Methodologies Of Portfolio Credit Risk Models

Published Date: 02 Nov 2017

Source: Kealhofer and Bohn (2001)

To explain it further, Kealhofer and Bohn (2001) assert that shareholders of a firm can be seen as holders of a call option on the firm’s asset value with a strike price equal to its liabilities. So, the shareholders can choose to exercise the option and pay the debt value or choose to default and pay a rate of the debt value to the lender considering whether the option is in the money or out of the money. However, based on the studies of Oldrich Vasicek and Stephen Kealhofer (VK), Moody’s KMV uses an extended version of Black-Scholes-Merton (BSM) formulation. The differences between these two models are summarized in table 1.

Table 1. Comparison of BSM and VK EDF models

Source: Bohn (2006)

From table 1 it may be not clear

CreditRisk+

In contrast to the previously presented models, CreditRisk+ is a pure actuarial model (Crouhy et al., 2001). In other words, the framework tries to model the default rate distributions, and via probability density function of number of total defaults it finds the analytical loss distribution of a bond or a loan portfolio.

CreditRisk+ of Credit Suisse First Boston (CSFB) is a default-mode type model – it does not involve rating migrations- but analogous to KMV Portfolio Manager, it is a framework that models default rates in a continuous manner. In addition, CreditRisk+ underlines the fact that it is not possible to know the precise time of a default or the exact number of total defaults in a credit portfolio within a certain time horizon, since default events are consequences of several different successive events (CSFB, 1997). Thus, CreditRisk+ tries to determine the distribution of number of defaults over a period by defining default rates and their volatilities but does not deal with the time of the defaults. CreditRisk+ requires four types of input; obligors’ credit exposures, default rates and their volatilities, and recovery rates. Furthermore, to determine default probabilities of the obligors and their volatilities over time, CSFB (1997) suggests the use of credit spreads in the market or use of obligors’ ratings as a proxy (by deriving a common default rate and volatility for each credit rating from historical data of rating changes). Therefore, CreditRisk+ tries to combine the discrete nature of rating transitions with the continuous nature of default rates while CSFB (1997) emphasizes that one-year default rates change simultaneously with the state of economy and several other factors resulting in a deviation from average default rates.

The aim of the framework is to find the total loss distribution. In 2002, Bessis refers to that CreditRisk+ easily generates a portfolio’s analytical loss distribution under several assumptions as regards the loss distributions of portfolio segments and their dependency structure.

METHODOLOGICAL APPROACH

In this section of thesis the short description about data sample of it is given. Also I give a clear explanation of CreditMetrics, KMV Portfolio Manager and CreditRisk+ methodologies. The exact step-by-step procedure of models calibration is given in this part of thesis. The combination of these models will be used to describe the credit quality of the chosen portfolio.

Data sample selection and description

In the following few paragraphs you will find short description about the data which was used in order to describe the credit quality of the small chosen portfolio using three main credit risk management models.

To begin with, a list of all firms listed in NASDAQ OMX Baltic, Baltic main list was taken into account. Baltic main list [1] consists of total 36 firms, 18 of them are traded in OMX Vilnius stock exchange, 13 in OMX Tallinn stock exchange and the remaining the remaining five are traded in OMX Riga stock exchange. However, the aim was to create a small portfolio consisting from 15 to 20 stocks listed in NASDAQ OMX Baltic, so 6 firms were taken form OMX Vilnius (Lithuania), 6 – from OMX Tallinn (Estonia), and five from OMX Riga (Latvia) in order to keep all portfolio within NASDAQ OMX Baltic main list. The final portfolio consists of 17 firms. Data was selected for five years period starting from 1st of January 2007 and finishing with 31st of December 2012, the total number of 1521 daily equity value data was collected for each of selected company. Indices data for the same range of time was also collected to be used as regressors for the calculations. It was decided to choose OMX Baltic Benchmark (OMXBB) index to represent all three Baltic States selected firms belongs to, OMX Vilnius index to represent Lithuania, OMX Riga index to represent Latvia and OMX Tallinn index to represent Estonia.

Borrowings, Non-Current liabilities and Current liabilities data were extracted for all selected companies from the annual financial reports of 2011. Year of 2011 were used in order to get as much comparable results as possible because not all of selected companies have submitted their annual financial reports for the year of 2012 yet.

To conclude, we have small portfolio consisting of 17 firms, downloaded daily equity value data from NASDAQ OMX Baltic web page. Also we have four indices to be used as a regressors, and some data from each of the selected firms statements of financial position.

Methodologies of Portfolio Credit Risk Models

In the following three subsections the detailed description of three main credit risk management models is given. It is attempted to be as clear as possible while describing the steps which are necessary to be done in each of the models.

CreditMetrics

As it was mentioned before in literature review part, the correlations are highly important in this model. Therefore, in CreditMetrics model correlations between firms are specified by systematic factors. Those systematic factors are something like global parameters that can influence firms which have any relations, direct or indirect, to each other. There also are non-systematic factors which do not have correlation between two or more firms because they are firm specific. Furthermore, it is assumed that normalized asset returns (R) (for calculation see equation 1) are standard normally distributed, this assumption holds to systematic and non-systematic risk factors too. It is essential, that non-systematic risk can be diversified while systematic risk cannot.

(1)

R – Normally distributed asset returns;

Zi – systematic risk factors;

ϵ – idiosyncratic risk of a firm;

É‘ - factor loading for systematic risk factor;

b – factor loading for idiosyncratic risk.

It can happen that systematic risk factors will be distributed with any dependence with each other. In that case, the idiosyncratic risk factor loading is calculated by the equation 2, for maintenance of standard normal distribution.

(2)

Calibration of the model to the real data is a trivial task after the correlations between systematic factors are determined. However, it is critical to detect how much of the equity movements can be explained by which factors. RiskMetrics Group (2007) in its CreditMetricsTM Technical Document emphasizes that in their model risk comes not only default but also from changes in value due to up(down) grades. This means that it is important to estimate the credit quality migration. The examples of how credit quality can change in one year period are given in Figure 2. Then, calibration consists of assessment of the rating asset return thresholds and the factor loadings.

Figure 2. Examples of credit quality migrations (one-year risk horizon)

Source: RiskMetrics Group, 2007.

Let us denote ZDef, ZCCC, ZB, ZBB, ZBBB, ZA, ZAA, ZAAA as asset return thresholds for a BB [2] rated obligor. If, for example, R [3] < ZDef then our BB rated obligor goes into default, it is because it shows that asset returns of our obligor have fallen dramatically. If ZDef<R< ZCCC, then it is an indication that our obligor should be downgraded to CCC. Let’s say that ZDef is equal to -60%, then 60% or bigger fall in asset value of our obligor would mean the default of the obligor.

We assume that our R is normally distributed. This means that we can calculate the probabilities of default of obligor or downgrade of obligor:

Pr – probability of asset return threshold;

R – normally distributed asset return;

Φ – cumulative distribution for the standard normal distribution;

σ – volatility [4] (or standard deviation).

As it is already clear that there is a connection between asset returns and credit rating through asset return thresholds, this connection can be represented in scheme (see Figure 3). In the figure we see the normally distributed asset returns of our BB obligor and the standings of credit ratings if asset returns are increasing or decreasing. If, for instance, our obligors asset returns are between ZBBB and ZA the our obligor should be upgraded to BB, it is clearly seen in figure 3.

Figure 3. Distribution of asset returns with rating change thresholds.

Source: RiskMetrics Group, 2007

Next step is to calculate the asset return thresholds; the following equation is for this calculation:

(3)

Φ-1 – the inverse function of cumulative distribution for the standard normal distribution.

Here Φ-1 multiplied by Pr{Def} shows the bottom level under which a standard normal distributed random variable falls with the probability Pr{Def}. Then we have the value solved for ZDef we may use the probability Pr{CCC} to calculate the asset return threshold ZCCC, this value and probability Pr{B} should be used to calculate ZB, and so on to calculate all asset return thresholds till ZAA. We do not need to calculate the value of asset return threshold ZAAA because any value bigger than ZAA leads to an upgrade to AAA.

The final step in our calculations is the determination of factor loadings. We must to do it in such manner that factor loadings would encompass real correlation structure and standard normal assumption of asset return would remain. The steps needed for such a determination are as follows:

Determination of the variance-covariance matrix of systematic factors. For this we need to analyze time series data of systematic factors;

To determine the accuracy of each factor explanation power over normalized asset returns of each obligor in the portfolio. This is done using regression analysis. To be more specific we need to use R2 statistics [5] of these regressions to know the percentage part of explained normalized asset return.

The factor loadings for systematic risk factors in CreditMetrics are defined by the following equation:

(4)

Ri – R2 obtained for the ith obligor;

wik – allocation of assets in variance-covariance matrix of the ith obligor to kth factor;

σk – standard deviation of the kth factor;

σi2 – variance of ith obligor.

Systematic factor loading ɑik is a real level of variation explained by that factor. Regarding that the factors together can explain Ri portion of the asset return variation and that because of the correlation between these factors and the asset allocations, we cannot assign Ri’s equally among systematic factors. The variance of ith obligor is calculated as follows:

(5)

Then calculating factor loadings for idiosyncratic risk factors we assume that normalized asset returns have the variance equal to one. So the equation for idiosyncratic risk factor is:

(6) (7)

This is how the multi-factor default structure is modeled. To conclude, this requires weekly or even better daily stock return data of the obligors within a portfolio. Also variance-covariance matrix of the systematic risk factors is needed mostly for factor loadings calculations. So, by applying forward yield curves for each rating group and the multifactor default structure CreditMetrics is able to estimate credit portfolio value and the unexpected loss of a credit portfolio when carrying out Mark-to-Market simulations.

KMV Portfolio Manager

Then compared to CreditMetrics, KMV Portfolio Manager is using beta distribution for recovery rates. It means that the mean parameter is user defined. Also, the framework provides absolute risk contributions of obligor, while CreditMetrics provides marginal risk contributions of obligors in a portfolio.