Var methodology for non-Gaussian finance

The aim of this book is to present recent results concerning one of the most popular risk indicators called Value at Risk or simply VaR. This indicator was born with the supervising rules of Basel I and Basel II for banks and also reused for building the future rules of Solvency II for insurance companies to measure their financial solidity. VaR can be used for only one risk but also from a global point of view for the balance sheet. Its statistical meaning is nothing else than the quartile at a certain level of confidence close to one (for example 0.95 or even 0.995) of the distribution function of the considered risk. So, it gives an amount of equities the bank or insurance company can use if their losses are larger than what is called the expected loss. It is not difficult to understand that the institutions have a strong interest to find good hedging with a minimum amount of equities dedicated to these eventual “unexpected losses”. The problem of VaR evaluation comes from the fact that it is not easy to know the distribution function of the considered risk, except with its estimation by simulation. Nevertheless this approach cannot lead in general to an approach with a stochastic model much more rich to study the future evolution. That is why in Basel I and II, the VaR indicator was building under the assumption that the considered risk has a normal distribution giving so nice formulas for the VaR computation. Unfortunately the values given in practical situations were often too low to give the theoretical hedging of the risk and that is why the regulator retains as final value the triple of the theoretical one! Moreover the recent financial and economic crises reinforce this insufficiency of this approach and a lot of academic works propose alternatives more or less realistic but the most important fact is that we pass to a so-called Gaussian approach to a non-Gaussian one so that we can expect more interesting results. But there is a challenge: the new VaR approaches must not only give more realistic results but also must be able to “predict” the short future in view that the hedging will be efficient enough for at least one or several years. This is particularly important for insurance companies because they are involved in long term activities.