Modelling the risk of underfunding in ALM models

Abstract

Asset and Liability Management (ALM) models have become well established decision tools for pension funds. ALMs are commonly modelled as multi-stage, in which a large terminal wealth is required, while at intermediate time periods, constraints on the funding ratio, that is, the ratio of assets to liabilities, are imposed. Underfunding occurs when the funding ratio is too low; a target value for funding ratios is pre-specified by the decision maker. The risk of underfunding has been usually modelled by employing established risk measures; this controls one single aspect of the funding ratio distributions. For example, controlling the expected shortfall below the target has limited power in controlling shortfall under worst-case scenarios. We propose ALM models in which the risk of underfunding is modelled based on the concept of Second Order Stochastic Dominance (SSD). This is a criterion of ranking random variables - in our case funding ratios - that takes the entire distributions of interest into account and works under the widely accepted assumptions of decision makers being rational and risk averse. In the proposed SSD models, investment decisions are taken such that the resulting short-term distribution of the funding ratio is non-dominated with respect to SSD, while a constraint is imposed on the expected terminal wealth. This is done by considering progressively larger tails of the funding ratio distribution and considering target levels for them; a target distribution is thus implied. Different target distributions lead to different SSD efficient solutions. Improved distributions of funding ratios may be thus achieved, compared to the existing risk models for ALM. This is the first contribution of this thesis. Interesting results are obtained in the special case when the target distribution is deterministic, specified by one single outcome. In this case, we can obtain equivalent risk minimisation models, with risk defined as expected shortfall or as worst case loss. This represents the second contribution. The third contribution is a framework for scenario generation based on the "Birth, Immigration, Death, Emigration" (BIDE) population model and the Empirical copula; the scenarios are used to evaluate the proposed models and their special cases both in-sample and out-of-sample. As an application, we consider the planning problem of a large DB pension fund in Saudi Arabia.