Making real options analysis more accessible for climate change adaptation. An application to afforestation as a flood management measure in the Scottish BordersRuth Dittricha, Adam Butlerb, Tom Ballc, Anita Wrefordd, Dominic Morane

a University of Portland, 5000 N Willamette Blvd, Portland, OR 97203, U.S., phone: +1-503-943-8000; email: dittrich@up.edu

b BioSS, James Clerk Maxwell Bldg, King’s Buildings, Mayfield Rd, Edinburgh EH9 3JZ, UK United Kingdom, email: adam.butler@bioss.ac.uk

c University of Winchester, Department of Geography, Sparkford Road, Winchester SO22 4NR, United Kingdom; phone: +44 1962 675129; email: tom.ball@winchester.ac.uk

d Lincoln University, PO Box 85084, Lincoln 7647, New Zealand; phone +64 3 230376; email: anita.wreford@lincoln.ac.nz

e Global Academy of Agriculture and Food Security, University of Edinburgh, Easter Bush Campus, Midlothian, EH25 9RG, United Kingdom, phone: +44 131 651 7439, email: dominic.moran@ed.ac.uk

AbstractClimate change uncertainty makes decisions for adaptation investments challenging, in particular when long time horizons and large irreversible upfront costs are involved. Often the costs will be immediate and clear, but the benefits may be uncertain and only occur in the distant future. Robust decision-making methods such as real options analysis (ROA) handle uncertainty better and are therefore useful to guide decision-making for climate change adaptation. ROA allows for learning about climate change by developing flexible strategies that can be adjusted over time. Practical examples of ROA to climate change adaptation are still relatively limited and tend to be complex. We propose an application that makes ROA more accessible to policy-makers by using the user-friendly and freely available UK climate data of the UKCP09 weather generator, which provides projections of future rainfall, deriving

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transition probabilities for the ROA in a straightforward way and demonstrating how the analysis can be implemented in spreadsheet format using backward induction. The application is to afforestation as a natural flood management measure (NFM) in a rural catchment in Scotland. The applicability of ROA to broadleaf afforestation as a NFM has not been previously investigated. Different ROA strategies are presented based on varying the damage cost from flooding, fixed cost and the discount rate. The results illustrate how learning can lower the overall investment cost of climate change adaptation but also that the cost structure of afforestation does not lend itself very well to ROA.

Keywords: climate change adaptation, economic appraisal, real options, flooding, afforestation

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1 Introduction

Climate change will increasingly lead to impacts associated with financial burdens. The 2014 IPCC summary for policy-makers (IPCC, 2014b) identifies increased economic loss from inland flooding as one of eight key risks of climate change, with potentially severe consequences for humans and socio-ecological systems. The identification of key risks is intended to help policy-makers prioritise adaptation investments, however it is challenging to make precise investment recommendations to reduce vulnerability to flood risk because of the uncertainty involved.

Investment costs may be immediate and clear while the benefits are uncertain and may only accrue in the distant future. The uncertainty stems from a number of sources: natural variability in the climate system; downscaling climate models used to explore climate change impacts on flooding (Towler et al., 2010, Otto et al., 2018); the unknown extent of climate change mitigation in the future; socio-economic changes which may increase or decrease the value of assets at flood risk; and the preferences of future generations add to the uncertainty (Dessai, van der Sluijs, 2007, Burke et al., 2016). Thus, there is a dilemma when it comes to climate change adaptation. As the effects of climate change are uncertain, decision-makers may be reluctant to invest in flood protection measures with possibly high and irreversible costs. Yet at the same time, inaction and under-investment may lead to potentially severe flood damages, and delayed action and disaster relief may be even more costly.

Robust decision-making tools to guide investment under uncertainty

Increasingly, robust decision-making tools are recommended in situations of uncertainty such as in the context of climate change adaptation to flooding (Dittrich, Wreford & Moran, 2016). These tools aim to incorporate uncertainty in adaptation investment appraisal by selecting projects that meet their purpose across a variety of plausible futures (for an overview of different robust methods see Dittrich, Wreford & Moran (2016)).

Real options analysis (ROA) is one robust decision-making tool that extends the principles of cost-benefit analysis of a ‘now or never’ decision by allowing for learning. The associated actions can be adjusted over time when additional information about climate change impacts becomes available. ROA originates

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from financial options (Black, Scholes, 1972) and has been further developed for financial investment and engineering projects since the 1990s (Trigeorgis, 1995). Real options can be “on” or “in” a project, or both (Wang, De Neufville, 2005). ‘‘On’’ a project has a time implication: delaying or modifying part or all of an investment until new information becomes available, for instance to expand or decrease. Real options ‘‘in’’ projects are technical engineering and design adjustments, which enable flexibility in operations that require the characterisation of interdependency/path-dependency amongst options (Cardin et al., 2013, Wang, De Neufville, 2005).

ROA is suited for (partly) irreversible investments with long lifetimes and sensitivity to climate conditions, when there is a significant chance of over- or underinvesting combined with an opportunity cost to waiting; i.e. if there is a need for action in the present. If the investment was completely reversible, i.e. no sunk cost was incurred, there would be no value in delaying the investment or setting it up with flexibility. However, most investments include fixed costs such as planning costs. Fixed costs are also the reason why incremental investments, e.g. annual reaction to observed changes in the climate, are inadvisable, as with every investment, fixed costs will have to be paid and cannot be recovered. Given that flood and water management infrastructure often has characteristics described above, ROA is an appropriate tool for analysis in this field.

The learning in ROA is based on an uncertain underlying parameter. In the context of flooding and climate change this will for example be precipitation or sea level rise. Due to climate change, hydrological variables are no longer reliably constant, and past hydrologic data do not necessarily provide a good indicator of future conditions, i.e. non-stationarity applies (Milly et al., 2008). Therefore, an annual exceedance probability (AEP)1 based on historical data will not deliver the required standard over time, for example, a 1% AEP may become a 1.5% AEP event in the future. In ROA, the uncertainty of the hydrological variable - at least with respect to climate change - is assumed to resolve with the passage of time due to increasing knowledge. ROA takes advantage of this assumption that the uncertainty is dynamic rather than deep and provides strategies that can be adapted in a changing context.

Case studies include investment in flood management for coastal and riverine flooding, both real “on” options (Scandizzo, 2011, Linquiti, Vonortas, 2012,

1 The annual exceedance probability (AEP) indicates the probability of occurrence of a flood in any given year.

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Abadie, de Murieta & Galarraga, 2017) and real “in” options such as for the Thames Estuary, England (Woodward et al. 2011). Gersonius et al. (2013) investigated the added value of real “in” options with respect to investments in urban drainage infrastructure in West Garforth, England. Several studies have examined water resource management under climate change. Jeuland and Whittington (2014) combined a ROA and robust decision-making approach (Lempert, Schlesinger, 2000) to guide water resources infrastructure investments and operating strategies for dams along the Blue Nile in Ethiopia. Haguma et al. (2014) optimised for the long-term planning of water resources systems and the mid-term operations for optimum hydropower production in Quebec, Canada. Case studies in South Korea used the ROA approach for the adaption of hydropower plants (Kyeongseok et al., 2016), urban infrastructure (Kyeongseok, Sooji & Hyoungkwan, 2017) drainage infrastructure (Taeil, Kim & Hyoungkwan, 2014). An application to the sequential expansion of rainwater harvesting systems reusing sceptic tanks in South Korea was demonstrated by Byungil et al. (2014). Finally, van der Pol et al. determined phased investments for dikes (2014) and cost-effective storage basins in a Dutch polder (2015). All studies show that flexible strategies are economically superior to inflexible strategies. De Neufville and Scholtes (2011) estimated that flexibility (for real “in” options) can bring expected performance improvements ranging between 10 and 30% compared to standard design and evaluation approaches.

While there is ample interest in robust-decision-making tools such as ROA on the policy level, it has not been widely used in actual policy making, possibly because it is relatively complex to implement. It requires an understanding of financial theory and relatively advanced mathematical techniques such as dynamic programming (van der Pol, van Ierland & Weikard, 2014) or genetic algorithms (Gersonius et al., 2013). Furthermore, data on the change of the uncertain parameter and transition probabilities are required, which may not be easily obtained.

In this article, we propose a ROA to make the tool more accessible to policy makers, and potentially to increase its use for climate change adaptation to flood risk. Here, accessible refers to ease of implementation: We have simplified the analysis using a spreadsheet approach and demonstrated it with an application to afforestation as a natural flood risk management (NFM) measure in a small catchment in Scotland. NFM are defined as techniques that aim to work with natural hydrological and morphological processes, features and characteristics to manage the sources and pathways of flood waters (SAIFF,

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2011, Forbes, Ball & McLay, 2015). This paper makes two contributions: first, we provide a more accessible approach to ROA to improve the tool for practitioners. We describe a general methodology to derive a ROA investment strategy in a spreadsheet format that can be applied to different climate variables such as rainfall or sea level rise. This includes in particular a novel and simple approach to derive the probabilities of different climate outcomes using the freely available UKCP09 climate data. Second, we explore the applicability of ROA to broadleaf afforestation as a NFM, which has not been previously investigated. To do this we develop a flexible real ‘on’ options strategy for planting broadleaf forest in a rural catchment in Scotland with the aim of minimising the total cost of the system, while avoiding a flood with an annual exceedance probability (AEP) of 5%.

Section 2 sets out the case study area, describes how broadleaves can work as a flood management measure and outlines the methodology for our case study. Section 3 presents the results followed by a discussion and conclusion in section 4.

2. Methodology for the case study

The case study area is the Eddleston Water catchment of 69km2 in the Scottish Borders, UK. The Eddleston Water is a small tributary of the River Tweed, flowing 17km north to south before reaching the main river Tweed in the town of Peebles. The village of Eddleston (940 inhabitants) and further downstream the town of Peebles (7850 inhabitants) which are both situated at the Eddleston Water are at risk of riverine flooding. A number of NFM have been implemented in the case study area to reduce the risk of flooding (and improve water quality) including afforestation with broadleaves (Tweed Forum, 2015). It should be noted that in many places NFM are human-made and alter the existing landscape: natural here does not imply that the features are already in place in nature, rather they are considered natural as opposed to ‘hard’ engineering

measures such as embankments or dikes. We use the term NFM throughout the text with this interpretation in mind as this is how it is used in the policy literature (Environment Agency, 2017, SEPA, 2015, European Commission, 2011). NFM is widely recognised as an option to reduce flooding alongside ‘hard’ engineering solutions for both fluvial and pluvial flooding, whilst achieving multiple benefits throughout the catchment such as ecosystem

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services including provision of habitat (Dadson et al., 2017). NFM is of increasing policy interest across Europe because of its potential to buffer the effects of climate change (European Commission, 2009, European Commission, 2011, Scottish Government, 2009, McVittie et al., 2018). Here we focus on afforestation as an example of NFM. Over time, trees develop a complex root system (growing and dying) creating preferential pathways for water flow and promoting higher infiltration rates (Schwärzel, Ebermann & Schalling, 2012, Archer, Quinton & Hess, 2002). Combined with higher rates of interception and evapotranspiration, this results in reduced runoff and sediment production (Calder, 1990). The performance of afforestation measures in reducing the flood peak in a specific catchment depends on several factors, notably degree of forest cover (Calder, Aylward, 2006), species, the previous land use and soil characteristics (Hümann et al., 2011) as well as spatio-temporal variations in rainfall and runoff (Pattison, Lane, 2012). For more information on the evidence of afforestation as a flood management measure, see Dadson et al. (2017) and Iacob et al. (2014).

We aim to identify the sequencing of planting forest that minimizes total cost from a 5% AEP2. This objective takes into account the cost from flooding but also the cost of planting trees. Maintaining a 5% AEP over time requires increasing flood protection measures at the same rate as the return period changes. The aim is to avoid both under and over-investment, which either results in a flood protection standard below 5% AEP, or flood regulation capacity above the required standard. We present several scenarios to illustrate how the strategy changes by varying inputs. These include 1) a base case, and relative to the base case 2) increased discount rate, 3) increased fixed cost, 4) high damage cost 5) very high damage cost. The decision problem can be structured in terms of the following stages (Gersonius et al., 2013):

1. Specify the decision-tree2. Identify the potential options 3. Formulate the optimisation objective4. Solve the optimisation problem

2 The 5% AEP was chosen as 1) no flooding occurs for a rainfall event with higher AEP and 2) flooding can be averted by afforestation in the catchment for all peak flows of a 5% AEP under different climate change outcomes.

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2. 1 Specify the decision tree

The decision problem can be depicted in a decision tree (Figure 1), the branches representing potential climate change paths, i.e. the expected change in rainfall intensity for different futures. The nodes describe the flood management measure implemented (hectare planted) depending on the different climate outcomes. We specify a decision tree with two decision points (2016, 2040) and four potential outcomes at each decision point and stopping planting in 2080. This is a compromise between adequately representing the climate uncertainty by choosing three points throughout the 21st century while reducing the complexity of the calculations by limiting the decision points. It is possible to choose other time slices depending on the specificities of each case study but any additional decision-point will increase the complexity significantly.

Figure 1 Decision tree for a real options analysis

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Determining the different branches, i.e. the climate change paths, is one of the main challenges in applications of real options to climate change. ROA assumes the realization of those branches occur with a certain probability, however, is not clear that probabilities for climate change outcomes on a case study level can be determined yet, due to the uncertainties described previously. Some authors (Kyeongseok et al., 2016, Gersonius et al., 2013, Linquiti, Vonortas, 2012, Scandizzo, 2011) assumed that climate change follows the stochastic process Geometric Brownian Motion (GBM) or the binomial options approach (Byungil et al., 2014, Abadie, de Murieta & Galarraga, 2017) both used in finance applications (Cohen, Black & Scholes, 1972, Cox, Ross & Rubinstein, 2002). Others used a moving window approach (van Der Pol et al., 2015). We applied an approach related to Woodward et al. (2011), which used the underlying distribution of the UKCP09 climate change data (Murphy et al., 2009) and is thus solidly based on the behaviour of climate projection models. We decided to use the UKCP09 as it is a publicly available repository that provides projections of future rainfall in the UK, based on perturbing the existing Weather Generator3 according to the probabilistic projections for climate change under different scenarios. This means the output rainfall can be directly fed into a hydrological model without further processing. In addition, the user interface is user friendly and does not require expert knowledge. The data is conditional on the high, medium and low GHG emission scenarios, corresponding to the 2001 IPCC Special Report on Emissions Scenarios (Watson, Albritton, 2001). As no information is available on the likelihood associated with the climate change scenarios, we have chosen the medium scenario to represent the central outcome. The A1B scenario is probably closest to the Representative Concentration Pathway RCP6.0 (IPCC, 2014a). However, given the recent evidence on future global emissions (IPCC, 2018, Le Quéré et al., 2016), we must assume that the A1B scenario is likely to be at the lower end of potential climate change outcomes, and we may underestimate the need for adaptation. Part of the rainfall data analysis process is illustrated in the flow chart of Figure 2.

3Weather generator use weather data and random number sampling to produce long time series of statistically plausible daily and hourly weather data.

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Figure 2 Calculation of transition probabilities and rainfall intensity of a 5% AEP from 2040 to 2080.

We downloaded 40 sets (with the same model ID) of rainfall data for the baseline period (1961-1990), the 2040s and the 2080s for our catchment location. The rainfall data for each time slice comes in 30-year hourly time series with 100 plausible climates in each set. This results in a 1200 (years) x 100 (plausible climates) matrix for each time slice (baseline, 2040, 2080). The data was analysed by fitting the generalized extreme value distribution to annual maxima (we call this the 'AM' method) (Coles, 2001) in the R package extRemes (Gilleland, 2015). We obtained 100 different rainfall intensities (based on the 100 plausible climates) of the 5% AEP flood event for the future periods. For the baseline period, only one rainfall intensity was obtained for all realisations, as they all originate from the same distribution and its variability reflects natural rather than climate change variability. The baseline period data characterises the current rainfall intensity of a 5% AEP, and is represented by the initial event node in the decision tree in Figure 1. To obtain the other 20 rainfall event nodes (4 for 2040 and 16 for 2080) in the decision tree as shown in the figure, the 100 return levels of the 5% AEP obtained for each 2040 and 2080 were respectively split into quartiles, and the mean return level estimate of each quartile represents the event nodes, i.e. the rainfall intensity of a 5% rainfall event in different futures. Thus, the range of climate change uncertainty is characterised by the distribution of the 100 return periods for the 5% AEP.

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The four blocks of the four event nodes in 2080 are identical, as these are the outcomes projected by the UKCP09 data for 2080.

We next determine the transition probabilities going from one branch to another. For each return level estimate associated with the 5% AEP, we know which quartile bin (labelled A to D, where A correspond to the 25th quartile and D to the 100% quartile) it belongs to, for the baseline period, 2040 and 2080. We assume that all baseline period runs are in one bin A. For a particular return level estimate, we might say that it is AB, i.e. implying that this particular return period lay in the 25th quartile of the 2040 distribution and within the 50th quartile of the 2080 distribution. As a result we obtain a list of 100 two letter-codes, characterising each of the runs, whose frequencies are entered in a transition matrix as shown in figure 2. Each row is scaled to sum to one by dividing each combination by the sum of its row to obtain the transition probabilities. For example, going from 2040 to 2080, we observe in figure 2 that 12 runs that were in the 25th quartile in 2040 (A) remain in the 25th quartile of the distribution (A) in 2080. Given that the total number of runs in this row is 24, we can determine the transition probability by calculating 12/24 =0.5. We can see that extreme outcomes such as moving from A to D (prob = 0) or D to A (prob = 0.04) are less likely than staying on the same climate path such as BB (prob = 0.36) or CC (prob = 0.32).

The use of UKCP09 is limited to the UK, however in many locations similar data will be available. What is needed to apply our method are a range of plausible climate outcomes (e.g. 100) of the variable of interest (e.g. rainfall) for different time slices (e.g. today, 2040 and 2080). In order to derive the transition probabilities as suggested here, it is also necessary to take note of the quartiles/bins in which the climate outcomes fall in the different time slices.

2.2 Identify the potential options

The range of options are available for building in decision flexibility depend on the specific problem. Here, flexibility comes from sequencing the planting of different hectares of the NFM measure (afforestation) with the full catchment and maximum afforestation corresponding to 6900 ha. Figure 3 shows the reduction in peak flow by implementing different levels of forest cover based on different rainfall intensities for our case study. Based on scenario runs of the hydrological model, we fitted functions to describe the relationship between

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forest cover and peak flow. For more information on the hydrological model, see Dittrich et al. (2018). The results can be directly used in the cost function.

Figure 3 Rainfall intensity (mm/h) and resulting peak flow (m3/s-1) for the baseline period and three

afforestation scenarios (2010, 4415, 6900 ha).

2.3 Formulate the optimisation objective

C=minzt

∑y t=1

Y t

I ( zt )+O(x¿¿ t)+D(x t)

(1+δ )y t− y1(1)¿

where C is the net present cost of total investment, operation and maintenance as well as damage costs. Investment costs are described by function I(zt), annual operation, opportunity and maintenance costs by function O(xt ), and damage

cost by D(xt). Costs are discounted at rate δ based on the recommendation of

the UK Green Book (HM Treasury, 2003). Damage occurs if an insufficient level of trees were planted. The decision variable is zt (e.g. investment in additional afforestation at the decision nodes and xt is the stock variable, the total forested area at year t. Additional investment zt is realised by planting forest zt at three decision points, at time t = 1, 2, 3 (corresponding to 2016, 2040 and 2080), i.e.:

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x t+1=x t+z t (2)

To avoid flooding, the peak flow needs to be below ∝ .

q tu(x t+ zt)≤∝ (3)

where ∝is a pre-defined standard and q tuis an estimate of the return level

associated 5% AEP of flow at time t based on a particular climate u. ∝ reflects

the threshold below which no flooding occurs. It is also possible to choose an exceedance probability for a specific water level (Van der Pol et al., 2015).

We apply figures from the case study areas to inform our cost functions. The costs for implementing the afforestation measures can be divided into investment cost I(z) and maintenance as well as opportunity costs O(x). Investment costs include fixed costs such as facilitation services (for example to negotiate with land owners) as well as fees and the variable planting costs and have been found to be logarithmic, i.e. the cost increase at a decreasing rate and doubling the forest size will lead to less than double the cost.

I ( z )={a+b ln ( z ) If z ≥10 If z<1

(4)

Maintenance costs m refer to thinning every five years and are assumed to be constant and linear depending on the hectare size. Opportunity costs n refers here to forgone use of land for sheep grazing, which is (mostly) the land use of the (modelled) afforested areas and is also assumed to be linear.

O ( x )=mx+nx(5)

2.4 Solving the optimisation problem

We show that an optimisation problem of this degree of complexity can be solved in a spreadsheet using backward induction starting with the last decision to be made and working progressively back in time from there.

The cost parameters used for the case study were obtained from actual figures from planting different plots of trees in the case study area and are specified with further parameters in Table 1. The damage cost for a 5% AEP rainfall event under the different quartiles was obtained from Dittrich et al. (2016). QMS (2014)

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figures on sheep profitability, suggest a net margin of £26 per ewe for improved pasture for the past years, which we adjust for 2016. We further assume that 1.5 ewes can be fed on one hectare in the case study area per grazing season based on land use data (Scottish Government, 2015).

We assume that the trees immediately have their full flood regulation effect but in practice, it can take between 5 -15 years for a full effect on the hydrological cycle (Farley, Jobbágy & Jackson, 2005).

Parameter Value (in 2012 prices where

applicable)

Constant (for fixed cost) £8600

Constant (for variable cost) £5200

Maintenance cost (per ha) £280

Opportunity cost (per ha) £40

Discount rate (until project year 30) 3.5%

Discount rate (after project year 30) 3%

Constraint for flood reduction ∝ 36 m3/s-1

Table 1 Case study parameters

In order to apply backward induction, we need to calculate in a first step the net cost for all 256 (44) paths if they were going to be implemented. Figure 4 shows part of the decision tree to illustrate this initial step. If path 1, which is a dotted line, eventuated, sufficient trees would be planted in 2016 (investment decision I1a) to prevent flooding of a 5% AEP year rainfall event associated with bin A (25th quartile of the distribution) in 2040. Getting to 2040, the 5% AEP rainfall event turns out to correspond to bin A (outcome N1a), so no further trees need to be planted to correct for a wrong decision in 2016 (I1=N1), and no damage cost has been incurred. Instead, further trees are planted to prevent flooding of a 5% AEP rainfall event in 2080 corresponding to bin A (25th quartile) in 2080 (I2a). In 2080, this choice turns out to be correct (N2a) and no further trees need to be planted (I2=N2). In 2080, we assume the uncertainty will have resolved and the final planting decision based on the climate outcome in 2080 can be made. The net cost of path 1 (P1 cost in figure 4) will therefore be the discounted cost of planting I1a + I2a. This includes the maintenance and opportunity costs, which depend on hectare planted. The cost of I3a is not incurred, as no additional trees are required as the correct outcome was anticipated in 2040. The net cost of path 2 (P2 in Figure 4) is identical to path 1 with the only difference that in 2080, the 5% AEP rainfall event corresponds to

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bin B (50th quartile) (N2 ≠I2). This has two implications: first additional trees would need to be planted in 2080 to make up the difference between the amount of trees planted in 2040 and the actual trees needed in 2080 to prevent the 5% AEP rainfall event. Second, damage from flooding occurs and causes additional cost.

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Figure 4 Illustration of backward induction for the decision problem

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In a second step, we carry out the backward induction. It is backward as it starts with the analysis of the last decision to be made and works from there to the initial planting decision. Our last and in this application second decision relates to the investment made in 2040 adapting to possible outcomes in 2080. This step tells us which investment decision to make at I2 depending on the state we reach in 2040. By 2040, we will have planted for one of the four quartiles and one of four possible outcomes of the rainfall intensity (the 25th, 50th, 75th or 100th quartile) will have been realized. This implies 16 possible states in 2040. We now have the choice of planting I2 which corresponds to the 25th, 50th, 75th 100th quartiles of the rainfall intensity associated with the 5% AEP in 2080. For example, as illustrated in Figure 4, if we had planted for the 25 th quartile in 2016 and the rainfall intensity associated with the 25th quartile had actually realised in 2040, we can make the investments I2a (25th quartile),....I2d (100th quartile) in a next step. The outcomes N2a (25th quartile),..., N2d (100th quartile) might occur in 2080 with the probabilities pAA2080, pAB2080, pAC2080 and pAD2080 respectively (A stands for climate outcomes in the 25th percentile, B for the 50th quartile etc, see also Figure 2.) Therefore, the expected cost for planting for the 25th quartile of the rainfall intensity in 2080 equals pAA2080P1 + pAB2080P2+ pAC2080P3 + pAD2080P4. P1 to P4 stand for the total net cost of each path calculated in step 1. This process needs to be carried out for the three other states we may be in 2040 (50th, 75th or 100th quartile) to decide on I2. We will always choose planting for the quartile with the lowest expected cost. The same procedure of obtaining the lowest net cost is carried out for the investment decision I1 in 2016. Specifically, we find the lowest net cost by multiplying the probabilities pAA2040…pAD2040 with the respective lowest outcome for I2 and comparing them for each quartile.

3. Results and discussion

Initially only investment I1 is implemented in 2016. Subsequently a set of further measures can be implemented during the second period starting in 2040 determined by the climate outcome of the first period. Thus, the optimal investment decision today is influenced by the possibility of the decision-maker to adjust their decision at a future moment in time based on the change of the peak flow of a 5% AEP in the future. If it is observed, for example, that peak flow has increased more over time than what had originally been planned and planted for, the strategy can be adjusted through increased planting in the most cost-effective way. The adjustment cannot undo the cost of past flood damages, however, it will avoid those costs in the future. Conversely, if peak flow did not increase significantly over time, valuable resources were saved by not planting excessively.

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Table 2 provides the suggested strategies zt for the different cases. The planting strategies zt correspond to the quartiles of the rainfall distribution for a 5% AEP event in 2016 and 2040. In the base case (scenario 1), the initial decision would be to plant for what corresponds to the 25th quartile of the rainfall distribution for a 5% AEP in 2016. This is the lower end of the distribution which implies planting only relatively little. The planting decisions in 2040 depend on the climate outcome in 2040, i.e. which peak flows have been observed over time by 2040. For all possible outcomes in the base case, the second choice will be to plant again for the 25th quartile. This signifies some additional planting as the 25th quartile of the 2080 flows are higher than the 25 th quartile of the 2040 flows. The second scenario uses a 7% discount rate for the first 30 years of the project (instead of 3.5%) and 6% from year 30 onwards (instead of 3 %). The resulting strategies are identical to the base case. The same strategy is recommended for scenario 3 which uses higher fixed cost (£200,000 instead of £8600). The planting strategies zt only change when increasing the damage cost from the flood event. Scenario 4 uses tenfold higher damage costs which leads to an initial planting that prevents flooding under the 50th quartile of the rainfall distribution in 2040 signalling to plant more trees as in the other scenarios. Independent of the outcome in 2040, it is suggested to plant for the 25th quartile going forward. Increasing the damage cost even more under scenario 5 (a twentyfold increase in damage cost relative to the base case) leads to a more cautious policy of planting more early on. In 2016, it is suggested to plant for the 50th quartile and if the 5% AEP rainfall event in 2040 corresponds to the 25th or 50th quartile of the distribution, the next step is to plan again for the 50th quartile (of the 2080 distribution). If the outcomes in 2040 are the 75th or the 100th quartiles of the distribution, planting for the 25thquartile (of the 2080 distribution) is suggested. This may be surprising, however, if in 2040, the rainfall event corresponds to the 75 th or 100th quartile, this will lead to additional planting in 2040 as in 2016, the planting was only sufficient to prevent the 5% AEP flood event under the 50 th percentile. As a consequence of the additional planting in 2040 (based on 2040 flows), there will be less additional planting necessary going towards 2080. For instance, the 100 th quartile flows for 2040 exceed the 25th quartile flows of 2080.

The resulting strategies mirror the trade-off between planting and maintenance cost versus damage cost that ‘pull’ the investment strategies in opposite directions. The planting and maintenance cost are mostly irreversible which provides an incentive to postpone planting unless the expected damage cost is higher than the investment for the afforestation. Higher damage cost leads to an earlier investment (more planting of trees) due to the risk of flooding and associated higher damage cost. The higher the damage cost, the more significant the planting becomes as can be seen when comparing strategy 4 (high damage cost) and strategy 5 (very high damage cost). In the base case, the damage cost is too low to result in anything but planting very little (the 25th quartile

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at each decision point). The base case is driven by planting cost and more importantly by the high maintenance cost relative to the damage cost. There is thus an incentive to postpone investments as much as possible. For our case study area, the estimated damage costs are almost certainly below the actual damage cost as we only have damage estimates available for the small village of Eddleston (940 inhabitants). Further downstream, a number of households in the town of Peebles (approximately 8000 inhabitants) are also positively affected by the planting of trees, however we have no data available on the damage avoided in Peebles to include in the study. Thus, scenarios 4 or 5 may be more realistic than the base case. Another factor that leads towards investing later is the discounting. The later the investment occurs, the more the costs will be discounted and the lower the total cost. This is the reason why scenario 2 with the higher discount rate provides the same strategy of planting as little as possible as in the base case. This illustrates the importance of discount rate choice as has been discussed extensively for the Stern Review in the context of climate change (Stern, 2007).

Higher fixed (irreversible) cost as in scenario 3 can lead to a higher initial investment to avoid having to replant several times and incurring the fixed cost several times. However, in our scenario 3, the increased fixed costs (from £8600 to £200,000) do not appear to play a major role, as there are only two investment decisions when fixed costs occur (2016 and 2040) and they are low relative to the maintenance cost. A planting strategy with additional decision nodes allowing for more frequent planting would likely result in a different strategy when using high fixed costs, however this would increase the complexity of the problem to be solved substantially and it is also not clear that increasing the planting decision points is necessarily more realistic from a policy-making point of view.

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Decision Point

1) Base Case(see Table 1 for

parameters)

2) Discount Rate(7% until year 30, 6% after year 30)

3) High Fixed Cost(£200,000)

4) High Damage Cost(10x increase)

5) Very High Damage Cost(20x increase)

2016Strategy Strategy Strategy Strategy Strategy

25th 25th 25th 50th 50th

2040

Outcome Strategy Outcome Strategy Outcome Strategy Outcome Strategy Outcome Strategy

25th 25th 25th 25th 25th 25th 25th 25th 25th 50th

50th 25th 50th 25th 50th 25th 50th 25th 50th 50th

75th 25th 75th 25th 75th 25th 75th 25th 75th 25th

100th 25th 100th 25th 100th 25th 100th 25th 100th 25th Expected Cost in £ 1,766,000 567,000 1,970,000 3,317, 000 4,996,000% exp. savings relative

to planting for the worst case

scenario

83% 89% 81% 68% 52%

Table 2 Under the base case, the higher discount rate and the high fixed cost, it is the most cost-efficient strategy to plant for the 25 th quartile of the rainfall distribution under climate change in both 2016 and 2040. Under the high damage cost case, the 2016 planting strategy suggests planting for the 50th quartile of the distribution, in 2040, independent of the outcome in 2040, the strategy suggests planting for the 25th quartile. Under the very high damage cost, the strategy is to plant for the 50th quartile in 2016 and in 2040 again for the 50th quartile if the observed outcomes were the 25th and 50th quartiles of the distribution in 2040, but for the 25th quartile if the outcomes were the 75th or 100th quartiles of the rainfall distribution under climate change .

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The expected cost of the flexible strategies is shown in Table 2 and varies strongly depending on the scenario. Scenario 2 with high discount rates provides the lowest expected value precisely because of the high discount rates. The higher the damage cost, the higher the cost of the flexible strategy as more trees need to be planted to prevent flooding. Also, higher fixed cost leads to a more costly investment. The cost savings relative to the worst-case scenario are substantial for all scenarios illustrating the value of learning. The worst-case scenario implies planting the maximum amount at both decision points and avoids all flooding. Expected cost savings relative to the worst-case scenario vary from 52% to 83% depending on the scenario. The high savings are driven by the fact that the flexible strategy suggests planting for the 25th quartile (very little climate change impacts) both in 2016 and 2040 independent of actual climate change outcomes whereas the worst-case scenario sets out a planting strategy for the 100th quartile (very severe climate change impacts.). The higher the damage cost, the lower the expected cost savings between flexible and worst case strategy as waiting for more information about the state of the climate is traded off with potentially higher damage cost. Higher fixed cost also decreases the cost savings as these fixed costs incur independent of the extent of forest planted. In particular in scenario 1 base case, we accept incurring flood damage, as this is cheaper than planting more trees to prevent flooding (based on actual damage cost). This demonstrates implicitly that the costs of afforestation do not actually exceed the benefits (which is damage avoided) as the analysis weighs the cost of planting/maintenance against the cost from potential damage. We observe that the planting costs are low but the maintenance cost (based on per hectare subsidies currently paid to farmers for maintenance of managed forest) are significant over time (£4k/year for the flexible strategy and £133k/year for the worst case scenario) and drive the investment strategy. Thus, while afforestation may be cheaper initially compared to ‘hard engineered’ measures, if the maintenance cost cannot be brought down, they may not necessarily be a more cost-effective strategy than hard engineered measures for flood regulation only. Other NFM, for example retention ponds, will have a different cost structure with significantly lower maintenance cost and might thus be more suited for flood protection and the application of ROA. However, afforestation provides ecosystem services such as carbon sequestration, recreation and habitat beyond flood regulation (Willis et al., 2003). These benefits were beyond the scope of the analysis but might alter the outcome of the analysis towards earlier investment and would provide a better estimate of the benefits accrued (see Dittrich et al. (2018) for a CBA on the eco-system services of afforestation in the case study area).

We believe that the approach described to determine the best flexible strategy can be carried out by policy-makers if they have access to perturbed rainfall data and we thus

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provide a valid contribution to applied adaptation to climate change in the context of flooding.

In the UK, perturbed rainfall data are freely available in a user-friendly way through the UKCP09 weather generator data and in the future likely with the updated UKCP18 (Lowe et al., 2018). Both datasets can be modified for different climate change scenarios based on policy-makers’ interests. We downloaded a large amount of data sets to ensure that we found the correct distribution of the rainfall but a good approximation can also be found with fewer sets. The analysis with the AM method is a standard analysis among hydrologists and can likely be carried out in the (flood) infrastructure department of a public authority together with the changes in peak flow due to flood management measures, in particular if the relationship between the measure analysed and peak-flow is well established (e.g. for retention reservoirs). With the results of the AM method, the transition probabilities can be easily calculated as described in section 2.1, which is often a major challenge in ROA. The cost of the measure can be obtained through quotes from different contractors. There might be historical data for the damage cost (under different peak flows) or it will require a damage analysis (in the UK for the example with the Multi-Coloured Handbook (MCH) (Penning-Rowsell et al., 2010). The backward induction can be carried out in an excel spreadsheet based on the description in section 2.4. Taken together, the steps are labour-intensive but can be carried out without in-depth knowledge of advanced programming. It is easily conceivable to apply our approach to sea-level rise or drought by using different climate variables from the UKCP09. Certainly, each case study will have its own specificities and challenges such as the choice and performance of the flood management measure but we believe that the described steps of 1) establishing a decision tree and deriving transition probabilities 2) biophysical analysis with and without the adaptation management measure 3) damage analysis with and without the measure 4) determining the cost of implementing the measure and 5) backward induction are generally applicable. Indeed, the greater challenge for any policy-maker may be to initiate policies that have a lifetime long beyond the current four to five years of a policy cycle.

A few caveats need to be mentioned. To assume that the AEP changes significantly with a time step rather than slowly over time is simplistic. The analysis could be enhanced by using different forms of uncertainty resolution. However, we believe the information gained by this may not necessarily outweigh the added complexity given that we have aimed to provide a relatively accessible approach to ROA. The results would also be affected (towards later investment) if we included delayed flood regulation benefits from afforestation as trees take time to grow, and the multiple eco-system benefits of afforestation beyond flood regulation (towards earlier investment). However, in order to illustrate the mechanisms that lead to earlier or later investment, we carried out the

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scenario analysis. Finally, the analysis does not include possible changes in land-use (Ball, Green, 2007), which may influence the flood patterns. For example, if the hills in the catchment were transformed from pastures to wheat fields, run-off would likely increase and require more planting of trees.

Going forward, it would be of interest to simplify the spreadsheet approach more by providing a mask that allows to input pay-offs and probabilities to generate the resulting strategies directly. In addition, a dialogue with decision-makers will be helpful to discuss their needs and requirements when developing further simplified tools of robust decision-making.

4. Conclusion

We have shown an accessible application of ROA as a climate change adaptation strategy using afforestation as a flood management measure in a case study. We believe that our simplifications may generate greater interest among policy-makers for ROA which is well-suited for uncertainty but has been too complex to be applied widely thus far. The strength of our approach is the ease of implementing backward induction in a spreadsheet format and a straightforward derivation of transition probabilities to describe the decision tree which remains challenging in ROA. Specifically, our analysis requires as inputs perturbed rainfall data freely available from the UKCP09 weather generator, analysis of changes of peak flow under the measure implemented, cost structures for the measures to be implemented (including opportunity cost) and damage costs under different outcomes. In our case study, the aim was to minimise the life cycle cost of a system to prevent flooding of 5% AEP using afforestation as a flood management measure. The results of the economic analysis in the base case show that the least cost option is to plant for the most conservative climate change outcome in 2016 and in 2040, independent on the climate change reality of 2040 due to the high maintenance cost in the system, which incentivises postponing those costs as much as possible and accepting the cheaper flood damages. The result indicates that afforestation as a flood management measure for the 5% AEP in the case study area does not pass the cost-benefit test. However, when considering higher damage costs, the strategy changes to a more cautious approach to avoid the increased damage cost. Overall, the strategy developed here is significantly cheaper than the planting for the worst-case scenario and shows for different configurations the potential for learning under climate change uncertainty as a way to allocate resources more efficiently.

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Acknowledgments

We acknowledge the part played by the Tweed Forum along with Scottish Government, SEPA, Scottish Borders Council, British Geological Survey and Dundee University which form the Project Board for the case study described and the local community, land owners and managers without whose assistance the study would not have been possible.

Funding information European Union, Seventh Framework Programme, 2007–2013, Grant/Award Number: 266018

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