Optimal Consumption and Investment Problem Incorporating Housing and Life Insurance Decisions: The Continuous Time Case

• Published date: 01 March 2020
• Author: Shang‐Yin Yang,Ko‐Lun Kung
• DOI: http://doi.org/10.1111/jori.12270
• Date: 01 March 2020

OPTIMAL CONSUMPTION AND INVESTMENT PROBLEM

INCORPORATING HOUSING AND LIFE INSURANCE

DECISIONS:THE CONTINUOUS TIME CASE

Ko-Lun Kung

Shang-Yin Yang

ABSTRACT

This study considers the optimal consumption-investment-insurance

problem incorporating housing decisions of a household when interest

rates and labor income are stochastic. Under the complete market

assumption, we derive the closed-form solution of the optimal insurance

demand, portfolio choice, and housing consumption. We calibrate the model

using data from the financial market of Taiwan. We find that the insurer’s

pricing strategy has a significant impact on the household’s consumption

pattern. Specifically, additional loading in insurance premium allows the

life-cycle model to produce hump-shaped consumptions of both perishable

goods and housing. Loading also creates an unfair background risk to

households. However, we only find a small portfolio risk reduction, because

households optimally choose a large coverage to mitigate the mortality

exposure. This suggests empirical background risk studies overestimate the

risk reduction when insurance is available.

INTRODUCTION

Households are exposed to a mix of risks, each with some unique characteristics. This

study considers how a household should do when facing a variety of risks. Common

asset holdings, such as stocks, bonds, and residential properties (houses), are

subjected to unique shocks and correlated shocks across the multiple markets. The

most valuable asset for many households is human capital, that is, the present value of

Ko-Lun Kung is an Assistant Professor in the Department of Risk Management and

Insurance, Feng Chia University, Taichung, Taiwan. Kung can be contacted via e-mail:

klkung@mail.fcu.edu.tw. Shang-Yin Yang is an Associate Professor in the Department of

Finance, Tunghai University, Taichung, Taiwan. Yang can be contacted via e-mail:

shangyin@thu.edu.tw. We thank the editor and two anonymous referees, Marie Kratz,

Christian Gollier, and seminar participants at 2014 IME conference, 2014 APRIA conference,

and the Insurance Risk and Finance Research Conference, Nanyang Technology University for

their valuable comments. All remaining errors are our own. The authors acknowledge the

financial support from MOST (project numbers MOST 106-2410-H-035-043 and MOST 106-

2410-H-029-020).

©2019 The Journal of Risk and Insurance. Vol. 9999, No. 9999, 1–29 (2019).

DOI: 10.1111/jori.12270

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Vol. 87, No. 1, 143–171 (2020).

future labor income from the breadwinner’s work. If the breadwinner dies, the

household immediately loses a large portion of its wealth. As losing the largest

component of wealth would be financially disastrous for a family, it is natural that

households consider purchasing life insurance to hedge against the mortality risk.

The complexity of the risk faced by households poses a challenge to modeling of the

household’s life-cycle decisions (Campbell, 2006). Therefore, researchers sometimes

work with a more restrictive model. For example, life insurance is important for

households to ensure the family’s well-being when the human capital is subjected to

mortality risk, but the few articles that focus on the role of life insurance in the

household life-cycle model either incorporate a simplistic financial market or resort to

time-consuming numerical techniques to solve the model.

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On the other hand, there

are ample studies focus on the optimal consumption-portfolio choices in a more

realistic financial market and labor income setting, but they contradict the empirical

observation by neglecting the possibility of death or the life insurance at all, thus

omitting an important part of household decisions.

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Our article is an attempt to bridge this gap between the life insurance demand and

the life-cycle literature with a tractable model that considers the joint demands of

life insurance and financial assets. We solve a feature-rich life-cycle model that

extends Huang, Milesvky, and Wang (2008) and Kraft and Munk (2011). We

analyze the household’s optimization problem with an altruistic bequest motive in

the sense of Barro (1974) and Becker (1974). The breadwinner not only cared for the

household when she was alive, but also cared about the utility of rest of the family

(heirs) after she dies. The household thus has an incentive to leave a legacy to heirs

by purchasing life insurance. The household faces a realistic financial market that

allows for stochastic labor income, stochastic interest rate, and stochastic housing

price and rent. Our closed-form solutions for optimal perishable and housing

consumptions, portfolio choices, and life insurance purchase enable us to analyze

the interaction with mortality risk and multiple financial risks in the household’s

life-cycle decision.

We use the life-cycle model to emphasize how realistic pricing of insurance can affect

the portfolio choice and consumption behavior of a household, which is often

overlooked in the literature. In the real world, almost every life insurance product is

sold with an additional loading to cover the necessary administrative cost and profit.

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Richard (1975) and Campbell (1980) explore the optimal consumption-insurance problem.

Pliska and Ye (2007) extend Richard (1975) to allow random lifetime for breadwinner. Huang

and Milevsky (2008) and Huang, Milevsky, and Wang (2008) focus on HARA and CRRA risk

preferences, respectively. Pirvu and Zhang (2012) consider asset with mean-reverting return.

Love (2010) and Hambel et al. (2016) offer a realistic model in discrete time and numerically

solve the life insurance demand with the family structure.

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See Bodie, Merton, and Samuelson (1992), Cocco, Gomes, and Maenhout (2005), and Munk

and Sorensen (2010) for models with labor income. Kraft and Munk (2011) extend Munk and

Sorensen (2010) by including optimal housing decisions. More complex life-cycle models with

housing and labor income are often solved numerically, for example, Yao and Zhang (2005)

and van Hemert (2010).

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