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5 Conclusion
Following the subprime mortgage crisis, market participants raised criticisms that the failure of rating agencies to properly assesses the risky structured instruments.
Therefore, the first part of this paper explores how securitization process to affect the risk profile of tranches. The essence of securitization is that by pooling a variety of assets to reduce the level of concentration of certain asset classes to attained diversification. Under the homogeneous portfolio assumption, the increasing in the number of mortgages will not affect the conditional expected loss of the underlying asset pool but it substantially increase the exposure of tranches to systematic risk.
This result is consistent with the finding of Coval, Jurek, and Stafford (2009 a) who indicate that the securitization process leads the default risk of tranche concentrate in systematically adverse economic states. Ceteris paribus, we further observe that the increase in correlation between mortgages will both raise the underlying asset pool and tranche’s exposure to systematic risk and the systematic risk sensitivity of tranches is much higher than underlying asset pool.
Typically, MBSs is made up of hundreds or thousands of residential mortgages.
Since most of the idiosyncratic risk is diversified away, then losses of tranches are driven by the systematic risk exposure. As a result, the dependence structure between mortgages has become the main ingredient in determining the risk profile of a tranche.
Once the underlying asset pool is highly concentrated in regions due to an unequal distribution of exposures to several regions, this will amplify the effect of dependence structure on loss distribution of underlying asset pool and tranches. In the assessing of concentration risk, we adopt a model-based approach which can deal more explicitly with exposure distribution and dependence structure. We explore how the attachment point of AAA rated tranche is affected under different risk scenarios and with it to evaluate the whether the AAA ratings assigned to structured instruments by rating agencies were reasonable.
We find that the impact of name concentration on attachment point is important for smaller underlying asset pool, and region concentration risk is the main contribution to attachment point for underlying asset pool of all sizes. For example, when we alter the distribution of mortgages from uniformly distributed in five regions to concentrate in a region, the attachment point of AAA will increase 58.14%.
Moreover, other things being equal, attachment point increases with correlations. The rating are difficult to justify when the stochastic recovery model is combined with a high-default-rate situation, the AAA ratings seem too high.
We also observe that the region concentration can indeed be reduced by
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distributing the mortgages into more regions. However, this benefit tends to decrease as the increase of correlation between regions. Especially when the correlation approaches one, then the attachment point is irrespective of the number of regions.
Finally, for the comparison of the effect of different type concentration risk (the concentration risk is induced by exposure distribution across names or regions, intra- and inter-region correlation, respectively) on attachment point, we find that the effect of dependence structure is larger than weight distribution.
To sum up, concentration risk is indeed an important determinant for estimating the attachment point of tranches. In most cases, we can via increasing number of mortgages or widely distributed regions to reduce the concentration risk. For structured financial instrument, pooling of numerous assets allows for broad diversification of idiosyncratic risks. As for the efficiency of increasing the number of regions is affected by the level inter-region correlation. These results suggest that the correlation-induced concentration should not be ignored.
AAA tranches created from subprime residential mortgages convinced investors that the tranche were almost completely free of risk. However, the one-dimensional nature of credit ratings based on expected loss or probability of default cannot fully gauge the riskiness of the structured instruments. On the other hand, the rating of structured instruments involved the application of a model rather than the traditional analysis and judgment methodology. The essence of pooling and tranching make the ratings are highly sensitive to the parameter estimation and modeling assumption. A modest imprecision in evaluating underlying default risks or correlations can dramatically alter the ratings of structured instruments. Therefore, investors should concern about not only the credit rating but also appreciate these securities’ intrinsic risk characteristics.
This paper complements the issue of modeling risk in estimating the internal credit enhancement of structured instruments. The modeling risk mentioned here is the model uncertainty arises from the departure from ASRF framework. We elaborate how this uncertainty is reflected in the design of attachment point of tranches and evaluate the reasonability of ratings. However, the assumption of systematic and idiosyncratic risk factor to be normally distributed is a double-edged sword. The advantage of using normal distribution is that we can combine the methodological framework of Pyktin (2004) to derive an analytic and tractable expression for the quantification and analysis of concentration risk. The disadvantage of normal distribution is that it is not enough to portray heavy-tailed of default correlation in reality. Hence, the extension of using double-t model (Hull and White, 2004) or normal inverse gamma distribution (Kalemanova, Schmid, and Werner, 2007) allows
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the following researcher to address issues of fat tails beyond those allowed by the normal distribution.
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【Essay I】References
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Adelino, M. (2009). Do investors rely only on ratings? The case of mortgage-backed securities. Job Market Paper, MIT Sloan School of Management and Federal Reserve Bank of Boston.
Benmelech, E., and Dlugosz, J. (2009). The credit rating crisis. NBER
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Carling, K., Ronnegard, L., and Roszbach, K. (2006). Is firm interdependence within industries important for portfolio credit risk?.
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Coval, J. D., Jurek, J. W., and Stafford, E. (2009b). The economics of structured finance. Journal of Economic Perspectives, 23(1), 3-26.
Düllmann, K., and Masschelein, N. (2010). Sector Concentration in Loan Portfolios and Economic Capital. SSRN Working Paper Series.
Gordy, M. B. (2003). A risk-factor model foundation for ratings-based bank capital rules. Journal of Financial Intermediation, 12(3), 199-232.
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Kalemanova, A., Schmid, B., and Werner, R. (2007). The normal inverse Gaussian distribution for synthetic CDO pricing. Journal of Derivatives, 14(3), 80-94.
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Rossi, S. P., Schwaiger, M. S., and Winkler, G. (2009). How loan portfolio diversification affects risk, efficiency and capitalization: A managerial behavior model for Austrian banks. Journal of Banking and Finance, 33(12), 2218-2226.
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Moody’s KMV Research Paper.
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Appendix 1.A
In single correlation model, the asset price 𝑥𝑖 is:
𝑥𝑖 = 𝑎𝑖𝑀̅ + √1 − 𝑎𝑖2𝜁𝑖
where 𝑀̅ is the single systematic factor. Therefore, the default in single factor granularity portfolio is:
𝐿̅ = μ(𝑀̅) = ∑𝑁 𝑤𝑖(1 − 𝑅𝑖)
̅
𝑖=1
𝑝̂𝑖(𝑀̅)
Where 𝑝̂𝑖(𝑀̅) = 𝑁 (𝑁−1(𝑝𝑖)−𝑎𝑖𝑀̅
√1−𝑎𝑖2 ) (𝐴1) 𝑀̅ is the single systematic factor, and 𝑎𝑖is the correlation of the 𝑖th asset to 𝑀̅. We can numerically approximation the value-at-risk of single factor granularity portfolio, however, approximating 𝑉𝑎𝑅𝑞(𝐿̅) to 𝑉𝑎𝑅𝑞(𝐿) in multi-factor correlation model is needed. Assume 𝑀̅ is a linear combination by “between-sector” factor, 𝑀𝑏𝑝, and
“within-sector” factor, 𝑀𝑤𝑝,𝑢:
𝑀̅ = 𝑏0𝑀𝑏𝑝+ ∑𝑛𝑢=1𝑏𝑢𝑀𝑤𝑝,𝑢 with ∑𝑛𝑢=0𝑏𝑢2 = 1 𝑓𝑜𝑟 𝑢 ∈ {1, … , 𝑛}
Next, we construct the correlation between composite factor, {𝑀𝑢}𝑢=1,…,𝑛, and 𝑀̅ to determine parameter 𝑎𝑖and 𝑏𝑢such that 𝑉𝑎𝑅𝑞(𝐿̅) sufficiently approximate 𝑉𝑎𝑅𝑞(𝐿)。
The composite factor of the 𝑖th asset in single correlation model is :
𝑀𝑖 = 𝜌̅ 𝑀𝑖̅ + √1 − 𝜌̅𝑖2𝜂𝑖 (𝐴2) where 𝜂𝑖 follows standard normal distribution and is independent to𝑀̅ .𝜌̅ is the 𝑖 correlation between composite factor, 𝑀𝑖, and single systematic factor, 𝑀̅:
𝜌̅ ≡ 𝐶𝑜𝑟(𝑀𝑖 𝑖, 𝑀̅) = 𝑏0√𝛼 + 𝑏𝑢√1 − 𝛼 for all 𝑖 ∈ 𝑢 By (A2), the asset price 𝑥𝑖 can be written in:
𝑥𝑖 = √𝜌𝜌̅ 𝑀𝑖 ̅ + √1 − (√𝜌𝜌̅ )𝑖 2𝜍𝑖, 𝑖 = 1, ⋯ , 𝑁
where 𝜍𝑖 follows standard normal distribution and is independent to𝑀̅, meanwhile, conditional default of portfolio is:
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𝐸[𝐿|𝑀̅] = ∑ 𝑤𝑖(1 − 𝑅𝑖)𝑁 (
𝑁−1(𝑝𝑖) − √𝜌𝜌̅ 𝑀𝑖̅
√1 − (√𝜌𝜌̅ )𝑖 2 )
𝑁̅
𝑖=1
(𝐴3)
From equation (A1) and (A3), if the condition below satisfies, then 𝐿̅ = 𝐸[𝐿|𝑀̅].
𝑎𝑖 = √𝜌𝜌̅ = 𝑏𝑖 0√𝛼𝜌 + 𝑏𝑢√(1 − 𝛼)𝜌 for all 𝑖 ∈ 𝑢
Therefore, parameter 𝑎𝑖 is determined by {𝑏𝑢}𝑢=0,…,𝑘. To make sure 𝑉𝑎𝑅𝑞(𝐿̅) in single factor granularity can be approximated to 𝑉𝑎𝑅𝑞(𝐿) in multi-factor correlation model. We need to choose the optimal {𝑏𝑢}𝑢=0,…,𝑛 by maximizing the correlation between 𝑀̅ and 𝑀𝑖, this means:
𝑚𝑎𝑥{𝑏𝑢} (∑𝑁 𝑐𝑖𝐶𝑜𝑟(𝑀̅, 𝑀𝑖)
̅
𝑖=1
) such that ∑ 𝑏𝑢2 = 1
𝑛
𝑢=0
From Pykhtin (2004), we derive:
{
𝑏0 = ∑ (𝑐𝑖 𝜆) √𝛼
𝑁̅
𝑖=1
𝑏𝑢 = ∑ (𝑐𝑖
𝜆) √1 − 𝛼 𝑢 = 1, … 𝑛
𝑁̅
𝑖=1
𝑐𝑖 = 𝑤𝑖(1 − 𝑅𝑖) ∙ 𝑁 [𝑁−1(𝑝𝑖) + √𝜌𝑁−1(𝑞)
√1 − 𝜌 ]
where 𝜆 is lagrange multiplier which makes 𝑏𝑢satisfies the constraint ∑𝑘𝑢=0𝑏𝑢2 = 1.
In this article, we assume assets in the same sector are homogeneous. This means we can derive 𝐿̅ in every sector since there are same probability of default, 𝑝𝑢, recovery of rate, 𝑅𝑢, and √𝜌 and 𝑎𝑢 separately correspond to composite facto, 𝑀𝑢, and system single factor, 𝑀̅, in the same sector.
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by considering concentration risk:𝑉𝑎𝑅𝑞(𝐿) = 𝑉𝑎𝑅𝑞(𝐿̅) + ∆𝑉𝑎𝑅𝑞∞+ ∆𝑉𝑎𝑅𝑞𝐺𝐴 (𝐵1) By assumptions 1 and 2, we rewrite every proportion in the equation above:
𝑉𝑎𝑅𝑞(𝐿̅) = ∑ 𝜔𝑢(1 − 𝑅𝑢)
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= − 1
2𝑁𝑝̂′(𝑚̅ ) {
{(1 − 𝑅)2[𝑝̂(𝑚̅ ) − 𝑁2(𝑁−1[𝑝̂(𝑚̅ )], 𝑁−1[𝑝̂(𝑚̅ )], 𝜌𝑖𝑖𝑀)] + 𝜎𝑖2𝑝̂(𝑚̅ )}
− 𝑝̂′(𝑚̅ ) {
(1 − 𝑅)2 [
1 − 2𝑁 (
𝑁−1[𝑝̂(𝑚̅ )] − 𝜌𝑖𝑖𝑀̅𝑁−1[𝑝̂(𝑚̅ )]
√1 − (𝜌𝑖𝑖𝑀̅)2 )]
+ 𝜎𝑖2 }
(𝑝̂′′(𝑚̅ ) 𝑝̂′(𝑚̅ ) + 𝑚̅ )
}
||
𝑚̅ =𝑁−1(1−𝑞)
(𝐵4)
where 𝑎 = √𝜌√𝑛1+ (1 −1
𝑛) 𝛼；𝑝̂(𝑚̅ ) = 𝑁 (𝑁−1(𝑝)−𝑎𝑚̅
√1−𝑎2 )；
𝑝̂′(𝑚̅ ) = − 𝑎
√1−𝑎2 𝑛 (𝑁−1(𝑝)−𝑎𝑚̅
√1−𝑎2 )；
𝑝̂′′(𝑚̅ ) = − 𝑎2
1−𝑎2
𝑁−1(𝑝)−𝑎𝑚̅
√1−𝑎2 𝑛 (𝑁−1(𝑝)−𝑎𝑚̅
√1−𝑎2 )；
𝜇′(𝑚̅ ) = (1 − 𝑅)𝑝̂′(𝑚̅ )；𝜇′′(𝑚̅ ) = (1 − 𝑅)𝑝̂′′(𝑚̅ )；𝜌𝑖𝑗𝑀̅ = {
𝜌−𝑎2
1−𝑎2 if 𝑖 = 𝑗
𝛼𝜌−𝑎2
1−𝑎2 if 𝑖 ≠ 𝑗
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How Loan Portfolio Diversification Affects U.S. Banks’ Return and Risk: Correlation and Contagion Perspectives.
1 Introduction
One fundamental implication of traditional portfolio theory is that diversification reduces the variance of returns for a portfolio of assets. Drawing on the analogy, the banking theory suggests that diversification reduces potential bank failures; and based on a delegated monitoring argument, it is optimal for financial intermediaries to be as diversified as possible (see, for example, Diamond, 1984; Diamond and Dybvig, 1986;
Boyd and Prescott, 1986). However, recent studies find aggressive strategies of diversification to be responsible for the banks’ increased risk and impaired return, and suggest that more careful assessments need to be made of the costs and benefits of diversification (Acharya et al., 2006; Berger et al., 2010; Tabak et al., 2011).
This study provides an alternative assessment of the costs of diversification in the presence of dependent relationships among the loan assets, namely, the correlation structure among loan assets within the portfolio and the contagion channels between loan assets cross the border of the portfolio. Based on data collected from the syndicated loan portfolio compositions of individual U.S. banks during 1987-2014, we examine how loan portfolio diversification, in the presence of asset correlation and contagion, affects bank profitability and riskiness.
The dependent relationships under consideration are important for several reasons. First, the HHI is well known for its limitations for neglecting asset correlation. Diez-Canedo (2005) finds that asset correlation affects diversification and proposes a correlation adjusted concentration index in place of HHI. Düllamnn and Masschelein (2010) show that quantifying concentration risk (due to imperfect diversification) without a proper assessment of asset correlation could lead to biased estimates of the economical capital requirements for financial institutions. The economic capitals of portfolios with embedded correlated structures are found to be substantially higher than the ones without. These findings underline the necessity of taking asset correlation into account when measuring diversification. In this paper, asset correlation is referred to as the intra-portfolio correlation with respect to a loan portfolio.
Second, the integration of interbank markets of recent decades provides greater scope of risk sharing, yet at the same time, a channel for cross border contagion (Bonfiglioli, 2008; Fecht, Gruner, and Hartmann, 2012). Cross border contagion results in the amplification of the banks’ idiosyncratic risk exposures which are otherwise assumed fully diversifiable under the Asymptotic Single Risk Factor (ASRF)
‧
framework (Vasicek, 1987; Gordy, 2003) of Basel II IRB guidelines. The conditional independence assumptiom1 behind ASRF attributes the sources of default clustering to observable macroeconomic factors. However, Das, Duffie, Kapadia, and Saita (2007) find evidence of the ASRF failing to fully explain default clustering, suggesting the presence of contagion or frailty2.
One line of the researches of credit contagion is to consider the information transfer effect. Lang and Stulz (1992) and Jorion and Zhang (2007) examine the intra-industry information transfer effects due to bankruptcy events. They find that Chapter 11 bankruptcies have negative impacts on industry peers. Hertzel and Officer (2012) further the investigation by considering how newly issued industry-specific bank loans are affected by the bankruptcy events of their rivals. The other approach to credit contagion is through contractual linkages. This effect arises when the default of one of the parties causes a ripple effect among its business partners. Jarrow and Yu (2001) provide a reduced-form framework where market-wide risk factors and firm-specific counterparty risk interact to affect bond prices. To empirically examine the contractual linkage contagion, Jorion and Zhang (2009) use the bankruptcy file to identify creditors of the filing firms. They find that bankruptcy announcements induce negative abnormal equity returns and an increase in CDS spreads of the creditors.
Hertzel, Li, Officer, and Rodgers (2008) examine the contagion effects of bankruptcy filing and pre-filing distress along the supply chain. Supplier abnormal returns around both the distress and bankruptcy filing of a major customer are significantly negative on average. They further observe that contagion effects spread beyond reliant suppliers to their respective industries. In summary, subsequent to the bankruptcy announcement or financial distress of a firm, its creditors will have a direct loss on current credit exposure. In addition, market participants update their beliefs on the financial conditional of creditors with the expectation that a client’s default will also affect future earnings, i.e., they may further impair by the negative information about sales prospects and this effect can even spread to creditors’ industry. These related studies imply that contagion can hinge either upon information transfer effect or on the capital connections between firms. The question they try to answer is whether idiosyncratic shocks from one particular asset are propagated to the other assets via these transmission channels. Since a complex web of contractual relationships is always present therein with any loan portfolio. Contractual relationships bring about directional information transfer effects and counterparty risk. We argue in this paper that contractual relationships plays a role in the dependence structure, and affects the banks’ risk-return profiles.
1 Otherwise known as doubly stochastic assumption.
2 Unobservable explanatory variables.
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To sum up, assets are correlated because they are jointly exposed to the same macroeconomic factors plus possible other factors, which may include the firm-specific business ties or information transfer. Hence, as regards the dependent relationships which may be caused by both systematic and idiosyncratic structure. The idiosyncratic dependence structure can be used to depict the contractual relationships between assets.
As refer to dependence structure, intra-portfolio correlation is a common indicator for identifying the dependence characteristics of a portfolio and measuring the costs of diversification. However, correlation is composed of the covariance between assets normalized by the square root of the variance of the individual assets involved.
The covariance captures the dependence between assets arising from correlated factors. For example, when a portfolio is composed of two assets and the dependence between these two assets originates only from macroeconomic factors, i.e., there exist no contractual relationship between these two underlying assets. We further assume that each underlying asset has its own distinct counterparties which are not included in the portfolio. The portfolio covariance only measure the strength and the direction of the relationship between these two underlying assets arising from systematic factors, but ignore the idiosyncratic dependence structure of each underlying asset.
The higher the strength of connection between underlying asset and its counterparty, the higher the probability of the value fluctuation of the counterparty may through the idiosyncratic dependence structure to have influence on underlying asset’s value as to the return distribution of the portfolio. However, the higher the strength of connection between underlying asset and its counterparty which is outside the scope of the portfolio will not have any influence on the portfolio covariance measure, but that will cause the higher volatility of underlying asset return and results in lower level of correlation. From the perspective of portfolio management, when the correlation between underlying assets is positive, the higher the volatility of the underlying asset return, the wider the range of uncertainty on the portfolio return distribution. This implies a lower level of diversification. Hence, the only use of intra-portfolio correlation as diversification measure will bias the estimates. Studying the dependence structure is important especially for some underlying assets with extensive linkages would tend to be more prone to external shocks than assets with less.3 In this paper, we take the idiosyncratic dependence structure into consideration
3 Phylaktis and Xia (2009) examine whether unexpected shocks form a particular market, or group of markets, are propagated to the sectors in other countries. The result confirm the sector heterogeneity of
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in the measurement of the degree of diversification.
In the first part of the paper, we extend the standard factor model to include additional latent factors, which are used to depict the idiosyncratic infectious effect.
Furthermore, according to the position of each underlying asset’s counterparties which are included in the portfolio or not, we decompose the idiosyncratic dependence structure into inner and outer transmission channels, respectively. In this framework, we are able to show the impact of different types of transmission channels on the measure of correlation and the degree of diversification.
In the second part we undertake an empirical investigation of how loan diversification affects banks’ profitability and riskiness. To prudent measure the extent of diversification, especially from the perspective of asset dependence structure, we decompose the dependence measure into two categories. The first category aims at intra-portfolio correlation, which measures the dependence between assets within the portfolio. The second category is aimed at the idiosyncratic part of the dependence structure of the underlying assets. However, in this paper we endeavor to investigate the unidirectional contagion effect which is focus on the risk that the fluctuation of an asset which is not included in the portfolio through outer transmission channels triggers the co-movement of its counterparty within the portfolio. Hereafter, we call it customer contagion effect.4 Hence, as proxies of diversification, we consider a traditional exposure weight measure and two dependence measures: the HHI, the intra-portfolio correlation and customer contagion. Each type of measures has different definitions. For example, while the HHI considers diversification as equal exposure to every industry and the intra-portfolio correlation measure uses the equally weighted average pairwise correlation across industries held in the portfolio to construct proxy of diversification, respectively.5 However, before we discuss the customer contagion effect, we have to define what exactly contagion is and how to measure it. In this paper, we follow the definition of contagion as excess correlation,
contagion. One point to note is the information technology stands out as a sector, as it is more globally integrated regardless of its geographic location. On the other hand, utilities sector is less affected by global shocks. This implies that there are sectors that can still provide a channel for achieving the benefits of international diversification during crises despite the prevailing contagion at the market level.
4 Hertzel et al. (2008) indicated that supplier abnormal returns around both the distress and bankruptcy filing of a major customer are significantly negative on average, i.e., the contagion effect are mainly propagated from the customer to its supplier.
5 Campbell, Lettau, Malkiel and Xu (2001) use the equally weighted average pairwise correlation
5 Campbell, Lettau, Malkiel and Xu (2001) use the equally weighted average pairwise correlation