This book provides the first comprehensive overview of granularity theory and illustrates its potential for risk analysis in finance and insurance.

The Granularity Principle

The recent financial crisis has heightened the need for appropriate methodologies to control and regulate risks in financial markets. The balance sheets of banks and insurance companies contain large portfolios of individual risks that correspond to financial securities, such as stocks and corporate or sovereign bonds, as well as individual contracts, such as corporate loans, household mortgages, and life insurance contracts. Risk analysis in such large portfolios is made difficult by the nonlinearities of the risk models, the dependencies between the individual risks, and the large sizes of the portfolios, which can include several thousand assets and contracts. The nonlinearities are induced, for instance, by the qualitative nature of the risks associated with default, rating migration, and prepayment for credit portfolios, or with mortality and lapse for life insurance portfolios. The dependencies between the individual securities and contracts are caused by systematic risk factors that affect the random payoffs of the individual assets. Systematic risks cannot be diversified even when the size of the portfolio becomes infinitely large. The consequence of these difficulties is that standard portfolio risk measures, such as the Value-at-Risk (VaR), cannot be computed analytically for realistic risk models. The portfolio VaR corresponds to the quantile of the portfolio loss distribution at a given percentile level; that is, the loss that is exceeded only with a given small probability.