Economic Capital Modeling : Closed-Form Approximation of Incremental Economic Capital for Real-Time Applications

Economic capital (ECap) modeling is a fundamental part of Pillar II of the Basel framework. Indeed, 'sophisticated' financial institutions need to have in place internal models for the assessment of the level of the overall capital buffer which is deemed sufficient to cover the risk of their business activities. On top, ECap models are also frequently used for pricing purposes on an ex-ante basis: financial institutions need to know the incremental economic capital (IECap), i.e. the size by which the overall capital buffer needs to be increased after addition of e.g. a single new loan to the existing portfolio. This is important in order to be able to price such additional loan accordingly. Finally, ECap contributions (ECapC) are also required ex-post in order to break down the overall capital buffer to the individual obligors, products etc. within the portfolio. Simulation of IECap and ECapC can be computationally expensive and unstable, but it appears that closed form approximations provide accurate, consistent and quick solutions in many cases. The formula introduced here is based on the multi-factor approximation from [Pykhtin, 2004] applicable to a default-mode Merton type model. As such, default correlations between obligors (stemming from a multi-factor-model) are taken into account, but the formula also captures specific amortization schedules and loss given default (LGD) values for each individual position -- a feature which is of practical relevance, but often neglected in standard default-mode models. For the time being, credit-risk mitigants such as the existence of guarantors on individual loans, are not captured by the formula, neither are credit migrations or correlations between default probabilities and LGDs. Our formula allows for approximation of all ECap contributions without extra computational cost. After calculation of the ECap of the original portfolio, IECap can be computed within few seconds and more accurately than in standard linear approximations based on ECap contributions