Due to different subordination levels, clearly the equity, junior mezzanine, senior mezzanine, junior senior and super senior tranches of LCDSs have quite different risk characteristics. Similar to CDO tranches, the LCDS tranches can be measured in various ways. In this Chapter, we use the three risk measures presented in Gibson (2004) and Chiang, M., Yueh, M., and Lin, A. (2009):
I. The tranche's expected loss (EL) from defaults in the reference portfolio occurring up to the maturity of the LCDS.
II. A level of loss due to default that is one standard deviation above the tranche's expected loss. This is one measure of unexpected loss (UL).
III. The sensitivity of the tranche's value to a change in credit spreads on the names in the reference portfolio as well as the hedge ratios, Delta and Gamma.
These risk measures are defined and computed in the following sections with loss distribution developed using the model described in Chapter 3.
4.1 Credit Risk Measures—Expected loss
Expected losses for tranches can be simply computed through the pricing procedure of LCDX tranche swap, and is widely accepted as a risk measure for credit derivatives.
Since the expected loss for LCDX tranches are express in absolute numbers, it is not able to present the relative risks for tranches of different subordination levels. Thus, here we introduce two more comparable risk measures, the expected loss percentage and leverage of expected loss to address the risk characteristics of specific tranches.
First, we can obtain the expected loss for tranches through the construction of the loss distribution for the reference entity. Then, we define the expected loss percentage as the ratio of expected loss and the notional for a given tranche. Also, the leverage of expected loss is defined as the ratio of the expected loss percentage for a given tranche divided by the expected loss percentage of the whole reference portfolio.
The computations are illustrated below.
expected loss percentage for tranche k = 𝐸[𝐿𝑡
𝑘 ]
𝐷 𝑘 −𝐶 𝑘 (4.1) leverage of expected loss for tranche k = 𝐸[𝐿𝑡
𝑘 ]
𝐷 𝑘 −𝐶 𝑘
/
𝐴𝐸[𝐿𝑡(𝑘)]𝑘 (4.2) Where the 𝐸[𝐿 𝑘 𝑡 ] is the expected loss for tranche k at time t, and 𝐸[𝐿𝑡] is the
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expected loss for the total reference portfolio at time t. Besides, 𝐶 𝑘 and 𝐷 𝑘 are the attachment point and detachment point for tranche k. 𝐴(𝑘) = 𝐷 𝑘 − 𝐶 𝑘 , which is the notional of tranche k.
The leverage ratio determines the relative risk between different tranches. For instance, if the leverage ratio for a given tranche is 15, it supposes that the credit risk for each dollar invested in the tranche is 15 times higher than the credit risk for each dollar invested in the total reference portfolio.
4.2 Credit Risk Measures—Unexpected loss
The unexpected loss is the loss beyond expectation. The amount of the unexpected loss is usually higher than the expected loss by a large proportion. According to Gibson (2004), the unexpected loss of a given tranche is defined as the expected loss of the tranche plus one standard deviation of losses of the tranche. Through the construction of the loss distribution, we can obtain the standard deviation of losses for tranche k under time t as:
𝑆𝐷𝑡(𝑘)= 𝐸 𝐿(𝑘)𝑡 − 𝐸[𝐿 𝑘 𝑡 ] 2 (4.3)
and the unexpected loss for tranche k at time t as
𝐿(𝑘)𝑡
= 𝐸 𝐿 𝑘 𝑡 + 𝑆𝐷𝑡(𝑘) (4.4)
Similarly, the unexpected loss percentage and leverage of unexpected loss can be easily computed as shown in the previous section.
unexpected loss percentage for tranche k =
𝐿 (𝑘)𝑡𝑖
𝐷 𝑘 −𝐶 𝑘 (4.5)
leverage of unexpected loss for tranche k = 𝐿𝑡
(𝑘)
𝐷 𝑘 −𝐶 𝑘
/
𝐿 𝑡𝐴𝑘 (𝑘) (4.6)
4.3 Hedge Ratios
In order to conduct the hedge ratios for loan credit derivatives and their tranched products, first we have to define the Market-to-Market value of the tranches. From a
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investor point of view, Market-to-Market value of a given tranche is defined as the present value of the income of the tranche (premium leg) minus the present value of the loss upon default (default leg).
𝑀𝑇𝑀(𝑘) = 𝑃𝐿(𝑘)− 𝐷𝐿(𝑘) (4.7) where 𝑀𝑇𝑀(𝑘) is the Market-to-Market value of tranche k and 𝑃𝐿(𝑘), 𝐷𝐿(𝑘) are the premium leg and default leg for tranche k.
Then, the sensitivity of the tranche's value to a change in credit spreads on the names in the reference portfolio can be computed , which is expressed below.
𝜕 𝑀𝑇𝑀(𝑘)
𝜕𝑆0 ≈ 𝑀𝑇𝑀 𝑘 𝑆0+ 1𝑏𝑝 − 𝑀𝑇𝑀 𝑘 (𝑆0) (4.8) where 𝑆0 is the tranche spread at time 0.
The initial tranche spread S0 for a given tranche k is the spread that enables the Market-to-Market value of the tranches to equal zero under the information given at time 0. However, as the market spread changes through time, tranches will be affected by significantly different amounts according to their subordination levels. The purpose for tranche investors to involve in hedging is to avoid the Market-to-Market value of the tranches from being affected by the change in market spreads. Using the sensitivity of the tranche's value to a change in credit spreads on the names in the reference portfolio illustrated above, we can further compute the hedge ratios, Delta and Gamma for different tranches.
Tranche Delta and tranche Gamma are the most widely used hedge ratios for tranched products by market practitioners. The tranche Delta is defined as ratio between the sensitivity of the a given tranche's value to a change in credit spreads and the sensitivity of the index value to a change in credit spreads. Whereas, the change in credit spreads can be categorized into (1) Average spread movement of the reference portfolio (index) (2) spread movement of a single LCDS. We can define DeltaLCDSind
and DeltaSingleLCDS for tranche Deltas of category (1) and (2). From a tranche investor point of view, if the tranche Delta is negative, it means that the investor should short the LCDX index to hedge his/her position; in other words, to buy protection. Thus, in this example, the investor should pay premium of the credit spread each year as hedging cost.
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𝛿𝐿𝐶𝐷𝑆𝑖𝑛𝑑 𝑘 = − 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑛𝑔𝑒 𝑖𝑛 𝑀𝑇𝑀 𝑘 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑒 𝑖𝑛𝑑𝑒𝑥 𝑣𝑎𝑙𝑢𝑒
= −
1 𝐴 𝑘
𝜕𝑀𝑇𝑀 𝑘
𝜕𝑠 1𝑏𝑝 1
𝑖𝑛𝑑𝑒𝑥 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙
𝜕𝑖𝑛𝑑𝑒𝑥 𝑣𝑎𝑙𝑢𝑒
𝜕𝑠 1𝑏𝑝
(4.9)
𝛿𝑆𝑖𝑛𝑔𝑙𝑒𝐿𝐶𝐷𝑆 𝑘 = − 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑛𝑔𝑒 𝑖𝑛 𝑀𝑇𝑀 𝑘 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑒 𝐿𝐶𝐷𝑆 𝑣𝑎𝑙𝑢𝑒
= −
1 𝐴 𝑘
𝜕𝑀𝑇𝑀 𝑘
𝜕𝑠 1𝑏𝑝 1
𝐿𝐶𝐷𝑆 𝑛𝑜𝑡𝑖𝑜𝑛𝑎𝑙
𝜕𝐿𝐶𝐷𝑆 𝑃𝑉
𝜕𝑠 1𝑏𝑝
(4.10)
From above, we know that the tranche Delta represents the ratio between the sensitivity of the a given tranche's value to a 1bp average movement in credit spreads and the sensitivity of the index value to a 1bp average movement in credit spreads.
Now, using the tranched Deltas, we can compute the tranche Gamma for a given tranche k, expressed as the change in tranched Delta when there is a 1 bp average movement in credit spreads.
𝛾(𝑘)= 𝜕𝛿 𝑘
𝜕𝑠 1𝑏𝑝 (4.11) where 𝛿 𝑘 is the tranche Delta for tranche k.
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