Giorgio Consigli

Scenarios for Multistage Stochastic Programs

A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative scenarios (or a representative scenario tree) relative to the problem being modeled.

The bond--stock yield differential as a risk indicator in financial markets

The trading prices for securities in financial markets can exhibit sudden shifts or reversals in direction. In this paper a methodology for asset price dynamics is presented where the diffusive component is combined with a risk process. The risk process accommodates deviations from an equilibrium process and reversions. The bond‐stock yield differential is considered as a risk factor affecting the risk process. An approach using a “peaks over threshold” technique and conditional maximum likelihood is used to estimate parameters in the model. Numerical results for the period 1985‐2004 in the US market validate the effectiveness of the model.

The Predictive Ability of the Bond-Stock Earnings Yield Differential Model

In this article, the authors survey the bond-stock prediction model for five worldwide equity markets. This model is useful for predicting the time-varying equity risk premium (ERP) and for strategic asset allocation of bond-stock equity mixes. The focus is on the models economic and financial implications and its application to the study of stock market strategies and corrections. The model has two versions. The first model, formulated more than 35 years ago by Ziemba and Schwartz, is the difference between the most liquid long bond, usually the 10 or 30-year bond, and the trailing equity yield. The idea is that asset allocation between stocks and bonds is related to their relative yields. When the bond yield is too high, a shift out of stocks into bonds can cause an equity market correction. This model predicted the 1987, 2000, and 2002 corrections in the United States and the 1990 correction in Japan. The second model and equivalent version, the Fed model, uses the ratio, or equivalently the logs, of the two yields, and has its origins in reports and statements from the Federal Reserve System under Alan Greenspan dating from about 1996. The ERP can thus be negative or positive and is, therefore, partially predictable. Despite its predictive ability, the bond-stock model has been criticized as being theoretically unsound because it compares a nominal quantity (the long-bond yield) with a real quantity (the earnings yield on stocks). Theoretical models of fairly priced equity indices can be derived and compared to actual index values to ascertain danger levels.

Asset-Liability management for individual investors

Publisher Summary This chapter considers theoretical and practical developments that are currently driving the remarkable growth of individual asset liability management (ALM) applications as part of the fund management industry worldwide. A personal financial planning is a decision problem faced by an individual whose aim is to manage his consumption and investment decisions to achieve a set of real or financial targets, given his current and expected income, over a long-term horizon. ALM has emerged as an ideal framework to address this type of decision problem under uncertainty, in which the achievement of a strategic objective is made conditional on the effective management of assets and liabilities over time. The individual problem can be regarded as an extension of a personal investment consumption model with a limited number of investment opportunities and a rich set of individual and regulatory constraints with a long-term objective. The peculiarity of the individual ALM problem comes from the extent and implications of a modeling approach, which in principle is expected to capture the different features of the management of a financial position with a typically long-term horizon, up to and sometimes beyond retirement for an investor whose preferences may very well change over the planning horizon. The stochastic programming approach to ALM has thus emerged as an effective and appropriate way to address and analyze the personal financial planning problem. The generality of the individual problem from the financial point of view is considered here looking at several application areas, such as private banking, pension fund management, personal financial planning, and wealth management.

Path-dependent scenario trees for multistage stochastic programmes in finance

The formulation of dynamic stochastic programmes for financial applications generally requires the definition of a risk–reward objective function and a financial stochastic model to represent the uncertainty underlying the decision problem. The solution of the optimization problem and the quality of the resulting strategy will depend critically on the adopted financial model and its consistency with observed market dynamics. We present a recursive scenario approximation approach suitable for financial management problems, leading to a minimal yet sufficient representation of the randomness underlying the decision problem. The method relies on the definition of a benchmark probability space generated through Monte Carlo simulation and the implementation of a scenario reduction scheme. The procedure is tested on an interest rate vector process capturing market and credit risk dynamics in the fixed income market. The collected results show that a limited number of scenarios is sufficient to capture the exposure of the decision maker to interest rate and default risk.

Heavy-tailed distributional model for operational losses

We examine the statistical properties of operational losses obtained from a large European bank using an actuarial-type framework. The simplistic assumption of a Poisson frequency distribution fails and we show that the frequency process follows closely a non-homogeneous Poisson process with a deterministic intensity of the form of a continuous cdf-like function. Further, operational losses are modeled using a variety of distributions. We address the problems of (1) reporting bias; (2) supplementing internal data with external data; (3) tail estimation; and (4) mixing the distributions of the body and the tail, and propose practical solutions to such problems. Finally, our empirical findings are consistent with other studies reporting very heavy-tailed loss distributions with the tail index below unity.

Multi-Period Risk Measures and Optimal Investment Policies

This chapter provides an in-depth overview of an extended set of multi-period risk measures, their mathematical and economic properties, primarily from the perspective of dynamic risk control and portfolio optimization. The analysis is structured in four parts: the first part reviews characterizing properties of multi-period risk measures, it examines their financial foundations, and clarifies cross-relationships. The second part is devoted to three classes of multi-period risk measures, namely: terminal, additive and recursive. Their financial and mathematical properties are considered, leading to the proposal of a unifying representation. Key to the discussion is the treatment of dynamic risk measures taking their relationship with evolving information flows and time evolution into account: after convexity and coherence, time consistency emerges as a key property required by risk measures to effectively control risk exposure within dynamic programs. In the third part, we consider the application of multi-period measures to optimal investment policy selection, clarifying how portfolio selection models adapt to different risk measurement paradigms. In the fourth part we summarize and point out desirable developments and future research directions. Throughout the chapter, attention is paid to the state-of-the-art and methodological and modeling implications.

Applying stochastic programming to insurance portfolios stress-testing

The introduction of the Solvency II regulatory framework in 2011 and unprecendented property and casualty (P/C) claims experienced in recent years by large insurance firms have motivated the adoption of risk-based capital allocation policies in the insurance sector. In this article, we present the key features of a dynamic stochastic program leading to an optimal asset-liability management and capital allocation strategy by a large P/C insurance company and describe how from such formulation a specific, industry-relevant, stress-testing analysis can be derived. Throughout the article the investment manager of the insurance portfolio is regarded as the relevant decision-maker: he faces exogenous constraints determined by the core insurance division and is subject to the capital allocation policy decided by the management, consistently with the companys risk exposure. A novel approach to stress-testing analysis by the insurance management, based on a recursive solution of a large-scale dynamic stochastic program, is presented.

Dynamic Portfolio Management for Property and Casualty Insurance

Recent trends in the insurance sector have highlighted the expansion of large insurance firms into asset management. In addition to their historical liability risk exposure associated with statutory activity, the growth of investment management divisions has caused increasing exposure to financial market fluctuations. This has led to stricter risk management requirements as reported in the Solvency II 2010 impact studies by the European Commission. The phenomenon has far-reaching implications for the definition of optimal asset–liability management (ALM) strategies at the aggregate level and for capital required by insurance companies. In this chapter we present an ALM model which combines in a dynamic framework an optimal strategic asset allocation problem for a large insurer and property and casualty (P&C) business constraints and tests it in a real-world case study. The problem is formulated as a multistage stochastic program (MSP) and the definition of the underlying uncertainty model, including financial as well as insurance risk factors, anticipates the model’s application under stressed liability scenarios. The benefits of a dynamic formulation and the opportunities arising from an integrated approach to investment and P&C insurance management are highlighted in this chapter.

Optimal Financial Decision Making Under Uncertainty

We use a fairly general framework to analyze a rich variety of financial optimization models presented in the literature, with emphasis on contributions included in this volume and a related special issue of OR Spectrum. We do not aim at providing readers with an exhaustive survey, rather we focus on a limited but significant set of modeling and methodological issues. The framework is based on a benchmark discrete-time stochastic control optimization framework, and a benchmark financial problem, asset-liability management, whose generality is considered in this chapter. A wide set of financial problems, ranging from asset allocation to financial engineering problems, is outlined, in terms of objectives, risk models, solution methods, and model users. We pay special attention to the interplay between alternative uncertainty representations and solution methods, which have an impact on the kind of solution which is obtained. Finally, we outline relevant directions for further research and optimization paradigms integration.