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Quantification and Analysis of Concentration Risk in Structured Products: The Case of Mortgage Backed Securities
1 Introduction
The rating agencies have encountered serious criticism for overvaluing structured products and underestimating the complexity embedded in structured scheme during the subprime crisis. In particular, Coval, Jurek, and Stafford (2009b) indicate that the modest imprecision in parameter estimates can lead to variation in the default risk of structured finance securities that is sufficient to cause a security rated AAA to have a severe downgrade. Hull and White (2010) further show that the model uncertainty about the choice of correlation model has great impact on the rating of structured instruments. These results indicate that the rating of structured instruments is highly sensitive to parameter estimation and model assumption. One of the most frequently stated benefits of structured instruments is risk diversification, which leads market participants and academic researches inherently ignore the role of concentration and may cause a biased estimation. In this paper, we undertake a model-based methodology for measuring concentration risk which can deal more explicitly with exposure distribution and dependence structure. We take concentration risk into account in the evaluating the reasonability of the AAA ratings assigned to structured instruments.
Let us explain the issue of concentration risk in more detail based on the concept of capital requirement in Basel II. In the portfolio risk factor frameworks that underpin the Internal Rating Based (IRB) risk weights of Basel II, credit risk in a portfolio arises only from two sources, systematic and idiosyncratic. The risk weight formulas for the computation of regulatory capital in IRB are based on the so called Asymptotic Single Risk Factor (ASRF) framework developed by Gordy (2003). This model was constructed under two main assumptions. First, portfolios must be asymptotic fine-grained, which is called granularity assumption, in the sense that the largest individual exposures account for a smaller share of total portfolio exposure, idiosyncratic risk has been fully diversified away, then the portfolio is immune from idiosyncratic risk. Second, dependence across exposures can be described by one-factor Gaussian copula model. Under these assumptions, capital charges are driven entirely by the systematic risk.
However, ASRF framework comes along with some drawbacks as it makes us hard to recognize the diversification effects. Since such a perfectly fine-grained portfolios do not exist due to the lumpy distributions of exposure sizes and a finite number of obligors in real portfolio. This form of credit concentration risk is called
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lack of granularity or name concentration. Thus any capital charges computed under the granularity assumption must underestimate the required capital for a real finite portfolio. The impact of undiversified idiosyncratic risk on capital charges can be assessed via a methodology knows as granularity adjustment. The basic concepts and approximate form for the granularity adjustment were first introduced by Gordy (2003). It was then substantially refined and derived analytical granularity adjustment term by Wilde (2001) and Martin and Wilde (2002).
Empirical researches by Zeng and Zhang (2001), Carling, Ronnegard, and Roszbach (2004) and McNeil and Wendin (2006) suggest that, all else equal, the credit losses associated with exposures to obligors in the same industry are more highly correlated with one another than those associated with exposures to obligors in different industries. A portfolio which can be considered as almost perfectly granular and therefore non-concentrated in the sense of name concentration might be highly concentrated in the sense of sector concentration due to excessive exposure to a single sector or to several highly correlated sectors. Hence, the other drawback of the ASRF model is the assumption of single risk factor can lead to a biased estimation of risk for portfolios with unequally distributed sector structure, which is called sector concentration. Pykhtin (2004) presents an analytical method which builds on earlier work by Wilde (2001) and Martin and Wilde (2002) on granularity adjustments in the ASRF model for dealing with an unbalanced distribution across names and sectors. In other words, Pykhtin model offers a measurement for both name and sector concentration.
There are two empicial papers that consider the impact of concentration risk on economic capital. Gordy and Lütkebohmert (2007) and Düllmann and Masschelein (2010) measure the potential impact of name and sector concentration on the economical capitals of bank loan portfolios in Germany, respectively. They all indicate that concentration risk can meaningfully increase economic capital.
Furthermore, Düllmann and Masschelein (2010) show that the analytic approximation formula developed by Pykhtin (2004) estimate economic capital reasonably well or err on the conservative side.
Furthermore since we have relaized that how concentration risk can potentially impact the loss distribution of a portfolio and the corresponding capital requirement.
The essence of structured finance is the pooling of assets and the subsequent issuance of a prioritized capital strucure of claims, i.e., these claims are issued against the underlying asset pool. Any imprecsion of portray the characteristic of the loss distribution of the underlying asset pool can have a greate impact on the rating or pricing of structured instruments. Hence, in the following, we focus on the
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concentration issue in the context of rating.
Benmelech and Dlugosz (2010) point out that there are 36,346 structured finance securities rated by Moody’s were downgraded (one-third of downgraded tranches bore the highest credit rating level) in 2007 and 2008. Nearly 62% of downgrades can be attributed to securities backed by home equity loans (HELs) or first mortgages.
Collateralized debt obligations (CDOs) backed by asset backed securities (ABSs) accounted for a large share of the downgrades and some of the most severe downgrades. ABS CDOs accounted for 42% of the total write-downs of financial institutions around the world. Therefore, both academics and practitioners blame for the sophistication of structured finance securities and doubt the righteousness and rationality of credit rating agencies.
Nonetheless, in practice, many investors rely heavily on credit ratings for pricing and risk assessment of credit sensitive securities. Recent research by Adelino (2009) examines whether yield spreads on each tranche at issuance contained information, in addition to that in their ratings, that would be useful in predicting performance. He finds only AAA rated tranche did not. Coval, Jurek, and Stafford (2009a) indicate that the securitization process substitute risks that are largely diversifiable for risks that are highly systematic. Moreover, as the result of prioritization scheme used in structuring claims confines senior tranche losses to systematically worse economic states. In particular, the increasing number of assets in the underlying portfolio shifts payoffs to tranche from states with high marginal utility to concentrate on states with low marginal utility. The state payoff transition and the prioritization scheme will cause the senior tranche undercompensated for the highly systematic nature of the risks it bears. In practical, they find that the market prices of senior CDX tranches are significantly higher than risk-matched financial instruments but approximately same as rating-matched ones which implies investors overly rely on credit rating and neglect underlying risk characteristic of senior tranches.
However, since investors are not familiar with the complexity of securitization process as to the risk characteristic of structured finance securities, they are inclined to believe AAA rated securities to be riskless. On the other hand, securitization obey the rule of originate to distribute to create value from spread arbitrage that the higher the principal was allocated to AAA tranche the larger the spreads between weighted average interest earned on the underlying assets and weighted average interest paid on the securitized products. Then issuers have incentive to cater to investor demand and enlarge arbitrage spread by carving out large portions of their deals as AAA.
Moody’s structured product ratings address the expected loss of the securities.
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Standard & Poor’s (S&P) and Fitch are based on probability of default given by their model. Based on the rating rulings, as a tranche was designed, the attachment point (i.e. the minimum loss on the underlying pool that affects the tranche) and the detachment point (i.e. the minimum pool loss that wipes out the entire tranche) are chosen so that the resulting expected loss or default probability of the tranche matches the level required for a desired rating. In this paper we obey the S&P criterion given the probability of default for AAA bond to derive theoretical attachment point (the attachment point that we report reflect the total subordination) of AAA rated tranche.
Specifically, we compared the theoretical subordination with the practical average subordination level of AAA rated tranches to evaluate whether the AAA ratings assigned to structured instruments by rating agency were reasonable. Furthermore, the estimation of the attachment point for AAA rated tranche is equivalent to calculate the Value at Risk (VaR) of a portfolio with the confidence level is given by the survival probability of AAA rated bond. If the theoretical attachment point of AAA rated tranche is higher than practical ones which demonstrates the existing subordination not afford to maintain the AAA rating and reflects the AAA ratings assigned to structured instruments cannot be justified.
Structural instruments are far more complex than bonds. To analysis of the risk profiles embedded in structural instruments, we focus on mortgages backed securities (MBSs), which are created from subprime residential mortgages. Hull and White (2010) also evaluated ratings for this type of products. However, they assumed that a mortgage pool is sufficient large that a large portfolio assumption applies to neglect the name concentration risk and the one-factor Gaussian copula model is used to depict the dependent structure across mortgages which ignore the sector concentration.
We do not make these assumptions. In this paper, we adopt multi-pool correlation model in Hull and White (2010) which allows us to distinguish mortgages into different regions. Since the estimation of attachment point of tranches is equivalent to the VaR of the underlying asset pool, we adopt the multi-factor analytical methodology proposed by Pykhtin (2004), which is mainly used to calculating the economic capital (where the economic capital is defined within the VaR paradigm), to take the name and sector concentration into consideration in calculating attachment point of AAA rated tranche.
To evaluate concentration risk more prudent, this paper comprise two different perspectives. In first part, we pay attention to the unbalanced distributions of exposure sizes across names and regions in the mortgage pool to emphasis on the exposure weight distribution. As to the second part, holding the exposure weight across names and regions fixed, we investigate the impact of concentration risk arise from varieties
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of dependence structure on ratings.
Herfindahl-Hirschman Index (HHI) is the most commonly used model-free concentration measure of portfolio in practical. The HHI is defined as the sum of squares of exposures as a fraction of total portfolio. However, HHI cannot distinguish between a portfolio which is highly concentrated towards an obligor (sector) with a high correlation with other obligors (sectors), and another portfolio which is equally highly concentrated, but towards an obligor (sector) which is only weakly correlated with other obligors (sectors). Therefore, we based on the multi-factor model for quantifying concentration risk which can not only deal more explicitly with exposure distribution but also the dependence structure.
We find that in the process of securitization, although increasing the number of mortgages in the underlying portfolio will not affect the conditional expected loss of the underlying portfolio, it raise the sensitivity of tranches to systematic risk. Ceteris paribus, both of the level of underlying asset pool and tranches exposure to systematic risk will increase with the increasing of correlation between mortgages. These results imply that, as the pooling allows for broad diversification of idiosyncratic risk which leads the expected loss of AAA rated tranche is highly sensitive to systematic risk.
Hence the dependence structure will play an important role in determining risk profile of tranches, especially for an excessive exposure to several highly correlated regions will amplify the effect of dependence structure on evaluating the tranche’s ratings.
Moreover, the fewer the number of mortgages in the portfolio, the higher the proportion in subordinates are contributed by idiosyncratic risk under a given dependence structure. This means that the consideration of name concentration is important for senior tranche backed by fewer mortgages. However, region concentration can meaningfully increase attachment point of AAA rated tranche for all sizes, where as we change portfolio exposure from uniformly distributed into five regions to total concentrated in one-region will lead the attachment point increased by 58.14%. Next, we evaluate concentration risk origin from changing dependence structure. The numerical results show that when correlations are low, the effect of systematic factor is in low level and relatively enhance the effect of idiosyncratic risk on the attachment point of AAA tranche. On the country, with the increasing correlation would rise the sensitive of tranches to systematic risk and the correlation triggered concentration risk would be reflected in enlarging attachment point of AAA tranche.
Additionally, we further examine the interaction effect of exposure distribution across regions and dependence structure on tranche’s risk profiles. We find that the
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widely distributed across regions of mortgages, the lower attachment points of tranches have, which means by distributing mortgages into more regions can achieve the goal of decreasing concentration risk. However, higher correlation between regions will erode the benefit through it. If the regions are perfectly correlated, then the attachment point of AAA tranches is irrespective of the distribution of regions.
Finally, once the recovery rate is negatively correlated with default probability (stochastic recovery rate model), most of theoretical attachment points of tranche will much higher than practical level in market. This suggests that AAA tranche should increase credit enhancement to support its credit rating.To sum up, we infer that the increasing in the numbers of mortgages and regions (evenly distributed mortgages across names and regions) in the underlying asset pool provide a way to lower the extent of concentration risks. Furthermore, the concentration of mortgages on several regions will enlarge the effect of dependence structure on the portfolio loss distribution, and relatively, the increasing in correlation will diminish the benefit of distributing mortgages into more regions.
The paper is organized as follow. In Section 2, we present descriptions of internal credit enhancement and subordination of structure securities. The methodology framework is in Section 3. We construct a multi-factor model involving concentration risk of single asset and sectors, and then calculate the theoretical attachment point for given credit ratings. In Section 4, we provide numerical results and analysis.
Considering concentration risk, we evaluate theoretical attachment points of mortgages backed securities under different scenarios and infer the rationality of AAA ratings assigned to MBSs. Summary and conclusion are in Section 5.