Asset and liability management under a continuous-time mean-variance optimization framework. (English) Zbl 1151.91493

Summary: Asset and liability (AL) management under the mean–variance criteria refers to an optimization problem that maximizes the expected final surplus subject to a given variance of the final surplus or, equivalently, minimizes the variance of the final surplus subject to a given expected final surplus. We employ stochastic optimal control theory to analytically solve the AL management problem in a continuous-time setting. More specifically, we derive both the optimal policy and the mean–variance efficient frontier by a stochastic linear quadratic control framework. Then, the quality of the derived optimal AL management policy is examined by comparing it with those in the literature. We further discuss consequences of a discrepancy in objectives between equity holders and investors of a mutual fund. Finally, the optimal funding ratio, i.e., the wealth-to-liability ratio, is determined.


91G10 Portfolio theory
93E20 Optimal stochastic control
Full Text: DOI


[1] Bielecki, T.R.; Jin, H.; Pliska, S.R.; Zhou, X.Y., Continuous-time mean – variance portfolio selection with bankruptcy prohibition, Mathematical finance, 15, 213-244, (2005) · Zbl 1153.91466
[2] Chen, A.; Jen, C.; Zionts, S., The optimal portfolio revision policy, Journal of business, 44, 51-61, (1971)
[3] Josa-Fombellida, R.; Rincon-Zapatero, J.P., Optimal risk management in defined benefit stochastic pension funds, Insurance: mathematics and economics, 34, 489-503, (2004) · Zbl 1188.91202
[4] Kell, A.; Müller, H., Efficient portfolios in the asset liability context, Astin bulletin, 25, 33-48, (1995)
[5] Li, D.; Chan, T.F.; Ng, W.L., Safety-first dynamic portfolio selection, Dynamics of continuous, discrete and impulsive systems, 4, 585-600, (1998) · Zbl 0916.90023
[6] Li, D.; Ng, W.L., Optimal dynamic portfolio selection: multiperiod mean – variance formulation, Mathematical finance, 10, 387-406, (2000) · Zbl 0997.91027
[7] Leippold, M.; Trojani, F.; Vanini, P., A geometric approach to multiperiod Mean variance optimization of assets and liabilities, Journal of economic dynamics & control, 28, 1079-1113, (2004) · Zbl 1179.91234
[8] Markowitz, H.M., The optimization of a quadratic function subject to linear constraints, Naval research logistics quarterly, 3, 111-133, (1956)
[9] Norberg, R., Ruin problems with assets and liabilities of diffusion type, Journal of stochastic processes and their applications, 81, 255-269, (1999) · Zbl 0962.60075
[10] Sharpe, W.F.; Tint, L.G., Liabilities-a new approach, Journal of portfolio management, 16, 5-10, (1990)
[11] Yong, J.; Zhou, X.Y., Stochastic controls: Hamiltonian systems and HJB equations, (1999), Springer-Verlag · Zbl 0943.93002
[12] Zhou, X.Y.; Li, D., Continuous-time mean – variance portfolio selection: A stochastic LQ framework, Applied mathematics and optimization, 42, 19-33, (2000) · Zbl 0998.91023
[13] Zhu, S.S.; Li, D.; Wang, S.Y., Risk control over bankruptcy in dynamic portfolio selection: A generalized mean – variance formulation, IEEE transactions on automatic control, 49, 447-457, (2004) · Zbl 1366.91150
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.