Expected Loss Measurement

In Table 5.1 below, we show the expected loss at maturity for the Legacy LCDX tranche swap as well as the Bullet LCDX tranche swap, using the same modeling parameters.

Table 5.1 Expected Loss for Legacy and Bullet LCDX tranche swap

Expected Loss the names that should be canceled in the cancellation case, will still have a chance to default before maturity under simulation.

Second, since the notional of different tranches is not the same, we cannot measure the relative risk between tranches using the expected loss of the tranche. For example, consider the Legacy LCDX , we find out that the expected loss for the super senior tranche is higher than the junior senior tranche, which is 273,340 dollars and 339,800 respectively. Even though the expected loss for the super senior tranche is higher, if we take the notional amount of the tranches into account, we will find out that the risk for the super senior tranche is apparently smaller.

Despite that rating agencies widely use expected loss as a one of the important risk measure, from this example, we noticed that expected loss is not an adequate risk measure for comparing risk among tranches. Thus, here we introduce the concept of the expected loss percentage.

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Table 5.2 Expected Loss Percentage for Legacy and Bullet LCDX tranche swap

Expected Loss Percentage (%)

Reference Portfolio

Equity Tranche

Junior Mezz

Senior Mezz

Junior Senior

Super Senior Legacy LCDX 4.50 50.27 25.25 15.38 9.11 0.40

Bullet LCDX 4.57 50.60 25.68 15.85 9.42 0.42

Table 5.2 above shows the expected loss percentage for LCDX tranche swap. For the Legacy LCDX, the equity tranche may suffer from a 50.27% loss at maturity while the super senior tranche will only have a 0.4% loss and 0.45% loss for the whole reference portfolio. This describes the high risk and high return characteristic for tranched products since the spread for equity tranche and super senior tranche is 1415.31bp and 8.66bp respectively. (here we quote all tranches in running spread for ease of comparison)

Similar to the result for expected loss, the expected loss percentage for the Bullet LCDX is higher than the Legacy LCDX for all of the tranches.

Next, using the expected loss percentage for the reference portfolio, we can further compute the leverage ratio for the tranches. In table 5.3, we show the leverage ratio of the tranches for Legacy LCDX and the Bullet LCDX under different time periods.

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Table 5.3 Leverage Ratio of Expected 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 17.76 17.96 2.17 2.02 0.80 0.72 0.31 0.24 0.01 0.01 The leverage ratio measures the expected loss percentage of the tranche compared to the expected loss percentage to the reference portfolio. From Table 5.3, for Legacy LCDX, the leverage ratio for the equity tranche starts off at 17.76 at the 3 month period and then decreases to 11.17 at maturity. On the other hand, the leverage ratio for the more senior tranches increases as time goes to maturity. This means that the equity tranche bears more risk during the initial period than the later period of the contract, while more senior tranches are just the opposite, which bears more risk during the later period of the contract. The explanation for this opposite trend is that the equity tranche bears the first 5% cumulative default loss, and then until the equity tranche is all wipe out, the junior mezzanine will bear the cumulative default loss which exceed the 5% credit enhancement. And it works similarly onto more senior tranches as more names default. Since senior tranches are credit enhanced, they only

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bears default loss at the later period of the contract. Thus, leverage ratios for senior tranches appear to increase throughout the time period of the contract, while leverage ratios for the equity tranche would decrease.

Moreover, this pattern we found from Table 5.3 also shed some light on the hedging issue for LCDX tranche swap investors. For equity investors who bear more risk in the initial period of the contract should hedge the risk from early defaults because they have to avoid losses on notional before they receive enough premium to compensate the losses. On the other hand, for senior tranche investors who bear risk in the later period of the contract should hedge the risk of late defaults due to increasing leverage ratio throughout the time period of the contract.

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. However, they have some slight differences. For the equity tranche, the leverage ratio for the Bullet LCDX starts at 17.96, which is slightly higher than 17.76 of the Legacy LCDX at the 3 month period, and then it decrease at a faster pace than the Legacy LCDX to arrive at 11.07, which is therefore smaller than 11.17 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 means that the change in risk throughout the term of the contract will have a slightly wider extent for the Bullet LCDX than the Legacy LCDX.