that A is positive definite, the Cholesky decomposition is unique24. When solving equations such as: Ax=b the algorithm goes as follows:
LLTx=b solve LTy=b solve Lx=y 24 http://eti.pg.edu.pl/documents/174618/23783336/Cholesky%20factorization%20of% 20matrices%20in%20parallel%20and%20ranking%20of%20graphs.pdf
To obtain the standardized residuals the calculations go as follows: Ht=LLT
Lzt = t
zt = L−1t
Note H
Table 20: AEX Mean Equations 2000-2017 Parameters Coefficients Standard Error
φ11,t−3 -0.06648** 1.45E-02 φ11,t−4 0.03697* 1.45E-02 φ11,t−5 -0.06462** 1.45E-02 φ11,t−8 0.05833** 1.50E-02 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 19.516 (0.997) Ljung-Box2(10) Q-Stat = 1239.26 (2.20E-16)
Table 21: AEX BEKK(1,1) 2000-2017 Parameters Coefficients Standard
Error a011** 0.00315 2.76E-04 a021* 0.00116 3.91E-04 a022** 0.00435 1.31E-03 a11** 0.34331 1.89E-02 a21 -0.01436 3.05E-02 a12 0.01089 9.41E-03 a22** 0.30645 3.75E-02 b11** 0.91361 1.03E-02 b21 0.00315 1.54E-02 b12 -0.00262 3.90E-03 b22** 0.93231 2.42E-02 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 39.126 (0.509) Ljung-Box(10)2 Q-Stat =38.299 (0.547)
Table 22: OSX Mean Equations 2000-2017 Parameters Coefficients Standard Error
φ12,t−1 0.07948** 1.17E-02 φ21,t−1 0.04416* 1.93E-02 φ21,t−2 -0.04153* 1.39E-02 φ21,t−3 0.0504* 1.78E-02 φ11,t−5 -0.04945* 1.50E-02 φ12,t−6 -0.0295* 1.17E-02 φ21,t−7 0.04249* 1.93E-02 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 16.216 (0.977) Ljung-Box2(10) Q-Stat = 1175.179 (2.20E-16)
Table 23: OSX BEKK(1,1) 2000-2017 Parameters Coefficients Standard
Error a011** 0.00349 3.50E-04 a021 0.00172 4.87E-04 a022** 0.00415 1.16E-03 a11** 0.31209 2.15E-02 a21 0.00159 2.23E-02 a12 0.00838 1.12E-02 a22** 0.30294 3.75E-02 b11** 0.92622 1.14E-02 b21 -0.00015 1.08E-02 b12 -0.00384 4.93E-03 b22** 0.93277 2.33E-02 **Significance at 1% Ljung-Box(10) Q-Stat = 29.854 (0.879) Ljung-Box(10)2 Q-Stat = 37.844 (0.219)
Table 24: OMX Mean Equations 2000-2017 Parameters Coefficients Standard Error
φ11,t−2 -0.05723** 1.46E-02 φ11,t−3 -0.04581** 1.46E-02 φ11,t−5 -0.05981** 1.45E-02 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 38.031 (0.559) Ljung-Box2(10) Q-Stat = 1278.194 (2.20E-16)
Table 25: OMX BEKK(1,1) 2000-2017 Parameters Coefficients Standard
Error a011** 0.00366 5.30E-04 a021* 0.00111 3.93E-04 a022** 0.00437 2.20E-03 a11** 0.32735 2.68E-02 a21 -0.00397 2.77E-02 a12 0.01925 1.17E-02 a22** 0.30441 5.87E-02 b11** 0.92102 1.60E-02 b21 -0.00113 1.54E-02 b12 -0.00541 4.70E-03 b22** 0.93273 3.92E-02 **Significance at 1% * Significance at 5% Ljung-Box(10) Q-Stat = 44.274 (0.296) Ljung-Box(10)2Q-Stat = 31.560 (0.488)
Note I
The constant correlation test (asymptotically χ2(s+1)) of Engle & Sheppard (2001) has the following hypothesis:
H0 : Rt = ¯R
and
Ha : vech(Rt) = vech( ¯R) + β1vech(Rt−1) + β2vech(Rt−2) + ... + βpvech(Rt−p)
The methodology of the test is as follows. Fit the univariate GARCH models on the series, obtain Ht and then the vector of the jointly standardized residuals: zt = H
−1/2
t t where
Ht = DtRD¯ t and zt = yt at the moment. Then zt = ¯R−1/2Dt−1yt.
An artificial regressions is then performed on the outer and lagged product of these residuals plus a constant which will give:
Yt = α + β1Yt−1+ β2Yt−2+ ... + βsYt−s
where Yt = vechu[( ¯R−1/2D−1t yt)( ¯R−1/2D−1t yt)
0
− Ik] and k = 2
If the coefficients of the regression are significantly different from 0, then the null hy-pothesis of constant correlation can be rejected.
