Reconciling Credit Correlations Andrew Chernih
∗
Luc Henrard
†
Steven Vanduffel
May 11, 2010
Abstract The credit crisis has created a new impetus for regulators to analyse the framework for determining regulatory capital requirements, in particular the assessment of credit risk will be challenged. Confronted with a lack of default statistics it is common practice by industry practitioners to apply a financial approach, also known as Merton’s model of the firm, and we remark that the latter approach also underpins modern solvency standards such as Basel II and Solvency II. However, while Merton’s theory is an academic beauty its implementation does not make full use of available default statistics but relies on the concept of so-called asset correlations instead. We study the different estimates used for asset correlations that have been mentioned in the literature and analyse to which extent these estimates are in line with each other, with available default statistics as well as with our own findings. Our results are in line with the majority of the literature but deviate away from the results reported by some major software providers as well as the Basel II and Solvency II figures. We offer several explanations as an attempt to reconcile the differences and point to several other features that should not be overlooked when building credit portfolio models. Key words: Solvency II, Basel II, Credit risk, dependence, tail-end correlations, worst-case scenario
∗
BNP Paribas Fortis. E-mail:
[email protected] BNP Paribas Fortis. E-mail:
[email protected] ‡ Vrije Universiteit Brussel (VUB). E-mail:
[email protected] †
1
Electronic copy available at: http://ssrn.com/abstract=950483
‡
1
Introduction and Context
In the last decade banks and insurance companies have been investing significantly in the design and implementation of risk management systems. However the financial crisis increases the pressure on financial institutions as well as regulators to challenge and to potentially revise all concepts and models used, and especially credit risk models are in the scope of such a critical review process. As far as credit risk is concerned, it is clear that different obligors usually operate in related socio-economic environments meaning that, at least to some extent, their assets are impacted similarly, pointing to a positive dependence between the default events. Hence one cannot readily resort to the law of large numbers to claim that for sufficiently large portfolios the actual defaults will always be “as expected” hence that no capital will be needed. In contrast capital is necessary to absorb adverse deviations from what is expected during the horizon at hand. To further elaborate on the need for capital, let us consider an infinitely large homogeneous portfolio for a given time horizon, that is, a portfolio with an infinite number of obligors with identical characteristics. Hence all single default probabilities as well as the pairwise default probabilities, denoted by qsingle and qpair respectively, will be similar across the different obligors involved. Furthermore, also the so-called exposures at default (i.e. the maximum losses in case of a default) and the loss given defaults (i.e. the percentage of effective loss upon default) will be equal to each other. These will be denoted by EAD and LGD respectively. By additionally assuming that the EAD and LGD are fixed numbers, and no random variables, it is easily shown that the standard deviation of the portfolio loss S expressed as a fraction of the total amount at risk (also called the Portfolio Unexpected Loss) and denoted by Stdev(S), is equal to Stdev(S) = ρD · Var(L). (1) Here, L is the individual obligor loss random variable (also expressed as a fraction) with variance Var(L) given by Var(L) = LGD2 · qsingle · (1 − qsingle ),
(2)
whereas ρD ≥ 0 is the default correlation (for the given time horizon) between the different obligors: 2 qpair − qsingle . ρ = qsingle · (1 − qsingle ) D
2
Electronic copy available at: http://ssrn.com/abstract=950483
(3)
Hence combining (1), (2) and (3), we observe that Stdev(S) depends on the loss given defaults (LGD), the individual default probabilities (qsingle ) and the pair-wise default probabilities (qpair ). While the process for estimating the qsingle and the LGD may be fairly under control, the intrinsic lack of sufficient default statistics puts a burden on the estimation of qpair , and consequently casts doubt on the accuracy of the estimate for Stdev(S) and ultimately the required capital. Confronted with the apparent lack of default data regulators and financial institutions resort to other techniques to derive the pairwise default probability qpair . In particular it appears (see also Deloitte & Touch’s global Risk Management Survey (2004)) that they often rely on “Merton’s model of the firm” which uses the intuitive idea of a default modelled as an adverse event triggered by asset value moving in the wrong direction. Technically Merton’s model amounts to assuming that for the time horizon under consideration the asset (log-)returns are multivariate normally distributed implying that a default for the i−th borrower (i = 1, 2, ..., n) occurs when the change in asset value Ni drops below a certain threshold. Hence, in the context of our full homogeneous portfolio the multivariate normal character of the asset returns implies that one can express Ni as: Ni = ρA M + 1 − ρA i , (4) so that qsingle = Pr(Ni < c) = Pr( ρA M + 1 − ρA i < c).
(5)
Here, M and the i (i = 1, 2, ..., n) are independent standard normally distributed random variables representing the common systematic factor (also called the state of the economy) and the idiosyncratic risks, respectively. Furthermore ρA ≥ 0 is the (asset) correlation coefficient, c is the constant threshold value and Φ(·) is the distribution function of a standard normally distributed random variable with inverse Φ−1 (·). Of course, joint defaults will be driven by the correlated asset returns and it is easily shown that qpair = ΦρA (Φ−1 (qsingle ), Φ−1 (qsingle ),
(6)
where the notation ΦρA (·, ·) is used to denote the distribution function of a bivariate standard normal random couple with correlation coefficient ρA , see also Crouhy et al. (2000) for more details. Note that in equation (5) the probability qsingle ( = Φ(c)) does not depend on the precise state of the 3
economy M and is therefore unconditional. In contrast, equation (5) can also be used to define conditional probabilities, denoted by qsingle (m), as follows: c − ρA m qsingle (m) = Pr(i < ) 1 − ρA Φ−1 (qsingle ) − ρA m = Φ( ). (7) 1 − ρA The probability qsingle (m) reflects the probability that there will be a default, given that the (future) state of the economy M is known to be equal to m. It is important to observe that under Merton’s firm value model, equation (6) provides an explicit relation between the (unconditional) parameters qsingle , qpair and ρA , which allows one to “translate” default correlations into “equivalent” asset correlations and vice versa. Indeed if default statistics have been used to derive estimates for qsingle and qpair (and thus also the default correlation ρD ), then equation (6) can be used to back out the implied asset correlation ρA . On the other hand if qsingle and ρA are available one can use equation (6) to derive qpair , and thus by equation (3) also the implied default correlation ρD . In this respect we remark that in the literature on credit correlations it is now common to use asset correlations (and not default correlations) to discuss and to compare the different findings. Merton’s firm value model can also be used to derive the loss distribution of the infinitely large homogeneous portfolio exactly and consequently also the maximal loss at a given confidence level p, i.e. the so-called Portfolio Value-at-Risk further denoted by VaRp (S), can be determined. Indeed, using the fact that conditionally on the state of the economy M the default events are independent it follows from the law of large numbers essentially that VaRp (S) = VaRp (E(S|M )) = VaRp (LGD · qsingle (M )),
(8)
so that we find that
Φ−1 (qsingle ) + ρA · Φ−1 (p) VaRp (S) = LGD · Φ( ) 1 − ρA
(9)
The latter formula can be traced back to Vasicek (2001) and is broadly used. In fact many credit risk portfolio models rely on (variants of) Merton’s 4
model of the firm as summarised above. For example Basel II relies on (9) to determine the required capital banks need for their credit portfolios, see Basel Committee on Banking Supervision (page 64-66, 2006) as well as Basel Committee on Banking Supervision (2005). Also the upcoming Solvency II framework is likely to use formula (9), with a minimum level of 50% for the asset correlations, to determine capital charges to cover for exposure to reinsurance or derivative counterparts. We refer to Committee of Insurance and Occupational Pension Supervisors (2008) for an overview of the technical Solvency II guidelines as well as to Doff (2008) for a detailed critical analysis of the whole Solvency II framework. We would also like to point out that under the stated assumptions of an infinitely large homogeneous portfolio formula equation (8) holds generally, so that in these instances capital formulae will always be determined by the properties of the random variable qsingle (M ) essentially. In fact under Merton’s paradigm qsingle (.) is determined by (7) whereas M is a standard normal random variable representing the uncertain future state of the economy, and this gives rise to formula (9) for VaRp (S). In contrast if other assumptions are made with regards to qsingle (.) and M then other capital formulae will obviously result. For example, let us assume that defaults indeed occur according to Merton’s paradigm but that in addition one is capable to predict with certainty the state of the economy, i.e. one knows ex-ante that M will be equal to some m∗ . Then formula (8) will still hold but now we find that VaRp (S) = VaRp (LGD · qsingle (m∗ )) = LGD · qsingle (m∗ ),
(10)
so that VaRp (S) is nothing more than a (conditional) expected loss in this instance. The example makes clear that Value-at-Risk and capital calculations are intimately connected with the design of the rating system used to generate default probabilities. If a so-called through-the-cycle rating system is put in place which generates (unconditional) probabilities qsingle as in (5), where M is unknown, then formula (9) is appropriate to determine Valueat-Risk. In contrast if one has a perfect point-in-time rating system in place were M is known with certainty then formula (10) is correct. In fact, when capital is taken as the difference between the Value-at-Risk and the expected loss, no capital will then even be needed because “everything will evolve as expected”, see also Cornaglia and Morone (2009) for some recent work on this. In this paper we use Merton’s paradigm as the basis and the results of the analysis have to be understood in this context. While it is fair to state that Merton’s idea has enhanced the understanding and the modelling of credit portfolio risk significantly, it needs to be stressed 5
that its implementation in practice relies on asset value data which is not directly observable but can only be derived - through a series of other models and assumptions - from equity value data. Indeed while under the Merton paradigm default statistics are likely to be used to derive estimates for the single default probabilities qsingle , the estimation of the qpair (see (6)) and thus also the portfolio unexpected loss Stdev(S) as well as the portfolio value at risk VaRp (S) (see (8)) is not backed by default statistics fully, but relies on asset value data, see also Crouhy et al. (2000). Nevertheless, whilst the wider availability of asset value data together with Merton’s appealing theory may provide some feeling of comfort one should also keep in mind that ultimately the performance of any credit default risk model needs to be assessed using default statistics. In this respect we agree with Frye (2008) who stated that “many naïve risk managers rely on asset correlations without checking the results”. Hence it appears of interest to review the different estimates used for asset correlations that have been mentioned in the literature, to analyse to which extent these estimates agree with each other, and to offer some potential explanations for observed differences. At the same time some of these explanations shed light on other features that should not be overlooked when building credit portfolio models - there is more than correlations only - and we provide some insight with respect to the impact these may have on the results. As far as we know this is the first study that provides a global overview of the literature findings with respect to the level of asset correlations, and we will compare the results of our study, which is based on a large sample of monthly asset value data, with a series of other studies. These other studies can be largely separated into two categories. The first category uses observed default data to calculate single and pairwise default frequencies from which default correlations can be derived directly. Papers in this category include Gordy (2000), Frey and McNeil (2003), Dietsch and Petey (2004), Jobst and de Servigny (2005) and de Servigny and Renault (2002). The second category of papers is using asset or equity values to obtain estimates for the asset correlations. Papers in the latter category include Duellmann et al. (2006), Lopez (2002), Pitts (2004), Zeng & Zhang (2001) and Akhavein et al. (2005). The last two papers report the findings from Moody’s KMV (MKMV) and Fitch Ratings respectively. We find that the results of our study are broadly in line with the majority of the literature and there are several plausible reasons already documented in the literature to explain observed differences. On the other hand all these 6
results deviate away from the results reported by MKMV and Fitch Ratings as well as the Solvency II and Basel II figures. We offer some potential explanations as an attempt to reconcile differences. Firstly, we provide evidence that credit risk modelling is subject to significant modelling risk implying that some stakeholders for reasons of conservatism simply prefer to use higher correlations than what is actually observed. In particular this may explain the high level of correlations used in Basel II and Solvency II. Secondly, we analyse two sources of dependence that may exist in a credit portfolio and that are typically overlooked by models, i.e. dependent LGD’s and the presence of group effects. Consequently, this may also explain to some extent why some software providers, rating agencies as well as regulators use higher than observed correlations. The paper is further structured as follows. Section 2 gives an overview of the results that appeared in the literature whereas Section 3 describes the results we obtain. Section 4 discusses all the various results and offers some explanations as an attempt to reconcile the observed differences. Finally, Section 5 concludes.
2 2.1
Asset Correlations from the Literature Asset Correlations using Default Data
In Table 1 we report the asset correlations from a variety of studies which have used observed default data. When the equivalent asset correlations were not mentioned in the study explicitly we have used equation (6) to back these out from the reported default correlations. We point out that all (inherent) default correlations are unconditional, and also that the above mentioned studies have used a variety of methods to estimate these. For example both Gordy (2000) and Cespedes (2000) appear to first estimate the standard deviation of the portfolio loss corresponding to the population at hand, and next rely on equation (1) to derive the default correlation. Other studies such as Hamerle et al. (2003a) also use a straightforward model-free technique; for more details see Lucas (1995). In addition to what is mentioned in Table 1 we note that Hamerle et al. (2003a) used the same data as in Boegelein et al. (2002) which comprised default data from Canada, France, Germany, the UK, Italy, Japan, South Korea, Singapore, Sweden and the US. Furthermore, we remark that there is also a study by Vassiliev (2006) which uses default data from UBS to calculate 7
Source Study
Default Data Source
Results
Gordy (2000, Table 2)
S&P
1.5% - 12.5%
Cespedes (2000)
Moody’s
10%
Hamerle et al. (2003a)
Unknown
max of 2.3%
Hamerle et al. (2003b, Table 1)
S&P 1982-99
0.4% - 6.04%
Frey & McNeil (2003, Table 1)
S&P 1981 - 2000
6.5% - 6.9% - 9.1%
Dietsch & Petey (2004)
Coface 1994-2001
0.12% - 10.72%
Jobst & de Servigny (2004)
S&P 1981-2003
4.7% - 14.6%
Duellmann & Scheule (2003)
DB 1987 - 2000
0.5% - 6.4%
Jakubik (2006)
BF 1988 - 2003
5.7%
Table 1: Asset correlations derived from default data S&P: Standard and Poor’s DB: Deutsche Bundesbank BF: Bank of Finland asset correlations which “are in the range reported in external studies (e.g., Dietsch and Petey, 2004)”. From Table 1 we can conclude that the different results, while using different estimation techniques to cope with default statistics from North America, Canada, Switzerland, France, Germany, Finland, the UK, Italy, Japan, South Korea and Singapore covering different data periods as well, yield broadly consistent correlation estimates in the range of approximately 1% - 12%. Note that the default data used in these studies was typically available at a 1 year horizon meaning that the implied asset correlations correspond to a one time year horizon in these instances as well.
2.2
Asset Correlations using Asset Value Data
Using asset value data as input, one can directly estimate asset correlations in a straightforward way (and also move to default correlations if necessary). There have been several studies which have used asset value data from various sources and the results are presented in Table 2.
8
Source Study
Asset Data Source
Results
Duellmann et al. (2006)
MKMV Credit Monitor
10.1%
Zeng & Zhang (2001)
MKMV source
9.46%-19.98%
Akhavein et al. (2005)
Equity
20.92%-24.09%
Lopez (2002)
MKMV Portfolio Manager
11.25%
De Servigny & Renault
Equity
6%
Table 2: Asset correlations derived from asset value data Note that, in contrast to studies based on default data, most asset value studies appear to use a weekly or monthly horizon to calculate asset returns and to derive asset correlations. Duellmann et al. (2006) however calculate 2 year asset correlations using rolling 24-month time windows. Since their study only used 8 years of asset return data (which means 4 distinct 24-month periods), this may be insufficient data for correlation estimates. Akhavein et al. (2005, Fitch Ratings) use equity values and a proprietary software to report 5 year correlations. Zeng & Zhang (2001, MKMV) do not release full details regarding how they build their asset correlation model nor the data used and it is designed to be used in conjunction with their Portfolio Manager software. Lopez (2002) also does not disclose details of the asset data used, only that MKMV Portfolio Manager was used for the analysis. Presumably MKMV Portfolio Manager is based on the same raw data (MKMV Credit Monitor) we also used in our analysis but there is no formal guarantee of this. We also point out that De Servigny & Renault (2002) effectively utilised equity correlations to obtain an average correlation of 6%. In addition to the information comprised in Table 2 we can mention that Basel II uses asset correlations ranging from 8 to 24%, the exact value depending on individual firm characteristics such as rating and asset size, to determine the required capital banks need for their corporate credit portfolios, see also Basel Committee on Banking Supervision (page 64-66, 2006). Finally, the upcoming Solvency II framework is likely to set a minimum level of 50% for the asset correlations when determining capital charges to cover for exposure to reinsurance or derivative counterparts, see Committee of Insurance and Occupational Pension Supervisors (2008). The results in Table 2 suggest that as compared to using default data the use of asset value data to estimate asset correlations appears to give rise to slightly higher estimates in general, see also Frye (2005) for some comments in the same vein. Gordy and Heitfield (2002) have shown that using the 9
specifications of Merton’s model in the statistical procedure for estimating asset correlations based on default data is likely to yield higher results in general, see also Miu and Ozdemir (2007) for related work in the context of estimating default probabilities. Hence using model-free methods to estimate asset correlations based on default data may be subject to some downward bias hence explaining why the asset correlations from Table 1 tend to be lower in general. Nevertheless, such downward bias is only really correct and meaningful if the real default data process is indeed (broadly) in line with Merton’s theory which is, due to scarcity of observed default data, difficult to prove in a statistically meaningful way. In other words, if real defaults do not occur according to the Merton mechanism then there is no real guarantee that the estimates in Table 1 are too low, and one should also keep in mind that model-free procedures are by nature more robust against model specification. Finally, we also notice that the studies from Zeng & Zhang (2001, MKMV) and Akhavein et al. (2005, Fitch) show significantly larger results than any other study reported so far.
3 3.1
Our Asset Correlations Data used
The source of the asset value data we used is MKMV Credit Monitor. Several papers have used the same raw data source, such as Pitts (2004) and Duellmann et al. (2006). In both papers it is stated that this asset data should be corrected for the impact of corporate actions and potential data errors. Therefore, the data was cleaned to remove outliers, asset values were adjusted for debt issues and buybacks and those months where asset value data were not available were removed. Then the monthly returns were calculated as the ratio of the ending (market) asset value minus the value of liabilities issued during that month to the starting (market) asset value. This meant that for each of the 20,144 companies, we had up to 107 months of return data available in the time period March 1998 to March 2007. To the best of our knowledge, this is the largest sample that has ever been used for an asset correlation study. When there was at least 40 months of data in common for a pair of companies, a correlation was calculated. The companies were aggregated into 336 clusters based on asset size, activity sector, probability of default and world region. Previous studies have shown some evidence that these are 10
factors that may differentiate asset correlation; see amongst others Lopez (2002), Duellmann and Scheule (2003) and Dietsch and Petey (2004).
3.2
Our main findings
With our 336 asset clusters as described above, the average intra-cluster asset correlation was 11.1% (this is obtained by calculating for each cluster the average of all pairwise correlations that exist within such a cluster and next averaging out across all such clusters) and the average inter-cluster asset correlation was 6.3% (this is obtained by calculating for each pair of clusters the average of all pairwise correlations that exist between pairs of assets belonging to two different clusters, and next taking the average across all combinations of two different clusters). A graph of the inter-cluster correlation grouped by asset size band and client rating is provided in Figure 1. As expected, the correlations are increasing in asset size (because the higher the asset size the more systemic the obligor’s risk profile typically becomes). Moreover in line with common expectations we see that correlations are decreasing in probability of default (because there is an inverse relation default probability and asset size in general).
4
Discussion
The results we obtain from using the monthly asset return data are consistent with the majority of the values from literature and it is fair to state that the results that have appeared in the literature are very closely aligned, with the exception of Akhavein et al. (2005, Fitch Ratings) who reports equity correlations as a substitute for asset correlations using return data spanning a 5-years horizon, and Zeng & Zhang (2001, MKMV). As such, on the one hand there appears to be a growing consensus in literature on the range for asset correlations as such, on the other hand the question rises why Fitch Ratings, MKMV as well as Basel II and Solvency II are using significant different values. Hereafter we offer a series of possible explanations. At the same time some of these explanations shed light on other features that should not be overlooked when building credit portfolio models - correlations are not the only component - and we provide some insight with respect to the impact these may have on the results. 11
AAA - BB+
BB - B-
B+
B-C
8.00%
Average Asset Correlation
7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% VERY SMALL
SMALL
MEDIUM
LARGE
Asset Size Bands
Figure 1: Asset correlations by S&P rating and asset size band. The four default probability groups, corresponding to S&P ratings, are in increasing order: blue, green, yellow and red.
4.1
Correlations do not measure dependence fully
Let us first remark that any given set of assumptions regarding the LossGiven-Defaults, Exposures-at-Default, single default probabilities and asset or default correlations will never be complete enough to estimate the portfolio loss distribution with certainty. Indeed, since correlations are related to individual and pairwise but not multiple default probabilities they do not provide a full picture of the dependence that exists within a portfolio. In fact all joint default probabilities need to be determined when estimating a credit loss distribution and the Portfolio Value-at-Risk in particular. Unfortunately however even estimating a triple-wise default probability based on default data is a cumbersome exercise meaning that in a credit context obtaining a full picture of dependence is far from being a trivial task, with negative impact on the potential accuracy of the estimated loss distribution. This is the reason why most companies make an implicit or explicit assump12
tion regarding the dependence structure, and the assumption of multivariate normally distributed asset returns as done in Merton’s model of the firm is just an (influential) example of such practice. Within this respect, we point out that it is even perfectly possible to construct models that preserve a given Stdev(S) while providing very different results for the Portfolio Value-at-risk VaRp (S). Indeed let us now assume that the normalised asset returns Ni evolve as follows: Ni = M · IM 1 − qsingle we will find that the portfolio Value at Risk at the p−confidence interval, i.e. the VaRp (S), is exactly equal to the maximal loss one can possibly suffer, i.e. when all obligors default! For example, let us consider a homogeneous portfolio with qsingle = 1.5%, LGD = 100% and ρA = 7%. Then using Monte Carlo simulation we find that Φ(−d) is approximately equal to 99%. Hence, when p > 99% we find under the model specification (11) for the asset returns that VaRp (S) = 100%, whereas using the (Merton) specification (4) we find using formula (9) that VaRp (S) = 6.12%. Hence, while the parameters for LGD, EAD, qsingle and ρA are exactly the same under both models, the results produced are substantially different, see also Chernih et al. (2008) for some more information in the same vein. Confronted with significant intrinsic model error some stakeholders may for reasons of conservatism just prefer to use higher correlations than what is actually observed. In particular this may explain the high level of correlations used in Basel II and Solvency II.
4.2
Effect of dependent loss given defaults
Furthermore, even the calculation of the portfolio unexpected loss is impacted by (explicit or implicit) modelling assumptions significantly. Indeed, the computation of portfolio unexpected loss is dependent on whether the loss given defaults are deterministic or stochastic, dependence between these, and 13
possible dependence between default events and loss-given-defaults. We will focus on the issue of dependent loss given defaults. To this end let us consider again an infinitely large homogeneous portfolio. However, while we still assume that the EAD’s are fixed we will consider three different methods to deal with the LGD’s. The first - and most sophisticated - model assumes stochastic dependent loss given defaults, the second model assumes stochastic independent loss given defaults and the third assumes deterministic loss given defaults. It is very likely that the first model enables one to provide the most accurate description of the portfolio risk. Indeed, different LGD’s are likely to be positively related to each other because they are driven by asset value processes which are likely to be positively correlated as well. All other things being equal using the second and third model for LGD will result in a lower estimate for the portfolio unexpected loss, i.e. Stdev(S), than the first model. Or, to achieve the best estimate of the unexpected loss (defined as the result from the first model) the other approaches will need to use higher asset correlations than the ones used in the first model. This naturally leads to the question: how much higher? From Dhaene et al. (2005) we have the following equation for the loss correlation ρL between the individual random credit losses: ρL = where
A+B Var(L)
(12)
LGD 2 A = ρD · qsingle · (1 − qsingle ) + qsingle ρ · Var(LGD)
(13)
B = ρD · qsingle · (1 − qsingle ) · E2 (LGD).
(14)
and Here ρLGD is the correlation between the loss given defaults for two distinct credit risks and E(LGD) is the expected value of the random loss given default for and Var(LGD) is its variance. Furthermore, the variance of an individual loss, i.e. Var(L), will now be given by: Var(L) = E2 (LGD) · qsingle · (1 − qsingle ) + qsingle · Var(LGD).
(15)
For the infinitely large portfolio as described above, it can be shown that the Portfolio Unexpected Loss expressed as a fraction of the total amount at risk equals: Stdev(S) =
ρL · Var(L). 14
(16)
We can calculate Stdev(S) with the first model and then calculate what the required asset correlation is for models two and three to reach the same Stdev(S). Note that (16) reduces to (1) when ρLGD = 0 meaning that for an infinitely large portfolio it makes absolutely no difference whether loss given defaults are stochastic and independent or deterministic. Hence the required asset correlations for models two and three will be the same. We calculated the “corrected” asset correlations for various values of qsingle , σ 2 (LGD) and ρLGD . The E(LGD) was fixed at 50%. 8 results are presented in Table 3. For example the first row considers a portfolio of obligors with a qsingle 0.21%, ρLGD of 25% and σ 2 (LGD) of 25%. The original (estimated) asset correlation is 13.96% but an asset correlation of 16.84% needs to be used to keep the same value of Stdev(S). qsingle 0.21%
ρLGD
Var(LGD)
Original asset Required asset correlation
correlation
25%
25%
13.96%
16.84%
0.21% 100%
25%
13.96%
23.32%
0.21%
25%
4.2%
13.96%
14.48%
0.21% 100%
4.2%
13.96%
15.94%
9.75%
25%
25%
8.45%
16.88%
9.75% 100%
25%
8.45%
37.53%
9.75%
25%
4.2%
8.45%
9.93%
9.75% 100%
4.2%
8.45%
14.18%
Table 3: With an infinitely large portfolio of obligors with a given qsingle , ρLGD , V ar(LGD) and asset correlation, what is the corrected asset correlation we need to use if we assume ρLGD = 0 to keep the same Portfolio Unexpected Loss
From Table 3 we observe that relative change from the true asset correlation to the corrected asset correlation (to account for dependent LGD’s implicitly) can be tremendous. The effect of dependent LGD may also explain the possible need to use higher asset correlations in portfolio models. Whilst one might claim to be measuring asset correlations, in fact this is used as an input to calculate the 15
overall credit loss distribution. That is, whilst best estimates of asset correlations might typically be in the region of 0%-10%, to obtain the best estimates of credit loss distributions might require using higher asset correlations (to account for other sources of dependencies). For example, in the MKMV Portfolio Manager software, the loss given defaults appear to be always modelled as mutually independent random variables which might explain a need for an upward adjustment of the originally derived asset correlations to prevent potential underestimation of the Portfolio Unexpected Loss.
4.3
Group effects
Another source of dependence that may exist in a portfolio and that is overlooked by most models is that of group effects. A credit risk portfolio often has individual policies which have strong legal or economical ties. A conglomerate (mother company) may be composed of different legal entities (daughter companies) and a default of the former may lead to the default of all others. In fact, such dependence considerations are often made when assessing the default probabilities within a group of related companies: the default of a daughter company may be prevented by the presence of the mother company and as such one often attributes, in the presence of group-effects, a better rating (and hence a lower default probability) to the different daughter companies as compared to a stand-alone assessment for the rating. Unfortunately, while such group effects are often considered to model the individual default probabilities they are almost never considered when modelling the dependence which leads to underestimating the risk. Indeed Vanduffel et al. (2008) used comonotonic theory (see Vanduffel (2005)) to extend a celebrated actuarial collective risk model (see Panjer (1981)), known in Industry as CreditRisk+ for such group effects. Their numerical examples showed that such group effects can have a significant impact on loss distribution, especially in the tail. Group effects are likely to increase the need for capital but most credit risk models fall short in capturing such effects. Hence some stakeholders may opt to consider the effect implicitly by increasing the levels of the asset correlations artificially.
4.4
Effect of Horizon
The correlation estimates reported in the literature do not necessarily all correspond to the same risk horizon and this may create bias when comparing 16
the results. Indeed a priori, it is not obvious that observed asset returns involve the same correlation across different horizons. While this would be true when asset returns are multivariate normally distributed this is by no means a general truth. Also default correlations may vary depending on the length of the horizon used. The effect of horizon has been studied for equity correlations (e.g. Koyluoglu et al. (2003)) but less so for asset correlations. In large part this is due to data limitations. Default data is generally only available to use at a 1- or 5-year level. In Jobst and De Servigny (2005) whilst default correlations were observed to increase with increasing horizon (using 1, 3 and 5 year time periods) the probability of defaults also increased keeping the asset correlation broadly constant. However in de Servigny and Renault (2002) some evidence for increasing asset correlations was found. Moving from an one year time period to a longer time period shows inconclusive evidence. Using asset returns requires the use of shorter time periods due to less data being available. Given that it is generally advised to use at least 50 values to estimate a correlation, using annual returns is impossible with the number of years of asset return data available. The commonly used options are to calculate either weekly or monthly asset returns and then use these as annual asset correlations. Some studies such as Duellmann et al. (2006) use rollingwindow time periods to calculate correlations, that is, where overlapping time periods are used however this does not increase the effective dimension of the calculations. Using rolling 24-month time periods with 8 years of data may lead to more observations, but consecutive observations are now built on almost identical data, differing only by one month at the beginning and one month at the end. One still has only 4 distinct 24-month periods. Our study was based on monthly asset returns and the results suggest that using monthly asset correlations as a proxy for asset correlations gives results that are in line with default correlations observed from default data. Generally speaking, we believe that the statistics used to parameterise the model should be aligned with the risk horizon chosen. The impact of horizon on asset correlations is certainly a topic that requires further research.
4.5
Effect of rating system
From the introduction we recall that rating systems and capital frameworks should be aligned in principle. Indeed, in the context of an infinitely large portfolio, equation (8) will hold more generally and reveals the close connection between the Portfolio Value-at-Risk VaRp (S), and the stochastic default probability qsingle (M). Rating systems may differ with respect to the 17
way m → qsingle (m) is modelled as well as with respect to the modelled stochasticity of M . For example, under a perfect point-in-time rating the stochasticity of M vanishes so that no deviations around the expected losses will be observed (for an infinitely large portfolio), and no capital is needed to account for possible shocks (as it will be accounted for in the provisions). In fact formula (9) is only fully appropriate when Merton’s model of the firm has been used to generate (unconditional) probabilities qsingle . Financial institutions often rely on internal rating systems to determine default probabilities, and on (Merton’s) formula (9) to determine capital requirements. As the underpinning frameworks used do not comply with each other per se they may for reasons of conservativeness opt for increasing the levels of the asset correlations used in formula (9) artificially. This may also contribute to explaining why Basel II and Solvency II are using higher asset correlations than observed, but still does not seem to provide a satisfactorily explanation for the higher levels of asset correlations used in Moody’s MKMV.
5
Final Remarks
The use of observed default data is a priori the best source to estimate single and pairwise default frequencies, and also default correlations can then be derived from this. However, confronted with a lack of sufficient default data, banks, insurance companies as well as software providers often resort to equity or asset correlations. These correlations are then transformed into default correlations by means of the so-called Merton Model of the firm. Whilst the additional sophistication and availability of asset value data may then provide some feeling of comfort, this comes at the cost of significant model risk. Indeed asset data itself is not observable but needs to be derived from equity data using option models, see Crouhy et al. (2000). The discussion on which kind of asset correlations to be used and in how much detail they need to be modelled is in our view a bit off topic. Credit risk portfolio models are by nature subject to tremendous model risk anyway, and correlations are only one piece of a very difficult puzzle which unfortunately can never be fully completed. Within this respect we believe that complex models are by no means a guarantee for more accurate results. For example in view of the considerations made before it appears fair to say that the added value of a sophisticated multi-factor model for asset correlations on the accuracy of capital calculations is questionable. 18
In our opinion credit portfolio models should be kept as simple and transparent as possible, capturing essential drivers such as correlation, dependent LGD’s, and group effects in a rather straightforward and consistent way, and making use of default (loss) statistics as much as possible (including the use of externally available default data). An example of a candidate credit portfolio model that appears to meet several of these criteria is the (one-factor) CreditRisk+ model, introduced in the industry by Crédit Suisse and in actuarial circles known as an example of a collective risk model. Indeed, the natural parameterisation of this model is based on default statistics, calculations can be done analytically, and features such as group dependence and dependent LGD’s can be readily accounted for, see for example Vanduffel et al. (2008) and Vandendorpe et al. (2008) for some further information.
6
Acknowledgement
The authors would like to thank Alan Pitts and Karl Rappl (UBS), Jon Frye (Federal Reserve Bank of Chicago), Ivan Goethals (ING) and Bruno de Cleen (Rabobank) for helpful discussions and comments on an earlier draft.
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