Life Insurance Completeness: A Path to Hedging Mortality and Achieving Financial Optimization

Abstract

This paper explores optimal consumption and investment strategies for agents facing mortality risk within a complete financial market. Departing from traditional frameworks, we leverage state-dependent utility theory, discounted by the state–price process, to compare consumption streams and utilize life insurance as a strategic hedging instrument. To model the ability of insurance companies to hedge the mortality risk of consumer pools, we introduce the concept of life insurance completeness, allowing individuals to achieve optimal consumption even in scenarios involving negative wealth. Our model relaxes the stringent integrability conditions commonly imposed in the literature, offering a more economically grounded approach to valuation and hedging. We derive a general solution to the optimization problem using martingale techniques under minimal assumptions, demonstrating that life insurance primarily serves as a mortality risk hedge rather than a bequest motive. This perspective resolves longstanding theoretical and empirical challenges, notably the annuity puzzle, by illustrating that optimal consumption and investment, in the absence of labor income, do not necessitate annuities or other life insurance policies. Our key contributions include (1) extending valuation frameworks to encompass prepaid insurance and less restrictive integrability criteria, (2) establishing life insurance completeness for effective mortality risk hedging, (3) demonstrating the feasibility of optimal consumption under negative wealth and state-dependent preferences, and (4) offering a resolution to the annuity puzzle that aligns with empirical observations.

1. Introduction

Optimal consumption and investment decisions in the presence of mortality risk are fundamental in financial planning, yet traditional models often overlook this critical factor. This paper addresses the challenge of integrating life insurance into a complete financial market, providing a comprehensive solution for agents facing financial and mortality risks.

Building on state-dependent utility theory as proposed by Londoño (2009), we extend the classical consumption–investment problem by incorporating life insurance to hedge against mortality risk. While previous works, such as Merton’s seminal contributions (Merton 1969, 1971, 1973), laid the foundation for portfolio management, they did not account for life insurance decisions or mortality risk. Subsequent efforts, including those by Richard (Richard 1975) and others, incorporated life insurance but used restrictive assumptions and deterministic models.

The classical problems of optimal consumption and investment without mortality risk of Merton (1969, 1971) have been extended and generalized by Cox and Huang (1989); Karatzas et al. (1987); Ocone and Karatzas (1991) among others.

Despite these advancements, several limitations persist in classical equilibrium and consumption models, including numerical difficulties and discrepancies between empirical data and theoretical predictions, most notably illustrated by the equity premium and risk-free rate puzzles. To address these gaps, scholars have proposed alternative approaches, such as habit formation, recursive utilities, and transaction costs, yet these models still face significant challenges.

As an alternative to address those theoretical and empirical problems, some literature in finance has focused on state-dependent utilities to explain the behavior of individual consumers and investors ((Londoño 2009) and references therein). Londoño’s framework proposes preferences on future flows of money discounted by the state–price process, which were initially investigated by Steffensen (2001). In this context, the individual’s objective is to maximize

$$E\left({\int }_0^T{U}_1\left(s,H\left(s\right)c\left(s\right)\right)dt+U_2\left(H\left(T\right)X\left(T\right)\right)\right),$$

where $H\left(·\right)$ is the state price process. By considering a more general valuation and arbitrage theory, Londoño (2008) developed a martingale methodology to obtain complete solutions to the problem in a general setting. Londoño (2009, 2020) addressed the meaning of the preference structure in those settings. In particular, Londoño (2020, Remark 17) showed that those preference structures are a way to compare personal consumption and wealth with society consumption. Also, Londoño (2020, Section 5.2) reasonably explained the equity premium puzzle and interest rate puzzle.

As a natural extension of optimization for discounted by the state price of consumption and investment, we address the problem of maximizing

$$E\left({\int }_0^{\tau }{U}_1\left(s,H\left(s\right)c\left(s\right)\right)ds+U_2\left(\tau ,H\left(\tau \right)\left(X\right(\tau )+D(\tau \left)\right)\right)\right),$$

where $\tau \in \left[0,T\right]$ is the uncertain lifetime, $c\left(·\right)$ is the rate of the consumption process, $X\left(·\right)$ is the current wealth that is financed (within the financial market) with an initial wealth x minus the value of insurance y at time 0, has a premium rate of $p(·)$, stream rate of payment $i(·)$ and pays $D\left(t\right)$ in case of premature death. We also assume that $U_1\left(·,·\right),U_2\left(·,·\right)$ are his/her time-dependent preferences for consumption and bequest discounted by the state price process, respectively. To the best of our knowledge, Steffensen (2001) was the first attempt to address optimization problems with discounted consumption and wealth by the state–price process to solve insurance and pension problems. Gomez (2020); Londoño and Søe (2024) also addressed the problem of optimal consumption and investment discounted by the state–price process to solve insurance problems and Londoño and Søe (2024) obtained solutions using dynamic programming.

In our setting, the mathematical setup is a consistent preference structure (see Definition 8). In this setting, a consumer computes their consumption and investment strategies as living a very long life. In the case of premature death, his preference for unrealized consumption becomes a legacy preference (see Remark 6). An example of this preference structure is an iso-elastic preference structure with some consistency conditions on the coefficients of the preference for consumption and preference for legacy (see Example 2). We explain the utility of legacy as a utility for unrealized consumption, and this approach is consistent with previous studies that show that consumer behavior is explained by uncertain lifetime, as seen in (Davies 1981; Hurd 1989). Also, Hurd (1987); Laitner and Juster (1996) showed evidence for the lack of bequest motives or limited impact of bequest motives on consumption and saving.

To solve the problem of optimal consumption and investment (see Theorems 3 and 4), we introduce a concept of life insurance completeness. Life insurance completeness refers to agents’ (insurance companies) ability for any insurance policy to access a pool of individuals with independent and identical mortality distributions who buy the same insurance. Theoretically, a pool with infinitely many consumers is necessary, and a large enough pool of consumers is needed in practical applications. For a complete life insurance market, any insurance agent can hedge (with a pool of consumers) any life insurance within the financial market. In this paper, life insurance is a contract with a price (used to hedge the contract’s obligations) used by the insurance agent to hedge the contract. Also, the insurance agent receives a premium rate of $p\left(t\right)$, pays a stream rate of payment $i\left(t\right)$, and pays any obligation in the insurance contract at the time of death $D\left(t\right)$ (to the holder of debts of the consumer or siblings). See Definition 2, Remark 2, and Proposition 1. In general, in the context of this paper, a consumer cannot hedge their mortality within the financial market unless they access a life insurance market with insurance agents who have access to a pool of consumers with similar tastes and mortality. With complete life insurance markets (and complete financial markets), consumers can hedge various life insurance contracts and optimize their consumption and investment. For recent approaches on the estimation of the value (or best estimate) of liabilities, see (Artzner et al. 2023) and references therein.

Although insurance is not required to achieve optimality when a positive endowment process is not assumed (see Theorem 3), consumers generally cannot hedge their mortality risk or achieve optimal consumption without a life insurance market or life insurance agents. The optimal solution (see Theorem 4) necessitates that consumers hold debts during the early stages of their productive life. The existence of a robust insurance market (refer to Remark 2 and Proposition 1) enables agents to access credit, ensuring that their debts are covered in the event of premature death. However, in the framework presented in this paper, life insurance primarily serves as a tool to hedge against mortality risk. Consequently, the choices made by consumers do not stem from optimal behavior, as multiple options can lead to similar optimal outcomes (see Theorem 4). Our model differs from the classical Yaari lifecycle model (Yaari 1965) as it more accurately reflects the behavior of individuals who tend to under-annuitize during retirement. In scenarios without labor income (for instance, after retirement), our model indicates that optimal consumption and investment decisions do not require any insurance, given a naturally plausible hypothesis. This aspect of our model contributes to explaining the annuity puzzle (refer to Remark 8). For a recent overview of the annuity puzzle, see (Alexandrova and Gatzert 2019).

Our main contribution in this paper is demonstrating that, within a well-structured framework of utility functions for discounting consumption and wealth based on state price processes, the approach of life insurance completeness provides an elegant solution to longstanding issues in actuarial science. For instance, Gomez (2020); Londoño and Søe (2024) have tackled these issues, but they assumed that the financial market is complete concerning mortality risk, which means that mortality risk is hedgeable within the financial markets. Even in scenarios without labor income, their solutions require life insurance. In contrast, our paper introduces the concept of life insurance completeness, suggesting that, even when financial risks are not hedgeable in financial markets, mortality risk can be mitigated by pooling similar policies through insurance companies. Furthermore, this approach allows for an optimal solution that does not require insurance in the absence of income, thereby explaining the annuity puzzle.

We extend a valuation and hedging framework to a natural non-arbitrage theory with credit limits that are bounded relative to society’s wealth. This framework introduces two critical concepts not fully explored in the literature. First, we relax the stringent conditions commonly imposed in the recent literature on hedging and valuation. In particular, many works assume strong boundness conditions for processes such as interest rates, insurance premium payout ratios, and asset prices, which are not always economically motivated. Our approach, drawing on the concept of “state tameness”, provides a more economically grounded framework for valuation, avoiding the artificial imposition of strict $L^2$ integrability conditions (see Remark 3). Second, we expand the scope of life insurance valuation to include prepaid insurance, a feature largely ignored in the existing literature. Recent models often assume life insurance is strictly proportional to premium payments, thus excluding scenarios wherein insurance is fully paid upfront. Our methodology incorporates this possibility, offering a more comprehensive framework for the valuation and hedging of life insurance instruments and a generalization of European contingent claims with random expiration dates (see Remark 4).

Using the latter framework, we provide a general solution to the problem of optimal consumption and investment with mortality risk, assuming financial completeness and life insurance completeness. We do not require the boundness of the mortality time, and our approach also works on bounded and unbounded mortality times. The coefficients of the evolution of prices (diffusion and volatility of the price process), interest rate, dividend rate, and endowment processes require minimal conditions on boundness. We assume that the later coefficient processes are just progressive measurable processes with some minimal conditions on boundness necessary for some definitions to make sense.

This paper is organized as follows. In Section 2, we begin by reviewing the foundational theories of state tameness, the valuation of European contingent claims, and state completeness, as developed by Londoño (2004, 2008). These concepts provide the necessary theoretical framework for our analysis. Building on this foundation, Section 3 introduces the concept of an actuarial market and defines the types of life insurance processes relevant to our study. Here, we formalize the notion of a complete life insurance market (see Definition 2) and establish the conditions under which life insurance processes can be valued and hedged within this framework (see Proposition 1). In Section 4, we shift our focus to the preferences and optimization problems central to this paper. We first review the types of preferences considered (see Definition 8) and then formulate the key optimization problems (see Problems 1 and 2). Using the theoretical tools developed in the preceding sections, we derive solutions to these problems (see Theorems 3 and 4) and illustrate their application through a detailed example (Example 2). Finally, Section 5 summarizes our key findings, discusses their implications, and identifies open questions for future research.

2. Financial Market

For completeness, we explicitly state all the hypotheses usually used for financial market models with a finite set of continuous assets defined on a Brownian filtration. Hereafter, we follow the notation in Karatzas and Shreve (1998) as closely as possible. We begin with the complete probability space $\left(\Omega ,\mathcal{F},P\right)$ where $\Omega$ is the canonical space of d-dimensional continuous functions with 0 value at $t=0$ $\Omega =C([0,T_M):{\mathbb{R}}^d;0)$, $W\left(t\right)={\left(W_1\left(t\right),\dots ,W_d\left(t\right)\right)}^{⊺},0\le t<T_M$ is the coordinate mapping $W\left(t\right)\left(\omega \right)=\omega \left(t\right)$. All economic activity will be assumed to take place on a time horizon $\left[0,T_M\right)$, where $0<T_M\le \infty$. We emphasize that we permit $T_M=\infty$; in fact, we address most results in this paper to this latter case. Define $\left({\mathcal{F}}_t\right)$ to be the filtration generated by $W\left(·\right)$, and let ${\mathcal{F}}_{t^{-}}=\sigma \left({\cup }_{0\le s<t}{\mathcal{F}}_s\right)$ for any $0\le t\le T_M$. Let P the Wiener measure on ${\mathcal{F}}_{T_M^{-}}$, and let $\mathcal{F}$ the completion of ${\mathcal{F}}_{T_M^{-}}$ under P; namely, $\mathcal{F}=\sigma ({\mathcal{F}}_{T_M^{-}}\cup \mathcal{N})$ where $\mathcal{N}$ is the collection of P-null subsets of ${\mathcal{F}}_{T_M^{-}}$.

For each $T<T_M$ we define ${\mathcal{N}}_T$ the $\sigma$-algebra of null sets of ${\mathcal{F}}_T$ under the Wiener measure on ${\mathcal{F}}_T$. Define the augmented filtration $\left({\mathcal{F}}_t^T\right):=\left(\sigma ({\mathcal{F}}_t\cup {\mathcal{N}}_T)\right)$. All processes in this paper are processes defined on $(\Omega ,\mathcal{F},P)$ that are restrictedly progressive measurable or restrictedly adapted processes (Karatzas and Shreve 1998). For a restrictedly progressive measurable (restrictedly adapted), we understand a process $X\left(t\right)$ of $\mathcal{F}$ measurable variables, such that for any $T<T_M$ there exist $T<T^{\prime }<T_M$ such that $\left\{X\right(t),0\le t\le T\}$ is $\{{\mathcal{F}}_t^{T^{\prime }},0\le t\le T\}$ progressive measurable (adapted).

We assume a bounded from below risk-free rate process $r\left(·\right)$, an n-dimensional mean rate of return process (after dividends) $b\left(·\right)$, an n-dimensional dividend rate process $\delta \left(·\right)$, and an $\left(n\times d\right)$-matrix-valued volatility process $\sigma \left(·\right)$; we also assume that $r\left(·\right)$, $b\left(·\right)$, $\delta \left(·\right)$, $\sigma \left(·\right)$ are RCLL progressively restrictedly measurable processes. We emphasize that we allow for stochastic risk-free rate, mean rate of return, dividend rate, and volatility processes.

We assume that for any $0<T<T_M$

$${\int }_0^T\left(\left|r\left(s\right)\right|+∥b\left(s\right)∥+∥\delta \left(s\right)∥+\sum _{i,j}{\sigma }_{i,j}^2\left(s\right)\right)ds<\infty .$$

As usual, we assume that the price of the money market process, $B\left(·\right)$, according to the equation

$$dB\left(t\right)=B\left(t\right)r\left(t\right)dt,\forall t\in \left[0,T_M\right).$$

(1)

We introduce n stocks with prices-per-share $S_1\left(t\right),\dots ,$ $S_n\left(t\right)$ at time t and with $S_1\left(0\right),\dots ,S_n\left(0\right)$ positive constants. The process’s $S_1\left(·\right),\dots ,$ $S_n\left(·\right)$ are continuous, strictly positive, and satisfy stochastic differential equations

$$d{S}_i\left(t\right)=S_i\left(t\right)\left[b_i\left(t\right)dt+\sum _{j=1}^d{\sigma }_{i,j}\left(t\right)d{W}_j\left(t\right)\right],\forall t\in \left[0,T_M\right),i=1,\dots ,n.$$

(2)

We refer to the described financial market as $\mathcal{M}:=\left(r\left(·\right),b\left(·\right),\delta \left(·\right),\sigma \left(·\right),S\left(0\right)\right).$

Given $\mathcal{M}$, we introduce a portfolio process $\left({\pi }_0\left(·\right),\pi \left(·\right)\right)$ consisting of an RCLL restrictedly progressively measurable, real-valued process ${\pi }_0\left(·\right)$ and a restrictedly progressively measurable ${\mathbb{R}}^n$-valued process $\pi \left(·\right)={\left({\pi }_1\left(·\right),\dots ,{\pi }_n\left(·\right)\right)}^{⊺}$ such that for any $T<T_M$

$$\begin{array}{ccc}\hfill {\int }_0^T\left(\left|{\pi }_0\left(s\right)+{\pi }^{⊺}\left(s\right){\mathbf{1}}_n\right|\left|r\left(s\right)\right|\right)ds& <& \infty ,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\int }_0^T\left|{\pi }^{⊺}\left(s\right)\left(b\left(s\right)+\delta \left(s\right)-r\left(s\right){\mathbf{1}}_n\right)\right|ds& <& \infty ,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill {\int }_0^T{∥{\sigma }^{⊺}\left(s\right)\pi \left(s\right)∥}^{2}ds& <& \infty .\hfill \end{array}$$

(5)

So, we can define the gains process $G\left(·\right)$ associated with $\left({\pi }_0\left(·\right),\pi \left(·\right)\right)$ by

$$\begin{array}{cc}\hfill G\left(t\right):=& {\int }_0^t\left({\pi }_0\left(s\right)+{\pi }^{⊺}\left(s\right){\mathbf{1}}_n\right)r\left(s\right)ds+{\int }_0^t{\pi }^{⊺}\left(s\right)\left(b\left(s\right)+\delta \left(s\right)-r\left(s\right){\mathbf{1}}_n\right)ds\hfill \\ \hfill & +{\int }_0^t{\pi }^{⊺}\left(s\right)\sigma \left(s\right)dW\left(s\right),\hfill \end{array}$$

(6)

for all $t\in \left[0,T_M\right)$.

We also define a cumulative income process $\left\{\Gamma \left(t\right);0\le t<T_M\right\}$ (representing the cumulative wealth received or expended by an investor on the time interval $[0,t]$) as an RCLL process of bounded variation, beginning at zero. Let $\Gamma \left(·\right)={\Gamma }_{+}\left(·\right)-{\Gamma }_{-}\left(·\right)$ be the representation of $\Gamma \left(·\right)$ as the difference of its positive variation processes ${\Gamma }_{+}\left(·\right)$ and its negative variation processes ${\Gamma }_{-}\left(·\right)$. Following the standard literature, see, e.g., (Karatzas and Shreve 1998), the wealth process $X\left(·\right)$ associated with $\left(\Gamma \left(·\right),\pi \left(·\right)\right)$ and initial wealth $x_0$ is the stochastic process

$$\begin{array}{ccc}\hfill \gamma \left(t\right)X^{x_0,\Gamma ,\pi }\left(t\right)& =& x_0+{\int }_{[0,t)}\gamma \left(s\right)d\Gamma \left(s\right)+{\int }_0^t\gamma \left(s\right){\pi }^{⊺}\left(s\right)\sigma \left(s\right)dW\left(s\right)\hfill \\ & & +{\int }_0^t\gamma \left(s\right){\pi }^{⊺}\left(s\right)\left(b\left(s\right)+\delta \left(s\right)-r\left(s\right){\mathbf{1}}_n\right)ds,\hfill \end{array}$$

where $\gamma \left(t\right)={\left(B\left(t\right)\right)}^{-1}$.

Moreover, we define a restrictedly progressively measurable market price of risk process $\theta \left(·\right)={\left({\theta }_1\left(·\right),\dots ,{\theta }_d\left(·\right)\right)}^{⊺}$ with values in ${\mathbb{R}}^d$ as the unique process satisfying $\theta \left(t\right)\in {ker}^{\perp }\left(\sigma \left(t\right)\right)$ for all $t\in \left[0,T_M\right),$ and

$$\left(b\left(t\right)+\delta \left(t\right)-r\left(t\right){\mathbf{1}}_n\right)-{\mathrm{proj}}_{ker\left({\sigma }^{⊺}\left(t\right)\right)}\left(b\left(t\right)+\delta \left(t\right)-r\left(t\right){\mathbf{1}}_n\right)=\sigma \left(t\right)\theta \left(t\right)a.s.$$

(7)

where we can assume that the process $\theta \left(·\right)$ is restrictedly progressively measurable, and we can assume that the process is RCLL (Karatzas and Shreve 1998). Moreover, we assume that, for any $0\le T<T_M$, $\theta \left(·\right)$ verifies

$${\int }_0^T{∥\theta \left(s\right)∥}^{2}ds<\infty a.s.$$

(8)

In addition, we define for all $t\in [0,T_M)$ a state price process by

$$H\left(t\right):=\gamma \left(t\right)Z_0\left(t\right),$$

(9)

where

$$Z_0\left(t\right):=exp\left(-{\int }_0^t{\theta }^{⊺}\left(s\right)dW\left(s\right)-{\displaystyle \frac{1}{2}}{\int }_0^t{∥\theta \left(s\right)∥}^{2}ds\right).$$

(10)

The name “state price process” is usually given to the process defined by Equation (9) when the market is a standard financial market (Karatzas and Shreve 1998). In that case, the process $Z_0$ is a martingale, and $Z_0\left(T\right)$ is indeed, a density. However, in this setting, we allow that $E\left(Z_0\left(T\right)\right)<1$.

2.1. State Tameness and State Arbitrage

We establish our notions of tameness and arbitrage following Londoño (2004, 2009).

Let $\Gamma \left(·\right)$ be a cumulative income process. If the portfolio process $\left({\pi }_0\left(·\right),\pi \left(·\right)\right)$ is $\Gamma$-financed, and the associated process for each $0\le T<T_M$ $H\left(t\right)X\left(t\right)-{\int }_{\left(0,t\right]}H\left(s\right)d\Gamma \left(s\right)$ $0\le t\le T$ is uniformly bounded from below; we say that the portfolio is $\Gamma$-state-tame. In particular, a self-financed $\left(\Gamma \left(·\right):=0\right)$ portfolio process $\left({\pi }_0\left(·\right),\pi \left(·\right)\right)$ is said to be state-tame if the discounted gains process for any $T<T_M$, $\left\{H\left(t\right)G\left(t\right),t\in \left[0,T\right]\right\}$ is almost surely bounded from below by a real constant that does not depend on t. In a financial market $\mathcal{M}$, we say that a given self-financed, state-tame portfolio process $\pi \left(·\right)$ is an arbitrage opportunity and there exist $0\le T<T_M$ where the associated gains process $G\left(·\right)$ of (6) satisfies $H\left(T\right)G\left(T\right)\ge 0$ almost surely and $H\left(T\right)G\left(T\right)>0$ with positive probability. A financial market $\mathcal{M}$ in which no such arbitrage opportunities exist is said to be state-arbitrage-free.

The following theorem is a characterization for the nonexistence of state-arbitrage opportunities.

Theorem 1.

A market $\mathcal{M}$ is state-arbitrage-free if and only if for all $t\in \left[0,T_M\right)$ the market price of risk $\theta \left(t\right)$ satisfies

$$b\left(t\right)+\delta \left(t\right)-r\left(t\right){\mathbf{1}}_n=\sigma \left(t\right)\theta \left(t\right)a.s.$$

(11)

Remark 1.

We observe that, if $\theta \left(·\right)$ satisfies Equation (11); then, for any initial capital x; and cumulative income process $\Gamma \left(·\right)$, we have

$$\begin{array}{cc}H\left(t\right)X\left(t\right)=x+{\int }_{[0,t)}H\left(s\right)d\Gamma \left(s\right)\hfill & \\ & \hfill +{\int }_0^{t}H\left(s\right){\left(\sigma {\left(s\right)}^{⊺}\pi \left(s\right)-X\left(s\right)\theta \left(s\right)\right)}^{⊺}dW\left(s\right),\end{array}$$

(12)

where $t\in \left[0,T_M\right)$.

Proof. 

See the proof of Theorem 1 in Londoño (2008). □

2.2. Completeness of Financial Markets

We proceed to present our notions of completeness. Let $\mathcal{M}$ be a state-arbitrage-free financial market.

An European state-contingent claim (ESCC) with expiration date $T<T_M$ is a couple of RCLL process $(\Gamma (t),D(t\left)\right)$, $0\le t<T$ where $\Gamma \left(t\right)={\Gamma }_{+}\left(t\right)-{\Gamma }_{-}\left(t\right)$ is of finite variation, ${\Gamma }_{+}(·)$ and ${\Gamma }_{-}(·)$ are the positive and negative variation of $\Gamma (·)$ and $D\left(t\right)$ is a semi-martingale such that $H\left(t\right)D\left(t\right)-{\int }_{\left[0,t\right)}H\left(s\right)d{\Gamma }_{-}\left(s\right)$ is (uniformly) bounded from below and

$$E\left[H\left(T\right)D\left(T\right)+{\int }_{\left[0,T\right)}H\left(s\right)d|\Gamma |\left(s\right)\right]<\infty .$$

(13)

In addition, an European state-contingent claim $\Gamma \left(·\right)$ with expiration date T is called attainable if there exists a $\left(-\Gamma \right)$-state-tame portfolio process $\pi \left(·\right)$ with

$$X^{x,\pi ,-\Gamma }\left(T^{-}\right)=D\left(T\right)$$

(14)

where

$$x:=E\left[H\left(T\right)D\left(T\right)+{\int }_{[0,T)}H\left(s\right)d\Gamma \left(s\right)\right].$$

(15)

We say that a financial market $\mathcal{M}$ is state-complete if every European state-contingent claim is attainable.

The following theorem provides a characterization for state-complete financial markets.

Theorem 2.

A financial market $\mathcal{M}$ is state-complete if and only if the volatility matrix $\sigma \left(t\right)$ has maximal rank for Lebesgue a.e. $t\in \left[0,T_M\right)$ almost surely.

Proof. 

See the proof of Theorem 2 in Londoño (2008). □

3. Actuarial Market

We assume a canonical actuarial market ${\mathcal{M}}_{\mathcal{A}}$ consisting a financial market $\mathcal{M}$ as above, and a probability space ${\Omega }_{\mathcal{A}}=\Omega \times [0,T_M)$, with join probability $P_{\mathcal{A}}=P\otimes P_{\mathcal{L}}$ where $P_{\mathcal{L}}$ is the mortality probability of a given population with density $f\left(t\right)$, $0\le t<T_M$ with ${lim}_{t\to T_M^{-}}f\left(t\right)=0$. We define the time of death $\tau (\omega ,s)=s$ for all $(\omega ,s)\in {\Omega }_{\mathcal{A}}$ and then $P_{\mathcal{L}}[s,t]=P_{\mathcal{A}}\left(\{s\le \tau \le t\}\right)={\int }_s^{t}f\left(u\right)du$ for $0\le s\le t<T_M$. In this setting, $T_M\le \infty$ would represent the maximum life span, including the case $T_M=\infty$. We notice that the maximum life span does not need to be achieved. The assumption on $T_M$ includes finite life spans. However, it also allows for distributions on life span with a neglectable but positive probability of occurrence for the unbounded time of death, such as the linear or power hazard rate distribution. We notice that these distributions include (but are not limited to) the cases of unbounded time of death, which are unrealistic for assuming life distributions but are useful for modeling. We also notice that our modeling assumption implies independence of financial risk and mortality risks. We assume a (deterministic) and continuous hazard function $\lambda >0$ of $\tau$ (with values in ${\mathbb{R}}^{++}$) as the function with ${\int }_0^{T_M}\lambda \left(s\right)ds=\infty$, and ${\int }_0^t\lambda \left(s\right)ds<\infty$ for $t<T_M$ where

$$f\left(t\right):=\lambda \left(t\right)exp\left(-{\int }_0^t\lambda \left(u\right)du\right).$$

(16)

and with conditional survival probability corresponding to $\tau$ given by

$$F\left(t\right):=P_{\mathcal{A}}\left(\tau >t\right)=exp\left(-{\int }_0^t\lambda \left(u\right)du\right).$$

(17)

For each $T<T_M$, we define the actuarial filtration $\left({\mathcal{F}}_{t,T}^{\mathcal{A}}\right)=(\sigma ({\mathcal{F}}_t\cup {\mathcal{N}}_T)\otimes \sigma \left(\mathcal{B}[0,t]\right))$ for $0\le t<T_M$ (where $\mathcal{B}[0,t]$ is the Borel $\sigma$-algebra of $[0,t]$) and extend $W\left(t\right)$, etc., in the natural way; in other words, $W\left(t\right)(\omega ,s)=W\left(t\right)\left(\omega \right)$ for $t\le s$ and $W\left(t\right)(\omega ,s)=W\left(s\right)\left(\omega \right)$ for $t\ge s$. Similarly, for any RCLL restrictedly adapted (restrictedly progressive) measurable process $X\left(t\right)$, we extend the process to ${\Omega }_{\mathcal{A}}$ as $X^{\mathcal{A}}\left(t\right)(\omega ,s)=X\left(t\right)\left(\omega \right)$ for $t\le s$ and $X^{\mathcal{A}}\left(t\right)(\omega ,s)=X\left(s\right)\left(\omega \right)$ for $t\ge s$. It is clear that, for any $T<T_M$, $X^{\mathcal{A}}\left(t\right)$ is a $\left({\mathcal{F}}_{t,T}^{\mathcal{A}}\right)$ adapted process, and that $\tau \wedge T$ is a $\left({\mathcal{F}}_{t,T}^{\mathcal{A}}\right)$ stopping time. In our setting, we only assume diversifiable mortality risk, expressed by the fact that $\lambda \left(t\right)$ is deterministic. It is worth noting that we do not allow for general stopping times with respect to the actuarial filtration because, in our study, we do not allow for dependence of $\tau$ on the behavior of prices of the financial market. Our case is indeed a particular case of general stopping times with respect to the actuarial filtration.

Moreover, we will say that $X\left(t\right)$ defined on ${\Omega }_{\mathcal{A}}=\Omega \times [0,T_M)$ is an actuarial process if an only if $X\left(t\right)(\omega ,s)=X\left(s\right)\left(\omega \right)$ for $s\le t$, and $X\left(t\right)(\omega ,s)=X\left(t\right)(\omega ,s^{\prime })$ for $0\le t\le s\le s^{\prime }<T_M$. Any actuarial process uniquely defines a financial process on $(\Omega ,\mathcal{F},P)$. We will say that the actuarial process $X\left(t\right)$ is a restrictedly measurable process if its corresponding financial process is restrictedly measurable. This later approach to the definition of $X\left(t\right)$ on ${\Omega }_{\mathcal{A}}$ allows us to decompose the mortality risk and the financial risk explicitly using the canonical representation of independent probabilities, keeping the independence structure clear instead of being an obscure relation on an abstract probability space. This approach is aligned with the approach of local times for stochastic flows presented by Kunita (1990) and makes the definition of the stopped process $X^{\tau }\left(t\right)=X(t\wedge \tau )$ for a mortality time $\tau$ a simple one. In other words, if the (realized sample) is $(\omega ,s)$, where $\omega$ represents all the randomness in the financial market of a realization a Brownian motion (since we are assuming the canonical Brownian motion as the generator of randomness of the financial market) and s is the time of death, then it is clear that $X\left(t\right)$ should explain the behavior of the underlying process until the time of death s in an adapted way (and only up to that time).

Whenever it is clear from the context, we will denote as $X^{\mathcal{A}}\left(t\right)$ the extension to ${\Omega }_{\mathcal{A}}$ of the process $X\left(t\right)$, which we will also denote as $X\left(t\right)$ hoping that the meaning would be apparent from the context.

By definition, it is clear that any restrictedly measurable financial process (defined on $(\Omega ,\mathcal{F},P)$) induces a restrictedly measurable actuarial process, and reciprocally, any restrictedly actuarial process induces a restrictedly financial process.

Let ${\mathcal{M}}_A$ be an actuarial market, where we assume that $\mathcal{M}$ is a state-arbitrage-free and state complete financial market with a time of death $\tau$ as defined above. We consider a policyholder with a life insurance policy issued at time $t=0$ and with death time $0<\tau <T_M\le \infty$. We notice that, in the setting of this paper, $T_M$ is never achieved by any time of death and is just a bound for those times (which includes the case where there are no bounds).

Definition 1.

Let $\left(i\right(t),p(t),D(t\left)\right)$, $0\le t<\tau$ be a triple of non-negative RCLL restricted progressive measurable actuarial processes where $D(·)$ is assumed to be a continuous semi-martingale process. We assume that the triple $\left(i\right(t),p(t),D(t\left)\right)$ is defined on $({\Omega }_{\mathcal{A}},\mathcal{F}\otimes \mathcal{B}([0,T_M),P_{\mathcal{A}})$ with financial market $\mathcal{M}$ as above. Assume that, for any $T<T_M$, the process

$$Z\left(t\right)=H\left(t\right)D\left(t\right)-{\int }_0^{t}H\left(u\right)p\left(u\right)du0\le t\le T$$

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is uniformly bounded from the below continuous process such that

$$\underset{0\le t<T_M}{sup}E\left[H\left(t\right)D\left(t\right)+{\int }_0^{t}H\left(u\right)(i\left(u\right)+p\left(u\right))du\right]<\infty .$$

(19)

We say that a triple as above is a stream rate of payment process, premium rate of payment process, and life insurance process, and we call the process

$$Y\left(t\right)=H(t\wedge \tau )D(t\wedge \tau )+{\int }_0^{t\wedge \tau }H\left(u\right)(i\left(u\right)-p\left(u\right))du,0\le t<T_M$$

(20)

the discounted payoff process for insurance. The meaning of the triple of processes $\left(i\right(t),p(t),D(t\left)\right)$ for $0\le t\le \tau$ is as follows: $i\left(t\right)$ is a payment stream process, $p\left(t\right)$ is the premium paid during the lifetime of insurance, and $D\left(t\right)$ is the final insurance payment. The insurer would pay $D\left(t\right)$ to holders that die on time t as long as $0\le t<T_M$. We notice that the insurer does not know in advance which holder dies at the time interval $[t,t+\delta )$, but it knows that the proportion of holders that dies is $P_{\mathcal{A}}(\tau \in [t,t+\delta ))$.

Our interpretation of the premium payment, stream payment, and life insurance process is natural. A consumer buys at time 0 insurance, pays premium payments at a rate of $p\left(t\right)dt$, and receives a flow of insurance payments at a rate of $i\left(t\right)dt$. Moreover, in case of premature death at time t, the insurance company transfers to the policy owners $D\left(t\right)$ dollars to pay outstanding debts and gives the remaining amount to siblings. The process $i\left(t\right)$ can be used to model an insurance disability to be paid before death or to model annuities. We also notice that the role of p and i are not symmetric (see Equations (18)–(20)); therefore, $i\left(t\right)$ cannot be absorbed within $p\left(t\right)$.

Next, we introduce a definition that encapsulates the modeling idea of insurance agents (insurance companies) capable of hedging the mortality risk of individuals:

Definition 2.

Assume the notation of Definition 1; we say that the market is a complete life insurance market for the mortality distribution F if for any triple $\left(i\right(t),p(t),D(t\left)\right)$ there exist an infinity set of instances of the insurance with independent distribution mortality F. In this definition, we assume that $\left(i\right(t),p(t),D(t\left)\right)$ is a stream payment, premium rate, and life insurance.

Remark 2.

For the sake of simplicity, we assume that there exists T with $0<\tau \le T<T_M$. Assume a complete life insurance market. If ${\Delta }_n=\{t_0=0<,\cdots <t_n=T\}$ is a partition of $[0,T]$, with norm $|{\Delta }_n|={max}_i|t_{i+1}-t_i|$, then any insurance company can hedge the insurance for people that die in $(t_i,t_{i+1}]$ with approximated $E\left[Y\left(t_i\right)\right]$ dollars at time 0. It follows that an approximation of the value of the insurance is

$$\sum _{i=0}^{n}E\left[Y\left(t_i\right)\right]P(t_i<\tau \wedge T\le t_{i+1})\to {\int }_0^{T}E\left[Y\left(t\right)\right]dF\left(t\right)=E_{\mathcal{A}}\left[Y\left(\tau \right)\right]$$

as the norm of the partition $|{\Delta }_n|\to 0$. Clearly, under state completeness, the mean hedging portfolio is the financial process (defined on $(\Omega ,\left({\mathcal{F}}_t\right),P$)) which at time s has value

$$\sum _{t_i\ge s}{\pi }^{t_i}\left(s\right)P(t_i<\tau \le t_{i+1})\to {\int }_s^T{\pi }^t\left(s\right)dF\left(t\right)=\pi \left(s\right),$$

where for each t, ${\pi }^t$ is the (mean) hedging portfolio (for the insurance policy) of $D\left(t\right)$ at time t, and with cumulative income process $\Gamma \left(u\right)={\int }_0^u(p\left(s\right)-i\left(s\right))ds$ for $0\le u\le t$. It follows that the value of insurance is $E_{\mathcal{A}}\left[Y\left(\tau \right)\right]$ (which is the amount of money that the insurance is paid at the beginning of the contract) and the hedging portfolio is $\pi \left(s\right)={\int }_s^T{\pi }^t\left(s\right)dF\left(t\right)$ (which is a ${\mathcal{F}}_t$ adapted process defined on Ω). Moreover, the seller (of insurance) of the contract pays at a mean insurance rate of $D\left(t\right)f\left(t\right)dt$ life insurance at time t.

Below, we give another approach to the same valuation and hedging problem. Assume a triple of the premium rate, stream rate, and life insurance with a given value of insurance y. Assume that the insurance company has access to the population, or at least it has access to a large number of clients with the same mortality distribution as the whole population (in the sense that the law of large numbers holds reasonably well). Hence, with value y as the initial payment to the insurance company, the insurance company also receives as income $p\left(t\right)dt$ at time t from the population pool $F\left(t\right)$ that is still alive and pays a stream rate at time t of $i\left(t\right)dt$ to the population pool $F\left(t\right)$ that is still alive. The insurance company would pay insurance at time t for $D\left(t\right)$ to the population $f\left(t\right)dt$ that dies on the time interval $dt$. The value for this process according to the results of Section 2.2, and Equation (29) is

$$\begin{array}{c}E\left[{\int }_0^T\left(f\left(t\right)H\left(t\right)D\left(t\right)-H\left(t\right)p\left(t\right)F\left(t\right)+i\left(t\right)H\left(t\right)F\left(t\right)\right)dt\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=E\left[{\int }_0^{T}f\left(t\right)H\left(t\right)D\left(t\right)dt\right]-E\left[{\int }_0^T\left(H\left(t\right)p\left(t\right)F\left(t\right)-i\left(t\right)H\left(t\right)F\left(t\right)\right)dt\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=E_{\mathcal{A}}\left[H\left(\tau \right)D\left(\tau \right)\right]-E_{\mathcal{A}}\left[{\int }_0^{\tau }\left(p\left(t\right)H\left(t\right)-i\left(t\right)H\left(t\right)\right)dt\right]=E_{\mathcal{A}}\left[Y\left(\tau \right)\right].\hfill \end{array}$$

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where the value is in the sense that, with initial capital $E_{\mathcal{A}}\left[Y\left(\tau \right)\right]$, it is possible in the financial market, with access to a pool of individuals with the same mortality distribution to hedge a stream of insurance $f\left(t\right)D\left(t\right)dt$ at time t. This stream of insurance is also financed by an income rate $\left(p\right(t)-i(t\left)\right)dt$ coming from the pool of consumers.

It follows that, under the assumption of state-complete financial markets and complete life insurance markets, the value (or best estimate) of the life insurance $D\left(t\right)$ with the rate of premium payments $p\left(t\right)$ and rate of stream payment $i\left(t\right)$ for $0\le t\le \tau$, and time of death $\tau$ (with distribution $F\left(t\right)$) is given by Equation (21).

This value represents the average value of the insurance. Assuming both complete financial markets and complete life insurance markets, the concept of value is analogous to the notion of value in complete financial markets. In simpler terms, the average value of insurance is the amount required (or charged) by the insurance company to each individual to hedge the payments (incoming and outgoing) associated with the policy defined by $\left(i\right(t),p(t),D(t\left)\right)$, for $0\le t<\tau$. The insurance company would utilize the contributions from a pool of clients with a similar mortality structure to trade in the financial market and use these initial payments to cover the payouts to the pool of clients defined by the insurance policy.

It is customary to refer to the quantity defined by Equation (21) as the value of the financial instrument. In the actuarial literature, it is typically called the best estimate of the insurance contract. However, to maintain consistency with financial and economic terminology, we should continue to refer to it as the value of life insurance.

We point out that, in general, for a single individual who does not have access to actuarial markets and insurance products, it is, in general, impossible to hedge his/her mortality risk. It is only by participation in the actuarial market and the existence of insurance agents (insurance companies that have access to pools of individuals with the same mortality distribution) that consumers can hedge their mortality risk. We establish the previous result as a proposition:

Proposition 1.

Assume the notation of Definition 1, and assume that the market is a complete life insurance market for the mortality distribution F. Then, the triple $\left(i\right(t),p(t),D(t\left)\right)$ is hedgeable (by insurance agents with access to a pool of consumers with the dependent instances of the same mortality distribution) for an initial capital (value) of $E_{\mathcal{A}}\left[Y\left(\tau \right)\right]$.

In this paper, it is natural to study only insurance whose premium rate is limited by income, as explained below.

Definition 3.

We say that a non-negative restrictedly progressive measurable actuarial process $ϵ\left(t\right),0\le t\le \tau$ defined on ${\Omega }_{\mathcal{A}}$, is an income process if the discounted income ${\int }_0^{s}H\left(t\right)ϵ\left(t\right)dt$ is bounded and the value of income $x^{ϵ}:=E_{\mathcal{A}}\left({\int }_0^{\tau }H\left(s\right)ϵ\left(s\right)ds\right)$ is finite.

We say that a non-negative restrictedly progressive measurable actuarial process $c\left(t\right),0\le t\le \tau$ defined on ${\Omega }_{\mathcal{A}}$ is a consumption rate process if ${\int }_0^{\tau }c\left(t\right)dt<\infty$.

We say that the triple $\left(i\right(t),p(t),D(t\left)\right)$ of stream rate of payment process, premium rate process, and life insurance process is feasible for the income process $ϵ\left(t\right),0\le t\le \tau$ if

$$i\left(t\right)+ϵ\left(t\right)\ge p\left(t\right)\mathrm{for}0\le t\le \tau .$$

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Definition 4.

A family of consumption/portfolio/stream rate of payment/premium rate of payment/life insurance processes $\left(c,\pi ,i,p,D\right)$ is called admissible at (initial physical wealth) $x\in \mathbb{R}$ (with a rate of income $ϵ\left(t\right),0\le t\le \tau$), and written $\left(c,\pi ,i,p,D\right)\in {\mathcal{A}}^{\tau }(x,ϵ)$, if $\left(i\right(t),p(t),D(t\left)\right)$ is feasible for the income process $ϵ\left(t\right),0\le t\le \tau$,

$$0\le y=E_{\mathcal{A}}\left[Y\left(\tau \right)\right]\le x+E_{\mathcal{A}}\left({\int }_0^{\tau }H\left(s\right)ϵ\left(s\right)ds\right)$$

(23)

and $(X^{x-y,{\Gamma }_{\Sigma },\pi }(·),d{\Gamma }_{\Sigma })$ is a hedgeable (in the financial market) wealth income structure that satisfies

$$X^{x-y,{\Gamma }_{\Sigma },\pi }\left(t\right)+D\left(t\right)\ge 0\mathrm{for}\mathrm{all}0\le t\le \tau .$$

(24)

where $d{\Gamma }_{\Sigma }\left(s\right)=(ϵ\left(s\right)-c\left(s\right)+i\left(s\right)-p\left(s\right))ds$.

The condition defined by Equation (24) is the classical institutional requirement motivated by the fact that siblings do not inherit debts with future labor as collateral.

It is worth noting that Equation (23) is natural, in the sense that it says that the insurance that a consumer can buy has a price bounded by the total wealth of a consumer, at time 0. In our approach, the insurance is similar to forward instruments. The latter holds since the contract conditions are set at the time of purchase and not resettled for the life of the insurance instrument.

A completely different approach will arise if we assume that agents can resettle their debts and insurance as time evolves. In this latter case, the condition defined by Equations (23) and (24) would need to be changed by the following equation:

$$\begin{array}{c}H\left(t\right)X^{x-y,{\Gamma }_{\Sigma },\pi }\left(t\right)+\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{E}_{\mathcal{A}}\left[1_{\{\tau \ge t\}}\left(H\left(\tau \right)D\left(\tau \right)+{\int }_t^{\tau }H\left(s\right)(ϵ\left(s\right)+i\left(s\right)-p\left(s\right))ds\right)\mid {\mathcal{F}}_t^{\mathcal{A}}\right]\ge 0\end{array}$$

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for $0\le t\le \tau$. This latter equation says that the current value of debts is limited by the value of future income when the insurance is bought.

Also, for technical reasons, we define the class of admissible portfolios when there is no income (and therefore, there is no need for life insurance to serve as collateral for premature death):

Definition 5.

A consumption/portfolio process $\left(c,\pi \right)$ is called admissible at $x\in \mathbb{R}$ and written $(c,\pi )\in {\mathcal{B}}^{\tau }\left(x\right)$, if $X^{x,{\Gamma }_c,\pi }(·)\ge 0$ for all $0\le t\le \tau$, where $d{\Gamma }_c\left(t\right)=-c\left(t\right)dt$.

Remark 3.

The recent literature on the valuation and hedging of future labor income (the value of human capital) and life insurance instruments assumes strong hypotheses regarding the boundness of various processes. For instance, works such as Pliska and Ye (2007), Ye (2007), and Ye (2019) assume that the interest rate, the insurance premium payout process, the income process, the drift process, and the insurance premium payout ratio process are uniformly bounded and that the volatility process remains uniformly away from zero. In fact, when general results are obtained for the hedging and valuation of life insurance instruments in continuous time, the wealth process, the consumption process, the life insurance premium, and the amount of risky assets are required to be in $L^2$ (Ye 2019). These conditions are usually imposed to avoid arbitrage opportunities, but they are artificial and lack economic motivation aside from allowing the mathematical framework to function. The hypotheses used in this paper are derived from the concept of state tameness introduced by Londoño (2004), which has a clear economic interpretation that the current value of debts, in relative terms (with respect to the total wealth of society), is bounded (Londoño 2020). A review of arbitrage theory using the concept of state tameness without imposing strong $L^2$ conditions is discussed by Londoño (2004)

Remark 4.

The recent literature on the valuation and hedging of life insurance instruments (Ye 2019) assumes that life insurance is proportional to the premium payment rate, making it impossible to account for prepaid insurance. Specifically, Ye (2019) assume that the insurance amount is given by $D\left(t\right)=p\left(t\right)/\eta \left(t\right)$ where $p\left(t\right)$ represents the life insurance premium, and $\eta \left(t\right)$ is a uniformly bounded, predictable insurance premium-payout ratio process, with premiums being paid continuously. This assumption prevents modeling scenarios wherein prepaid insurance is offered in the market (i.e., insurance instruments fully paid at time 0 when the contract is established).

Our approach extends the valuation and hedging of financial instruments and can be seen as a generalization of European contingent claims with independent random expiration dates (see Londoño (2004) for a discussion on the valuation and hedging of European Contingent Claims with stopping times as expiration dates). Our methodology accommodates prepaid insurance and includes standard results on the valuation of life insurance.

4. Utility Maximization

This paper mainly focuses on portfolio processes obtained from optimal consumer behavior, as explained below.

Next, we review the definitions and properties of the type of utilities that we use in this paper.

Definition 6.

Consider a function $U:\left(0,\infty \right)\mapsto \mathbb{R}$ continuous, strictly increasing, strictly concave, and continuously differentiable, with $U^{\prime }(\infty )={lim}_{x\to \infty }{U}^{\prime }\left(x\right)=0$ and $U^{\prime }(0+)\triangleq {lim}_{x\downarrow 0}{U}^{\prime }\left(x\right)=\infty$. Such a function will be called a utility function.

For every utility function $U(·)$, we denote by $I(·)$ the inverse of the derivative $U^{\prime }(·)$; both of these functions are continuous, strictly decreasing and map $(0,\infty )$ onto itself with $I(0+)=U^{\prime }(0+)={lim}_{x\to 0^{+}}{U}^{\prime }\left(x\right)=\infty$, $I(\infty )={lim}_{x\to \infty }I\left(x\right)=U^{\prime }(\infty )=0$. We extend U by $U\left(0\right)=U\left(0^{+}\right)$, and keep the same notation to the extension to $[0,\infty )$ of U hoping that it will be clear to the reader to which function we are referring. It is a well-known result that

$$\underset{0<x<\infty }{max}\left(U\left(x\right)-xy\right)=U\left(I\left(y\right)\right)-yI\left(y\right),0<y<\infty .$$

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Definition 7.

Consider a couple of continuous functions $U_1,U_2:[0,T_M)\times \left(0,\infty \right)\mapsto \mathbb{R}$, such that $U_1(t,·)$ and $U_2(t,·)$ are utility functions in the sense of Definition 6 for all $t\in [0,T_M)$. Moreover, if $I_1(t,x)\triangleq {(\partial U_1(t,x)/\partial x)}^{-1}$, and $I_2(t,x)\triangleq {(\partial U_2(t,x)/\partial x)}^{-1}$, where ${(\partial U_1(t,·)/\partial x)}^{-1}$ and ${(\partial U_2(t,·)/\partial x)}^{-1}$ denotes the inverse of the derivative of $U_1(t,·)$ and $U_2(t,·)$, respectively, it follows that $I_1$ and $I_2$ are continuous functions. We call a couple of functions above a state preference structure. In addition, we assume the following notations for any preference structure:

$$\mathcal{X}(t,T,y)\triangleq I_2\left(T_{,}y\right)+{\int }_t^T{I}_1(s,y)ds.$$

(27)

for $0\le t\le T<T_M$.

We extend $U_1$ and $U_2$ by defining $U_1(t,0)=U(t,0^{+})$, for all $0\le t\le T_M$ and $U_2\left(0\right)=U_2\left(0^{+}\right)$, and we keep the same notation to the extension of $U_1$ and $U_2$ to $[0,T_M)\times [0,\infty )$. Discussion of those utility functions defined above is given in Londoño (2009). We notice that the interpretation is natural. $U_1(t,·)$ is the utility for consumption of $c_t$ dollars (as seen at time 0) discounted by the state price at time t and $U_2(t,·)$ is the utility on discounted wealth (as seen as time 0) by the state price when premature death occurs at time t.

Natural classes of utility structures are those whose utility for terminal wealth corresponds to the utility of future consumption that is not realized due to premature death. The precise definition is as follows:

Definition 8.

A state preference structure is time consistent if for any $0\le T^{\prime }\le T<T_M$

$$I_2(T^{\prime },y)-I_2(T,y)={\int }_{T^{\prime }}^T{I}_1(s,y)ds$$

Moreover, we say that the state preference structure has integrable inverse marginal utilityif for each y, ${sup}_{0\le t<T_M}{I}_2(t,y)\vee {\int }_0^{T_M}{I}_1(t,y)dt<\infty$.

If $U_1,U_2$ is a time-consistent state utility preference structure, the functions $\mathcal{X}(t,T,y)$ and ${\mathcal{X}}^{-1}(t,T,y)$ do not depend on $0<T<T_M$.

Remark 5.

Assume that $U_1,U_2$ is a time-consistent state utility preference structure, and $\mathcal{X}(t,T,y)$ and the inverse (with respect to the spatial variable y), ${\mathcal{X}}^{-1}(t,T,y)$ does not depend on $0\le T<T_M$. Namely, for any $T,T^{\prime }\in [0,T_M)$, $\mathcal{X}(t,T,y)=\mathcal{X}(t,T^{\prime },y)$ for $0\le t\le T\wedge T^{\prime }$. In this latter case, we denote the common value as $\mathcal{X}(t,y)$, and the common value of its inverse as ${\mathcal{X}}^{-1}(t,y)$.

Example 1.

Let $c>0$, $0<\alpha <1$, and $0<\beta <1$. Define $U_1(t,x)=c{e}^{-\beta t}{x}^{\alpha }$, and $U_2(t,x)=d{e}^{-\beta t}{x}^{\alpha }$ where $d=c{((1-\alpha )/\beta )}^{1-\alpha }$. It is straightforward to see that $U_1,U_2:[0,\infty )\times (0,\infty )\to (0,\infty )$ is a time-consistent utility structure with integrable inverse marginal utility.

If $\left(U_1,U_2\right)$ is a preference structure, we propose as the natural problem of optimal consumption and investment with insurance the following optimization problems. Problem 1 addresses optimal consumption without labor income and mortality risk. Problem 2 addresses optimal consumption with labor income and mortality risk.

Problem 1.

Find an optimal consumption and portfolio processes $\left(c,\pi \right)\in {\mathcal{B}}^{\tau }\left(y\right)$ for the problem

$$\begin{array}{cc}{V}_2^{\tau }\left(y\right):=\hfill & \\ & \hfill \underset{(c,\pi )\in {\mathcal{B}}^{\tau }\left(y\right)}{sup}{E}_{\mathcal{A}}\left[{\int }_0^{\tau }{U}_1\left(s,H\left(s\right)c\left(s\right)\right)ds+U_2\left(\tau ,H\left(\tau \right)(X^{y,{\Gamma }_c,\pi }\left(\tau \right)\right)\right]\end{array}$$

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over the class

$$\begin{array}{c}{\mathcal{B}}^{\tau }\left(y\right):=\hfill \\ \hfill \hspace{1em}\left\{(c,\pi )\in {\mathcal{B}}^{\tau }\left(y\right):E_{\mathcal{A}}\left[{\int }_0^{\tau }\left(U_1^{-}(s,H\left(s\right)c\left(s\right)\right)ds+U_2^{-}\left(\tau ,H\left(\tau \right)\left(X^{y,{\Gamma }_c,\pi }\left(\tau \right)\right)\right)\right]<\infty \right\}\end{array}$$

Problem 2.

Given an initial physical wealth $x\ge 0$, and rate of income $ϵ\left(t\right),0\le t\le \tau$, find an optimal family of admissible consumption/portfolio/stream rate of payment/premium rate of payment/life insurance processes $\left(c,\pi ,i,p,D\right)\in {\mathcal{A}}^{\tau }\left(x,ϵ\right)$ (where ${\mathcal{A}}^{\tau }\left(x,ϵ\right)$ is given by Definition 4) for the problem

$$\begin{array}{cc}{V}^{\tau }\left(x,ϵ\right):=\hfill & \\ & \hspace{1em}\hspace{1em}\hspace{1em}\underset{(c,\pi ,i,p,D)\in {\mathcal{A}}^{\tau }(x,ϵ)}{sup}{E}_{\mathcal{A}}\left[{\int }_0^{\tau }{U}_1\left(s,H\left(s\right)c\left(s\right)\right)ds+\right.\hfill \\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.U_2\left(\tau ,H\left(\tau \right)(X^{x-E_{\mathcal{A}}\left[Y\left(\tau \right)\right],{\Gamma }_{\Sigma },\pi }\left(\tau \right)+D\left(\tau \right))\right)\right]\hfill \end{array}$$

over the class

$$\begin{array}{c}{\mathcal{A}}^{\tau }(x,ϵ):=\left\{(c,\pi ,i,p,D)\in {\mathcal{A}}^{\tau }(x,ϵ):\right.\hfill \\ \hspace{1em}{E}_{\mathcal{A}}\left[{\int }_0^{\tau }\left(U_1^{-}\left(s,H\left(s\right)c\left(s\right)\right)ds+U_2^{-}\left(\tau ,H\left(\tau \right)\left(X^{x-E_{\mathcal{A}}\left[Y\left(\tau \right)\right],{\Gamma }_{\Sigma },\pi }\left(\tau \right)+D\left(\tau \right)\right)\right)\right]<\infty \right\}\hfill \end{array}$$

where $Y\left(t\right)$ is the discounted payoff process as defined by Equation (20), $X^{x-E_{\mathcal{A}}\left[Y\left(\tau \right)\right],{\Gamma }_{\Sigma },\pi }$ wealth process associated with $\left({\Gamma }_{\Sigma },\pi \right)$ and initial wealth $x-E_{\mathcal{A}}\left[Y\left(\tau \right)\right]$ and $d{\Gamma }_{\Sigma }$ is given in Definition 4.

The following remark helps to understand that the consistency property for preferences refers to the fact that value to wealth non-consumed is the preference of wealth kept for future consumption:

Remark 6.

The definition of consistency for preferences of Definition 8 implies that the solution for the classical problem of optimal intermediate consumption for deterministic horizon time does not depend on the terminal time, as explained next. Assume a preference structure $(U_1(t,·),U_2(T_1,·))$ and $(U_1(t,·),U_2(T_2,·))$ for $T_1<T_2\le T$. Assume that $ϵ\left(t\right)$ is an endowment process for $0\le t\le T$, and $({\xi }_1\left(t\right),c_1\left(t\right))$ and $({\xi }_2\left(t\right),c_2\left(t\right))$ are the optimal wealth and consumption processes for the problem of optimal consumption and investment for discounted consumption and wealth by state–price process for terminal time $T_1$, and $T_2$, respectively. It follows that $c_1\left(t\right)=c_2\left(t\right)$ for $0\le t\le T_1$. The optimization problem is similar to Problem 2.

Let us fix an income process $ϵ\left(·\right)$ with current value of income $x^{ϵ}:=E_{\mathcal{A}}\left({\int }_0^{\tau }H\left(s\right)ϵ\left(s\right)ds\right)$. By using the Fubini–Tonelli theorem, we can check that $x^{ϵ}=E\left({\int }_0^{T_M}F\left(s\right)H\left(s\right)ϵ\left(s\right)ds\right)$.

Indeed, for any couple of continuous integrable adapted processes $Y\left(t\right),y\left(t\right)$ for $0\le t<T_M$

$$E_{\mathcal{A}}\left[Y\left(\tau \right)+{\int }_0^{\tau }y\left(s\right)ds\right]=E\left[{\int }_0^{T_M}f\left(s\right)Y\left(s\right)ds\right]+E\left[{\int }_0^{T_M}F\left(s\right)y\left(s\right)ds\right].$$

(29)

The previous remark allows us to solve Problem 2 and Problem 1. For the solution of Problem 2, we adapt the solution to the equivalent problem for deterministic terminal times $\tau$. The solution to Problem 2 is identical to the solution of consumption and terminal wealth assuming a physical capital equal to the value of future labor income given by Londoño (2009) with deterministic terminal times. The solution obtained for Problem 2 does not require the existence of insurance companies and only requires the knowledge of agents of their current value of future income.

Theorem 3.

Assume a time-consistent state preference structure with utilities $U_1,U_2$, with integrable inverse marginal utility. Then, the optimal wealth and an optimal couple of consumption and portfolio processes $(c,\pi )\in {\mathcal{B}}^{\tau }\left(y\right)$ for an individual for the Problem 1 and $0\le t<\tau$ are given by

$$X^{y,{\Gamma }_c,\pi }\left(t\right)\triangleq H^{-1}\left(t\right)\mathcal{X}(t,{\mathcal{X}}^{-1}(0,y))t\le \tau$$

(30)

and

$$c\left(t\right)\triangleq H^{-1}\left(t\right)I_1(t,{\mathcal{X}}^{-1}(0,y))t<\tau$$

(31)

and

$$\pi \left(t\right)\triangleq X^{y,{\Gamma }_c,\pi }\left(t\right){\left(\sigma {\sigma }^{⊺}\right)}^{-1}(b\left(t\right)+\delta \left(t\right)-r\left(t\right)1_n)t\le \tau$$

(32)

Proof. 

We observe that, for $ϵ>0$,

$$\begin{array}{c}{E}_{\mathcal{A}}\left[H\left(\tau \right)X^{y,{\Gamma }_c,\pi }\left(\tau \right)+{\int }_0^{\tau }H\left(s\right)c\left(s\right)ds\right]=\hfill \\ E\left[{\int }_0^{T_M}f\left(s\right)H\left(s\right)X^{y,{\Gamma }_c,\pi }\left(s\right)ds+{\int }_0^{T_M}F\left(s\right)H\left(s\right)c\left(s\right)ds\right]=\\ E\left[{\int }_{T_M-ϵ}^{T_M}f\left(s\right)H\left(s\right)X^{y,{\Gamma }_c,\pi }\left(s\right)ds+{\int }_{T_M-ϵ}^{T_M}F\left(s\right)H\left(s\right)c\left(s\right)ds\right]+\\ \hfill I_2(T_M-ϵ,{\mathcal{X}}^{-1}(0,y))F(T_M-ϵ)+{\int }_0^{T_M-ϵ}{I}_1(s,{\mathcal{X}}^{-1}(0,y))ds\to y\mathrm{as}ϵ\to 0\end{array}$$

since $f\left(t\right)\to 0$ as $t\to T_M$, and since the preference structure has integrable marginal utility. The proof of the latter statement is a straightforward consequence from the definitions, Londoño (2009, Theorem 3), and the identity from Equation (29).

Let

$$u=E\left[{\int }_0^{T_M}F\left(s\right)U_1(s,H\left(s\right)c\left(s\right))ds+{\int }_0^{T_M}f\left(s\right)U_2(s,H\left(s\right)X^{y,{\Gamma }_c,\pi }\left(s\right))ds\right]$$

If $(c^{\prime },\psi )\in {\mathcal{B}}^{\tau }\left(y\right)$, and assume that $u<\infty$ and $ϵ>0$, it follows that

$$\begin{array}{c}E\left[{\int }_0^{T_M-ϵ}F\left(s\right)U_1\left(s,H\left(s\right)c^{\prime }\left(s\right)\right)+f\left(s\right)U_2\left(s,H\left(s\right)X^{y,{\Gamma }_{c^{\prime }},\psi }\left(s\right)\right)ds\right]\le \hfill \\ {\mathcal{X}}^{-1}(0,y)\left(E\left[{\int }_0^{T_M-ϵ}F\left(s\right)H\left(s\right)c^{\prime }\left(s\right)ds\right]+E\left[{\int }_0^{T_M-ϵ}f\left(s\right)H\left(s\right)X^{y,{\Gamma }_{c^{\prime }},\psi }\left(s\right)ds\right]\right)+\\ E\left[{\int }_0^{T_M-ϵ}F\left(s\right)U_1(s,H\left(s\right)c\left(s\right))ds+{\int }_0^{T_M-ϵ}f\left(s\right)U_2(s,H\left(s\right)X^{y,{\Gamma }_c,\pi }\left(s\right))ds\right]\\ \hfill -E\left[{\int }_0^{T_M-ϵ}f\left(s\right)H\left(s\right)X^{y,{\Gamma }_c,\pi }\left(s\right)ds+{\int }_0^{T_M-ϵ}F\left(s\right)H\left(s\right)c\left(s\right)ds\right]{\mathcal{X}}^{-1}(0,y)\end{array}$$

(33)

and similarly, and using $u<\infty$ and Equation (26), it follows that

$$\begin{array}{c}\hfill E\left[{\int }_{T_M-ϵ}^{T_M}F\left(s\right)U_1\left(s,H\left(s\right)c^{\prime }\left(s\right)\right)+f\left(s\right)U_2\left(s,H\left(s\right)X^{y,{\Gamma }_{c^{\prime }},\psi }\left(s\right)\right)ds\right]\to 0\end{array}$$

(34)

as $ϵ\to 0$. It follows by Equations (34) and (33) that

$$\begin{array}{c}{E}_{\mathcal{A}}\left[{\int }_0^{\tau }{U}_1\left(s,H\left(s\right)c^{\prime }\left(s\right)\right)ds+U_2\left(\tau ,H\left(\tau \right)X^{y,{\Gamma }_{c^{\prime }},\psi }\left(\tau \right)\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}E\left[{\int }_0^{T_M}F\left(s\right)U_1\left(s,H\left(s\right)c^{\prime }\left(s\right)\right)+f\left(s\right)U_2\left(s,H\left(s\right)X^{y,{\Gamma }_{c^{\prime }},\psi }\left(s\right)\right)ds\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}={\mathcal{X}}^{-1}(0,y)\left(E_{\mathcal{A}}\left[{\int }_0^{\tau }H\left(s\right)c^{\prime }\left(s\right)ds\right]+E_{\mathcal{A}}\left[H\left(\tau \right)X\left(\tau \right)ds\right]\right)+\\ \hspace{1em}\hspace{1em}\hspace{1em}{E}_{\mathcal{A}}\left[{\int }_0^{\tau }{U}_1(s,H\left(s\right)c\left(s\right))ds\right]+E_{\mathcal{A}}\left[U_2(\tau ,H\left(\tau \right)X^{y,{\Gamma }_c,\psi }\left(\tau \right))\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}-y{\mathcal{X}}^{-1}(0,y)\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le E_{\mathcal{A}}\left[{\int }_0^{\tau }{U}_1(s,H\left(s\right)c\left(s\right))ds\right]+E_{\mathcal{A}}\left[U_2(\tau ,H\left(\tau \right)X^{y,{\Gamma }_c,\psi }\left(\tau \right))\right]\end{array}$$

(35)

The latter inequality proves optimality. Finally, Equation (12) implies that $\pi \left(t\right)$ is the desired portfolio. □

Theorem 4.

Assume a state preference structure with utilities $U_1,U_2$ with integrable inverse marginal utility and assume a complete financial and complete life insurance market. Then, the optimal consumption/portfolio/stream rate of payment/premium rate of payment/life insurance processes $(c,\pi ,i,p,D)\in {\mathcal{A}}^{\tau }(x,ϵ)$ for Problem 2 is given by

$$c\left(t\right)\triangleq H^{-1}\left(t\right)I_1(t,{\mathcal{X}}^{-1}(0,x+x^{ϵ}))t\le \tau$$

(36)

$$X^{x-y,{\Gamma }_{\Sigma },\pi }\left(t\right)+D\left(t\right)\triangleq H^{-1}\left(t\right)\mathcal{X}(t,{\mathcal{X}}^{-1}(0,x+x^{ϵ}))t\le \tau$$

(37)

where $(i,p,D)$ is any triple of premium rate of payment, stream rate of payment, and life insurance processes with discounted payoff value $0\le Y\left(0\right)=y\le x+x^{ϵ}$. It is assumed $(c,\pi ,i,p,D)\in {\mathcal{A}}^{\tau }(x,ϵ)$, and $X^{x-y,{\Gamma }_{\Sigma },\pi }\left(t\right)$ is a hedgeable wealth process (within the financial market and $d\Gamma =\left(ϵ\right(s)+i(s)-c(s)-p(s\left)\right)dt$, and portfolio $\pi \left(t\right)$). In particular, an optimal triple of premium rate of payment, stream rate of payment, and life insurance processes with 0 value is given by $i\left(t\right)=0$, $p\left(t\right)=F\left(t\right)ϵ\left(t\right)$, and

$$D\left(t\right)=H^{-1}\left(t\right)E\left[{\int }_t^{T_M}F\left(u\right)H\left(u\right)ϵ\left(u\right)du\mid {\mathcal{F}}_t\right].$$

(38)

and with an optimal portfolio given by

$$\begin{array}{c}\pi \left(t\right)\triangleq H^{-1}\left(t\right){\left(\sigma {\sigma }^{⊺}\right)}^{-1}\sigma {\phi }^{⊺}\left(t\right)+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(\mathcal{X}(t,{\mathcal{X}}^{-1}(0,x+x^{ϵ}))-D\left(t\right)\right){\left(\sigma {\sigma }^{⊺}\right)}^{-1}(b\left(t\right)+\delta \left(t\right)-r\left(t\right)1_n)t\le \tau \hfill \end{array}$$

(39)

where $\phi \left(t\right)$ is the process arising as a stochastic integral of the martingale given by

$$x^{ϵ}+{\int }_0^t\phi \left(t\right)dW\left(t\right)=E\left[{\int }_0^{T_M}F\left(u\right)H\left(u\right)ϵ\left(u\right)du\mid {\mathcal{F}}_t\right].$$

Proof. 

We assume that $0<T<T_M$ exists with $\tau \le T$. We observe that a proof similar to the proof of Theorem 3 implies that

$$\begin{array}{c}{E}_{\mathcal{A}}\left[H\left(\tau \right)\left(X^{x,{\Gamma }_{\Sigma },\pi }\left(\tau \right)+D\left(\tau \right)\right)+{\int }_0^{\tau }H\left(s\right)c\left(s\right)ds\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}{\int }_0^{T}f\left(s\right)I_2(s,{\mathcal{X}}^{-1}(0,x+x^{ϵ}))ds+{\int }_0^{T}F\left(s\right)I_1(s,{\mathcal{X}}^{-1}(0,x+x^{ϵ}))ds=x+x^{ϵ}.\hfill \end{array}$$

(40)

If the premium rate of payment is $p\left(t\right)=F\left(t\right)ϵ\left(t\right)$, the stream rate of payment is $i\left(t\right)=0$ and the life insurance process is given by Equation (38), then $(X^{x-y,{\Gamma }_{\Sigma },\pi }\left(t\right),c+p,ϵ+i)$ is hedgeable with initial wealth $x-y$, where y is the value of insurance since

$$\begin{array}{c}H\left(t\right)X\left(t\right)+{\int }_0^{t}H\left(s\right)(c\left(s\right)+p\left(s\right)-ϵ\left(s\right)-i\left(s\right))ds=\hfill \\ \hfill I_2(t,{\mathcal{X}}^{-1}(0,x+x^{ϵ}))+{\int }_0^t{I}_1(s,{\mathcal{X}}^{-1}(0,x+x^{ϵ}))ds-E\left[{\int }_0^{T}F\left(u\right)H\left(u\right)ϵ\left(u\right)du\mid {\mathcal{F}}_t\right]\end{array}$$

(41)

is a martingale. We also observe that a straightforward computation shows that $(0,p,D)$ is a stream payment rate, premium rate, and life insurance process with 0 value.

Moreover, for any family of consumption/portfolio/stream rate of payment/premium rate of payment/life insurance processes $\left(c^{\prime },{\pi }^{\prime },i^{\prime },p^{\prime },D^{\prime }\right)\in {\mathcal{A}}^{\tau }(x,ϵ)$ with wealth $X^{\prime }\left(t\right)\equiv X^{x-y^{\prime },d{\Gamma }_{{\Sigma }^{\prime }},{\pi }^{\prime }}$ where $d{\Gamma }_{{\Sigma }^{\prime }}\left(s\right)=(ϵ\left(s\right)-c\left(s\right)+i^{\prime }\left(s\right)-p^{\prime }\left(s\right))ds$ and $y^{\prime }=E_{\mathcal{A}}\left[Y^{\prime }\left(\tau \right)\right]$,

$$\begin{array}{c}{E}_{\mathcal{A}}\left[H\left(\tau \right)\left(X^{\prime }\left(\tau \right)+D^{\prime }\left(\tau \right)\right)+{\int }_0^{\tau }H\left(s\right)c^{\prime }\left(s\right)ds\right]=\hfill \\ E_{\mathcal{A}}\left[H\left(\tau \right)X^{\prime }\left(\tau \right)+{\int }_0^{\tau }H\left(s\right)\left(c^{\prime }\left(s\right)-ϵ\left(s\right)-i^{\prime }\left(s\right)+p^{\prime }\left(s\right)\right)ds\right]+\\ \hspace{1em}\hspace{1em}{E}_{\mathcal{A}}\left[H\left(\tau \right)D^{\prime }\left(\tau \right)+{\int }_0^{\tau }H\left(s\right)(i^{\prime }\left(s\right)-p^{\prime }\left(s\right))ds\right]+E_{\mathcal{A}}\left[{\int }_0^{\tau }H\left(s\right)ϵ\left(s\right)\right]\le x+x^{ϵ}\hfill \end{array}$$

(42)

where the last inequality follows since $X^{\prime }$ is hedgeable (within the financial market), then

$$\begin{array}{c}{E}_{\mathcal{A}}\left[H\left(\tau \right)X^{\prime }\left(\tau \right)+{\int }_0^{\tau }H\left(s\right)\left(c^{\prime }\left(s\right)-ϵ\left(s\right)-i^{\prime }\left(s\right)+p^{\prime }\left(s\right)\right)ds\right]=\hfill \\ \hspace{1em}\hspace{1em}{\int }_0^{T}f\left(t\right)E\left[H\left(t\right)X^{\prime }\left(t\right)+{\int }_0^{t}H\left(s\right)\left(c^{\prime }\left(s\right)-ϵ\left(s\right)-i^{\prime }\left(s\right)+p^{\prime }\left(s\right)\right)ds\right]dt\le x-y^{\prime }.\hfill \end{array}$$

(43)

The proof of theorem is a consequence of Equations (40)–(43). Equation (39) is a consequence of Equation (12).

The more general proof for $\tau <T_M$ is similar to that for Theorem 3. □

Remark 7.

Assume that $(c,\pi ,i,p,D)\in {\mathcal{A}}^{\tau }(x,ϵ)$ is the optimal solution for Problem 2, where $(i,p,D)$ is any triple of premium rate of payment, stream rate of payment, and life insurance processes with discounted payoff value $0\le Y\left(0\right)=y\le x+x^{ϵ}$ with $(c,\pi ,i,p,D)\in {\mathcal{A}}^{\tau }(x,ϵ)$, and $X^{x-y,{\Gamma }_{\Sigma },\pi }\left(t\right)$ is a hedgeable wealth process with income $d\Gamma =\left(ϵ\right(s)+i(s)-c(s)-p(s\left)\right)dt$. Then, a review of the proof of Theorem 4 shows that the optimal portfolio is given by

$$\begin{array}{c}\pi \left(t\right)\triangleq H^{-1}\left(t\right){\left(\sigma {\sigma }^{⊺}\right)}^{-1}\sigma \phi \left(t\right)+\hfill \\ \hspace{1em}\hspace{1em}\left(H^{-1}\left(t\right)\mathcal{X}(t,{\mathcal{X}}^{-1}(0,x+x^{ϵ}))-D\left(t\right)\right){\left(\sigma {\sigma }^{⊺}\right)}^{-1}(b\left(t\right)+\delta \left(t\right)-r\left(t\right)1_n)t\le \tau \hfill \end{array}$$

(44)

where $\phi \left(t\right)$ is the process arising as a stochastic integral of the martingale given by

$$y-{\int }_0^t\phi \left(t\right)dW\left(t\right)=Y\left(t\right)+{\int }_0^{t}H\left(s\right)ϵ\left(s\right)ds.$$

where $Y\left(t\right)$ is the discounted payoff process for insurance.

Remark 8.

The solution—specifically, the consumption and optimal portfolios—to Problem 1 as provided by Theorem 3 is identical (until the time of death) to the solution for consumption and terminal wealth assuming that physical capital equals the value of future labor income, as described by Londoño (2009) with deterministic terminal times. This solution for Problem 1 does not depend on the existence of insurance companies but only requires agents to know the current value of their future income.

Problem 2 becomes straightforward once the solution for Problem 1 is established. Generally, consumers purchase insurance that allows them to consume as if they have physical capital equal to the value of future labor income. However, their physical wealth combined with the insurance enables them to borrow against their future labor income in a way that ensures the insurance can cover any debts in the event of death. In this scenario, insurance companies facilitate optimal consumption even when physical capital is negative, provided that total wealth (defined as future labor income plus physical wealth) is positive.

Example 2.

Assume a financial market with a price process $S\left(t\right)$ with a constant mean rate of return b, constant dividend δ, constant volatility σ, and constant risk-free rate r. We assume a linear hazard rate $\lambda \left(t\right)={\lambda }_0+{\lambda }_{1}t$ with ${\lambda }_0,{\lambda }_1>0$, an endowment process $ϵ\left(t\right)={ϵ}_0{e}^{\kappa t}{1}_{t<T}$ for $\kappa ,T>0$, where T represents the retirement date. We assume a time consistent preference structure given by $U_1(t,x)=c{e}^{-\beta t}{x}^{\alpha }$ and $U_2(t,x)=c{((1-\alpha )/\beta )}^{1-\alpha }{e}^{-\beta t}{x}^{\alpha }$ for $0<\alpha <1$ and $\beta >0$, and $c>0$. Natural values of the parameter α are between 0 and 1, as discussed by Londoño (2020, Remark 18). It follows that

$$\begin{array}{cc}\hfill I_1(t,x)& ={\left(\alpha c\right)}^{1/(1-\alpha )}{e}^{\beta t/(\alpha -1)}{x}^{1/(\alpha -1)}and,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill I_2(t,x)& ={\displaystyle \frac{(1-\alpha )}{\beta }}{\left(\alpha c\right)}^{1/(1-\alpha )}{e}^{\beta t/(\alpha -1)}{x}^{1/(\alpha -1)}\hfill \end{array}$$

(46)

Moreover, it follows that the state–price process is $H\left(t\right)=e^{-{\displaystyle \frac{1}{2}}{\theta }^{2}t-rt-\theta W\left(t\right)}$ with $\theta =(b+\delta -r)/\sigma$. It follows that the optimal consumption is

$$c\left(t\right)=(x+x^{ϵ})\left({\displaystyle \frac{\beta }{1-\alpha }}\right)exp\left(-{\displaystyle \frac{\beta }{1-\alpha }}t\right)H^{-1}\left(t\right)t<\tau$$

and the optimal wealth plus insurance is

$$X^{x,{\Gamma }_{\Sigma },\pi }\left(t\right)+D\left(t\right)=(x+x^{ϵ})exp\left(-{\displaystyle \frac{\beta }{1-\alpha }}t\right)H^{-1}\left(t\right)t\le \tau$$

where x is the initial wealth and $x^{ϵ}$ is the current value (at time 0) of future wealth. Under the current hypothesis, the survival function is $F\left(t\right)=exp(-{\int }_0^t\lambda \left(s\right)ds)=exp(-{\lambda }_{0}t-{\lambda }_1{t}^2/2)$ and the current value of future wealth at time 0 is

$$x^{ϵ}={ϵ}_0{\int }_0^{T}exp\left((\kappa -{\lambda }_0-r)t-{\displaystyle \frac{{\lambda }_1^2}{2}}{t}^2\right)dt$$

An optimal life insurance process is given by

$$D\left(t\right)={ϵ}_0{H}^{-1}\left(t\right)\left({\int }_t^{T}exp\left((\kappa -{\lambda }_0-r)u-{\displaystyle \frac{{\lambda }_1^2}{2}}{u}^2\right)du\right)1_{t<T\wedge \tau }$$

with optimal premium rate of payment $p\left(t\right)={ϵ}_0(1-e^{-{\lambda }_{0}t-{\lambda }_1{t}^2/2})e^{\kappa t}{1}_{t<T\wedge \tau }$ and optimal stream rate of payment $i\left(t\right)=0$.

5. Conclusions

This study presents an innovative approach to modeling optimal consumption and investment in the presence of mortality risk, operating within a complete financial market framework under relaxed assumptions. It extends the framework of state-dependent utility theory, discounted by state–price processes, to compare consumption streams and incorporates life insurance as a hedging tool. A key element of our analysis is the concept of life insurance completeness, which plays a crucial role in hedging pooled mortality risks and enables consumers to optimize consumption even when facing negative wealth. Our research offers several important insights:

• Enhanced Valuation and Hedging: We extend valuation and hedging methodologies to a broad class of instruments that include mortality risk, such as prepaid insurance, and alleviate restrictive integrability requirements, thereby enriching the universe of allowable policies.
• Mortality Risk Management through Life Insurance Completeness: We introduce the concept of life insurance completeness, which encapsulates the ability of insurance companies to hedge mortality risk by pooling clients with similar mortality characteristics. This framework allows for the valuation and hedging of life insurance contracts within a complete financial market, enabling consumers to achieve optimal consumption and investment strategies.
• Feasibility of Optimal Consumption under Financial Constraints: We show that consumers can use state-dependent preferences to compare consumption streams and achieve optimal consumption, even when facing negative wealth, provided they have access to a robust life insurance market. This result is particularly relevant for understanding the behavior of individuals during the early stages of their productive lives, when debt accumulation is typical.
• Empirical Alignment with the Annuity Puzzle: By treating life insurance primarily as a tool to hedge mortality risk rather than as a bequest motive, our model aligns with empirical evidence showing that annuities (or other life insurance policies) are not necessary to achieve optimal consumption in the absence of a steady income stream.

The problem addressed in this paper assumes that mortality and preference structures are defined at time t = 0, when consumers determine their optimal “algorithm”, which adapts to incoming information at any future time without resetting the problem. Future research directions include exploring the dynamics of consumer behavior as mortality distributions and preference parameters evolve (e.g., due to aging or medical advancements). Also of interest are the implications of these dynamic approaches to consumption and investment for intertemporal equilibrium, both with and without production. For initial attempts in this latter direction without accounting for mortality risk, see (Londoño 2020, 2025). Finally, incorporating systematic or aggregate mortality risks (Milevsky and David Promislow 2001) presents an intriguing avenue for future research. These risks refer to factors (such as epidemics or medical technological innovations) that lead to changes in mortality distributions and cannot be diversified through the pooling of independent contracts. Of particular interest is the interaction between these systematic risks and financial markets, for which one can see (Deelstra et al. 2024) and references therein.