# Copula-based factor model for credit risk analysis

## Abstract

A standard quantitative method to assess credit risk employs a factor model based on joint multivariate normal distribution properties. By extending the one-factor Gaussian copula model to produce a more accurate default forecast, this paper proposes the incorporation of a state-dependent recovery rate into the conditional factor loading and to model them sharing a unique common factor. The common factor governs the default rate and recovery rate simultaneously, implicitly creating their association. In accordance with Basel III, this paper shows that the tendency toward default during a hectic period is governed more by systematic risk than by idiosyncratic risk. Among those considered, the model with random factor loading and a state-dependent recovery rate is shown to be superior in terms of default prediction.

### Keywords

Factor model Conditional factor loading State-dependent recovery rate### JEL classification

C38 C53 F34 G11 G17## 1 Introduction

The global economy has repeatedly witnessed clusters of default events, such as the burst of the dotcom bubble in 2001 and the global financial crisis from 2007 to 2009. Clusters of default events have been blamed on the role played by systematic risk in leading to default. To reveal this role, numerous studies emphasize 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 common method of capturing obligors’ shared behavior through a joint common factor and of reducing the dimension of dependence parameters, which benefits bond portfolio management. However, it is also relatively common to see certain unrealistic settings in this method, such as constant and linear dependence structures with thin tails of embedded risk factor distribution.

The factor copula model imposes a dependence structure on common factors and on the variables of interest. In measuring credit risk using systematic factors, the factor loading represents the sensitivity of the nth obligor to the systematic factor. All the correlations between obligors thus arise from their dependence on the common factor, and the common factor thus plays a major role in determining their joint dependence. By incorporating factor copula model into credit risk modeling, we can decompose a latent variable into its systematic and idiosyncratic components, which are independent of one another. A latent variable typically acts as a proxy for a firms’ assets or liquidation value (Andersen and Sidenius 2004). Default is triggered by company asset values falling below a threshold that corresponds to a fraction of company debt (Merton 1974). In this model, credit risk is measured by a Gaussian random default variable derived from firm asset value that is latent and modeled by a factor copula framework. The implied firm value from the model ideally projects the default time we desire; thus, the lower the firm value, the shorter default the time is.

A constant factor loading assumption embedded in a one-factor Gaussian model is inconsistent with the fact that the loading on common factors varies over time, which hampers the measurement of the dependency structures of obligors. In fact, this observation is at the core of research on the mispricing of structured products (Choroś-Tomczyk et al. 2013, 2014). Longin and Solnik (2001) and Ang and Chen (2002) argue that a “correlation breakdown” structure acts better in the dependence specification. In particular, if we set the factor loading to be constant, we may underestimate default risk as the market turns downward. Our simulation and empirical evidence show that a greater factor loading in a market downturn leads to a higher contribution of common factors on firm value.

Annual defaulted corporate bond recoveries

Year | Bond | |||||
---|---|---|---|---|---|---|

Sr. Sec. (%) | Sr. Unsec. (%) | Sr. Sub. (%) | Sub. (%) | Jr. Sub. (%) | All Bonds (%) | |

1997 | 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 |

Andersen and Sidenius (2004) address the fact that both default events and recovery rates are driven by a single factor but with an independence assumption between default and recovery rate, although there are reasons to doubt this assumption. Chen (2010) demonstrates that recovery rates are strongly negatively correlated with default rates (which is given as −0.82). As a consequence, the dependence between them relies on the common factor, which is represented by the macroeconomic state. We claim that the common factor (the market) governs the default rate and recovery rate simultaneously, implicitly creating their association. One of our purposes is to build a tractable model that can reflect the obligors’ behavior in reacting to the impact from the market. In addition, we show that systematic risk plays a critical role in credit measurement and prediction, and it contributes more to a firm’s credit risk during a market downturn than during a tranquil period. In this sense, the factor loading on the common factor is conditional on market states. This conditional specification enables risk managers to be alerted regarding the deteriorating credit risk conditions when the market turns downward, which prevents underestimating the default probability.

We extend the one-factor Gaussian copula model in two ways. First, to improve the factor loading of Andersen and Sidenius (2004) given a two-point distribution, we apply the state-dependent concept from Kim and Finger (2000) with specific distributions to characterize the correlations in hectic or quiet periods. This concept potentially captures two typical features of equity index distributions: fat tails and a skew to the left. However, for a two-point distribution setting, it is difficult to decide on the threshold level of the two-point distribution and on a time to be chosen arbitrarily. Second, by relaxing the constant recovery rate that is naively presumed by both scholars and practitioners, our state-dependent recovery rate model allows the systematic risk factor to determine loss given default (LGD), as suggested by Amraoui et al. (2012). In addition, it restricts the recovery rate, as a percentage of the notional is bounded on [0,1] to achieve the tractable and numerically efficient missions. In summary, our contributions include incorporating the state-dependent recovery rate into the conditional factor copula model, and we model them by sharing their unique common factor. The common factor governs the default rate and recovery rate simultaneously, while creating their association 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 we evaluate their performances on the accuracy of forecasting default in the following year. By mapping the various factor copula models developed in the literature to the competing models, this comparison fosters a discussion on model performance. Therefore, to achieve a broader and robust comparison, we group the factor copula models developed in the literature into four competing models: (1) the FC model, i.e., the standard one-factor Gaussian copula model with a constant recovery rate (Van der Voort 2007; Rosen and Saunders 2010); (2) the RFL model, i.e., a one-factor Gaussian copula model in which the factor loadings are tied to the state of the common factor and the recoveries are assumed to be constant (Kalemanova et al. 2007; Chen et al. 2014); (3) the RR model, i.e., a standard one-factor Gaussian copula model in which the recoveries are related to the macroeconomic state (Amraoui and Hitier 2008; Elouerkhaoui 2009; Amraoui et al. 2012); and (4) the RRFL model, i.e., a conditional factor loading specification together with a state-dependent recovery rate, which is the model that we are developing. If the empirical results show that it shows superior performance in predicting default, then the outstanding performance of our refined RRFL model will be clear.

In the FC model, we estimate the Pearson’s correlation coefficient between each obligor and the common factor and set the recovery rate as constant. This is a conventional model used to measure capital requirements 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 asymmetric systematic impact from the market. The RR model uses the method proposed by Amraoui et al. (2012) to investigate the effects of the stochastic recovery rate. It allows the LGD function to be driven by the common factor and the hazard rate, while maintaining constant factor loadings. In the RRFL model, we incorporate the conditional factor loading into the state-dependent recovery rate and model them by sharing the unique common factor. To evaluate whether these two specifications significantly improve the default prediction, we use the dataset of daily stock indices of the S&P 500 to represent the market (common factor) and the respective stock prices of the defaulting companies for the period of five years before the default year from the Datastream database. In theory, stock returns should reflect the credit risk information of each firm, based on Merton (1974). Moreover, Xiang et al. (2015) document that strong evidence of time-varying credit risk links to equity markets.

Our default data analysis contains 2008 and 2009 data, as collected by Moody’s report. We use Moody’s Ultimate Recovery Database (URD), which is the ultimate payoff that obligors can obtain when the defaulting company emerges from bankruptcy or is liquidated rather than the post-default trading price that is proposed by Carty et al. (1998). These authors examine whether the trading price represents a rational forecast of actual recovery and find that it does not. For this period, we employ a state-dependent concept to capture the asymmetric impact from the common risk factor. As a result, both conditional factor loading and state-dependent recovery rates improve the calibration of our default prediction. The conventional factor copula underestimates the impact of systematic risk and portfolio credit loss when the market is in a downturn. We find that incorporating factor loading into the state-dependent recovery rate improves the accuracy of the default prediction. This result is consistent with the goal of Basel III, which emphasizes the role of systematic risk on overall financial stability and default risk. In our later empirical analysis, we concentrate on senior unsecured bonds because there is a rich data source available.

The remainder of the study is organized as follows. Section 2 describes the goal of Basel III. We present a general framework and the standard one-factor Copula in Sect. 3. Furthermore, we extend the standard one-factor Copula using conditional factor loading and the state-dependent recovery model. Section 4 describes the dataset. In Sect. 5, we offer empirical evidence. Section 6 presents our conclusions.

## 2 Systematic risk in Basel III

As highlighted by Basel III, several aspects of systemic risk are crucial to the financial markets. First, a bank can trigger a shock throughout a system, and the shock can spill over to its counterparties (Drehmann and Tarashev 2013). Second, procyclicality can also destabilize all the systemic risk. Borrowers cannot offer more funding, as their collateral assets have depreciated due to weak economic conditions. Third, as Basel II focused on minimizing the default probability of individuals, this accord failed to guarantee a stable financial system due to its inattention to systemic risk. The new Basel accord is thus expected to emphasize the role of systemic risk.

The systematic factor is an important driver of systemic risk and likely constitutes a serious 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 impedes the financial system, whereas the latter refers to the commonality in the risk exposures of financial institutions. Their model assumes that systemic risk can have systematic and idiosyncratic components. Systemic risk is understandably heightened by systematic risk. A bank is characterized as a systemically important (too-big-to-fail) financial institution; its default would lead to a dramatic impact on systemic risk. This is the very outcome that Basel III attempts to regulate and prevent. In our paper, our model proposes that the contribution of systematic risk is higher than that of the idiosyncratic component and that this dominance is characterized by a higher factor loading on systematic risk during a market downturn. We therefore see that the contribution of systematic risk to credit risk varies with time and market conditions. In this regard, one concern is the interconnection between credit risk and market risk. Notably – and importantly –the points discussed above determine the sufficiency of capital requirements in the banking industry.

To obtain sufficient capital requirements, the recovery rate is one of the determinant variables in the credit risk estimation. Thus, in a recession period, recovery rates tend to decrease while default rates tend to rise. As such, increasing capital requirements under this condition seems advisable. Most early academic studies on credit risk assume that recovery rates are deterministic (Schönbucher 2001; Rosen and Saunders 2010), or they are stochastic but independent of default probabilities (Jarrow et al. 1997; Andersen and Sidenius 2004). Neglecting the stochastic nature of the recovery rate and the interdependence between recovery rates and default rates results in a biased credit risk estimation (Altman et al. 2005).

To adhere to the spirit of Basel III, our study extends the previous literature in two ways. First, we highlight that systematic risk is a predominant factor in a recession period and provide an analysis that measures the proportional contribution of systematic risk against that of an idiosyncratic component. Second, we propose a methodology in which recovery rates and default rates are correlated by sharing a unique factor, both of which are state-dependent. Our model design, model simulation and empirical results offer several justifications for the goals of Basel III.

## 3 Methodology

### 3.1 Default modeling

*N*, which represents the number of assets. Specifically, we use a non-standardized Gaussian model to represent the deteriorating market condition by presuming a negative mean value together with a higher volatility. The model is based on decomposing a latent variable \(U_{i}\) for obligor

*i*into systematic factor

*Z*and idiosyncratic component \(\varepsilon_{i}\):

*Z*

*∼*N(

*µ,*\(\sigma^{2}\)) and \(\varepsilon_{i}\) have zero-mean unit variance distributions. In a Gaussian context,

*Z*and \(\varepsilon_{i}\) are orthogonal and \(\varepsilon_{i}\) is mutually uncorrelated. In an empirical study, \(U_{i}\) is a proxy of respective stock return, which is systematically related to a common factor,

*Z*(Choi and Jen 1991). The distribution of vector

*U*can be described by a copula function that joins two marginals,

*Z*and \(\varepsilon_{i}\). The correlation coefficient \(\rho_{ij}\) between \(U_{i}\) and \(U_{j}\) can be described by their \(\alpha_{i}\) and \(\alpha_{j}\):

*N*parameters \(\alpha_{i}\): \(i = 1, \ldots ,N\) must be estimated. We express the covariance matrices between \(U_{i}\) and \(U_{j}\) using a factor model,

*t*, \(I\left\{ {\tau_{i} \le t} \right\}\), by projecting \(U_{i}\) into \(\tau_{i}\). \(U_{i}\) here can be viewed as the proxies for a firm’s asset and liquidation value (Andersen and Sidenius 2004). In this regard, the lower asset value of the firm is, the shorter the time to default, \(\tau_{i}\). More precisely, \(U_{i} \le F^{ - 1} \left\{ {P_{i} \left( t \right)} \right\}\) leads to \(\tau_{i} \le t\), where \(P_{i} \left( t \right)\) is a hazard rate and marginal probability that obligor

*i*defaults before

*t*, and \(F^{ - 1}\)(·) donates the inverse cdf of any distribution. The default indicator then can be written as

*i*, \(G_{i} , i = 1, \ldots ,N\), we aggregate them as total portfolio loss,

*L*, as follows:

### 3.2 Conditional default model

In accordance with the spirit of Basel III, the systematic latent factor, *Z*, representing the general economic condition that characterizes the systematic credit risk, influences the default probability \(P_{i} \left( t \right)\) and the recovery rate \(R_{i} = 1 - G_{i}\). Given *Z*, the conditional default probability may be written as \(P_{i} \left( {Z|S = H,Q} \right)\) and conditional LGD, \(G_{i} \left( {Z|S = H,Q} \right)\), as a function of *Z*, and it is state-dependent, \(S \in \left\{ {H, Q} \right\}.\) H and Q represent the hectic and quiet periods, respectively.

*t*. Avoiding such a structure that may be too rigid, we assume the two asset returns,

*Z*(the common factor proxied by USD S&P 500) and \(U_{i}\) (firm stock price), to have a mixture of bivariate normal distribution (see “Appendix 1”) to obtain the estimation of \(\alpha_{i}^{H}\) and \(\alpha_{i}^{Q}\). Given the conditional factor loading, \(\alpha_{i}^{H}\),\(\alpha_{i}^{Q}\), the conditional default model is defined as follows:

*Z*, and the corresponding factor loading govern the conditional default probability, which is consistent with empirical findings (Andersen and Sidenius 2004; Bonti et al. 2006). Notably, \(\alpha_{i}^{S}\) is state-dependent instead of a constant setting in the previous literature (Andersen and Sidenius 2004; Amraoui et al. 2012). Ang and Chen (2002) set the probability of both regimes equally (\(\omega = 0.5\)); however, we instead estimate it from the historical data of the S&P 500 Index return proxied for systematic risk,

*Z*, P(S = H) = \(\omega\), P(S = Q) = 1 − \(\omega\) using expectation–maximization (EM) algorithm.

*i*, in relation to the common factor

*Z*and the marginal default probability \(P_{i}\). The state-dependent LGD is expressed as

In Eqs. (9, 10), \(0 \le \bar{R}_{i} \le R_{i} \le 1\) indicates a downward shift of \(\bar{R}_{i}\) to \(R_{i}\), such that \(\bar{R}_{i} = R_{i} - \nu\) and \(R_{i} \ge \nu > 0\). \(\nu\) is the size of the downward shift. By assuming that the expected loss in name *i* remains unchanged, we set \(\left( {1 - R_{i} } \right)P_{i} = \left( {1 - \bar{R}_{i} } \right)\bar{P}_{i}\). Please see the proof in A.1 in Amraoui et al. (2012). \(\varPhi ( \cdot )\) denotes a Gaussian distribution and \(\bar{P}_{i}\) is the adjusted default probability calibrated proposed by Amraoui and Hitier (2008). The LGD function, \(G_{i} \left( {Z |S = H,Q} \right)\), can essentially be obtained under formula (9,10). Numerous studies show that recoveries decline during recessions (Altman et al. 2005; Bruche and González-Aguado 2010). Consistent with the spirit of Eq. (6, 7), we design \(\alpha_{i}^{H}\),\(\alpha_{i}^{Q}\), and the factor loadings in Eq. (9,10) are therefore conditional and state-dependent. \(\bar{R}_{i}\) is a lower bound for \(G_{i} \left( {Z|S = H,Q} \right)\). Moreover, a partial derivative of the LGD function with respect to *Z* is less than zero, as shown by property 3.2 in Amraoui et al. (2012), which means that \(G_{i} \left( {Z|S = H,Q} \right)\) is decreasing in *Z*. Assuming \(\alpha_{i}^{H}\) > \(\alpha_{i}^{Q}\) means that a higher factor loading that is typically accompanied by a bad market condition on *Z* tends to increase LGD. In this regard, “Appendix 2” can be referenced for greater detail. The magnitude of LGD is not only influenced by *Z* but also sensitive to the factor loading under *Z*, which is one of our main findings and contributions to the literature. In addition, recovery rates are also linked to the probability of default and are negatively correlated (see Altman et al. 2005; Khieu et al. 2012). With *Z*, \(P_{i}\) and the estimated conditional factor loading \(\alpha_{i}^{H}\),\(\alpha_{i}^{Q}\), we obtain the state-dependent recovery rate, \(R_{i} \left( {Z|S = H,Q} \right)\), and state-dependent LGD, \(G_{i} \left( {Z|S = H,Q} \right) = 1 - R_{i} \left( {Z|S = H,Q} \right)\).

The detail of proof is set forth in “Appendix 3”.

### 3.3 Monte Carlo simulation

In this section, we investigate default prediction performance by establishing a simulation of realistic scenarios. The default probability and recovery rate functions are governed by systematic factors produced by different regimes. Indeed, they are crucial elements in evaluating the accuracy of the default prediction. Our interest is to see whether the designs of conditional factor loadings and state-dependent recovery rates contribute to the default prediction.

#### 3.3.1 One-factor non-standardized Gaussian copula

We simulate a one-factor non-standardized Gaussian copula subject to different states. As described in Eqs. (6) and (7), we generate systematic factor *Z* by non-standardized Gaussian distribution with different volatilities and independent \(\varepsilon_{i}^{{\prime }}\) s to reflect the nature of distinct variations exhibited in different market conditions.

*Z*, is presumed to distribute as \({\text{N}}\left( { - 0.03, 3.05} \right)\) estimated in 2008 and 2009, while \(\varepsilon_{i} \sim{\text{N}}\left( {0,1} \right)\) represents idiosyncratic risk.

*Z*and \(\varepsilon_{i}\) generated 10,000 scenarios. Given any of the generated systematic factor random variables,

*Z*, and using Bayes’ rule, we calculate the conditional probability that date

*t*belonged to the hectic is \(\pi \left( {Z = z} \right)\) using its counterpart, unconditional probability \(\omega\), as a formula (13).

*φ*

^{H},

*θ*

^{Q}represent in the hectic (H) and the quiet (Q) periods.

*φ*(·) is a normal distribution. Plugging

*α*

_{i}

^{H},

*α*

_{i}

^{Q}shared with the same simulated

*Z*random variables, conditional

*U*

_{i}|S is generated as developed in Eqs. (6, 7). These simulated random variables together with the published hazard rates

*P*

_{i}(

*t*) ideally produce the simulated default times.

#### 3.3.2 Default time

*t*= 1, which represents the time interval of 1 year, so that \(\tau_{i} < 1\) is referred to as a default event in the

*i*th obligor. The hazard rate \(P_{i}\) is the probability of occurrence of the default event within one year. \(\tau_{i}\) represents the default time of the

*i*th obligor. More precisely, the expected value of \({\text{I}}(\tau_{i} < 1)\) is P \((\tau_{i} < 1)\) and referred to as \(P_{i}\), see Franke et al. (2011) Chapter 22, which can be connected to the firm’s stock return or firm’s value, and \(U_{i}\) leads to \(P_{i} = {\text{E}}[{\text{I}}\{ U_{i} < {{\varPhi }}_{i}^{ - 1} \left( {P_{i} } \right)\} ]\), where \({{\varPhi }}_{i}\) denote the Gaussian cdf of \(U_{i}\). By applying generated \(U_{i}\) from the conditional factor model into the definition of the survival rate, we have generated the default time, \(\tau_{i}\), derived from \(1 - \exp \left( { - P_{i} \tau_{i} } \right) = {{\varPhi }}\left( {U_{i} } \right)\) (Hull 2006). To remain in the state-dependent environment, the conditional default time for each obligor is generated by formula (14).

*i*will default during the first year, conditional on no earlier default, and is obtained from Moody’s. It is the cumulative of the default rates during the first year. Equation (14) states that as \(U_{i} |_{S}\) becomes larger, \(\tau_{i} | {\text{S}}\) will become longer. The larger \(U_{i}\) reduces the tendency of default and postpones the default time, \(\tau_{i} | {\text{S}}\).

#### 3.3.3 State-dependent recovery rate simulation

In the third step, we consider a more realistic situation by simulating recovery rates, as described in our settings. The adjusted default probability \(\bar{P}_{i}\) is calibrated using hazard rate \(P_{i}\) from Moody’s report. \(\bar{R}_{i}\) is a lower bound for the state-dependent recovery rate [0,1]; therefore, we set \(\bar{R}_{i} = 0\) in the simplest case. With \(\alpha_{i}^{H}\),\(\alpha_{i}^{Q}\), *Z*, \(\bar{P}_{i}\), the simulated state-dependent recovery rates are obtained using formula (9, 10).

#### 3.3.4 Loss function

Given the simulated *Z* random variables, conditional probability \(\pi \left( {Z = z} \right)\) naturally provides better information than unconditional probability \(\omega\) does. By the given formula (15), we compare the theoretical loss amounts across four models with the realized loss values, and evaluate the performance of the default prediction by the mean of square error.

#### 3.3.5 Absolute error

## 4 Data

### 4.1 Financial return data

### 4.2 Data description

We use the list of default companies for 2008 through 2009 published by Moody’s annual report since this is a rich source of available data. In total, we obtained 341 defaults with corporate bond recovery rates from Moody’s URD covering the period from 1987 to 2007. We focus on senior unsecured bonds because of their wide use in financial contracts, regulatory rules, and the risks associated with measuring for assets under the standardized approach of Basel II (Pagratis and Stringa 2009). We also collected the credit rating of obligors from Moody’s to measure the hazard rate. Although there are 94 and 247 defaulting firms in 2008 and 2009, the observations were reduced due to missing stock prices and credit ratings of obligors’ bonds. If there were insufficient reported stock prices of defaulting subsidiary companies, we used the stock prices of parent companies instead. In all cases, 31 and 64 sampling firms were collected in 2008 and 2009, respectively.

Estimate mixture of normal distribution by employing an EM algorithm

Model | Probability | Mean | STD |
---|---|---|---|

Period | 2003–2007 | ||

Unconditional | 100.00% | −0.01 | 0.99 |

Conditional on quiet | 21.97% | 0.09 | 0.24 |

Conditional on hectic | 78.03% | −0.03 | 1.12 |

Period | 2004–2008 | ||

Unconditional | 100.00% | 0.04 | 0.99 |

Conditional on quiet | 24.91% | 0.19 | 0.26 |

Conditional on hectic | 75.09% | −0.01 | 1.14 |

As presented in Table 2, the volatility of the hectic distribution is larger than that of the quiet distribution, and the mean of the hectic distribution is smaller than that of the quiet distribution, reflecting the fat tails and right skew that are consistent with Kim and Finger (2000).

## 5 Empirical result

### 5.1 Conditional factor loading estimation

In our approach, we consider this asymmetric correlation structure under real market conditions to implement the conditional default model developed in Sect. 3.2. As shown in Figs. 1 and 2, the factor loadings \(\alpha_{i}\) in state H are higher than those in state Q. As factor loadings become higher in state H, the correlation coefficient \(\rho_{ij}\) between firm *i* and *j* defined in Eq. (2) is expected to increase in this market condition. Therefore, obligors tend to co-move more closely during hectic periods than during quiet periods.

### 5.2 State-dependent recovery rate estimation

*Z*on the state-dependent recovery rate, we use Fig. 3 to 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 Sect. 3.2. It can be observed that as the effect of the systematic factor on the recovery rate is positive, the recovery rate gets higher as

*Z*grows. Because the slope of this curve is influenced by estimated \(\alpha_{i}^{H}\),\(\alpha_{i}^{Q}\) corresponds to formula (9, 10), the slopes behave differently in the four panels but stay monotonically positive. We also depict the stochastic recovery rates in red ‘+’ estimated and simulated through the Amraoui et al. (2012) model, in comparison with blue ‘*’, which is simulated in our model. Taking (c) E*TRADE as an example, compared with the simulated recovery rates based on Eqs. (9) and (10), we note those generated from Amraoui et al. (2012) by assuming constant factor loadings tend to produce higher recovery rates in the market downturn and lower rates in the booming market. This evidence suggests that the recovery rate may be overestimated in a bearish market but underestimated in a bullish market if constant factor loading is assumed. As a consequence, it is highly possible to underestimate credit loss in a bearish market and overestimate it in a bullish market. Similarly, the evidence from (a) Glitnir Banki (b) Lehman Brothers Holdings, Inc. and (d) Idearc, Inc. are comparable and consistent. Notably, the impact of the systematic factor on the recovery rate seems nonlinear, as it is higher in the market downturn but relatively mild in the booming market, and its marginal slope decreases abruptly when the index return decreases; however, the marginal slope decelerates when the index return becomes positive. This simulation result is in accordance with the Moody’s report in Table 1. From 2004 to 2006, the annual recovery rates of senior unsecured bond increase slowly. As the crisis begins in August 2007, the recovery rate drops dramatically. By capturing the correlation structure, \(\alpha_{i}^{H}\) > \(\alpha_{i}^{Q}\), as shown in (a), (b), (c) and (d), we find this asymmetric pattern, which is more consistent with reality.

### 5.3 Empirical results of absolute errors

To gauge the conditional factor loading and state-dependent recovery rate approaches for default prediction, we propose four models: (1) the FC model, i.e., the standard one-factor Gaussian copula model with a constant recovery rate developed by Van der Voort (2007) and Rosen and Saunders (2010); (2) the RFL model, i.e., the one-factor Gaussian copula model in which factor loadings are tied to the state of the common factor and the recoveries assumed as constant, as proposed by Kalemanova et al. (2007) and Chen et al. (2014); (3) the RR model, i.e., the standard one-factor Gaussian copula model but with the recoveries related to the macroeconomic state (Amraoui and Hitier 2008; Elouerkhaoui 2009; Amraoui et al. 2012); and (4) the RRFL model, i.e., a conditional factor loading specification together with a state-dependent recovery rate. We address the question of whether the two specifications, conditional factor loading and the state-dependent recovery rate model, are meaningful and significant in explaining the gap between expected and actual loss value. To check the predictive ability of the different models, we report the AE and MAE estimated from Sect. 3.3.5.

The mean of actual portfolio loss, expected portfolio loss and AE, MAE (in million)

FC | RFL | RR | RRFL | |
---|---|---|---|---|

2008 | ||||

Actual portfolio loss | 2035.02 | 2035.02 | 2035.02 | 2035.02 |

Expected portfolio loss | 1070.57 | 1085.67 | 1537.46 | 1567.66 |

AE | 964.45 | 949.35 | 497.56 | 467.36 |

MAE | 31.11 | 30.62 | 16.05 | 15.08 |

Expected portfolio loss/actual portfolio loss (%) | 52.61 | 53.35 | 75.55 | 77.03 |

2009 | ||||

Actual portfolio loss | 3853.10 | 3853.10 | 3853.10 | 3853.10 |

Expected portfolio loss | 2033.25 | 2064.47 | 3318.25 | 3380.69 |

AE | 1819.85 | 1788.63 | 534.85 | 472.41 |

MAE | 28.43 | 27.95 | 8.36 | 7.38 |

Expected portfolio loss/actual portfolio loss (%) | 52.77 | 53.58 | 86.12 | 87.74 |

We compare the four competing models of each obligor and choose the best model for achieving the minimum AE and MAE. We find that including the conditional factor loading (RFL model) instead of the Pearson correlation (FC model) does not significantly improve the estimations in 2008 and 2009. Table 3 shows that introducing the state-dependent recovery rate (RR model) leads to a promising improvement over the standard model the (FC model). We interpret this to mean that the setting of a stochastic recovery rate seems necessary, which brings a remarkable improvement to the default prediction, which is consistent with Altman et al. (2005) and Ferreira and Laux (2007). Compared with the RR model, the RRFL model includes conditional factor loading in default probabilities and a state-dependent recovery rates function and produces considerably more modest improvements.

We propose two specifications on factor loading and recovery rates across four models. If we assume that default probabilities are a function of two-state correlation constructs but that recovery rates are not, the specification is only identified as concentrated on factor loading. In this case, the recovery rates do not contain information about the state of the business cycle. Conversely, if we assume that recovery rates vary, but factor loading is fixed, then the refinement occurs only by means of variations in the recovery rate. Since the RRFL model with both specifications is superior to the other three competing models, and there is no redundant specification in this study. In this regard, we extend the models proposed by prior studies (Kalemanova et al. 2007; Van der Voort 2007; Amraoui and Hitier 2008; Elouerkhaoui 2009; Amraoui et al. 2012; Rosen and Saunders 2010; Chen et al. 2014), which leads to more accurate default predictions in one year.

### 5.4 Basel III: relative contribution

^{°}line represents the proportion of systematic risk that is equal to that of idiosyncratic risk. If the scatter points are located in the ‘A, B, C, D’ zones, the contribution of systematic risk to default risk is greater than that of idiosyncratic risk. On the other hand, if the scatter points are settled in the ‘a, b, c, d’ areas, the contribution of the systematic component is less than that of idiosyncratic risk. For example, the effect of systematic risk on default risk will become larger when point ‘Y’ moves to point ‘X’. Most studies focus on either systematic (King and Khang 2005; Uhde and Michalak 2010) or firm-specific components (Goyal and Santa-Clara 2003; Ferreira and Laux 2007), and a limited number of studies compare the influence of both of them.

By simulating \(Z \sim {\text{N}}\left( { - 0.03, 3.05} \right)\), each simulated *Z* random variable can therefore be mapped into a specific conditional probability of being in a hectic state in Eq. (13). We gather the scatter plots into three groups here. The first group (marked as ‘+’ in red) includes only the simulated *Z* r.v. with projecting conditional probabilities above the 75% quartile, and indicates that they are generated in distress. The second group (marked as ‘*’ in blue) includes the *Z* r.v. with projecting conditional probabilities below the 25% quartile to indicate that they are generated in a bullish atmosphere. The third group (marked as ‘x’ in yellow) collects the rest. With regard to the tranquil scenarios (‘blue’ points) in 2008, most observations were located in the area in which the relative contribution of idiosyncratic risk is larger than that of the economy-wide component, where credit risk was mainly driven by the idiosyncratic component before the subprime crisis, as reported in Rodríguez-Moreno and Peña (2013), who found that idiosyncratic components were larger than systematic risk before the subprime crisis and were extracted from the CDX-IG-5y using high-frequent measures. At the beginning of the financial crisis, systematic risk skyrocketed. Intuitively, systematic risk increases sharply due to the larger factor loadings when the market is in hectic scenarios. Our result shows that systematic risk was higher than the idiosyncratic component in the hectic scenarios (‘red’ points) in 2008; in the quiet scenarios, however, firm-specific factors are more important at some points, as noted by Rodríguez-Moreno and Peña (2013). Similarly, it has been shown that the relative contribution of the systematic component explains a higher proportion of obligor asset value in 2009.

More visibly, the 3D plot identifies the relationship among the level of average \(U_{i} |_{S}\), which is referred to as the mean of firms’ value, systematic and idiosyncratic component. Each observation in Fig. 5 reflects its mean of \(U_{i} |_{S} \quad i = 1, \ldots ,N\) in each simulated day in 2008 and 2009, respectively. Figure 5 shows that the points in the hectic period marked as red ‘+’ indicates a negative shock from systematic risk, which lowers the average asset value of obligors; specifically, most observations show the negative impact of systematic shock, which accounts for a substantially larger proportion of firms’ value substantially. Note that it is easy to drive the default event since it lowers the firms’ value significantly. On the other hand, the points in quiet days marked as blue ‘*’ indicate a positive shock from the systematic component. However, the negative shock from firm-specific factors may compromise the benefit from economy-wide components that lowers the level of average \(U_{i} |_{S}\) at some points.

Our model emphasizes the importance of systematic risk, which explains most obligors’ default behavior, particularly in hectic periods, which is one of the important features of Basel III (Tarashev et al. 2010; Uhde and Michalak 2010; Schwerter 2011). To be specific, we measure and demonstrate the contribution of overall systematic risk to each asset, and identify the impact direction from systematic and idiosyncratic risk. Moreover, this analysis can be applied to a variety of systematic risk measures. In this sense, portfolio managers should be aware of the systematic risk that can substantially influence the value of portfolios. We propose that the regulatory tool of Basel III could be estimated with such contributions. A related question is how these measures can aid policymakers. The measures in this paper can be used as a tool to prevent systematic crises, and our model can be used as an early warning system that will alert regulators when an individual bank is in trouble and to intervene before a crisis occurs.

### 5.5 Robustness test

*s*is CDS spread. We consider the latest one-year prior to the default year CDS quotes of obligors provided from Datastream. We also use a credit spread, which is the yield on an annual par yield bond issued by the obligors over one-year LIBOR (London Interbank Offered Rate) if the obligor does not have CDS data. Theoretically, the CDS spread is close to the credit spread (Hull and White 2000; Hull et al. 2004). By plugging in the recovery rate,

*R*, obtained from the Moody’s report, we compute the average default intensity, \(\bar{\kappa }\), per year conditional on no earlier default instead of \(P_{i}\). Compared with \(P_{i}\) from the Moody’s annual report, a CDS spread with active trading activity reflects the market assessments of default risk in a timely fashion. In this regard, the proposed models that incorporate the hazard rate implied in CDS spreads may yield a better prediction.

The actual portfolio loss, expected portfolio loss, AE, and MAE (in million) for robustness

FC | RFL | RR | RRFL | |
---|---|---|---|---|

2008 | ||||

Actual portfolio loss | 1489.81 | 1489.81 | 1489.81 | 1489.81 |

Expected portfolio loss | 920.68 | 930.11 | 1245.14 | 1258.17 |

AE | 569.13 | 559.70 | 244.67 | 231.64 |

MAE | 22.76 | 22.39 | 9.79 | 9.27 |

Expected portfolio loss/actual portfolio loss (%) | 61.80 | 62.43 | 83.58 | 84.45 |

2009 | ||||

Actual portfolio loss | 2707.30 | 2707.30 | 2707.30 | 2707.30 |

Expected portfolio loss | 1776.77 | 1784.18 | 2381.91 | 2402.54 |

AE | 930.52 | 923.11 | 325.39 | 304.76 |

MAE | 22.16 | 21.98 | 7.75 | 7.26 |

Expected portfolio loss/actual portfolio loss (%) | 65.63 | 65.90 | 87.98 | 88.74 |

## 6 Conclusion

This paper proposes a refined factor copula model to assess and predict credit risk. On the basis of our estimated model, we find that systematic risk plays a simultaneously critical role in governing default rates and recovery rates simultaneously. Our simulation results show that recoveries vary with the returns of the S&P 500 and that the impact of systematic factors on the recovery rate is asymmetric by finding a higher factor loading in hectic periods than in tranquil periods. Among the various factor copula models developed in the past and in the current literature as the competing models, the model with conditional random factor loading and a state-dependent recovery rate turns out to be the best performing. In other words, our refined model contributes to studies that have been mapped to three 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 risk to each firm’s value, and we also identify the relative roles of both systematic and idiosyncratic risk. Moreover, this analysis can be applied to a variety of systematic risk measures, and it aids regulators in preventing a systematic crisis. In addition, by investigating the effect of state-dependent recovery rates on the loss function, we suggest that banks should apply this capital requirement issue to ensure its sufficiency.

In further research, we plan to go beyond this study in several ways. First, other copula functions can be modeled to capture various dependence structures. Second, the marginal distribution can be considered in a more general way to capture a fat-tail feature. We will leave these issues for future studies.

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