Dynamic asset–liability management in a Markov market with stochastic cash flows

Abstract

This paper provides a general model to investigate an asset–liability management (ALM) problem in a Markov regime-switching market in a multi-period mean–variance (M–V) framework. Emphasis is placed on the stochastic cash flows in both wealth and liability dynamic processes, and the optimal investment and liquidity management strategies in achieving the M–V bi-objective of terminal surplus are evaluated. In this model, not only the asset returns and liability returns, but also the cash flows depend on the stochastic market states, which are assumed to follow a discrete-time Markov chain. Adopting the dynamic programming approach, the matrix theory and the Lagrange dual principle, we obtain closed-form expressions for the efficient investment strategy. Our proposed model is examined through empirical studies of a defined contribution pension fund. In-sample results show that, given the same risk level, an ALM investor (a) starting in a bear market can expect a higher return compared to beginning in a bull market and (b) has a lower expected return when there are major cash flow problems. The effects of the investment horizon and state-switching probability on the efficient frontier are also discussed. Out-of-sample analyses show the dynamic optimal liquidity management process. An ALM investor using our model can achieve his or her surplus objective in advance and with a minimum variance close to zero.

Keywords:

• Stochastic cash flow
• Asset–liability management
• Multi-period mean–variance model
• Markov regime-switching
• Efficient investment strategy

JEL Classification:

• G11
• C61
• G23

Acknowledgement

We thank two anonymous referees for their insightful comments. This paper is supported by the National Natural Science Foundation of China (Nos. 71471045, 61472089), NSFC-Guangdong Joint Found (U1501254), the Natural Science Foundation of Guangdong Province: Great Incubation Project (No. 2014A030308008), the Science and Technology Planning Project of Guangdong Province (No. 2013B051000076), the Philosophy and Social Science Foundation of Guangzhou (No. 14G42), the China Postdoctoral Science Foundation (Nos. 2014M560658, 2015T80896), the Key Technology Research and Development Programs of Guangdong Province, the Characteristic and Innovation Foundation of Guangdong Colleges and Universities (Humanity and Social Science Type), the Humanities and Social Science Research Foundation of the National Ministry of Education of China (No. 15YJAZH051) and the Hong Kong RGC under grants 519913, 15209614 and 15224215.

Notes

¹ See for example, Bajeux-Besnainou and Portait (1998), Li and Ng (2000), Zhu et al. (2004), Bielecki et al. (2005), Fu et al. (2010), Chiu and Wong (2011), Vigna (2014) and Cui et al. (2014).

² For example, Leippold et al. (2004, 2011), Chen and Yang (2011), Chiu and Wong (2012) and Yao et al. (2013b) have adopted dynamic M–V frameworks to study ALM problems. Another crucial branch of ALM is in the expected utility maximization framework. Please see Rudolf and Ziemba (2004) and Hoevenaars et al. (2008) and references therein.

³ Some of these applications include, but are not limited to, Zhou and Yin (2003) for a continuous-time Markowitz M–V problem; Costa and Araujo (2008) and Costa and Oliveira (2012) for multi-period M–V portfolio selection problems; Çanakoğlu and Özekici (2010) for multi-period HARA utility maximization portfolio selection; Liu (2011) for continuous-time utility maximization portfolio choice; Jiang and Pistorius (2012) for the optimal dividend distribution problem; Elliott and Siu (2011) for American options pricing; and Gourieroux et al. (2014) for bond pricing. For parameter estimation in the regime-switching model, Hardy (2001) develops a maximum likelihood estimation method using the data regarding S&P 500 and TSE 300 indices.

⁴ In this study, we focus on the optimal liquidity level. Regarding the influence of market frictions on the capital structure, please see Leary and Roberts (2005) and Tsyplakov (2008).

⁵ We also select different investment periods as investment horizon analysis in the next subsections.

⁶ The average interest rates are calculated on the total un-matured interest-bearing debt. See, http://www.treasurydirect.gov/govt/rates/pd/avg/avg.htm.

⁷ We tried alternative data sets by selecting other risky assets. Results are upon requests. However, we argue that this result does not affect the conclusions in the following parts.

⁸ Please see Remark 1 (Section 2), we give some specific examples on different cases of cash flows that an ALM investor could meet.