New research directions in engineering economics--modeling dependencies with copulas.

Understanding and quantifying dependencies among variables often arises in stochastic capital investment and real option analysis. In such modeling situations, Pearson product moment linear correlation is widely used as a dependence measure. Linear correlation has several limitations. More recently, copulas have been used in financial economics and insurance to model dependencies. We contribute to the engineering economy literature by introducing copulas to model dependent risks that will enhance research and practice. We demonstrate the usefulness of copula-based sampling in simulation of project risk and regression analysis for forecasting. We also discuss potential copula research in engineering economic analysis.

INTRODUCTION

Engineering economics deals with the evaluation of engineering proposals. In the design of systems, products, and services, engineers face two interconnected environments: the physical environment and the economic environment (Thuesen and Fabrycky 1984). The physical environment is to a greater extent more certain than the economic environment, which depends on the collective behavior of individuals (market forces). Risks and uncertainties associated with physical designs and markets cannot be ignored but have to be evaluated. Engineering economists have applied and/or developed tools such as decision tree analysis (DTA), chance constrained programming, stochastic programming, stochastic dominance, Bayesian analysis, and Monte Carlo and simulation to deal with risks and uncertainty arising in physical and economic environments. More recently, real option analysis (ROA) have been added to the tool kit to value managerial flexibilities in strategic investments (Park and Herath, 2000).

Financial engineering deals with the design and analysis of financial instruments (contracts) for risk management in complex markets. Financial engineering has looked at risk and uncertainty in terms of efficient transfer and sharing of risks through hedging and insurance products. More specifically, financial engineering deals with decisions regarding pricing, hedging, trading, and portfolio management. The basic set of skills required for assessing risks and uncertainties in engineering economics and financial engineering is grounded in probability theory, advanced statistics, data analysis, Monte Carlo and simulation techniques, and mathematical programming. In fact, they both share a common skill base. However, there is one fundamental difference in the approaches to deal with risk and uncertainty. An engineering economics approach is based on operational hedging (Birge, 2007)--construction of physical facilities (e.g., a grain silo to hedge price risk), technical uncertainty resolution in Bayesian context, phased investment analysis using DTA, assessment of project risks using Monte Carlo simulation, and project comparison using the stochastic dominance approach. Engineering economists have shown relatively less attention to management of risks and uncertainties in engineering projects that arise from material procurement in commodity markets and foreign exchange via hedging and insurance strategies.

Copulas are now being used in financial economics and insurance to model dependencies that arise in pricing financial instruments and insurance products for hedging and managing risks and uncertainties. Copulas have been applied to model extreme events, in value-at-risk models, pricing derivatives (credit and spread options), and insurance contracts. In this article, we introduce copula techniques to enhance the research and practice of engineering economics. Correlation (linear dependence) is found to be insufficient to describe the complete dependence structure between random variables (Cherubini and Luciano, 2002; Frees and Valdez, 1998; Amershi et al., 2006; Armstrong et al., 2004; Andersen and Sidenius, 2004; Klugman and Parsa, 1999; Bennett and Kennedy, 2004; Hull and White, 2004; Clemen and Reilly, 1999). More specifically, correlation as a dependence measure has the following drawbacks. First, correlation is the correct dependence measure only if the marginal and joint distributions of the random variables are normal. (1) Empirical research in finance and insurance indicates that marginal distributions of asset price returns, lifetimes, and financial losses seldom fall in this class (Ross, 1999; Frees and Valdez, 1998; Amershi et al., 2006). There is ample evidence that many financial return series exhibit tail dependencies (heavy-tailed and skewed). Second, the widely used Pearson product moment linear correlation ignores any nonlinear dependencies. Third, correlation is not invariant under strictly increasing transformation of the random variables.

Understanding and quantifying dependencies among variables is often required in the economic analysis of engineering projects. In risk simulation, comparison of risky projects, and real option analysis, one has to often deal with two or more random variables interacting simultaneously. For example, in order to develop cash flow models it is necessary to model sums, differences, products, or quotients of random variables. If the random variables are statistically independent, it is easy to calculate their first and second moments (i.e., expected value and variance). There are also known procedures to compute statistical moments when the random variables are not statistically independent. Often, however, more details are required. Instead of the simple measures, mean and variance, one would like to know the exact probability distribution for the sums, differences, products, and quotients. In some instances, such as when marginal distributions are normal, obtaining the joint distribution is easy. This is not the case if the two random variables are statistically dependent and two marginal distributions are of two different types (i.e., X is normal and Y is uniform or continuous empirical distribution, etc.) Copulas provide an alternate method to construct multivariate distributions being given the marginal distributions of any type.

How has the issue of dependence among random variables been addressed in the engineering economics literature when we need to aggregate variables? Pearson product moment correlation (linear correlation; i.e., r) is widely used as a dependence measure. When it is difficult to obtain the exact distributions for linear combinations of random variables, analysts use statistical moments (Park and Sharp-Bette, 1990). In risk simulation, for example, often independent random variables are used, but then the models may suffer from not considering dependencies (Kelton et al., 2004; Kelton and Law, 2000; Park, 1993). There are number of ways to consider dependencies in traditional risk simulation. A multivariate distribution that captures the dependency (using correlation) may be fitted and used for generating random variates from the distribution. In other instances, one may be able to specify a formula that captures the association between random inputs either using a regression model or fitting an exact equation (Kelton et al., 2004; Park and Sharp-Bette, 1990).

While there is hardly any research on the above issues in engineering economic analysis, limitations of correlation as a dependence measure and the need to combine different types of marginal distributions affect risk simulation, comparison of risky projects, and real options in both theory and practice. In this article, we introduce the copula-based technique to model dependent risks in engineering economic analysis. Since this is the first article to introduce copulas to engineering economics, the article is written primarily as a tutorial using a risk simulation example. Since copulas are new to the readers of the engineering economics, we have attempted to introduce this technique using standardized notation and a textbook-style example for intuitive clarity. Our article makes several key contributions. First, we introduce copulas to the engineering economic literature and demonstrate the copula methodology in two application areas: risk simulation and forecasting. We build upon the sampling procedures for dependent variables by using copulas to capture nonlinear dependencies and illustrate how copulas can be used to define regressions for forecasting. Second, we identify and discuss potential research areas in engineering economics where copulas can be used, opening up several new avenues of research with implications for practice. The article is organized as follows. The following section presents commonly used classical dependence measures, covariance and correlation. In the next section, we show how to construct copulas and discuss other measures of dependence that capture nonlinearities. Then we provide a survey of some well-known copula examples and procedures to identify the best fit copula, followed by an illustration of copula-based simulation and quantile regression. The article concludes by discussing potential copula applications in engineering economic analysis.

CLASSICAL MEASURES OF DEPENDENCE