Annals of Operations Research

, Volume 258, Issue 2, pp 479–502 | Cite as

The optimal harvesting problem under price uncertainty: the risk averse case

Article
  • 147 Downloads

Abstract

We study the exploitation of a one species, multiple stand forest plantation when timber price is governed by a stochastic process. Our model is a stochastic dynamic program with a weighted mean-risk objective function, and our main risk measure is the Conditional Value-at-Risk. We consider two stochastic processes, geometric Brownian motion and Ornstein–Uhlenbeck: in the first case, we completely characterize the optimal policy for all possible choices of the parameters while in the second, we provide sufficient conditions assuring that harvesting everything available is optimal. In both cases we solve the problem theoretically for every initial condition. We compare our results with the risk neutral framework and generalize our findings to any coherent risk measure that is affine on the current price.

Keywords

Multistage stochastic programming Optimal harvesting  Forestry Coherent risk measures 

References

  1. Alvarez, L., & Koskela, E. (2005). Wicksellian theory of forest rotation under interest rate variability. Journal of Economic Dynamics and Control, 29(3), 529–545.CrossRefGoogle Scholar
  2. Alvarez, L., & Koskela, E. (2006). Does risk aversion accelerate optimal forest rotation under uncertainty? Journal of Forest Economics, 12(3), 171–184.CrossRefGoogle Scholar
  3. Alvarez, L., & Koskela, E. (2007). Taxation and rotation age under stochastic forest stand value. Journal of Environmental Economics and Management, 54, 113–127.CrossRefGoogle Scholar
  4. Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.CrossRefGoogle Scholar
  5. Blomvall, J., & Shapiro, A. (2006). Solving multistage asset investment problems by the sample average approximation method. Mathematical Programming, 108(2–3), 571–595.CrossRefGoogle Scholar
  6. Boychuk, D., & Martell, D. (1996). A multistage stochastic programming model for sustainable forest-level timber supply under risk of fire. Forest Science, 42(1), 10–26.Google Scholar
  7. Brazee, R., & Mendelsohn, R. (1988). Timber harvesting with fluctuating prices. Forest Science, 34(2), 359–372.Google Scholar
  8. Carmel, Y., Paz, S., Jahashan, F., & Shoshany, M. (2009). Assessing fire risk using monte carlo simulations of fire spread. Forest Ecology and Management, 257(1), 370–377.CrossRefGoogle Scholar
  9. Clarke, J., & Harry, R. (1989). The tree-cutting problem in a stochastic environment: The case of age-dependent growth. Journal of Economic Dynamics and Control, 13(4), 569–595.CrossRefGoogle Scholar
  10. Dixit, A., & Pindyck, R. (1994). Investment under uncertainty. Princeton: Princeton University Press.Google Scholar
  11. Gjolberg, O., & Guttormsen, A. (2002). Real options in the forest: What if prices are mean-reverting? Forest Policy and Economics, 4(1), 13–20.CrossRefGoogle Scholar
  12. Gong, P. (1998). Risk preferences and adaptive harvest policies for even-aged stand management. Forest Science, 44(4), 496–506.Google Scholar
  13. Gong, P., & Löfgren, K. (2003). Risk-aversion and the short-run supply of timber. Forest Science, 49(5), 647–656.Google Scholar
  14. Gong, P., & Löfgren, K. (2008). Impact of risk aversion on the optimal rotation with stochastic price. Natural Resource Modeling, 21(3), 385–415.CrossRefGoogle Scholar
  15. Haneveld, W. K. K., Streutker, M. H., & van der Vlerk, M. H. (2010). An alm model for pension funds using integrated chance constraints. Annals of Operations Research, 177(1), 47–62.CrossRefGoogle Scholar
  16. Hildebrandt, P., Kirchlechner, P., Hahn, A., Knoke, T., & Mujica, R. (2010). Mixed species plantations in Southern Chile and the risk of timber price fluctuation. European Journal of Forest Research, 129(5), 935–946.CrossRefGoogle Scholar
  17. Insley, M., & Rollins, K. (2005). On solving the multirotational timber harvesting problem with stochastic prices: A linear complementarity formulation. American Journal of Agricultural Economics, 87, 735–755.CrossRefGoogle Scholar
  18. Kim, Y. H., Bettinger, P., & Finney, M. (2009). Spatial optimization of the pattern of fuel management activities and subsequent effects on simulated wildfires. European Journal of Operational Research, 197(1), 253–265.CrossRefGoogle Scholar
  19. Knoke, T., Stimm, B., Ammer, C., & Moog, M. (2005). Mixed forests reconsidered: A forest economics contribution on an ecological concept. Forest Ecology and Management, 213(1), 102–116.CrossRefGoogle Scholar
  20. Lembersky, M., & Johnson, K. (1975). Optimal policies for managed stands: An infinite horizon Markov decision process approach. Forest Science, 21(2), 109–122.Google Scholar
  21. Lönnstedt, L., & Svensson, J. (2000). Non-industrial private forest owners’ risk preferences. Scandinavian Journal of Forest Research, 15(6), 651–660.CrossRefGoogle Scholar
  22. Maller, R., Müller, G., & Szimayer, A. (2009). Ornstein–Uhlenbeck processes and extensions. Handbook of financial time series (pp. 421–438). Berlin, Heidelberg: Springer.CrossRefGoogle Scholar
  23. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.Google Scholar
  24. Mosquera, J., Henig, M., & Weintraub, A. (2011). Design of insurance contracts using stochastic programming in forestry planning. Annals of Operations Research, 190(1), 117–130.CrossRefGoogle Scholar
  25. Ollikainen, M. (1993). A mean-variance approach to short-term timber selling and forest taxation under multiple sources of uncertainty. Canadian Journal of Forest Research, 23(4), 573–581.CrossRefGoogle Scholar
  26. Philpott, A., de Matos, V., & Finardi, E. (2013). On solving multistage stochastic programs with coherent risk measures. Operations Research, 61(4), 957–970.CrossRefGoogle Scholar
  27. Piazza, A., & Pagnoncelli, B. (2014). The optimal harvesting problem with price uncertainty. Annals of Operations Research, 217(1), 425–445.Google Scholar
  28. Piazza, A., & Pagnoncelli, B. K. (2015). The stochastic mitra-wan forestry model: Risk neutral and risk averse cases. Journal of Economics, 115, 175–194.Google Scholar
  29. Plantinga, A. (1998). The optimal timber rotation: An option value approach. Forest Science, 44(2), 192–202.Google Scholar
  30. Rapaport, A., Sraidi, S., & Terreaux, J. (2003). Optimality of greedy and sustainable policies in the management of renewable resources. Optimal Control Applications and Methods, 24(1), 23–44.CrossRefGoogle Scholar
  31. Reeves, L., & Haight, R. (2000). Timber harvest scheduling with price uncertainty using markowitz portfolio optimization. Annals of Operations Research, 95(1–4), 229–250.CrossRefGoogle Scholar
  32. Rockafellar, R., & Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk, 2, 21–42.CrossRefGoogle Scholar
  33. Roessiger, J., Griess, V., & Knoke, T. (2011). May risk aversion lead to near-natural forestry? A simulation study. Forestry, 84(5), 527–537.CrossRefGoogle Scholar
  34. Ruszczynski, A., & Shapiro, A. (2005). Conditional risk mappings. Mathematics of Operations Research, 31, 544–561.CrossRefGoogle Scholar
  35. Shapiro, A. (2009). On a time consistency concept in risk averse multistage stochastic programming. Operations Research Letters, 37(3), 143–147.CrossRefGoogle Scholar
  36. Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on stochastic programming: Modeling and theory (Vol. 9). Philadelphia: SIAM.CrossRefGoogle Scholar
  37. Tahvonen, O., & Kallio, M. (2006). Optimal harvesting of forest age classes under price uncertainty and risk aversion. Natural Resource Modeling, 19(4), 557–585.CrossRefGoogle Scholar
  38. Thomson, T. (1992). Optimal forest rotation when stumpage prices follow a diffusion process. Land Economics, 68(3), 329–342.CrossRefGoogle Scholar
  39. Valladão, D. M., Veiga, Á., & Veiga, G. (2014). A multistage linear stochastic programming model for optimal corporate debt management. European Journal of Operational Research, 237(1), 303–311.CrossRefGoogle Scholar
  40. Yoshimoto, A., & Shoji, I. (1998). Searching for an optimal rotation age for forest stand management under stochastic log prices. European Journal of Operational Research, 105(1), 100–112.CrossRefGoogle Scholar
  41. Zheng, H. (2009). Efficient frontier of utility and cvar. Mathematical Methods of Operations Research, 70(1), 129–148.CrossRefGoogle Scholar
  42. Zhong-wei, W., & Yan, P. (2009) Measurement of forest fire risk based on var. In International conference on management science and engineering, 2009. ICMSE 2009. IEEE.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Escuela de Negocios, Universidad Adolfo IbáñezSantiagoChile
  2. 2.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile

Personalised recommendations