Portfolio selection with commodities under conditional asymmetric dependence and skew preferences

Transcription

1 Portfolio selection with commodities under conditional asymmetric dependence and skew preferences Carlos González-Pedraz a, Manuel Moreno b and Juan Ignacio Peña a a Department of Business Administration, Universidad Carlos III de Madrid 893 Getafe, Spain b Department of Economic Analysis and Finance, Universidad de Castilla La Mancha 457 Toledo, Spain Current version: January 4, Abstract: This paper addresses the portfolio selection problem when commodities are included in the investment opportunity set. We model the joint assets distribution within a flexible multivariate setting including conditional multivariate copulas with time-varying parameters. We allow for conditional means, variances, skewness, and extreme outcomes both in the dependence structure and in the marginal distributions. We generalize the investor s standard mean-variance preferences allowing for skew preferences. The empirical application is based on S&P 5 stock index and two commodities: crude oil and gold, using weekly data from June 99 to September. The results are compared with respect to the standard Markowitz portfolio allocation. We find significant economic differences between the standard and the generalized approach in the optimal portfolio s structure. Keywords: Portfolio selection, commodity futures, conditional copulas, and skew preferences. JEL classification: C46, G, and G3. addresses of the authors: C. González-Pedraz, cugonzal@emp.uc3m.es; M. Moreno, manuel.moreno@uclm.es; and J.I. Peña, ypenya@eco.uc3m.es.

2 . Introduction Commodities attract the attention of many financial investors, who perceive them as a new asset class. Historically, fully collateralized futures contracts on commodities have shown Sharpe ratios close to those of equities and zero or even negative correlation with stock returns (Bodie and Rosansky (98)), Erb and Harvey (6) and Gorton and Rouwenhorst (6)). Therefore, according to the traditional portfolio theory, commodities should increase diversification when included in equity portfolios and therefore they may help to reduce portfolio risk for a given expected return level. Specifically, an equity investor should include commodities in his portfolio, for example by taking positions in commodity futures, to enhance the portfolio s risk/return profile. Based on this economic argument, in the last years there has been a large investment flow into commodity indexes and related instruments (see Tang and Xiong ()). Despite the growing focus on commodities as investments vehicles, most published works are based on the standard unconditional Markowitz s framework. This is unlikely to be the appropriate setting for commodities because of their returns specific distributional characteristics (Casassus and Collin-Dufresne (5); Kat and Oomen (7); Routledge, Seppi, and Spatt ()). This paper contributes to the literature presenting a more general and realistic model to be used in the optimal portfolio selection process. If commodities belong to the set of investment opportunities, we focus on two points in order to determine the optimal portfolio choice: first, the joint distribution of commodities and stock returns and second, the preferences of the investors trading with commodities and stocks. With respect to the first point, this paper provides a novel approach to model the join distribution of returns using a flexible copula model that allows for skewness and time-varying investment opportunities. Concerning the second point, we consider that investor s utility function depends on the first three moments of the portfolio returns distribution generalizing the traditional Markowitz s set-up. Thus, we introduce aversion to downside risk in order to stress the relevance of using a flexible, possibly asymmetric, multivariate model. The reason to focus on downside risk is the well-known loss aversion argument that is pervasively found in the behavioral finance literature and which has received considerable empirical support (Shefrin ()). On the basis of our theoretical approach, in the empirical part of the paper we employ weekly data on the S&P5 stock index and in two futures contracts (crude oil and gold) for the period 99 to to examine three primary issues, () Are there asymmetric dependence among commodities and equity returns () Are there discrepancies in the optimal portfolio allocations between our approach and the traditional Markowitz benchmark? (3) Do these discrepancies translate into economically relevant performance differences among methods? (4) Finally, what is the key factor explainig these discrepancies?. We find the following results: () We find that there is evidence of the presence of skewness and tail dependence () We find substantial discrepancies among the

3 optimal weights of traditional portfolio choice and time-varying strategies. (3) These differences in portfolio weights translate generally in better Sharpe ratios. (4) The key factors explaining the differences in allocations and performance are, by this order, (i) the proper specification of timevarying univariate behavior in terms of volatility, skewness, and fat tails, (ii) the dynamics in the dependence among marginal functions, and (iii) the asymmetric and extreme dependence. The remainder of the paper is organized as follows. Section formulates the investor s preferences and the optimal portfolio problem. Section 3 presents the joint conditional model and the estimation methodology. Section 4 shows: first, the basic statistical properties of the commodity futures returns we employ in the paper; presenting empirical evidence supporting the elliptical hypothesis, asymmetric dependence, and time-varying correlation; then, presents in two stages the results of the estimation and allocation process.finally, Section 5 concludes.. Portfolio choice with commodity futures and skew preferences In this section, we present the allocation problem when commodity futures are part of the investment opportunity set. From the point of view of the methodology, we believe that including commodity futures in the investment opportunity set implies two main effects on the portfolio s choice specification. The first of these effects has to do with the statistical properties of commodity returns, which make them different from other assets. It is well studied that commodity returns exhibit a strong departure from a pure Gaussian distribution. In particular, it is well documented the presence of jumps (positive and negative) in the data generating process of commodity returns (Casassus and Collin-Dufresne (5)). As a result, when adding to a traditional portfolio (mainly formed by fixed-income and equity), commodity assets constitute a significant source of not-null skewness. Thus, their presence seems to increase the importance of including the third moment in the portfolio selection problem. The second effect has to do with the way futures exchanges operate. No money changes hands when futures are sold or bought; just a margin is posted to settle gains and losses. Since there is not any upfront payment when taking a position in futures contracts, it is not clear how to define the rate of return. That is, considering the collateral of futures contracts will affect the way their rates of returns are computed, and also the budget constraint of the investor s decision process. We can argue that these stylized statistical properties are related with the fundamentals behind the price formation of commodity prices. For instance, the presence of jumps can be explained by the highly convex relationship between commodity prices and inventories (Routledge, Seppi, and Spatt ()). As a consequence, shocks in supply, demand, or both can have a very large impact on prices, specially, when the market supply-demand balance is particularly tight.

4 For simplicity, and following the common approach in this literature, we assume that the futures position are fully collateralized; that is, the initial margin deposit corresponds to the whole notional value of the futures contract. Taking into account the latter considerations, our specification of the investor s problem with commodity futures is an extension of previous models of portfolio selection with skewness but where rates of returns and budget constraints are determined by the particular characteristics of commodity futures contracts. Thus, we reconcile the view of collateralized futures contracts as investments with the theories of portfolio selection with higher moments. Formally, our portfolio choice problem can be formulated in terms of an investor who maximizes her expected utility at period t + h by building at time t a portfolio that includes two group of assets: one group with n commodity futures contracts, and other group with N n spot contracts, such as, a risk-free asset or a risky equity index. We assume that the initial margin deposit of a fully collateralized futures position indicates the initial capital investment related with that position (long or short). Therefore, the gross return of a short or long position in the commodity futures contract i at time t + h is given by ( + Rt+h) i = F t+h i Ft i ( + R f t+h ), i = N n +,..., N. () where Ft i and Ft+h i are the futures settlement prices at times t and t + h, and ( + Rf t+h ) is the gross return on the risk-free asset in the h-period, which is the interest earned on the initial margin deposit. Finally, given a set of investment opportunities, the wealth at time t+h (of an investor with unit initial wealth) will be given by the gross return of the portfolio over the h-period, +R t+h (ω t ), defined as + R t+h (ω t ) = + N j= ω j t (exp(rj t+h ) ), () where ω t = (ωt,..., ωt N n, ωt N n+,..., ωt N ) is the vector of portfolio weights (for spot and futures contracts), chosen at time t; and r j t+h = log( + Rj t+h ) is the continuously compounded return of asset j over the h-period. The maximization problem is subject to the next non-linear constraint that, at time t, the sum of portfolio weights must be equal to one, that is, N n j= ω j t + N ωt i i=n n+ =. (3) Note we take the absolute value of the weights of commodity futures as both long and short positions in futures require the same initial margin. Thus, a short position in futures contracts cannot finance an increase in the holdings of other assets. 3

5 Now, we present the utility function that we use to describe the investor s preferences. The investor s utility function depends on the first three moments of the portfolio returns. Then, the investor s objective consists in choosing a portfolio allocation that maximizes the expected portfolio return in equation (), penalized for: (i) risk (that is, variance of portfolio s returns) and (ii) negatively skewed portfolio returns. Hence, the expected value of this utility function at time t can be written formally as E t [U( + R t+h (ω))] = E t [R t+h (ω)] φ v Var t [R t+h (ω)] + φ s Skew t [R t+h (ω)] (4) where E t ( ), Var t ( ), and Skew t ( ) are the conditional first three moments of the portfolio returns; and φ v and φ s are the aversion coefficients to risk (variance) and extreme negative outcomes (negative skewness), respectively. At the same time, by adding aversion towards negative skewness, we reflect the possibility that an investor might accept a lower expected return if there is a chance of high positive skewness, in the form of large positive jumps for example. Or from the other point of view, our investor is eager to decrease the chance of large negative deviations, which could reduce their portfolio value and their future consumption. 3. A multivariate conditional copula model with asymmetry In this section, we describe the model for the multivariate distribution of risky assets returns, which are represented by the vector r t IR d, where d N is the number of risky factors. In our approach, we model asset returns by means of a conditional multivariate copula model with asymmetry. A multivariate model is constructed by the marginal functions that describe each univariate variable and a joint dependence function, that establishes the relationships between these variables. Using copula functions, we can disentangle the dependence structure between individual assets returns from the different behavior of their univariate processes. So, this methodology gives us the enough flexibility to model, first, the marginal functions of each return in an independent way, and, then, the copula that governs the whole dependence structure. For a thorough exposition of copula theory, we refer to Embrechts, Lindskog, and McNeil (3) and Nelsen (6), among others. Extensions to conditional copulas can be found in Patton (6b). We follow these works in our presentation here. Formally, a copula C is defined as a d-dimensional distribution function on the unit interval [, ] d, that is, a joint distribution with standard uniform univariate distribution functions. Consider a multivariate conditional distribution F t (rt,..., rt d ) constituted by marginal distributions F i,t (rt), i where i =,..., d, and the subscript t denotes that joint and marginal distributions are conditioned on the information set F t available at time t. Then, there exist a function C 4

6 that maps the domain [, ] d towards the interval [, ] called the copula, such that, ) F t (rt,..., rt d ) = C t (F,t (rt ),..., F d,t (rt d ). (5) Using the expression (5), any copula C t can be employed to define a joint distribution F t (r t ) with the desired marginal distributions F,t,..., F d,t.these results constitute the d-dimensional version for conditional copulas of Sklar (959) Theorem. 3.. Modeling the univariate processes In order to define our flexible multivariate distribution, we begin with the formal description of the marginal distributions of the assets returns, r t. In general, the marginal distribution function consists of a conditional parametric distribution and a particular dynamic econometric specification for the conditional distribution parameters. Our choice of the univariate conditional distribution is based on previous work of Harvey and Siddique (999) and Jondeau and Rockinger (3), which allow the distribution parameters to depend on their lagged values and other regressors, in a similar way to ARCH-like processes. Formally, our univariate conditional model is expressed as follows: p r t = µ + β X t + Φ j r t j + D t z t (6) j= σ i,t = α i + α i,+ σ i,t z i,t l {zi,t >} + α i, σ i,t z i,t l {zi,t <} + α i σ i,t, (7) z i,t f(z i,t ν i,t λ i,t ) (8) ( ) ν i,t = Λ ν δ i + δ i z i,t + δλ i ν (ν i,t ), (9) ( ) λ i,t = Λ λ ζ i + ζ i z i,t + ζλ i (λ i,t ). () λ In equation (6), we are defining the dynamics of the conditional mean, where z t = (z i,t ) i=,...,d denotes the vector of innovations; µ IR d, D t is the diagonal matrix of conditional volatilities, that is, D t = diag(σ,t,..., σ d,t ), β is a m d matrix, and Φ j are square matrices of dimension d and constant coefficients. In equation (7), we express the dynamics of the conditional variance, where σ i,t = Var (r i,t). In order to guarantee a positive and stationary volatility, the parameters in (7) have to satisfy the next constraints: α i >, α i,+, α i,, α i, α i + αi, + αi,+ <. In our univariate models, the conditional means of the returns are linear functions of previous values of the returns up to certain lag, p, and a vector of m market factors, X t, that act as It is just a direct application of the concept of conditional copula (Patton (6b), Theorem ) to a multivariate case (Nelsen (6), Theorem..9). 5

7 exogenous regressors. Previous returns allow for possible serial correlation in returns, while the exogenous regressors allow for possible predictability in returns. As explicative factors, we include the momentum, basis, risk-free rate, default spread, and growth in open interest; all of them have been used to predict commodity and stock returns by previous studies (Hong and Yogo ()). As it is traditionally used in financial literature, in order to consider a possible leverage effect in the variance of returns (Black (976)), this paper employs a leveraged GARCH process (Campbell and Hentschel (99)) for the conditional variance. For this process, the variance of returns at time t is a linear function of the value of variance at time t, and reflects a non-linear or asymmetric response to positive and negative shocks occurred in the previous period. In order to assess the economic importance of higher moments and conditional dependence we need a univariate conditional distribution, f(zt), i that can capture these features. For that purpose, we employ as conditional distribution in equation 8 the generalized or skewed t distribution f(z i,t ν i,t, λ i,t ), introduced by Hansen (994) with time-varying degrees of freedom, ν i,t, and skewness parameter, λ i,t in equation (9) and (). In Appendix A, we define the functional form of this distribution, together with its density and inverse function. In equations (9) and (), δ i, δi, δi, ζi, ζi and ζi are constant parameters, and Λ ν,λ( ) denotes the modified logistic map designed to keep the degrees of freedom, ν i,t, and the asymmetry parameter λ i,t in their respective domains of definition, that is in the intervals (, ) and [4, ), respectively. This general specification allows for time-varying skewness and kurtosis and nests some particular models. For instance, if the asymmetry parameter goes to zero, we obtain the symmetric Student s t distribution; if degrees of freedom tend to infinity, we converge to the Gaussian case. 3.. Modeling the copula functions A useful corollary from Sklar s theorem states that the copula C t of a joint distribution can be obtained explicitly in terms of the multivariate distribution F t and its marginal functions by evaluating the joint distribution F t at Fi,t (u i), where Fi,t is the inverse or quantile function of the univariate distribution F i,t, u i belongs to [, ], and i =,..., d. That is, C t (u,..., u d ) = F t (F,t (u ),..., F d,t (u d)). () Equation () shows how we can extract implicit copulas from multivariate distributions. In particular, in this paper, we will consider three dependence functions: the Gaussian copula, the t copula, and the skewed t copula. They are obtained applying equation () from the multivariate Gaussian, Student s t, and generalized hyperbolic skewed t distributions, respectively. We choose these copulas because they can model different types of dependence between extreme realizations, as can be seen in the contour plots and probability density functions included in Figure. Appendix B provides the functional forms of these copulas. 6

8 It is well known that the Gaussian copula is completely determined by the correlation coefficients. In this case, the correlation matrix defines a linear dependence between the individual variables, so the Gaussian copula does not allow for extreme or asymmetric dependence. The t copula also depends on a correlation matrix and is also defined by an extra parameter, the degrees of freedom, which describes to what extent there is extreme dependence between the individual processes. Figure shows that the t copula assigns more weight to the extreme values than the Gaussian one. However, both are elliptically symmetric copulas. Finally, the skewed t copula can capture asymmetric and extreme dependence between the changes in the state variables, by means of a vector of asymmetry parameters with dimension equal to the number of marginal distributions.through the vector of asymmetry parameters, the skewed t copula assigns more dependence to one tail than the other. We want to remark that these implicit copulas have straightforward generalizations for more than two dimensions. This is a very useful property, specially for skewed t copula, since the other asymmetric alternatives, such as the Archimedean copulas, do not have a so clear or simple multivariate specifications. 3 Thus, inspired by the dynamic conditional correlation model of Engle () and extending it to other type of dependence beyond the Gaussian copula, we propose the following dynamics for the correlation parameters: ρ ij t = Λ [,] ( ω + ω M M z i,t m z j,t m + ω m= M M m= ) x i,t m x j,t m + ω ρ ij t. () where z i,t = F i, (u i ), x i,t = z i,t l {zi,t <}, and M is the number of previous lags we take into account as forcing variable Estimation Dividing the multivariate distribution into two components, one for the marginal distributions and the other for the dependence structure, allows us to implement a two-step estimation of the parameters employed in the model. In a first step, we obtain the maximum likelihood estimates for the individual processes; and, then, the parameters estimates related with the copula function. 4 Formally, this procedure can be expressed as follows. Suppose we have a sample of log-returns 3 Following some assumptions in the dependence structure, other works use multivariate generalizations of Archimedean copulas. For example, Garcia and Tsafack () and Stefanova (9) employ parsimonious specifications of these copulas. 4 This two-stage estimation is the natural choice when dealing with copulas or certain multivariate GARCH models. At this respect, we refer to Engle and Sheppard (), Jondeau and Rockinger (6), Patton (6a,b), and Lee and Long (9), among others. 7

9 {r} t=,...,t IR d and we want to find the set of parameters estimates θ that maximizes the loglikelihood function of the multivariate model parameterized by the parameter set θ. That is, we have the next log-likelihood function T L(θ ; r,..., r T ) = log f t (r t ; θ), (3) t= where f t (r t ; θ) is the probability density function of the multivariate model parameterized by θ. Therefore, maximum likelihood estimates parameters satisfy that θ = arg max L. However, according to Sklar s theorem summarized in equation (5), we can split the set of parameters, θ, into the parameters of the d marginal distributions θ M = (θ,m... θ d,m ), and the set of parameters of the copula function θ C. Therefore, following equation (5), the conditional density function associated with the conditional joint distribution function C t (u t,..., u d t ; θ C ) is given by f t (r t ; θ) = d F t (r t ; θ) r t... rd t d = f i,t (rt i ; θ i,m ) c t (u i t,..., u d t ; θ C ), (4) i= where u i t = F i,t (r i t ; θ i,m ) are the marginal distributions, f i,t (r i t ; θ i,m ) are the corresponding marginal density functions, and c t (u i t,..., u d t ; θ C ) = d C t (u t ;θ C ) is the copula density function, being u u t = t... ud t (u t,..., u d t ). According to equation (4), we see that the log-likelihood function in equation (3) can be divided into two terms: d T T L(θ M, θ C ; r t,..., r T ) = log f i,t (rt i ; θ i,m ) + log c t (u t,..., u d t ; θ C ) i= t= t= d = L i (θ i,m ; r,..., r T ) + L C (θ C ; θ M, r,..., r T ). (5) i= where L i and L C are the log-likelihood functions for the marginal model i and for the copula, respectively. So, we can adopt a two-stage procedure to estimate sequentially θ i,m = arg max L i and θ C = arg max L C. 5 This estimation method is also known in the literature as the inference functions for margins method. Patton (6a) shows that one-step maximum likelihood estimators and the two-stage estimators are equally asymptotically efficient. 6 However, some remarks have to be considered. First, related with the univariate processes, the quality of the copula estimation will strongly depend on the goodness-of-fit of the assumed parametric functions for the marginal distributions. Second, related with the copula, for symmetric and skewed t copulas, we need to find extra parameters apart from the correlation coefficients. In these cases, since the objective function often falls in 5 Appendix B provides the explicit expressions of the copula densities, c t (u t ; θ C ), we consider. 6 In addition, Patton (6a) shows that this two-step procedure allows for estimating a multivariate model for time series of different lengths with a weakly loss of efficiency. 8

10 local maximums, convergence difficulties may arise when maximizing directly the log-likelihood function. We avoid this problem by using a global optimization approach, such as simulation annealing, as a robustness check Empirical Application 4.. Data For the empirical application, we focus on two commodities: crude oil and gold. We use weekly settlement prices from the WTI crude oil futures traded at NYMEX and the futures contract of.995 fine gold bar traded at COMEX. For both commodities, we employ the futures contracts with highest liquidity, measured by trading volume. We use the S&P 5 index to compute the market portfolio returns. All prices are obtained from Datastream. Our sample ranges from June, 99 to September 8, ; with a total of 99 weekly observations. We use data from June 99 to June 6 to fit the model. The data from June 6 to September is employed for the out-of-sample exercice. Table reports the main statistics for the returns of these assets. Normality of the unconditional distribution of returns is strongly rejected in the whole period, as well as, in both sub-samples. The table shows particularly large differences between the in-sample and out-of-sample data, specially, in terms of unconditional skewness and correlation coefficients between commodities and the index. Thus, this situation may represent a good bench test to check the flexibility of the time-varying copula model we propose. Since we are interested in checking the presence of asymmetric interactions between commodities and equity, we apply a test of elliptical symmetry and compute the exceedence correlation function. Finally, we analyze the dependence structure between returns in the time domain by means of their respective cross-correlograms. In Figure, we show qq-plots for the McNeil, Frey, and Embrechts (5)statistic, see also (Li, Fang, and Zhu, 997). The curvature in the qq-plot suggests that the vector of returns is not elliptically distributed. We also implement a Kolmogorov-Smirnov test with this data. The resulting statistic is equal to.74, larger than the critical value (.48), therefore, we reject the elliptical hypothesis. In a second analysis, we pay attention to the possibly asymmetric interactions between the assets returns of the portfolio. Following Longin and Solnik (), Ang and Chen (), and Patton (4), we use the exceedence correlation for measure this asymmetry in the dependence. 7 Demarta and McNeil (5) proposes some alternatives to the direct maximization of the likelihood function. 9

11 In Figure 3 we plot exceedence correlation for each pair of returns of the portfolio as a function of the quantile q. The shapes of these plots depend on the bivariate distribution between the given returns, that is allow us to size the degree of asymmetry in the dependence function of the returns. According to plots in Figure 3, the dependence structure between pairs of returns seems to be strongly asymmetric. 4.. Estimation of the conditional copula model In this section we present the estimation results for the previously proposed model for the multivariate dependence between commodities returns and the equity index returns. Table 3 presents estimates for the conditional univariate model with asymmetry, in which variance, tail-behavior, and skewness are time-varying. The copula model requires an appropriate specification of the marginal distributions. To evaluate the goodness-of-fit of the models for marginal distributions we conduct the test proposed by Diebold, Gunther, and Tay (998). This test checks if the probability integral transforms behave like i.i.d. uniformly distributed variables between and ; otherwise, the copula model will be misspecified. In Table 4 we present the results for the skewed t distribution model with time-varying parameters. This model passes the goodness-of-fit test for all the univariate returns. In Figure 4, we compare graphically the goodness-of-fit of the time-varying skewed t univariate model with a TARCH model with Gaussian innovations. The univariate asymmetric model seems to perform better. In the case of commodities returns, the estimates for the volatility dynamics show that positive returns have a stronger effect on variance than negative ones, i.e. we find a negative leverage effect. However we do not find leverage effects in equity returns. The degrees of freedom and skewness parameter present both a rather persistent dynamics as expected. Second, we focus on the dependence function. For that purpose, we estimate different types of copulas. Since the univariate or marginal distributions are different, in the sense that they have different degrees of freedom, and different asymmetry parameters, we obtain a multivariate distribution that does not correspond strictly to a Gaussian or t copula. In the literature, this multivariate distribution is termed meta-copula. Henceforth, we use the term copula in the general sense. Table 5 shows the maximum log-likelihood for the three copulas considered, that is, for the Gaussian, t, and skewed t copulas, and for their time-varying extensions. The log-likelihood ratio tests are also reported. According to the later comparison criteria, we can see that the skewed t copula collects a significant part of the remaining asymmetry in the dependence structure between commodity returns and the index returns. The presence of tail-dependence is also not-negligible even taking into account that we have modeled part of the tail behavior in the heavy-tail univariate distributions.

12 Summing up, the skewed t copula provide the more informative measure of the dependence between commodities and equity-index returns, even, when we model the skewness and excess kurtosis in the marginal distributions. Therefore (possibly time-varying) tail thickness and asymmetry are key factors which are not taken into account in the standard Markowitz approach. The extent to which these factors have significant economic impact in the portfolio choice decision is addressed in the next section Optimal portfolio results Now we proceed with the analysis of the optimal portfolio choice. The investment set is formed by three risky assets: oil, gold, and an S&P 5. We consider seven different strategies: (i) an equally weighted portfolio, (ii) the unconditional multivariate Gaussian distribution, (iii) the multivariate TARCH with Constant Conditional Correlation (CCC) and Gaussian innovations, (iv) the multivariate TARCH with Dynamic Conditional Correlation (DCC) and Gaussian innovations (v) the conditional Gaussian copula (vi) the conditional t copula, and (vii) the conditional skewed t copula. The latter three conditional copulas have generalized Student s t marginal distribution functions with time-varying moments. For each period, we use the models estimated previously to forecast the moments of the assets returns and then we maximize the expected utility using Monte Carlo simulations. We get for each period, the portfolio weights, and therefore the optimal portfolio return series. In Table 6, we report summary statistics and investment ratios of the realized optimal portfolio for each strategy and for different parameterizations of the risk aversion. Table 7 shows the performance measures (in basis points per week) of these realized portfolio returns over the allocation (out-of-sample) period against the equally weighted portfolio. In Table 8, we show the optimal weights summary statistics over the out-of-sample period for the most general model, the conditional skewed t copula model. In Table 9 we compare the optimal weights summary statistics among different strategies for a fixed level of risk aversion. A graphical representation of these results is presented in Figures 7 and 8. According to the investment ratios and the performance measures, we obtain the next conclusions: first, the univariate higher moments seem to be a key feature in terms of better investment ratios and out-of-sample performance of the optimal portfolios; second, the dynamics in the dependence among marginal functions has also an important role in the allocation decision process, and, finally, the asymmetric and extreme dependence modeled in conditional t copulas make a difference for certain levels of risk aversion. As robustness check, we repeat the analysis and compute the investment ratios and performance measures of the optimal portfolios when a risk-free asset is present in the set of investment opportunities (see Table ). Results are supportive to the latter conclusions. Finally, a last ro-

13 bustness check is carried out. It consists in a test of superior performance ability of the seven strategies as a whole (see Hansen (5)). With this test we accomplish a reality check with a stationary bootstrap using the realized utility as performance (or loss function). Results are shown in Table, and supports the conclusion that benchmarks models such as the equally weighted and the unconditional multivariate Gaussian strategies do not perform as well as the best competing alternative strategy. 5. Conclusion This paper addresses the portfolio selection problem when commodities are included in the investment opportunities set. This paper contributes to the literature presenting a more general and realistic model to be used in the optimal portfolio selection process. If commodities belong to the set of investment opportunities, we focus on two the crucial issues that condition the optimal portfolio choice: first, the joint distribution of commodities and stock returns and second, the preferences of the investors trading with commodities and stocks. With respect to the first point, we model the joint assets distribution within a flexible multivariate setting including conditional multivariate copulas with time-varying parameters. We allow for conditional means, variances, skewness, and extreme outcomes both in the dependence structure and in the marginal distributions. Concerning the investor s objective function, we generalize the investor s standard mean-variance preferences allowing for skew preferences. Despite the growing focus on commodities as investments vehicles, few works have focused on their returns specific distributional characteristics and their consequences on the portfolio selection problem. The empirical application is based on S&P 5 stock index and two commodities: crude oil and gold, using weekly data from June 99 to September. With respect to the marginal distributions, we find evidence against a symmetric behavior for the conditional distribution of univariate commodity and equity-index returns. In the case of commodities returns, the estimates for the volatility dynamics show that positive returns have a stronger effect on variance than those returns in the case of the equity index. In addition, the estimates for degrees of freedom and skewness parameter present a rather persistent dynamics. Concerning the joint dependence, we find that the skewed t-copula provide a more informative measure of the dependence between commodities and equity-index returns, even, when we have modeled the skewness and excess kurtosis in the marginal distributions. Including models consistent with these results can be of crucial importance in the portfolio selection problem, since skewness and the dependence structure are both key elements in the investor s objective function. The results can be compared with respect to the standard Markowitz portfolio

14 allocation Since, modeling both individual skewness and asymmetric dependence will enable the investor in commodities to better control for undesired negatively skewed portfolio returns, we find significant economic differences between the standard and the generalized approach in the optimal portfolio s structure and performance. Technical appendices A. The generalized t univariate distribution function In this appendix, we summarize some useful results concerning the skewed t univariate distribution introduced by Hansen (994), and posteriorly analyzed by Jondeau and Rockinger (3), among others. The next presentation is based on these works. Consider a variable z that follows a skewed t distribution, then its density function, f(z), is defined as [ ( ) ] (ν+)/ b c + b z+a ν λ z < a/b, f(z) = [ ( ) ] (ν+)/ b c + b z+a ν +λ z a/b, (A.a) where < ν < +, and < λ <. The constants a, b and, c are given by ( ) Γ ν+ a = 4λ c (ν )/(ν ), b = + 3λ a, and c = ( π(ν ) Γ ν ), (A.b) where Γ(z) is the Gamma function defined as Γ(z) = t z e t dt for R(z) >, and satisfies the next recurrence formula: Γ(z + ) = zγ(z) = z! (for more details about the Gamma function, see (Abramowitz and Stegun, 965)). According to Jondeau and Rockinger (3, Proposition ), we can express the cumulative distribution function (c.d.f.) of the skewed t distribution, F (p), as a function of the c.d.f. of the Student s t distribution as follows ( ( λ) T ν F (p) = ( + λ) T ) bp+a ν λ ) bp+a +λ ( ν ν p < a/b, p a/b, (A.) where T (p) is the standard Student s t distribution defined as ( ) p Γ ν+ ( ) (ν+)/ T (p) = πν Γ ( ) ν + x dx. (A.3) ν Finally, the inverse of the distribution function of a skewed t requires the inversion of the standard Student s t distribution, and it can be expressed as [ F b ( λ) ( ) ] ν ν (u) = T u λ, ν a [ b ( + λ) ( ) ] ν ν T u+λ +λ, η a 3 u < λ, u λ.. (A.4)

15 where u [, ]. We need the later expression for simulating our univariate random variables using the inverse c.d.f. technique and a generator of uniform random numbers. B. Copula functions In this section, we describe the copula functions that we use to model the dependence between asset returns in our portfolio selection problem. Since the skewed t copula is less common in the financial literature, we describe it in more detail than the Gaussian and t copulas. We follow mainly Embrechts, Lindskog, and McNeil (3) and Demarta and McNeil (5). B.. Gaussian copula Let u i [, ] and u = (u,..., u d ) be a set of numbers. Then, we define the multivariate Gaussian copula as C Gauss (u; P) = Φ P (Φ (u ),..., Φ (u d )) where Φ denotes the inverse of the univariate standard normal distribution function, and Φ P (B.) denotes the joint distribution function of the d-variate standard normal distribution with correlation matrix P. 8 The density function of the multivariate Gaussian copula is given by c Gauss (u; P) = P / exp ( ( ) ) z P II d z where z i = Φ (u i ) and z = (z,..., z d ). (B.) B.. t copula The unique t copula of a d-variate t distribution can be expressed as C t (u; ν, P) = t ν,p (t ν (u ),..., t ν (u n )) (B.3) where t ν is the inverse function of a standard univariate Student s t distribution with ν degrees of freedom and t ν,p is the joint distribution function of a standard d-variate t distribution with ν degrees of freedom and correlation matrix P. The density function of the t-copula is given by ( ) c t (u; ν, P) = P Γ ν+d / Γ ( ) ν d ) d Γ ( ν+ di= ( + z i ν ) (ν+)/ ( + z P z ν ) (ν+d)/ (B.4) 8 MMF Notice that the copula function of a d-variate normal distribution with a certain mean and variance is the same than the copula of a d-variate standard normal distribution. The reason is that any strictly increasing transformation of the marginal distributions leaves the copula function invariant. This argument is valid for any copula. 4

16 where z i = t ν (u i ). B.3. Skewed t copula A special class of multivariate normal mixtures distribution is defined when the mixing variable follows an inverse gamma distribution (see Demarta and McNeil (5)). In that case, the resulting mixture distribution is named general hyperbolic skewed t distribution with asymmetry vector γ. We refer to its copula as a skewed t copula, which is defined as C skew (u; ν, γ, P) = t ν,γ,p (t ν,γ (u ),..., t ν,γ d (u n )) (B.5) where t ν,γ,p is the d-variate skewed t distribution with correlation matrix P, degrees of freedom ν, and asymmetric vector γ = (γ,..., γ d ), and t ν,γ i is the inverse function of the i-th univariate margin of the skewed t distribution with asymmetry parameter γ i. The density function of the d-variate skewed t copula is given by K c skew ν+d (u; ν, γ, P) = ( (ν + z P z)γ P γ di= K ν+ P / ( Γ( ν ) ν/ ) ( (ν + z P z)γ P γ ) ν+d ( ) ( ) ν+ (ν + zi )γ i (ν + zi )γ i e z iγ i ) d di= ( + z i ν ) (ν+)/ ( + z P z ν e z P γ ) (ν+d)/. (B.6) where z i = tν,γ i (u i ) and K η is the modified Bessel function of the second kind with order η defined as K η (x) = y η e x/(y+y )dy. (B.7) In Abramowitz and Stegun (965), we can find the main properties of the modified Bessel functions, some of them are useful for the numerical implementation of these functions. References Abramowitz, M., and I. A. Stegun (965). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, New York. Ang, A., and J. Chen (). Asymmetric correlations of equity returns, Journal of Financial Economics, 63(3), Black, F. (976). Studies of stock price volatility changes, in Proceedings of the 976 meetings of the business and economic statistics section, American Statistical Association, pp

17 Bodie, Z., and V. I. Rosansky (98). Risk and return in commodity futures, Financial Analysts Journal, 36(3), Campbell, J., and L. Hentschel (99). No news is good news:: An asymmetric model of changing volatility in stock returns, Journal of Financial Economics, 3(3), Casassus, J., and P. Collin-Dufresne (5). Stochastic convenience yield implied from commodity futures and interest rates, Journal of Finance, 6(5), Demarta, S., and A. J. McNeil (5). The t copula and related copulas, International Statistical Review, 73(), 3. Diebold, F. X., T. A. Gunther, and A. S. Tay (998). Evaluating density forecasts with applications to financial risk management, International Economic Review, 39(4), Embrechts, P., F. Lindskog, and A. McNeil (3). Modelling dependence with copulas and applications to risk management, in Handbook of heavy tailed distributions in finance, ed. by S. T. Rachev, pp Elsevier/North-Holland, Amsterdam. Engle, R. F. (). Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, Journal of Business and Economic Statistics, (3), Engle, R. F., and K. Sheppard (). Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH, NBER Working Paper. Erb, C. B., and C. R. Harvey (6). The strategic and tactical value of commodity futures, Financial Analysts Journal, 6(), Garcia, R., and G. Tsafack (). Dependence structure and extreme comovements in international equity and bond markets, Journal of Banking and Finance, 35, Gorton, G., and K. G. Rouwenhorst (6). Facts and fantasies about commodity futures, Financial Analysts Journal, 6(), Hansen, B. E. (994). Autoregressive conditional density estimation, International Economic Review, 35(3), Hansen, P. R. (5). A test for superior predictive ability, Journal of Business and Economic Statistics, 3(4), Harvey, C. R., and A. Siddique (999). Autoregressive conditional skewness, Journal of Financial and Quantitative Analysis, 34(4),

18 Hong, H., and M. Yogo (). What does futures market interest tell us about the macroeconomy and asset prices?, Journal of Financial Economics, forthcoming. Jondeau, E., and M. Rockinger (3). Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements, Journal of Economic Dynamics and Control, 7(), (6). The copula-garch model of conditional dependencies: an international stock market application, Journal of International Money and Finance, 5(5), Kat, H. M., and R. C. A. Oomen (7). What every investor should know about commodities. Part II: multivariate return analysis, Journal of Investment Management, 5(3), 6 4. Lee, T. H., and X. Long (9). Copula-based multivariate GARCH model with uncorrelated dependent errors, Journal of Econometrics, 5(), 7 8. Li, R. Z., K. T. Fang, and L. X. Zhu (997). Some qq-probability plots to test spherical and elliptical symmetry, Journal of Computational and Graphical Statistics, 6(4), Longin, F., and B. Solnik (). Extreme correlation of international equity markets, Journal of Finance, 56(), Mardia, K. (97). Measures of multivariate skewness and kurtosis with applications, Biometrika, 57(3), McNeil, A. J., R. Frey, and P. Embrechts (5). Quantitative risk management: concepts, techniques, and tools. Princeton University Press, New Jersey. Nelsen, R. B. (6). An introduction to copulas. Springer, New York. Patton, A. J. (4). On the out-of-sample importance of skewness and asymmetric dependence for asset allocation, Journal of Financial Econometrics, (), (6a). Estimation of multivariate models for time series of possibly different lengths, Journal of Applied Econometrics, (), (6b). Modelling asymmetric exchange rate dependence, International Economic Review, 47(), Politis, D., and J. Romano (994). The stationary bootstrap, Journal of the American Statistical Association, pp Routledge, B. R., D. J. Seppi, and C. S. Spatt (). Equilibrium forward curves for commodities, Journal of Finance, 55(3),

19 Shefrin, H. (). Beyond greed and fear: Understanding behavioral finance and the psychology of investing. Oxford University Press, New York. Sklar, A. (959). Fonctions de répartition à n dimensions et leurs marges, Publications de l Institut Statistique de l Université de Paris, 8, 9 3. Stefanova, D. (9). Dynamic hedging and extreme asset co-movements, Working Paper VU University Amsterdam. Tang, K., and W. Xiong (). Index investment and financialization of commodities, NBER Working Paper. 8

20 Table. Descriptive univariate statistics for oil, gold, and equity weekly returns This table reports sample statistics of the weekly returns for the crude oil futures (NYMEX), gold futures (COMEX), and equity index (SP5). The full sample period ranges from June 99 to September, and include 56 observations. The in-sample period runs from June 99 to June 6 (836 observations), and the out-of-sample period from June 6 to September ( observations). Mean, Std. Dev., Min., Max., and VaR 5% are expressed in weekly percentage. JB and KS refer to the Jarque-Bera and Kolmogorov-Smirnov normality test statistics. LB() and LM() are the Ljung-Box and the Lagrange-Multiplier test statistics, both conducted using lags, for testing the presence of autocorrelation in returns and squared returns, respectively. p-values are reported in parentheses. Full sample period In-sample period Out-of-sample period Interval -Jun-99 / 8-Sep- -Jun-99 / -Jun-6 -Jun-6 / 8-Sep- Assets oil gold equity oil gold equity oil gold equity Mean (.34) (.8) (.45) (.85) (.353) (.45) (.99) (.4) (.766) Std. Dev Min Max VaR 5% Skewness (.) (.97) (.) (.) (.9) (.5) (.489) (.5) (.) Kurtosis (.) (.) (.) (.) (.) (.) (.) (.) (.) JB (.) (.) (.) (.) (.) (.) (.) (.) (.) KS p-val (.) (.) (.) (.) (.) (.) (.) (.) (.) LB() (.) (.4) (.) (.6) (.) (.7) (.) (.73) (.3) LM() (.) (.) (.) (.) (.) (.) (.) (.) (.) 9

21 Table. Descriptive multivariate statistics for oil, gold, and equity weekly returns This table reports the descriptive multivariate statistics of crude oil, gold, and equity index weekly returns. The full sample period ranges from June 99 to September, and include 56 observations. The in-sample period runs from June 99 to June 6, and the out-of-sample period from June 6 to September. Panel A shows the sample correlation for each period. In Panel B, we present the results of the Engle and Sheppard () test for constant correlation. p-values are reported in parentheses and shows the probability of constant correlation. In Panel C, we report the results of the Mardia (97) test of joint normality, which is based on the multivariate measures of skewness and kurtosis, denoted by s 3 and k 3. In Panel C, we also report the results of the Li, Fang, and Zhu (997) test of ellipticity. In Panel D, we present the estimates of the upper and lower tail dependence parameters, τ U and τ D, of the symmetrized Joe-Clayton (SJC) copula (Patton (6b)) for each pair of asset returns: oil-gold (o-g), oil-equity (o-e), and gold-equity (g-e). Full sample period In-sample period Out-of-sample period -Jun-99 / 8-Sep- -Jun-99 / -Jun-6 -Jun-6 / 8-Sep- Panel A: Unconditional Correlation i oil gold equity oil gold equity oil gold equity ρ i,oil (.) (.9) (.) (.79) (.) (.) ρ i,gold Panel B: Test of dynamic correlation (.698) (.) (.4) s lags stat (p-val.) (.) (.) (.) (.9) (.3) (.48) (.3) (.) (.4) Panel C: Mardia s test and test of ellipticity s 3 k 3 T ellip s 3 k 3 T ellip s 3 k 3 T ellip coeff stat (p-val.) (.) (.) (.) (.) (.) (.) (.) (.) (.) Panel D: Tail dependence estimates of SJC copula for each pair of returns o-g o-e g-e o-g o-e g-e o-g o-e g-e τ U (p-val.) (.57) (.8) (.) (.67) (.88) (.74) (.773) (.368) (.684) τ L (p-val.) (.) (.3) (.93) (.3) (.9) (.767) (.) (.) (.4)

22 Table 3. Results for the marginal distribution models This table reports the maximum likelihood parameter estimates of the marginal distribution model for oil, gold, and equity-index returns with generalized Student s t distribution and time-varying moments. Parameters of the mean, variance, degrees-of-freedom, and asymmetry equations are defined in equations (6), (7), (9), and (), respectively. The results corresponds to the estimation period from June 99 to June 6 (836 observations). The p-values of the estimates appear in parentheses and are computed using the robust standard errors. logl is the sample log-likelihood of the marginal distribution model. oil gold equity coeff. (p-val.) coeff. (p-val.) coeff. (p-val.) mean equation µ ( ).4 (.36).474 (.37).3 (.37) basis -.3 (.6) momentum -.75 (.53) MA(r f ) -. (.3) r t -.8 (.) r t -.4 (.3) r t 3.5 (.) variance equation α ( ).5 (.75).3 (.).6 (.) α +.85 (.).83 (.). (.956) α.6 (.3).9 (.89).45 (.) α.9 (.).87 (.).89 (.) degrees-of-freedom equation δ. (.).5 (.4) -. (.54) δ (.576) (.).38 (.49) δ 5.54 (.7) (.) (.383) δ.998 (.).9 (.).966 (.) asymmetry parameter equation ζ ( ).85 (.77).88 (.) -.5 (.99) ζ (.33) -.93 (.4) (.798) ζ.8 (.7).59 (.) (.34) ζ.998 (.). (.).98 (.) logl,55.4,6.7,49.6