*Risk*, 15 (12) (2002), 160–162; reprinted in *Risk* 20 (7) (2007), 130–133; reprinted in A. Lipton (ed.), *Theory and Practice of Credit Risk Modelling*, London: Risk Books, 2008.

The amount of capital necessary to support a portfolio of debt securities depends on the probability distribution of the portfolio loss. Consider a portfolio of loans, each of which is subject to default resulting in a loss to the lender. Suppose the portfolio is financed partly by equity capital and partly by borrowed funds. The credit quality of the lender's notes will depend on the probability that the loss on the portfolio exceeds the equity capital. To achieve a certain credit rating of its notes (say Aa on a rating agency scale), the lender needs to keep the probability of default on the notes at the level corresponding to that rating (about .001 for the Aa quality). It means that the equity capital allocated to the portfolio must be equal to the percentile of the distribution of the portfolio loss that corresponds to the desired probability.

In addition to determining the capital necessary to support a loan portfolio, the probability distribution of portfolio losses has a number of other applications. It can be used in regulatory reporting, measuring portfolio risk, calculation of value-at-risk (VaR), portfolio optimization and structuring, and pricing debt portfolio derivatives such as collateralized debt obligations (CDO).

In this chapter, we derive the distribution ...

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