Hedging Analysis

When it comes to hedging LCDX tranche loss, tranche investors can choose to sell the LCDX index swap or a single-name CDS depending on the risk one wants to avoid.

Using the LCDX index swap as a hedging instrument means that the investor wants to hedge the risk that the spread of all the underlying assets change at the same time due to economic condition changes; and using a single-name CDS means that the investor wants to hedge the risk of spread change or default for a certain underlying CDS. As a

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result, the two hedging instruments differs a lot by their hedging costs and outcomes.

Nowadays, many standardized index products of LCDS and CDS have been introduced to the market making it easier for investors of LCDX tranche swap to hedge their positions. For simplicity, here we consider only using the LCDX index swap as the hedging instrument. By the definitions introduced in Chapter 4, we can compute the following Greek Ratios, tranche Delta and tranche Gamma.

Table 5.7 shows the tranche Delta at the initial period of the contract for each of the five tranches. First, we assume the average credit spread of the reference portfolio moves up by 1 bp, then we can see that all the five tranches result in a negative change of MTM. Next, we divide the change of MTM for each of the five tranches by the amount of the change of MTM on the total reference portfolio to calculate the percentage of MTM change. Last, we divide the percentage of MTM change for each of the five tranches by the percentage of MTM change on the total reference portfolio and eventually we can obtain the tranche Delta.

Table 5.7 Greek Ratios — Tranche DELTA

time (yrs)

Change of MTM (thousands)

Percentage of MTM

change (%) Delta

Legacy Bullet Legacy Bullet Legacy Bullet Equity -15.7248 -20.8126 -3.1450 -4.1625 -10.5852 -10.6767 Junior Mezz -6.2219 -8.2750 -2.0740 -2.7583 -6.9805 -7.0750 Senior Mezz -5.7217 -7.7850 -1.4304 -1.9462 -4.8144 -4.9920 Junior Senior -2.3993 -4.0667 -0.7998 -1.3556 -2.6918 -3.4769 Super Senior -4.2048 -5.0981 -0.0495 -0.0600 -0.1665 -0.1538 Total -29.7110 -38.9871 -0.2971 -0.3899 0.0000 0.0000 When the average credit spread of the reference portfolio moves up by 1 bp, the equity tranche is affected the most, then the effect become smaller for mezzanine tranches, and the smallest for senior tranches, where the percentage of MTM change is even smaller than the total reference portfolio.

For Legacy LCDX, the tranche Delta for the equity tranche is -10.59, which means that in order to hedge the risk of the change in the spread of the underlying

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assets, equity tranche investors have to sell the LCDX index swap for an amount of 10.59 times the notional of the equity tranche. As a result, the hedging cost can be express as the amount of spread the investor have to pay every quarter to involve in a hedging position.

Table 5.8 Greek Ratios — Tranche GAMMA

time (yrs)

Delta by 1bp change in spread

Delta by 10bp

change in spread Gamma Legacy Bullet Legacy Bullet Legacy Bullet Equity -10.5852 -10.6767 -10.4233 -10.2922 0.0180 0.0427 Junior Mezz -6.9805 -7.0750 -7.0135 -6.9278 -0.0037 0.0163 Senior Mezz -4.8144 -4.9920 -4.9738 -4.8940 -0.0177 0.0109 Junior Senior -2.6918 -3.4769 -3.3540 -3.4113 -0.0736 0.0073 Super Senior -0.1665 -0.1538 -0.1713 -0.1839 -0.0005 -0.0033

Besides the tranche Delta, we also compute the tranche Gamma shown in Table 5.8.

Tranche Gamma stands for the change in tranche Delta when the credit spread of the underlying assets increase by 1 bp. Tranche Gamma is also a important ratio to consider when we deal with hedging issues.

Results from Table 5.8 show that the tranche Gamma for the equity tranche is positive which indicates that as the credit spread widens, the absolute value of the tranche Delta decreases. This can be explain by when spread widens, realized loss reduces the uncertainty of future default for the equity tranche, thus lowers the amount of hedging position needed. On the other hand, the tranche Gamma for mezzanine and senior tranches are all negative which indicates that as the credit spread widens, the absolute value of the tranche Delta increases. In this case, as the spread widens, the probability of losses increase, thus increase the amount of hedge position needed.

Furthermore, if we compare the tranche Gamma for Legacy and Bullet LCDX, we can find out that the tranche Gamma for the junior mezzanine tranche, senior mezzanine tranche, and junior senior tranche are positive for the Bullet LCDX, while

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they are negative for the Legacy LCDX. This means that unlike the Legacy LCDX, the Bullet LCDX will experience a decrease in the amount of hedge position for these tranches as the spread widens.

When there exists a LCDX index swap of the reference portfolio, tranche investors can use the index to conduct a complete hedge. Nevertheless, using the index swap as a hedging instrument could be expensive. The next part we will analyze its hedging cost and outcomes.

We show the result of tranche Delta hedging for Legacy LCDX in Table 5.9 and the result of tranche Delta hedging for Bullet LCDX in Table 5.10.

Table 5.9 Result of Tranche Delta Hedging for Legacy LCDX tranche swap

Equity Junior

Mezz

Senior Mezz

Junior Senior

Super Senior

Initial Position (M) 5 3 4 3 85

Spread (bp) 1418.84 561.59 322.45 185.36 8.57 Tranche Delta -10.59 -6.98 -4.81 -2.69 -0.17

PV01 0.031450 0.020740 0.014304 0.007998 0.000495 Hedge Position (M) 52.93 20.94 19.26 8.08 14.15

Change in Initial

Position (1bp up) -157248.31 -62219.31 -57216.57 -23993.13 -42048.29 Change in Hedge

Position (1bp up) 157248.31 62219.31 57216.57 23993.13 42048.29 Tranche Value 180158.58 -40937.52 -63594.90 -25146.77 -68641.65 Net Tranche Income

(bp) 360.32 -136.46 -158.99 -83.82 -8.08 Hedging Cost (bp) 1058.52 698.05 481.44 269.18 16.65

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Table 5.10 Result of Tranche Delta Hedging for Bullet LCDX tranche swap

Equity Junior The second row in the table shows the initial position of the five tranches in million dollars, and the third row shows the computed spread for the tranches. In this case, we can see that the spread of the super senior tranche for Legacy LCDX is higher than the Bullet LCDX. The fourth row is the tranche Delta from Table 5.7, and PV01 in the fifth row is represents the change in MTM base on one dollar notional of the tranche when the average credit spread of the reference portfolio increase by 1 bp. In order to calculate the hedge position in the sixth row, we multiply the tranche Delta by the initial position. Next, in the seventh and eighth row, we compute the change in initial position and change in hedge position by multiplying the initial position by the PV01 of the tranches and multiplying the hedge position by the PV01 of the reference portfolio respectively. Last, we compute the tranche value in the ninth row which is the net tranche income (tranche spread times the notional of the tranche) minus the hedging cost (the spread of the LCDX index swap times the hedge position of the tranche). Also, we can divide the net tranche income by the notional of the tranche to arrive at net tranche return which is shown in the tenth row. Finally, the hedging cost shown in the last row can be computed as the tranche spread minus the net tranche return.

Consider the equity tranche for Legacy LCDX, the tranche Delta is -10.59 which means that in order to hedge the default risk of the 5 million dollar notional,

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equity investors have to sell the LCDX index swap for an amount of 53.38 million dollar. From the table, we see that the loss of initial position as the spread of the reference portfolio increase by 1 bp is offset by the increased value in the hedge position which is equal to 208,126.29 dollars. Thus, using the tranche Delta, we can assure a complete hedge of our initial tranche position. In terms of hedging costs, the equity tranche receives 1418.84 bp a year, but pays 10.59 times the spread of the LCDX index (100bp) every year. After involved in hedging, equity tranche can receive only 360.32 bp, while the hedging cost is 1058.52bp. Therefore, the tranche Delta hedge is a very expensive hedging strategy.

Now we consider the equity tranche for the Bullet LCDX, which the equity tranche receives 1433.29 bp a year, but pays 10.68 times the spread of the LCDX index (100bp) every year. After involved in hedging, equity tranche can receive only 365.62 bp, while the hedging cost is 1067.67bp. Comparing both the hedging results for Legacy and Bullet LCDX, we find out that since the tranche spread is higher for the Bullet LCDX, the hedging costs for Bullet LCDX is also higher. However, it is not true for all tranches. Since the super senior tranche will suffer from notional reduction when cancellation events occur for Legacy LCDX, the tranche spread might happen to be higher than the tranche spread of the Bullet LCDX under some conditions.

Recall that we set our correlation parameter ρ = 0.5 and υ = -0.5, which means that we assume a high correlation between the underlying asset value and the macro factor and a high negative correlation between default and cancellation. In this case, the spread of super senior tranche for Legacy LCDX is higher, resulting in a higher hedging cost for Legacy LCDX too.

To sum up, although selling the LCDX index swap at the amount of the tranche Delta can achieve complete hedging, it also results in great hedging costs, causing negative net tranche return for the mezzanine and senior tranches. Only the equity tranche can still receive positive return after tranche Delta hedging. Therefore, we suggest tranche investor to sell single-name LCDSs to rather than the LCDX index swap to hedge the tranche loss because the tranche Delta computed using the single-name LCDSs is smaller. Generally, the sensitivity of the MTM of a tranche to the change of spread of a single-name LCDS is very small, usually smaller than 1; as a result, using single-name LCDSs to hedge tranche loss could be more efficient.

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