Present values are also linear functions of a default surface (in a very explicit form provided herein) provided the joint distribution of sequential defaults is exchangeable.

## The exchangeable recovery assumption

By this we mean the following. Let \(R_k\) denote the stochastic recovery of reference entity \(k\). The recovery model is a joint distribution \( P(R_1 = r_1,..,R_n = r_n )\). It is said to be exchangeable if\begin{equation} \label{exchangeable} P( R_1 = r_1,..,R_{n} = r_n ) = P( R_{\pi(1)} = r_1, .. ,R_{\pi(n)} = r_n ) \end{equation} for any permutation \(\pi\). Here \(R_{(i)}\) is the \(i\)’th order statistic. The recovery of the first reference entity to default is \(R_{(1)}\). The recovery of the second reference entity to default is \(R_{(2)}\) and so forth.

It may be said that we are modeling the sequence of recovered amounts independent of the details of each reference entity. A limitation, certainly, although the class of such model is still quite interesting. It includes the ubiquitous constant recovery assumption, as well as models that lead to rather more sensible results for supersenior tranches.

Condition (\ref{exchangeable}) implies that the marginal distribution of recovery for every reference entity is identical, but it does not imply they are independent. A finite version of de Finetti’s theorem does, however, imply that all such models are mixtures of i.i.d. models.

## Decomposition into simple contracts

In what follows we assume a reference portfolio of \(N\) assets and \(M\) different tranches. Suppose we are interested in the implications of \(K\) different default time joint distributions over these \(N\) reference obligations.Let \(\{\theta\} \in \Theta\) denote the parameters for the exchangeable recovery model and \(\{\omega\}_{k=1}^K \in \Omega\) denote the parameters for the default time model. For fixed \(\theta\) the present values for the \(M\) different tranches under \(K\) different default model assumptions comprise an \(M,K\) matrix. We shall develop matrix expressions for the same, as follows: \begin{eqnarray} \label{decomp} \left[ PV_{credit}(\theta,\omega) \right]_{M, K} & = & \left[ C(\theta)\right]_{M, N} \left[D(\omega) \right]_{N, K} \nonumber \\ \left[ PV_{premia}(\theta,\omega)\right]_{M,K} & = & \left[p \right]_{M,1}\left[ e\right]_{1,K} \cdot \left[ P(\theta) \right]_{M,N+1} \left[ A(\omega) \right]_{N+1, K} + \left[ U \right]_{M,1} \left[ e\right]_{1,K} \end{eqnarray} where \(e\) is a vector of ones used to replicate \(U\) and \(p\). In turn \(U\) is a known, upfront amount paid by the purchaser of protection. The vector \(p\) comprises the agreed running premia (typically \(100\) or \(500\) basis points) for the tranches in questions. The dot denotes elementwise multiplication.

Definitions of other matrices will follow but point is that parameters \(\theta\) and \(\omega\) appear separately on the right hand side, not together. So any tensor product of assumptions regarding recovery and default can be computed efficiently.

We might call \(C(\theta)\) the credit protection matrix. The entry \(C_{m,n}\) is the average proportion of the \(n\)'th credit event payment which will be paid to the buyer of protection in tranche \(m\), where said average is taken over realizations of the recovery model with parameter \(\theta\).

Similarly, the entry \(P_{m,n}\) in what we might call the premia matrix \(P(\theta)\) is the mean proportion of the annuity for the \(n\)'th to default “tranchelet” which will accrue to the holder of tranche \(m\). Here again we refer to the mean over all realizations of the recovery model when the parameter is set to \(\theta\). It is convenient to allow the special case \(n=N+1\). We define \(P_{m,N+1}\) to be the proportion of the annuity for the \(m\)'th tranche that will be paid regardless of whether defaults occur or not. The value may well be zero, but typically will not be zero for the most senior tranche due to “writedown from above”.

In (\ref{decomp}) the matrix \(A(\omega)\) has entries \(A_{n,k}\). The entry \(A_{n,k}\) denotes the present value of a contract which pays the holder $1 per year up until the \(n\)'th default - computed under the assumptions of the \(k\)'th default time model. For the special case \(n=N+1\), \(A_{N+1,k}\) is the present value of a contract paying $1 per year up until the deal maturity regardless of default.

Similarly \(D(\omega)\) with entries \(D_{n,k}\) comprises the present value of elementary contract which pays the holder $1 at the time of the \(n\)'th default under the assumptions of the \(k\)'th default time model.

## Pricing of simple contracts

Next we consider the pricing of our simple risky annuities \(A\) and default count bets \(D\) given a default surface \(S\) of dimension \(N+1 \times T\). We claim these matrices can be computed as follows: \begin{eqnarray} \label{simple} D(\omega) & = & \left(\left(L S W\right)\cdot \left(R\Lambda'W\right)\cdot \left(R\Gamma'G\right)\right)F/10000 \nonumber \\ A(\omega) & = & \left(\left(H\left(\left(1-LS\right)Z\right)W\right)\cdot \left(H\left(R\Lambda'W\right)\right)\right)F \end{eqnarray} where \(\Gamma\) is a vector of times and \(\Lambda\) a vector of risk free discount factors.The remaining matrices are constant. $$ L = \left[ \begin{array}{cccc} 1 & 0 &\dots & 0 & \\ 1 & 1 & \dots & 0 & \\ \vdots & \vdots & \ddots & 0 & \\ 1 & 1 & 1 & 1 & \end{array} \right]_{N+1, N+1} $$ $$ F = \left[ \begin{array}{cccc} 1 & 1 &\dots & 1 \\ 0 & 1 & \dots & 1 \\ \vdots & \vdots & \ddots & 1 \\ 0 & 0 & 0 & 1 \end{array} \right]_{T+1, T+1} $$ $$ H = \left[ \begin{array}{cccc|c} 1 & 0 &\dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right]_{N, N+1} $$ $$ R = \left[ \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \end{array} \right]_{N+1,1} $$ $$ G = \left[ \begin{array}{cccc} -1 & 0 &\dots & 0 \\ 1 & -1 & \dots & 0 \\ 0 & 1 & \ddots & 0 \\ \vdots & \vdots & \ddots & -1 \\ 0 & 0 & 0 & 1 \end{array} \right]_{T, T-1} $$ $$ W = \left[ \begin{array}{cccc} 1/2 & 0 &\dots & 0 \\ 1/2 & 1/2 & \dots & 0 \\ 0 & 1/2 & \ddots & 0 \\ \vdots & \vdots & \ddots & 1/2 \\ 0 & 0 & 0 & 1/2 \end{array} \right]_{T, T-1} $$ $$ Z = \left[ \begin{array}{ccccc} 1 & -1 & 0 & \dots & 0 \\ 0 & 1 & -1 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & -1 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right]_{T, T} $$ Combining (\ref{decomp}) and (\ref{simple}) we can price all tranches for all default models using only matrix computations. The \(\theta\) dependent matrices \(C\) and \(P\) in (\ref{decomp}) can be computed offline.

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