Unexpected Loss Measurement

The unexpected loss is the loss beyond expectation. The unexpected loss is usually higher than the expected loss by a large proportion. From the settings in Chapter 4, we define the unexpected loss of a given tranche as the expected loss of the tranche plus one standard deviation of losses of the tranche. Other common risk measures, such as Value-at-Risk (VaR) are not suitable for the measure of the unexpected loss of tranches because there is great possibility that the equity tranche will experience a total wipe out. Thus, the unexpected loss used here will be more appropriate.

In Table 5.4 below, we show the unexpected loss at maturity for the Legacy LCDX as well as the Bullet LCDX, using the same modeling parameters.

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Table 5.4 Unexpected Loss for Legacy and Bullet LCDX tranche swap

Unexpected Loss Legacy LCDX 11169.87 4282.69 1956.39 1951.51 1092.71 1886.56

Bullet LCDX 11322.33 4300.91 1976.14 1987.17 1114.16 1943.95

Before discussing the unexpected loss for tranches, it is worth noting that this risk measure is generally not additive. That is, if we sum up all the unexpected loss for all tranches, it will not equal to the unexpected loss for the reference portfolio.

First, from Table 5.4 above, we can see that unexpected loss is greater than expected loss for both Legacy and Bullet LCDX. Consider the reference portfolio for Legacy LCDX, the expected loss is 4,499,460 dollars while the unexpected loss is 11,169,870 dollars, which is 2.48 times higher.

Second, similar to the results for expected losses, the Bullet LCDX has a higher unexpected loss than the Legacy LCDX for all tranches. However, in order to measure the relative risk between tranches, we again have to divide the unexpected loss by the notional of the tranche to compute the unexpected loss percentage for LCDX.

Table 5.5 Unexpected Loss Percentage for Legacy and Bullet LCDX tranche swap

Unexpected Loss unexpected loss, risk for super senior tranche increased 4 times and more, while the risk for equity tranche only increase by 70%. The reason is that if unexpected defaults

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happens, especially at the later of the contract when the notional for the equity tranche has already been reduced by previous defaults, it lowers the credit enhancement and makes the mezzanine and senior tranches vulnerable to unexpected future losses.

Again, here we compare the unexpected loss percentage for Legacy and Bullet LCDX and we can find similar results as the expected loss percentage that the unexpected loss percentage for Bullet LCDX is higher than the Legacy LCDX in terms of all tranches.

Table 5.6 Leverage Ratio of Unexpected Loss for Legacy and Bullet LCDX tranche swap

time (yrs)

Equity Junior Mezz Senior Mezz Junior Senior Super Senior Legacy Bullet Legacy Bullet Legacy Bullet Legacy Bullet Legacy Bullet 0.25 12.36 12.59 5.05 4.90 2.95 2.82 1.81 1.65 0.07 0.07

0.5 11.62 11.82 5.19 5.23 3.15 3.07 2.03 1.94 0.09 0.08 0.75 11.08 11.22 5.37 5.40 3.29 3.30 2.17 2.08 0.10 0.10 1 10.63 10.78 5.50 5.50 3.43 3.45 2.28 2.24 0.11 0.11 1.25 10.29 10.40 5.60 5.59 3.56 3.57 2.38 2.38 0.12 0.12 1.5 9.99 10.09 5.67 5.67 3.68 3.67 2.47 2.47 0.13 0.12 1.75 9.72 9.81 5.72 5.72 3.77 3.75 2.56 2.57 0.13 0.13 2 9.47 9.55 5.76 5.79 3.86 3.83 2.65 2.65 0.14 0.14 2.25 9.25 9.31 5.78 5.81 3.93 3.90 2.72 2.72 0.15 0.14 2.5 9.05 9.10 5.81 5.83 3.99 3.98 2.79 2.77 0.15 0.15 2.75 8.87 8.90 5.83 5.85 4.04 4.04 2.85 2.83 0.16 0.16 3 8.70 8.72 5.83 5.86 4.09 4.09 2.92 2.90 0.16 0.16 3.25 8.54 8.55 5.84 5.87 4.13 4.13 2.98 2.96 0.17 0.17 3.5 8.39 8.39 5.85 5.87 4.18 4.18 3.02 3.02 0.17 0.17 3.75 8.25 8.24 5.85 5.87 4.21 4.22 3.07 3.06 0.18 0.18 4 8.12 8.10 5.85 5.86 4.25 4.26 3.11 3.11 0.18 0.18 4.25 8.00 7.96 5.85 5.85 4.28 4.30 3.15 3.15 0.19 0.19 4.5 7.88 7.84 5.85 5.84 4.31 4.33 3.19 3.20 0.19 0.19 4.75 7.78 7.71 5.84 5.83 4.34 4.36 3.23 3.24 0.19 0.20 5 7.67 7.60 5.84 5.82 4.37 4.39 3.26 3.28 0.20 0.20 Similar to the leverage ratio for expected loss, the leverage ratio for unexpected loss is express as the percentage loss of a given tranche based on the percentage loss of the reference portfolio. From Table 5.6, we find out that the leverage ratio of unexpected

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loss is lower than the leverage ratio of expected loss for the equity tranche in terms of all time periods, but they both decreased throughout the term of the contract. On the other hand, for the mezzanine tranches and senior tranches, the leverage ratio of unexpected loss is higher than the leverage ratio of expected loss in terms of all periods; however, they both increased throughout the term of the contract. At maturity, the leverage ratio of unexpected loss for the following tranches are 7.67, 5.84, 4.37, 3.26, and 0.2. Compared to the leverage ratio for expected loss, we see that the leverage ratio for unexpected loss in different tranches have become closer, it means that the difference of risk among tranches have narrowed, that is, when there is unexpected loss, risk will be transferred from equity to more senior tranches.

This result tells us that when default occurs, the realized loss of the reference portfolio has different level of affects on the relative risk of the five tranches. For the equity tranche, realized loss reduced the notional of the tranche, but it also decreased the uncertainty of future losses of the equity tranche. And for mezzanine tranches, realized loss brings down the subordination level causing a decrease of the protection for mezzanine tranches. Thus, the relative risk for mezzanine tranches raise until the tranches had realized a certain amount of loss that lowers its uncertainty for future loss.

Furthermore, if we compare the results for the Legacy LCDX and the Bullet LCDX, we find out that both LCDX share the same risk characteristics mentioned above. Again, we find some slight differences. For the equity tranche, the leverage ratio for the Bullet LCDX starts at 12.59, which is slightly higher than 12.36 of the Legacy LCDX at the 3 month period, and then it decrease at a faster pace than the Legacy LCDX to arrive at 7.60, which is therefore smaller than 7.67 of the Legacy LCDX at maturity. Similar result can also be found in the mezzanine, and senior tranches; however the leverage ratios for Bullet LCDX seem to increase slightly faster than the Legacy LCDX throughout the time period in this case. This is the same as the pattern we discovered for the leverage ratio of expected loss.