Management of interest rate and foreign exchange risk is commonplace. The management of credit risk through the use of large credit derivative portfolio trades is growing rapidly and is the main driving force behind the growth of the credit derivative market. This article looks at the application and valuation of these portfolio derivatives.
The last ten years have seen the growth of ‘portfolio credit derivatives’. These derive from the concept of a collateralized debt obligation (CDO) but, instead of bonds or loans underlying the portfolio, they may include the risk synthetically in the form of default swaps. Thus, the underlying risk portfolio on which default protection is obtained may comprise tens to hundreds of individual reference names (such as Fiat).
The key concept in portfolio protection is the ‘tranching’ of risk into a ‘first-loss piece’, a ‘second-loss piece’ etc. Typically there are between three and seven tranches of risk. Some or all tranches may be rated. The most senior (lowest risk/last-loss pieces) would be AAA, the most junior (first-loss piece/equity piece) is typically unrated but, if it were rated, it would be well below investment grade. Market pricing for individual tranches has been driven by the rating of each tranche (the equity piece is often retained by the originator). The underlying portfolio is typically five-years and between €1-10 billion total exposure.
Users and uses
While the single-name default swap market is the credit derivative’s volume product, it is the portfolio product that has potentially the most economically important role and generates much of the secondary trading in single-name default swaps. For example:
- A commercial bank may have a portfolio of several thousand-loan risks and may need to release capital (either regulatory or economic). By obtaining protection through a portfolio CD, referencing (some or all of) these names, capital is released without having to sell the underlying assets.
- A medium sized bank’s credit portfolio may have concentration risk —typically there is geographical risk (a large percentage of borrowers tend to be close to the head office), industry risk (there may be concentration on the chemical industry) and small number risk (half the exposure may be to ten names only). A portfolio trade may figure in the solution to the diversification of this risk
- An insurance or investment credit portfolio may be invested in Asian risks and wish to swap a portfolio of Asian risk for a portfolio of European risks
- Individual corporates may have large exposures to other corporations via trade finance agreements. A ‘credit bank’ can be used to create a portfolio of credit risks from several such corporations, which can then be further diversified synthetically, and protection obtained in tranched form with the originators taking the first loss piece. This can require less capital on their part, be cheaper and obtain risk diversification benefits.
Elements of valuation
One key variable in the design of portfolio trades is the way defaults (and losses) impact both the outstanding notional on each tranche and the premium being paid on that tranche: the ‘waterfall structure’. The waterfall is far from standardized, has a significant impact on pricing and needs accurate modeling in the valuation of individual tranches.
We have mentioned the rating of tranches and the market’s historical tendency to price off the rating. A ‘financial economics’ approach to the valuation of these tranches would be to price off the hedges — in this case the single-name default swaps underlying the portfolio. All factors relevant to the valuation of single-name default swaps are important — primarily the assessment of the default and the recovery risk and the impact of uncertainty in these variables on the values of each tranche
Two further factors deserve more detailed discussion: liquidity and correlation.
A large commercial bank may have exposure via loans to 5,000 individual names. Of these perhaps 20% trade in the secondary loan or the bond market, though perhaps less than 500 trade regularly enough to be called liquid. Perhaps 250 of the names trade in the default swap market. Yet the bank may wish to Rut (say) 1000 names in the portfolio CDO. How are these illiquid risks to be priced?
As a portfolio trade where the entire risk is transferred to a single counterparty, the key element is diversification. Coupled with a proxy for default risk —which might be the originating bank’s internal rating — this can provide sufficient information for a reasonable guess at the value of the entire portfolio. However, when it comes to valuing individual tranches this is not good enough — the equity piece is driven by the high risk names and the presence of five or ten high risk illiquid names needs to be recognized in the valuation of the tranche. Price-specific information (current bond spreads or new loan spreads) for the risk should be used where available or, when resort has to be made to internal rating to assess risk, recognition should be made of the fact that all ‘BBB’ names (for example) do not trade on the same spread.
The situation becomes even more complex when there are elements of concentration of risk. In any case, the uncertainties in valuation should be tested through stressing the unknown inputs.
When buying risk related to the first (or last) to default of two or more names the `correlation’ of these names becomes important. Two correlations are potentially relevant. Of lesser importance (it turns out) is the correlation between the spreads — or default rates — on the names. For an investment bank, hedging a tranche of a portfolio CDO via single-name default swaps, the hedge ratios will change as spreads change and the correlation of spreads could bring a profit or a loss as the bank re-hedges its risk. But for five-ten-years trades with mostly investment grade equivalent names, this is not a key element. What does have a big impact, however, is the tendency for the default of one name to be related to the defaults of other names. Imagine a situation where there is no correlation between the names. Defaults occur in an intuitive manner and the equity piece takes the brunt of the losses with the senior piece rarely being affected by defaults at all: almost all of the value lies in the equity piece. But now take the extreme case where defaults are highly correlated. Most of the time there are no losses, but occasionally everything defaults together. The senior piece —being larger than the equity piece — is much more valuable.
How do we assess and model this `default correlation’? In reality, what drives this correlation is external events —a US recession, the telecoms funding crisis, the Asian crisis, etc — which have different impacts on different names but whose general effect is to make spreads (and default rates) jump. It is this suddenly higher set of default rates that gives rise to the observed tendency for defaults to occur together.
In practice the form of the joint default distribution is unknown. A standard trick — using a ‘Copula’ function —is used to introduce correlation in a way that is consistent with the known single-name default intensities.
Standard valuation practice
The industry standard models for portfolio pricing are:
- A ratings-driven approach — assigning a rating and pricing off the rating
- An arbitrage-free pricing model based on pair-wise correlations for the underlying names.
The former does not guarantee that the sum of the tranche values equals the sum of the single-name default swap values, and has problems coping with different waterfall structures. Its use has seen the growth of ‘arbitrage CDOs’ where the investment bank makes a profit out of buying tranched protection, particularly the mezzanine pieces, at lower cost than it can sell the CDS protection.
The latter is a more sophisticated model whose advantages are that it avoids arbitrages, and it is tractable (that is, fast to implement). However, the drawbacks are:
- Defaults within this model are triggered by the defaults of individual names — something that is not generally observed in practice
- In the usual implementation a ‘normal Copula’ is used to generate correlated default times. The correlations in this implementation should be ‘asset’ correlations (total assets of the firm). Asset correlations are not directly observable and often equity correlations are used instead, although this is not theoretically justified within the model’s own framework
- Under our view of the world, where default rates are driven by sudden external shocks, we might expect similar names to experience proportionate jumps in default rates. The Normal Copula does not reflect this, but some other Copula functions do.
Portfolio credit derivatives are enormously valuable instruments for the management of credit risk within the banking, insurance, corporate and investment community. The standard modeling techniques can be improved by an alternative means of introducing correlation and by the use of correlations other than equity correlation. Modeling uncertainties remain and a key element in the pricing and risk control of these products will be testing the impact of alternative assumptions.