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Radon–Nikodym Derivative see Equivalence of Probability Measures

Random Factor Loading Model (for Portfolio Credit)

Consider a portfolio of N risky assets, all assumed (for simplicity) to generate a $1 loss at the time of default. Let τi denote the random default time for asset i, such that the total portfolio loss L(T) on the horizon [0, T] is

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From credit default swap (CDS) or bond markets, we can normally extract risk-neutral survival probabilities

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for all T; this information locks risk-neutral expected portfolio losses at

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To be able to construct the entire distribution of L(T)—and not just its first moment—we need additional information about the default codependencies among the N assets.

The default codependence model that we consider here is in the class of factor models, in the sense that codependence is induced solely by a scalara random variable Z—the so-called systematic factor—that affects all assets through a factor “loading” function. Conditional on Z, all N default times τi are assumed to be independent of each other.

In practice, specification of factor loading is done ...

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