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Mathematics and Financial Economics

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13.03.2020

Consumption and portfolio decisions with uncertain lifetimes

verfasst von: Shou Chen, Richard Fu, Lei Wedge, Ziran Zou

Erschienen in: Mathematics and Financial Economics | Ausgabe 3/2020

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Abstract

We study the consumption and portfolio decisions by incorporating mortality risk and altruistic factor in the classical model of Merton (Rev Econ Stat 51:247–257, 1969; J Econ Theory 3:373–413, 1971) and Yaari (Rev Econ Stud 32(2):137–150, 1965). We find that besides the present-biased preference, the process of updating mortality information may be another underlying cause of dynamically time-inconsistent consumption behavior. We use the game-theoretic approach to obtain the extended Hamilton–Jacobi–Bellman equation. Furthermore, we obtain the closed-form solution for the logarithmic utility and explore comparative statics and implications for dynamic behavior. We present numerical results for the power utility that shows the sophisticated individual enjoys higher expected discounted utility than the naive. Our analytical solution enables us to generate a set of testable predictions that are consistent with existing empirical evidence. In particular, we show that for a moderate range of expected investment return, individuals will exhibit a “hump-shaped” consumption pattern, as widely documented in the empirical literature.

Vorheriger Artikel No arbitrage in continuous financial markets

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The bequest in our model is in the form of a consumption stream instead of a lump sum inheritance. The assumption of an inherited consumption stream can be justified by the increasing use of trust funds, which guarantee beneficiaries, such as the heirs, a steady stream of cash flows akin to the consumption stream in our model.

The individual’s optimal consumption and investment solutions are time inconsistent in the sense that they do not follow the Bellman optimality principle.

The corresponding time-consistent solution is also referred to as the Markov Perfect Equilibrium (MPE) in Karp [39] and Ekeland et al. [21].

Compared with the exponential discounting, hyperbolic discounting increases the instantaneous consumption rate, but does not affect the share of wealth invested in the risky asset.

Approximately 80% of the capital held by households is inherited [41]. Kuehlwein [42] shows that elderly households value bequests as highly as their own consumption. De Nardi [18] shows that the bequest motive is quantitatively important in explaining the wealth accumulation behavior of the richest in Sweden.

Ekeland et al. [21] and Ekeland and Pirvu [23] use a higher discount rate to discount one’s bequests to heirs than to discount one’s own consumption.

$D(t,\tau )$ of Eq. (10) is similar in form to the expectation of the stochastic hyperbolic discount function, proposed by Harris and Laibson [33]. However, $\pi (y)$ in Harris and Laibson [33] is constant. The experiments in McClure et al. [48] also show that when making an inter-temporal decision, the brain has two decision systems. The discount rate of the decision system responsible for discounting the far future utility is lower than the that of the decision system responsible for discounting the near future utility.

“Appendix A” gives the definition for a time varying discount function.

Gul and Pesendorfer [29] propose an alternative approach to model time-consistent preference suggesting that temptation but not preference change might be the cause of dynamically inconsistent behavior. Miao [51] adopts the Gul-Pesendorfer approach to solve the problem of the optimal option exercise by dynamic programming technique.

“Appendix A” gives the definition of non-stationarity.

If the hazard rate of death is constant, the discount function reflects only the time preference and the problem is the same as Ekeland and Pirvu [23], Ekeland and Lazrak [25], Ekeland et al. [24], Marín-Solano and Navas [46], Zou et al. [68], in which the solution depends only on the initial wealth.

$\Gamma (\cdot )$ is the gamma function that satisfies $\Gamma (\eta )=\int _0^\infty y^{\eta -1}e^{-y}dy$.

The distinction between naivety and sophistication, first proposed by Strotz [58], has been analyzed by Akerlof [2], O’Donoghue and Rabin [52], and among others.

Basak and Chabakauri [4] show that the optimal consumption policies are time inconsistent in their mean-variance model because of the adjustment of the variance term.

Ekeland and Pirvu [23], Marìn-Solano and Navas [46] investigate the time-consistent consumption and portfolio choices under time-invariant hyperbolic discounting from the game-theoretic point of view.

By letting $\tau =y+t$ and from Eq. (15), we obtain Eq. (16).

See for instance Ekeland and Lazrak [20, 25], Björk and Murgoci [8], Basak and Chabakauri [4], Björk et al. [10], and Björk et al. [11] and so on.

We could also argue the time-consistent consumption and portfolio rules based on a more complex characterization of utility, for example, Epstein-Zin-Weil (EZW) recursive preferences utility functions [22, 65], which separate risk aversion from the elasticity of intertemporal substitution of consumption. Although EZW preferences have been useful in explaining the behavior of financial markets, we cannot obtain closed-form analytical solutions.

For the actual functional form of A(t) and B(t), please see “Appendix E”.

$c^*_t(\tau )$ is the optimal instantaneous consumption rate at time $\tau $ from the perspective of time t and $c^*_\tau (\tau )$ is the actual instantaneous consumption rate at time $\tau $ for a naive Bayesian individual.

The equivalence of actual and time-consistent consumption and portfolio choices is also shown, under different contexts, in Phelps and Pollak [55], Barro [6], Marín-Solano and Navas [45, 46], and Palacios-Huerta and Pérez-Kakabadse [54].

Because of the individual’s finite lifetime, her wealth will be inherited by her heirs after her death, with the death time denoted by S. Following Merton [49], we assume that the bequest function takes the form of $H(S,w(S))=\xi ^b\frac{(w(S))^{1-b}}{1-b}$, which means $V(S,w(S))=\beta H(S,w(S))$ and $H(S)=\beta \xi ^b$, where $\xi =\frac{b}{\rho -(1-b)(r+\frac{a^2}{2b{\bar{\sigma }}^2})}$. Therefore, $\lim _{t\rightarrow \infty }h(t)=\beta \xi ^b$.

o(1) is infinitesimal and $\lim _{\epsilon \rightarrow 0}\frac{o(1)}{\epsilon }$=0.

To be specific, this equilibrium strategy is also a weak and regular equilibrium strategy. Readers are referred to He and Jiang [37] for detailed discussions of the weak and regular strategy.

It is necessary to use a numerical solution to analyze the dynamic behavior with a power utility function. We omit it here for the sake of brevity. The non-linear integro-differential Eq. (28) is similar to those (Equation 42) in Zou et al. [68]. Readers are referred to Zou et al. [68] for the numerical analysis.

For the sake of brevity, we do not repeat these assumptions verbatim in the following proposition statements.

The consumption amount ${\hat{C}}(t)$ = ${\hat{c}}(t){\hat{w}}(t)$ is the product of the instantaneous consumption rate and wealth.

We show that $\frac{1}{\rho }>\int _t^{\infty }e^{-\lambda (y^{\gamma }-t^{\gamma })}e^{-\rho (y-t)} dy$. By letting $y-t=m$, we have

$$\begin{aligned}&\int _t^{\infty }e^{-\lambda (y^{\gamma }-t^{\gamma })}e^{-\rho (y-t)} dy\\&\quad =\int _0^{\infty }e^{-\lambda ((m+t)^{\gamma }-t^{\gamma })}e^{-\rho m} dm\\&\quad<\int _0^{\infty }e^{-\lambda \gamma t^{\gamma }m}e^{-\rho m} dm\\&\quad =\frac{1}{\lambda \gamma t^\gamma +\rho }\\&\quad <\frac{1}{\rho } \end{aligned}$$

The third strict inequality holds because $(m+t)^{\gamma }-t^{\gamma }>\gamma t^{\gamma }m$ for $\gamma >1$, $m>0$ and $t>0$.

Since the life expectancy E[T] is ${(\frac{1}{\lambda })}^{\frac{1}{\gamma }}\Gamma (1+\frac{1}{\gamma })$, E[T] decreases with the parameter $\lambda $. Eq. (33) implies that a shorter lifetime is associated with an increasing instantaneous consumption rate and therefore a longer expected lifespan is associated with a decreasing instantaneous consumption rate. Since E[T] is not monotonic with the parameter $\gamma $, we do not explore the comparative statics of the time-consistent instantaneous consumption rate with regard to the parameter $\gamma $ on the time-consistent instantaneous consumption rate, ${\hat{c}}(t)$.

i.e., $\frac{\partial {\bar{t}}}{\partial R}>0$, $\frac{\partial {\bar{t}}}{\partial \beta }>0$, where ${\bar{t}}$ is the duration of the period for which the individual is a net saver and ${\bar{t}}$ is defined as the solution to $R=\frac{1}{(1-\beta )\int _t^\infty e^{-\lambda (y^\gamma -t^\gamma )}e^{-\rho (y-t)}dy+\frac{\beta }{\rho }}$.

The life expectancy is $(\frac{1}{\lambda })^{\frac{1}{\gamma }}\Gamma (1+\frac{1}{\gamma })=\Gamma (1.5)=0.8862$.

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Titel: Consumption and portfolio decisions with uncertain lifetimes
verfasst von: Shou Chen
Richard Fu
Lei Wedge
Ziran Zou
Publikationsdatum: 13.03.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Mathematics and Financial Economics / Ausgabe 3/2020
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-020-00263-0

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