Models and Solution Methods for Liability Determined Investment

Traditional Asset and Liability Management(ALM) models have been recently re- cast as Liability Driven Investment (LDI) models for making integrated financial decisions in pension schemes investment: matching and outperforming liabilities. LDI has become extremely popular as a decision tool of choice for pension funds. The last decade experienced a fall in the equity markets while bond yields reached low levels. New regulations were introduced whereby liabilities were hard to meet. In the case of a deficit the pension fund trustees and employers have to agree on extra contributions to fill the deficit within 10 years time. The UK Accounting standard FRS17 (since 2001, replacing SSAP24) requires the assets to be mea- sured by their market value and liabilities measured by a projected unit method and a discount rate reflecting the market yields then available on AA rated cor- porate bonds of appropriate currency and term (see Accounting Standards Board Financial Reporting Standard 17). Furthermore, deficit or surplus has to be fully included on the balance sheet. In the Netherlands and the Nordic countries LDI models have become established; the UK, Italy and few other European countries are close followers of this trend. Traditionally, assets and liabilities were considered separate. In asset management the aim was to maximize return for a given risk level. However, the matching of the liabilities was not taken into consideration. The main argument was that assets should be made to grow faster than liabilities. The modern integrated approach to LDI considers the cash flow streams for in- vested assets which can be fixed income portfolios enhanced by interest rate swaps and in some cases includes added swaptions. We present an asset and liability management (ALM) problem for LDI, which we model using three approaches: a deterministic model, a stochastic model incorpo- rating uncertainty and a chance-constrained stochastic model. In the deterministic model we look at the relationship between PV 01 matching and the required fund- ing. PV 01 is the change of the net present value of a bond due to 0:01% positive parallel shift in the yield curve. In the stochastic programming model we have two sources of randomness: liabilities and interest rates. We generate interest rate scenarios, look at the relationship between funding requirements and minimize the deviation of the PV (present value) matching of the assets and liabilities over time. In the chance-constrained programming model we limit the number of future de/cit events by introducing binary variables and a user specified reliability level. The last model has integrated chance constraints, which not only limits the events of underfunding, but also the amount of underfunding relative to the liabilities. Fur- thermore, a fixed mix model is introduced for testing purposes only.