Academic journal article Journal of Risk and Insurance

Portfolio Choice and Life Insurance: The CRRA Case

Academic journal article Journal of Risk and Insurance

Portfolio Choice and Life Insurance: The CRRA Case

Article excerpt

ABSTRACT

We solve a portfolio choice problem that includes life insurance and labor income under constant relative risk aversion (CRRA) preferences. We focus on the correlation between the dynamics of human capital and financial capital and model the utility of the family as opposed to separating consumption and bequest. We simplify the underlying Hamilton--Jacobi--Bellman equation using a similarity reduction technique that leads to an efficient numerical solution. Households for whom shocks to human capital are negatively correlated with shocks to financial capital should own more life insurance with greater equity/stock exposure. Life insurance hedges human capital and is insensitive to the family's risk aversion, consistent with practitioner guidance.

INTRODUCTION AND MOTIVATION

There is a glaring disconnect between the way most financial advisers sell life insurance and how they sell or promote investment products such as mutual funds, stocks, or bonds. Aside from the regulatory environment and the different licenses required--namely, securities as opposed to insurance--these two financial decisions are presented as if there were a "separation theorem" that justified their relative invariance. Moreover, although the concepts of risk tolerance, risk aversion, and utility are ubiquitous in the lingo and marketing material of the securities industry, the same terminology rarely enters the dialogue in the life insurance arena. This is quite odd since historically the economics of insurance was the breeding ground for much of the development in utility theory.

This observation is more than just anecdotal. The insurance literature's starting point for the optimal face amount of life insurance is the original work by Solomon Huebner in the early 1900s, based on the concept of a human life value (HLV). This idea is also at the heart of much forensic economics and litigation, where the courts must determine the value of a lost life, for the purposes of compensation. See Zietz (2003) or Todd (2004) for a comprehensive review of the academic literature on life insurance. Most financial planning and investment textbooks advocate something called a needs analysis, where the family determines how much would be required to meet long-term expenses and other financial goals if the primary source of income were lost. Other authors focus on the present value of lost wages. Regardless of the precise mechanics, rarely is this decision presented as a portfolio allocation problem akin to the investment in stocks and bonds. We believe it should. After all, life insurance is a hedge against the loss of human capital and portfolio hedging decisions should be made jointly, not independently.

Therefore, our objective in this article is to jointly analyze the decision of how much life insurance a family unit should have to protect against the loss of its breadwinner as well as how the family should allocate its financial resources between risk-free and risky assets vis-a-vis the dynamics of labor income and human capital in our (1) model.

We view this problem within the paradigm of portfolio choice and use the tools of financial economics pioneered by Merton (1969, 1971) to arrive at optimal controls for investment, consumption, and life insurance--but where the family unit is placed front and center.

A number of recent articles have extended the set of decisions included in the portfolio choice problem to highlight the interaction and the risks faced by the household, broadly defined. Thus, for example, Goetzmann (1993), Yao and Zhang (2005), and as Cocco (2005) focus on the role of the housing portfolio; Campbell and Cocco (2003) focus on optimal mortgage choices; Cairns, Blake, and Dowd (2006) examine portfolio choice in defined contribution pension plans; Sundaresan and Zapatero (1997) examine the role of defined benefit pensions; whereas Dybvig and Liu (2004) and Bodie et al. …

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