A Theory of Portfolio Choice for Heterogeneous Investors

Li, Mingzhe (2025): A Theory of Portfolio Choice for Heterogeneous Investors.

Abstract

The Merton framework has difficulty explaining empirical phenomena in household finance and life-cycle patterns. This paper proposes a continuous-time heterogeneous agent model with common noise to study portfolio choice for heterogeneous investors. Households pay an "asset-exposure premium" (AEP) based on their wealth for bearing common financial risk. The AEP effect serves as a non-monotonic third term in optimal portfolio choice. For households near the borrowing constraint, the AEP effect is large and negative, overwhelming other motives and explaining limited participation. As wealth increases, the AEP effect turns positive before disappearing at high wealth, resulting in a hump-shaped profile of risky holdings. The AEP enables households to identify the risk characteristics of their portfolio, explaining why the low wealth exhibits underdiversification and why middle-to-high wealth households hold diversified portfolios. Furthermore, this paper introduces a steady state and solves the degeneration problem of the wealth distribution in the continuous-time ABH framework.

• Item Type: MPRA Paper
• Language: English
• Keywords: portfolio choice; household finance; limited participation; wealth distribution; heterogeneous agent; continuous time; underdiversification; life cycle
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C73 - Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games; D - Microeconomics > D9 - Intertemporal Choice > D91 - Intertemporal Household Choice ; Life Cycle Models and Saving; E - Macroeconomics and Monetary Economics > E2 - Consumption, Saving, Production, Investment, Labor Markets, and Informal Economy > E21 - Consumption ; Saving ; Wealth; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions

References

Achdou, Y., Buera, F. J., Lasry, J.-M., Lions, P.-L., & Moll, B. (2014). Partial differential equation models in macroeconomics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2028), 20130397.

Achdou, Y., Han, J., Lasry, J.-M., Lions, P.-L., & Moll, B. (2022). Income and wealth distribution in macroeconomics: A continuous-time approach. The Review of Economic Studies, 89 (1), 45–86.

Addoum, J. M. (2017). Household portfolio choice and retirement. Review of Economics and Statistics, 99 (5), 870–883.

Ahn, S., Kaplan, G., Moll, B., Winberry, T., & Wolf, C. (2018). When inequality matters for macro and macro matters for inequality. NBER Macroeconomics Annual, 32 (1), 1–75.

Aiyagari, S. R. (1994). Uninsured idiosyncratic risk and aggregate saving. The Quarterly Journal of Economics, 109 (3), 659–684.

Amann, H., & Escher, J. (2009). Analysis iii. Springer.

Ameriks, J., & Zeldes, S. P. (2011). How do household portfolio shares vary with age? SSRN.

Anderson, E. W., Ghysels, E., & Juergens, J. L. (2005). Do heterogeneous beliefs matter for asset pricing? The Review of Financial Studies, 18 (3), 875–924.

Ando, A., & Modigliani, F. (1963). The” life cycle” hypothesis of saving: Aggregate implications and tests. The American Economic Review, 53 (1), 55–84.

Andrade, P., Gaballo, G., Mengus, E., & Mojon, B. (2019). Forward guidance and heterogeneous beliefs. American Economic Journal: Macroeconomics, 11 (3), 1–29.

Angerer, X., & Lam, P.-S. (2009). Income risk and portfolio choice: An empirical study. The Journal of Finance, 64 (2), 1037–1055.

Athreya, K., Ionescu, F., & Neelakantan, U. (2023). Stock market participation: The role of human capital. Review of Economic Dynamics, 47, 1–18.

Bali, T. G., Cakici, N., & Whitelaw, R. F. (2011). Maxing out: Stocks as lotteries and the cross-section of expected returns. Journal of Financial Economics, 99 (2), 427–446.

Barberis, N., & Huang, M. (2008). Stocks as lotteries: The implications of probability weighting for security prices. American Economic Review, 98 (5), 2066–2100.

Barberis, N. C. (2013). Thirty years of prospect theory in economics: A review and assessment. Journal of Economic Perspectives, 27 (1), 173–196.

Benhabib, J., Bisin, A., & Zhu, S. (2015). The wealth distribution in bewley economies with capital income risk. Journal of Economic Theory, 159, 489–515.

Benzoni, L., Collin-Dufresne, P., & Goldstein, R. S. (2007). Portfolio choice over the life-cycle when the stock and labor markets are cointegrated. The Journal of Finance, 62 (5), 2123–2167.

Bertaut, C. C., & Haliassos, M. (1997). Precautionary portfolio behavior from a life-cycle perspective. Journal of Economic Dynamics and Control, 21 (8-9), 1511–1542.

Betermier, S., Jansson, T., Parlour, C., & Walden, J. (2012). Hedging labor income risk. Journal of Financial Economics, 105 (3), 622–639.

Bewley, T. (1986). Stationary monetary equilibrium with a continuum of independently fluctuating consumers. In W. Hildenbrand & A. Mas-Colell (Eds.), Contributions to mathematical economics in honor of gerard debreu. North-Holland.

Bhamra, H. S., & Uppal, R. (2014). Asset prices with heterogeneity in preferences and beliefs. The Review of Financial Studies, 27 (2), 519–580.

Bharath, S. T., & Cho, D. (2023). Do natural disaster experiences limit stock market participation? Journal of Financial and Quantitative Analysis, 58 (1), 29–70.

Bigio, S., Silva, D., & Zilberman, E. (2025). Heterogeneous beliefs, asset prices, and business cycles (tech. rep.). National Bureau of Economic Research.

Bilal, A. (2023). Solving heterogeneous agent models with the master equation (tech. rep.). National Bureau of Economic Research.

Bilal, A., & Goyal, S. (2025). Some pleasant sequence-space arithmetic in continuous time (tech. rep.). National Bureau of Economic Research.

Blanchard, O. J. (2025). Convergence? thoughts about the evolution of mainstream macroeconomics over the last 40 years.

Blume, M. E., & Friend, I. (1975). The asset structure of individual portfolios and some implications for utility functions. The Journal of Finance, 30 (2), 585–603.

Bodie, Z., Merton, R. C., & Samuelson, W. F. (1992). Labor supply flexibility and portfolio choice in a life cycle model. Journal of Economic Dynamics and Control, 16 (3-4), 427–449.

Borovič ka, J. (2020). Survival and long-run dynamics with heterogeneous beliefs under recursive preferences. Journal of Political Economy, 128 (1), 206–251.

Boyer, B. H., & Vorkink, K. (2014). Stock options as lotteries. The Journal of Finance, 69 (4), 1485–1527.

Calvet, L. E., Campbell, J. Y., & Sodini, P. (2007). Down or out: Assessing the welfare costs of household investment mistakes. Journal of Political Economy, 115 (5), 707–747.

Calvet, L. E., Campbell, J. Y., & Sodini, P. (2009). Measuring the financial sophistication of households. American Economic Review, 99 (2), 393–398.

Campbell, J. Y. (1993). Intertemporal asset pricing without consumption data. The American Economic Review, 83(3), 487–512.

Campbell, J. Y. (2006). Household finance. The Journal of Finance, 61 (4), 1553–1604.

Campbell, J. Y. (2018). Financial decisions and markets: A course in asset pricing. Princeton University Press.

Campbell, J. Y., Cocco, J. F., Gomes, F. J., & Maenhout, P. J. (2001). Investing retirement wealth: A life-cycle model. In Risk aspects of investment-based social security reform (pp. 439–482). University of Chicago Press.

Campbell, J. Y., & Viceira, L. M. (1999). Consumption and portfolio decisions when expected returns are time varying. The Quarterly Journal of Economics, 114(2), 433–495.

Campbell, J. Y., & Viceira, L. M. (2001). Who should buy long-term bonds? American Economic Review, 91(1), 99–127.

Campbell, J. Y., & Viceira, L. M. (2002). Strategic asset allocation: Portfolio choice for long-term investors. Clarendon Lectures in Economic.

Cardaliaguet, P., Delarue, F., Lasry, J.-M., & Lions, P.-L. (2019). The master equation and the convergence problem in mean field games:(ams-201). Princeton University Press.

Cardaliaguet, P., & Porretta, A. (2020). An introduction to mean field game theory. In P. Cardaliaguet & A. Porretta (Eds.), Mean field games: Cetraro, italy 2019 (pp. 1–158). Springer International Publishing. https://doi.org/10.1007/978-3-030-59837-2 1

Carmona, R., Delarue, F., et al. (2018). Probabilistic theory of mean field games with applications i-ii. Springer.

Carroll, C. D., & Kimball, M. S. (1996). On the concavity of the consumption function. Econometrica: Journal of the Econometric Society, 981–992.

Catherine, S. (2022). Countercyclical labor income risk and portfolio choices over the life cycle. The Review of Financial Studies, 35 (9), 4016–4054.

Catherine, S., Sodini, P., & Zhang, Y. (2024). Countercyclical income risk and portfolio choices: Evidence from sweden. The Journal of Finance, 79 (3), 1755–1788.

Chang, Y., Hong, J. H., & Karabarbounis, M. (2018). Labor market uncertainty and portfolio choice puzzles. American Economic Journal: Macroeconomics, 10 (2), 222–262.

Cocco, J. F., Gomes, F. J., & Maenhout, P. J. (2005). Consumption and portfolio choice over the life cycle. The Review of Financial Studies, 18 (2), 491–533.

Coeurdacier, N., Rey, H., & Winant, P. (2011). The risky steady state. American Economic Review, 101 (3), 398–401.

Davis, S. J., & Willen, P. S. (2000). Occupation-level income shocks and asset returns: Their covariance and implications for portfolio choice.

De Long, J. B., Shleifer, A., Summers, L. H., & Waldmann, R. J. (1990). Noise trader risk in financial markets. Journal of Political Economy, 98 (4), 703–738.

Dend Haan, W. J. (1996). Heterogeneity, aggregate uncertainty, and the short-term interest rate. Journal of Business & Economic Statistics, 14 (4), 399–411.

Dimmock, S. G., Kouwenberg, R., Mitchell, O. S., & Peijnenburg, K. (2021). Household portfolio underdiversification and probability weighting: Evidence from the field. The Review of Financial Studies, 34 (9), 4524–4563.

Duffie, D., Geanakoplos, J., Mas-Colell, A., & McLennan, A. (1994). Stationary markov equilibria. Econometrica: Journal of the Econometric Society, 745–781.

Enke, B. (2024). The cognitive turn in behavioral economics. Unpublished manuscript. https://benjamin-enke. com.

Eraker, B., & Ready, M. (2015). Do investors overpay for stocks with lottery-like payoffs? an examination of the returns of otc stocks. Journal of Financial Economics, 115 (3), 486–504.

Evans, G. W., & Honkapohja, S. (2001). Learning and expectations in macroeconomics. Princeton University Press.

Evans, G. W., & Honkapohja, S. (2012). Learning and expectations in macroeconomics. In Learning and expectations in macroeconomics. Princeton University Press.

Evans, L. C. (2022). Partial differential equations (Vol. 19). American Mathematical Society.

Fagereng, A., Gottlieb, C., & Guiso, L. (2017). Asset market participation and portfolio choice over the life-cycle. The Journal of Finance, 72 (2), 705–750.

Farhi, E., & Werning, I. (2019). Monetary policy, bounded rationality, and incomplete markets. American Economic Review, 109 (11), 3887–3928.

Fernández-Villaverde, J., Hurtado, S., & Nuno, G. (2023). Financial frictions and the wealth distribution. Econometrica, 91 (3), 869–901.

Fernández-Villaverde, J., Nuño, G., & Perla, J. (2024). Taming the curse of dimensionality: Quantitative economics with deep learning (tech. rep.). National Bureau of Economic Research.

Fiegenbaum, A., & Thomas, H. (1988). Attitudes toward risk and the risk–return paradox: Prospect theory explanations. Academy of Management Journal, 31 (1), 85–106.

Fudenberg, D., & Levine, D. K. (1993). Self-confirming equilibrium. Econometrica: Journal of the Econometric Society, 523–545.

Futia, C. A. (1982). Invariant distributions and the limiting behavior of markovian economic models. Econometrica: Journal of the Econometric Society, 377–408.

García-Schmidt, M., & Woodford, M. (2019). Are low interest rates deflationary? a paradox of perfect-foresight analysis. American Economic Review, 109 (1), 86–120.

Goetzmann, W. N., & Kumar, A. (2008). Equity portfolio diversification. Review of Finance, 12 (3), 433–463.

Gollier, C., & Pratt, J. W. (1996). Risk vulnerability and the tempering effect of background risk. Econometrica: Journal of the Econometric Society, 1109–1123.

Gomes, F. (2020). Portfolio choice over the life cycle: A survey. Annual Review of Financial Economics, 12 (1), 277–304.

Graaff, J. d. V. (1950). Mr. harrod on hump saving. Economica, 81–90.

Grandmont, J. M. (1977). Temporary general equilibrium theory. Econometrica: Journal of the Econometric Society, 535–572.

Grandmont, J. M. (1989). Temporary equilibrium. In General equilibrium (pp. 297–304). Springer.

Grossman, S. J., & Shiller, R. J. (1982). Consumption correlatedness and risk measurement in economies with non-traded assets and heterogeneous information. Journal of Financial Economics, 10 (2), 195–210.

Gu, Z., Lauriere, M., Merkel, S., & Payne, J. (2024). Global solutions to master equations for continuous time heterogeneous agent macroeconomic models. arXiv preprint arXiv:2406.13726.

Guiso, L., Jappelli, T., & Terlizzese, D. (1996). Income risk, borrowing constraints, and portfolio choice. The American Economic Review, 158–172.

Haliassos, M., & Michaelides, A. (2003). Portfolio choice and liquidity constraints. International Economic Review, 44 (1), 143–177.

Han, J., Jentzen, A., & E, W. (2018). Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115 (34), 8505–8510.

Han, J., Yang, Y., et al. (2021). Deepham: A global solution method for heterogeneous agent models with aggregate shocks. arXiv preprint arXiv:2112.14377.

Heaton, J., & Lucas, D. (1997). Market frictions, savings behavior, and portfolio choice. Macroeconomic Dynamics, 1 (1), 76–101.

Heaton, J., & Lucas, D. (2000). Portfolio choice and asset prices: The importance of entrepreneurial risk. The Journal of Finance, 55 (3), 1163–1198.

Hicks, J. R. (1946). Value and capital: An inquiry into some fundamental principles of economic theory. OUP Catalogue.

Huang, J. (2023). Breaking the curse of dimensionality in heterogeneous-agent models: A deep learning-based probabilistic approach. Available at SSRN 4649043.

Hubener, A., Maurer, R., & Mitchell, O. S. (2016). How family status and social security claiming options shape optimal life cycle portfolios. The Review of Financial Studies, 29 (4), 937–978.

Huggett, M. (1993). The risk-free rate in heterogeneous-agent incomplete-insurance economies. Journal of Economic Dynamics and Control, 17 (5-6), 953–969.

Huggett, M., & Kaplan, G. (2016). How large is the stock component of human capital? Review of Economic Dynamics, 22, 21–51.

Jagannathan, R., Kocherlakota, N. R., et al. (1996). Why should older people invest less in stocks than younger people. Federal Reserve Bank of Minneapolis Quarterly Review, 20, 11–20.

Javed, K., & Sutton, R. S. (2024). The big world hypothesis and its ramifications for artificial intelligence. Finding the Frame: An RLC Workshop for Examining Conceptual Frameworks.

Ji, S., Peng, S., Peng, Y., & Zhang, X. (2022). Solving stochastic optimal control problem via stochastic maximum principle with deep learning method. Journal of Scientific Computing, 93 (1), 30.

Jouini, E., & Napp, C. (2007). Consensus consumer and intertemporal asset pricing with heterogeneous beliefs. The Review of Economic Studies, 74 (4), 1149–1174.

Kahneman, D., & Tversky, A. (2013). Prospect theory: An analysis of decision under risk. In Handbook of the fundamentals of financial decision making: Part i (pp. 99–127). World Scientific.

Kai-Ineman, D., Tversky, A., et al. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47 (2), 363–391.

Kamada, Y. (2010). Strongly consistent self-confirming equilibrium. Econometrica, 78 (2), 823–832.

Kelly, M. (1995). All their eggs in one basket: Portfolio diversification of us households. Journal of Economic Behavior & Organization, 27 (1), 87–96.

Krusell, P., & Smith, A. A., Jr. (1998). Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy, 106 (5), 867–896.

Kumar, A. (2009). Who gambles in the stock market? The Journal of Finance, 64 (4), 1889–1933.

Kunita, H. (1990). Stochastic flows and stochastic differential equations (Vol. 24). Cambridge University Press.

Lasry, J.-M., & Lions, P.-L. (2007). Mean field games. Japanese Journal of Mathematics, 2 (1), 229–260.

Lindahl, E. (1939). Theory of money and capital. Allen; Unwin; Holt, Rinehart; Winston.

Ljungqvist, L., & Sargent, T. J. (2018). Recursive macroeconomic theory. MIT press.

Marcet, A., & Sargent, T. J. (1989). Convergence of least-squares learning in environments with hidden state variables and private information. Journal of Political Economy, 97 (6), 1306–1322.

McDermott, R., Fowler, J. H., & Smirnov, O. (2008). On the evolutionary origin of prospect theory preferences. The Journal of Politics, 70 (2), 335–350.

Melcangi, D., & Sterk, V. (2025). Stock market participation, inequality, and monetary policy. Review of Economic Studies, 92 (4), 2656–2690.

Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Economics and Statistics, 247–257.

Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3 (4), 373–413.

Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica: Journal of the Econometric Society, 867–887.

Merton, R. C. (1992). Continuous-time finance.

Michaelides, A., & Zhang, Y. (2017). Stock market mean reversion and portfolio choice over the life cycle. Journal of Financial and Quantitative Analysis, 52 (3), 1183–1209.

Moll, B. (2025). The trouble with rational expectations in heterogeneous agent models: A challenge for macroeconomics. arXiv preprint arXiv:2508.20571.

Moll, B., & Ryzhik, L. (2025). Mean field games without rational expectations. arXiv preprint arXiv:2506.11838.

Parker, J. A., Schoar, A., Cole, A., & Simester, D. (2025). Household portfolios and retirement saving over the life cycle. The Journal of Finance.

Parker, J. A., & Vissing-Jørgensen, A. (2009). Who bears aggregate fluctuations and how? American Economic Review, 99 (2), 399–405.

Peng, S. (1992). Stochastic hamilton–jacobi–bellman equations. SIAM Journal on Control and Optimization, 30 (2), 284–304.

Phelps, E. S. (1962). The accumulation of risky capital: A sequential utility analysis. Econometrica: Journal of the Econometric Society, 729–743.

Piazzesi, M., & Schneider, M. (2016). Housing and macroeconomics. Handbook of Macroeconomics, 2, 1547–1640.

Polkovnichenko, V. (2005). Household portfolio diversification: A case for rank-dependent preferences. The Review of Financial Studies, 18 (4), 1467–1502.

Poterba, J. M. (2001). Demographic structure and asset returns. Review of Economics and Statistics, 83 (4), 565–584.

Poterba, J. M., & Samwick, A. (2001). Household portfolio allocation over the life cycle. In Aging issues in the united states and japan (pp. 65–104). University of Chicago Press.

Pratt, J. W., & Zeckhauser, R. J. (1987). Proper risk aversion. Econometrica: Journal of the Econometric Society, 143–154.

Ramey, V. A. (2016). Macroeconomic shocks and their propagation. Handbook of macroeconomics, 2, 71–162.

Sargent, T. J. (2025). Macroeconomics after lucas. Journal of Political Economy, 0 (ja), null. https://doi.org/10.1086/737999

Schenk-Hoppé, K. R., & Schmalfuss, B. (2001). Random fixed points in a stochastic solow growth model. Journal of Mathematical Economics, 36 (1), 19–30.

Shiller, R. J., Fischer, S., & Friedman, B. M. (1984). Stock prices and social dynamics. Brookings Papers on Economic Activity, 1984 (2), 457–510.

Stokey, N. L. (2008). The economics of inaction: Stochastic control models with fixed costs. Princeton University Press.

Thaler, R. H., & Johnson, E. J. (1990). Gambling with the house money and trying to break even: The effects of prior outcomes on risky choice. Management Science, 36 (6), 643–660.

Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty, 5 (4), 297–323.

Viceira, L. M. (2001). Optimal portfolio choice for long-horizon investors with nontradable labor income. The Journal of Finance, 56 (2), 433–470.

Vissing-Jørgensen, A. (2002). Limited asset market participation and the elasticity of intertemporal substitution. Journal of Political Economy, 110 (4), 825–853.

Woodford, M. (2013). Macroeconomic analysis without the rational expectations hypothesis. Annu. Rev. Econ., 5 (1), 303–346.