Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-17T21:18:14.568Z Has data issue: false hasContentIssue false

Optimal consumption, investment, and insurance under state-dependent risk aversion

Published online by Cambridge University Press:  23 January 2023

Mogens Steffensen
Department of Mathematical Sciences, University of Copenhagen. Universitetsparken 5, 2100 København, Denmark
Julie Bjørner Søe*
Department of Mathematical Sciences, University of Copenhagen. Universitetsparken 5, 2100 København, Denmark Mancofi A/S, Carl Jacobsens Vej 20, 2500 København, Denmark
*Corresponding author. E-mail:


We formalize a consumption–investment–insurance problem with the distinction of a state-dependent relative risk aversion. The state dependence refers to the state of the finite state Markov chain that also formalizes insurable risks such as health and lifetime uncertainty. We derive and analyze the implicit solution to the problem, compare it with special cases in the literature, and illustrate the range of results in a disability model where the relative risk aversion is preserved, decreases, or increases upon disability.

Research Article
© The Author(s), 2023. Published by Cambridge University Press on behalf of The International Actuarial Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Aase, K.K. (2017) The investment horizon problem: A possible resolution. Stochastics: An International Journal of Probability and Stochastic Processes, 89, 115141.CrossRefGoogle Scholar
Asmussen, S. and Steffensen, M. (2020) Risk and Insurance, vol. 96. Probability Theory and Stochastic Modelling. Springer.CrossRefGoogle Scholar
Choi, K.J. and Koo, H.K. (2005) A preference change and discretionary stopping in a consumption and porfolio selection problem. Mathematical Methods of Operations Research, 61, 419–435.CrossRefGoogle Scholar
Dhaene, J., Kukush, A., Luciano, E., Schoutens, W. and Stassen, B. (2017) On the (in-)dependence between financial and actuarial risks. Insurance: Mathematics & Economics, 52, 522–531.Google Scholar
Doctor, O. (2021) Application of generalized geometric itô-lévy process to investment-consumption-insurance optimization problem under inflation risk. Journal of Mathematical Finance, 11, 163–175.CrossRefGoogle Scholar
Hambel, C., Kraft, H., Schendel, L. and Steffensen, M. (2016) Life insurance demand under health shock risk. Journal of Risk and Insurance, 84, 1171–1202.Google Scholar
Hoem, J.M. (1988) The versatility of the markov chain as a tool in the mathematics of life insurance. Transactions of the 23rd International Congress of Actuaries, 3, 171–202.Google Scholar
Jarrow, R. and Li, S. (2021) Concavity, stochastic utility, and risk aversion. Finance and Stochastics, 25, 311–330.CrossRefGoogle Scholar
Karni, E. (1983) Risk aversion for state-dependent utility functions: Measurement and applications. International Economic Review, 24, 637–647.CrossRefGoogle Scholar
Kraft, H. and Steffensen, M. (2008a) The policyholder’s static and dynamic decision making of life insurance and pension payments. Blätter der DGVFM, 29, 211–244.CrossRefGoogle Scholar
Kraft, H. and Steffensen, M. (2008b) Optimal Consumption and Insurance: A Continuous-time Markov Chain Approach. ASTIN Bulletin, 38, 231–257.CrossRefGoogle Scholar
Kwak, M., Shin, Y.H. and Choi, U.J. (2011) Optimal investment and consumption decision of a family with life insurance. Insurance: Mathematics and Economics, 48, 176–188.Google Scholar
Lakner, P. and Nygren, L.M. (2006) Portfolio optimization with downside constraints. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 16, 283–299.CrossRefGoogle Scholar
Lichtenstern, A., Shevchenko, P.V. and Zagst, R. (2020) Optimal life-cycle consumption and investment decisions under age-dependent risk preferences. Mathematics and Financial Economics. doi: 10.1007/s11579-020-00276-9.Google Scholar
Merton, R. (1971) Optimal portfolio and consumption rules in a continuous-time model. Journal of Economic Theory, 3, 373–413.CrossRefGoogle Scholar
Merton, R.C. (1969) Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Economics and Statistics, 247257.CrossRefGoogle Scholar
Richard, S.F. (1975) Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics, 2, 187203.CrossRefGoogle Scholar
Steffensen, M. (2011) Optimal consumption and investment under time-varying relative risk aversion. Journal of Economic Dynamics and Control, 35, 659–667.CrossRefGoogle Scholar
Wang, H., Wang, R., Wei, J. and Xu, S. (2019) Optimal investment-consumption-insurance strategy in a continuous-time self-exciting threshold model. Communications in Statistics – Theory and Methods, 48, 3530–3548.CrossRefGoogle Scholar
Wang, N.W., Jin, Z., Siu, T.K. and Qui, M. (2021) Household consumption-investment-insurance decisions with uncertain income and market ambiguity. Scandinavian Actuarial Journal, 2021, 832–865.CrossRefGoogle Scholar
Wei, J.W., Cheng, X., Jin, Z. and Wang, H. (2020) Optimal consumption–investment and life-insurance purchase strategy for couples with correlated lifetimes. Insurance: Mathematics and Economics, 91, 244–256.Google Scholar
Yaari, M.E. (1965) Uncertain lifetime, life insurance, and the theory of the consumer. Review of Economic Studies, 32, 137–150.CrossRefGoogle Scholar