Models and model value in stochastic programming

Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44253/1/10479_2005_Article_BF02031741.pd

Real Options using Markov Chains: an application to Production Capacity Decisions

In this work we address investment decisions using real options. A standard numerical approach for valuing real options is dynamic programming. The basic idea is to establish a discrete-valued lattice of possible future values of the underlying stochastic variable (demand in our case). For most approaches in the literature, the stochastic variable is assumed normally distributed and then approximated by a binomial distribution, resulting in a binomial lattice. In this work, we investigate the use of a sparse Markov chain to model such variable. The Markov approach is expected to perform better since it does not assume any type of distribution for the demand variation, the probability of a variation on the demand value is dependent on the current demand value and thus, no longer constant, and it generalizes the binomial lattice since the latter can be modelled as a Markov chain. We developed a stochastic dynamic programming model that has been implemented both on binomial and Markov models. A numerical example of a production capacity choice problem has been solved and the results obtained show that the investment decisions are different and, as expected the Markov chain approach leads to a better investment policy.Flexible Capacity Investments, Real Options, Markov Chains, Dynamic Programming

STOCHASTIC DYNAMIC MODELING: AN AID TO AGRICULTURAL LENDER DECISION MAKING

Factors affecting a lenderÂ’s decision to grant farmers operating credit in North Dakota are quantified in an intertemporal loan profitability model using stochastic dynamic programming. Experimental data obtained from a panel of lenders demonstrates the sensitivity of an optimal policy to changes in a lenderÂ’s discount rate, a borrowerÂ’s repayment status, and patronage. The value of credit scoring models that appraise a borrowerÂ’s credit worthiness also is determined.Agricultural Finance,

Employees Provident Fund (EPF) Malaysia: Generic models for asset and liability management under uncertainty

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We describe Employees Provident Funds (EPF) Malaysia. We explain about Defined Contribution and Defined Benefit Pension Funds and examine their similarities and differences. We also briefly discuss and compare EPF schemes in four Commonwealth countries. A family of Stochastic Programming Models is developed for the Employees Provident Fund Malaysia. This is a family of ex-ante decision models whose main aim is to manage, that is, balance assets and liabilities. The decision models comprise Expected Value Linear Programming, Two Stage Stochastic Programming with recourse, Chance Constrained Programming and Integrated Chance Constraints Programming. For the last three decision models we use scenario generators which capture the uncertainties of asset returns, salary contributions and lump sum liabilities payments. These scenario generation models for Assets and liabilities were developed and calibrated using historical data. The resulting decisions are evaluated with in-sample analysis using typical risk adjusted performance measures. Out- of- sample testing is also carried out with a larger set of generated scenarios. The benefits of two stage stochastic programming over deterministic approaches on asset allocation as well as the amount of borrowing needed for each pre-specified growth dividend are demonstrated. The contributions of this thesis are i) an insightful overview of EPF ii) construction of scenarios for assets returns and liabilities with different values of growth dividend, that combine the Markov population model with the salary growth model and retirement payments iii) construction and analysis of generic ex-ante decision models taking into consideration uncertain asset returns and uncertain liabilities iv) testing and performance evaluation of these decisions in an ex-post setting.This stuyd is funded by the Universiti Teknologi MARA Malaysia

Accounting for Uncertainty Affecting Technical Change in an Economic-Climate Model

The key role of technological change in the decline of energy and carbon intensities of aggregate economic activities is widely recognized. This has focused attention on the issue of developing endogenous models for the evolution of technological change. With a few exceptions this is done using a deterministic framework, even though technological change is a dynamic process which is uncertain by nature. Indeed, the two main vectors through which technological change may be conceptualized, learning through R&D investments and learning-by-doing, both evolve and cumulate in a stochastic manner. How misleading are climate strategies designed without accounting for such uncertainty? The main idea underlying the present piece of research is to assess and discuss the effect of endogenizing this uncertainty on optimal R&D investment trajectories and carbon emission abatement strategies. In order to do so, we use an implicit stochastic programming version of the FEEM-RICE model, first described in Bosetti, Carraro and Galeotti, (2005). The comparative advantage of taking a stochastic programming approach is estimated using as benchmarks the expected-value approach and the worst-case scenario approach. It appears that, accounting for uncertainty and irreversibility would affect both the optimal level of investment in R&D –which should be higher– and emission reductions –which should be contained in the early periods. Indeed, waiting and investing in R&D appears to be the most cost-effective hedging strategy.Stochastic Programming, Uncertainty and Learning, Endogenous Technical Change

Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models

We develop numerically stable and accurate stochastic simulation approaches for solving dynamic economic models. First, instead of standard least-squares methods, we examine a variety of alternatives, including least-squares methods using singular value decomposition and Tikhonov regularization, least-absolute deviations methods, and principal component regression method, all of which are numerically stable and can handle ill-conditioned problems. Second, instead of conventional Monte Carlo integration, we use accurate quadrature and monomial integration. We test our generalized stochastic simulation algorithm (GSSA) in three applications: the standard representative agent neoclassical growth model, a model with rare disasters and a multi-country models with hundreds of state variables. GSSA is simple to program, and MATLAB codes are provided.Stochastic simulation; generalized stochastic simulation algorithm (GSSA), parameterized expectations algorithm (PEA); least absolute deviations (LAD); linear programming; regularization.

Multistage Stochastic Portfolio Optimisation in Deregulated Electricity Markets Using Linear Decision Rules

The deregulation of electricity markets increases the financial risk faced by retailers who procure electric energy on the spot market to meet their customers’ electricity demand. To hedge against this exposure, retailers often hold a portfolio of electricity derivative contracts. In this paper, we propose a multistage stochastic mean-variance optimisation model for the management of such a portfolio. To reduce computational complexity, we perform two approximations: stage-aggregation and linear decision rules (LDR). The LDR approach consists of restricting the set of decision rules to those affine in the history of the random parameters. When applied to mean-variance optimisation models, it leads to convex quadratic programs. Since their size grows typically only polynomially with the number of periods, they can be efficiently solved. Our numerical experiments illustrate the value of adaptivity inherent in the LDR method and its potential for enabling scalability to problems with many periods.OR in energy, electricity portfolio management, stochastic programming, risk management, linear decision rules

Heterogeneous-Expectations Model of the Value of Bonds Bearing Call Options

This paper develops a dynamic programming model of the optimal refunding strategy and the corresponding value of a callable bond. The model differs from previous work on this subject primarily in that it explicitly admits the possibility of differences between the issuer's expectations of future interest rates and an investor's corresponding expectations. This generalization facilitates the application of the model to determine what a specific bond (issued, for example, by a particular corporation) is worth to any given investor. Additional analytical features of the model, which differ from corresponding aspects of some previous models, include the use of a stochastic discounting rate and the use of continuous distributions to characterize the relevant interest rate expectations. For the bond issuer, his own expectations (together with the bond's coupon and call features) suffice to indicate the critical refunding yield as well as the expected value of the bond in each time period until the bond matures. For an investor, however, the analytical solution of the model and the illustrative numerical examples presented in the paper show that the issuer's expectations and the investor's own both matter if the two differ.