Dan Rosen

Credit risk optimization with Conditional Value-at-Risk criterion

Abstract.This paper examines a new approach for credit risk optimization. The model is based on the Conditional Value-at-Risk (CVaR) risk measure, the expected loss exceeding Value-at-Risk. CVaR is also known as Mean Excess, Mean Shortfall, or Tail VaR. This model can simultaneously adjust all positions in a portfolio of financial instruments in order to minimize CVaR subject to trading and return constraints. The credit risk distribution is generated by Monte Carlo simulations and the optimization problem is solved effectively by linear programming. The algorithm is very efficient; it can handle hundreds of instruments and thousands of scenarios in reasonable computer time. The approach is demonstrated with a portfolio of emerging market bonds.

Beyond VaR: from measuring risk to managing risk

The paper examines tools for managing, as opposed to simply monitoring, a portfolios value-at-risk (VaR). These tools include the calculation of VaR contribution, marginal VaR and trade risk profiles. We first review the parametric, or delta-normal versions of these tools and then extend them to the simulation based, or nonparametric case. We analyze two sample portfolios: one, consisting of foreign exchange contracts, is well-suited for parametric analysis while the other, which contains European options, is best addressed with simulation based methods. The limitations of the simulation based approach, due to the potential effects of sampling error, are also discussed.

The practice of portfolio replication. A practical overview of forward and inverse problems

Portfolio replication is a powerful tool that has proven in practice its applicability toenterprise‐wide risk problems such as static hedging in complete and incomplete marketsand markets that gap; strategic asset and capital allocation; benchmark tracking; design ofsynthetic products; and portfolio compression. In this paper, we revise the basic principlesbehind this methodology, currently used by financial institutions worldwide, and presentseveral practical examples of its application. We further show how inverse problems infinance can be naturally formulated in this framework. In contrast to mean‐variance optimization,the scenario approach allows for general non-normal, discrete and subjectivedistributions, as well as for the accurate modeling of the full range of nonlinear instruments,such as options. It also provides an intuitive, operational framework for explaining basicfinancial theory.

Managing Risk with Expected Shortfall

This paper examines tools for managing a portfolio’s risk as measured by expected shortfall, which has been proposed as an alternative to the more widely-used Value-at-Risk. These tools include the calculation of risk contributions, marginal risk, best hedge positions and trade risk profiles. We first derive the parametric, or delta-normal, versions of these tools and then extend them to the simulation-based, or non-parametric, case. We analyze two sample portfolios: one, consisting of foreign exchange contracts, is well-suited for parametric analysis while the other, which contains European options, is best addressed with simulation-based methods. While expected shortfall and Value-at-Risk are constant multiples of each other in the parametric case, expected shortfall tends to provide more robust estimates of relevant risk analytics under the simulation-based approach.

Efficient risk/return frontiers for credit risk

We construct efficient credit risk frontiers for a portfolio of bonds issued in emerging markets, using not only the variance but also quantile-based risk measures such as expected shortfall, maximum (percentile) losses and unexpected (percentile) losses. Our results demonstrate that minimizing variance yields portfolios that are far from efficient with respect to the standard quantile-based measures of credit risk.

Beyond VaR: parametric and simulation-based risk management tools

Financial institutions worldwide have devoted much effort to developing enterprise-wide systems that integrate financial information across their organizations to measure their institutions risk. Probabilistic measures such as value-at-risk (VaR), are now widely accepted by both financial institutions and regulators for assigning risk capital and monitoring risk. Since development efforts have been driven largely by regulatory and internal requirements to report risk numbers, tools needed to understand and manage risk across the enterprise have generally lagged behind those designed to measure it. The paper presents an extended simulation based risk management toolkit developed on top of the analytical tools presented by R. Litterman (1996; 1997). Simulation based tools provide additional insights when the portfolio contains nonlinearities, when the market distributions are not normal or when there are multiple horizons. In particular, these tools should prove very useful not only for market risk, but also for credit risk, where the exposure and loss distributions are generally skewed and far from normal. We also demonstrate that simulation based tools can be used, sometimes even more efficiently, with other risk measures in addition to VaR. Indeed, they also uncover limitations of VaR as a coherent risk measure, as demonstrated by P. Artzner et al. (1998). We focus in particular on constructing useful visualizations of risk, as provided by triangular decompositions and trade risk profiles, and calculating relevant risk measures such as marginal VaR and VaR contributions.

Clptimization As A Tool In Finance

In practice, even the most efficient, liquid markets behave in a discrete manner, exhibiting gaps in the underlying variables. Most modern finance, however, both theory and practice, has been built around continuous models that assume unlimited trading in frictionless markets. Even in markets that exhibit smooth behavior, replication of instruments with discontinuous characteristics, such as certain exotic options, cannot be done effectively with methods that presuppose continuity. Just as serious, is the fact that the cost of implementing dynamic hedging strategies under real conditions in real markets is not known or control I able, pr i mar i I y because the assumptions of con t i n u i ty , i nf i n i te I iqu id i ty and frictionless trading are far from real. Moreover, problems with high dimensionality and non-normal distributions have become increasingly necessary for risk management and difficult to solve without some discretization. Not only does the replication framework based on scenario optimization explicitly model discrete markets we observe in practise, where trading may be costly and liquidity limited, but it is also a powerful numerical tool for problems in high dimensions, with high non-linearities and deviations from normality. We discuss several real-world applications, including static hedging of complex derivatives and portfolios, hedging basis risk of fixed income portfolios, and computation of implied probabilities in option pricing.