Knightian uncertainty embedded in stock returns causes rising demand for life insurance, as the uncertainty averse agent seeks alternative investment channels. Life insurance demand of middle-aged agent is more sensitive to the uncertainty. Stock return uncertainty reduces the agent’s total wealth and subsequently the propensity of wealthy agent serving as an insurance seller. Rising demand and falling supply of life insurance imply that life insurance is more expensive in the presence of stock return uncertainty. Sensitivity of life insurance demand to the mortality rate and key stock return characteristics also changes with the uncertainty.

Insurance market and stock market are closely connected in many ways [

We find that optimal insurance demand increases with the level of uncertainty. The agent shifts some of his investment in the stock to life insurance, confirming that the insurance market and the stock market, to some degree, substitute each other. Life insurance is used as a way to circumvent the uncertainty embedded in the stock. This effect is more prominent for the middle-aged agent. Younger agent’s demand for life insurance is low in the first place. Elder agent consumes more, repressing demand for life insurance when it is close to the end of his financial planning horizon. As a result, the demand of younger and elder agents for life insurance is less sensitive to stock return uncertainty.

When the agent is endowed with a sufficiently high level of initial wealth, he optimally supplies insurance [

The sensitivity of insurance demand with respect to the mortality rate may change in the presence of stock return uncertainty. In the absence of uncertainty, an increase in the mortality rate leads to lower insurance demand. However, facing stock return uncertainty, the agent might demand more for insurance as the mortality rate increases. The rationale is that the mortality rate plays two roles in insurance decision making. On the one hand, it adversely affects the insurance payout ratio, so a higher mortality rate reduces the utility brought by life insurance. On the other hand, the mortality rate affects the probability of the agent obtaining life insurance payment within the financial planning horizon; thereby, the agent with a higher mortality rate is more willing to buy life insurance because there is a greater chance to receive the payment. When stock return uncertainty is low, the first effect dominates, while when the uncertainty is sufficiently high, the second effect is more prominent.

Our discoveries echo the phenomena found in empirical research, especially on the relationship between the stock market and the insurance market. Jawadi et al. [

Our work contributes uniquely to the life insurance literature [

The remainder of the paper is organized as follows:

This section introduces our model. It first describes the economy, followed by stock return uncertainty and the agent’s objective function that admits stock return uncertainty.

Consider a simple continuous-time rational expectations model as in Merton [

Life insurance provides a lump sum payment when the agent deceases. Following Richard [

We do not require

Based on the above expressions, the mortality rate is given by

Rewrite the mortality rate

The life insurance premium

Expected stock returns are difficult to estimate [

Under Novikov’s condition,

The agent also receives labor income

The agent’s utility consists of two parts: utility

The uncertainty averse agent optimizes his utility based on the worse-case alternative model by solving the following Max-Min problem:

The third term of equation (

In the presence of stock return uncertainty, the agent’s utility function is

Substituting equations (

Let

This section presents the solutions to the model with general utility and then the solutions to the CRRA utility model.

We use the dynamic programming method to obtain the Hamilton–Jacobi–Bellman (HJB) equation. According to the optimality principle, the value function in equation (

By Ito’s lemma, the Dynkin operator follows that

We first solve the minimization part of equation (

For an uncertainty averse agent with a general utility

The optimal insurance demand

The agent balances the optimal consumption

When the agent has a CRRA utility function,

Equation (

Substitute equations (

We identify and distinguish three types of wealth: the current wealth, the labor wealth, and the total wealth. We will solve the problems above using these definitions. Denote

The agent’s current wealth

We conjecture the objective function as

Based on the above results, we have the following proposition.

For the uncertainty averse agent with CRRA utility

The corresponding legacy is

Using

According to equation (

Substitute

Given

This ODE has a general solution. The boundary condition that

Note

To study the effect of uncertainty embedded in stock returns on insurance investment decisions at different ages, we consider the path-dependent wealth dynamics. According to equation (

We obtain the following proposition.

The CRRA agent has the following wealth process:

Substitute

Using equation (

Substituting equation (

Matching equation (

Thus,

We have proved that the solution to the total wealth satisfies the differential function in equation (

When

If the agent’s total wealth exceeds a certain threshold, that is,

It is sensible that the agent does not leave a legacy

This section estimates wealth-measured utility loss when the agent follows a suboptimal strategy; that is, he applies the optimal strategy under the uncertainty-free model in the uncertainty aversion model. It provides a yardstick to measure the improvement in utility by undertaking robust optimal decisions. To more intuitively illustrate the utility loss, we use the indifference curve to transform the implied utility loss into percentage of wealth.

We solve equations (

The suboptimal control functions are used in the HJB function in equation (

Solving the model yields the worst-case adjustment to the expected stock return:

Following Branger and Larsen [

Using the definition of utility loss and equation (

This section conducts numerical analysis to examine the qualitative implications of stock return uncertainty. The benchmark parameter values are given in Table

Benchmark parameter values.

Parameter | Value |
---|---|

0.1 | |

0 | |

0.04 | |

0.2 | |

0.03 | |

4 | |

80 |

The parameter values in Table

Figure

Total wealth at different ages.

Change in total wealth with

Figure

Optimal consumption.

Figure

Optimal investment.

Figure

Demand of insurance at different levels of uncertainty.

Figure

Demand of insurance for different financial planning horizons. (a) Insurance demand (

Lemma

Buying vs. selling insurance.

Figure

Supply of insurance and

Stock market uncertainty also remarkably affects the relationships between the agent’s life insurance decision and other structural factors. Figures

Insurance demand and

Insurance demand and

Figure

Insurance demand and

Figure

Insurance demand and

Figure

Uncertainty-induced utility loss.

This paper formulates a continuous-time rational expectations model to examine the effects of stock return uncertainty on life insurance. The model considers uncertainty aversion as the agent suspects that stock return in the stochastic process is potentially misspecified, and it imposes an uncertainty penalty to the objective function in reflecting his skeptical and conservative perspective. Facing stock return uncertainty, the agent shifts some of the investment in the stock to life insurance, confirming that the insurance market and the stock market are partially supplementary. Life insurance is used as a way to circumvent the uncertainty embedded in the stock. Overall, the agent would behave more conservatively in the insurance market facing stock return uncertainty. Agents would demand more insurance at a falling supply, implying that insurance premium might increase in equilibrium. We leave it to future research to develop an equilibrium model to explore such implications rigorously.

All data generated or analyzed during this study are included in this article.

The authors declare that they have no conflicts of interest.

The authors acknowledge financial supports from the National Natural Science Foundation of China (No. 71471099) and Tsinghua University Research Grant (No. 2019THZWLJ14).