**Suggested Citation:**"Appendix C - Monte Carlo Simulation and Value-at-Risk Analysis." National Academies of Sciences, Engineering, and Medicine. 2019.

*Climate Resilience and Benefit–Cost Analysis: A Handbook for Airports*. Washington, DC: The National Academies Press. doi: 10.17226/25497.

**Suggested Citation:**"Appendix C - Monte Carlo Simulation and Value-at-Risk Analysis." National Academies of Sciences, Engineering, and Medicine. 2019.

*Climate Resilience and Benefit–Cost Analysis: A Handbook for Airports*. Washington, DC: The National Academies Press. doi: 10.17226/25497.

**Suggested Citation:**"Appendix C - Monte Carlo Simulation and Value-at-Risk Analysis." National Academies of Sciences, Engineering, and Medicine. 2019.

*Climate Resilience and Benefit–Cost Analysis: A Handbook for Airports*. Washington, DC: The National Academies Press. doi: 10.17226/25497.

**Suggested Citation:**"Appendix C - Monte Carlo Simulation and Value-at-Risk Analysis." National Academies of Sciences, Engineering, and Medicine. 2019.

*Climate Resilience and Benefit–Cost Analysis: A Handbook for Airports*. Washington, DC: The National Academies Press. doi: 10.17226/25497.

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79 Chapter 2 provided an initial discussion of Monte Carlo simulation techniques and associated VaR analysis. Where appropriate, much of that material is repeated here, along with more detailed discussion, to ensure that a complete description of the techniques is accessible in one place. Monte Carlo Analysis In the context of climate change, the uncertainty surrounding future climate events can often be characterized in terms of the percentage chance (or probability) of an event occurring. For example, one may have estimates that there is currently a 2% probability of a significant flood event, and this percentage will increase evenly by 0.1% per year over the next 30 years. This is enough information to perform a Monte Carlo simulation. The basic idea would be to draw a random number (by convention, between 0 and 1) for each year, with the value determining whether a storm surge event occurs in that year based on its probability. For example, a random number between 0 and 0.02 would indicate that the flood event occurs, while any number drawn higher than that would indicate no flood event. After going through all 30 years, one would have completed one simulation showing a possible future path for such flood events. This process then could be repeated multiple times (drawing new random numbers each time), thereby generating many different simulations representing possible future events. Another possibility is that, rather than having estimates of a percentage probability of a single event occurring, one instead has projections of, say, maximum daily temperatures from a num- ber of different climate models. If temperatures exceed some threshold, flight departures at the airport may be disrupted for certain aircraft types given the length of the runway. Here the range of variation and uncertainty depends not only on which future year one is looking at, but also which model is being used. The idea in this case would be to sample from the different models, yielding projections of the number of days that exceed the temperature threshold. If there were 10 different models, then one could assign Model #1 to the interval 0.0â0.1 for sampling pur- poses, Model #2 to the interval 0.1â0.2, and so on. Again, by repeating the random draws many times, the result will be multiple iterations that show many possible future outcomes for the number of times that daily high temperature exceeds the threshold. But how does this all fit into a BCA or FFA? Consider the following simplified example: sup- pose an airport is considering building a runway extension to handle the extra takeoff length required on days when the temperature exceeds 110Â°F. As with any BCA, the goal is to evaluate the present value of benefits and costs over time. Presumably, the costs of the runway extension can be accurately projected (say $X) based on construction and maintenance estimates. The harder part is to identify and quantify the benefits. For simplicity, suppose that without the runway extension, carriers will experience schedule delays or weight restrictions estimated A P P E N D I X C Monte Carlo Simulation and Value-at-Risk Analysis

80 Climate Resilience and BenefitâCost Analysis: A Handbook for Airports to cost $Y per day if the temperature exceeds 110Â°F. With the extension, flights can operate normally without any service disruptions. The suggested approach would be to run Monte Carlo simulations of the maximum daily temperature projections from the different available climate models. Suppose the analyst decides to do 5,000 simulations over a 30-year analysis period. So for Simulation #1, it may turn out that the Year 1 projection is for 5 days in excess of 110Â°F; in Year 2, there are 12 such days, and so on. After going through all 30 years, one can then compute the base-case damages that would occur without the project; the scenario damages (if any) that would occur with the project; and the construction, maintenance, and operation costs of the project. From these numbers, one can also compute the NPV of the project and a benefitâcost ratio. The entire process would be repeated 5,000 times,18 each time generating a new set of results that will vary depending on the number and timing of high-temperature days in the future. Some of the runs will have much lower than average high-temperature days; others will be just the opposite. But collectively, the results should accurately reflect the range and likelihood of high temperatures as projected across all the different climate models. One could then compute a mean value and standard deviation across the 5,000 NPVs or benefitâcost ratios. This is valuable information for decision makers and provides estimates of not only the average expected NPV or benefitâcost ratio but also the likely range of outcomes as measured by the standard deviation. Note that the Monte Carlo simulation approach could be used just as well in a financial feasibility study, the only difference being the inclusion or exclusion of certain benefits or costs, depending on whether they accrue to the airport itself. (This topic is discussed further in Chapter 7.) At this point, it is important to note that using a probabilistic benefits approach as described here is not the method that the FAA is accustomed to when assessing requests for AIP funding. Rather, as noted in ACRP Synthesis 13, its general approach is that a benefit either will or will not be realized with certainty, and therefore will or will not be included in a BCA (Landau and Weisbrod 2009). However, as was shown previously, there are cases where the FAA accepts BCAs where benefits are estimated over multiple scenarios reflecting different assumptions about uncertain future events. Moreover, the FAAâs BCA guidance document explicitly addresses the issue of uncertainty and suggests that Monte Carlo methods (what it calls probabilistic or stochastic models) may be used to âgenerate quickly hundreds or thousands of scenarios based on the specified probability distributions of uncertain variablesâ (FAA 1999b, p. 89). In essence, the Monte Carlo simulation approach described here is a formal method for con- sidering multiple scenarios; it can be thought of as a robust way of handling uncertainty about future events that may or may not occur. Given the uncertainty inherent in long-term climate predictions and the fact that when a given climate event will occur is likely to be highly uncertain, a probabilistic approach really is the only reasonable and valid way to account for climate risks. No single, certain alternative provides a realistic estimate of the likely benefits from an invest- ment designed to enhance climate resilience. Value at Risk As a natural extension, one can also use the results from the simulations to look at VaR, which is a concept that originated in the financial industry in the late 1980s. The idea was to estimate the likelihood of a financial firmâs maximum loss during a relatively short period of time. Because these financial institutions managed large and highly diverse portfolios, it was often difficult to fully understand all of the risks they were exposed to.

Monte Carlo Simulation and Value-at-Risk Analysis 81 A sample VaR for the portfolio of a large financial institution might show that it has a 1% chance of losing $100 million or more in a day given its portfolio and the historic price movements in the underlying individual stocks and bonds. In conventional usage, one would say that the companyâs 1% VaR is $100 million. This is a useful metric because it gives managers and regulators a way to assess how much capital a firm should have on hand to cover maximum daily losses. The Securities and Exchange Commission instituted capital requirements for financial firms in 1980 sufficient to cover, with 95% confidence, the losses that might be incurred during the time it would take to liquidate a securities firm (30 days) (Holton 2002). One important difference between VaR as applied to climate risk and that applied to conven- tional financial risk is that the relevant time periods involving climate change are decades long. However, the information VaR provides to managers is similar: it provides a means for deciding how much risk the enterprise is willing to accept, in this case through the distribution of poten- tial losses from climate change. In the present context, the output from a Monte Carlo BCA can be transformed into a VaR analy- sis in a straightforward way. Recall from earlier discussion that a traditional BCA compares benefits of the project (measured as the discounted present value of the dollar reduction in damages) to the costs of the mitigation project (including construction, maintenance, and operation). For purposes of a VaR analysis, rather than focusing on the benefitâcost ratio of a project, one can look at the results in a slightly different way and consider the net impacts for both the base case and the scenario. For the base case, net impacts are simply the present value of the dollar damages incurred if the project is not undertaken. For the scenario, net impacts are the present value of the remain- ing damages not mitigated by the project plus the present value of the investment costs (includ- ing construction, maintenance, and any other relevant costs) for the project. For VaR purposes, each of these impacts will be represented as negative dollar quantities. One could then plot these two quantities on a graph; if the scenario value is more negative than the base-case value, this indicates that the project did not pay off. This would be repeated for each Monte Carlo simulation, resulting in a new pair of net impacts under the base case and scenario. To assess these results across all the simulations, they can be sorted based on the differ- ence between the two values and then plotted along a percentage scale. The result is a VaR graph such as the one shown in Exhibit C-1. Based on the varying benefit results from the Monte Carlo simulations, the blue line in the chart shows that if the airport does nothing, it faces a 10% chance of incurring damages (in the form of delay costs) of at least around $25 million (where the blue line passes the 10% point on the horizontal axis) and could incur damages of more than $50 million. On the other hand, if it does undertake the mitigation project, it must pay the investment costs and incur any remaining delay impacts; these two factors combined could total as much as about $30 million (left extremity of chart for the red line). But also note that the range of potential net impacts is much larger under the baseline case (from about $5 millionâ$50 million in damages) than under the scenario case ($10 millionâ $30 million in damages and project costs). The chart also shows that there is about a 50% chance that the NPV of the project would be positive (indicated by the point at which the two curves intersect). It is important to properly interpret the meaning of these results. Facing a 10% chance of incurring damages of at least $25 million means that in 10% of the simulations, the present value of damages was $25 million or worse. Remembering that each simulation represents a set of future outcomes running from 2020 through 2090, these will include many different spe- cific outcomes that vary across the years. In some simulations, there may be a small number of unusually hot years early on, resulting in a few highly valued delays (because they are discounted less when occurring early). In many others, the high temperatures will have been estimated to occur in later years, but they are likely to occur more often, resulting in more lower-valued

82 Climate Resilience and BenefitâCost Analysis: A Handbook for Airports delays. So it is important to recognize that the 10% chance of damages includes many differ- ent potential outcomes; it does not refer to an annual probability of occurrence, but rather the overall likelihood (over the entire analysis period) that the airportâs users would face $25 million or more of delay costs (in present value terms) under the base case. This provides a different perspective from simply focusing on the average NPV or average benefitâcost ratio from the simulations. The airport can use the results to help it decide between the risky, but higher potential payoff of doing nothing, and the certain cost of investing in the miti- gation project that reduces but does not completely eliminate its exposure. If desired, the results could also be displayed in alternate waysâfor example, by highlighting the worst or best possible results (shown at the extreme left or right for the baseline and scenario cases in Exhibit C-1), or by presenting results for every decile (e.g., the benefitâcost ratio at 10% likelihood, 20% likelihood, and so on). Outside Examples Several examples exist of using VaR as a means to assess the global impacts of climate risk. One analysis estimates that the expected âclimate value at riskâ of global financial assets today is 1.8%, assuming a âbusiness-as-usualâ global emissions path (Dietz 2016). Taking a repre- sentative estimate of global financial assets, this amounts to around $2.5 trillion. Using a simi- lar methodology, the Economist Intelligence Unit (2015) estimates that expected climate VaR would be about $7 trillion, assuming private investor discount rates, but $43 trillion using lower government discount rates. There also have been various studies of VaR applied to specific assets and locations. One such study described the use of VaR to test the viability of drought derivatives or hedges that had been proposed for financial markets as a means for farmers (and others) to offset some of the risks of climate change in Switzerland (Torriani 2008). Another developed climate VaR estimates with respect to the production and income for coffee production in Veracruz, Mexico (Estrada et al. 2012). Finally, a VaR analysis was used as part of a risk assessment of sea level rise due to climate change for 19 large European coastal cities (Abadie et al. 2016). Exhibit C-1. Value-at-risk comparison.