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2.4.3 Andersen's recursive formula

The recursive approach described in [1] builds on the fact that the portfolio loss function can only take a limited number of values. This set of values depends in turn on the obligors' individual loss given default levels L_n, ?n ? {1, .., N}. We now assume that all those loss levels can be expressed as multiples of a loss unit l:

?n ? {1, .., N}, ?a_n ? N, L_n = a_nl

?i ? {0,1, ..,

XN k=1

ák}, P (L = il) = i+8 Pz--z(L^(N) = L il)dö(z)

We further assume that all N obligors are ranked. The possible values for the loss function L are restricted to the following subset:

{m

L ? E _ájkl, m ? {1,..,N}, {j1,..,j_m} ? {1,.., N} ? {0} k=1

The power of Andersen's recursive algorithm is that it allows to compute the loss distribution while assuming that the pool of obligors results from the sequential addition of all obligors upon one specific ranking order. Let j ? {1, .., N} refer to the first j obligors added to the pool and L⁽j⁾ the discretized loss function associated

with that sub-pool. We can then express the loss distribution of L⁽j⁾ as a function of L(j-1).

Proposition 3. Andersen's recursive formula

Let j ? {1, .., N} and L⁽⁰⁾ = 0. Assume the loss distribution function L(j-1) conditional on the common factor Z is known. Let QZ denote the risk neutral probability measure conditional on the factor Z and pj the default probability of j^th obligor conditional on factor Z. Then we have the following recursive result:

?i ? {0, 1, .., _XN ák},

k=1

QZ (-⁰⁾ = il) = (1 - pj)QZ(L^(j-1) = il) + pjQZ(L^(j-1) = (i - áj)l)

Proof. Let Dj,j?{1,..N} denote the default indicator variable of j^th obligor conditional on the factor Z. Using the conditional independence of Dj,j?{1,..N}, we can then write for all j ? {1, ..,N} and for all i ? {0, .., Ejk=1 ák}:

QZ(L⁽j⁾ = il) = QZ(L⁽j^-1) = (i - áj)l,Dj = ¹) + QZ(Lj-1 = il, Dj = 0)

= QZ(L⁽j^-1) = (i - áj)l)QZ(Dj = 1) + QZ(L⁽j^-1) = il)QZ(Dj = 0)

= pjQZ(L(j-1) = (i - áj)l) + (1 - pj)QZ(L(j-1) = il)

Andersen's recursive formula evaluated at rank N thus provides the conditional loss distribution L^(N). The last step in the computation of the unconditional discretized loss distribution L is to integrate the conditional loss distribtion against the density function of the factor's standard gaussian law: