Pension Design and Risk Sharing: Mixed Solutions Between Defined Benefit and Defined Contribution for Public Pension Schemes

Abstract

In classical pension design, there are essentially two kinds of pension schemes: Defined Benefit (DB) and Defined Contribution (DC) plans. Each scheme corresponds to a different philosophy of spreading risk between the stakeholders: in a DB, the main risks are taken by the organizer of the plan, while in a DC, the affiliates must bear all the risks. Especially when applied to social security pension systems, this traditional view can in both cases lead to unfair intergenerational equilibrium. The purpose of this chapter is to present alternative architectures based on a mix between DB and DC in order to achieve both financial sustainability and social adequacy. An example of this approach is the so-called Musgrave rule, but other risk-sharing approaches will be developed in a pay-as-you-go philosophy. These principles will be illustrated by the Belgian proposition of reform of the first pillar, based on a points system with a simultaneous automatic adaptation mechanism of the retirement age, the contribution rate, the replacement rate, and the indexation rate.

Appendix: Another Family of Pension Schemes Defined through a Convex Invariant

The family of pension plans (indexed by parameter α) defined in Eq. (14.22) can be viewed as a ‘line segment’ linking DC (corresponding to the beginning of the segment: α = 0) to DB (corresponding to the end of the segment: α = 1). Of course, the straight line is not the only way of connecting these two points: we can obtain infinitely many such curves. Formally, any regular function

$$ f:{\left(0,1\right)}^2\times \left[0,1\right]\to \left(0,1\right):\left(x,y,\alpha \right)\mapsto {f}_{\alpha}\left(x,y\right) $$

such that f₀(x, y) = y and f₁(x, y) = x leads to a family of pension plans by defining, for some fixed α,

$$ f\left({\delta}_t,{\pi}_t,\alpha \right)={C}_{\alpha} $$

for some constant C_α ∈ (0, 1). Of course, we get back to Eq. (14.22) choosing f(x, y, α) = αx + (1 − α)y.

Defining f(x, y, α) = x^αy^1 − α leads to an interesting family of pension plans:

$$ {\delta}_t^{\alpha}\ {\pi}_t^{1-\alpha}={C}_{\alpha}\kern1em \iff \kern0.75em \alpha\;\log\;{\delta}_t+\left(1-\alpha \right)\;\log {\pi}_t=\log\;{C}_{\alpha}. $$

The result is a log-linear combination of the replacement and contribution rates, i.e. the equivalent of (14.22) for the logarithm of the rates instead of the rates themselves.

Similarly to (14.23), in this case, we obtain from the budget equation the following expressions for the contribution and replacement rates:

$$ \left\{\begin{array}{l}{\pi}_t={\delta}_t\ {D}_t\\ {}{\delta}_t^{\alpha}\ {\pi}_t^{1-\alpha}={C}_{\alpha}\end{array}\Rightarrow \left\{\begin{array}{l}{\delta}_t={C}_{\alpha}{D}_t^{\alpha -1}\\ {}{\pi}_t={C}_{\alpha}{D}_t^{\alpha}\end{array}.\right.\right. $$