A Consistent Approach To The Term Structure of Correlation
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Derivatives

A Consistent Approach To The Term Structure of Correlation

Common market best practice for pricing off-the-run or bespoke collateralized debt obligation tranches involves mapping implied base correlation surfaces calibrated from actively traded tranches such as those on the CDX or iTraxx.

For this week's Learning Curve in PDF format, click here.

Common market best practice for pricing off-the-run or bespoke collateralized debt obligation tranches involves mapping implied base correlation surfaces calibrated from actively traded tranches such as those on the CDX or iTraxx. This was outlined in an earlier Learning Curve (DW, 11/18/05).

Over the last few years, research in this area has resulted in a constant stream of improvements and refinements of this process. Much of this research and refinement has concentrated on the capital structure dimension of the correlation surface.

In particular, a plethora of new methods for mapping attachment and detachment points have evolved, each with particular strengths and weaknesses. All these approaches are incremental improvements on the original method of mapping the expected loss of a bespoke tranche to the matching base correlation of a standard tranche with the same expected loss. This allows base correlations to be applied to portfolios that differ from that of the standard tranches. The success of the mapping depends on how similar in certain dimensions the bespoke portfolio is to that of the standard tranche and how sophisticated the mapping process.

 

The Maturity Dimension

As the market has matured, tranches across the maturity spectrum have become more actively traded. Five, seven, and 10-year tranches are now commonly quoted on the standard indices. The maturity has added another dimension to the correlation skew to form a correlation surface.

The common market practice has been to calibrate a term structure of base correlation for each of the standard indices using the market quotes for the index and related tranches. As part of this process, a number of choices such as dealing with the index basis, the mapping method to be used, interpolation methods and extrapolation methods need to be made. Using this maturity-matching approach, each maturity is calibrated separately. Specifically, to calibrate the base correlation for a seven-year tranche, only seven-year tranche quotes are used.

 

 

 

 

In particular, for a seven-year tranche of notional N, the year base correlation using an expected loss mapping is calculating by solving the previous equation.

In the equation, where LossPV7 is the present value of the loss leg, S is the quoted tranche spread, and Annuity017 is the present value of the premium leg assuming a premium of 1.

 

A Consistent Term Structure Of Correlation

Using the current maturity-matching method, key information about the timing of default implied by the quotes on the five-year standard tranches are ignored for the seven-year tranche. The market has started to adopt an extension of the base correlation framework which includes this term-structure information.

Using the term-structure approach we include the information from the five-year tranche in the pricing of the seven-year tranche. In particular, when we price a seven-year tranche, we use the base correlation implied from the five-year tranches for calculating expected losses to five years, and the base correlation implied from the seven-year tranches for expected losses from five to seven years.

More formally, for a seven-year tranche of notional N, the five to seven-year base correlationusing an expected loss mapping is calculated by solving:

 

 

 

 

 

Where LossPV7 is the present value of the loss leg, S is the quoted tranche spread, and Annuity017 is the present value of the premium leg assuming a premium of 1.

This methodology is used consistently during both the calibration and the pricing phase.

At first glance, the impact of the improved term-structure approach may be expected to be small. In practice, however, the impact can be significant. In particular:

* Pricing of non-standard tranches can differ significantly;

* Hedge ratios and risk sensitivities can differ significantly;

* Pricing for term-structure sensitive products such as interest-only, forward-starting tranches or tranche options can differ significantly;

* The differences are largest for equity and junior mezzanine tranches which are more sensitive to correlation than senior tranches;

* The differences are largest for longer dated tranches;

* The differences are largest for low correlation environments where equity tranches are more sensitive to correlation, and

* The differences are largest for correlation environments where the term structure is steep.

 An example of calibrating base correlation surfaces on the CDX NA IG Series 7 using both maturity matching and term-structure approaches is given. It can be seen that the two approaches differ for the longer dated maturities.

In pricing and calibrating the seven-year correlation, the five-year correlation is used which is lower than the seven-year correlation. The term structure approach therefore pushes risk out of the equity tranches into the mezzanine tranches as higher correlation lowers risk on equity tranches and reduces the break-even spread.

 

Analyzing A Bespoke Tranche

Intuitively, using the term structure approach changes the timing of expected losses when pricing a tranche. To see this we will analyze five and 10-year USD100 million notional, 1-4% bespoke tranches on the CDX NA Series 7 index portfolio.

When comparing the expected losses of the five and 10-year bespoke tranches using both the maturity matching and term structure approaches, we can clearly see the shifting of the expected loss over the life of the tranches. We can also see that the expected loss of the five-year tranche overlays that of the 10-year tranche only when the term structure approach is used. When we use the maturity matching approach, we are using inconsistent expected losses when pricing and analyzing the five and 10-year tranches.

Comparing the pricing and sensitivities of the 10-year bespoke tranche when priced using both maturity matched and term structure approaches shows significant pricing and sensitivity differences.

In conclusion, the term structure approach for base correlation surfaces is an important incremental improvement over commonly used methods for dealing with the maturity dimension of base correlation surfaces. It is particularly important for longer dated equity and junior mezzanine tranches in environments where the correlation term structure is steep.

 

 

 

This week's Learning Curve was written by Rohan Douglas, ceo of Quantifiand adjunct professor of Polytechnic University in New York.

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