An improved closed-form solution for the constrained minimization of the root of a quadratic functional

The problem of minimizing the root of a quadratic functional, subject to a system of affine constraints, occurs in investment portfolio selection, insurance risk theory, tomography, and other areas. We provide a solution that improves on the current published solution by being considerably simpler in computational terms. In particular, a succession of partitions and inversions of large matrices is avoided. Our solution method employs the Lagrangian multiplier method and we give two proofs, one of which is based on the solution of a related convex optimization problem. A geometrically intuitive interpretation of the objective function and of the optimization solution is also given.     

Exponential smoothing methods in pension funding

‘Smoothed-market’ methods are used by actuaries,when they value pension plan assets, in order to dampen thevolatility in contribution rates recommended to plan sponsors.A method involving exponential smoothing is considered. Thedynamics of the pension funding process is investigated in thecontext of a simple model where asset gains and losses emergeas a result of random rates of investment return and where thegains and losses are spread. It is shown that smoothing marketvalues up to a point does improve the stability of contributionsbut excessive smoothing is inefficient. It is also shown thatconsideration should be given to the combined effect of theasset valuation and gain and loss adjustment methods. Practicaland efficient combinations of gain/loss spreading periods andasset value smoothing parameters are suggested.