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In this work, we consider a risk-averse ultimate pit problem where the grade of the mineral is uncertain. We derive conditions under which we can generate a set of nested pits by varying the risk level instead of using revenue factors. We propose two properties that we believe are desirable for the problem: risk nestedness, which means the pits generated for different risk aversion levels should be contained in one another, and additive consistency, which states that preferences in terms of order of extraction should not change if independent sectors of the mine are added as precedences. We show that only an entropic risk measure satisfies these properties and propose a two-stage stochastic programming formulation of the problem, including an efficient approximation scheme to solve it. We illustrate our approach in a small self-constructed example, and apply our approximation scheme to a real-world section of the Andina mine, in Chile.

Portfolio selection problems have been thoroughly studied under the risk-and-return paradigm introduced by Markowitz. However, the usefulness of this approach has been hindered by some practical considerations that have resulted in poorly diversified portfolios, or, solutions that are extremely sensitive to parameter estimation errors. In this work, we use sampling methods to cope with this issue and compare the merits of two approaches: a sample average approximation approach and a performance-based regularization (PBR) method that appeared recently in the literature. We extend PBR by incorporating three different risk metrics—integrated chance-constraints, quantile deviation, and absolute semi-deviation—and deriving the corresponding regularization formulas. Additionally, a numerical comparison using index-based portfolios is presented using historic data that includes the subprime crisis.

Defined contribution (DC) pension plans have been gaining ground in the last 10–20 years as the preferred system for many countries and other agencies, both private and public. The central question for a DC plan is how to invest in order to reach the participant's retirement goals. Given the financial illiteracy of the general population, it is common to offer a default policy for members who do not actively make investment choices. Using data from the Chilean system, we discuss an investment model with fixed contribution rates and compare the results with the existing default policy under multiple objectives. Our results indicate that the Chilean default policy has good overall performance, but specific closed-loop policies have a higher probability of achieving desired retirement goals and can reduce the expected shortfall at retirement.

We propose a class of partially observable multistage stochastic programs and describe an algorithm for solving this class of problems. We provide a Bayesian update of a belief-state vector, extend the stochastic programming formulation to incorporate the belief state, and characterize saddle-function properties of the corresponding cost-to-go function. Our algorithm is a derivative of the stochastic dual dynamic programming method.

Continuous gas lift is a popular method to enhance productivity in offshore oil platforms. We propose a steady-state two-stage stochastic programming model to maximize production, where the first-stage injection level determines the production potential, while recourse actions ensure capacity and platform constraints for each uncertainty realization. In particular, we develop a concave approximation of the performance curve that incorporates uncertainty in the water cut (WC) and gas–oil ratio (GOR). We generate WC and GOR realizations using a two-step data-driven approach: we extrapolate the trends using a $$\ell _1$$ -filter, and bootstrap historical deviations to generate future realizations of WC and GOR. We present numerical results for the sample average approximation of the problem and assess the solution quality using standard techniques in the literature. Our numerical results suggest that taking uncertainty into account in the problem can lead to considerable gains.

Several studies discovered that private teachers have a higher probability of losing their jobs than public school teachers during the Global Financial Crisis. This appears to be the case in many low- and middle-income countries throughout the current crisis or covid-19 pandemic. This phenomenon is generating a lot of problems, particularly as voiced out by private school teachers in the context of developing countries. This motivates the researcher to investigate and discuss the problems faced by private school teachers in the Philippines in the wake of the Covid-19 crisis. Specifically, the researcher described their motivation factors teaching in the private school and their challenges during a Covid-19 pandemic. The researcher used a descriptive phenomenological research approach and an in-depth interview. Using purposive sampling, seven (7) participants were selected to participate in the study. To preserve the confidentiality of the research participants, ethical measures were also implemented. The responses of the participants were thematically analyzed using Colaizzi���s methods of descriptive phenomenology. Results revealed 3 themes on the motivation factors: Passion in Teaching, Teaching as a Calling and Promotion in Public School and 5 themes related to challenges which include Financial Constraints, Mental Health Issues, Working beyond Contract, Lack of Teaching Resources and Poor Relationship with the School Heads. Indeed, private teachers are also encountering problems that need to be addressed by the national government. As voiced out by the participants they received nothing from the government as compared to the public teachers during the pandemic. This study recommends that the national government must also look at how to help private teachers who are affected by the pandemic, such as financial assistance. 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1. Arturo Cifuentes 2. Bernardo Pagnoncelli 1. To order reprints of this article, please contact David Rowe at d.rowe{at}pageantmedia.com or 646-891-2157. Molecular biology is undergoing a major shakeup. The cell-as-machine concept, the foundation on which the discipline has built

The alternate direction method of multipliers (ADMM) has received significant attention recently as a powerful algorithm to solve convex problems with a block structure. The vast majority of applications focus on deterministic problems. In this paper we show that ADMM can be applied to solve two-stage stochastic programming problems, and we propose an implementation in three blocks with or without proximal terms. We present numerical results for large scale instances, and extend our findings for risk averse formulations using utility functions.

We propose a multistage stochastic programming model to optimally allocate cargo to the passengers network in order to maximize profit, taking into account incomes, costs and penalties for not delivering cargo that was previously accepted. Flights have a discrete number of possible capacity outcomes, with known probabilities, and uncertainty is represented by a scenario tree. The resulting problem is a large-scale linear program, and we use decomposition techniques to solve it, leveraging on the problem structure in order to be able to find good quality solutions. Our numerical experiments are based on a real network of a major commercial airline.

In defined contribution (DC) pension schemes, the regulator usually imposes asset allocation constraints (minimum and maximum limits by asset class) in order to create funds with different risk–return profiles. In this article, we challenge this approach and show that such funds can exhibit erratic risk–return profiles that deviate significantly from the intended design. We propose to replace all minimum and maximum asset allocation constraints by a single risk metric (or measure) that controls risk directly. Thus, funds with different risk–return profiles can be immediately created by adjusting the risk tolerance parameter accordingly. Using data from the Chilean DC pension system, we show that our approach generates funds whose risk–return profiles are consistently ordered according to the intended design, and outperforms funds created by means of asset allocation limits.

We propose an algorithm based on infeasible irreducible subsystems to solve binary linear chance-constrained problems with random technology matrix. By leveraging on the problem structure we are able to generate good quality upper bounds to the optimal value early in the algorithm, and the discrete domain is used to guide us efficiently in the search of solutions. We apply our methodology to individual and joint binary linear chance-constrained problems, demonstrating the ability of our approach to solve those problems. Extensive numerical experiments show that, in some cases, the number of nodes explored by our algorithm is drastically reduced when compared to a commercial solver.

International audience; We model the relation between an aggregator and consumers joining a coalition to reduce the risk resulting from the unpredictability of their base load demand, as a Stackelberg game formulated as a mathematical bilevel program with private information on the consumers' reservation prices. At the upper-level of the Stackelberg game, the aggregator optimizes his daily price profile so as to reach a net targeted profit which is the maximum value guaranteeing that no consumer will leave the coalition - to contract with a conventional retailer considered here as a fixed alternative - while meeting fairness criterion imposed by the cost-sharing mechanism. At the lower-level, the consumers are asked to provide in day ahead an estimate of their base load hourly demand profile and to schedule their shiftable loads depending on the price signal sent by the aggregator. We provide algorithms that determine the unique price profile and consumer shiftable load schedules as functions of the reservation price estimates. The Stackelberg game between the aggregator and the consumers being repeated for a period of time, the aggregator has the possibility to update his estimates of the reservation prices relying on a feedback function which depends on the percentage of activated loads. A randomized algorithm for consumers' reservation price learning based on regret minimization is provided. For four cost-sharing mechanisms such as uniform allocation, stand-alone cost, Shapley value, separable and non-separable costs, we determine the closed form of the aggregator's optimal net targeted profit guaranteeing the stability of the coalition. We also determine conditions guaranteeing the core non-emptiness and prove that for a profit-maximizing aggregator, the stand-alone cost is always preferable to the Shapley value, which coincides with the uniform allocation. Furthermore, the optimal size of the coalition - in terms of the aggregator's profit - can be determined analytically when the Shapley value is implemented as cost-sharing mechanism. The results are illustrated on a case study where we show that there exists an optimal net targeted profit below which the consumers energy bill is lower when joining the aggregator than with the conventional retailer. Coalition dynamics is also analyzed numerically depending on the consumer inertia in their energy supplier choice process, for each cost-sharing mechanism.

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.

We propose a two-step hybrid investment strategy suitable for pension funds. Our method consists of an active component (an optimization-based approach to decide the asset allocation), followed by a passive strategy (an index-based approach). We test our strategy with data from the Chilean pension system using two different risk metrics and we show that our approach, in three out of five cases, yields results that are better than those generated by the Chilean fund administrators. In the two cases where our approach underperformed we show that it was the result of excessively tight constraints set up by the regulator.

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.

Previous studies have suggested using gamma cameras with cadmium-zinc-telluride (CZT) detectors to quantify myocardial blood flow (MBF) and flow reserve (MFR). In this study, we aimed to evaluate the feasibility and accuracy of MFR quantification using a CZT camera compared to coronary angiography.Forty-one participants referred for coronary angiography underwent a rest/stress one-day myocardial perfusion imaging protocol using a CZT gamma camera. Rest and stress dynamic phases were followed by acquisition of traditional perfusion images and time-activity curves were generated. Angiographic and perfusion results were compared to MFR.Patients with abnormal perfusion presented reduced MFR (2.01 [1.48-2.77] vs. 2.94 [2.38-3.64], P = 0.002), and reduced stress MBF. Patients with high-risk CAD had lower global MFR compared to patients without obstructive disease (1.99 [1.22-2.84] vs. 2.89 [2.22-3.58], P = 0.026). Obstructed vessels showed lower regional MFR when compared to non-obstructed (1.81 [1.19-2.67] vs. 2.75 [2.13-3.42], P 0.001). A regional MFR of 2.2 provided a sensitivity of 63.2% and specificity of 74.1% to identify an obstructive lesion in the corresponding artery.In patients undergoing invasive coronary angiography for the evaluation of CAD, quantifying MBF and MFR in a CZT gamma camera is feasible and reflects underlying disease. In these patients, reduced regional MFR suggests the presence of obstructive lesion(s).

Mexico and Chile have Defined Contributions (DC) pension systems. In both cases affliates are offered several investment funds with, allegedly, different risk-return profiles. Analyzing actual return data for the April 2008-March 2018 period, and using a number of risk (and return) related metrics, we reach quite different conclusions in relation to such funds. In the case of Mexico, the funds delivered returns according to their intended risk profile, and they are consistently ranked correctly in terms of absolute risk, risk-adjusted returns, and cumulative returns. Chilean funds, on the other hand, exhibited erratic risk-return patterns, with the most conservative fund outperforming the riskiest fund in terms of cumulative returns. Overall our analysis is an indictment on the idea of using asset allocation limits to control portfolio risk (Chile), and supports the view that risk is much better controlled using an overall portfolio-level risk metric (Mexico). Since most pension funds still rely heavily on asset-class limits to manage risk, our results should serve as a serious warning against the danger of relying on this practice.

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.

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.

In defined contribution (DC) pension schemes, the regulator usually imposes asset allocation constraints (minimum and maximum limits by asset class) in order to create funds with different risk-return profiles. In this article we challenge this approach and show that such funds exhibit erratic risk-return profiles that deviate significantly from the intended design. We propose to replace all minimum and maximum asset allocation constraints by a single risk metric (or measure) that controls risk directly. Thus, funds with different risk-return profiles can be immediately created by adjusting the risk tolerance parameter accordingly. We demonstrate the effectiveness of this approach with data from the Chilean DC pension system. Specifically, we show that our approach generates funds whose risk-return profiles are consistently ordered according to the intended design and outperform funds created by means of asset allocation limits.

In 1980 Chile switched from a state-managed defined-benefit pension system to a defined-contribution scheme based on individual capital accounts. The new system was further refined in 2002 with the introduction of five investment funds, with, allegedly, different risk-return profiles. The funds differ in their portfolio composition which is driven by strict minimum and maximum limits (mostly related to stocks and bonds), dictated by the regulator. We have examined the performance of these funds over a fifteen-year period looking at their returns and actual risk profiles, aided by three rank-order metrics. Unfortunately, our results are unambiguously distressing: while the regulator succeeded in creating five funds with clearly different risk profiles, their risk-adjusted returns as well as their cumulative (absolute) returns are completely at odds with the desired goal. In fact, during long stretches of time the funds exhibited a performance that was exactly the opposite of what it was intended: an indictment on the idea of controlling portfolio risk via asset allocation limits.

We study chance-constrained problems in which the constraints involve the probability of a rare event. We discuss the relevance of such problems and show that the existing sampling-based algorithms cannot be applied directly in this case, since they require an impractical number of samples to yield reasonable solutions. We argue that importance sampling (IS) techniques, combined with a Sample Average Approximation (SAA) approach, can be effectively used in such situations, provided that variance can be reduced uniformly with respect to the decision variables. We give sufficient conditions to obtain such uniform variance reduction, and prove asymptotic convergence of the combined SAA-IS approach. As it often happens with IS techniques, the practical performance of the proposed approach relies on exploiting the structure of the problem under study; in our case, we work with a telecommunications problem with Bernoulli input distributions, and show how variance can be reduced uniformly over a suitable approximation of the feasibility set by choosing proper parameters for the IS distributions. Although some of the results are specific to this problem, we are able to draw general insights that can be useful for other classes of problems. We present numerical results to illustrate our findings.

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.

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.

We propose a model to assess the credit risk features of fixed income portfolios assuming they can be characterized by two parameters: their default probability and their default correlation. We rely on explicit expressions to assess their credit risk and demonstrate the benefits of our approach in a complex leveraged structure example. We show that using expected loss as a proxy for credit risk is misleading as it does not capture the dispersion effects introduced by correlation. The implications of these findings are relevant for improving current risk management practices and for regulation purposes.