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A Review of Perturbative Approaches for Robust Optimal Portfolio Problems

  • Fabio Trojani
  • Paolo Vanini
Part of the Applied Optimization book series (APOP, volume 74)

Abstract

Only a few intertemporal optimal consumption and portfolio problems in partial and general equilibrium can be solved explicitly. It is illustrated in the paper that perturbation theory is a powerful tool for deriving approximate analytical solutions for the desired optimal policies in problems where general state dynamics are admitted and a preference for robustness is present. Starting from the perturbative approach proposed recently by Kogan and Uppal it is demonstrated how robust equilibria for some formulations of a preference for robustness in the literature can be solved. A crucial requirement for this approach is the existence of a known functional form for the candidate model solutions, a condition which is not satisfied by some models of a preference for robustness. For these cases, recent results by Trojani and Vanini can be used to obtain a perturbative solution to the Bellman equation of the relevant benchmark model and to give some formal conditions under which the perturbative solution converges to the correct one.

Keywords

Financial Equilibrium Merton’s Model Model Misspecification Perturbation Theory Robust Decision Making 

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References

  1. Anderson, E.W., (1998), Uncertainty and the Dynamics of Pareto Optimal Allocations, University of Chicago, Ph.D. Thesis.Google Scholar
  2. Anderson, E.W., L.P. Hansen, and T.J. Sargent (1998), Risk and Robustness in General Equilibrium, Preprint University of Chicago.Google Scholar
  3. Anderson, E.W., L. P. Hansen, and T.J. Sargent (2000), Robustness, Detection and the Price of Risk, Preprint University of Chicago.Google Scholar
  4. Campbell, J. Y., (1993), Intertemporal Asset Pricing without Consumption Data, American Economic Review, 83, 487–512.Google Scholar
  5. Campbell, J. Y., and L. M. Viceira (1998), Who Should Buy Long Term Bonds, NBER Working Paper No 6801.Google Scholar
  6. Campbell, J. Y., and L. M. Viceira (1999), Consumption and Portfolio Decisions when Expected Returns are Time Varying, Quarterly Journal of Economics, 114, 433–495.CrossRefGoogle Scholar
  7. Chacko, G. and L. M. Viceira (1999), Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets, Working Paper, Harward University.Google Scholar
  8. Chen, Z. and L. G. Epstein, 2000, Ambiguity, Risk and Asset Returns in Continuous Time, mimeo, University of Rochester.Google Scholar
  9. Duffie, D., and L. Epstein (1992a), Stochastic Differential Utility, Econometrica, 60, 353–394.CrossRefGoogle Scholar
  10. Duffie, D., and L. Epstein (1992b), Asset Pricing with Stochastic Differential Utility, Review of Financial Studies, 5, 411–436.CrossRefGoogle Scholar
  11. Duffie, D., P. Geoffard, and C. Skiadas (1994) Efficient and Equilibrium Allocations with Stochastic Differential Utility, Journal of Mathematical Economics, 23, 133–146.CrossRefGoogle Scholar
  12. Duffie, D., and C. F. Huang (1985), Implementing Arrow-Debreu Equilibrium by Continuous Trading of Long Lived securities, Econometrica 53, 1337–1356.CrossRefGoogle Scholar
  13. Duffie, D., and W. Zame (1989), The Consumption Based Capital Asset pricing Model, Econometrica 57, 1279–97.CrossRefGoogle Scholar
  14. Dumas, B., (1989), Two-Person Dynamic Equilibrium in the Capital Market, Review of Financial Studies, 2, 157–188CrossRefGoogle Scholar
  15. Epstein, L., and T. Wang (1994), Intertemporal Asset Pricing Under Knightian Uncertainty, Econometrica, 62, 283–322.CrossRefGoogle Scholar
  16. Epstein, L., and J. Miao (2001), A Two-Person Dynamic Equilibrium under Ambiguity, mimeo, University of Rochester.Google Scholar
  17. Epstein, L., and M. Schneider (2001), Recursive Multiple Priors, mimeo, University of Rochester.Google Scholar
  18. Epstein, L., and M. Schneider (2001a), Learning under Ambiguity, work in progress.Google Scholar
  19. Fleming, W. H., and P. E. Souganidis (1989), On the Existence of Value Functions of Two-Player, Zero Sum Stochastic Differential Games, Indiana University Mathematics Journal, 38, 293–314.CrossRefGoogle Scholar
  20. Gilboa, I. and D. Schmeidler, (1989), Maxmin Expected Utility with non-Unique Prior, Journal of Mathematical Economics, 18, 141–153.CrossRefGoogle Scholar
  21. Hansen, L.P., T.J. Sargent and D., Jr., Tallarini, (1999), Robust Permanent Income and Pricing, Review of Economic Studies, 66, 873–907.CrossRefGoogle Scholar
  22. Hansen, L.P., T.J. Sargent, G. A. Turmuhambetova, and N. Williams, (2001) Robustness and Uncertainty Aversion, Preprint University of Chicago.Google Scholar
  23. Judd, K. L. and S. Guu,(1996), Approximation, Perturbation and Projection Methods for Economic Growth Models, in H. Amman, D. Kendrick, and J. Ust eds., Hanbook of Computational Economics, Elsevier, Amsterdam.Google Scholar
  24. Judd, K. L., (1998), Numerical Methods in Economics, MIT Press, Cambridge, MA.Google Scholar
  25. Karatzas, I., P. Lakner, J.P. Lehoczky and S. E. Shreve, (1990), Existence and Uniqueness of Multi-agent Equilibrium in a Stochastic, Dynamic Consumption/Investment Model, Math. Operations Research 15, 80–128.Google Scholar
  26. Kim, T. S., and E. Omberg (1996), Dynamic Nonmyopic Portfolio Behaviour, Review of Financial Studies, 9, 141–161.CrossRefGoogle Scholar
  27. Knight, F., (1921). Risk, Uncertainty and Profit. Boston, Mass: Houghton Mifflin. Reprint, London: London School of Economics, 1946.Google Scholar
  28. Kogan, L. and R. Uppal, (2000), Risk Aversion and Optimal Portfolios in Partial and General Equilibrium Economics, U.B.C., Preprint.Google Scholar
  29. Lei, C. I., (2001), Why Don’t Investors Have Large Positions in Stocks? A Robustness Perspective, Ph. D. Thesis, Chicago University.Google Scholar
  30. Maenhout, P., (1999), Robust Portfolio Rules and Asset Pricing, Ph.D. Thesis, Harvard University, Cambridge, MA.Google Scholar
  31. Marinacci, M., (1999), Limit Laws for Non-Additive Probabilities and their Frequentist Interpretation, Journal of Economic Theory, 84, 145–195.CrossRefGoogle Scholar
  32. Merton, R.C., (1969), Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case, Review of Economics and Statistics 51, 247–257.CrossRefGoogle Scholar
  33. Merton, R.C., (1971). Optimum Consumption and Portfolio Rules in a Continuous-Time Model, Journal of Economic Theory, 3, 373–413.CrossRefGoogle Scholar
  34. Renyi, A., (1961). On Measures of Entropy and Information, in: Proceedings of the Fourth Berkeley Symposium in Mathematical Statistics. Berkeley: University of California Press.Google Scholar
  35. Renyi, A., (1971). Probability Theory. Amsterdam: North Holland.Google Scholar
  36. Schroder, M. and C. Skiadas (1999), Optimal Consumption and Portfolio Selection with Stochastic Differential Utility, Journal of Economic Theory, 89, No. 1, 68–126.CrossRefGoogle Scholar
  37. Trojani, F. and P. Vanini (2001), A Note on Robustness in Merton’s Model of Intertemporal Consumption and Portfolio Choice, Forthcoming, Journal of Economic Dynamics and Control.Google Scholar
  38. Trojani, F. and P. Vanini (2001a), Perturbative Solutions of Hamilton-JacobiBellman Equations in Non-Homothetic Robust Decision Making, Preprint, University of Southern Switzerland, Lugano.Google Scholar
  39. Trojani, F. and P. Vanini (2001b), Risk, Robustness and Knightian Uncertainty in Heterogeneous-Agents Financial Equilibria, Preprint, University of Southern Switzerland, LuganoGoogle Scholar
  40. Uppal, R. and T. Wang (2001), Model Misspecification and Under-Diversification, U. B. C., preprint.Google Scholar
  41. Wang, J., (1996), The Term Structure of Interest Rates in a Pure Exchange Economy with Heterogenous Investors, Journal of Financial Economics, 41, 75–110.CrossRefGoogle Scholar
  42. White, H., (1996), Estimation, Inference and Specification Analysis. Econometric Society Monographs, No. 22, Cambridge University Press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Fabio Trojani
    • 1
  • Paolo Vanini
    • 2
    • 3
  1. 1.Institute of FinanceUniversity of Southern SwitzerlandLuganoSwitzerland
  2. 2.Zürcher KantonalbankZürichSwitzerland
  3. 3.Institute of FinanceUniversity of Southern SwitzerlandLuganoSwitzerland

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