Dynamic Portfolio Choice with Stochastic Wage and Life Insurance

Abstract

We study optimal insurance, consumption, and portfolio choice in a framework where a family purchases life insurance to protect the loss of the wage earner's human capital. Explicit solutions are obtained by employing constant absolute risk aversion utility functions. We show that the optimal life insurance purchase is not a monotonic function of the correlation between the wage and the financial market. Meanwhile, the life insurance decision is explicitly affected by the family's risk preferences in general. The model also predicts that a family uses life insurance and investment portfolio choice to hedge stochastic wage risk.

Notes

The optimal investment under CARA utility functions generally is a constant independent of the wealth level, which is a limitation of CARA utility functions.

Table 1 of Huang and Milevsky (2008) shows that the life insurance purchases I*_t/λ_{y + t} under the cases ρ = −0.5, 0, 0.5 are different for varying risk aversion (where ρ is the correlation between the financial market and the wage). The case c_l = c_d = 0 corresponds to the result in Table 1 of Huang et al. (2008), which shows the purchases are identical across risk aversion.

We solve for this case in Section A.4 of the Appendix. The results show that allowing for the purchase of life insurance after the wage earner has retired will only slightly affect the optimal policies before the retirement date, which is the time period of primary interest in this article, while all patterns found in the current setting will still hold.

The terminal utility here stands for a wealth constraint, which is necessary for a valid consumption/portfolio/insurance strategy when negative exponential utility functions are employed and the time horizon is finite. If power utility functions are employed or if the time horizon is infinite, wealth constraints are imposed implicitly, and a terminal utility may not be needed explicitly.

With similar analysis, it is interesting to observe that the life insurance amount may be a monotonically increasing function of ρ if the risk-free return r is greater than the risky return μ, because may be lower than − 1 and is therefore increasing over [ − 1, 1] as a convex quadratic function of ρ, given r > μ. However, the assumption r > μ is not reasonable in reality.

In fact, we can find it directly that , where the const and are independent of γ, given ρ = ±1 or σ_Y = 0.

Although we further discuss the case of γ₁ ≠ γ₂ below, detailed discussion on this issue here is beyond the scope of this article.

If σ_Y = 0, there is no risk from labor income; i.e., the labor income is spanned by the risk-free asset.

It is called the “positive wealth effect,” which is discussed in empirical studies including Gandolfi and Miners (1996) and Eisenhauer and Halek (1999).

We note that for other ages not shown in Table 1 that the amount spent on life insurance might be shifted for different sets of parameters of the model. But we shall expect the same pattern, i.e., that families with middle-age wage earners spend more on life insurance in a constant absolute risk aversion world. We also check an alternative mortality rate, λ_n(t) = 0.005 + 0.001125(n + t), used in Pliska and Ye (2007), and we observe the same pattern.

We suggest that the setting of the planning time horizon is what gives the death benefit this feature of a risky asset. We note that for different planning horizons other than T, the argument does not apply because the death benefit is received at the end of time and no further planning is needed. Under such settings, the relation may be positive, as in Bayraktar and Young (2013) who use death as the terminal horizon.