Value‐at‐Risk bounds with two‐sided dependence information

Value‐at‐Risk (VaR) bounds for aggregated risks have been derived in the literature in settings where, besides the marginal distributions of the individual risk factors, one‐sided bounds for the joint distribution or the copula of the risks are available. In applications, it turns out that these improved standard bounds on VaR tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute VaR bounds when besides the marginal distributions of the risk factors, two‐sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds that we derive by means of the Monge–Kantorovich transportation duality. In several applications, we illustrate that two‐sided dependence information typically leads to strongly improved bounds on the VaR of aggregations.     

Option prices and risk‐neutral densities for currency cross rates

The theoretical relationship between the risk‐neutral density (RND) of the euro/ pound cross rate and the bivariate RND of the dollar/euro and the dollar/pound rates is derived; the required bivariate RND is defined by the dollar‐rate marginal RNDs and a copula function. The cross‐rate RND can be used by banks, international businesses, and central bankers to assess market expectations, to measure risks, and to value options, without relying on over‐the‐counter markets, which may be either non‐existent or illiquid. Empirical comparisons are made between cross‐rate RNDs estimated from several data sets. Five one‐parameter copula functions are evaluated and it is found that the Gaussian copula is the only one‐parameter copula function that is ranked highly in all of the comparisons we have made. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:324–360, 2010     

Dynamic hedging with futures: A copula‐based GARCH model

In a number of earlier studies it has been demonstrated that the traditional regression‐based static approach is inappropriate for hedging with futures, with the result that a variety of alternative dynamic hedging strategies have emerged. In this study the authors propose a class of new copula‐based GARCH models for the estimation of the optimal hedge ratio and compare their effectiveness with that of other hedging models, including the conventional static, the constant conditional correlation (CCC) GARCH, and the dynamic conditional correlation (DCC) GARCH models. With regard to the reduction of variance in the returns of hedged portfolios, the empirical results show that in both the in‐sample and out‐of‐sample tests, with full flexibility in the distribution specifications, the copula‐based GARCH models perform more effectively than other dynamic hedging models. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1095–1116, 2008     

Long‐period Positions in Multi‐sectoral Cobb–Douglas Economies

The present paper considers a constant returns to scale non‐joint production economy with one primary factor (labour). It follows from the non‐substitution theorem that macroeconomic variables are completely determined by the wage–profit distribution in a long‐period position. We investigate the special case in which production in each sector as well as consumption are characterized by means of Cobb–Douglas functions. Despite the non‐linearity, this parametrization allows us to compute sectoral and macroeconomic variables in simple and intuitive equations. As noted in earlier literature, many variables behave differently than would be expected on basis of a one‐sectoral Solow growth model.     

Empirical evidence on the dependence of credit default swaps and equity prices

We investigate the common practice of estimating the dependence structure between credit default swap prices on multi‐name credit instruments from the dependence structure of the equity returns of the underlying firms. We find convincing evidence that the practice is inappropriate for high‐yield instruments and that it may even be flawed for instruments containing only firms within a sector. To do this, we model individual credit ratings by univariate continuous time Markov chains, and their joint dynamics by copulas. The use of copulas allows us to incorporate our knowledge of the modeling of univariate processes, into a multivariate framework. However, our test and results are robust to the choice of copula. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:695–712, 2009     

Cross‐commodity analysis and applications to risk management

The understanding of joint asset return distributions is an important ingredient for managing risks of portfolios. Although this is a well‐discussed issue in fixed income and equity markets, it is a challenge for energy commodities. In this study we are concerned with describing the joint return distribution of energy‐related commodities futures, namely power, oil, gas, coal, and carbon. The objective of the study is threefold. First, we conduct a careful analysis of empirical returns and show how the class of multivariate generalized hyperbolic distributions performs in this context. Second, we present how risk measures can be computed for commodity portfolios based on generalized hyperbolic assumptions. And finally, we discuss the implications of our findings for risk management analyzing the exposure of power plants, which represent typical energy portfolios. Our main findings are that risk estimates based on a normal distribution in the context of energy commodities can be statistically improved using generalized hyperbolic distributions. Those distributions are flexible enough to incorporate many characteristics of commodity returns and yield more accurate risk estimates. Our analysis of the market suggests that carbon allowances can be a helpful tool for controlling the risk exposure of a typical energy portfolio representing a power plant. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:197–217, 2009     

The realized volatility of FTSE‐100 futures prices

Five‐minute returns from FTSE‐100 index futures contracts are used to obtain accurate estimates of daily index volatility from January 1986 to December 1998. These realized volatility measures are used to obtain inferences about the distributional and autocorrelation properties of FTSE‐100 volatility. The distribution of volatility measured daily is similar to lognormal while the volatility time series has persistent positive autocorrelation that displays long‐memory effects. The distribution of daily returns standardized using the measures of realized volatility is shown to be close to normal, unlike the unconditional distribution. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:627–648, 2002     

Estimating the Cost‐of‐Equity Capital Using Empirical Asset Pricing Models

Empirical asset pricing models seek to capture characteristic‐based patterns in the cross‐section of average stock returns. I propose a new approach for constructing these models, and investigate its performance with respect to estimating the cost‐of‐equity capital. Using a model that accounts for the cross‐sectional relation between five characteristics and average stock returns, I obtain cost‐of‐equity estimates that outperform those produced by the Fama‐French five‐factor model in out‐of‐sample tests. Because the proposed approach builds directly on standard cross‐sectional regression techniques, it provides complete flexibility in choosing the firm characteristics used to formulate the cost‐of‐equity estimates.     

Dealing with downside risk in a multi‐commodity setting: A case for a “Texas hedge”?

This study analyzes the problem of multi‐commodity hedging from the downside risk perspective. The lower partial moments (LPM₂)‐minimizing hedge ratios for the stylized hedging problem of a typical Texas panhandle feedlot operator are calculated and compared with hedge ratios implied by the conventional minimum‐variance (MV) criterion. A kernel copula is used to model the joint distributions of cash and futures prices for commodities included in the model. The results are consistent with the findings in the single‐commodity case in that the MV approach leads to over‐hedging relative to the LPM₂‐based hedge. An interesting and somewhat unexpected result is that minimization of a downside risk criterion in a multi‐commodity setting may lead to a “Texas hedge” (i.e. speculation) being an optimal strategy for at least one commodity. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:290–304, 2010     

Multi‐period hedge ratios for a multi‐asset portfolio when accounting for returns co‐movement

This study presents a model to select the optimal hedge ratios of a portfolio composed of an arbitrary number of commodities. In particular, returns dependency and heterogeneous investment horizons are accounted for by copulas and wavelets, respectively. A portfolio of London Metal Exchange metals is analyzed for the period July 1993–December 2005, and it is concluded that neglecting cross correlations leads to biased estimates of the optimal hedge ratios and the degree of hedge effectiveness. Furthermore, when compared with a multivariate‐GARCH specification, our methodology yields higher hedge effectiveness for the raw returns and their short‐term components. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:182–207, 2008     

A copula‐based regime‐switching GARCH model for optimal futures hedging

The article develops a regime‐switching Gumbel–Clayton (RSGC) copula GARCH model for optimal futures hedging. There are three major contributions of RSGC. First, the dependence of spot and futures return series in RSGC is modeled using switching copula instead of assuming bivariate normality. Second, RSGC adopts an independent switching Generalized Autoregressive Conditional Heteroscedasticity (GARCH) process to avoid the path‐dependency problem. Third, based on the assumption of independent switching, a formula is derived for calculating the minimum variance hedge ratio. Empirical investigation in agricultural commodity markets reveals that RSGC provides good out‐of‐sample hedging effectiveness, illustrating importance of modeling regime shift and asymmetric dependence for futures hedging. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:946–972, 2009     

Jumping hedges: An examination of movements in copper spot and futures markets

Price risk is an important factor for both copper purchasers, who use the commodity as a major input in their production process, and copper refiners, who must deal with cash‐flow volatility. Information from NYMEX cash and futures prices is used to examine optimal hedging behavior for agents in copper markets. A bivariate GARCH‐jump model with autoregressive jump intensity is proposed to capture the features of the joint distribution of cash and futures returns over two subperiods with different dominant pricing regimes. It is found that during the earlier producerpricing regime this specification is not needed, whereas for the later exchange pricing era jump dynamics stemming from a common jump across cash and futures series are significant in explaining the dynamics in both daily and weekly data sets. Results from the model are used to under‐take both within‐sample and out‐of‐sample hedging exercises. These results indicate that there are important gains to be made from a time‐varying optimal hedging strategy that incorporates the information from the common jump dynamics. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:169–188, 2006     

Density forecasts of crude‐oil prices using option‐implied and ARCH‐type models

The predictive accuracy of competing crude‐oil price forecast densities is investigated for the 1994–2006 period. Moving beyond standard ARCH type models that rely exclusively on past returns, we examine the benefits of utilizing the forward‐looking information that is embedded in the prices of derivative contracts. Risk‐neutral densities, obtained from panels of crude‐oil option prices, are adjusted to reflect real‐world risks using either a parametric or a non‐parametric calibration approach. The relative performance of the models is evaluated for the entire support of the density, as well as for regions and intervals that are of special interest for the economic agent. We find that non‐parametric adjustments of risk‐neutral density forecasts perform significantly better than their parametric counterparts. Goodness‐of‐fit tests and out‐of‐sample likelihood comparisons favor forecast densities obtained by option prices and non‐parametric calibration methods over those constructed using historical returns and simulated ARCH processes. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:727–754, 2011     

RISK‐AVERSION,OPTIMAL LEVERAGE AND THE INVESTMENT–UNCERTAINTY RELATION

We develop a partial equilibrium dynamic model in which firms are risk‐averse. We analyse the determinants of the investment–uncertainty relationship by means of numerical techniques. When firms can borrow ‘outside’ resources at the riskless rate, an increase in price volatility depresses investment for realistic parameter values. In our model, portfolio considerations play an important role. When the marginal revenue of capital becomes more uncertain, the risk‐averse firm's owners reduce their ‘short position’ in the risk‐free asset, thus diminishing the firm's debt level. The contraction in leverage reduces the expected returns on investment because the expected marginal revenue product is higher than the user cost of capital. In turn, the reduction in expected yields tends to depress investment.     

Asymmetric pricing of implied systematic volatility in the cross‐section of expected returns

Assuming a symmetric relation between returns and innovations in implied market volatility, Ang, A., Hodrick, R., Xing, Y., and Zhang, X. (2006) find that sensitivities to changes in implied market volatility have a cross‐sectional effect on firm returns. Dennis, P., Mayhew, S., and Stivers, C. (2006), however, find an asymmetric relation between firm‐level returns and implied market volatility innovations. We incorporate this asymmetry into the cross‐sectional relation between sensitivity to volatility innovations and returns. Using both portfolio sorting and firm‐level regressions, we find that sensitivity to VIX innovations is negatively related to returns when volatility is rising, but is unrelated when it is falling. The negative relation is robust to controls for other variables, suggesting only the increase in implied market volatility is a priced risk factor. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:34–54, 2011     

Hedging and value at risk: A semi‐parametric approach

The non‐normality of financial asset returns has important implications for hedging. In particular, in contrast with the unambiguous effect that minimum‐variance hedging has on the standard deviation, it can actually increase the negative skewness and kurtosis of hedge portfolio returns. Thus, the reduction in Value at Risk (VaR) and Conditional Value at Risk (CVaR) that minimum‐variance hedging generates can be significantly lower than the reduction in standard deviation. In this study, we provide a new, semi‐parametric method of estimating minimum‐VaR and minimum‐CVaR hedge ratios based on the Cornish‐Fisher expansion of the quantile of the hedged portfolio return distribution. Using spot and futures returns for the FTSE 100, FTSE 250, and FTSE Small Cap equity indices, the Euro/US Dollar exchange rate, and Brent crude oil, we find that the semiparametric approach is superior to the standard minimum‐variance approach, and to the nonparametric approach of Harris and Shen (2006). In particular, it provides a greater reduction in both negative skewness and excess kurtosis, and consequently generates hedge portfolios that in most cases have lower VaR and CVaR. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:780–794, 2010     

Time variation in the correlation structure of exchange rates: high‐frequency analyses

The correlation structure of asset returns is a crucial parameter in risk management as well as in theoretical finance. In practice, however, the true correlation structure between the returns of assets can easily become obscured by time variation in the observed correlation structure and in the liquidity of the assets. We employed a time‐stamped high‐frequency data set of exchange rates, namely, the US$–deutsche mark and the US$–yen exchange rates, to calibrate the observed time variation in the correlation structure between their returns. We also documented time variation in the liquidity structure of these rates. We then attempted to link the observed correlations with the liquidity via an application of an illiquid trading model first developed by Scholes and Williams (1976). We show that the observed correlation structure is strongly biased by the liquidity and that it is possible to effect at least a partial rectification of the otherwise downward‐biased observed correlation. The rectified sample correlation is, therefore, more appropriate for input into models used for forecasting, option pricing, and other risk management applications. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:127–144, 2001     

Minimum‐variance futures hedging under alternative return specifications

It is widely believed that the conventional futures hedge ratio, is variance‐minimizing when it is computed using percentage returns or log returns. It is shown that the conventional hedge ratio is variance‐minimizing when computed from returns measured in dollar terms but not from returns measured in percentage or log terms. Formulas for the minimum‐variance hedge ratio under percentage and log returns are derived. The difference between the conventional hedge ratio computed from percentage and log returns and the minimum‐variance hedge ratio is found to be relatively small when directly hedging, especially when using near‐maturity futures. However, the minimum‐variance hedge ratio can vary significantly from the conventional hedge ratio computed from percentage or log returns when used in cross‐hedging situations. Simulation analysis shows that the incorrect application of the conventional hedge ratio in crosshedging situations can substantially reduce hedging performance. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:537–552, 2005     

THE TWO‐SECTOR VON THÜNEN ORIGINAL MARGINAL PRODUCTIVITY MODEL OF CAPITAL; AND BEYOND

This article considers a two‐sector model of scalar capital from the perspectives of smoothly differentiable neoclassical technologies and also non‐differentiable technologies based on discrete alternative Leontief–Sraffa techniques. The analysis shows that in these Thünen‐like scenarios without joint production the real wage and the interest rate are necessarily in an inverse Ricardian tradeoff. This complements the findings of Samuelson and Etula (2006 , Japan and the World Economy, 18, pp. 331–356) and completes the analysis of single homogeneous scalar capital.     

Option pricing with a non‐zero lower bound on stock price

Black, F. and Scholes, M. (1973) assume a geometric Brownian motion for stock prices and therefore a normal distribution for stock returns. In this article a simple alternative model to Black and Scholes (1973) is presented by assuming a non‐zero lower bound on stock prices. The proposed stock price dynamics simultaneously accommodate skewness and excess kurtosis in stock returns. The feasibility of the proposed model is assessed by simulation and maximum likelihood estimation of the return probability density. The proposed model is easily applicable to existing option pricing models and may provide improved precision in option pricing. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:775–794, 2005