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In this paper a case study is presented to propose an alternative mechanism to include the impact of climate change into the hydropower projects' feasibility valuation. We start from independent engineer historical energy generation simulations, therefore applying mixing unconditional disturbance and extreme value theory, a new path that satisfy a return level specification is created. New path is used to analyze the effect of extreme events on the internal rate of return of the project. This mechanism could be used also to execute a simple sensitivity test that it's done with an educated guess. Keywords: extreme value theory, generalized Pareto distribution, return level, mixing unconditional disturbances, climate change, and stress testing. JEL Classification: C, G. 1. Introduction With a changing climate, the resource potential for hydropower could change due to: a) Changes in river flow (runoff) related to changes in local climate, particularly in precipitation and temperature in the catchment area; b) Changes in extreme events (floods and droughts) may increase the cost and risk for the hydropower projects; and c) Changes in sediment loads due to changing hydrology and extreme events; more sediment could increase turbine abrasions and decrease efficiency, and increased sediment load could also fill up reservoirs faster and decrease the live storage, reducing the degree of regulation and decreasing storage services (IPCC 2011). Also, many of the current climate change studies indicate that the frequency in the occurrence of extreme events will increase in the future (IPCC 2007). In this paper, we will analyze the effect of extreme events, which alter annual energy generation of a hydropower, on the internal rate of return of the project. 2. Case study The case study refers to a hydropower plant of 20.0 MW installed capacity developed in Central America. Following table summarize annual energy generation (GWh) estimated for an international prestige independent engineer using historical daily streamflow records: Table 1. Historical annual energy generation simulations 1976 88.1 1988 93.2 2000 89.4 Average 1977 77.9 1989 83.5 2001 105.9 94.1 1978 90.2 1990 96.8 2002 110.7 St. Dev.: 1979 84.0 1991 93.2 2003 103.9 10.3 1980 93.5 1992 105.9 2004 97.3 Min.: 1981 100.8 1993 85.3 2005 107.4 77.5 1982 97.8 1994 90.2 2006 82.9 Max.: 1983 98.9 1995 77.5 2007 101.2 122.1 1984 88.2 1996 78.3 2008 122.1 1985 89.8 1997 84.6 2009 95.5 1986 96.1 1998 102.6 2010 109.1 1987 89.9 1999 80.1 2011 95.9 2.1. Extreme events Extreme events occur when a risk takes values from the tail of its distribution (McNeil 1999). The Central American Bank for Economic Integration (CABEI) is Central America's main multilateral bank and financial arm. Issue 2(6) Volume III Winter 2012 Let X = (X1, ..., Xn) be independent identically distributed random variables with a unknown distribution function F. The sample maximum, Mn, with n the size of the block is defined Mn= max (X1, ..., Xn). Under the Fisher - Tippett Theorem the sequence of normalized maxima converges in distribution: H(x) = exp ( -1(1 + ( ))-1/ ) for 0 where is the location parameter, is the scale parameter and is the shape parameter. Using the extRemes Toolkit developed by Eric Gilleland, within statistical software R, we apply the Block Maxima method an estimated a Generalized Extreme Value Distribution (GEV). As we are interest in the minimum annual energy generation, we must first transform the data: -Max(-X1, ..., -Xn) = Min(X1, ..., Xn). Estimated GEV has parameters: µ=-96.5439, =11.06785 and = -0.50838 2.1.1. Return level The return level is the level expected, on average, to be exceeded in one out of k periods of length n. The return period is the amount of time expected to wait for particular return level to be exceed; return period is the inverse of the probability of an event (e.g. a called "100 years event" has a 1% probability of exceed the record level in a given year). Return level is simply the calculation of quantiles from the Generalized Extreme Events Distribution, specifically: Pr (Mn ) = 1/k (1 1/k) ( 1- ( - ln (1 1/k))- ) for 0 The estimated 100 years return level (R100) is -76.8, with 95% confidence interval of (-78.94354, 71.30753), meaning, on average, only once in one hundred years the annual generation will be below that level. Figure 1 shows return level plot. Figure 1. Return Level Plot for the Historical Annual Energy Generation Simulations 2.2. Modeling impact of climate change The main premise to modeling the impact of climate change is the assumption that a "100 years event" turns into a "much lower years event", in this case, the probability of exceed the record level in a given year will Journal of Environmental Management and Tourism increase from 1% to 25%, from 1 event every 100 years to 25 events every 100 years. Therefore, we have to create a new annual energy generation path that computes a 4 years return level (R4) equal to -76.8. 2.2.1. New path construction Tompkin, and D'Ecclesia (2006) introduce the Mixing Unconditional Disturbances (MUD) model where simulations of path are obtained re-writing history, under this approach parameter estimation and distributional assumption are not required and the statistical characteristics of the original path are conserved. Given the historical series returns for a variable Xt, for t = 0,...,T, the unconditional mean µ, and standard deviation , are estimated. Normalizing the sequence of the variable yields: Zt = Xt - µ / where Zt is the series of standardized "disturbances" from 1 to T. By design, resulting disturbances have a mean of 0 and standard deviation of 1. The simulated variable Xt at each time t > 0 are obtained using the standardized disturbances, to generate new path we "freeze" the Zt and use formulation: Xt = Zt * + µ We look for a simulated new path (Figure 2) that search a lower average annual energy generation and a higher standard deviation, and also that compute the required return level specification. G W h Figure 2. New Historical Annual Energy Generation Simulations Path Next table summarize the new annual energy generation (GWh) path estimated: Table 2. New Historical Annual Energy Generation Simulations Path 1976 88.1 1988 92.7 2000 87.5 Average 1977 76.7 1989 81.9 2001 105.7 915 1978 90.2 1990 96.5 2002 110.9 St. Dev.: 1979 83.3 1991 92.5 2003 103.3 20.7 1980 93.9 1992 106.5 2004 95.8 Min.: 1981 101.9 1993 83.6 2005 106.9 53.0 1982 98.5 1994 88.9 2006 79.9 Max.: 1983 99.6 1995 74.9 2007 99.9 134.1 1984 87.5 1996 75.7 2008 123.0 1985 89.2 1997 82.5 2009 93.4 1986 96.2 1998 102.2 2010 108.3 1987 89.2 1999 77.4 2011 93.7 Figure 3 depicts comparative histogram of original and new historical annual energy generation simulations paths . 3.5% D e n s i t y 3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% 60 70 80 90 100 GWh 110 120 130 140 Original Path New Path Figure 3. Annual Energy Generation Simulations Histogram Plot For the new path, estimated GEV has parameters: µ=-97.74382, =20.98933 and =-0.37610. Issue 2(6) Volume III Winter 2012 The 4 years return level (R4) recorded is -76.8, with 95% confidence interval of (-84.36307,-69.44988). Figure 4 presents return level plot for the new path. Figure 4. Return level plot for the new historical annual energy generation simulations path 2.2.2. Internal rate of return To compute the internal rate of return of the project, first we assumed a total investment cost of US$60.0 million; second we use annual energy generation simulations (Table 1 & 2) to estimate: i) annual income as a multiplication of annual energy generation times a monomic price of US$120.0 / MWh adjusted by an annual increase of 1.5% (inflation rate), and ii) annual expense as a multiplication of annual energy generation times an operating and maintaining cost of US$20.0 / MWh; third no capital expenses, taxes and change in working capital is considered. Figure 5 shows annual cash flows for original and new path simulations. 30.0 U S $ m i l l i o n 10.0 0.0 -10.0 -20.0 -30.0 -40.0 -50.0 -60.0 Old Path New Path 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Year Figure 5. Cash flow simulations plot The impact in the internal rate of return of the project is around 150 bps, decreasing 8.8% from 16.7% to 15.2%. Similar impact results, a reduction in the IRR single value between 6% and 16% was obtained by Harrison et. al. (2003). Additionally, if we assume an equity contribution of US$18.0 million (30% of total investment cost), and a senior debt of US$42.0 million (70% of total investment cost) to be paid under following conditions: 15 years tenor, 8% interest rate, and "mortgage style" payments for an annual US$4.9 million debt service payment, therefore the impact in the internal rate of return of investors is around 375 bps, decreasing 14.5% from 25.9% to 22.2%. Figure 6 presents annual free cash flows for original and new path simulations. Journal of Environmental Management and Tourism 20.0 U S $ 15.0 5.0 m 0.0 i -5.0 l l -10.0 i o -15.0 n -20.0 Year Old Path New Path Figure 6. Free cash flow simulations plot As a result of the approach, climate risk is reflected in a reduction of project's cash flow and investors' free cash flow, however, selection of discount rate to resolve about the feasibility of the project is a final subjective decision from risks' takers. 2.2.3. Stress testing Stress results helps asses risk taken versus risk appetite, identify major contributors to overall event risk exposure, and uncover hidden sources of risk (Schachter 1998). Stress tests are inevitably subjective because they depend on scenarios chosen by the stress tester. As a result, the value of the stress testing depends critically on the choice of scenarios and therefore on the skill of the modeler. (Aragones et al. 2000). The most common stress tests involve the determination of the impact of a move in a particular risk factor. In the case of hydropower projects valuation, a simple sensitivity test changing average annual energy generation (e.g. ± 10.0%) is frequently done. The alternative mechanism proposed to include the impact of climate change into the hydropower projects' feasibility valuation, could be used to execute a simple sensitivity test that it's done with an educated guess. 3. Conclusions and extensions In this document, we propose a new approach to include the impact of climate change into the hydropower projects' feasibility valuation applying mixing unconditional disturbance and extreme value theory based on the main assumption that a "100 years event" turns into "much lower years event" and evaluating these impact in the internal rate of return. The obtained results with this new technique could provide a simple sensitivity test, too. We presented here only one particular scenario of the many possible climate change impacts, future lines of research could evaluated multiple clime change scenarios and/or multiple return level specifications. References  Aragones, J.R., Blanco, C., and Dowd, K. 2001. Incorporating stress tests into market risk modeling. Derivatives Quarterly, Spring: 44-49.  Edenhofer, O. et al. 2011. Renewable Energy Sources and Climate Change Mitigation. Special Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, ISBN: 978-1107607101  Harrison, G., Whittington, B., and Wallace, R. 2003. Climate change impacts on financial risk in hydropower projects. Institute of Electrical and Electronics Engineers - Power Systems, 18 (4): 1324-1330.  Parry, M.L., Canziani, O.F., Palutikof, J.P., van der Linden P.J., and Hanson C.E., Eds. 2007. Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, UK, 976 pp  McNeil, A.J. 1999. Extreme Value Theory for Risk Managers. Internal Modelling and CAD II. RISK Books, 93113. Issue 2(6) Volume III Winter 2012  Schachter, B. 1998. The Value of Stress Testing in Market Risk Management. Derivatives Risk Management Service.  Tompkins, R.G., and D'Ecclesia, R.L. 2006. Unconditional Return Disturbances: A Non-parametric Simulation Approach. Journal of Banking & Finance: 30(1), 287-314.
Journal of Environmental Management and Tourism – de Gruyter
Published: Dec 1, 2012
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