Consumption, Investment and Life Insurance Strategies with Heterogeneous Discounting

Abstract

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent's preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the timeconsistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically. AbstractEn aquest treball s'analitza l'efecte que comporta l'introducció de preferències inconsistents temporalment sobre les decisionsòptimes de consum, inversió i compra d'assegurança de vida. En concret, es pretén recollir la creixent importància que un individu dóna a la herència que deixa i a la riquesa disponible per a la seva jubilació al llarg de la seva vida laboral. Amb aquesta finalitat, es parteix d'un model estocàstic en temps continu amb temps final aleatori, i s'introdueix el descompte heterogeni, considerant un agent amb una distribució de vida residual coneguda. Per tal d'obtenir solucions consistents temporalment es resol una equació de programació dinàmica no estàndard. Per al cas de funcions d'utilitat del tipus CRRA i CARA es troben solucions explícites. Finalment, els resultats obtinguts s'il·lustren numèricament.

Preference heterogeneity and its equilibrium path

“…Equation (15) illustrates that present bias ( small 1 or large 1 ) increases consumption, as has been demonstrated by Harris and Laibson. 2 c(t) W(t) Equation (16) shows that due to present bias ( small 1 or large 1 ), consumption under the SNHQH discounting is larger than that under NC discount function (e.g., Karp,11 Marín-Solano and Navas, 12 Marín-Solano and Shevkoplyas, 5 Marín-Solano and Navas 13 ).…”

“…Reduction experiment is a commonly used validation. Specifically, the SNHQH discount function (1) is designed in such a way that, by controlling parameters, the SNHQH discount function (1) and its nonstandard HJB (8) can simultaneously reduce to those in many papers (e.g., de-Paz et al, 14 de-Paz et al, 15 Karp,11 Marín-Solano and Patxot, 1 Marín-Solano et al, 19 Marín-Solano and Navas, 12 Marín-Solano and Shevkoplyas, 5 Marín-Solano and Navas, 13 Peng and Hager, 20 Peng and Hager 21 ). Both these papers and the present article use a game-theoretic approach in the sense that no functional assumption has been imposed.…”

“…(ii) continuous-time nonstandard optimization can explore a multiself intrapersonal game (e.g., Ekeland and Pirvu, 8 Ekeland and Lazrak, 9 Ekeland et al, 4 Marín-Solano and Shevkoplyas, 5 Marín-Solano and Patxot, 1 Basak and Chabakauri, 3 Björk et al 10 ). (iii) discrete-time nonstandard optimization arms HJBs with parameters such that users can easily create visual graphs (e.g., Karp,11 Marín-Solano and Navas, 12 Marín-Solano and Navas, 13 de-Paz et al, 14 de-Paz et al 15 ). However, due to lack of explicit parameters in HJBs, most existing HJBs cannot capture present bias, nor can easily visualize the results.…”

“…Being a powerful method to quantify selves' belief, behavioral dynamic programming (e.g., de-Paz et al, 14 de-Paz et al, 15 Caliendo, 16 Caliendo and Aadland 17 ) is performed in a series of stages. The first stage is the establishment of a dynamic programming equation (i.e., behavioral equation), representing the mutual initial plans of the naive and precommited decision-makers.…”

Optimal Consumption and Portfolio Decision with Convertible Bond in Affine Interest Rate and Heston’s SV Framework

“…By complicated deliberation and calculation, Zhao et al [8] got the optimal investment-consumption policies with nonexponential discounting and logarithmic utility. In order to further investigate investment-consumption problems, Palacios-Huertay and Prez-Kakabadsez [9] and de-Paz et al [10] introduced discounting function into stochastic hyperbolic discounting function and heterogeneous discounting function, respectively. Kronborg and Steffensen [11] devoted themselves to an inconsistent investment-consumption problem and received some instructive results.…”