Abstract
This entry considers the problem of a typical pension fund that collects premiums from sponsors or employees and is liable for fixed payments to its customers after retirement. The fund manager’s goal is to determine an investment strategy so that the fund can cover its liabilities while minimizing contributions from its sponsors and maximizing the value of its assets. We develop robust optimization and scenario-based stochastic programming approaches for optimal asset-liability management, taking into consideration the uncertainty in asset returns and future liabilities. Our focus is on computational tractability and ease of implementation under conditions typically encountered in practice, such as asymmetries in the distributions of asset returns. Computational results from tests with real and generated data are presented to illustrate the performance of these models.
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Appendix
Appendix
Proof of Theorem 1.
Let us derive the robust counterpart of the first constraint in the optimization problem formulation \( (\mathcal{P}_{sym}^{R}) \). The derivation of the robust counterparts of the remaining constraints is similar. The first constraint can be rewritten as
The robust counterpart of the original constraint can be written as
Intuitively, we would like the constraint to be satisfied even if the uncertain coefficients take their worst-case values within the uncertainty set. In this case, the worst-case value of the uncertain expression is obtained when \( \tilde{\boldsymbol{\alpha }}'\boldsymbol{\kappa } \) is at its minimum value for values of \( \tilde{\boldsymbol{\alpha }} \) within the uncertainty set. (If it was at its maximum value, it would be easier, not harder, for the inequality to be satisfied because of the negative sign in front of the vector product.) We solve the inner minimization problem first. The inner problem is in the explicit form
Let π ≥ 0 be a Lagrangian multiplier for the constraint in the optimization problem above. The Lagrangian function is
The first-order optimality condition is
and the complementarity condition is
Using the optimality conditions stated in (4.23) and (4.24), we can find the optimal value of the random parameter within the uncertainty set. Note that in (4.24), π ≠ 0. In addition,
From (4.23) and (4.25), we obtain
In addition, it can be easily shown that
Using (4.27) and (4.25), the Lagrangian multiplier is found as \( \pi = \sqrt{\boldsymbol{\kappa }'\boldsymbol{\Xi }^{o}\boldsymbol{\kappa }} \). Substituting this in (4.23) confirms that the optimal solution of the inner problem is
The robust counterparts of the other constraints that contain uncertain coefficients, namely the liability and the cash balance constraints, can be derived in a similar manner. As a result, the robust model for \( (\mathcal{P}^{R}) \) can be obtained as stated in Theorem 1.
Proof of Theorem 2.
The theorem follows from (4.17) and the fact that, given the representation of the uncertain coefficients as linear combinations of factors, the constraints can be written in the form (4.15). To see how the constraint \( \tilde{\boldsymbol{\alpha }}'\boldsymbol{\kappa } -\lambda W_{T}\gamma _{T}-\nu \geq 0 \), for example, can be written in the form (4.15), note that it can be written in terms of the uncertain factors \( \tilde{\mathbf{z}}^{\alpha } \) as
Given (4.17), the robust counterpart of the constraint
when the uncertain factors \( \tilde{\mathbf{z}}^{\alpha } \) vary in uncertainty set \( \mathcal{A}\mathcal{U}^{o} \) is
The robust counterparts of the remaining constraints can be derived in a similar manner.
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Pachamanova, D., Gülpınar, N., Çanakoğlu, E. (2017). Robust Approaches to Pension Fund Asset Liability Management Under Uncertainty. In: Consigli, G., Kuhn, D., Brandimarte, P. (eds) Optimal Financial Decision Making under Uncertainty. International Series in Operations Research & Management Science, vol 245. Springer, Cham. https://doi.org/10.1007/978-3-319-41613-7_4
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