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Equilibrium in securities markets with heterogeneous investors and unspanned income risk

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  • Christensen, Peter Ove
  • Larsen, Kasper
  • Munk, Claus

Abstract

In a finite time horizon, incomplete market, continuous-time setting with dividends and investor incomes governed by arithmetic Brownian motions, we derive closed-form solutions for the equilibrium risk-free rate and stock price for an economy with finitely many heterogeneous CARA investors and unspanned income risk. In equilibrium, the Sharpe ratio is the same as in an otherwise identical complete market economy, whereas the risk-free rate is lower and, consequently, the stock price is higher. The reduction in the risk-free rate is highest when the more risk-averse investors face the largest unspanned income risk.

Suggested Citation

  • Christensen, Peter Ove & Larsen, Kasper & Munk, Claus, 2012. "Equilibrium in securities markets with heterogeneous investors and unspanned income risk," Journal of Economic Theory, Elsevier, vol. 147(3), pages 1035-1063.
    • Handle: RePEc:eee:jetheo:v:147:y:2012:i:3:p:1035-1063
    DOI: 10.1016/j.jet.2012.01.007
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    1. Orazio P. Attanasio & Monica Paiella, 2011. "Intertemporal consumption choices, transaction costs and limited participation in financial markets: reconciling data and theory," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 26(2), pages 322-343, March.
    2. Constantinides, George M & Duffie, Darrell, 1996. "Asset Pricing with Heterogeneous Consumers," Journal of Political Economy, University of Chicago Press, vol. 104(2), pages 219-240, April.
    3. Den Haan, Wouter J., 2001. "The importance of the number of different agents in a heterogeneous asset-pricing model," Journal of Economic Dynamics and Control, Elsevier, vol. 25(5), pages 721-746, May.
    4. Wang, Neng, 2004. "Precautionary saving and partially observed income," Journal of Monetary Economics, Elsevier, vol. 51(8), pages 1645-1681, November.
    5. Darrell Duffie & Chi-Fu Huang, 2005. "Implementing Arrow-Debreu Equilibria By Continuous Trading Of Few Long-Lived Securities," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 4, pages 97-127, World Scientific Publishing Co. Pte. Ltd..
    6. Steven J. Davis & Felix Kubler & Paul Willen, 2006. "Borrowing Costs and the Demand for Equity over the Life Cycle," The Review of Economics and Statistics, MIT Press, vol. 88(2), pages 348-362, May.
    7. Kjetil Storesletten & Chris Telmer & Amir Yaron, 2007. "Asset Pricing with Idiosyncratic Risk and Overlapping Generations," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 10(4), pages 519-548, October.
    8. Krusell, Per & Smith, Anthony A., 1997. "Income And Wealth Heterogeneity, Portfolio Choice, And Equilibrium Asset Returns," Macroeconomic Dynamics, Cambridge University Press, vol. 1(2), pages 387-422, June.
    9. Krueger, Dirk & Lustig, Hanno, 2010. "When is market incompleteness irrelevant for the price of aggregate risk (and when is it not)?," Journal of Economic Theory, Elsevier, vol. 145(1), pages 1-41, January.
    10. Weil, Philippe, 1989. "The equity premium puzzle and the risk-free rate puzzle," Journal of Monetary Economics, Elsevier, vol. 24(3), pages 401-421, November.
    11. Lynch, Anthony W. & Tan, Sinan, 2011. "Labor income dynamics at business-cycle frequencies: Implications for portfolio choice," Journal of Financial Economics, Elsevier, vol. 101(2), pages 333-359, August.
    12. Malinvaud, E, 1973. "Markets for an Exchange Economy with Individual Risks," Econometrica, Econometric Society, vol. 41(3), pages 383-410, May.
    13. Calvet, Laurent E., 2001. "Incomplete Markets and Volatility," Journal of Economic Theory, Elsevier, vol. 98(2), pages 295-338, June.
    14. Judd, Kenneth L., 1985. "The law of large numbers with a continuum of IID random variables," Journal of Economic Theory, Elsevier, vol. 35(1), pages 19-25, February.
    15. Wei Xiong & Hongjun Yan, 2010. "Heterogeneous Expectations and Bond Markets," The Review of Financial Studies, Society for Financial Studies, vol. 23(4), pages 1433-1466, April.
    16. Lucas, Deborah J., 1994. "Asset pricing with undiversifiable income risk and short sales constraints: Deepening the equity premium puzzle," Journal of Monetary Economics, Elsevier, vol. 34(3), pages 325-341, December.
    17. Aiyagari, S. Rao & Gertler, Mark, 1991. "Asset returns with transactions costs and uninsured individual risk," Journal of Monetary Economics, Elsevier, vol. 27(3), pages 311-331, June.
    18. Telmer, Chris I, 1993. "Asset-Pricing Puzzles and Incomplete Markets," Journal of Finance, American Finance Association, vol. 48(5), pages 1803-1832, December.
    19. Luis M. Viceira, 2001. "Optimal Portfolio Choice for Long‐Horizon Investors with Nontradable Labor Income," Journal of Finance, American Finance Association, vol. 56(2), pages 433-470, April.
    20. Jérôme B. Detemple & Ren Garcia & Marcel Rindisbacher, 2003. "A Monte Carlo Method for Optimal Portfolios," Journal of Finance, American Finance Association, vol. 58(1), pages 401-446, February.
    21. Harald Uhlig, 1996. "A law of large numbers for large economies (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 41-50.
    22. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    23. Heaton, John & Lucas, Deborah J, 1996. "Evaluating the Effects of Incomplete Markets on Risk Sharing and Asset Pricing," Journal of Political Economy, University of Chicago Press, vol. 104(3), pages 443-487, June.
    24. Henderson, Vicky, 2005. "Explicit solutions to an optimal portfolio choice problem with stochastic income," Journal of Economic Dynamics and Control, Elsevier, vol. 29(7), pages 1237-1266, July.
    25. Wang, Neng, 2006. "Generalizing the permanent-income hypothesis: Revisiting Friedman's conjecture on consumption," Journal of Monetary Economics, Elsevier, vol. 53(4), pages 737-752, May.
    26. Milton Friedman, 1957. "Introduction to "A Theory of the Consumption Function"," NBER Chapters, in: A Theory of the Consumption Function, pages 1-6, National Bureau of Economic Research, Inc.
    27. George M. Constantinides & John B. Donaldson & Rajnish Mehra, 2002. "Junior Can't Borrow: A New Perspective on the Equity Premium Puzzle," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 117(1), pages 269-296.
    28. David K. Levine & William R. Zame, 2002. "Does Market Incompleteness Matter?," Econometrica, Econometric Society, vol. 70(5), pages 1805-1839, September.
    29. Neng Wang, 2003. "Caballero Meets Bewley: The Permanent-Income Hypothesis in General Equilibrium," American Economic Review, American Economic Association, vol. 93(3), pages 927-936, June.
    30. Hui Guo & Robert F. Whitelaw, 2006. "Uncovering the Risk–Return Relation in the Stock Market," Journal of Finance, American Finance Association, vol. 61(3), pages 1433-1463, June.
    31. Svensson, Lars E. O. & Werner, Ingrid M., 1993. "Nontraded assets in incomplete markets : Pricing and portfolio choice," European Economic Review, Elsevier, vol. 37(5), pages 1149-1168, June.
    32. repec:bla:jfinan:v:59:y:2004:i:4:p:1481-1509 is not listed on IDEAS
    33. Lars Nielsen & Maria Vassalou, 2006. "The instantaneous capital market line," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 28(3), pages 651-664, August.
    34. Edmond Malinvaud, 1974. "The Allocation of Individual Risks in Large Markets," International Economic Association Series, in: Jacques H. Drèze (ed.), Allocation under Uncertainty: Equilibrium and Optimality, chapter 8, pages 110-125, Palgrave Macmillan.
    35. Pierre-Olivier Gourinchas & Jonathan A. Parker, 2002. "Consumption Over the Life Cycle," Econometrica, Econometric Society, vol. 70(1), pages 47-89, January.
    36. Joao F. Cocco, 2005. "Consumption and Portfolio Choice over the Life Cycle," The Review of Financial Studies, Society for Financial Studies, vol. 18(2), pages 491-533.
    37. John Y. Campbell, 2006. "Household Finance," Journal of Finance, American Finance Association, vol. 61(4), pages 1553-1604, August.
    38. Ralph S. J. Koijen & Theo E. Nijman & Bas J. M. Werker, 2010. "When Can Life Cycle Investors Benefit from Time-Varying Bond Risk Premia?," The Review of Financial Studies, Society for Financial Studies, vol. 23(2), pages 741-780, February.
    39. Milton Friedman, 1957. "A Theory of the Consumption Function," NBER Books, National Bureau of Economic Research, Inc, number frie57-1.
    40. Bodie, Zvi & Merton, Robert C. & Samuelson, William F., 1992. "Labor supply flexibility and portfolio choice in a life cycle model," Journal of Economic Dynamics and Control, Elsevier, vol. 16(3-4), pages 427-449.
    41. George M. Constantinides, 2002. "Rational Asset Prices," Journal of Finance, American Finance Association, vol. 57(4), pages 1567-1591, August.
    42. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    43. Bernard Dumas & Alexander Kurshev & Raman Uppal, 2009. "Equilibrium Portfolio Strategies in the Presence of Sentiment Risk and Excess Volatility," Journal of Finance, American Finance Association, vol. 64(2), pages 579-629, April.
    44. Szpiro, George G, 1986. "Measuring Risk Aversion: An Alternative Approach," The Review of Economics and Statistics, MIT Press, vol. 68(1), pages 156-159, February.
    45. Munk, Claus & Sørensen, Carsten, 2010. "Dynamic asset allocation with stochastic income and interest rates," Journal of Financial Economics, Elsevier, vol. 96(3), pages 433-462, June.
    46. Duffie, Darrell & Jackson, Matthew O., 1990. "Optimal hedging and equilibrium in a dynamic futures market," Journal of Economic Dynamics and Control, Elsevier, vol. 14(1), pages 21-33, February.
    47. Wang, Neng, 2009. "Optimal consumption and asset allocation with unknown income growth," Journal of Monetary Economics, Elsevier, vol. 56(4), pages 524-534, May.
    48. Christian Gollier & Richard Zeckhauser, 2005. "Aggregation of Heterogeneous Time Preferences," Journal of Political Economy, University of Chicago Press, vol. 113(4), pages 878-896, August.
    49. Michael Gallmeyer & Burton Hollifield, 2008. "An Examination of Heterogeneous Beliefs with a Short-Sale Constraint in a Dynamic Economy," Review of Finance, European Finance Association, vol. 12(2), pages 323-364.
    50. Alexander David, 2008. "Heterogeneous Beliefs, Speculation, and the Equity Premium," Journal of Finance, American Finance Association, vol. 63(1), pages 41-83, February.
    51. John Heaton & Deborah Lucas, 2000. "Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk," Journal of Finance, American Finance Association, vol. 55(3), pages 1163-1198, June.
    52. Tano Santos & Pietro Veronesi, 2006. "Labor Income and Predictable Stock Returns," The Review of Financial Studies, Society for Financial Studies, vol. 19(1), pages 1-44.
    53. Huggett, Mark, 1993. "The risk-free rate in heterogeneous-agent incomplete-insurance economies," Journal of Economic Dynamics and Control, Elsevier, vol. 17(5-6), pages 953-969.
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    5. Zhao, Yang & Yao, Yuan & Wang, Mingtao, 2024. "Risk-free rate puzzle: An explanation of the heterogeneity of consumer risk attitudes under China's income gap," International Review of Economics & Finance, Elsevier, vol. 89(PB), pages 940-960.
    6. Martin Herdegen & Johannes Muhle-Karbe, 2018. "Stability of Radner equilibria with respect to small frictions," Finance and Stochastics, Springer, vol. 22(2), pages 443-502, April.
    7. Jiun-Hua Su, 2019. "Model Selection in Utility-Maximizing Binary Prediction," Papers 1903.00716, arXiv.org, revised Jul 2020.
    8. Akbas, Ferhat & Boehmer, Ekkehart & Jiang, Chao & Koch, Paul D., 2022. "Overnight returns, daytime reversals, and future stock returns," Journal of Financial Economics, Elsevier, vol. 145(3), pages 850-875.
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    10. Keisuke Kizaki & Taiga Saito & Akihiko Takahashi, 2023. "A multi-agent incomplete equilibrium model and its applications to reinsurance pricing and life-cycle investment (Forthcoming in "Insurance: Mathematics and Economics")," CARF F-Series CARF-F-576, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    11. Kim Weston & Gordan Žitković, 2020. "An incomplete equilibrium with a stochastic annuity," Finance and Stochastics, Springer, vol. 24(2), pages 359-382, April.
    12. Luis Escauriaza & Daniel C. Schwarz & Hao Xing, 2020. "Radner equilibrium and systems of quadratic BSDEs with discontinuous generators," Papers 2008.03500, arXiv.org, revised May 2021.
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    14. Tyler Abbot, 2017. "General Equilibrium Under Convex Portfolio Constraints and Heterogeneous Risk Preferences," Papers 1706.05877, arXiv.org, revised Jun 2018.
    15. Su, Jiun-Hua, 2021. "Model selection in utility-maximizing binary prediction," Journal of Econometrics, Elsevier, vol. 223(1), pages 96-124.
    16. Liang, Xiaoqing & Young, Virginia R., 2018. "Annuitization and asset allocation under exponential utility," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 167-183.
    17. Jin Hyuk Choi & Kasper Larsen, 2013. "Taylor approximation of incomplete Radner equilibrium models," Papers 1310.2973, arXiv.org, revised Sep 2014.
    18. Xiaoyu Deng & Jing Tian & Rong Chen, 2019. "Effect of Social Security System on Consumption through Income and Uncertainty: Evidence from China," Sustainability, MDPI, vol. 11(7), pages 1-16, March.
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    More about this item

    Keywords

    Unspanned income; Heterogeneous preferences; Continuous-time equilibrium; Risk-free rate puzzle; Equity premium; Incomplete markets; Brownian motion;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets

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