Insurance: Mathematics and Economics

TLDR

This paper investigates a continuous‑time mean‑variance asset‑liability management problem in which all assets can be risky and liabilities are endogenous and controllable, making the analysis more challenging than models with exogenous liabilities. The study aims to establish existence and uniqueness of the Riccati‑type equation solution and to derive closed‑form efficient strategies and the mean‑variance frontier for this asset‑liability model. The authors employ the Khatri‑Rao product technique, stochastic control theory, Lagrange multipliers, the Hamilton‑Jacobi‑Bellman equation, and illustrate the results with numerical examples.

Abstract

a b s t r a c t This paper investigates a continuous-time mean-variance asset-liability management problem with endogenous liabilities in a more general market where all the assets can be risky. Different from exogenous liabilities that cannot be controlled, the endogenous liabilities can be controlled by various financial instruments and investors' decisions. For example, a company can raise fund by issuing different kinds of bonds. Types and quantities of the bonds are controlled by the company itself. Investors optimize allocation not only for their assets, but also for their liabilities under our model. This makes the analysis of the problem more challenging than in the setting based on exogenous liabilities. In this paper, we first prove the existence and uniqueness of the solution to the associated Riccati-type equation by using the Khatri-Rao product technique and the relevant stochastic control theory; we then derive closed form expressions of the efficient strategy and the mean-variance efficient frontier by using the Lagrange multiplier method and the Hamilton-Jacobi-Bellman equation approach, and we next discuss two degenerated cases; finally, we present some numerical examples to illustrate the results obtained in this paper.