Optimal life insurance with no-borrowing constraints: duality approach and example

Abstract

We solve an optimal portfolio choice problem under a no-borrowing assumption. A duality approach is applied to study a family’s optimal consumption, optimal portfolio choice, and optimal life insurance purchase when the family receives labor income that may be terminated due to the wage earner’s premature death or retirement. We establish the existence of an optimal solution to the optimization problem theoretically by the duality approach and we provide an explicitly solved example with numerical illustration. Our results illustrate that the no-borrowing constraints do not always impact the family’s optimal decisions on consumption, portfolio choice, and life insurance. When the constraints are binding, there must exist a wealth depletion time (WDT) prior to the retirement date, and the constraints indeed reduce the optimal consumption and the life insurance purchase at the beginning of time. However, the optimal consumption under the constraints will become larger than that without the constraints at some time later than the WDT.

Keywords:

• portfolio choice
• constraint
• life insurance
• martingale
• duality approach

JEL Classifications:

• D91
• G11

Acknowledgements

We would like to thank Moshe A. Milevsky and the seminar participants for their helpful comments. All errors are solely our responsibility.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

Preliminary results of the present paper with title Optimal Consumption and Portfolio Choice with Life Insurance under Uncertainty and Borrowing Constraints were reported at the 16th International Congress on Insurance: Mathematics and Economics, June 28–30, 2012, Hong Kong; SIAM Conference on Financial Mathematics and Engineering (FM12), July 9–11, 2012, Minneapolis, USA.

Additional information

Funding

The present paper is partially supported by NSFC [grant number 71271127/G0115].