Abstract
Recent work uses Markowitz’s mean-variance framework to identify efficient allocations of conservation activity across subregions of a planning landscape to minimize climate change–induced uncertainty in future conservation benefits. We replace variance with a downside measure of uncertainty and compare the resulting portfolios of conservation activity and uncertainty–expected value results against those generated by the mean-variance approach. Results illustrate that conservation agents can manage climate changed–induced uncertainty in future conservation outcomes more efficiently using our framework when they are particularly averse to downside uncertainty and when predicted future conservation outcomes within subregions across their planning horizon exhibit skewed distributions. (JEL D81, Q57)
I. INTRODUCTION
Uncertainty poses problems for conservation planning and natural resource management. For example, climate change–driven uncertainty in future conservation outcomes makes it difficult for conservation planners to undertake systematic conservation planning; protecting places that currently have high conservation value does not ensure that the future conservation value of those reserves will be high. Modern portfolio theory (MPT) has been used in several branches of economic research on natural resources management under uncertainty, including biodiversity conservation, salmon management, forest restoration, and invasive species control (Figge 2004; Koellner and Schmitz 2006; Crowe and Parker 2008; Kennedy, Balasubramanian, and Crosse 2009; Richardson et al. 2009; Moloney et al. 2011; Moore et al. 2010; Schindler et al. 2010; Schloss et al. 2011; Ando and Mallory 2012a; Yemshanov et al. 2014). The classic mean-variance (MV) portfolio framework from finance identifies diversified portfolios of environmental investments that are Pareto efficient in the sense that uncertainty in total outcome is reduced with the smallest loss in the expected value (EV) of returns possible (in finance such portfolios are referred to simply as “efficient”; we will use this terminology the same way).
However, tools that minimize variance may fail to identify the best strategies for environmental management when conservation returns are nonnormally distributed and decision-makers are particularly averse to deviations below a benchmark return (Dunkel and Weber 2012; Ando and Mallory 2012b). This paper addresses that limitation of MPT by developing a tractable approach to managing downside uncertainty in future natural resources management outcomes and answering several key questions related to that approach: When is it important for analysts to use a downside measure of uncertainty in portfolio analysis, and how do management decisions and outcomes change as a result of managing downside instead of symmetric uncertainty? For concreteness, this paper focuses on application of portfolio theory to conservation theory, but the methods are applicable to any use of portfolio theory in natural resources management.
The use of variance in the MV approach is an appropriate measure of uncertainty for settings in which returns are normally distributed and investors are not particularly averse to downside deviations in expected outcomes (Tsiang 1974; Estrada 2007). However, variance can be invariant to the skewness of distributions when returns follow nonnormal patterns. A tool that minimizes variance implicitly assumes that both upside and downside deviations are equally undesirable. Even when returns are multivariate normal, variance still captures the wrong definition of uncertainty for a conservation agent that is only concerned with underperformance of a portfolio below a certain benchmark return that is different than the mean (Rom and Ferguson 1993). Some decision-makers may prefer to use downside measures of uncertainty to identify efficient portfolio allocation strategies (Roy 1952; Harlow 1991). We do not make a normative argument for such preferences, but develop tools that can be used by decisionmakers who display them.
Several downside measures of uncertainty can be used to evaluate portfolio allocation decisions when investors are averse to “the probability of shortfall below some benchmark level of return” (Harlow 1991).1 These alternative measures have been widely used for decision-making related to financial and real estate investment and corporate expansions and strategic investment decisions (Jorion 1997; Duffie and Pan 1997; Dowd 1998; Bertsimas, Lauprete, and Samarov 2004). A commonly used downside measure of uncertainty that encompasses a broad range of investor risk preferences is lower partial moments (LPM). LPM is a measure of the shape of the returns distribution below a certain threshold (Bawa 1975; Fishburn 1977; Nawrocki 1991, 1992, 1999). We use second-degree LPM (LPM2) with current conservation outcomes as our threshold; thus, the uncertainty to be minimized includes only the chances of future conservation outcomes that are below current levels—outcomes that increase overall downside uncertainty. Mean-LPM2 (MLPM2) portfolios accommodate a wider range of investor risk preferences and nonnormal distributions and are consistent with both first-order and second-order stochastic dominance rules; MV portfolios are consistent only with first-order stochastic dominance rules (Porter and Gaumnitz 1972; Joy and Porter 1974; Bawa 1975, 1978; Bawa and Lindenberg 1977).2
Numerous organisms, communities, and ecosystems are exhibiting widespread shifts in ranges and phenological changes in response to climate change (Parmesan 2006; Dawson et al. 2011). While climate change is an imminent threat to biodiversity, ecosystems, habitats, and ecological communities, it is difficult to predict with certainty its impact on the future spatial patterns of outcomes associated with these natural resources. If conservation planners want to minimize the chances of large unanticipated losses, they have to deal explicitly with this spatial uncertainty. Studies show that explicitly incorporating uncertainty in conservation planning tools can lead to different management and conservation decisions (Harwood and Stokes 2003; Doyen and Béné 2003; Lande, Engen, and Saether 2003; McCarthy et al. 2005; Regan, Hope, and Ferson 2002; Grafton, Kompas, and Lindenmayer 2005; Regan et al. 2005).
Many methodologies ranging from purely statistical analysis to more complex and dynamic modeling techniques are used to predict future conservation outcomes for different climate change scenarios (Dawson et al. 2011). Irrespective of the methodology used, downside uncertainty measures may better capture the variability in predictions of future geographic patterns of conservation outcomes when conservation planners are concerned with the possibility of a decrease in conservation benefits from current levels. We use a case study of the current and potential future status of 147 bird species’ abundance levels in the eastern United States (Matthews et al. 2011) to illustrate the differences in the combinations of uncertainty and EV of returns that are possible in the portfolios that result from the MLPM2 and MV approaches to designing efficient portfolios of conservation activity.
This paper advances the use of portfolio allocation methods in conservation planning decisions in several ways. We incorporate the use of a downside uncertainty measure in efficient spatial targeting of conservation activity to accommodate nonnormal returns distributions and investor averseness to the probability of a loss below current levels of conservation returns. We draw on previous work in finance to develop a computationally tractable method for using an optimization algorithm that is intuitively similar to the MV framework to solve the mean downside uncertainty optimization problem. We use the return distributions associated with bird abundance levels in the eastern United States to illustrate when it is most important to use a downside measure of uncertainty in selecting conservation reserves. We also estimate the magnitude of differences in the uncertainty–EV of returns levels when a conservation agent uses LPM2 rather than variance to evaluate efficient land conservation portfolio in the presence of skewed environmental outcomes under uncertain climate change scenarios. The methods used in this study are applicable to a wide range of resource economics studies ranging from conservation problems that deal with uncertain species or populations outcomes to systematic conservation planning problems over a spatially uncertain planning horizon.
II. METHODS
We use portfolio theory to design efficient portfolios of conservation activity that enable a conservation agent to achieve the best possible trade-offs between climate change uncertainty and EV of conservation returns across a planning landscape. The traditional approach to conservation planning implicitly invests resources in the set of lands that maximizes the EV of conservation returns subject to budget constraints. With climate change, there is then a great deal of uncertainty associated with that strategy. Such uncertainty cannot be reduced without accepting at least a slightly lower EV of returns from the conservation actions one takes; portfolio theory helps the planner to reduce outcome uncertainty without reducing the EV of returns more than is strictly necessary to accomplish a desired reduction in uncertainty.
Figure 1 illustrates this idea conceptually. Every portfolio represents a mixture of conservation investments that has a probability distribution function (pdf) with several moments: EV, variance, and skewness of returns. The solid line in panel A of Figure 1 shows the EV of returns and standard deviation (SD) of all portfolios that have Pareto efficient combinations of those two moments that a conservation agent can achieve using the MV portfolio optimization framework; hence, it is called the efficient frontier. The traditional conservation planning approach, which does not worry about climate change uncertainty in the portfolio design process, will identify a conservation portfolio that achieves the highest possible EV of returns, albeit with maximum levels of variance (point A in Figure 1).3 The agent could choose a portfolio with less variance by diversifying the investment, but that choice must be made carefully. Portfolio optimization that uses an MV framework enables a conservation agent to identify an efficient set of portfolios such that, for a given level of EV of returns, there are no other portfolios with lower variance, and for a given level of variance, there are no other portfolios with higher EV of returns. For example, point A (associated with some portfolio A, which has the pdf labeled PA in panel B) provides high EV of returns but has a high variance. A conservation agent can use the portfolio optimization approach to reduce some of the variance in total outcome by accepting a lower EV of returns and choosing a portfolio at a lower point on the efficient frontier (e.g., point B). Every portfolio that lies below the efficient frontier is suboptimal; each is dominated by a different portfolio that provides greater EV of returns for the same level of variance.
Conceptual Figure of Portfolio Optimization and Downside Uncertainty: A, Efficient Frontier; B, Probability Distribution Function Distributions
A limitation of the MV portfolio optimization framework is that it cannot always distinguish between portfolios that have pdfs of returns with very different shapes when it identifies portfolios that are efficient. For example, the two hypothetical portfolios B1 and B2 are associated with the same point B on the efficient frontier in SD/EV space, since they have identical means and SDs. However, the downside uncertainty associated with the positively skewed portfolio B1 is lower than the downside uncertainty associated with the normally distributed portfolio B2 (as illustrated in their respective pdfs, labeled PB1 and PB2 in panel B of Figure 1); note that the pdf of B1 does not have a long negative tail, so there is no chance of a catastrophic outcome with such a portfolio. Some investors might have a strong preference for a portfolio like B1 over a portfolio like B2 (Harlow 1991). Thus, the traditional measure of uncertainty in finance, variance, is not always appropriate in making efficient investment decisions. A MLPM2 portfolio optimization framework can help conservation agents concerned with managing downside uncertainty as measured by LPM2 (rather than simple variance) to identify efficient portfolios that account for the shapes of their probability distributions.
Consider a conservation agent’s problem of allocating available conservation dollars across N subregions to achieve a target conservation outcome and efficiently manage climate change uncertainty. There are conservation returns associated with each subregion i. We define overall returns in each subregion i as a conservation benefit to conservation cost ratio denoted as Bi. Conservation benefits represent the outcome associated with the existence of biodiversity, a particular species, or a particular ecosystem service. Conservation costs represent the cost of doing conservation activity in each subregion i; incorporating this feature into the return criteria ensures that conservation resources are not directed to expensive subregions when the same resources can achieve higher conservation benefits in lower-cost subregions (Naidoo et al. 2006; Wilson et al. 2006). In the context of this study, we use individual bird abundance levels as a measure of conservation benefits and the average cost of land as a proxy for conservation cost.
Due to climate change, the future returns in each subregion cannot be known with certainty. We adapt portfolio analysis from a financial setting to a conservation setting as follows: (1) the N subregions are analogous to financial assets such as stocks and bonds, and (2) the range of expected conservation outcomes under different climate scenarios for each subregion is analogous to the historical returns associated with each financial asset. Information about the joint distributions of the conservation outcomes in the subregions can be used to design efficient conservation portfolios.
The conservation agent’s objective is to allocate funds across these N subregions in order to (1) minimize the uncertainty associated with the spatial variation in conservation returns from this portfolio of conservation activities and (2) achieve a desired level of conservation returns. Below, we define MV and MLPM2 portfolio optimization frameworks to identify efficient portfolio allocation of conservation activity across the N subregions.
In one set of analyses, we derive efficient portfolios using the MV framework (Ando and Mallory 2012a), where we use variance as a measure of uncertainty. We first define the average return for each subregion i:
[1]
where Bit represents the conservation returns for subregion i for climate scenario t when T climate change scenarios are available. The variance for each subregion i is
[2]
The MV approach exploits information about the covariances in predicted returns between subregions. This relationship is represented by a variance-covariance matrix. The covariance (Cov) between two subregions, i and j, is
[3]
where ρi,j is the correlation coefficient between the two subregions. The total symmetric uncertainty of a portfolio of investments in subregions i and j is
[4]
where ω is a vector of portfolio weights for the N subregions and Σ is a variance-covariance matrix of conservation returns,
[5]
The EV of returns for the entire portfolio is
[6]
The conservation agent’s optimization problem is to minimize portfolio uncertainty subject to three constraints:
[7]
In the equation above, the three constraints ensure that the full amount set aside for conservation is invested (portfolio weights add up to one), there is no short selling (efficient portfolio weights are all greater than or equal to zero), and the EV of returns of the portfolio is equal to a target level rp.
In another set of analyses, we replace variance with LPM2 as the measure of uncertainty. We define the downside uncertainty associated with returns Bi in each subregion i as
[8]
This is the LPM2i for a single subregion, i, with a reference rate R and with T climate change scenarios. The reference rate is arbitrary and can vary based on a conservation planner’s preferences. It acts as a cutoff rate such that any conservation outcome below this rate represents a loss in overall conservation activity. In this study, we use the current observed conservation returns for each subregion as the reference rate, Ri; thus, equation [8] becomes
[9]
To identify an efficient portfolio of conservation activity that minimizes downside uncertainty associated with achieving a target EV of returns, we need to calculate a co-lower partial moments (CLPM) matrix; CLPM is a statistic that captures the comovements in deviations away from the reference rate between subregions. As defined by Scherer (2007), the CLPM of subregion i with subregion j is
[10]
where dit = 1 when subregion i is below R. Similarly, the CLPM of the return on subregion j with subregion i is
[11]
Here, dit = 1 when subregion j is below R. In general, dit≠djt, because it is not necessary that the two returns are simultaneously below the reference rate, R; thus, CLPM is not symmetric. To arrive at a symmetric CLPM matrix, we use the following definition (Nawrocki 1991):
[12]
where ρi,j is the correlation coefficient between two subregions. The total downside uncertainty of a portfolio of investments in subregions is
[13]
where ω is a vector of subregion weights and L is a CLPM matrix of conservation returns,
[14]
The portfolio optimization problem is then similar to the MV framework shown in equation [7]:
[15]
While we use LPM2 (which is a second-degree lower partial moment) in this paper, higher degrees of LPM can be used to accommodate larger uncertainty aversion. In particular, conservation planners interested in finding an efficient portfolio of conservation activity in settings with ecosystem services that are at risk of extinction may want to use a higher-degree of LPM to capture strong aversion to very poor outcomes that may lead to irreversible loss.
III. DATA
We could carry out an analysis of conservation planning for hypothetical species in a hypothetical landscape. However, using real predictions of species distributions under varied climate change scenarios gives us a more realistic context in which to demonstrate our methodology. Hence, we use predicted bird abundance levels that are readily and publicly available from the U.S. Department of Agriculture Forest Services, Northern Research Station (Matthews et al. 2007 and ongoing), though this is not intended to be an endorsement of the particular methodology used in that paper. Future analysts could use very different approaches to generate the natural resource outcome predictions under varied climate scenarios that are needed to use the portfolio tools we describe.
Studies find evidence of significant bird species responses to climate change, which include shifts in migratory patterns and arrival dates, mismatches in the timing of resource availability, northward expansion of wintering and breeding in North America, and earlier nesting times (Beaumont, McAllan, and Hughes 2006; Jonzén et al. 2006; Waite and Strickland 2006; Both et al. 2006). Matthews et al. (2011) expect climate change to alter future bird abundance levels in the eastern United States significantly. Their study illustrates that bird distributions are driven by climate conditions as well as tree species patterns that directly affect the birds’ habitats. They develop a statistical model that predicts the relative abundance of birds in the eastern United States based on climate, elevation, and tree species distribution patterns. The statistical model is used to make future projections for bird abundance levels, assuming that the birds do not adapt to climate change and that the currently assumed relationships between the predictor variables and the bird abundance levels remain unchanged over the next 30 years. Additionally, the model predictions provided by the study are relevant only to the extent that the predictor variables included in the statistical model are representative of the key variables that determine bird abundance levels.
Matthews et al. (2011) use the statistical model to predict future bird habitats for three general circulation climate models: HadleyCM3 (HADCM3) model, Parallel Climate Model (PCM), and Geophysical Fluid Dynamics Laboratory model (GFDL), and two emissions scenarios: A1f1 and B1. A1f1 represents an intensive emissions scenario with little effort for CO2 mitigation, and B1 represents a low-emissions scenario with significant effort for CO2 mitigation. They provide six future outlooks (HADCM3-A1f1, PCM-A1f1, GFDL-A1f1, HADCM3-B1, PCM-B1, and GFDL-B1) for the area-weighted abundance levels for 147 bird species in five subregions in the eastern United States: South East, South Central, Great Plains, North Central, and North East, as shown in Figure 2. Of the 147 birds, the predicted future abundance levels for 28 birds are shown to increase under all climate change scenarios. Thus, there is no downside uncertainty associated with these birds (i.e., the future abundance levels for these birds will never be less than the current levels), and we exclude these birds from our analyses.
Eastern United States Divided into Five Subregions
In the context of our study, the hypothetical conservation planner seeks to allocate the total conservation budget among these five subregions to form an efficient portfolio of conservation activity designed to preserve target bird abundance levels. The future bird abundance level in each subregion is uncertain and will be determined by which of the six climate change scenarios comes to pass. The planner can estimate the relative likelihood of each of the six possible outcomes and use the resulting probability distribution to represent the uncertainty associated with holding a portfolio of conservation activity in the five subregions. We assign equal probability weights to each of the six climate scenarios such that returns associated with each future climate model and emissions scenario are equally likely. Returns in the individual subregions are calculated as the ratio of conservation benefit to conservation cost.4 The predicted abundance levels for each bird represent the conservation benefits. The average cost of land for each subregion represents the cost of doing conservation activity in that subregion (shown in Table 1); these estimates are based on land values presented by Schlenker, Hanemann, and Fisher (2005). We conduct Mardia’s test of multivariate normality to determine whether the joint distributions of returns in the subregions for each bird are multivariate normal (Mardia 1970). We find the distributions of returns are not multivariate normal for all 119 birds, based on the results of Mardia’s kurtosis test.5
Average Cost of Land in the Five Regions in the Eastern United States
For illustrative purposes, we focus our discussion on three representative birds out of the 119 in the dataset: the red-headed woodpecker, the pileated woodpecker, and the yellow-bellied sapsucker. These are forest birds found across different parts of North America, with significant presence in the eastern United States. The study by Matthews et al. (2011) predicts that these birds will experience changes in abundance levels under available climate change scenarios. This is of particular concern because the International Union for the Conservation of Nature (IUCN) accords a “near threatened” status to the red-headed woodpecker; the IUCN status for the pileated woodpecker and the yellow-bellied sapsucker is “least concern.” Figures 3, 4, and 5 graphically illustrate the abundance values for the three birds across the eastern United States based on current and future modeled climate scenarios. The North Central, Great Plains, and South Central subregions are likely to experience high abundance rates, whereas the North East subregion is expected to experience low abundance levels of the red-headed woodpecker under all climate scenarios (as seen in Figure 3). Abundance levels for the pileated woodpecker are highest in the South Central, South East, and North Central subregions (as seen in Figure 4) and lowest in the Great Plains subregion. Abundance levels for the yellow-bellied sapsucker are highest in the North East and North Central subregions and almost negligible in the South Central and South East subregions (as seen in Figure 5). Table 2 shows the individual subregion and climate scenario conservation returns for each of the three birds.6
Modeled Abundance Levels for the Red-headed Woodpecker: A, Current Levels across the Eastern United States; B, Under the Six Modeled Climate Scenarios
Modeled Abundance Levels for the Pileated Woodpecker: A, Current Levels across the Eastern United States; B, Under the Six Modeled Climate Scenarios
Modeled Abundance Levels for the Yellow-bellied Sapsucker: A, Current Levels across the Eastern United States; B, Under the Six Modeled Climate Scenarios
Current, Future, and Average Returns for Red-headed Woodpecker, Pileated Woodpecker, and Yellow-bellied Sapsucker
IV. RESULTS
We solve for two sets of efficient portfolios—MV-efficient (minimizing variance) and MLPM2-efficient (minimizing LMP2)—for a range of target EV of returns. Using the investment weights associated with each portfolio in the two sets, we calculate the EV, SD, and downside deviation (DSD, defined as the square root of the portfolio’s LPM2) of portfolio returns, using the values in Table 2, Appendixes A1 and A2, and equations [4], [6], and [13]. The resulting values of EV of returns and uncertainty form efficient frontiers that are upward sloping; to achieve a higher EV of returns, a conservation agent must accept more uncertainty. We plot EV and DSD of returns for both sets of efficient portfolios in EV-DSD space; this permits direct comparison of the outcomes of the portfolio allocations that are generated by the two frameworks.
For many conservation targets, the MLPM2 framework yields an MLPM2-efficient frontier that lies to the left of the MV-efficient frontier. Thus, for the same level of EV of conservation returns, MLPM2 portfolios reduce the downside uncertainty by focusing on the deviations in future returns below current levels. The differences in portfolio allocations and uncertainty-EV trade-offs between the two frameworks are mainly driven by large deviations in returns from current levels that result in substantially different downside and symmetric uncertainty. Differences in portfolio allocation frameworks are especially large for birds whose subregion-specific average returns do not vary substantially. In such scenarios, there is much that diversification can do to reduce uncertainty or increase EV of returns, and thus, use of different measures of uncertainty will lead to substantially different outcomes. When there are only one or two subregions with high average returns relative to the other subregions, there is little that diversification can do to reduce uncertainty without sacrificing much EV of returns, and hence use of either framework will result in similar outcomes.
We use stylized features of the results to illustrate three characteristic differences between the portfolio outcomes obtained from the MV and MLPM2 frameworks for all 119 birds. We illustrate these stylized features graphically in Figure 6, which plots two efficient frontiers in EV-DSD space. The dominant efficient frontier in Figure 6 represents an MLPM2-efficient frontier; the MV-efficient frontier lies below it. The first stylized feature is the maximum percentage difference in EV of returns for the same level of downside uncertainty across all the birds if a conservation agent uses LPM2 rather than variance to design an efficient conservation strategy. In Figure 6, (A–B)/B represents this maximum percentage difference between the two frameworks; A–B is the maximum absolute difference in EV of returns for the same level of DSD between the MLPM2 and MV frameworks. A second stylized feature of the result is the maximum percentage difference in downside uncertainty (or DSD) between MLPM2 and MV portfolios. The ratio (D–C)/ D represents this maximum percentage difference in uncertainty between the MLPM2 and MV portfolio allocation frameworks, where D–C represents the largest possible difference in DSD between the MLPM2 and MV efficient frontier for the same level of EV of returns. A third stylized feature of the result is the overlap between the MLPM2- and MV-efficient frontiers. This overlap represents the extent of the efficient frontier for which use of either uncertainty measure yields similar portfolio allocation decisions and downside uncertainty-EV relationships. In Figure 6, the ratio E/F represents the normalized overlap between the two efficient frontiers, where E is the extent of DSD for which both MV and MLPM2 frontiers overlap and F is the range of DSD for the MLPM2-efficient frontier.
Stylized Results
Table 3 illustrates the maximum, minimum, and average values for the three stylized features of the results for all 119 birds. On average, portfolio DSD can be reduced by 73% and portfolio EV of returns can be increased by 53% when conservation agents averse to downside uncertainty use MLPM2 portfolios to guide their planning decisions. The MLPM2 portfolio can achieve a maximum possible reduction in DSD of 100% over an MV portfolio for the same level of EV of returns when more subregions exhibit skewed returns distributions and/or the returns in one or more individual subregions are largely skewed. Under similar circumstances, the EV of returns can be increased by a maximum of 100% for the same level of DSD through use of an MLPM2 framework. Conversely, when the returns within individual subregions are relatively less skewed and/or there are small differences in the average outcomes across subregions, the potential of the MLPM2 portfolio to augment EV of returns or reduce DSD decreases to 11% and 17%, respectively. The MLPM2 portfolios recommend larger conservation activity in the high-return subregions than MV portfolios. This finding is similar to previous results in finance where MLPM2 portfolios represent larger allocations to safer investment options such as bonds that increase downside protection while offering the same or greater EV of returns (Harlow 1991).
Summary of Stylized Results
The average normalized overlap between the two efficient frontiers is 21%. The overlap between the MV- and MLPM2-efficient frontiers is zero when the returns within individual subregions are highly skewed. This overlap reaches a maximum of 92% as more individual subregions have normally distributed returns (though statistically the returns are still jointly nonnormal) and when several subregions have negligible bird abundance levels such that the conservation portfolio diversification process has to necessarily include the remaining one or two higher conservation return subregions.
Figure 7 illustrates the MV- and MLPM2-efficient frontiers for the red-headed woodpecker. A conservation agent that is averse to deviations in returns below current levels can reduce DSD by as much as 60% by choosing an MLPM2 portfolio of conservation activity rather than an MV portfolio for the same EV of returns. Similarly, a conservation agent can achieve up to 40% higher EV of returns from an MLPM2 portfolio for the same level of DSD as an MV portfolio. The MLPM2 portfolios include larger conservation activity in the high-return North Central subregion. While the MV and MLPM2 portfolios have identical allocations in the high uncertainty–high EV subregion for many birds, for the redheaded woodpecker these two efficient frontiers coincide at only one point. This point represents the maximum possible EV of return for which both frameworks recommend full investment in the high-return North Central subregion. In the case of the pileated woodpecker, this overlap is 85%, indicating that any benefit to be gained from use of an MLPM2 portfolio occurs only in the low uncertainty–low EV neighborhood (as shown in Figure 8). Portfolio allocations in the low uncertainty–low EV neighborhood based on the MLPM2 framework reduce DSD by as much as 69% for the same level of EV of returns and increase EV of returns by as much as 14% for the same level of DSD as MV portfolios for the pileated woodpecker.7
Efficient Frontiers for the Red-headed Woodpecker Derived Using the MV and MLPM2 Frameworks in Mean–Downside Deviation Space
Efficient Frontiers for the Pileated Woodpecker Derived using the MV and MLPM2 Frameworks in Mean–Downside Deviation Space
The differences in portfolio allocation strategies between the MV and MLPM2 frameworks and the associated differences in expected uncertainty–EV of return trade-offs are driven largely by the extent of skewness in distribution of returns in individual subregions and the available range of returns across the five subregions. When returns in different subregions are similar, the conservation agent has more choice of subregions to include in the conservation portfolio than if there are some subregions with very high returns and other subregions with low or negligible returns. For the latter scenarios, optimal portfolio allocation strategy will abandon the low-return subregions and focus resources in the high-return subregions.8 The differences in portfolio allocation strategy are even larger when the returns within individual subregions have skewed distributions owing to large deviations in future predicted outcomes relative to current outcomes for different climate change scenarios.
The panels of Figure 9 plot the variation in efficient portfolio weights for red-headed woodpecker conservation in each subregion as the portfolio EV of return increases. Subregions such as Great Plains and North Central (Figure 9, panels C and E), in which returns associated with the red-headed woodpecker are very high and are expected to increase from current levels for most climate scenarios, are always allocated higher weights under the MLPM2 portfolio allocation strategy. These subregions have low downside uncertainty, since the probability of deviations below current levels is low. However, lower weights are allocated to these subregions under the MV approach because the variance of returns (which captures positive as well as negative deviations from the mean) is large in those subregions. Conversely, the weight allocation to the South Central subregion is greater for the MV framework compared to the MLPM2 framework (Figure 9, panel B). While the average return in this subregion is higher than in some of the other subregions, the probability that future returns are lower than the current returns is also larger than in other subregions, resulting in high downside uncertainty and thus a lower weight allocation under the MLPM2 framework for this subregion.
Change in Optimal Portfolio Weights for Each Region for an Increase in Portfolio Returns for the Red-headed Woodpecker: A, South East; B, South Central; C, Great Plains; D, North East; E, North Central
The differences in portfolio allocation strategies between the two frameworks are prominent for the lower and intermediate levels of EV of returns, where there is much that diversification can do to reduce uncertainty. For high EV of returns, the portfolio allocation strategies based on either measure of uncertainty are the same. Even when there is zero overlap between the two efficient frontiers, the two efficient frontiers always meet at the point of highest EV of return (and highest uncertainty). When the target EV of return is high, the optimal conservation strategy is to focus in the one subregion that has the largest average returns, rather than a diversified strategy that includes all subregions. For example, to obtain an EV of return of 1,740 for the redheaded woodpecker, both portfolio allocation strategies recommend 100% conservation activity in the North Central subregion, since this is the subregion with the highest EV of return. When there are large differences in EV of return across subregions, there is little that diversification can achieve at the higher end of the efficient frontier; this yields concentrated and similar portfolio allocations irrespective of the measure of uncertainty used.
While minimizing downside uncertainty instead of variance can yield large changes in strategy and outcome for many conservation targets, we do find a large overlap between the two efficient frontiers for some birds. This overlap occurs for birds with very low average returns in several subregions and less skewed distributions within the remaining high average return subregions. In such scenarios, both MV and MLPM2 frameworks identify exactly the same portfolio allocations for a large portion of the efficient frontier in the high uncertainty–high EV neighborhood such that the uncertainty-EV combinations are the same. The only differences in uncertainty-EV profiles for such scenarios are found in the very low EV of return–low uncertainty neighborhood. The yellow-bellied sapsucker is a good example of a bird for which the results of the MV and MLPM2 are almost identical for a large part of the efficient frontier; those results are shown in Figure 10.9 We find similar results for 30 other birds.
Efficient Frontiers for the Yellow-bellied Sapsucker Derived Using the MV and MLPM2 Frameworks in Mean–Downside Deviation Space
V. CONCLUSION
This paper advances the systematic conservation planning literature by developing a tractable approach to evaluate efficient spatial targeting of conservation activity that minimizes downside uncertainty from climate-induced variations in future spatial patterns of conservation-related outcomes. We identify the characteristics of return distributions that lead to differences in efficient portfolio allocations based on the use of downside versus symmetric measures of uncertainty, using a case study of the potential changes in bird abundance levels in the eastern United States. We further highlight the differences in the magnitude of the uncertainty–EV of return combinations possible from portfolios of conservation activity derived using the two alternative measures of uncertainty. While the specific results associated with the case study of protecting habitat for individual bird abundance levels in the eastern United States are sensitive to the accuracy of the data provided by Matthews et al. (2011), the general findings are applicable to a broad range of systematic conservation planning and environmental management problems.
We find that replacing variance with a downside measure of uncertainty leads to significant differences in portfolio allocations and uncertainty-EV frontiers when return distributions for subregions of a broader landscape exhibit skewed distribution patterns in future conservation outcomes. Investments in subregions that are likely to experience positively skewed returns (perhaps due to likely increases in conservation returns associated with climate change) reduce portfolio downside uncertainty; thus, such subregions are allocated greater portfolio weights under an MLPM2 diversification strategy than under an MV diversification strategy. On the other hand, investments in subregions with negatively skewed returns increase portfolio downside uncertainty; such subregions are allocated low portfolio weights by the MLPM2 approach. Even if returns are multivariate normal, a mean–downside uncertainty framework is more suitable when conservation agents are especially averse to deviations below a benchmark return that is different than the average return. These differences in diversification strategies to achieve target conservation goals are prominent for low uncertainty–low EV ranges of portfolio outcomes. Conversely, when conservation EV targets are high and there are large differences in average returns between subregions, the choice of uncertainty measure does not lead to significant differences in portfolio allocations.
Our study illustrates that when there are large differences in average returns across subregions, but returns in any given subregion do not vary much with climate change scenario, analyses using both measures of uncertainty identify identical portfolio allocation strategies and there are no differences in the uncertainty-EV outcomes. However, when average returns in each subregion are relatively similar and fluctuations in returns within each subregion are large, the choice of uncertainty measure significantly alters the uncertainty-EV profile associated with portfolio allocation decisions. For a conservation agent that is averse only to deviations below the target level, choosing a portfolio allocation strategy that uses a downside measure of uncertainty can substantially lower the uncertainty by allocating lower investments to subregions that are more likely to experience negative climate-induced changes in conservation outcomes. Alternatively, for the same level of downside uncertainty, a conservation agent can achieve a significantly higher EV of return by allocating greater portfolio weights to subregions that are likely to benefit from potential climate change events.
Many modifications of the work presented in this paper are possible. For example, we assume, somewhat arbitrarily, that the conservation agent in our illustrative examples is interested in planning a reserve network to achieve conservation benefits that are based on abundance levels of a single bird, rather than one that seeks to conserve a large proportion of multiple birds. However, conservation planners who are interested in preservation of habitat for multiple birds can use some form of weighted index that characterizes the species distribution of multiple birds as a measure of benefit (Weitzman 1992; Solow, Polasky, and Broadus 1993). We also weigh the conservation returns associated with each bird in each subregion equally when calculating total EV of return. However, the choice of weights may vary with the specific conservation planning objectives and the main drivers of conservation returns. If much of the returns from bird conservation are attributable to bird-viewing activities, then it may be more appropriate to weigh bird-specific conservation returns by human population in each subregion. Mallory and Ando (2014) provide extensive study of the importance of how benefits are measured in portfolio analysis; such an extensive treatment of this issue is beyond the scope of this paper. Finally, we use a second-degree LPM to measure downside uncertainty in this paper. A higher degree of LPM could be used instead to represent greater uncertainty aversion.
Recent studies highlight the difficulties of decision-making under climate change uncertainty due to the deep and irreducible nature of the uncertainties associated with the pace and implications of climate change (Hallegatte et al. 2012; Xepapadeas and Yannacopoulos 2013; Heal and Millner 2013). Traditional decision-making tools such as portfolio optimization that require the conservation agent to know the probability distribution over future climate outcomes can result in misallocation of resources and failure to achieve desired conservation objectives. However, planning recommendations based on such techniques can alleviate some of the uncertainty associated with deep climate uncertainty and achieve better results than all- or-nothing conservation plans (Lempert et al. 2004; Mallory and Ando 2014).
This paper provides an important practical advance in the general methodology of using portfolio theory to efficiently diversify investments for a broad range of goals in environmental and natural resource management and planning. Many decision-makers want to avoid uncertain bad outcomes but would not mind the occasional unexpectedly good result. Our work provides a useful tool for such decision-makers to efficiently reduce the risk they dislike, especially when they are significantly risk-averse, and the investments among which they can allocate their resources have the following features: average returns are strong for multiple investments (so diversification is possible without reducing the portfolio’s expected returns too much), and the distributions of some of the investments’ returns over climate outcomes are skewed (so it makes a difference to minimize downside uncertainty instead of classic symmetric variance). Future research can adapt and apply this approach to problems other than conservation, helping planners and policy makers to make many kinds of choices that yield outcomes that are robust to irreducible climate change uncertainty.
Acknowledgments
We are grateful to Phil Garcia, Mindy Mallory, Paul Ferraro, Kathy Baylis, Alex Winter-Nelson, two anonymous referees, and members of the UIUC Program in Environmental and Resource Economics seminar group for helpful comments. Part of the work for this paper was done by Payal Shah while a Ph.D. student at the University of Illinois at Urbana-Champaign. This paper was based in part on work supported by USDA-NIFA Hatch project #ILLU-470-316, by Cooperative Agreement Number G12AC20056 from the U.S. Geological Survey, and by the National Science Foundation under Grant #1339944. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the USDA, USGS, or NSF.
APPENDIX A
Variance-Covariance Matrix of Returns for Red-headed Woodpecker, Pileated Woodpecker, and Yellow-bellied Sapsucker
CLPM of Return Values for the Red-headed Woodpecker, Pileated Woodpecker, and Yellow-bellied Sapsucker
Optimal Portfolio Weights, Uncertainty, and EV (Expected Value) of Returns for the Yellow-bellied Sapsucker
Optimal Portfolio Weights, Uncertainty, and Expected Value (EV) of Returns for the Red-headed Woodpecker
Optimal Portfolio Weights, Uncertainty, and EV of Returns for the Pileated Woodpecker
APPENDIX B
Acronyms
Footnotes
The authors are, respectively, research scientist, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan; and professor, Department of Agricultural and Consumer Economics, University of Illinois at Urbana-Champaign, and University Fellow, Resources for the Future, Washington, D.C.
↵1 Two popular downside measures of uncertainty are value-at-risk (VaR) and conditional-value-at-risk (CVaR). However, computational difficulties and large data requirements make the use of VaR and CVaR difficult in conservation settings with uncertain climate change scenarios where the number of observations is usually small.
↵2 Stochastic dominance is a risk analysis tool that uses the entire cumulative probability distribution of an investment to determine if one investment is superior to another (Porter and Gaumnitz 1972). The main advantages of stochastic dominance rules are (1) they can be used to evaluate trade-offs between uncertainty and EV of returns for all probability distributions, and (2) they include all possible risk-averse utility assumptions.
↵3 A recent study by Mohd, Mohamad, and Mohamed (2013) uses the median-variance approach to improve traditional MV portfolio optimization when returns follow nonnormal distributions. They find that the median-variance portfolio has a lower coefficient of variation (the ratio of standard deviation to mean returns) than the MV portfolio. Thus, the median-variance portfolio optimization technique can help conservation planners better manage symmetric uncertainty. However, because the median-variance method uses variance to measure uncertainty, it will not effectively choose portfolios that minimize downside rather than symmetric uncertainty. A conservation planner that uses a median-variance portfolio model may still identify areas with very high degrees of positive skewness as regions that increase overall uncertainty; a mean–downside uncertainty model (such as the MLPM2) would consider these regions as uncertainty-reducing areas and thus allocate larger investments to such areas. Conversely, portfolio allocations based on a median-variance model may allocate larger investments to areas with high negative skewness relative to mean-downside uncertainty portfolios, even though the risks of catastrophic losses associated with such areas are large.
↵4 We also separately weight the returns in each subregion by population and size of subregion. We find that using population- or area-weighted returns does not qualitatively change the results we present in the paper.
↵5 In a separate research paper, we also conduct Mardia’s test of multivariate normality for a different set of spatial data that provide predictions of future conservation outcomes for a range of climate change scenarios. This dataset is provided by the U.S. Geological Survey, which uses WETLANDSCAPE, a climate-driven, process-based, deterministic simulation model, which generates predictions for the Cover Cycle Index (CCI) across the Prairie Pothole Region (PPR) for different climate change scenarios. The CCI acts as a measure of conservation outcome for the PPR. We find that the distributions of CCI outcomes across the PPR are also not multivariate normal.
↵6 The variance-covariance matrix and the symmetric CLPM matrix for the three birds are given in Appendix A.
↵7 Appendix A, Tables A4A and A5A illustrate the SD, EV of returns, and optimal portfolio allocations for conservation activity associated with the red-headed woodpecker and the pileated woodpecker, respectively, based on an MV framework. Appendix A, Tables A4B and A5B illustrate the DSD, EV of returns, and optimal portfolio allocations for conservation activity associated with the red-headed woodpecker and the pileated woodpecker, respectively, based on an MLPM2 framework.
↵8 Returns are calculated as conservation benefits to conservation cost ratio, where individual bird abundance levels represent conservation benefits and the average cost of purchasing land in each of the five subregions is used as a proxy for the cost of doing conservation activity in that subregion. Thus, a low-return subregion may be associated with a low abundance level and/or a high average cost of land. If the low-return subregion is associated with low abundance levels, then it is possible that an optimal portfolio allocation strategy implies abandoning this low-return subregion (with low abundance levels) in order to focus resources on high-return subregions. Alternately, because we take the economic cost of doing conservation activity into our conservation return calculation, a high-abundance subregion with a very high average cost of land may be identified as a low-return subregion and thus may be abandoned to focus on subregions with lower abundance levels but even lower average land costs.
↵9 Appendix A, Table A3A illustrates the SD, EV of returns, and optimal portfolio allocations for conservation activity associated with the yellow-bellied sapsucker based on an MV framework. Table A3B illustrates the DSD, EV of returns, and optimal portfolio allocations for conservation activity associated with the yellow-bellied sapsucker based on an MLPM2 framework.
References
In this issue
Jump to section
Related Articles
Cited By...
- No citing articles found.