## The Supervisory Formula Approach

Under the Supervisory formula (SF) approach, the capital charge for a securitized tranche depends on five key inputs The IRB capital charge had the underlying exposures not been securitized (KIRB) the tranche's credit enhancement level (L) thickness (T) the pool's effective number of exposures (N) and the pool's exposure-weighted average loss-given-default (LGD). The capital charge is calculated as follows Tranche's IRB capital charge the amount of exposures that have been securitized time the...

## PNx N1p

And, this may be inverted to give actual losses for different quantile where x is the proportion of portfolio losses and a is the quantile level. If, e.g., a is set to 99.9 percent, then the portfolio losses are equal to the amount in this formula, with the twin parameters set to typical levels. The Vasicek model can be used in the context of European mortgage securitizations to identify the likelihood of certain stressed scenarios. By setting the mean of the distribution of the Vasicek...

## Credit Analysis

Quantifying the amount of loss that a mortgage pool will experience in all economic scenarios is the key to modeling credit risk for ratings. To achieve this, analysts use varying stress assumptions to gauge mortgage pool performance in all types of economic environments. The basis for the stress scenario applied to each rating category can be found in the historical loss experience of the mortgage market. Based on studies of historical data, Standard & Poor's developed the criteria embedded...

## References

Chen (2005), Pricing equity default swaps, Risk, June. Alexander, C. (2001), Market Models A Guide to Financial Data Analysis, John Wiley & Sons. Bayliffe, D., and B. Pauling (2003), Long term equity returns, working paper, Tower Perrin. Black, F., and R. Jones (1986), Simplifying portfolio insurance, Journal of Portfolio Management. Black, F., and R. Rouhani (1989), Constant proportion portfolio insurance and the synthetic put option a comparison, in F. J. Fabozzi...

## Systematic Versus Idiosyncratic Risks

We have shown how the return of synthetic CDOs depends on spread movements throughout the life of the transaction and the interaction of hedging and realized spread correlation. Single name spread convexity, while providing an important measure of issuer risk, is not sufficient to fathom CDO trading risk-reward, as the mtm sensitivity to marketwide spread changes (index spread convexity) can have a different sign from the idiosyncratic spread convexity. If all spreads widen together by much...

## ST CDOs A Ratings Perspective

Before moving to the evolution on ST synthetic CDOs, we start by reviewing typical risk assessments conducted in a similar way by most rating agencies (RAs) on the standard, vanilla ST CDO product. RAs, such as Standard & Poor's, Moody's, Fitch, or DBRS, are typically interested in the risk a CDO investor is facing throughout the life of the transaction and base their opinions partly on model-based statistics. For example, Moody's rating is a so-called expected loss rating and, as a result,...

## Credit Event Risk Versus Credit Delta Risk

We have also shown that the positive carry synthetic CDO trades in which the traders wealth increases with time in the absence of any market moves, can be created with little CS01 risk, yet being long credit exposures insofar as the trades have a marginal default sensitivity VOD that is negative, i.e., a loss in the event of a default, for all the names in the CDO reference pool. Additionally, for these positive carry CS01-neutral trades, the loss due to default sensitivity VOD tends to be an...

## Regulatory Changes Parallel To Bis2

Two other regulatory changes are already putting their stamp on the structured finance market. One is the change in accounting practices, the other is the introduction of regulatory capital requirements for insurance companies and pension funds, loosely tailored after BIS1 rather than BIS2 . The accounting changes strike at the heart of securitization practices, affecting off-balance sheet treatment of securitization, accounting for securitization exposures, etc. Given the uncertainty about the...

## Leverage Super Senior Transactions

LSS structures are relatively new products offered in the synthetic CDO market. Their development in 2005 has resulted from a desire by protection The author would like to thank Sriram Rajan, Derek Ding, Benoit Metayer, Lapo Guadagnuolo, and Cian Chandler from Standard amp Poor's for many interesting discussions and numerical support. LSS could be cash funded, too, in which case a detailed modeling of excess spreads and IC OC tests would be required. buyers in the credit market to transfer...

## Figure 612

Quality of Fit of Various Copulas Based on Base Correlation. Implied base correlation iTraxx Burtshell Gregory Laurent 2005 0 -3 0 -6 0 -9 0 -12 0 -22 0 -3 0 -6 0 -9 0 -12 0 -22 Market Gaussian Clayton Student 12 T 4 - t 4 - Stoch. Gauss. MO What we can see is that by trying to fit each copula to the empirical conditional losses in the portfolio, we obtain very different results. In particular, we can observe on Figure 6.11 that neither of the Gaussian, Student-t, and Clayton copulas pick-up...

## Estimating Cumulative Default Rates and Transition Matrices

Stability of Default Rates and Transition Matrices over the Cycle Transition matrices appear to be dependent on the economic cycle, as downgrades and PDs increase significantly during recessions. Nickell etal. 2000 classify years between 1970 and 1997 in three categories growth, stability, and recession , according to GDP growth for the G7 countries. One of their observations is that for IG counterparts, migration volatility is much lower during growth periods than during recessions. Their...

## Corporate Mezzanine Loans

Corporate mezzanine loans are common to many European leveraged loan CDO transactions. These loans have a junior secured position and typically have two components to their interest payments a current-pay coupon and a PIK coupon. The latter coupon is structured in the loan documents to pay in kind from day 1 and accrues to principal in effect, it behaves like a zero coupon bond. Although a mezzanine loan typically has a 10-year tenor, it is quite likely that it will be refinanced within two to...

## Cross Subordination

One of the innovations in the CDO-squared market has been the introduction of so-called cross-subordination. This mechanism allows different bespoke CDOs to share the total subordination provided by all bespoke CDOs. For example, eight CDOs with attachment points and thicknesses of 10 million would create a total cross-subordination of 80 million. During the life of the transaction, if any CDO experiences losses greater than 10 million, these losses are not passed through to the CDO-squared...

## Scenario Loss Rate

The primary risk measure used in our analysis of CDO transactions is the scenario loss rate SLR , which is a quantile of the portfolio loss distribution consistent with a given rating and maturity. For example, if the rating quantile corresponding to a certain rating and maturity is 0.5 percent, the required percentile of the loss distribution will be 99.5 percent. It is important to note that the rating quantiles have been developed specifically for CDO tranches and are not identical to the...

## Review and Risk Tolerance

The analysis of structured finance products and portfolios is a complex undertaking. We highlight a number of criteria in no particular order Granular deals with strong credit quality are less susceptible to event risk of single-name exposures than nongranular deals. Historical evidence suggests that more granular, high quality ABS have experienced little spread volatility compared with low quality granular deals and nongranular deals. These observations are true across ABS capital structures....

## Credit Spread Sensitivity

As previously discussed, a CS01 based risk management framework can provide a good approximation for vanilla credits, while the nonlinearity introduced through tranching creates non monotonic mtm changes for spread changes e.g., market moves versus single-name, idiosyncratic moves . Here, we put the machinery developed in Chapter 7 into practice and examine model CDO trades. At inception, there is little or no CS01 risk, yet if spreads were to blowout on any issuer name, the trade incurs a mtm...

## The Cdo Evaluator Model

The main purpose of the CDO Evaluator model is the computation of the loss distribution of a portfolio of N assets. This is carried out by first simulating the default time of each asset. If the default occurs before the maturity of the CDO transaction, an asset-specific recovery is also computed. If the exposure to each asset at the time of default is known, then the complete distribution of portfolio losses can be computed. In addition to modeling the individual or univariate default and...

## Info

N lt b N lt b Nq, degree of freedom Nu N lt b N lt b Nq, degree of freedom Nu copula from a multivariate set of observations. There are different reasons to account for such a situation A copula summarizes in a stable way the dependencies between the margins. The existence of temporal dependencies in time series does not facilitate the identification of stable patterns. Longin and Solnik 2001 , for instance, identify different dependencies during periods with large movements in returns and more...

## Typologies Of Cdos

It is customary to classify CDOs depending on their function. In this case, usually consider CDOs are in balance sheets and arbitrage deals. The former type of transactions is typically used by financial institutions in order to rebalance their portfolio, whereas the latter focuses on the excess spread generated in the securitized pools because of diversification see Chapter 10 for further details . In the current analysis we focus on a different perspective, i.e., pricing techniques. As a...

## Technical Document Outline

The remainder of this part is divided into five sections. In the section The CDO evaluator model, the underlying mathematical model for CDO Evaluator version 3.0 is outlined, along with the assumptions required by the model for computation of the portfolio loss distribution. The sections Transition and default probabilities, Recoveries, and Correlation outline the data and methods used to estimate these assumptions, which are the main inputs required by CDO Evaluator. The section CDO risk...

## Figure 89b

Is relatively thin at high confidence levels compared to the index for the same amount of initial carry. Such a view is quite different from simply looking at the absolute carry. For instance, the carry-default profile of the different CDO strategies come out to be quite similar on the specific date shown here , despite the absolute carry numbers being widely different. Hence, a proper risk capital calculation based on default risk would render the carry per unit risk capital for these...

## Fundamental Theorems of Asset Pricing FTAP and Risk Neutral Measure

In many occasions in this book, we encounter the concept of risk-neutral measure and of pricing by discounted expectation. We will now summarize briefly the key results in this area. A more detailed and rigorous exposition can be found, for example, in Duffie 1996 . Intuitively, the price of a security should be related to its possible payoffs, to the likelihood of such payoffs, and to discount factors reflecting both the time value of money and investors risk aversion. Standard pricing models...

## Transition And Default Probabilities Rated Companies

For rated companies, we make use of our global CreditPro database of rating transitions and defaults over the period 1981-2003, which contains a ratings history of 9740 companies from January 1, 1981 to December 31, See also Drill-Down Approach for Synthetic CDO Squared Transactions, Standard amp Poor's Special Report, December 10, 2003, available to subscribers of RatingsDirect, our Web-based credit analysis system, at www.ratingsdirect.com. The criteria can also be found on our Web site at...

## Figure 810

Under Hedge CS01 Hedge Over Hedge Under Hedge CS01 Hedge Over Hedge those issuers can incur a significant loss. If all risk management is doing is staring at credit delta or CS01 or equivalent bond market value exposure , then CDO trading can simply become a pretext to sell default protection without any limit, or recognition of opportunities and risks. While the previous analysis dissects the residual risks in CS01 or delta-hedged trades, and presents interesting risk-return tradeoffs carry...

## Practical Risk Management I Pitfalls Of Monitoring Credit Delta Alone

The trading strategies introduced above are particularly interesting in the view of traditional risk management systems. Such systems typically do not address structured credit capital structures and the ensuing credit non-linearity. These risk systems were built for aggregating risks from vanilla credit products, such as corporate bonds or single name CDS, and are designed to monitor delta exposures bond equivalent market values as customary in big bond shops, or delta-notional, respectively ....

## Joint Default Probabilities JDP Approach

Based on the number of transitions to the default state D for sector i and j M'd and M'd, respectively and the total number of assets in sector i and j N' and N', respectively , the average JDP can be estimated as follows with T the maximum number of years and w t the weight at time t. To analyze the impact of the strong growth of the SF market, we estimate equally-weighted that is, w t 1 T and size-weighted that is, w t VNi t Nj t ytT 01 VN' t Nj t average JDP. Implied asset correlation, which...

## Building a Hazard Rate Term Structure

The standard assumption in credit markets is to assume that the hazard rate is a piecewise flat function of maturity, which is sensible given the limited number of observable points on the term structure of credit spreads. Given 1-, 3-, 5-, 7-, and 10-year default swap spread values, we would build a hazard rate term structure with five sections A01, A13, A35, A57, and A710 where Xkl is a short form of Xkl S t in which t denotes the time when the credit spread curve is available. Bootstrapping...

## Identifying the Impact of Cycles on the Tranching of Rated Transactions

Based on the work that has been performed in the past, we know from Bangia et al. 2002 that it is relevant to extract cumulative growth and recession default rates per rating category based on the approximation of first order Markovian transition matrices see Table 4.12 . Based on empirical findings, on an average, default based asset-implied correlation during growth periods is found equal to 4.15 percent, correlation during recession periods amounts to 9.22 percent, and overall average...

## Delta Hedging with a CDS Index Credit01 Sensitivity

In practice, an alternative to hedging each individual name by delta-amounts of single name CDS is to hedge by taking a position in a liquid index such as the CDX or ITraxx indices . The advantage of hedging with an index is that liquidity is very high and bid-ask spreads transaction costs are tight. However, the quality of the hedge depends on how similar the portfolio referenced by the CDO tranche is to the computation of the index. Formally, we define the Credit01 as the change in MtM dollar...

## Convexity of Delta Hedged Tranches

In practice, one is mostly concerned with the convexity of delta-neutral tranches, or portfolios of tranches, index, and single name positions when specific trading strategies are being developed. While a more elaborate discussion of specific strategies follows in the next chapter, we explore important convexity issues for simple delta-hedged equity and senior tranche positions next. Similarly to the definitions in Equations 9 and 10 , the convexity of a single name CDS can be defined as the...

## Credit Spread Convexity Gamma

While first order spread sensitivity is a very important measure of risk, the sensitivity of credit product spread changes beyond 1 bp also needs to be considered. This is especially true when hedging instruments have different leverage, i.e. hedging a tranche with an index, or an equity tranche with a mezzanine or senior tranche. Spread convexity of credit products usually refers to the MtM behavior as a function of the underlying level of credit spreads. Spread convexity, or gamma, of various...

## Figure

Generating the Loss Distribution of the Portfolio In the previous chapters, we have discussed in great detail how to estimate univariate survival probabilities Chapters 2 and 3 as well as recovery rates Chapter 3 and correlation Chapter 4 . Based on these three constituents, we can generate the loss distribution of the portfolio at a defined horizon. The loss distribution in the CDO portfolio is a key input to obtain the tranche loss distribution and, subsequently, the expected loss per...

## Combining Pd And Recovery Models

Investors in credit-risky debt are usually interested in the expected loss or the loss distribution of a given debt instrument. The latter one can be used, in its turn, as an input into a portfolio model for the computation of portfolio VaR, economic capital, or other risk characteristics of a credit portfolio. The loss distribution of a single credit can be computed by combining a PD model and a recovery model. Let us consider a debt instrument with risk factors x this denotes the vector of...

## Elliptical Copulas Gaussian and tCopulas

The Gaussian Copula As recalled earlier, copulas are multivariate distribution functions. Obviously, the Gaussian copula will be a multivariate Gaussian normal distribution. Using the notations of Equation 28b , we can write C , the n-dimensional Gaussian copula with covariance matrix X CGau uv , un NX N-1 u1 , , N-1 Un , 29 with N and denoting, respectively, the n-dimensional cumulative Gaussian distribution with covariance matrix, X and the inverse of the cumulative univariate standard normal...

## Arbitrage Cash CDOs

The arbitrage cash CDO sector includes a number of CDO types, widely differentiated by the type of exposure used to rampup the CDO collateral pool. Among them are cash CDOs comprising high grade and or mezzanine ABS cash CLO of leveraged loans and or middle market loans cash CDOs of insurance and bank trust preferred securities CDO of emerging markets exposures, both sovereign and corporate. Each of these subsectors follows the credit and technical dynamics of its respective market. A CDO...

## Recovery Risk

In the previous sections, we have reviewed various approaches to assess default risk. However, the credit risk that an investor is exposed to consists of default risk and recovery risk. The latter, which reflects the uncertainty associated with the recovery from defaulted debt, is the topic of this section. To date, much less research effort has been made toward modeling recovery risk than toward understanding default risk. Consequently, the literature on this topic is fairly small in volume...

## Statistical Pd Modeling And Credit Scoring

In order to quantify credit risk, practitioners often build models that provide PDs of specific obligors over a given period of time. Alternatively, one often assigns a so-called credit score to an obligor, e.g., a number between 1 and 10 with 1 corresponding to low risk and 10 corresponding to high risk of default. There are two fundamentally different approaches to modeling PDs or assigning credit scores Structural approach also called Merton model Both types of approaches, along with a...

## Performance Analysis for PD Models

There are a variety of measures that are commonly used to quantify the performance of PD models. Many, such as the Gini curve or cumulative accuracy curve CAP and receiver operator characteristic ROC , which we shall discuss next, analyze how a PD model ranks individual obligors. Other performance measures, such as the likelihood, which we shall also discuss next, do not explicitly focus on ranks but rather depend on the PD values that are assigned to obligors. A commonly used measure of...

## Cristina Polizu

As editors, we would like to thank all the contributors to this book Alexander Batchvarov, Sven Sandow, Philippe Henrotte, Astrid Van Landschoot, Olivier Renault, Vivek Kapoor, Varqa Khadem, Francis Parisi, Cristina Polizu, Aymeric Chauve, and William Perraudin. Our gratitude also goes to those who have helped us in carefully reading this book and providing valuable comments. We would like to thank in particular Jean-David Fermanian, Pieter Klaassen, Andre Lucas, Jean-Paul Laurent, Joao Garcia,...

## Overview Of Structured Finance Markets And Trends

The easiest way to highlight the development of the structured finance market is to quantify its new issuance volume. That volume has been steadily climbing all over the world, with U.S. leading, followed closely by Europe, and Japan and Australia a distant third and fourth. The rest of the world is now awakening to the opportunities offered by structured credit products to both issuers and investors and gearing up for a strong future growth. In that respect, it is worth mentioning Mexico,...