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Consumption and Portfolio Choice under Internal Multiplicative Habit Formation

Published online by Cambridge University Press:  27 September 2019

Servaas van Bilsen
Affiliation:
van Bilsen, S.vanBilsen@uva.nl, University of Amsterdam Department of Quantitative Economics
A. Lans Bovenberg
Affiliation:
Bovenberg, A.L.Bovenberg@uvt.nl, Tilburg University Department of Economics
Roger J. A. Laeven*
Affiliation:
Laeven, R.J.A.Laeven@uva.nl, University of Amsterdam Department of Quantitative Economics
*
Laeven (corresponding author), R.J.A.Laeven@uva.nl

Abstract

This paper explores the optimal consumption and investment behavior of an individual who derives utility from the ratio between his consumption and an endogenous habit. We obtain closed-form policies under general utility functionals and stochastic investment opportunities by developing a nontrivial linearization to the budget constraint. This enables us to explicitly characterize how habit formation affects the marginal propensity to consume and optimal stock–bond investments. We also show that in a setting that combines habit formation with Epstein–Zin utility, consumption no longer grows at unrealistically high rates at high ages and investments in risky assets decrease.

Type
Research Article
Copyright
Copyright © Michael G. Foster School of Business, University of Washington 2019

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Footnotes

We are very grateful to Hendrik Bessembinder (the editor) and Mark Schroder (the referee) for comments and suggestions that have significantly improved the paper. We are also grateful to Yacine Aït-Sahalia, Markus Fels (discussant), Glenn Harrison, Gur Huberman, Frank de Jong, Torsten Kleinow (discussant), Olivia Mitchell, Theo Nijman, Antoon Pelsser (discussant), Hato Schmeiser (discussant), and seminar and conference participants at the Australian Research Council (ARC) Centre of Excellence in Population Ageing Research (CEPAR) in Sydney, the Australasian Finance and Banking Conference, the Center for Economic Analysis of Risk (CEAR)/Munich Risk and Insurance Center (MRIC) Behavioral Insurance Workshop, the European Group of Risk and Insurance Economists (EGRIE) Annual Meeting, the Netspar International Pension Workshop, Oxford University, the Quantitative Methods in Finance Conference, Tilburg University, Tinbergen Institute, the University of Amsterdam, the University of Liverpool, the University of Pennsylvania Wharton School, and the Winter School on Mathematical Finance for their helpful comments and suggestions. An earlier version of this paper was circulated under the title “How to Invest and Spend Wealth in Retirement? A Utility-Based Analysis.” This research was supported in part by the Netherlands Organization for Scientific Research (NWO) under grant NWO VIDI 2009 (van Bilsen, Laeven) and by the European Commission under the seventh framework program (EU-MOPACT; van Bilsen, Bovenberg).

References

Abel, A. B.Asset Prices under Habit Formation and Catching up with the Joneses.” American Economic Review, 80 (1990), 3842.Google Scholar
Abel, A. B.Risk Premia and Term Premia in General Equilibrium.” Journal of Monetary Economics, 43 (1999), 333.CrossRefGoogle Scholar
Ang, A.; Bekaert, G.; and Lui, J.. “Why Stocks May Disappoint.” Journal of Financial Economics, 76 (2005), 471508.CrossRefGoogle Scholar
Bell, D. E.Regret in Decision Making under Uncertainty.” Operations Research, 30 (1982), 961981.CrossRefGoogle Scholar
Bell, D. E.Risk Premiums for Decision Regret.” Management Science, 29 (1983), 11561166.CrossRefGoogle Scholar
Bell, D. E.Disappointment in Decision Making under Uncertainty.” Operations Research, 33 (1985), 127.CrossRefGoogle Scholar
Berkelaar, A. B.; Kouwenberg, R.; and Post, T.. “Optimal Portfolio Choice under Loss Aversion.” Review of Economics and Statistics, 86 (2004), 973987.CrossRefGoogle Scholar
Bhamra, H. S., and Uppal, R.. “The Role of Risk Aversion and Intertemporal Substitution in Dynamic Consumption-Portfolio Choice with Recursive Utility.” Journal of Economic Dynamics and Control, 30 (2006), 967991.CrossRefGoogle Scholar
Bilsen, S. van; Laeven, R. J. A.; and Nijman, T. E.. “Consumption and Portfolio Choice under Loss Aversion and Endogenous Updating of the Reference Level.” Management Science, forthcoming (2020).Google Scholar
Bodie, Z.; Detemple, J. B.; Otruba, S.; and Walter, S.. “Optimal Consumption-Portfolio Choices and Retirement Planning.” Journal of Economic Dynamics and Control, 28 (2004), 11151148.CrossRefGoogle Scholar
Bodie, Z.; Merton, R. C.; and Samuelson, W. F.. “Labor Supply Flexibility and Portfolio Choice in a Life-Cycle Model.” Journal of Economic Dynamics and Control, 16 (1992), 427449.CrossRefGoogle Scholar
Bowman, D.; Minehart, D.; and Rabin, M.. “Loss Aversion in a Consumption-Savings Model.” Journal of Economic Behavior and Organization, 38 (1999), 155178.CrossRefGoogle Scholar
Brennan, M. J., and Xia, Y.. “Dynamic Asset Allocation under Inflation.” Journal of Finance, 57 (2002), 12011238.CrossRefGoogle Scholar
Campbell, J. Y.; Cocco, J.; Gomes, F.; Maenhout, P. J.; and Viceira, L. M.. “Stock Market Mean Reversion and the Optimal Equity Allocation of a Long-Lived Investor.” European Finance Review, 5 (2001), 269292.CrossRefGoogle Scholar
Campbell, J. Y., and Cochrane, J.. “By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior.” Journal of Political Economy, 107 (1999), 205251.CrossRefGoogle Scholar
Campbell, J. Y., and Deaton, A.. “Why Is Consumption So Smooth?Review of Economic Studies, 56 (1989), 357373.CrossRefGoogle Scholar
Campbell, J. Y., and Mankiw, N. G.. “The Response of Consumption to Income: A Cross-Country Investigation.” European Economic Review, 35 (1991), 723767.CrossRefGoogle Scholar
Campbell, J. Y., and Viceira, L. M.. “Consumption and Portfolio Decisions When Expected Returns Are Time Varying.” Quarterly Journal of Economics, 114 (1999), 433495.CrossRefGoogle Scholar
Carroll, C. D.Solving Consumption Models with Multiplicative Habits.” Economics Letters, 68 (2000), 6777.CrossRefGoogle Scholar
Carroll, C. D.; Overland, J.; and Weil, D. N.. “Comparison Utility in a Growth Model.” Journal of Economic Growth, 2 (1997), 339367.CrossRefGoogle Scholar
Carroll, C. D.; Overland, J.; and Weil, D. N.. “Saving and Growth with Habit Formation.” American Economic Review, 90 (2000), 341355.CrossRefGoogle Scholar
Chacko, G., and Viceira, L. M.. “Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets.” Review of Financial Studies, 18 (2005), 13691402.CrossRefGoogle Scholar
Chan, Y. L., and Kogan, L.. “Catching Up with the Joneses: Heterogeneous Preferences and the Dynamics of Asset Prices.” Journal of Political Economy, 110 (2002), 12551285.CrossRefGoogle Scholar
Chapman, D. A.Habit Formation and Aggregate Consumption.” Econometrica, 66 (1998), 12231230.CrossRefGoogle Scholar
Cocco, J. F.Portfolio Choice in the Presence of Housing.” Review of Financial Studies, 18 (2005), 535567.CrossRefGoogle Scholar
Cocco, J. F.; Gomes, F. J.; and Maenhout, P. J.. “Consumption and Portfolio Choice over the Life Cycle.” Review of Financial Studies, 18 (2005), 491533.CrossRefGoogle Scholar
Constantinides, G. M.Habit Formation: A Resolution of the Equity Premium Puzzle.” Journal of Political Economy, 98 (1990), 519543.CrossRefGoogle Scholar
Corrado, L., and Holly, S.. “Multiplicative Habit Formation and Consumption: A Note.” Economics Letters, 113 (2011), 116119.CrossRefGoogle Scholar
Cox, J. C., and Huang, C.. “Optimal Consumption and Portfolio Policies when Asset Prices Follow a Diffusion Process.” Journal of Economic Theory, 49 (1989), 3383.CrossRefGoogle Scholar
Cox, J. C., and Huang, C.. “A Variational Problem Arising in Financial Economics.” Journal of Mathematical Economics, 20 (1991), 465487.CrossRefGoogle Scholar
Crawford, I.Habits Revealed.” Review of Economic Studies, 77 (2010), 13821402.CrossRefGoogle Scholar
Crossley, T. F.; Low, H.; and O’Dea, C.. “Household Consumption through Recent Recessions.” Journal of Applied Public Economics, 34 (2013), 203229.Google Scholar
Deaton, A.Life-Cycle Models of Consumption: Is the Evidence Consistent with the Theory?” In Advances in Econometrics: Fifth World Congress, Vol. 2, Bewley, T. F., ed. Cambridge University Press (1987), 121148.CrossRefGoogle Scholar
Deaton, A.Understanding Consumption. Oxford, UK: Oxford University Press (1992).CrossRefGoogle Scholar
Detemple, J. B., and Zapatero, F.. “Asset Prices in an Exchange Economy with Habit Formation.” Econometrica, 59 (1991), 16331657.CrossRefGoogle Scholar
Detemple, J. B., and Zapatero, F.. “Optimal Consumption-Portfolio Policies with Habit Formation.” Mathematical Finance, 2 (1992), 251274.CrossRefGoogle Scholar
Duffie, D., and Epstein, L. G.. “Stochastic Differential Utility.” Econometrica, 60 (1992), 353394.CrossRefGoogle Scholar
Duffie, D., and Skiadas, C.. “Continuous-Time Security Pricing: A Utility Gradient Approach.” Journal of Mathematical Economics, 23 (1992), 107131.CrossRefGoogle Scholar
Dus, I.; Maurer, R.; and Mitchell, O. S.. “Betting on Death and Capital Markets in Retirement: A Shortfall Risk Analysis of Life Annuities.” Financial Services Review, 14 (2005), 169196.Google Scholar
Edwards, R. D.Health Risk and Portfolio Choice.” Journal of Business and Economics Statistics, 26 (2008), 472485.CrossRefGoogle Scholar
Epstein, L. G., and Zin, S. E.. “Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework.” Econometrica, 57 (1989), 937969.CrossRefGoogle Scholar
Flavin, M.Excess Sensitivity of Consumption to Current Income: Liquidity Constraints or Myopia?Canadian Journal of Economics, 18 (1985), 117136.CrossRefGoogle Scholar
Fuhrer, J. C.Habit Formation in Consumption and Its Implications for Monetary-Policy Models.” American Economic Review, 90 (2000), 367390.CrossRefGoogle Scholar
Gomes, F. J.; Kotlikoff, L. J.; and Viceira, L. M.. “Optimal Life-Cycle Investing with Flexible Labor Supply: A Welfare Analysis of Life-Cycle Funds.” American Economic Review, 98 (2008), 297303.CrossRefGoogle Scholar
Gomes, F. J., and Michaelides, A.. “Portfolio Choice with Internal Habit Formation: A Life-Cycle Model with Uninsurable Labor Income Risk.” Review of Economic Dynamics, 6 (2003), 729766.CrossRefGoogle Scholar
Gomes, F. J., and Michaelides, A.. “Optimal Life-Cycle Asset Allocation: Understanding the Empirical Evidence.” Journal of Finance, 60 (2005), 869904.CrossRefGoogle Scholar
Gómez, J.-P.; Priestley, R.; and Zapatero, F.. “Implications of Keeping-Up-with-the-Joneses Behavior for the Equilibrium Cross Section of Stock Returns: International Evidence.” Journal of Finance, 64 (2009), 27032737.CrossRefGoogle Scholar
Guasoni, P.; Huberman, G.; and Ren, D.. Shortfall Aversion. Working Paper No. 15–22, Columbia Business School (2015).CrossRefGoogle Scholar
Guillén, M.; Jørgensen, P. L.; and Nielsen, J. P.. “Return Smoothing Mechanisms in Life and Pension Insurance: Path-Dependent Contingent Claims.” Insurance: Mathematics and Economics, 38 (2006), 229252.Google Scholar
Guillén, M.; Nielsen, J. P.; Pérez-Marín, A. M.; and Petersen, K. S.. “Performance Measurement of Pension Strategies: A Case Study of Danish Life-Cycle Products.” Scandinavian Actuarial Journal, 2013 (2013), 4968.CrossRefGoogle Scholar
Gul, F.A Theory of Disappointment Aversion.” Econometrica, 59 (1991), 667686.CrossRefGoogle Scholar
Horneff, W. J.; Maurer, R. H.; Mitchell, O. S.; and Dus, I.. “Following the Rules: Integrating Asset Allocation and Annuitization in Retirement Portfolios.” Insurance: Mathematics and Economics, 42 (2008), 396408.Google Scholar
Jørgensen, P. L., and Linnemann, P.. “A Comparison of Three Different Pension Savings Products with Special Emphasis on the Payout Phase.” Annals of Actuarial Science, 6 (2012), 137152.CrossRefGoogle Scholar
Kahneman, D., and Tversky, A.. “Prospect Theory: An Analysis of Decision Under Risk.” Econometrica, 47 (1979), 263292.CrossRefGoogle Scholar
Karatzas, I.; Lehoczky, J. P.; and Shreve, S. E.. “Optimal Consumption and Portfolio Decisions for a ‘Small Investor’ on a Finite Horizon.” SIAM Journal of Control and Optimization, 25 (1987), 15571586.CrossRefGoogle Scholar
Karatzas, I., and Shreve, S. E.. Methods of Mathematical Finance, Applications of Mathematics: Stochastic Modelling and Applied Probability, Vol. 39, New York: Springer (1998).Google Scholar
Kőszegi, B., and Rabin, M.. “A Model of Reference-Dependent Preferences.” Quarterly Journal of Economics, 121 (2006), 11331165.Google Scholar
Kőszegi, B., and Rabin, M.. “Reference-Dependent Risk Attitudes.” American Economic Review, 97 (2007), 10471073.CrossRefGoogle Scholar
Kőszegi, B., and Rabin, M.. “Reference-Dependent Consumption Plans.” American Economic Review, 99 (2009), 909936.CrossRefGoogle Scholar
Kozicki, S., and Tinsley, P. A.. “Dynamic Specifications in Optimizing Trend-Deviation Macro Models.” Journal of Economic Dynamics and Control, 26 (2002), 15851611.CrossRefGoogle Scholar
Kraft, H., and Seifried, F. T.. “Stochastic Differential Utility as the Continuous-Time Limit of Recursive Utility.” Journal of Economic Theory, 151 (2014), 528550.CrossRefGoogle Scholar
Kreps, D. M., and Porteus, E. L.. “Temporal Resolution of Uncertainty and Dynamic Choice Theory.” Econometrica, 46 (1978), 185200.CrossRefGoogle Scholar
Laeven, R. J. A., and Stadje, M. A.. “Robust Portfolio Choice and Indifference Valuation.” Mathematics of Operations Research, 39 (2014), 11091141.CrossRefGoogle Scholar
Linnemann, P.; Bruhn, K.; and Steffensen, M.. “A Comparison of Modern Investment-Linked Pension Savings Products.” Annals of Actuarial Science, 9 (2014), 7284.CrossRefGoogle Scholar
Liu, J.Portfolio Selection in Stochastic Environments.” Review of Financial Studies, 20 (2007), 139.CrossRefGoogle Scholar
Loomes, G., and Sugden, R.. “Regret Theory: An Alternative Theory of Rational Choice under Uncertainty.” Economic Journal, 92 (1982), 805824.CrossRefGoogle Scholar
Loomes, G., and Sugden, R.. “Disappointment and Dynamic Consistency in Choice under Uncertainty.” Review of Economic Studies, 53 (1986), 271282.CrossRefGoogle Scholar
Maurer, R.; Mitchell, O. S.; Rogalla, R.; and Siegelin, I.. “Accounting and Actuarial Smoothing of Retirement Payouts in Participating Life Annuities.” Insurance: Mathematics and Economics, 71 (2016), 268283.Google Scholar
Maurer, R.; Mitchell, O. S.; Rogalla, R.; and Kartashov, V.. “Life Cycle Portfolio Choice with Systematic Longevity Risk and Variable Investment-Linked Deferred Annuities.” Journal of Risk and Insurance, 80 (2013), 649676.CrossRefGoogle Scholar
Maurer, R.; Rogalla, R.; and Siegelin, I.. “Participating Payout Annuities: Lessons from Germany.” ASTIN Bulletin, 43 (2013), 159187.CrossRefGoogle Scholar
Merton, R. C.Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case.” Review of Economics and Statistics, 51 (1969), 247257.CrossRefGoogle Scholar
Merton, R. C.Optimum Consumption and Portfolio Rules in a Continuous-Time Model.” Journal of Economic Theory, 3 (1971), 373413.CrossRefGoogle Scholar
Merton, R. C.The Crisis in Retirement Planning.” Harvard Business Review, 92 (2014), 4350.Google Scholar
Morningstar. “2017 Target-Date Fund Landscape. Answers to Frequently Asked Questions.” (2017).Google Scholar
Mossin, J.Optimal Multiperiod Portfolio Policies.” Journal of Business, 41 (1968), 215229.CrossRefGoogle Scholar
Muermann, A.; Mitchell, O. S.; and Volkman, J. M.. “Regret, Portfolio Choice, and Guarantees in Defined Contribution Schemes.” Insurance: Mathematics and Economics, 39 (2006), 219229.Google Scholar
Munk, C.Portfolio and Consumption Choice with Stochastic Investment Opportunities and Habit Formation in Preferences.” Journal of Economic Dynamics and Control, 32 (2008), 35603589.CrossRefGoogle Scholar
Pagel, M.Expectations-Based Reference-Dependent Life-Cycle Consumption.” Review of Economic Studies, 84 (2017), 885934.Google Scholar
Pliska, S. R.A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios.” Mathematics of Operations Research, 11 (1986), 371382.CrossRefGoogle Scholar
Quiggin, J.Regret Theory with General Choice Sets.” Journal of Risk and Uncertainty, 8 (1994), 153165.CrossRefGoogle Scholar
Samuelson, P. A.Lifetime Portfolio Selection by Dynamic Stochastic Programming.” Review of Economics and Statistics, 51 (1969), 239246.CrossRefGoogle Scholar
Samwick, A. A.Discount Rate Heterogeneity and Social Security Reform.” Journal of Development Economics, 57 (1998), 117146.CrossRefGoogle Scholar
Schroder, M., and Skiadas, C.. “Optimal Consumption and Portfolio Selection with Stochastic Differential Utility.” Journal of Economic Theory, 89 (1999), 68126.CrossRefGoogle Scholar
Schroder, M., and Skiadas, C.. “An Isomorphism Between Asset Pricing Models With and Without Linear Habit Formation.” Review of Financial Studies, 15 (2002), 11891221.CrossRefGoogle Scholar
Smith, W. T., and Zhang, Q.. “Asset Pricing with Multiplicative Habit and Power-Expo Preferences.” Economics Letters, 94 (2007), 319325.CrossRefGoogle Scholar
Sugden, R.An Axiomatic Foundation for Regret Theory.” Journal of Economic Theory, 60 (1993), 159180.CrossRefGoogle Scholar
Sundaresan, S. M.Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth.” Review of Financial Studies, 2 (1989), 7389.CrossRefGoogle Scholar
Tversky, A., and Kahneman, D.. “Advances in Prospect Theory: Cumulative Representation of Uncertainty.” Journal of Risk and Uncertainty, 5 (1992), 297323.CrossRefGoogle Scholar
Viceira, L. M.Optimal Portfolio Choice for Long-Horizon Investors with Nontradable Labor Income.” Journal of Finance, 56 (2001), 433470.CrossRefGoogle Scholar
Wachter, J. A.Portfolio and Consumption Decisions Under Mean-Reverting Returns: An Exact Solution for Complete Markets.” Journal of Financial and Quantitative Analysis, 37 (2002), 6391.CrossRefGoogle Scholar
Yao, R., and Zhang, H. H.. “Optimal Consumption and Portfolio Choices with Risky Housing and Borrowing Constraints.” Review of Financial Studies, 18 (2005), 197239.CrossRefGoogle Scholar