Bernardo K. Pagnoncelli

Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective

We discuss the incorporation of risk measures into multistage stochastic programs. While much attention has been recently devoted in the literature to this type of model, it appears that there is no consensus on the best way to accomplish that goal. In this paper, we discuss pros and cons of some of the existing approaches. A key notion that must be considered in the analysis is that of consistency, which roughly speaking means that decisions made today should agree with the planning made yesterday for the scenario that actually occurred. Several definitions of consistency have been proposed in the literature, with various levels of rigor; we provide our own definition and give conditions for a multi-period risk measure to be consistent. A popular way to ensure consistency is to nest the one-step risk measures calculated in each stage, but such an approach has drawbacks from the algorithmic viewpoint. We discuss a class of risk measures—which we call expected conditional risk measures—that address those shortcomings. We illustrate the ideas set forth in the paper with numerical results for a pension fund problem in which a company acts as the sponsor of the fund and the participants’ plan is defined-benefit.

Risk-Return Trade-off with the Scenario Approach in Practice: A Case Study in Portfolio Selection

We consider the scenario approach for chance constrained programming problems. Building on existing theoretical results, effective and readily applicable methodologies to achieve suitable risk-return trade-offs are developed in this paper. Unlike other approaches, that require solving non-convex optimization problems, our methodology consists of solving multiple convex optimization problems obtained by sampling and removing some of the constraints. More specifically, two constraint removal schemes are introduced, one greedy and the other randomized, and a comparison between them is provided in a detailed computational study in portfolio selection. Other practical aspects of the procedures are also discussed. The removal schemes proposed in this paper are generalizable to a wide range of practical problems.

The optimal harvesting problem under price uncertainty

In this paper we study the exploitation of a one species forest plantation when timber price is governed by a stochastic process. The work focuses on providing closed expressions for the optimal harvesting policy in terms of the parameters of the price process and the discount factor, with finite and infinite time horizon. We assume that harvest is restricted to mature trees older than a certain age and that growth and natural mortality after maturity are neglected. We use stochastic dynamic programming techniques to characterize the optimal policy and we model price using a geometric Brownian motion and an Ornstein–Uhlenbeck process. In the first case we completely characterize the optimal policy for all possible choices of the parameters. In the second case we provide sufficient conditions, based on explicit expressions for reservation prices, assuring that harvesting everything available is optimal. In addition, for the Ornstein–Uhlenbeck case we propose a policy based on a reservation price that performs well in numerical simulations. In both cases we solve the problem for every initial condition and the best policy is obtained endogenously, that is, without imposing any ad hoc restrictions such as maximum sustained yield or convergence to a predefined final state.

A Provisioning Problem with Stochastic Payments

We consider the problem of determining the minimal requirement one must establish in order to meet a series of future random payments. It is shown in a very general setting that this problem can be recast as a chance constrained model and how the technique of Sample Average Approximation can be employed to find solutions. We also use comonotonic theory to analyze analytical approximations in a restricted Gaussian setting. Our numerical illustrations demonstrate that the Sample Average Approximation is a viable and efficient way to solve the stated problem generally and outperforms the analytical approximations. In passing we present a result that is related to Stein’s famous lemma (Stein, 1981) and is of interest in itself.

The optimal harvesting problem under price uncertainty: the risk averse case

We study the exploitation of a one species, multiple stand forest plantation when timber price is governed by a stochastic process. Our model is a stochastic dynamic program with a weighted mean-risk objective function, and our main risk measure is the Conditional Value-at-Risk. We consider two stochastic processes, geometric Brownian motion and Ornstein–Uhlenbeck: in the first case, we completely characterize the optimal policy for all possible choices of the parameters while in the second, we provide sufficient conditions assuring that harvesting everything available is optimal. In both cases we solve the problem theoretically for every initial condition. We compare our results with the risk neutral framework and generalize our findings to any coherent risk measure that is affine on the current price.

A risk averse approach to the capacity allocation problem in the airline cargo industry

In air cargo transportation, capacity can be reserved via allotment, which are long-term contracts with fixed price, and free, which is the space not assigned to allotment contracts. In this later case, reservations are made closer to the departure date, and normally higher tariffs are charged. The demand, the tariff, and the show-up rate for the free mode are stochastic. We consider risk neutral and risk averse formulations, using the Conditional Value-at-Risk as a risk measure. We solve the resulting problems using the Sample Average Approximation and test our models with nine experiments representing different demand patterns using real data from a major airline.

Demystifying Credit Risk Derivatives and Securitization: Introducing the Basic Ideas to Undergraduates

Securitization has been a significant breakthrough in our ability to manage financial risk. In the same way that a futures contract permits exposure to price risk to be separated from ownership of a risky asset, securitization allows the separation of many types of risk exposure and other important characteristics from ownership of the securities in which they originate. Credit derivatives, in particular, can eliminate nearly all exposure to default risk for most investors in a pool of credit-risky bonds, even if the credit quality of the underlying securities is not strong. But the financial crisis of 2008 exposed several important facts: These instruments are complicated, their true risk characteristics had not been fully appreciated, and few people really understood them—apparently including some of those who were in the business of buying, selling, and creating them. This article is a basic primer on securitization of credit risk and the derivative securities that are created in the process. It provides an accessible overview that will be useful for teaching undergraduates about securitization of credit risk and as a general introduction to the subject for the non-technically-oriented reader.

Scenario reduction for stochastic programs with Conditional Value-at-Risk

In this paper we discuss scenario reduction methods for risk-averse stochastic optimization problems. Scenario reduction techniques have received some attention in the literature and are used by practitioners, as such methods allow for an approximation of the random variables in the problem with a moderate number of scenarios, which in turn make the optimization problem easier to solve. The majority of works for scenario reduction are designed for classical risk-neutral stochastic optimization problems; however, it is intuitive that in the risk-averse case one is more concerned with scenarios that correspond to high cost. By building upon the notion of effective scenarios recently introduced in the literature, we formalize that intuitive idea and propose a scenario reduction technique for stochastic optimization problems where the objective function is a Conditional Value-at-Risk. Numerical results presented with problems from the literature illustrate the performance of the method and indicate the cases where we expect it to perform well.

Credit-Risk Behavior of Homogeneous Portfolios: A Theoretical Result with Surprising Practical Implications

This article describes an analytical approach to examine the credit-risk behavior of a homogeneous portfolio. The authors demonstrate the usefulness of the approach using a synthetic index linked to high-yield corporate bonds (which resembles a synthetic CDO) and then analyze an actual synthetic CDO transaction. They show that the conventional approach to analyze these structures (Monte Carlo simulations combined with the Gaussian copula) fails to account for the tri-modal nature of the underlying portfolio default distribution, and consequently, risk assessments based on this method give a misguided view of the risk–reward profile of such portfolios. The authors further show that the benefits of portfolio diversification in the context of credit-risk portfolios are limited in high-correlation scenarios. These findings have important implications for risk managers and financial regulators.

Securitization and Poisoned Bottles: The Perfect Teaching Analogy

Securitization is a difficult topic to teach. Most students have preconceived ideas about it. Some of them are quite wrong. And frequently many students feel that there is almost something like “black magic” behind the concept that one can create AAA-securities out of risky assets. Additionally, the fact that securitization played an important role in the recent financial crisis, coupled with the sturdy popularity of certain securitization vehicles, makes a strong case to teach at least the basics of securitization to undergraduates majoring in economics. In this paper the authors introduce a novel way to present the topic in the classroom. The approach is based on a combination of intuition and some elementary formulas from probability theory. The authors illustrate the approach with a simple example.