This paper introduces multivariate dynamic copula models to account for the time-varying dependence structure in asset portfolios. We firstly enhance the fexibility of this structure by modeling regimes with multivariate mixture copulas. In our second approach, we derive dynamic elliptical copulas by applying the dynamic conditional correlation model (DCC) to multivariate elliptical copulas. The best-ranked copulas according to both in-sample fit and out-of-sample forecast performance indicate the importance of accounting for time-variation. The superiority of multivariate dynamic Clayton and Student-t models further highlight that positive tail dependence as well as the capability of capturing asymmetries in the dependence structure are crucial features of a well-fitting model for an equity portfolio.
working report - Former investigation (Approximation of Profit-and-Loss Distributions, Part I) introduces the application of the barycentric approximation methodology for evaluating profit-and-loss distributions numerically. Although, convergence of the quantiles is ensured by the weak convergence of the discrete measures, as proclaimed in Part I, recent numerical results have indicated that the approximations of the profit-and-loss distribution are less practical when the portfolio gets a reasonable complexity. This experience has revealed that the weak convergence of the probability measures appears not to be strong enough for evaluating quantiles numerically in a satisfactory way. Thereupon, the authors have focused on information offered by the barycentric approximation but still unused in the algorithmic procedure of Part I. It has been realized that the dual to the derived discrete probability measure helps evaluate the profit-and-loss distribution in a better way. In this Part II, the barycentric approximation technique is outlined and benchmarked with the intention to focus on the dual viewpoint for simplicial refinement. This technique poses no assumption on the risk factor space, except that the variance-covariance matrix of the risk factors exist. Therefore, it is applicable for general multivariate or empirical distributions. Furthermore, the technique provides approximation of the risk profile as well as of the risk factor distribution.Beforehand, various test environments are specified which help illustrate the sensitivity of value-at-risk numbers. These environments are characterized by the probability measure P of the risk factors and a risk profile g which represents the payoff structure of some portfolio. The corresponding numerical results illustrate the sensitivity of value-at-risk with respect to market volatility and correlation of risk factors. This provides information on the model risk one is exposed to within the value-at-risk approach.
The incorporation of single-factor interest rate models within the stochastic programming methodology is investigated and applied to multiperiod investment. Barycentric approximation is used for discretizing the stochastic factors and for generating scenario trees which take the various term structure movements into account. It is shown that employing the Vasicek model for the instantaneous rate process preserves convexity of the stochastic multistage program and, hence, guarantees information on the accuracy of the approximate investment strategies. To the contrary, the convexity of the program cannot be assessed if the square root process due to Cox-Ingersoll-Ross is used for the instantaneous rate. In this case, the approximate investment policies and their associated interest surplus may be accepted as estimates. Numerical results for 8-period and 6-period investment problems are discussed.
working report - Value functions (risk profiles) of financial instruments and the real distributions of risk factors are not available in analytically closed forms. These components have to be approximated. In this work, a new approach for risk measurement is introduced. The underlying methodology is based on the utilization of extremal measures for approximating the P&L distribution. A special class of "extremal measures" is employed which exploits the monotonicity of price sensitivities entailed by convexity. Clearly, in case the value functions have monotonous derivatives, the payoff-functions are convex or concave depending on whether a position is held short or long. The incorporated extremal measures provide approximations for both risk factor distribution and risk profiles, and allow for deriving an adequate approximation of the P&L distributions, in particual for appealing VaR-estimates. The basics of this approach are presented and first numerical results are tested against the currently apllied VaR-approaches and the simulation benchmarks established earlier in Allen (1994).
The design and analysis of efficient approximation schemes is of fundamental importance in stochastic programming research. Bounding approximations are particularly popular for providing strict error bounds that can be made small by using partitioning techniques. In this article we develop a powerful bounding method for linear multistage stochastic programs with a generalized nonconvex dependence on the random parameters. Thereby, we establish bounds on the recourse functions as well as compact bounding sets for the optimal decisions. We further demonstrate that our bounding methods facilitate the reliable solution of important real-life decision problems. To this end, we solve a stochastic optimization model for the management of non-maturing accounts and compare the bounds on maximum profit obtained with different partitioning strategies.
Institutionelle Investoren - Die gegenwärtig in der Praxis eingesetzten
Modelle zur Portfolioselektion basieren meistens auf deterministischen
Volatilitäten und Korrelationen sowie auf friktionslosen
Rahmenbedingungen. Damit wird ein hohes Modellrisiko eingegangen, das sich - wie die jüngsten Vermögensentwicklungen vieler institutioneller Investoren belegen - in einer ungenügenden Performance niederschlägt.
This paper shows how a mean variance criterion can be applied to a multi period setting in order to obtain efficient portfolios in an asset and liability context. The optimization model allows for rebalancing activities, transaction costs, stochastic volatilities for both assets and liabilities. Furthermore, a general framework for the projection of pension fund liabilities as well as for the generation of asset returns is given. In a further step the dynamics of the liability maturity structure is modeled as customized index, whose volatility and correlation with asset returns become integral components of the applied regime switching approach. The numerical results illustrate the diversification of the assets and its risk return pattern in dependency of the liability dynamics.
