Imagine a pool of defaultable instruments (bonds, loans, credit default swaps CDSs,...) from different firms is put together. The losses on the initial portfolio value due to the default of the underlying firms depend on the default probability of each firm and the losses derived from each default (losses given default). Additionally, the degree of dependence between the firms’ default probability, usually known as default correlation, plays an important role on the timing of the firms’ defaults (whether they tend to cluster or they are independent) and, as a consequence, on the distribution of the portfolio losses.
Next, imagine we, the owners of the portfolio, decide to buy protection against the possible losses due to the defaults of the underlying firms, but we can not sell the portfolio. One way to do it is buying Credit Default Swaps (CDSs) of each firm, but that’s not the way we are interested in here. We can sell the portfolio in tranches, i.e. we can buy protection for those losses in tranches. A Collateralized Debt Obligation (CDO) consists on tranching and selling the credit risk of the underlying portfolio. For example, a tranche with attachment points [KL,KU] will bear the portfolio losses in excess of KL percent of the initial value of the portfolio, up to a KL percent. The tranche absorbing the first losses, called equity tranche, is characterized by KL = 0 and KU > 0. The holders of a tranche characterized by attachment points [KL,KU] won’t suffer any loss as long as the total portfolio loss is lower than KL percent of its initial value. When the total portfolio loss goes above KL percent, the tranche holders are responsible for the losses exceeding KL percent, up to KU percent. Losses above KU percent of the initial portfolio value do not affect them. The lower attachment point KL of each tranche corresponds to the upper attachment point KU of the previous (more junior) tranche.
Obviously, the holders of each tranche (sellers of credit risk protection) have to be compensated for bearing those losses: they receive a periodic fee, called premium, until the maturity of the CDO (point in which they also stop being responsible for future losses in the portfolio.) The premium of the equity tranche will be the highest because its holders absorb the first losses of the portfolio. In order for the holders of more senior tranches to start suffering losses, the holders of more junior tranches would have already born all losses they were exposed to (KU - KL percent of the initial portfolio value). As a consequence, the higher the seniority of the tranche the lower the premiums holders receive.
The whole problem lies in determining the tranches’ premiums. They have to compensate tranche holders for the expected losses they will suffer and, therefore, they depend on the distribution of the portfolio losses which, as we argued above, depends on the underlying firms’ default probabilities, default correlations, and losses given default.
Our review of CDO pricing models focus on a particular branch of this literature: the ones based on structural models. The main distinguishing characteristic of such models with respect to the other credit risk modelling alternative, reduced form models, is the link they provide between the probability of default and the firms’ fundamental financial variables: assets and liabilities. The way structural models incorporate the dependence between the firms’ default probabilities (which is a key ingredient for CDO pricing) is by making such fundamental variables depend on a set of, generally unobserved, common factors.
Contents
1 Introduction
2 Structural model for credit risk: Merton (1974)
3 Vasicek asymptotic single factor model
4 CDOs
4.1 Mechanics
4.2 Pricing
4.3 Vasicek model: homogeneous large portfolio
4.4 Sensitivity of tranche premiums to the model parameters
4.5 Trading issues
- 4.5.1 Moral hazard
4.5.2 Types of CDOs
4.5.3 Correlation smile and base correlations
4.5.4 Exotic CDOs
4.6 Extensions of the Vasicek model
- 4.6.1 Homogeneous finite portfolio
4.6.2 General distribution functions
4.6.3 Heterogeneous finite portfolio
4.6.4 Stochastic default correlations
4.6.5 Multifactor models
4.6.6 Random loss given default
4.6.7 Totally external defaults
4.6.8 Focusing on a single tranche
4.7 Parameter calibration
- 4.7.1 Default probabilities
4.7.2 Loss given default
4.7.3 Correlation
Appendix
A Bank capital regulation
References
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Credit Risk Models IV: Understanding And Pricing CDOs
