Non-exponential Discounting Portfolio Management and insurance With Habit Formation

Non-exponential Discounting Portfolio Management and insurance With Habit Formation

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This paper studies the portfolio management problem for an individual with a non-exponential discount function and habit formation in finite time. The individual receives a deterministic income, invests in risky assets, and consumes and buys insurance continuously. The utility of the individual comes from excessive consumption, legacy and terminal wealth. The non-exponential discounting makes the optimal strategy taken by a naive person time inconsistent which means traditional optimal control strategy will change from time to time. Nash subgame perfect theory is used to derive the equilibrium strategy for a sophisticated person, therein being time consistent. We derive both a time-inconsistent strategy for the naive person and a time-consistent strategy for a sophisticated person by the method introduced in [5]. We calculate the analytical solution under the naive strategy and the equilibrium strategy in the CRRA case, and compare the results of the two strategies. By numerical simulation, we find that the sophisticated individual will spend less for consumption and insurance and save more compared with the naive person. The difference in strategies of the naive and sophisticated person decreases over time. In addition, if the individual of either type is more patient in the future or has a higher tendency to form a habit, he will consume less and buy less insurance, and the difference between two strategies will also be increased. The sophisticated person's consumption and habit level are initially lower than that of a naive person but are higher at later times.

[1] Detemple, J.B., Zapatero, F. (1992) “Optimal consumption-portfolio policies with habit formation.” Mathematical Finance, 2(4), 251-274

[2] Ekeland I., Mbodji O., Pirvu T.A. “Time-Consistent Portfolio Management.” SIAM Journal of Financial Math, 2012, 3, 1-32

[3] Björk T., Murgci A., Zhou, X.Y. (2014) “Mean-variance portfolio optimization with state-dependent risk aversion.” Mathematical Finance, 24(1), 1-24

[4] Wei J.Q., Li D.P., Zeng Y., Yang H.L.(2018) “Robust optimal consumption-investment strategy with non-exponential discounting.” Journal of Industrial And Management Optimization, In press

[5] Björk T., Khapko M., Murgoci A. “On time-inconsistent stochastic control in continuous time (2017)” Finance Stoch, 21, 331-360

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