Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time

Wei Liu,Youfa Sun,Xu Chen

doi:10.1515/math-2022-0007

Wei Liu, Youfa Sun + Show 1 more

Open Access

https://doi.org/10.1515/math-2022-0007

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Abstract

Abstract The asset-liability management problem with cash flow under an uncertain exit time has been investigated in this article, which is based on the fundamental framework of the mean-variance model in the multi-period version. The liability and random cash flow will affect asset optimization, while the investor may be forced to withdraw from investments with a random probability at each period in our model. The closed-form expressions for the mean-variance optimal portfolio selection and its corresponding efficient frontier are obtained by employing the mean-field formulation and dynamic programming approach. Moreover, some numerical examples are provided to illustrate the validity and accuracy of the theoretical results.

Highlights

With an explosive development of the economy in the recent 50 years, it is becoming more common that lots of private assets have been invested in the financial market
The optimal strategy of the model in Corollary 1 can be obtained according to Theorem 1, which is consistent with the results derived by Yao et al [11]
According to the data given in the study by Elton et al [17], we investigate a portfolio selection consisting of S&P 500 (SP), the index of emerging market (EM), and small stock (MS) of the U.S market

Summary

Introduction

With an explosive development of the economy in the recent 50 years, it is becoming more common that lots of private assets have been invested in the financial market. The mean-field formulation is a simple but powerful tool to derive the optimal strategy of a multi-period mean-variance portfolio selection problem. Yi et al [9] used the mean-field method to study the meanvariance model under the uncertain exit time condition but did not consider the cash flow and liability. Li and Xie [12] studied the optimal investment with stochastic income under the uncertain exit time They derived the analytical optimal strategy and explicit expression of the efficient frontier by using the Lagrange method and traditional dynamic programming with the additional conditions of endogenous liabilities if the investors exit the market randomly. Since the smoothing property is no longer valid on the variance term, we cannot decompose the nonseparable problem into a stage wise backward recursion formulation, which can be tackled with traditional dynamic programming method We solve it by employing the mean-field method

Mean-field formulation

The optimal strategy

Numerical example

Conclusion