Robust Approaches to Pension Fund Asset Liability Management Under Uncertainty

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.

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

$$ \displaystyle{\nu -\tilde{\boldsymbol{\alpha }}'\boldsymbol{\kappa } +\lambda \gamma _{T}W_{T} \leq 0.} $$

The robust counterpart of the original constraint can be written as

$$ \displaystyle{\nu -\text{Min}_{\tilde{\boldsymbol{\alpha }}\in \mathcal{S}\mathcal{U}^{o}}\left \{\tilde{\boldsymbol{\alpha }}'\boldsymbol{\kappa }\right \} +\lambda \gamma _{T}W_{T} \leq 0} $$

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

$$ \displaystyle{\begin{array}{rl} \text{Min}_{\tilde{\boldsymbol{\alpha }}}&\tilde{\boldsymbol{\alpha }}'\boldsymbol{\kappa } \\ \text{s.t.}&\left \|\left (\boldsymbol{\Xi }^{o}\right )^{-\frac{1} {2} }\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}]\right )\right \|_{2} \leq \varOmega ^{o}\end{array} } $$

Let π ≥ 0 be a Lagrangian multiplier for the constraint in the optimization problem above. The Lagrangian function is

$$ \displaystyle{ \mathcal{L}(\tilde{\boldsymbol{\alpha }},\pi ) =\tilde{\boldsymbol{\alpha }} '\boldsymbol{\kappa } +\pi \left (\left \|\left (\boldsymbol{\Xi }^{o}\right )^{-\frac{1} {2} }\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}]\right )\right \|_{2} -\varOmega ^{o}\right ). } $$

The first-order optimality condition is

$$ \displaystyle{ \frac{\partial \mathcal{L}} {\partial \tilde{\boldsymbol{\alpha }}} =\boldsymbol{\kappa } + \frac{\pi } {\left \|\left (\boldsymbol{\Xi }^{o}\right )^{-\frac{1} {2} }\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}]\right )\right \|_{2}}\left (\boldsymbol{\Xi }^{o}\right )^{-1}\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}]\right ) = 0 } $$

(4.23)

and the complementarity condition is

$$ \displaystyle{ \pi \left (\left \|\left (\boldsymbol{\Xi }^{o}\right )^{-\frac{1} {2} }\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}]\right )\right \|_{2} -\varOmega ^{o}\right ) = 0 } $$

(4.24)

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,

$$ \displaystyle\begin{array}{rcl} \left \|\left (\boldsymbol{\Xi }^{o}\right )^{-\frac{1} {2} }\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}]\right )\right \|_{2} =\varOmega ^{o}& &{}\end{array} $$

(4.25)

From (4.23) and (4.25), we obtain

$$ \displaystyle{ \left (\boldsymbol{\Xi }^{o}\right )^{-1}\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}]\right ) = -\frac{\varOmega ^{o}} {\pi } \boldsymbol{\kappa } \Rightarrow \tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}] = \left (-\frac{\varOmega ^{o}} {\pi } \right )\boldsymbol{\Xi }^{o}\boldsymbol{\kappa }. } $$

(4.26)

In addition, it can be easily shown that

$$ \displaystyle\begin{array}{rcl} \left \|\left (\boldsymbol{\Xi }^{o}\right )^{-\frac{1} {2} }\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}]\right )\right \|_{2}& =& \sqrt{\left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}] \right ) \left (\boldsymbol{\Xi }^{o } \right ) ^{-1 } \left (\tilde{\boldsymbol{\alpha }}-\mathrm{E}[\tilde{\boldsymbol{\alpha }}] \right )} \\ & =& \sqrt{\left (-\frac{\varOmega ^{o } } {\pi } \boldsymbol{\Xi }^{o}\boldsymbol{\kappa }\right )'\left (\boldsymbol{\Xi }^{o}\right )^{-1}\left (-\frac{\varOmega ^{o}} {\pi } \boldsymbol{\Xi }^{o}\boldsymbol{\kappa }\right )} \\ & =& \frac{\varOmega ^{o}} {\pi } \sqrt{\boldsymbol{\kappa }'\boldsymbol{\Xi }^{o}\boldsymbol{\kappa }}. {}\end{array} $$

(4.27)

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

$$ \displaystyle{ \tilde{\boldsymbol{\alpha }}'\boldsymbol{\kappa } =\mathrm{ E}[\tilde{\boldsymbol{\alpha }}]'\boldsymbol{\kappa } -\varOmega ^{o}\sqrt{\boldsymbol{\kappa }'\boldsymbol{\Xi }^{o}\boldsymbol{\kappa }}. } $$

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

$$ \displaystyle{ \tilde{\boldsymbol{\alpha }}'\boldsymbol{\kappa } -\lambda W_{T}\gamma _{T}-\nu =\mathop{\underbrace{ \hat{\boldsymbol{\alpha }}'\boldsymbol{\kappa } -\lambda W_{T}\gamma _{T}-\nu }}\limits _{f_{0}(\boldsymbol{\kappa },\gamma _{T})} +\sum _{ j=1}^{M+T}\mathop{\underbrace{\mathbf{e}'_{ j}\left (\boldsymbol{\Delta }^{\alpha }\right )'\boldsymbol{\kappa }}}\limits _{f_{j}(\boldsymbol{\kappa })} \cdot \tilde{z}_{j}^{\alpha } } $$

(4.28)

Given (4.17), the robust counterpart of the constraint

$$ \displaystyle{\tilde{\boldsymbol{\alpha }}'\boldsymbol{\kappa } -\lambda W_{T}\gamma _{t}-\nu \geq 0} $$

when the uncertain factors $\tilde{\mathbf{z}}^{\alpha }$ vary in uncertainty set $\mathcal{A}\mathcal{U}^{o}$ is

$$ \displaystyle\begin{array}{rcl} \hat{\boldsymbol{\alpha }}'\boldsymbol{\kappa } -\lambda W_{T}\gamma _{t}-\nu & \geq & \varOmega ^{o}\vert \vert \mathbf{u}^{\alpha }\vert \vert + \left (\mathbf{r}^{\alpha }\right )'\left (\overline{\mathbf{z}}^{\alpha }\right ) + \left (\mathbf{s}^{\alpha }\right )'\left (\underline{\mathbf{z}}^{\alpha }\right ) {}\\ u_{j}^{\alpha }& \geq & -p_{ j}\left (\mathbf{e}_{j}'\left (\boldsymbol{\Delta }^{\alpha }\right )'\boldsymbol{\kappa } + r_{ j}^{\alpha } - s_{ j}^{\alpha }\right ),\quad j = 1,\ldots,M + T {}\\ u_{j}^{\alpha }& \geq & q_{ j}\left (\mathbf{e}_{j}'\left (\boldsymbol{\Delta }^{\alpha }\right )'\boldsymbol{\kappa } + r_{ j}^{\alpha } - s_{ j}^{\alpha }\right ),\quad j = 1,\ldots,M + T {}\\ \mathbf{r}^{\alpha },\mathbf{s}^{\alpha }& \geq & 0 {}\\ \end{array} $$

The robust counterparts of the remaining constraints can be derived in a similar manner.