Design criteria for flood-defense structures based on probabilistic cost-benefit optimization with value at risk ( VaR ) methods . Application to the Choluteca River in Tegucigalpa ( Honduras )

A probabilistic CBA framework, combined with a Value at risk (VaR) methods, as applied in financial risk management, can be used to select the best mitigation scheme among several alternatives and reliability levels, based on a quantitative and objective procedure. The proposed method looks for the alternative that minimizes the accumulated maximum damage that can be produce by any particular sequence of events, over the life span of the structure, using numerical simulation and possibly including interactions among individual events (two large floods within a short time cannot damage twice the same assets). This is equivalent to a stochastic optimization problem, where the entity to be minimized are the maximum losses. The optimal alternative, based on a VaR criteria (including conditional VaR), differs largely from the one that maximizes the average NPV, and is more stable, compared with the average or a deterministic NPV. To demonstrate the proposed procedure, and show the differences among the three performance indicators (average NPV, VaR and CVaR), the case of the Choluteca River in Tegucigalpa, capital city of Honduras, are used, with real data of economic and human damages provided by a recent study by IDB.


Motivation
Extreme floods are ubiquitous and represent the most common natural catastrophes around the world [1]. In comparison with other hazards as earthquakes and landslides, floods are expected to increase in magnitude and frequency in many regions of the world, due to climate change and land-use transformation processes. The global investment needs in flood reduction schemes for the next decade are expected to be huge, and many of them will be aimed at developing Countries, where the financial conditions do not bode well for large public infrastructures. The problem of money allocation for flood reduction schemes will come more and more to the forefront, as the human and economic damages due to extreme events become more frequent and harsher, and their impacts become more global. In any case, there will probably be a need to increase and polish the ex-ante project evaluation and appraisal techniques for flood reductions programs, in order to optimize the allocation of economic resources, in a context of over-demand for this type of interventions.
There is a growing trend to analyze flood reduction schemes from the point of view of cost-benefit analysis (CBA), including as an income the spared losses and possibly other monetized public benefits as land value surpluses and indirect damage reduction (traffic interruptions, commercial activity, etc.). Many financial institutions, and in particular development banks as WB, IDB and ADB advocate for cost-benefit analyses of public infrastructures, including risk reduction schemes, in order to guarantee a coherence between the resourced needed and the resulting benefits; furthermore, CBA is a convenient tool to benchmark different alternatives, even different in nature, for a particular problem, and eventually select the fittest one.

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application of CBA to the evaluation of flood reduction schemes in several countries (mainly of middle and low income) over the last decade, and tries to reflect on some of the strengths and weaknesses of such approach for this particular type of projects. Although CBA is a simple and powerful tool, it has some limitations, when applied to evaluate and design flood mitigation alternatives for a particular site. Firstly, there is an inherent difficulty in dealing with human damages. The human dimension of flood risks is usually characterized in terms of affected population (with several possible definitions), refugees (people in need of shelter due to the destruction of their houses), number of casualties, deceased, etc. CBA has to express such losses in monetary terms, which opens a moral dilemma and leaves the choices of unit prices to the subjectivity of the practitioner. This issue will not be the main concern of this work, although some comments and suggestions will be given on how to address it.
A second drawback of the CBA approach is the stochastic nature of the natural phenomena triggering the flood risks: rainfall, discharges, storm surge and other sources are random processes, sometimes with complex cross-correlations, that should not be treated in a deterministic way. Since the timeline of damages is dictated by the stochastic sequence of extreme events, and the reduction or mitigation of such damages is the final goal of any intervention, ACB should implement the probabilistic nature of the hazards. Furthermore, ACB is based on the economic quantification of the value of time through a compound-interest discount rate, which makes more compelling the need to pay attention to the chronological structure of extreme events. In a standard CBA, both investments and revenues are treated in a deterministic way, and spared costs are introduced as a mean annualized damage, often obtained from the distribution of damages for selected events. This approach dodges the probabilistic nature of extreme phenomena and their highly variable patterns of occurrence in time. Since ACB is based on the timevarying cost of money through an interest rate, the exact sequence of events that a particular investment will withstand largely affects its financial profitability, measured as its net present value (NPV), which shows high volatility, depending on each realization of the series of floods, possibly fitted with a certain extreme value distribution.
Finally, there are other potential limitations of the CBA approach for flood mitigation appraisal, which will not be here addressed: This paper puts forward a methodology, based on Monte Carlo simulation of different realizations of sequences of extreme events, to perform a probabilistic CBA to evaluate and benchmark flood mitigation projects. It draws on several concepts and tools that have been used in the financial realm since the 198 ¶V EXW have seldom or not at all been applied for public infrastructures. Financing a mitigation scheme to reduce flood risks is itself a highly risky decision, albeit the use of public money and the difficulty in monetizing some of its benefits (human, social, environmental) make it less evident, or simply moves the decision making to the political arena. The proposed method tries to counteract these facts and give both decision-makers and politicians a more pragmatic view of the nature of the problem they face.
Currently, many of the economic appraisals of public infrastructures carried out by financing institutions around the world rely on an expert-based, scenario-driven approach, in which a particular investment generates a series of deterministic cash flows over time, which eventually can be summarized in a set of profitability parameters (net present value, internal return rate, equilibrium time, payout ratio, etc.) indicating how good the investment is. Here, it will argued that a value-at-risk (VaR) method could be more realistic and reflect in a better way the nature of the decisions under consideration, especially when assessing risk mitigation schemes. From this standpoint, the decision-maker should focus on minimizing the economic risks involved in a particular decision or, in other words, make sure that the maximum losses to be attained are minimum. In this framework, the key feature is the tail of the probability density function (PDF) of the net present value (NPV) for a particular alternative/scenario. The most effective alternative is the one that gets to push the right tail of the NPV distribution furthest towards the negative axis (reminder: losses are positives by convention), implying that the maximum failure or loss is minimum. Basically, we advocate for the use of new measure of profitability, a VaR or more exactly a variant of it (the conditional value at risk or CVaR), instead of the classic average NPV. This new measure requires the consideration of the cash flows and their discounted sums as probabilistic functions, making the procedure somehow more datademanding and cumbersome to apply, at least with a standard spreadsheet.   Table 1. Source: IDB [8].