Credit Portfolio Models and Securitizations
For securitizing retail portfolios, consumer lending or mortgages, a single factor model is often used with a uniform correlation, using simulations similar to those explained in the portfolio simulation chapter (Chapter 50). The most common model used by banks is a simulation using either the structural model or the timeintensity model. Timeintensity models would use various intensity of defaults across successive periods, in a piecewise way. Default correlation plays a major role in securitizations, since it drives the thickness of the fat tail of the loss distribution.
For addressing structuring with portfolio models, the waterfall of losses serves for allocating losses to each tranche. A simple passthrough model5 allocates each randomly generated loss to the various notes according to the seniority of notes. The passthrough model is an oversimplification because of the many covenants that apply for routing the losses and the
TABLE 58.6 Example of a structuring of notes
various triggers to be considered. The passthrough model allows easier generation of the loss distribution for each tranche and assigning an expected loss, which is an objective basis for assigning a rating.
The loss of any structured note depends upon the aggregated level of losses of portfolio of loans. Losses flow to structured notes according to their seniority level. Any note benefits from the protection of all subordinated notes, whose size determines the level of overcollateralization of the note. The total size of all subordinated notes under a specific structured note is a loss franchise of benefit to the next senior tranche.
The principle for allocating random losses to each tranche is simple under a passthrough model. A portfolio has a size of 100. There are five classes of structured notes, each having an equal size of 20 (Table 58.6). Tranches get more senior when moving up. The lowest tranche is "Sub 1," serving as "equity" for the others since it is hit by the first losses of the portfolio. The most senior tranche is Senior 3, on top.
The exercise is to allocate losses. The portfolio loss can range from 0 up to 100, although this 100 loss will never happen. The distribution of portfolio losses results from simulation or relies on the limit distribution (Chapter 49) for granular uniform retail portfolio. If the simulated loss is lower than 20, it hits only Sub 1. If it is above 20, the excess over 20 hits the upper tranche. Hence a 25 portfolio loss results in a total loss of 20 for Sub 1 and a loss of 25  20 = 5 for Sub 2. Following the same rationale, the allocation of a portfolio loss of 45 would be 20 for Sub 1, 20 for Sub 2 and 5 for Senior 1, and 0 for both Senior 2 and 3.
The loss distribution of each note has a lower bound of zero. For example, the Senior 1 note has a loss of zero when the portfolio loss is 40 and a loss of 1 when the portfolio loss reaches 41. The loss distribution of each note is also capped by its size. In our example, Senior 1 cannot lose more than 20. In order to reach this loss level, the portfolio loss should hit the value 60.
The random loss of the portfolio is L, the random loss for the note NA benefiting from the seniority level defined by its attachment point, "a," is LA. The size of the note NA is b  a, and it is equal to the maximum loss of the note. For any value of simulated portfolio loss L, the loss allocation for each note follows the waterfall. Figure 58.5 shows both portfolio loss and the loss for the note NA Analytically, the loss for the note is simply: LA — min[max(L  a, 0), b  a]. The first loss appears only when L passes the lower bound a. If the portfolio loss L gets bigger, the loss LA remains bounded by b  a. Any excess above this cap hits the next structured note. The expected loss6 results from the portfolio loss distribution truncated at the levels a and b.
FIGURE 58.5 Loss distributions of the portfolio and of a structured note
In order to illustrate the above methodology, we use the above simple example of a portfolio funded by five classes of structured notes. The portfolio characteristics are:
• 100 obligors
• a uniform unit exposure
• a uniform default probability of 5%
• recoveries of zero
• a uniform asset correlation of 30%.
There are five classes of structured notes, all equal to 20% of the portfolio, or 20, following the above example.
In order to determine expected losses, multiple simulations serve for generating the underlying asset loss distribution. The number of simulations is 1000. Allocating the portfolio loss to all structured notes at each run also generates the loss distributions of the structured notes. The expected loss of each note is simply the arithmetic average of losses hitting each tranche across all 1000 trials. Table 58.7 summarizes the results.
With 30% correlation, there is no loss in the 1000 trials for Senior 1 and Senior 2. The 3 most senior notes are investment grade, since the expected loss is 0.06% for Senior 1. The average simulated loss of the portfolio is 4.98%, close to the 5% value. For the subordinated notes, the
TABLE 58.7 Simulation of the expected loss of a portfolio and of the notes
Portfolio loss 
Senior 3 
Senior 2 
Senior 1 
Sub 2 
Sub 1 (equity) 

Maximum 
58.00% 
0.00% 
0.00% 
18.00% 
17.00% 
19.000% 
Mean 
4.98% 
0.00% 
0.00% 
0.06% 
0.21% 
3.71% 
expected loss is lower because each note loss is bounded upward and downward.