Stochastic Programming: Applications In Finance, Energy, Planning And Logistics

Applications in Finance, Energy, Planning and Logistics

Horand I Gassmann, William T Ziemba

• Publisher: WSPC
• Year: 2012
• Print ISBN: 9789814407502
• eBook ISBN: 9789814407526
• Topic: Scienze biologiche
• Subtopic: Scienza generale

This book shows the breadth and depth of stochastic programming applications. All the papers presented here involve optimization over the scenarios that represent possible future outcomes of the uncertainty problems. The applications, which were presented at the 12th International Conference on Stochastic Programming held in Halifax, Nova Scotia in August 2010, span the rich field of uses of these models. The finance papers discuss such diverse problems as longevity risk management of individual investors, personal financial planning, intertemporal surplus management, asset management with benchmarks, dynamic portfolio management, fixed income immunization and racetrack betting. The production and logistics papers discuss natural gas infrastructure design, farming Atlantic salmon, prevention of nuclear smuggling and sawmill planning. The energy papers involve electricity production planning, hydroelectric reservoir operations and power generation planning for liquid natural gas plants. Finally, two telecommunication papers discuss mobile network design and frequency assignment problems.

Contents:

• • Introduction and Summary
• Papers in Finance:
  • Longevity Risk Management for Individual Investors (Woo Chang Kim, John M Mulvey, Koray D Simsek and Min Jeong Kim)
  • Optimal Stochastic Programming-Based Personal Financial Planning with Intermediate and Long-Term Goals (Vittorio Moriggia, Giorgio Consigli and Gaetano Iaquinta)
  • Intertemporal Surplus Management with Jump Risks (Mareen Benk)
  • Jump-Diffusion Risk-Sensitive Benchmarked Asset Management (Mark Davis and Sébastien Lleo)
  • Dynamic Portfolio Optimization under Regime-Based Firm Strength (Chanaka Edirisinghe and Xin Zhang)
  • Options Portfolio Management as a Chance Constrained Problem (Dmitry Golembiovsky and Anatoliy Abramov)
  • Stochastic Models for Optimizing Immunization Strategies in Fixed-Income Security Portfolios under Some Sources of Uncertainty (Larraitz Aranburu, Laureano F Escudero, M Araceli Garín and Gloria Pérez)
  • Stochastic Programming and Optimization in Horserace Betting (William T Ziemba)
• Papers in Production Planning and Logistics:
  • Multi-Stage Stochastic Programming for Natural Gas Infrastructure Design with a Production Perspective (Lars Hellemo, Kjetil Midthun, Asgeir Tomasgard and Adrian Werner)
  • A Stochastic Programming Model for Optimizing the Production of Farmed Atlantic Salmon (Martin B Hæreid, Peter Schütz and Asgeir Tomasgard)
  • Prioritizing Network Interdiction of Nuclear Smuggling (Dennis P Michalopoulos, David P Morton and J Wesley Barnes)
  • Sawmill Production Planning under Uncertainty: Modelling and Solution Approaches (Masoumeh Kazemi Zanjani, Mustapha Nourelfath and Daoud Ait-Kadi)
• Papers on Energy:
  • An Electricity Procurement Model with Energy and Peak Charges (Andy Philpott and Geoff Pritchard)
  • A Stochastic Game Model Applied to the Nordic Electricity Market (Stein-Erik Fleten and Tek Tjing Lie)
  • Multi-Lag Benders Decomposition for Power Generation Planning with Nonanticipativity Constraints on the Dispatch of LNG Thermal Plants (Andre L Diniz and Maria E P Maceira)
• Papers on Telecommunications:
  • Stochastic Second-Order Cone Programming in Mobile Ad-Hoc Networks: Sensitivity to Input Parameters (Francesca Maggioni, Marida Bertocchi, Elisabetta Allevi, Florian A Potra and Stein W Wallace)
  • Stochastic Frequency Assignment Problem (Wadie Benajam, Alexei Gaivoronski and Abdel Lisser)

Readership: Advanced undergraduate students, graduate students and researchers who are interested in the applications of stochastic programming.

Chapter 1

Introduction and Summary

Summary of papers

Part I has papers on finance. The paper by Kim, Mulvey, Simsek and Kim presents an individual asset liability model for a retired couple where longevity risk is modeled using term life insurance. They present calculations for three cases including using expected lifetimes where the longevity risk is assumed away. The other two cases discuss a shift in expected lifetimes or a decease in retirement income. As expected, the optimal policy depends on these risks and the price of insurance.

Moriggia, Consigli and Iaquinta present a multi-period asset liability planning model for individuals with intermediate and long-term goals. The investments include mutual and pension funds, fixed income instruments, unit-linked annuities as well as complex insurance and retirement protection. Optimal strategies are computed for various levels of risk aversion considering inflation-adjusted income and savings conditions, family consumption and taxes. Various time-distributed investment and payout plans are considered.

Benk presents an intertemporal portfolio choice model with jump risks that can be applied to pension and life insurance funds, and private investors. Following the Rudolf and Ziemba (2004) (the full reference is given in Chapter 4 by Benk) model, these long-term investors maximize the intertemporal expected utility of the surplus of assets net of liabilities. Returns on liabilities are modelled with a pure-diffusion process and the returns on assets are assumed to follow a jump-diffusion process with two jump components. An investor's optimal portfolio consists of three funds: a market portfolio, a liability-hedging portfolio, and a riskless asset. In contrast to the results of Rudolf and Ziemba (2004), a market portfolio not only hedges diffusion risk, but it also hedges systemic risk and it takes into account idiosyncratic jump risk so that the investor is additionally protected against both a systemic risk and an idiosyncratic jump risk.

The paper by Davis and Lleo considers an asset management problem where the assets follow a jump-diffusion process. The objective is to outperform a benchmark as is common in the investment industry for mutual and other funds. They explore two sets of assumptions on the coefficients of the stochastic process that yield unique optimal investment strategies for the stochastic risk-sensitive control problem. The first set of assumptions relate to an affine function of a factor process with constant asset diffusions, and the asset jumps are independent of the factors, which are assumed to be Gaussian diffusion processes. In this case, the associated Bellman equation is a partial differential equation which has a unique classical solution. In the second set of assumptions, both the asset growth rates and volatility depend upon the factors which are jump-diffusion processes. Hence, this model has stochastic volatility. This yields a fully nonlinear controlled jump-diffusion, and the Bellman equation is a partial integer-differential equation [PIDE] for which no analytical solution exists. Proving that the Hamilton-Jacobi-Bellman [HJB] PIDE admits a unique classical (C1,2) solution requires the development of a more sophisticated argument combining viscosity solutions and classical solutions. The argument used in the derivation hinges on only three key points. First, the Lipschitz continuity of the value function provides the ability to rewrite the HJB PIDE as a PDE. Second, viscosity solutions give existence and uniqueness of a weak solution to both of these equations. A proof of existence by Fleming and Rishel based on a policy improvement originally due to Bellman completes the analysis by providing a smooth solution. The robustness of this approach is a clear advantage for control problems: solving benchmarked or ALM investment management problems is not more difficult than addressing an asset-only investment problem.

The paper by Edirisinghe and Zhang presents a dynamic portfolio optimization model for stock portfolio management when there is market uncertainty modeled with regimes. The portfolio of long and short stocks is determined from the regimes and fundamental business strength measured using data envelopment analysis. The data used to calibrate and test the model is for the forty year period 1971–2010. The out of sample test shows that the model is superior to sector based ETF portfolios and the market index for the January to June 2011 period.

Golembiovsky and Abramov present a modification of a standard option portfolio where the goal is to attempt to have excess returns over the risk free rate by taking a small risk. The risk is modeled using chance constraints so is a value at risk concept. Using the Black-Scholes formula, they evaluate European options over a two month horizon. Simulations show that the goal can be achieved.

Aranburu, Escudero, Garín and Pérez present several approaches for the stochastic optimization of immunization strategies. The risk averse measures used are two-stage and multiple stage stochastic dominance and a new multistage VaR stochastic dominance criterion. Scenarios are used and the models consider investing in a market with coupon bonds having different levels of default risk.

The paper by Ziemba demonstrates that racetrack betting is simply an application of portfolio theory. The racetrack offers many bets that involve the results of one to about ten horses. Each race is a special financial market with betting then a race that takes one or a few minutes. Unlike the financial markets, one cannot stop the race when one is ahead or having the market going almost 24/7. There is a well-defined end point. Like standard portfolio theory, the key issues are to get the means right. In this case, it is the probabilities that two, three or four horses finish in the first two, respectively three or four places, in the given order, and to bet well. For the latter, the Kelly capital growth criterion is widely used, which maximises the expected logarithm of final wealth. Transaction and price pressure odds changes fit well into the stochastic programming models. This paper relates the theory, computations and examples of real races and experiences for various bets such as win, place and show, exactas, triactors, superfectas, super hi five, place pick all, double, pick 3, 4, 5 and 6. Many great races can be seen free on the website www.chef-de-race.com.

Part II has papers relating to production planning and logistics. Hellemo, Midthun, Tomasgard and Werner describe a multi-stage model for the design and operation of a network of pipelines and other infrastructure components for the production of natural gas. The model distinguishes two time scales and corresponding uncertainty; the strategic level deals with major investment decisions, such as construction of pipelines and opening of new reservoirs, while the embedded operational level concerns decisions regarding the production, transportation and marketing of the recovered gas. While strategic decisions have implications from one stage to the next, the impact of the operational decisions is assumed to be confined within each strategic stage.

Hæreid, Schütz and Tomasgard build a model that can be used by farmers of salmon in helping to plan their production in the presence of uncertainty about both salmon prices and the development of the fish stock (i.e., growth, mortality, escapes, etc.). The objective is to maximize the farmer's expected profits by considering both the timing and the quantity of the salmon harvest and the introduction of new stock. Important constraints include regulatory limits on the allowable biomass as well as a requirement to keep a particular holding tank empty (fallow) for a certain period after the stock has been harvested.

The paper by Michalopoulos, Morton and Barnes discusses models for interdicting nuclear smugglers. They look at the one-country case where only border-points are potential locations for installations that can detect nuclear material. The original and very general aspect of this model is the fact that not only are the smuggling activities random from the view point of the authorities, but so is the budget for installing detection devises. This is important because the optimal set of check-points for a budget b is normally not a subset of those for a budget

for

> b. And as is often the case with public investments, decisions of which installations to build are made one by one as budgets are made available. So the output of the model is a priority list of installations which optimally (given a density function for the random budget) trades off the possible budgets.

The paper by Kazemi Zanjani, Nourelfath and Ait-Kadi discusses operational planning in the sawing units of sawmills under two types of uncertainty: Quality uncertainty stemming from the variation in log sizes and qualities and more traditional demand uncertainty. Bad sawing decisions not only lead to more backlog for high quality products but also more inventories of low quality products. Several models are discussed; two-and multi-stage stochastic programming as well as some robust formulations. In some models setup costs are considered, leading to mixed integer formulations. The setup of scenarios is discussed. Since the models are very hard to solve, two approximation schemes are developed, one based on the progressive hedging algorithm, the other on scenario updates.

Part III has papers on energy. Philpott and Pritchard present many different clever modelling ideas to help a resource intensive industry decide on the appropriate manufacturing schedule to take advantage of lower electricity costs while satisfying a known demand schedule. There are many sources of uncertainty in this model, including the price of electricity, which depends on both hydrological processes (for the production of hydro-energy) and the total system demand for electricity. A novel energy surcharge levied on the 100 highest demand periods (30-minute time intervals) during the previous year (collected ex-post) is also considered. The paper is ambitious and far-reaching.

The paper by Fleten and Lie uses a two-stage numerical oligopoly model to study market power in a mixed hydro and thermal system, with example data taken from the Nordic market. Uncertainty is related to water inflows to reservoirs. The point of the model is to understand if market power is exercised and to what extent prices differ from those that would have occurred in a fully competitive market. For the case at hand they find that market prices are about 7% above the competitive prices — a rather small difference. They also show that the largest producer in the region has incentives to reduce its thermal output in order to increase the spot prices. This type of model can be used by regulators to observe the large producers to prevent the exercise of potential market power.

The paper by Diniz and Maceira discusses a classical dispatch problem for a mixed hydro and thermal energy system with one additional difficulty. When thermal units are fueled by liquid natural gas (LNG) the dispatch decision can no longer be made in the same period as the other units (in a medium to long term model), but must be made many periods earlier. This is caused by the way logistics decisions about the LNG supply are made. This change creates algorithmic challenges in both Benders (L-shaped) decomposition and in dual dynamic programming. The paper illustrates that this difficulty can be overcome, and shows how to approach the difficulties. Illustrations are based on the Brazilian energy system.

Part IV has two theoretical papers that deal with telecommunication. The paper by Maggioni, Bertocchi, Allevi, Potra and Wallace uses second-order cone programming to analyze a problem from mobile communication networks. Uncertainty is primarily associated with the movements of the destination node of a message sent from a non-moving sender node. Scenarios are in the form of ellipses describing where the destination node might be based on where it was last observed. The main goal of the paper is very general: To understand when deterministic models are useful, and if they are, in what sense, and when they are not. In particular, a bad deterministic solution (in terms of a large Value of the Stochastic Solution) might have valuable information embedded in it even if it is very bad in its own right. This points to the fact that deterministic solutions are not just good or bad: There is a range of positions in between.

The paper by Benajam, Gaivoronski and Lisser uses stochastic programming and semidefinite approximations to model and analyze the frequency assignment problem from telecommunications. While the basic resource for mobile communications remains basically unchanged, the demand is growing rapidly. At the same time, the services face challenging randomness in demand and operational conditions. The main contribution of this paper is its positive answer to the question: Can adding randomness to the already challenging deterministic versions of the model, lead to models that are solvable? The conclusions are backed up with numerical illustrations of moderately sized problems.

PART I

Papers in Finance

Chapter 2

Longevity Risk Management for Individual Investors

Woo Chang Kim^*, John M. Mulvey^†, Koray D. Simsek^‡ and Min Jeong Kim^§

Summary

We model and numerically solve the optimal asset allocation problem of a retired couple with uncertain lifetime, in the presence of a life insurance policy. The couple maximizes expected utility over their joint lifetime by dynamically adjusting their asset allocation and purchasing term-life insurance. We conduct three numerical analyses: In the base case, we find optimal policies assuming the expected lifetimes are correct. The other two examples introduce longevity risk through either a shift in the expected lifetimes or an unexpected cut in retirement income. We find that optimal asset allocation policy depends on the presence and the type of these risks as well as the relative price of insurance. Furthermore, we show that a generalized linear policy is not likely to help under such circumstances.

1 Introduction

The objective of this study is to determine the optimal portfolio of the traditional asset classes (stocks and bonds) along with the life insurances and the pension plans in the retirement planning framework. The extended life span thanks to the advances in the medical sciences is good news for the mankind. However, it could affect the retirement planning for the individuals, and the pension plan management in a negative fashion. Importantly, ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Acknowledgements
• List of Contributors
• Preface
• Books and Collections of Papers on Stochastic Programming
• Contents
• 1. Introduction and Summary
• Part I. Papers in Finance
• Part II. Papers in Production Planning and Logistics
• Part III. Papers on Energy
• Part IV. Papers on Telecommunications
• Index