Basic CDS Index Analytics

The biggest difference between entering a portfolio of single-name CDS contracts and entering a CDS index contract is the protection premium. In the CDS portfolio, each single-name CDS has its own fair-market premium, while in the index all contracts share the same premium. The index premium is set on the launch date of the index to create an instrument with a net present value (NPV) near zero. Once spreads move, the index NPV becomes positive or negative.

The NPV of the index can be expressed as a running spread. The index spread is what market makers typically quote for investment-grade CDS indices. For the purpose of converting a CDS index NPV into a running premium or vice-versa, the market standard is to model the CDS index as a single-name CDS contract with a hypothetical reference entity. This hypothetical entity has a flat credit spread curve and is assumed to recover 40% in default. Market participants often use a simple CDS calculator, such as Bloomberg's CDSW, to convert the NPV of an index contract into a spread or vice-versa.

For example, consider the current CDX.NA.IG index contract. This contract matures on Sept. 20, 2009 and has a fixed premium of 60bps. The CDX.NA.IG index market on May 18, 2004 was 66/66.5bps. This implies that the dealer is willing to buy protection for an upfront payment of 19bps inclusive of accrued interest and a running payment of 60bps. Correspondingly, the dealer is willing to sell index protection for an upfront payment of 21bps and 60bps running.

The CDX.NA.IG index NPV can be calculated using Bloomberg by bringing up the CDX.NA.IG index (CDX CDS CORP <GO>), selecting the CDX.NA.IG.2 maturing on Sept. 20, 2009, and typing CDSW <GO>. The index market level is entered in the par CDS spread section of the Bloomberg screen. The index NPV is reported on the bottom of the screen.

The calculation to compute the unwind value of an existing CDS index position is identical. Suppose the investor had bought index protection at 66.5bps, making an upfront payment 21bps. If the index spread immediately widens to 70bps, the investor can unwind the position the same day, receiving a lump-sum payment of 38bps.

Figure 1 shows the historical NPV of the CDX.NA.IG.2 along with the index spread. In the beginning of the period the index spread was below the 60bps fixed premium, resulting in a positive NPV from the perspective of the protection seller. In early May, the index spread increased above the fixed premium, resulting in a negative index NPV.

There is a simple equation describing the relationship between the index NPV and spread. For an index with premium C, the equation is:

The risky DV01 of a credit curve, denoted as RDV01(S), is the risky present value of a 1bps coupon valued on the curve.

 

Calculating Intrinsic Value Of A CDS Index

The difference between a CDS index market spread and the intrinsic value implied by the underlying CDS contracts (the so-called intrinsic basis) is interesting for several reasons. If the intrinsic basis is large enough, investors could try to arbitrage it by simultaneously entering an index contract and offsetting single-name CDS contracts. Even if the basis is not sufficiently wide to warrant an arbitrage strategy, it may be a signal of the market's direction. Since the CDS indices typically lead the market, investors can get a sense of the momentum by tracking the intrinsic basis.

To compute the intrinsic value of a CDS index, we need all of the underlying single-name CDS curves. The first step is to compute the RDV01s for each underlying curve. The intrinsic NPV of a CDS index with N underlying curves is simply the sum of the single-name CDS contract NPVs:

where S_i is the fair-market premium of the n^th CDS contract, and C is the index premium. The single-name CDS contacts are typically off-market--have non-zero NPVs--since they all share the same index premium C.

To convert the intrinsic NPV into an index level, we again use a CDSW-style calculator in reverse: Solve for the spread level such that:

Then, the basis between the index and its intrinsic level is simply S - S^I.

Some market participants approximate the index level by taking a weighted average of the underlying single-name CDS premiums weighted by the relative RDV01s. This approximation neglects the fact that the reported index level is derived by converting the index NPV into a hypothetical single-name CDS premium.

 

Hedging The CDX.NA.IG With HVOL

Suppose we want to isolate the return component of the CDX.NA.IG (which we term the aggregate) that is not in the HVOL sub-index. We will call this artificial low volatility sub-index LVOL. In terms of notional, HVOL constitutes 24% (30 HVOL names/125 names in the aggregate) of the aggregate, while LVOL makes up the remaining 76%.

As a first step, we back out the implied LVOL spread from the aggregate and HVOL spread observed in the market. Using the basic index relationship:

We can observe or directly calculate S^A,S^HVOL, RDV01^A (S^A), and RDV01^HVOL(S^HVOL), allowing us to compute the aggregate and HVOL index NPVs. Subtracting 24% of the HVOL NPV from the aggregate NPV and dividing the result by 76% gives us the LVOL NPV.

Since the RDV01^LVOL(S^LVOL) is a function of S^LVOL, we can back out S^LVOL from RDV01^LVOL(S^LVOL) by solving a non-linear equation:

To find the HVOL weight that best hedges the HVOL component in the aggregate index, we decompose a change in value of the aggregate index into a change in value of the HVOL and LVOL components:

The change in the HVOL aggregate index component NPV can be written as:

The change in the traded HVOL index NPV can be written as:

The difference between SDV01^{HVOL in A} and SDV01^HVOL is caused by the difference in the coupon of the CDS contract. In the case of SDV01^{HVOL in A} the contract coupon is 60bps, while in the case of SDV01^HVOL it is 115bps.

The best HVOL index weight that removes the HVOL return component from the aggregate index is:

The ratio of the embedded HVOL SDV01 to the traded HVOL SDV01 is slightly less than one, because the coupon of the embedded HVOL is lower than the coupon of the traded HVOL.

 

This week's Learning Curve was written by Alex Reyfman, head of credit derivatives research, and Kristina Ushakova, analyst, at Bear Stearns in New York.