Copula-Based Factor Model for Credit Risk Analysis
CCopula-Based Factor Model for Credit Risk Analysis ∗ Lu, Meng-Jou † Chen, Cathy Yi-Hsuan ‡ H¨ardle, Karl Wolfgang § This version: October 7, 2020
Abstract
A standard quantitative method to access credit risk employs a factor model based on joint multi-variate normal distribution properties. By extending a one-factor Gaussian copula model to make a moreaccurate default forecast, this paper proposes to incorporate a state-dependent recovery rate into the con-ditional factor loading, and model them by sharing a unique common factor. The common factor governsthe default rate and recovery rate simultaneously and creates their association implicitly. In accordancewith Basel III, this paper shows that the tendency of default is more governed by systematic risk ratherthan idiosyncratic risk during a hectic period. Among the models considered, the one with random fac-tor loading and a state-dependent recovery rate turns out to be the most superior on the default prediction.
JEL classification:
C38, C53, F34, G11, G17
Keywords:
Factor Model, Conditional Factor Loading, State-Dependent Recovery Rate ∗ This is a post-peer-review, pre-copyedit version of an article published in Review of Quantitative Finance andAccounting. The final authenticated version is available online at: https://doi.org/10.1007/s11156-016-0613-x † Department of Information and Finance Management, Institute of Finance and Institute of Information Management,National Chiao-Tung University, No.1001 Daxue Rd, Hsinchu City, Taiwan. Ladislaus von Bortkiewicz Chair of Statistics,Humboldt–Universit¨at zu Berlin, C.A.S.E. – Center for Applied Statistics and Economics, Unter den Linden 6, 10099Berlin, Germany. E-mail: [email protected] . ‡ Corresponding author. Department of Finance, Chung Hua University, 707, WuFu Rd., Hsinchu 300, Taiwan.Ladislaus von Bortkiewicz Chair of Statistics, Humboldt–Universit¨at zu Berlin, C.A.S.E. – Center for Applied Statisticsand Economics, Unter den Linden 6, 10099 Berlin, Germany. E-mail: [email protected] . § Ladislaus von Bortkiewicz Chair of Statistics, Humboldt–Universit¨at zu Berlin, C.A.S.E. – Center for AppliedStatistics and Economics, Unter den Linden 6, 10099 Berlin, Germany. Sim Kee Boon Institute for Financial Eco-nomics, Singapore Management University Administration Building, 81 Victoria Street, 188065 Singapore. E-mail: [email protected] . a r X i v : . [ q -f i n . R M ] O c t Introduction
The global economy has repeatedly observed clusters of default events, such as the burst of the dotcombubble in 2001, and the financial crisis from 2007 to 2009. The clustered default has been attributedto systematic risk which plays a crucial role in the default event. To discover this issue, numerousstudies emphasise the role of systematic risk by employing a factor model (Andersen and Sidenius,2004; Pan and Singleton, 2008; Rosen and Saunders, 2010). The factor model is a prevalent way tocapture the obligors’ shared behaviour through a joint common factor, and to reduce the dimension ofdependence parameters which benefits bond portfolio management. However, one can still find someunrealistic settings on this method such as a constant and linear dependence structure with thin tailsof risk factor distribution embedded.The factor copula model imposes a dependence structure on common factors and the variablesinterested. In credit risk measurement, the factor loading represents the sensitivity of the n th obligorto the systematic factor. All the correlations between obligors arise from their dependence on thecommon factor. The common factor plays a major role in determining their joint dependence. Byapplying factor copula model into credit risk modelling, we are able to decompose a latent variableinto the systematic and the idiosyncratic component which are independent. A latent variable usuallyrepresents the proxy of firms’ assets or liquidation value (Andersen and Sidenius, 2004). Default istriggered by company asset values falling below a threshold, representing a fraction of company debt(Merton, 1974). In this model, credit risk is measured by a Gaussian random default variable gen-erated from firm asset value that is latent and modelled by a factor copula framework. The impliedfirm value from the model ideally projects the default time we desire; that is, a lower firm value is, ashorter default time is.A constant factor loading assumption embedded in a one factor Gaussian model is inconsistentwith the fact that the loading on common factors varies over time, which hampers the measurementof the dependency structures of obligors. This observation is in fact at the core of research on themispricing of structured products (Choro´s-Tomczyk et al., 2013; Choro´s-Tomczyk et al., 2014). Lon-gin and Solnik (2001) and Ang and Chen (2002b) argue that a “correlation breakdown” structureacts better in the dependence specification. Note that if we set the factor loading constant, we mayunderestimate the default risk as the market turns downward. Our simulation and empirical evidenceshow that a greater factor loading in market downturn leads to a higher contribution of common factoron firm value.In addition to the specification of factor loading, a critical and essential part in calculating theportfolio loss function is recovery rate. According to Table 1, a state-dependent recovery rate modelis suggested since the recovery rate seems to be subject to the market conditions; that is, higher ina bull market and lower in a bear market. By closely looking, one observes a lower average annualrecovery rate in the period 1998 to 2001 (internet bubble) and 2008 to 2009 (US subprime crisis)compared to the rest of the periods with bullish prospects. It is certain that the recovery rate in thebull market should not be lower than that in the bear market. Therefore, the recovery rate is likelyto vary with market conditions, which resembles the behaviour of the default rate. Notice that themarket condition is the unique common factor shared between recovery rate and default rate, andcauses their time variations. 2ondYear Sr. Sec. Sr. Unsec. Sr. Sub. Sub. Jr. Sub. All Bonds1997 75.5% 56.1% 44.7% 33.1% 30.6% 48.8%1998 46.8% 39.5% 45.0% 18.2% 62.0% 38.3%1999 36.0% 38.0% 26.9% 35.6% n.a. 33.8%2000 38.6% 24.2% 20.8% 31.9% 7.0% 25.1%2001 31.7% 21.2% 19.8% 15.9% 47.0% 21.6%2002 50.6% 29.5% 21.4% 23.4% n.a. 29.7%2003 69.2% 41.9% 37.2% 12.3% n.a. 41.2%2004 73.3% 52.1% 42.3% 94.0% n.a. 58.5%2005 71.9% 54.9% 32.8% 51.3% n.a. 56.5%2006 74.6% 55.0% 41.4% 56.1% n.a. 55.0%2007 80.6% 53.7% 56.2% n.a. n.a. 55.1%2008 54.9% 33.2% 23.3% 23.6% n.a. 33.9%2009 37.5% 36.9% 22.7% 45.3% n.a. 33.9%2010 62.5% 51.5% 37.5% 33.7% n.a. 51.8%2011 63.3% 41.3% 36.7% 35.4% n.a. 46.3%2012 51.2% 43.0% 33.7% 37.3% n.a. 44.7%2013 57.7% 43.8% 20.7% 26.4% n.a. 45.6% Table 1: Annual defaulted corporate bond recoveries
Annual corporate bond recovery rates based on post default trading price, Moody’s 27th annual default study.Note that Sr. Sec., Sr. Unsec., Sr. Sub., Sub., and Jr. Sub. represent senior secured, senior unsecured, seniorsubordinated, subordinated and junior subordinated, respectively.
Andersen and Sidenius (2004) address that both default events and recovery rates are driven bya single factor, but with an independent assumption between default and recovery rate. There arereasons to doubt this assumption. Chen (2010) demonstrates that the recovery rates are stronglynegatively correlated with default rates (is given as -0.82). As a consequence, the dependence betweenthem depends on the common factor represented by the state of macroeconomics. We claim that thecommon factor (market) governs the default rate and recovery rate simultaneously and creates theirassociation implicitly. One of our purposes is to build a tractable model that is capable of reflect-ing the obligors’ behaviour in reacting to the impact from the market. In addition, we show that asystematic risk plays a critical role in credit measurement and prediction, and contributes more to afirm’s credit risk in a market downturn than in a tranquil period. In this sense, the factor loading oncommon factor is conditional on market states. This conditional specification enables risk managersto be alerted regarding risk to the deterioration of the credit conditions when the market turns down,which avoids underestimating the default probability.We extend the one factor Gaussian copula model in two ways. Firstly, to improve the factorloading of Andersen and Sidenius (2004) given a two-point distribution, we apply the state-dependentconcept from Kim and Finger (2000) with the specific distributions to characterise the correlationsin hectic or quiet periods, respectively. It potentially captures two typical features of equity indexdistributions: fat tails and a skew to the left. However, for a two-point distribution setting, it isdifficult to decide on the threshold level of the two-point distribution, and on a time to be chosen arbi-trarily. Secondly, by relaxing the constant recovery rate presumed naively by academia and industry,3ur state-dependent recovery rate model permits that the systematic risk factor determines the LossGiven Default (LGD), as suggested by Amraoui et al. (2012). In addition, it restricts the recoveryrate, as a percentage of the notional is bounded on [0,1] to achieve the tractable and numericallyefficient missions. In summary, we contribute the incorporation of the state-dependent recovery rateinto the conditional factor copula model, and model them by sharing the unique common factor. Thecommon factor governs the default rate and recovery rate simultaneously, and creates their associ-ation implicitly. Our Monte Carlo simulation and empirical evidence appropriately reflect this feature.We propose four competing default models that have been widely applied to measure credit risk,and evaluate their relative performances on the accuracy of forecasting default in the following year.This comparison, by mapping the various factor copula models developed in the past and current liter-ature to the competing models, fosters the discussion on the model performance. Therefore, to achievea broader and robust comparison, we group the factor copula models developed in the literature intofour competing models: (1) The FC model: the standard one-factor Gaussian copula model with theconstant recovery rate (Van der Voort, 2007; Rosen and Saunders, 2010). (2) The RFL model: theone-factor Gaussian copula model with the factor loadings tied to the state of common factor andthe recoveries being assumed constant (Kalemanova et al., 2007; Chen et al., 2014). (3) The RRmodel: standard one-factor Gaussian copula model but the recoveries being related to the state of themacroeconomic state (Amraoui and Hitier, 2008; Elouerkhaoui, 2009; Amraoui et al., 2012), and (4)The RRFL model: a conditional factor loading specification together with a state-dependent recoveryrate, and this is the model what we are developing and contributing to. If further empirical resultsshow its best performance on default prediction, the outstanding performance of our refined RRFLmodel becomes very clear.In the FC model, we estimate the Spearman’s correlation coefficient between each obligor andcommon factor and set the recovery rate as constant. This is a conventional model to measure thecapital requirement in the Basel II accord. By relaxing the constant correlation in the RFL model,we suggest that the conditional factor loading plays a significant role in capturing an asymmetricsystematic impact from the market. The RR model uses the method proposed by Amraoui et al.(2012) to investigate the effect of stochastic recovery rate. It allows the LGD function to be drivenby the common factor and the hazard rate, but keeps factor loadings constant. In the RRFL model,we incorporate the conditional factor loading into state-dependent recovery rate and model them bysharing the unique common factor. To evaluate whether these two specifications carry significantimprovements to the default prediction, we use the data set of daily stock indices of the S&P 500 torepresent the market (common factor) and the respective stock prices of the default companies for theperiod of 5 years before the default year from the Datastream database.Our default data analysis contains 2008 and 2009, as collected by Moody’s report. We use Moody’sUltimate Recovery Database (URD) which is the ultimate payoff that obligors can obtain when thedefault emerges from bankruptcy or is liquidated instead of the post-default trading price as proposedby Carty et al. (1998). They examine whether the trading price represents a rational forecast of actualrecovery, and find that it is not a rational estimation of actual recovery. For this period, we employ astate-dependent concept in order to capture an asymmetric impact from the common risk factor. Asa result, we achieve the goal that both conditional factor loading and state-dependent recovery ratesimprove the calibration of our default prediction. The conventional factor copula underestimates theimpact of systematic risk and portfolio credit loss when the market is in downturn. We find that4he incorporation of factor loading into the state-dependent recovery rate improves the accuracy ofthe default prediction. This result is coherent with the goal of Basel III, which emphasises the roleof systematic risk on overall financial stability and systemic risk. In our later empirical analysis, weconcentrate on the senior unsecured bond, since there is a rich data source available.The remainder of the study is organised as follows. Section 2 describes the goal of Basel III. Wepresent a general framework and the standard one-factor Copula in section 3. Besides, we extend thestandard one-factor Copula by using the conditional factor loading and the state-dependent recoverymodel. Section 4 describes the data set. In the section 5, we offer empirical evidence. Section 6presents the conclusion.
As highlighted by Basel III, systemic risk is crucial in financial markets from several aspects. First,a bank can trigger a shock throughout a system and spill over to its counterparties (Drehmann andTarashev, 2013). Secondly, procyclicality could destabilise the whole systemic risk (Committee et al.,2009). The borrowers hardly fund more as their collateral assets have depreciated caused by weakeconomic conditions. Third, since Basel II focused on minimising the default probability of individu-als, this accord failed to guarantee a stable financial system due to a lack of concern for systemic risk.Therefore, a new Basel accord is expected to emphasize its role.Systematic factor is one of the important drivers of systemic risk and probably constituting a se-rious threat to systemic fragility (Schwerter, 2011; Uhde and Michalak, 2010). Tarashev et al. (2010)also distinguish between systemic risk and systematic risk. The former refers to the risk that impedesthe financial system, while the latter refers to the commonality in risk exposures of financial institu-tions. Their model assumes that systemic risk can have systematic and idiosyncratic components. Itis understandable that systemic risk is heightened by systematic risk. A bank is characterised as oneof systemically important (too-big-to-fail) financial institutions, its default would lead to a dramaticimpact on systemic risk. This is the very reason what Basel III attempts to regulate and prevent.Through our paper, our model proposes that the contribution of systematic risk is higher than thatof idiosyncratic component, and this dominance is characterised by a higher factor loading on system-atic risk during a market downturn. We, therefore, see the contribution of systematic risk on creditrisk varies with time and market conditions. In this regard, one shall concern on the interconnectionbetween credit risk and market risk. It is worth noting that the points mentioned above determinethe sufficiency of capital requirement in the banking industry.To obtain the sufficient capital requirements, recovery rate is one of determinant variables in creditrisk estimation. A real observation is that in a recession period, recovery rates tend to decrease whiledefault rates tend to rise. As such, increasing capital requirement under this condition seems necessary.Most early academic studies on credit risk assume that recovery rates are deterministic (Sch¨onbucher,2001; Rosen and Saunders, 2010), or they are stochastic but independent from default probabilities(Jarrow et al., 1997; Andersen and Sidenius, 2004). Neglecting the nature of stochastic in recoveryrate and the interdependence between recovery rates and default rates result in a biased credit riskestimation (Altman et al., 2005). 5o be close to the spirit of Basel III, our study extends the existing literature into two dimen-sions. First, we highlight that systematic risk is a predominant factor in a recession period, andproceed a relative contribution analysis to measure the proportional contribution from a systematicrisk in comparison with that from an idiosyncratic component. Second, we propose a methodology inwhich recovery rates and default rates are correlated by sharing a unique factor, and both are state-dependent. Our model design, the simulation and empirical results provide a bundle of justificationsfor the goals of Basel III.
Recognising the importance of systematic risk, one-factor Gaussian models have been considered animportant tool underlying the internal ratings based approach (Crouhy et al., 2000; Pykhtin and Dev,2002; Frey and McNeil, 2003) and used to price CDOs (Hull and White, 2004; Andersen and Side-nius, 2004; Choro´s-Tomczyk et al., 2013). It reduces the number of correlations being estimated from N ( N − by a multivariate Gaussian Model to N which represents the number of assets. Specifically, weuse a non-standardised Gaussian model to represent the deteriorating market condition by presuminga negative mean value together with a higher volatility. The model is based on decomposing a latentvariable U i for obligor i into the systematic factor Z and the idiosyncratic component ε it : U i = α i Z + (cid:113) − α i ε i i = 1 , . . . , N (1)where − ≤ α i ≤
1. Suppose that Z ∼ N ( µ, σ ) and ε i have zero-mean unit-variance distributions.In a Gaussian content, Z and ε i are orthogonal and ε i are mutually uncorrelated. The distribution ofvector U can be described by a copula function which joins two marginals, Z and ε i . The correlationcoefficient ρ ij between U i and U j can be described by their α i and α j : ρ ij = α i α j σ (cid:113) α i ( σ −
1) + 1 (cid:113) α j ( σ −
1) + 1 (2)where σ i = (cid:113) α i ( σ −
1) + 1 , σ j = (cid:113) α j ( σ −
1) + 1. As a consequence, the number of correlationsdescribing the dependency structure is reduced in size since only N parameters α i : i = 1 , . . . , N needto be estimated. We express the covariance matrices between U i and U j under a factor model,Σ ij = σ i σ j (cid:32) ρ ij ρ ji (cid:33) (3)The one-factor Gaussian copula model we consider is used to model the default indicators to time t ,6 { τ i ≤ t } , by projecting U i into τ i . U i here can be viewed as the proxies for firm asset and liquidationvalue (Andersen and Sidenius, 2004). In this regard, the lower asset value of firm the shorter time todefault, τ i . More precisely, U i ≤ F − { P i ( t ) } leads to τ i ≤ t , where P i ( t ) is a hazard rate and marginalprobability that obligor i defaults before t , and F − ( · ) donates the inverse cdf of any distribution.The default indicator then can be written as I { τ i ≤ t } = I (cid:2) U i ≤ F − { P i ( t ) } (cid:3) (4)Given the LGD for each i , G i , i = 1 , . . . , N , we aggregate them as total portfolio loss, L , as fol-lowing, L = N (cid:88) i =1 G i I { τ i ≤ t } = N (cid:88) i =1 G i I (cid:2) U i ≤ F − { P i ( t ) } (cid:3) (5) In accordance with the spirit of Basel III, the systematic latent factor, Z , representing the generaleconomic condition that characterises the systematic credit risk influences the default probability P i ( t )and the recovery rate R i = 1 − G i . So given Z , one may write the conditional default probability P i ( Z | S = H, Q ) and conditional LGD, G i ( Z | S = H, Q ) as a function of Z , and it is state-dependent,S ∈ { H,Q } . H, and Q represent the hectic and quiet periods, respectively.A higher factor loading, α i in equation (1) has been observed in hectic periods (Longin and Solnik,2001; Ang and Bekaert 2002a; Ang and Chen 2002b). This observation can be modelled by a regime-switching mechanism, requiring a globally valid time series structure for α i from t . Avoiding such apossible too rigid structure, we assume the two asset returns, Z common factor proxied by USD S&P500, U i (firm stock price) have a mixture of bivariate normal distribution (See Appendix A) to obtainthe estimation of α Hi and α Qi . Given the conditional factor loading, α Hi , α Qi , the conditional defaultmodel is defined as following, U i | S=H = α Hi Z + (cid:113) − ( α Hi ) ε i (6) U i | S=Q = α Qi Z + (cid:113) − ( α Qi ) ε i (7)Therefore, the state-dependent conditional default probability can be denoted by7 ( τ i < t | S) = F F − { P i ( t ) } − α Si Z (cid:113) − ( α Si ) = P i ( Z | S) S ∈ {
H,Q } (8)Given P i ( t ), if the factor loadings in hectic periods are greater than ones in quiet days, say α H > α Q ,and if the index return of S&P 500 is negative in a bad market condition, both conditions will resultin a higher conditional default probability in equation (8). From equation (8), the systematic risk, Z ,and the corresponding factor loading govern the conditional default probability, which is consistentwith empirical findings (Andersen and Sidenius, 2004; Bonti et al., 2006). It is worth pointing outthat α Si is state-dependent instead of a constant setting in previous literature (Andersen and Sidenius,2004; Amraoui et al., 2012). Ang and Chen (2002b) set probability of both regimes equally ( w = 0 . Z , P(S=H)= ω , P(S=Q)=1 − ω by Expectation-Maximization (EM) algorithm.Likewise, the recovery rates can be designed in this way by incorporating market condition as amain driver across different states. Based on the finding of Das and Hanouna (2009), recovery ratesare negatively correlated with probabilities of defaults and driven by market condition. By relaxingconstant recovery rates, we follow Amraoui et al. (2012) to connect recovery rates and default eventsvia a common factor, but extend their model to a conditional or state-dependent framework. Therecovery rate is governed by the state of economy, in addition, we incorporate a conditional correlationstructure, α Si , into stochastic recovery rate model, and set R i ( Z | S = H, Q ), of obligor i , in relation tothe common factor Z and the marginal default probability P i . The state-dependent recovery rate isexpressed as, G i ( Z | S=H) = (1 − R i ) F (cid:20) { F − (cid:0) P i (cid:1) − α Hi Z } / (cid:113) − ( α Hi ) (cid:21) F (cid:20) { F − ( P i ) − α Hi Z } / (cid:113) − ( α Hi ) (cid:21) (9) G i ( Z | S=Q) = (1 − R i ) F (cid:20) { F − (cid:0) P i (cid:1) − α Qi Z } / (cid:113) − ( α Qi ) (cid:21) F (cid:20) { F − ( P i ) − α Qi Z } / (cid:113) − ( α Qi ) (cid:21) (10)In equation (9, 10), 0 ≤ ¯ R i ≤ R i ≤ R i to R i , so that ¯ R i = R i − υ and R i ≥ υ > υ is size of downward shift. By assuming that expected loss in name i remainsunchanged, we set (1 − R i ) P i = (1 − ¯ R i ) ¯ P i . Please see the proof in A.1 in Amraoui et al. (2012). F ( · )denotes any distribution and P i is the adjusted default probability calibrated proposed by Amraouiand Hitier (2008). The LGD function, G i ( Z | S=H,Q) essentially can be obtained according to formula(9,10). Numerous studies show that recoveries decline during recessions (Altman et al., 2005; Bruche8nd Gonzalez-Aguado, 2010). Consistent with the spirit of equation (6,7), we design α H , α Q , the factorloading in equation (9,10) are therefore conditional and state-dependent. Moreover, a partial derivativeof LGD function with respect to Z is less than zero proved by property 3.2 in Amraoui and Hitier(2008), which means that G i ( Z | S=H,Q) is decreasing in Z . By assuming α H > α Q , which means thata higher factor loading that is usually accompanied by a bad market condition on Z tends to increaseLGD. The magnitude of LGD is not only influenced by Z but also sensitive to the factor loading under Z ; this is what we point out and contribute to the literature. In addition, recovery rates are also linkedto the probability of default and they are negatively correlated (see Altman et al., 2005; Khieu et al.,2012). With Z , P i and the estimated conditional factor loading α H , α Q , we obtain the state-dependentrecovery rate, R i ( Z | S=H,Q), and state-dependent LGD, G i ( Z | S=H,Q) = 1 − R i ( Z | S=H,Q).With these two specifications, the conditional default probability P i ( Z | S=H,Q) and conditionalLGD, G i ( Z | S=H,Q), conditional expected loss, therefore, is E ( L i | Z ) = ωG i ( Z | S=H) P i ( Z | S=H) + (1 − ω ) G i ( Z | S=Q) P i ( Z | S=Q) (11)where ω = P(S=H) and 1 − ω = P(S=Q). H and Q represent the hectic and quiet periods, respectively. In this section, we investigate the performance of default prediction by establishing a simulation ofrealistic scenarios. The default probability and recovery rate function are governed by systematicfactors generated from different regimes. Indeed, they are crucial elements in evaluating the accuracyof the default prediction. Our interest is to see whether the design of conditional factor loadings andstate-dependent recovery rates contribute to the default prediction.
We simulate one-factor non-standardised Gaussian copula subject to different states. As describedin equation (6) and (7), we generate systematic factor Z by non-standardised Gaussian distributionwith different volatilities and independent ε (cid:48) i s . To reflect the nature of distinct variations exhibitedin different market conditions.Through a mixture bivariate distribution setting in Appendix A, the conditional factor loadings, α Hi and α Qi are derived, in the one-factor non-standardised Gaussian copula model. We estimatethem from the daily stock returns of S&P 500 and of collected default companies during the crisis(2008-2009) period. The five-year period prior to the crisis period is the estimation period for theconditional factor loadings. The return of S&P 500 Index represented as a systematic factor, Z , ispresumed to distribute as N ( − . , .
05) estimated in 2008 and 2009, while ε i ∼ N (0 ,
1) representsidiosyncratic risk. Z and ε i are generated 1000 scenarios, respectively. Given any one of generatedsystematic factor random variables, Z , and using Bayes’ rule, we calculate the conditional probabilitythat date t belonged to the hectic is π ( Z = z ) by using its counterpart, unconditional probability ω ,as a formula (12). 9( S = H | Z = z ) = π ( Z = z ) = ωϕ ( z | θ H )(1 − ω ) ϕ ( z | θ Q ) + ωϕ ( z | θ H ) (12) θ H , θ Q represent the parameters of distribution in the hectic (H) and the quiet (Q) period. ϕ ( · ) is anormal distribution. Plugging α Hi , α Qi shared with the same simulated Z random variables, conditional U i | S is generated as developed in equation(6, 7). These simulated random variables together with thepublished hazard rates h i ( t ) ideally produce the simulated default times. Projecting U i simulated from section 3.3.1 to default time, τ i , stated in equation(4) provides the clueas to whether the firm defaults before time. We set t = 1, represents the time interval of 1 year, so that τ i < i th obligor. The hazard rate h i is the probability of occur-rence of the default event within one year. τ i is referred to default time of i th obligor. More precisely,the expected value of E [I( τ i < τ i <
1) or named as P i , see Franke et al. (2015) Chapter 22, thatcan be connected to the firm’s stock return or firm’s value, U i leads to P i = E [I { U i < F − i ( P i ) } ] where F i denote the cdf of U i . By applying generated U i from the conditional factor model into the definitionof the survival rate, we have generated default time, τ i , derived from 1 − exp( − P i τ i ) = F ( U i ) (Hull,2006). To keep on the state-dependent environment, the conditional default time for each obligor isgenerated by formula (13). τ i | S = − log { − F ( U i | S) } P i (13)where P i is the hazard rate or marginal probability that obligor i will default during the first year,conditional on no earlier default, and is obtained from Moody’s report. It is the cumulative of defaultrates during the first year. Equation (13) states that U i | S becomes larger, τ i | S will become longer.The larger U i reduces the tendency of default and postpones the default time, τ i | S. In the third step, we consider a more realistic situation by simulating recovery rates as described inour settings. The adjusted default probability ¯ P i is calibrated by using hazard rate P i from Moody’sreport. ¯ R i is a lower bound for state-dependent recovery rates [0,1], therefore, we set ¯ R i = 0 inthe simplest case. With α Hi , α Qi , Z, ¯ P i , the simulated state-dependent recovery rates are obtained byformula (9, 10). By changing scenarios to quiet and hectic states, we assume the exposure of each obligor is 100 millionand generate the expected loss under the given scenarios corresponding to formula (11).10 ( L i | Z ) = π ( Z = z ) G i ( Z | S=H) P i ( Z | S=H) + (1 − π ( Z = z )) G i ( Z | S=Q) P i ( Z | S=Q) (14)Given the simulated Z random variables, conditional probability π ( Z = z ) naturally provides betterinformation than unconditional probability ω does. By the given formula (14), we compare the theo-retical loss amounts across four models with the realised loss values, and evaluate the performance ofthe default prediction by the mean of square error. In step 5, the performance of the competing models: FC, RFC, RR, RRFC are evaluated here todecide which one is the best in predicting the default for the following year. Absolute Error (AE) hereis linked to the prediction performance and is defined asAE = (actual portfolio loss - expected portfolio loss) (15)where actual portfolio loss is from Moody’s report. Expected loss is estimated from equation (14),whereas in an unconditional default model, it is computed from formula (5). For each competingmodel, we generate 1000 scenarios, then, the mean of absolute error referred as MAE is calculated.One can expect that the best one is entailed on the minimum AE and MAE as well.
We use the list of default companies for 2008 through to 2009 published by Moody’s annual reportsince this is a rich available data source. In total, we obtained 341 defaults with corporate bond re-covery rates from Moody’s URD covering the period from 1987 to 2007. We focus on senior unsecuredbonds because of their wide use in financial contracts, regulatory rules, and the risk of measuring forassets under the standardised approach of Basel II (Pagratis and Stringa, 2009). We also collected thecredit rating of obligors from Moody’s report in order to measure the hazard rate. Although there are94 and 247 default firms in 2008 and 2009, the observations were reduced due to missing stock pricesand credit rates of obligors’ bonds. If there was a lack of stock prices of default subsidiary companies,we used stock prices of parent companies instead. In all cases, 31 and 62 sampling firms were collectedin 2008 and 2009, respectively.To estimate the conditional factor loadings of sampling firms, we collect the daily USD S&P500 return and the respective stock return of the default companies for a 5-year period prior to thedefault year from the Datastream database. USD S&P 500 Index here simply represents the commonsystematic risk. By assuming a mixture of bivariate normal distribution, we estimate the parametersincluding factor loadings by EM algorithm. Table 2 presents the results of EM algorithm.11odel Probability Mean STDPeriod 2003-2007Unconditional (one normal) 100.00% 0.03% 0.77%Conditional on quiet 58.68% 0.10% 0.43%Conditional on hectic 41.32% -0.08% 1.07%Period 2004-2008Unconditional (one normal) 100.00% 0.03% 0.83%Conditional on quiet 56.77% 0.10% 0.38%Conditional on hectic 43.23% -0.06% 1.17%
Table 2: Estimate mixture of normal distribution by employing an EM algorithm
STD represent standard deviation
As presented in Table 2, the volatility of the hectic distribution is larger than that of the quietdistribution, and the mean of the hectic distribution is smaller than that of the quiet distribution,reflecting the fat tails and a skew to the right which are consistent with Kim and Finger (2000).
Figure 1 and 2 shows that the majority of correlation coefficients or called factor loadings in factorcopula model during the hectic period is higher than in the quiet period. The proposed correlationstructure leads to more accurate and realistic implementations, and to avoid the underestimation offactor loading in a hectic period or the overestimation in a quiet period. These ideas are well knownin statistics and have already been applied to financial questions (Ang and Chen 2002b; Patton, 2004).In our approach, we consider this asymmetric correlation structure under real market conditionsto implement the conditional default model developed in Section 3.2. As shown in Figure 1 and 2, thefactor loadings α i in state H are higher than those in state Q. As factor loadings get higher in stateH, the correlation coefficient ρ ij between firm i and j defined in equation (2) is expected to increasein this market condition. Therefore, obligors tend to comove more closely during hectic periods thanduring quiet periods. To demonstrate the impact of market conditions measured by Z on the state-dependent recovery rate,in Figure 3 we depict the relationship between the state-dependent recovery rate and the S&P 500(the proxy for systematic factor Z) in blue ‘*’, which developed in section 3.2. One can observe thatthe effect of the systematic factor on the recovery rate is positive, the recovery rate gets higher as Z grows. Since the slope of this curve is influenced by estimated α Hi , α Qi corresponding to formula(9, 10), the slopes behave differently in the four panels but keep positive monotonically. We alsodepict the stochastic recovery rates in red ‘+’ estimated and simulated through Amraoui et al. (2012)model, in comparison with blue ‘*’, simulated from our model. Taking (c) E*TRADE as an example,12 igure 1: Conditional and unconditional factor loading comparision in 2008 The estimation of Conditional and Unconditional Factor Loading between S&P 500 and default companies in2008. we observe that compared with the simulated recovery rates based on equation (9) and (10), thosegenerated from Amraoui et al. (2012), by assuming constant factor loadings, tend to produce higherrecovery rates in the market downturn and lower ones in the booming market. This evidence suggeststhat the recovery rate may be overestimated in a bearish market but underestimated in a bullishmarket if the constant factor loading is assumed. As a consequence, an underestimation of credit lossin a bearish market but an overestimation in a bullish market are highly possible. Similarly, the evi-dence from (a) Glitnir banki (b) Lehman Brothers Holdings, Inc. and (d) Idearc, Inc. are comparableand consistent. Note that the impact of the systematic factor on recovery rate seems nonlinear, it ishigher in the market downturn but relatively milder in the booming market, and the marginal slopedecreases abruptly when the index return decreases, whereas the marginal slope decelerates when theindex return becomes positive. This simulation result is in accordance with Moody’s report in Table1. From 2004 to 2006, the annual recovery rate of senior unsecured bond increases slowly. Whenthe crisis started in August 2007, the recovery rate drops dramatically. By capturing the correlationstructure, α H > α Q , as shown in (a), (c) and (d), we find this asymmetric pattern which is moreconsistent with the reality.Having the simulated recovery rates from equations (9, 10), we are more interested in the relationbetween it and conditional default probability from equation (8). As can been seen in Figure 4, thesimulation result shows the downward trend between default probability and recovery rate consistedwith Altman et al. (2005) and Das and Hanouna (2009). It shows that the common factor governs thedefault rate and recovery rate simultaneously and creates their negative association implicitly. Alt-13 igure 2: Conditional and unconditional factor loading comparision in 2009 The estimation of Conditional and Unconditional Factor Loading between S&P 500 and default companies in2009. man et al. (2005) find that permitting a dependence between default rates and recovery rates increasesaround 29% in the Value at Risk compared with a model that assumes no dependence between defaultrates and recovery rates. 14 a) Glitnir banki: α = 0 . , α Q = 0 . , α H = 0 . (b) Lehman Bro.: α = 0 . , α Q = 0 . , α H = 0 . (c) E*TRADE: α = 0 . , α Q = 0 . , α H = 0 . (d) Idearc, Inc.: α = 0 . , α Q = 0 . , α H = 0 . Figure 3: The relationship between state-dependent recovery rate and index return of S&P 500, Z . Panel (a) and (b), ‘*’ in blue illustrates the pattern of state-dependent recovery rate of Glitnir banki andLehman Brothers Holdings, Inc. which incorporate conditional factor loading in 2008. ‘+’ in red plots therecoveries proposed by Amraoui et al. (2012). In panel (c) and (d), E*TRADE Financial Corp. and Idearc,Inc. in 2009.
To gauge the conditional factor loading and state-dependent recovery rate approaches for defaultprediction, we propose four models: (1) The FC model: the standard one-factor Gaussian copulamodel with the constant recovery rate developed by Van der Voort (2007) and Rosen and Saunders(2010) (2) The RFL model: the one-factor Gaussian copula model with the factor loadings tied tothe state of common factor and the recoveries being assumed as constant proposed by Kalemanovaet al. (2007) and Chen et al. (2014). (3) The RR model: standard one-factor Gaussian copula modelbut the recoveries being related to the state of the macroeconomic state (Amraoui and Hitier, 2008;Elouerkhaoui, 2009; Amraoui et al., 2012), and (4) The RRFL model: a conditional factor loadingspecification together with a state-dependent recovery rate. We address the question of whether thetwo specifications, conditional factor loading and the state-dependent recovery rate model are mean-15a) 2008 (b) 2009
Figure 4: The relationship between state-dependent recovery rates and default probabilities
By simulating Z ∼ N ( − . , . ingful and significant in explaining the gap between expected and practical loss value. In order to checkthe predictive ability of the different models, we report the AE and MAE estimated from section 3.3.5.Table 3 reports the AE between actual portfolio loss and expected portfolio loss constructed by31 and 62 observations in 2008 and 2009, respectively. In a comparison with four models, one canobserve that the estimate of expected portfolio loss in the RRFL model is highest and closest to thecorresponding actual one, which means the expected portfolio losses may be underestimated by theother three models. Especially, a modelling recovery rate in a stochastic fashion indeed contributes tocredit loss estimation.We compare the four competing models of each obligor and choose the best model which achievesthe minimum AE and MAE. It can be seen that including the conditional factor loading (RFL model)instead of the Spearman correlation (FC model) does not significantly improve the estimations in 2008and 2009. As can be seen in Table 4, we find that introducing the state-dependent recovery rate (RRmodel) leads to a promising improvement over the standard model (FC model). We interpret thisas saying that the setting of stochastic recovery rate seems necessary, which brings a remarkable im-provement on default prediction. This result is consistent with Altman et al. (2005) and Ferreira andLaux (2007). Compared with the RR model, the RRFL model includes conditional factor loading indefault probabilities and a state-dependent recovery rates function which produces much more modestimprovements.We propose two specifications on factor loading, and recovery rates across four models. If weassume that default probabilities are the function of two-state correlation constructers, but recoveryrates do not, the specification is only identified concentrated on factor loading. In this case, therecovery rates do not contain information about the state of business cycle. Conversely, if we assumethat recovery rates vary, but factor loading is fixed, then the refinement is only through the variationin the recovery rate. Since the RRFL model with both specifications is superior to the other threecompeting models, there is no redundant specification in this study. In this regard, we extend themodels proposed by prior literatures (Van der Voort, 2007; Rosen and Saunders, 2010; Kalemanova16t al., 2007; Chen et al., 2014; Amraoui and Hitier, 2008; Elouerkhaoui, 2009; Amraoui et al., 2012)which leads more accurate default prediction in one year. FC RFL RR RRFL2008Actual portfolio loss 2035.02 2035.02 2035.02 2035.02Expected portfolio loss 509.60 527.06 687.01 690.86AE 1525.42 1507.96 1348.01 1344.16MAE 47.12 47.67 42.13 42.01Expected portfolio loss/Actual portfolio loss 25.04% 25.90% 33.76% 33.95%2009Actual portfolio loss 4073.80 4073.80 4073.80 4073.80Expected portfolio loss 1203.56 1212.38 1769.14 1788.05AE 2870.24 2861.42 2304.67 2285.75MAE 43.49 43.35 34.92 34.63Expected portfolio loss/Actual portfolio loss 29.54% 29.76% 43.43% 43.89% Table 3: The mean of actual portfolio loss, expected portfolio loss and AE, MAE (in million)
This table reports the AE and MAE by comparing the four models: (1) The FC model: the standard one-factor Gaussian copula model with the constant recovery rate. (2) The RFL model: the one-factor Gaussiancopula model with the factor loadings tied to the state of common factor and the recoveries being assumed tobe constant. (3) The RR model: standard one-factor Gaussian copula model but the recoveries being relatedto the state of the macroeconomic state. and (4)The RRFL model: a conditional factor loading specificationtogether with a state-dependent recovery rate. This table also presents the difference between actual portfolioloss and expected portfolio loss as referred to AE, and divided by 31 and 62 observations in 2008 and 2009,respectively, as MAE. The percentage represents expected portfolio loss divided by the actual portfolio loss.
Since Basel III is proposed to control systematic risk (one of systemic risk measures) to achieve thegoal of overall financial stability, systematic risk has been considered one of the main causes of the2007-2009 crisis. In this section, we highlight the role of systematic risk and its impact to fit the goalsof Basel III. The aim of relative contribution analysis is to investigate the proportional contributionfrom systematic risk in comparison to that from the idiosyncratic component. By measuring thesystematic risk, α Si Z , and idiosyncratic risk, (cid:113) − ( α Si ) ε i , S ∈ { H,Q } from formula (6,7), we depicta scatter plot for simulated systematic risk (horizontal axis) and idiosyncratic risk (vertical axis) inFigure 5. As can be seen in the 2D plot in 2008, the 45 ◦ line represents the proportion of systematicrisk is equal to that of idiosyncratic risk. If scattered points are located in the ‘A, B, C, D’ zones, thecontribution of systematic risk on default risk is greater than idiosyncratic risk. On the other hand, ifscattered points are settled in the ‘a, b, c, d’ areas, the contribution of systematic component is lessthan idiosyncratic risk. For example, the effect of systematic risk on default risk will become largerwhen point ‘Y’ moves to point ‘X’. Most literature focus on either systematic (Huang et al., 2009;Acharya et al., 2010) or firm-specific components (Goyal and Santa-Clara, 2003; Ferreira and Laux,2007), and a limit number of studies compare the influence of both of them.By simulating Z ∼ N ( − . , . U i | S referred toas the mean of firms’ value, systematic and idiosyncratic component. Each observation in Figure4 reflects its mean of U i | S i = 1 , . . . , N in each simulating day in 2008 and 2009, respectively. Ascan be seen in Figure 5, the points in the hectic period marked as green circles indicates a negativeshock from systematic risk which lowers the average asset value of obligors; specifically, the majorityof observations show a negative impact of systematic shock which accounts for a larger proportionon the firms’ values substantially. Note that it is easy to drive the default event since it lowers thefirms’ value significantly. On the other hand, the points in quiet days marked as blue circles indicate apositive shock from the systematic component. However, the negative shock from firm-specific factorsmay compromise the benefit from economy-wide components that lowers the level of average U i | S atsome points.Our model emphasises the importance of systematic risk which explains most obligors default be-haviour particular in hectic periods, which is one of the important measures of Basel III (Schwerter,2011; Uhde and Michalak, 2010; Tarashev et al., 2010). To be specific, we measure and demonstratethe contribution of overall systematic risk to each asset, and identify the impact direction from sys-tematic and idiosyncratic risk. Moreover, it can be applied to a variety of systematic risk measures.In this sense, portfolio managers should be aware of the systematic risk which influences the value ofportfolios substantially. We propose that the regulatory tool of Basel III could be estimated accordingto such contributions. A related question is how these measures can aid policymakers. The measuresin this paper can be used as a tool to prevent systematic crisis. Our model can be used as an earlywarning system that will alert the regulators when an individual bank is in trouble and to intervenebefore the crisis happens. 18 a) (b) Figure 5: The 2D and 3D scatters plot of relative contribution
By simulating Z ∼ N ( − . , . α i Z , and idiosyncratic risk, (cid:112) − α i ε i . Each simulated Z random variable can therefore be mapped intoa specific conditional probability of being hectic state in Eq. (12). We depict the scatters in three groupshere. The first group (marked as ‘+’ in green) only includes the simulated Z r.v. with projecting conditionalprobabilities above 75%-quartile, and indicates that they are generated in distress. The second group (markedas ‘*’ in blue) includes the Z r.v. with projecting conditional probabilities below 25%-quartile to indicate thatthey are generated in a bullish atmosphere. The third group (marked as ‘x’ in red) collects the rest. In 3D plot,observations in hectic periods are marked as green circles. In quiet days are marked as blue circles, otherwiseas red circles. Since Table 3 reports that the expected portfolio loss is far away from the actual portfolio loss, wegauge that using bond credit rates as a measure of hazard rate has the disadvantage that they arereleased annually by Moody’s report. In this section, we use credit default swap (CDS) spread data asan alternative market-based measure of the company’s credit risk. A CDS spread is a financial swapagreement that the seller of CDS will compensate the buyer in the event of a loan default. Basically,the variation of CDS spread reflects the dynamic of risk condition or hazard rate implicitly. Thelarger the CDS spread is, the riskier the debtor is. Therefore, the hazard rate, ¯ κ , for a company canbe estimated by, 19 κ = s − R (16)where s is CDS spread. We consider the latest one-year CDS quotes of obligors before the defaultyear provided from Datastream. We also use a credit spread which is the yield on a annual par yieldbond issued by the obligors over one-year LIBOR (London Interbank Offered Rate) if the obligordoesn’t have CDS data. Theoretically, the CDS spread is very close to the credit spread (Hull andWhite, 2000; Hull et al., 2004). By plugging in the recovery rate, R , obtained from Moody’s report,we compute the average default intensity, ¯ κ , per year conditional on no earlier default instead of P i .Compared with P i from Moody’s annual report, a CDS spread with active trading activity reflectsmarket assessments of default risk in a timely fashion. In this regard, the proposed models with anincorporation of the hazard rate implied in CDS spreads may produce a better prediction.According to Table 4, the models with a hazard rate implied in a CDS spread seem to performbetter than those with a hazard rate from historical bond credit rates. By comparing Tables 3 and 4,generally, a CDS spread as the hazard rate measure reflects information more timely than the bondcredit rate does. As can be seen in Table 4, the RRFL model outperforms in robustness test. In bothTables, the RRFL model consistently outperforms, which produces the expected portfolio loss mostclosely to the actual portfolio loss. FC RFL RR RRFL2008Actual portfolio loss 1401.31 1401.31 1401.31 1401.31Expected portfolio loss 560.50 533.82 589.54 591.40AE 840.81 867.49 811.77 809.91MAE 35.03 36.15 33.82 33.75Expected portfolio loss/Actual portfolio loss 40.00% 38.09% 42.07% 42.20%2009Actual portfolio loss 2707.30 2707.30 2707.30 2707.30Expected portfolio loss 1457.07 1462.18 1677.89 1683.97AE 1250.23 1245.12 1029.42 1023.33MAE 29.77 29.65 24.51 24.37Expected portfolio loss/Actual portfolio loss 53.82% 54.01% 61.98% 62.20% Table 4: The actual portfolio loss, expected portfolio loss, AE, and MAE (in million) for robust-ness
This table reports the value of AE and MAE of four models by using market-based method during 2008 and2009. This table also shows the actual portfolio loss and expected portfolio loss of 24 and 42 observations in2008 and 2009. The percentage represents expected portfolio loss divided by the actual portfolio loss. Conclusion
This paper proposes a refined factor copula model for credit risk prediction. On the basis of our esti-mated model, we find that systematic risk plays a critical role in governing default rates and recoveryrates simultaneously. Our simulation results show that recoveries vary with the returns of the S&P500 and the impact of systematic factors on the recovery rate is asymmetric by characterising a higherfactor loading in hectic periods than in tranquil ones. Among the various factor copula models de-veloped in the past and current literature as the competing models, the one with conditional randomfactor loading and a state-dependent recovery rate turns out to be the most superior. In other words,our refined model contributes to literature that have been mapped to 3 groups of competing models(the FC, RFL, and RR models)As a response to Basel III, we measure and demonstrate the contribution of overall systematic riskto each firm’s value and identify the relative role of the systematic and idiosyncratic risk. Moreover, itcan be applied to a variety of systematic risk measures, and aids regulators in preventing a systematiccrisis. In addition, by investigating the effect of state-dependent recovery rates on the loss function,we suggest that banks should apply this issue on capital requirement to make sure of its sufficiency.In further research, we plan to go beyond this study in several ways. For instance, other copulafunctions can be modelled to capture various dependence structures. Secondly, the marginal distribu-tion can be considered in a more general way to capture a fat-tail feature. We will leave these issuesfor future studies.
Acknowledgements
This research was financially supported by the Deutsche Forschungsgemeinschaft (DFG) via SFB 649” ¨Okonomisches Risiko” and IRTG 1792 ”High-Dimensional Non-Stationary Times Series” is gratefullyacknowledged.
Appendix A Conditional Factor Loading
We assume the two asset returns Z (USD S&P 500), U i (firm stock price) to have a mixture of bivariatenormal distribution:( Z, U i ) ∼ N (cid:40) (cid:34) µ QZ µ Qi (cid:35) , (cid:34) ( σ QZ ) ( σ QZ )( α Q )( σ Qi )( σ QZ )( α Q )( σ Qi ) ( σ Qi ) (cid:35) (cid:41) P(S=Q) = 1 − ωN (cid:40) (cid:34) µ HZ µ Hi (cid:35) , (cid:34) ( σ HZ ) ( σ HZ )( α H )( σ Hi )( σ HZ )( α H )( σ Hi ) ( σ Hi ) (cid:35) (cid:41) P(S=H) = ω (A.1)where volatility in hectic periods is higher than in a quiet periods, ( σ Hi ) > ( σ Qi ) . α Q and α H arethe correlation coefficient between each obligor and the S&P 500 in quiet and hectic period proposedby Kim and Finger (2000), respectively. 21e estimate the unknown parameters ω , µ QZ , σ QZ , µ HZ , σ HZ from the marginal distribution of Z : N (cid:104) µ QZ , ( σ QZ ) (cid:105) P(S=Q) = 1 − ωN (cid:104) µ HZ , ( σ HZ ) (cid:105) P(S=H) = ω (A.2) References
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