
Thursday, 7th November 2024 
From copulas to CDOs  pricing tranches 
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In FINANCE February, Finbarr Murphy and Bernard Murphy looked at how one can price Basket Default Swaps (BDSs) using Gaussian copulas. Copula functions are a relatively new tool in finance, and they are used to construct multivariate distributions, and to investigate dependence structure between random variables. In this issue, they continue this discussion to demonstrate how one can use Copula functions to value multitranche synthetic Collateralised Debt Obligations (CDOs), and they point out some of the risks inherent in these instruments. 
The notional amount of outstanding Credit Default Swaps (CDSs) in June 2005 was $12.43 trillion1. This represents an annualised growth rate in excess of 100 per cent. The use of CDSs as the underlying instrument in Synthetic Collateralised Debt Obligations (CDOs) has fuelled much of this growth. Typically, synthetic CDOs are constructed to meet client risk preferences and are sold in different tranches. Risk adverse investors will seek tranches higher in the capital structure for lower risk and consequently lower expected returns. Conversely, those investors seeking a higher return with associated risk will source lower (equity tranches). In tandem with the explosive growth in CDS activity, CDO issuance has grown quickly in North American with $155 billion issued in 20052 and in Europe, issuance reached €243.5 billion in 20043.
The iTraxx Credit Default Swap index in Europe and the Dow Jones CDX index in the US have allowed for greater transparency and liquidity in the Synthetic CDO market.
Standardised tranched credit products from these indices (Tracers and TriBoxx) are now quoted by banks to clients. When a bank quotes a price on standardised tranches or on a bespoke structured tranche, by extension, it runs risk on the remaining capital structure. It is therefore critically important to measure the correlation between the constituent components in order to effectively risk manage a disparate portfolio of credit derivatives.
The standard market model for valuing default swaps on multiconstituent credit baskets is the Gaussian copula model that uses one parameter to describe the correlations between the individual names credit default times. This standard market approach also assumes these correlations are constant for the life of the basket swap. It is also assumed that the recovery rates and swap spreads are constant. This simple, standard approach allows dealers to quote in terms of implied correlation rather than a swap spread.
Pricing synthetic CDOs
A CDO is a transaction that securitises a diversified pool of debt assets. A synthetic CDO is one where the underlying portfolio consists of single name Credit Default Swaps (CDSs), hence the ‘synthetic’ precursor and the reason that a synthetic CDO is classified as a credit derivative. These single name CDSs are liquid instruments with a payout based on defined credit events. The spread of the CDS is, in part, defined by the default probability of the underlying asset or entity. Based on a time series of CDSs on a particular name, one can therefore create an implied default probability distribution.
A CDO can be valued as a swap transaction with the premium being described by setting the default payout equal to the periodic insurance payments. We now describe how a multitranche CDO with n reference assets can be valued.
Let us first assume that the jth asset has an exposure Ej and a recovery rate Rj. Nj(t) = 1 denotes the default indicator for j. The cumulative portfolio loss at t, is shown in formula 1.
A CDO tranche has attachment points at A and a detachment point at B. Default payment occurs above the A threshold and below the B threshold where 0 A B i=1Ni. When A = 0, the tranche is usually referred as the Equity Tranche. When A > 0 and B < i=1Ni we are usually referring to the Mezzanine Tranche(s) and when B = i=1Ni we are usually dealing with the Senior, or Super Senior Tranche.
As with a CDS, a CDO tranche spread is that amount, based on the notional, paid to the investor such that the premium leg(s) equal the default payment legs. Consider first the default leg. Let M(t) denote the cumulative loss on given tranche. We can summarise this loss in formula 2.
So, M(t) is a jump process where a payment of M(t+)M(t) occurs at each jump. Now, we can write the price of the default payment leg of the tranche as shown in formula 3, where B(0,t) is the discount factor to time t and EQ represents the expectation under the default probability Q.
Now turning to the premium payment, X, of a CDO tranche. Let ti denote the premium payment times where i = 1,…,T with T being the CDO maturity. Let i1,i denote the time period, [ti1,ti] and again, B(0,ti) is the discount factor to maturity ti.
The tranche premium is due on the outstanding notional amount between the attachment detachment points, A and B. At time ti, if the number of names in default, given by N(ti), is less than A, then the premium payment will be due on the full tranche notional. Similarly, if N(ti) B, the entire tranche has defaulted. Where A N(ti) < B, the premium is due on the outstanding tranche notional, BN(ti).
We can now write the expected discounted premium payment at ti in formula 4, where Q(N(ti) = k) represents the probability of k names being in default at time ti. We can calculate the entire premium leg by summing over the entire payment schedule. The above solution can be easily extended to include accrued premiums from the time of the last regular premium payment date to j, the default date of entity j between ti1 and ti.
Simulation methodology
We produce a random vector x1,…,xn, from the standard multivariate normal distribution with a mean of zero and a correlation specified by the matrix. Recall that a Copula function links the multivariate distribution to the individual marginal distributions.
Specifically, the Gaussian copula states in formula 5, where is the correlation matrix, 1 is the generalised inverse of the univariate normal distribution function and n is the multivariate normal distribution function. We can transform the vector x1,…,xn, into default times through the function of the marginal distributions. E.g. i = ln(xi).
Attributing losses to a tranche is cumulative up to the detachment point, B. Take a 100 name synthetic CDO where the recovery rate, Rj = 40% and a notional value of Hj for all names. The attachment point, A = 0% and the detachment point, B = 5%. The cumulative losses are given as shown in formula 6.
In this case, the equity tranche takes the losses of the 1st to 8th name to default and takes 33% of the loss on the 9th name to default. Similarly, assuming a mezzanine tranche with an attachment point at 5% and a detachment point at 15%, it will take 66% of the loss on the 9th tranche and takes all of the losses from the 9th to the 25th default. The senior tranche takes all subsequent losses.
Simulation results
We consider a Synthetic CDO with 125 underlying reference names with equal weight.
We assume a constant recovery rate of 40% for all reference entities and a deterministic, riskfree interest rate of 5%. We further suppose the CDO to be divided into multiple tranches, the base case assuming equity tranches with attachment points at 0, 3 and 6%.
The detachment points will be 3, 6 and 9% respectively. Finally, we assume a constant default intensity () of 0.01 for all reference names. 5000 simulations were conducted for all scenarios.
We begin by observing the loss distribution and spread for the three base tranches. Figure 1 below shows the loss distribution graph for the baseline scenario. Note the increasing probability of zero defaults as the correlation increases. This demonstrates an increasing marktomarket book value of a long equity tranche investor. Conversely, an increasing correlation tends to decrease the value of senior tranches as the likelihood of all reference obligations increases.
This spread reaction, to differing correlation values, is further demonstrated in Figure 2 below. We have graphed the spreads for [03], [36] and [69]% tranches against increasing correlation. The spread for the equity tranche decreases with increasing correlation but note the slightly convex shape to the higher [69]% tranche.
Conclusion
The standard Gaussian Copula model has drawbacks. It is argued that one should use stochastic models with negative correlation between recovery rates and default probabilities. The model assumes that default rates are constant and equal which are implied from CDS spreads. Another drawback is the assumed flat correlation structure across reference names which misrepresents the complex default relationship between entities and sectors. One of the strongest arguments against the model is the computational resource required to produce stable values, particularly when using Monte Carlo simulations.
Despite these drawbacks, the Gaussian copula model has the advantage of simplicity and tractability. The introduction of the onefactor approach has reduced the computational burden and extensions to the model allowing clustered correlation for inter and intrasector modelling, random recovery rates that allowed for correlation between recovery rates and default probabilities give more accurate results.
1 Source: International Swaps and Derivatives Association (ISDA), 2005 midyear survey of 86 firms
2 Source: American Securitization Forum
3 Source: European Securitisation Forum 
Finbarr Murphy and Dr. Bernard Murphy are lecturers in derivatives and financial engineering on the MSc in Financial Services programme in the Kemmy Business School, University of Limerick.


