EVALUACIÓN DE LOS PELLETS

ing a CDO

In this section we will review the practical example considered in [4]. It involves a synthetic CDO called EuroStoxx50, which is composed of 50 single name credit default swaps on credits that belong to the DJ EuroStoxx50 equity index. Each reference credit has a notional equal to 10 million euros, making the total nomi- nal amount of the collateral portfolio 500 million euros. This CDO comprises five tranches with the standard prioritised scheme: super senior tranche notes are paid before mezzanine and subsequent lower subordinations; after which any residual equity tranches are paid (refer to Chapter 2 for more details on CDO structuring).

Application to Pricing

Since one cannot observe a time to default series, one is forced to use alternative proxies to get the desired parameters both for the marginals and the copula itself. Spreads of the single-name credit defaults swaps (CDSs), which can be observed in the market, for each obligor were considered to be best proxies for such a task. For the joint default behaviour the authors considered the Gaussian, Student t, Clayton and Frank copulas. Although the suitability of the Gaussian copula to default modelling has been questionable and recently proved inappropriate, is a common starting point for copula analysis, beginning with [23]. Studenttcopula is a natural next step due to its tail dependence and asymptotical behaviour which tends towards the Gaussian copula. Clayton and Frank copulas are from the Archimedean family of copulas, which have recently become popular. For more properties of each of these copulas refer to Chapter 3.

Chapter 4 §4.4 Modelling Multiple Survival Times 47

In order to get the simulations for each obligor’s time to default, it is assumed that marginal densities follow an exponential distribution for which the hazard term has been derived from the CDS curve at a particular date for each obligor. Each hazard term is assumed as follows:

λt=

˜

p(0, t) 1−R,

where the recovery rateRis fixed to 30%, as is common practice in the market, and ˜

p(0, t) represents the CDS spread at a given time t. The intensity process obtained in this case is under the risk neutral measure, while using historical default rates would lead to the process under a historical measure.

As mentioned in [4], the above relationship can be derived by considering the cash- flows exchanged during a CDS and the fact that a fair valuation requires the total amount of payment received by the protection seller to equal the expected loss the seller would pay the buyer if a credit event occurs.

As discussed in Chapter 2, the relationship between the hazard rate and the cumu- lative default probability is given by

F(t) = 1−exp [ − ∫ t 0 λ(s)ds ] . (4.2)

In the real world we cannot deduce the hazard rate for all periods of time, but only for a finite set of times for which we have information on the CDS spread. In the example considered there are three observed points implied by the single name CDS premium at terms 1 year, 3 years and 5 years. In general, we can have N points in timet1, t2, . . . , tN. Hence, the hazard rate function we obtain is a stepwise constant

function of time based on the observable values ofλ.

Our cumulative distribution function for each obligor’s time to default becomes:

F(t) = 1−exp  ₋∑k j=1 λjδj  , wherehj =h(tj), δj =tj−tj−1 and k=          1, ift≤t1 2, ift1 < t≤t2 .. . N, ift > tN−1

This methodology has been used for each reference credit in the portfolio in order to obtain the draws for each time to default, after having generated the draws from the chosen copula.

Chapter 4 §4.4 Modelling Multiple Survival Times 48

Application to Risk Monitoring

For risk monitoring, the recovery rate across the portfolio was also changed in [4] into a random variable. It was generated from a Beta distribution, independent from the times to default and interest rates. The Beta distribution has a range 0≤x≤1 and shape parametersv >0,w >0. Its density is given by:

xv−1(1−x)w−1

B(v, w) ,

where B(v, w) = ∫₀1uv−1(1−u)w−1du. It is well-known that its mean is equal to

v/(v+w) and variance equal tovw/[v+w)2(v+w+ 1)]. The parametersv,wwere chosen by fixing the mean equal to 50% and variance equal to 30% (this data was reported in a Moody’s study on recoveries from defaulted corporate bonds).

Various levels of VaR (value at risk) were computed for each tranche’s cumulative loss and the results from different copulas were compared. The paper found that more losses were simulated from the Studenttcopula than the Gaussian or the two Archimedean copulas. The result is not surprising, especially when one notes that the chosen degrees of freedom in [4] was 8 for the Studentt copula. The lower the degrees of freedom, the higher the tail dependence, increasing the probability of multiple defaults. The Gaussian and Frank copulas do not have tail dependence. It would be of interest to extend this investigation to the conditional tail expectations, loss amounts or number of defaults.