AnalysisOptimizing agricultural land-use portfolios with scarce data—A non-stochastic model
Introduction
Modern financial theory is still largely based on the famous results that follow from Markowitz, 1952, Markowitz, 2010 theory of portfolio selection. The theory of portfolio selection is used to analyze and improve decision making in natural resources and the environment, for issues such as biodiversity conservation, forestry, grassland and fisheries management, and land allocation. Examples include analyzing common agricultural policy, conservation payments, irrigation, flood management, and optimization. For instance, Figge (2004) applied portfolio theory to develop a concept for valuing the benefits of biodiversity. In a marine case study on biological conservation, Halpern et al. (2011) adopted portfolio selection theory to analyze the impact of spatial variance in returns from natural resources on the equitable delivery of value to individuals and communities. Koellner and Schmitz (2006) contributed an application of the portfolio theory in grassland science using the reward-to-variability ratio for optimization (cf. Sharpe, 1994). There are also various applications of the portfolio theory in forest science; Hildebrandt and Knoke (2011) provide an overview on forest investment decisions under uncertainty, including applications of portfolio theory.
There are many other applications of portfolio selection theory in fishery science. For example, Griffiths et al. (2014) recently concluded that portfolio theory provides a straightforward method for characterizing the resilience of salmon ecosystems and their services. Moore et al. (2010) also used economic portfolio theory to simulate the impact of synchronization of salmon populations on the risk-adjusted performance of fish portfolios. Sanchirico et al. (2008) employed a portfolio framework to consider variance and covariance in gross fishing revenues when setting total allowable catch for individual species. Another example is the work of Edwards et al. (2004), who systematically combined various fish stocks into a portfolio that balances expected aggregate returns against risks. In an earlier study, Larkin et al. (2003) maximized unit returns for various pre-defined risk levels to show that the actual composition of fish resources is not part of the efficient portfolios (that is, portfolios that achieve maximum economic return for pre-defined risk levels).
This paper deals with the important problem of allocating scarce land to various land-use options, which has also been supported by portfolio theory in various studies. Land allocation is among the world's most pressing environmental issues, as confirmed by Wise et al. (2009). They conclude that allocating scarce land resources to competing ends, for instance, to balance climate protection and food production, will remain a major challenge of the 21st century. The allocation of land has long been studied in land-use economics, beginning with von Thünen's (1842) seminal work on distributing land-use options based on their land rent (see Samuelson, 1983, for an extensive review of Thünen's work). At present, Thünen's theory is applied in several international studies (e.g., Angelsen, 2010, Phelps et al., 2013). Macmillan (1992) has proposed an extension of the Thünen land rent model based on portfolio theory. Many studies have applied the classical land rent theory combined with portfolio selection theory. Roche and McQuinn (2004) and Havlik et al. (2008) used a portfolio theory framework to analyze the consequences of the Common Agricultural Policy in the European Union. Abson et al. (2013) linked landscape diversity and the resilience of agricultural returns, while Kaplan (1985) optimized the structural composition of farmlands. In other land-use studies, portfolio selection theory has been used to derive conservation payments (Benitez et al., 2006, Castro et al., 2013). Water management and scarcity issues have increasingly been studied using the theory of portfolio selection. For example, afforestation on marginal land in irrigated farming systems has been investigated (Djanibekov and Khamzina, 2014) as well as improved irrigation water management in uncertain conditions (Paydar and Qureshi, 2012). A further development of flood management (Aerts et al., 2008) and water planning (Marioni et al., 2011) represent other environmental studies based on portfolio selection theory.
While many studies in environmental and resource economics rely on the classical and prestigious theory of portfolio selection, this theory has some disadvantages. In Markowitz's (1952) model, the economic return is the expected value of a random portfolio return, where the associated risk is quantified by the variance of the return (Goldfarb and Iyengar, 2003). The theory of portfolio selection is thus based on using a stochastic model, and the associated mathematics to combine covariances of economic returns assume that these returns are normally distributed. However, in real applications, it is hard to obtain the actual distribution of returns (Yu and Jin, 2012) because historical data are often limited. To overcome the problem of missing data, simulation techniques such as Monte Carlo Simulation have been used, but these methods are data-demanding and often show that returns are not normally distributed (e.g., Knoke and Wurm, 2006).
The stability of estimates for covariances is another problem. Robust and reliable information on covariance that will also hold true for future developments of market and biophysical risks is almost impossible to obtain. Also, practitioners often abstain from using the results from portfolio optimizations because these results can be very sensitive to small perturbations in the input parameters of the problem (e.g., Goldfarb and Iyengar, 2003). Given these complications, alternative approaches that can still provide good results without this information on covariance could be advantageous.
To solve these problems, alternative programming techniques have been proposed to optimize portfolio composition (e.g., Bertsimas and Sim, 2004). These include non-stochastic convex programming that guarantees exact solutions, should those exist. This may be accomplished, for example, using linear programming (Ben-Tal and Nemirovski, 2000). Uncertainty can be included by using constraints that reflect plausible margins for possible parameter perturbations to achieve a robust optimization. Robust optimization, as defined here, searches for an acceptable result for all parameter perturbations considered.
Techniques that can produce robust results also exist in stochastic optimization, such as maximizing the value at risk as a variant of worst-case optimization (see Härtl et al., 2013, for an example in forest planning). However, in contrast to stochastic optimization, non-stochastic robust optimization weights all data perturbations equally and does not assign different probabilities, for instance, to different return deviations. Non-stochastic robust optimization also needs at least some specification of possible input data variations. However, the assumptions about data variation are not necessarily as detailed as those needed in stochastic mean-variance optimization. Thus, non-stochastic optimization may be less data-demanding than a classical portfolio optimization. Given this background, this method could be a good alternative to the existing economic land-use models based on classical portfolio selection theory, in cases where economic information about uncertainties is scarce.
A recent literature review on robust optimization shows that this technique is still popular and that its application is increasing (Gabrel et al., 2014). However, robust optimization has rarely been applied to problems in environmental and resource economics. One exception is the work of Palma and Nelson (2009), who apply robust optimization in forest resource management. However, this study does not analyze a portfolio-based problem of land allocation. Applying robust portfolio optimization to analyze the economics of land-use diversification is actually very rare. It will therefore be interesting to compare portfolio compositions achieved with a robust, non-stochastic optimization technique to those obtained by classical mean-variance optimization. A recent study on the economic attractiveness of producing organic bananas in Ecuador (Castro et al., 2015) will provide the data for such a comparison. The analysis will include economic aspects and the resulting degree of land-use diversification. Shannon's index for the various land-use portfolios can be used to compare differences in landscape diversity (Nagendra, 2002). Successfully applying non-stochastic optimization to land-use allocation problems could reduce data requirements, while resulting in meaningful land-use portfolios that may be more stable across changing assumptions about uncertainty. Such improvements in the modeling technique could increase the application of land-allocation models that can address uncertainties, while helping to adjust models to different spatial scales and environmental conditions. These models are an important tool for designing cost-effective land-use and environmental policies under the increasing uncertainty of future food markets and climate conditions (Knoke et al., 2013). The following questions will therefore be addressed: (i) How do land-use portfolios derived using robust optimizations differ from those that are classically derived by mean-variance optimization? (ii) How does the method of portfolio optimization influence the Shannon index? (iii) How do robust land-use portfolios perform in a mean-variance context?
Section snippets
A Stochastic and a Non-stochastic Model
Classical portfolio optimization derives an efficient frontier formed by portfolios that maximize the economic returns for pre-defined levels of economic uncertainty. The efficient frontier is based on “mean-variance” optimization. The standard deviation of the economic return is usually adopted as a measure for uncertainty. Using this theory for an optimized allocation of land to different land-use options in a Thünen framework (which is not spatially explicit in our examples), we formulate
Land-use Portfolios in a Stochastic Mean-variance Context
The analysis in this paper uses “classical” land-use portfolios as reference, derived from land rent for an Ecuadorian agricultural landscape, which Castro et al. (2015) computed using mean-variance optimization as per Markowitz, according to Eq. (1). The degree of diversification of land-use portfolios decreases as the level of accepted economic risk/uncertainty rises (Fig. 3).
Organic banana is an important part of the resulting land-use portfolios over a large range of uncertainty levels. The
Conclusions
The results of the analysis show that non-stochastic portfolio optimization leads to land-use portfolios that are more diverse than classical mean-variance portfolios. The robust portfolios show only small or (when very little prior information is available) moderate losses in expected economic returns when the parameters used for mean-variance optimization are assumed to be true. It may thus be concluded that non-stochastic robust portfolio modeling is a good alternative to classical
Acknowledgments
We are grateful to “Deutsche Forschungsgemeinschaft” (DFG) for financial support of the study (KN 586/5-2, KN 586/9-1, and KN 586/11-1) and to the members of the research group FOR 816 and the Platform for Biodiversity and Ecosystem Monitoring and Research in South Ecuador whose research initiative (http://www.tropicalmountainforest.org/) made the study possible. Furthermore, we thank Laura Carlson and Elizabeth Gosling for the language editing, and Dr. Martin Döllerer for immense help with the
References (46)
- et al.
Shortfall as a risk measure: properties, optimization and application
J. Econ. Dyn. Control.
(2004) - et al.
Portfolio management of wild fish stocks
Ecol. Econ.
(2004) - et al.
Recent advances in robust optimization: an overview
Eur. J. Oper. Res.
(2014) - et al.
Using portfolio theory to assess tradeoffs between return from natural capital and social equity across space
Biol. Conserv.
(2011) - et al.
Risk-sensitive planning support for forest enterprises: the YAFO model
Comput. Electron. Agric.
(2013) - et al.
Optimizing the shares of native tree species in forest plantations with biased financial parameters
Ecol. Econ.
(2009) - et al.
Investment decisions under uncertainty—a methodological review on forest science studies
For. Pol. Econ.
(2011) - et al.
Food production and climate protection—what abandoned lands can do to preserve natural forests
Glob. Environ. Chang.
(2013) Opposite trends in response for the Shannon and Simpson indices of landscape diversity
Appl. Geogr.
(2002)- et al.
Irrigation water management in uncertain conditions—application of modern portfolio theory
Agric. Water Manag.
(2012)
Extended lexicographic goal programming: a unifying approach
Omega
An empirical approach to ecosystem-based fishery management
Ecol. Econ.
Goal programming for decision making: an overview of the current state-of-the-art
Eur. J. Oper. Res.
Landscape diversity and the resilience of agricultural returns: a portfolio analysis of land-use patterns and economic returns from lowland agriculture
Agric. Food Secur.
Dealing with uncertainty in flood management through diversification
Ecol. Soc.
Policies for reduced deforestation and their impact on agricultural production
Proc. Natl. Acad. Sci.
Conservation payments under risk: a stochastic dominance approach
Am. J. Agric. Econ.
Robust optimization
Robust solutions of linear programming problems contaminated with uncertain data
Math. Program.
The price of robustness
Oper. Res.
Constructing uncertainty sets for robust linear optimization
Oper. Res.
Avoiding the loss of shade coffee plantations: how to derive conservation payments for risk-averse land-users
Agrofor. Syst.
Ecuadorian banana farms should consider organic banana with low price risks in their land-use portfolios
PLoS One
Cited by (55)
Using modern portfolio theory to enhance ecosystem service delivery: A case study from China
2023, Journal of Environmental ManagementOptimizing forest landscape composition for multiple ecosystem services based on uncertain stakeholder preferences
2023, Science of the Total EnvironmentAn efficient portfolio approach towards ecosystem-based fisheries governance in EU
2022, Fisheries ResearchThe contribution of bio-economic assessments to better informed land-use decision making: An overview
2022, Ecological Engineering