CHAPTER NINE
Managing Credit Risk—Advanced Topics
CHAPTER STRUCTURE
Section II Select Approaches and Models—The Credit Migration Approach
Section III Select Approaches and Models—The Option Pricing Approach
Section IV Select Approaches and Models—The Actuarial Approach
Section V Select Approaches and Models—The Reduced form Approach
Section VI Pricing Credit Derivatives
Section VII The Global Credit Crisis of 2007
Annexure I The Global Credit Crisis—A Brief Chronology of Events in 2007–08
KEY TAKEAWAYS FROM THE CHAPTER
Understand the issues in measuring credit risk
Understand how credit risk is measured through quantitative models
Find out how the popular industry sponsored credit risk models work
Understand how credit derivatives are priced and traded in developed markets
Understand how the global credit crisis happened
SECTION I
BASIC CONCEPTS
Traditional methods, such as the Altman's Z score and other credit scoring models (described in the previous chapters) try to estimate the probability of default (PD), rather than potential losses in the event of default (LGD). The traditional methods define a firm's credit risk in the context of its ‘failure’—bankruptcy, liquidation or default. They ignore the possibility that the ‘credit quality’ of a loan or portfolio of loans could undergo a mere ‘upgrade’ or ‘downgrade’ as described in the classification of loans in the previous chapter.
The credit risk of a single borrower/client is the basis of all risk modelling. In addition, credit risk models should also capture the ‘concentration risk’ arising out of portfolio diversification and correlations between assets in the portfolio.
Recall from the previous chapter that the expected loss (EL) for a bank, arising from a single borrower or a credit portfolio is the product of three factors—Probability of default (PD), loss given default (LGD) and exposure at default (EAD).
Typically, therefore, credit risk models are expected to generate (1) loss distributions for the default risk of a single borrower and (2) portfolio value distributions for migration (upgrades and downgrades of a borrower's creditworthiness) and default risks. It follows that all models require some common inputs, such as (1) information on the borrower(s), (2) credit exposures to these borrowers, (3) recovery rates (or LGD) and (4) default correlations (derived from asset correlations) to assess concentration risk in the credit portfolio.
Credit risk models have a wide range of applications. They are prevalently used for
 Assessing the ‘EL’ of a single borrower.
 Measuring the ‘economic capital’^{1} of a financial institution
 Estimating credit concentration risk
 Optimising the bank's asset portfolio
 Pricing credit instruments based on the risk profile
Figure 9.1 generalizes the components of credit risk models:
FIGURE 9.1 COMPONENTS OF CREDIT RISK MODELS
Estimating PD, EAD and LGD—The Issues
Recall our discussion on ‘Loan pricing’ in an earlier chapter, where we used a simplified model to estimate the risk premium to be embedded in the loan price, analogous to the EL formula above. In that model, we had assumed that the PD had to be estimated for the borrower, taking into account various risk factors, and superimposed on the recovery rate (RR) (in the event of default), and the likely exposure of the bank to the borrower over the period (assumed as 1 year). The shortfall in the contracted rate arising out of the borrower's default probability was called the ‘risk premium’. This was analogous to the ‘EL’ and, therefore, had to be compensated. The bank did this by simply adding the ‘risk premium’ to the loan price. In this case, we had assumed that the EAD and LGD remained constant at the estimated levels.
However, in practice, the above view is too simplistic. There are various issues in estimating the risk factors.
Let us look at each of the values that define EL.
 Assigning a PD to each customer in a bank's credit portfolio is far from easy. One way to assess the PD is based on the fact that a default has occurred according to the bank's internal definition or a legal definition adopted by the bank—i.e., the borrower has exceeded some default threshold. Alternatively, a credit rating migration approach can be used.
 There are two commonly used approaches to estimate default probabilities. One is to base the estimate on historical default experience—market data based on credit spreads of traded products or internally generated data, or use models, such as the KMV and others—which will be described later on in this chapter. A second approach is to associate default probabilities with credit ratings (described in an earlier chapter)—either internally generated customer ratings or those provided by external credit rating agencies, such as CRISIL (or S&P). The process of assigning a default probability to a rating is called ‘calibration’.
 The EAD is the quantity of exposure of the bank to the borrower. This is not merely the ‘outstanding’ amount in the bank's books, as we assumed in the case of loan pricing in the earlier chapter. In fact, the EAD comprises two major parts—‘outstandings’ and ‘commitments’. The ‘outstandings’ represent the portion of the loan already drawn by the borrower, and is shown as a funded exposure on the bank's assets. In case of default, the bank will stand exposed to the total amount of the ‘outstandings’. In the time before default, the borrower is also likely to have ‘commitments’ (undrawn portion of the bank's promised credit limit to the borrower). These commitments can again be classified into ‘undrawn’ and ‘drawn’. Historical default studies shows that borrowers tend to draw quickly on committed lines of credit in times of financial distress. Hence, the EAD should also include the ‘drawn’ portion of the commitment at the time of default. Since the borrower has the ‘right’ but not the ‘obligation’ to draw on undrawn commitments (the embedded ‘option’), we can consider the proportion of drawn to undrawn commitments as a random variable. Therefore, now EAD becomes the aggregate of the outstanding exposure + a fraction of the undrawn commitment likely to be drawn prior to default. In practice, banks will calculate this fraction as a function of the borrower's creditworthiness and the type of credit facility.
 One approach to estimate EAD is to assume it as being equal to ‘current exposure + “Loan Equivalency Factor (LEF)” × unutilized portion of the limit’.^{2} The LEF is the portion of the unutilized credit limit expected to be drawn down before default, and depends on the type of facility and the credit rating just before default, as stated in the preceding paragraph. The cited World Bank paper provides a simple method to approximate LEF as shown in Figure 9.2.^{3}
FIGURE 9.2 ONE METHOD OF APPROXIMATING LEF
Source: Wold Bank Policy, Research Working Paper
 In the case of derivatives, the potential future exposure depends on the movement in the credit quality/value of the underlying asset, and this is estimated using simulation techniques.
 Modelling EAD should also take into account the ‘covenants’ attached to the loan document. Most of these covenants represent ‘embedded options’ for the borrower or the bank. For example, a covenant may require that the borrower provides additional collateral security in times of impending financial distress. In such a case, the borrower firm will have more information about its likelihood of default than the bank, which watches for signals of incipient distress (as described in an earlier chapter). In cases where banks are permitted by the covenants to close undrawn commitments based on predefined early indicators of default, the banks have to move quickly before the borrower draws on the undrawn commitments. This aspect will be discussed in detail in the subsequent chapter on ‘Capital Adequacy’ and the Basel Committee regulations.
 The LGD is simply the proportion of EAD lost in case of default, and is typically estimated as (1recovery rate)—i.e., the actual ‘loss’ to the bank in case of default by the borrower. However, recovery rate (RR) is not static—it depends on the ‘quality’ of the collaterals (in case of decline in economic growth, the value of these securities could decline) and the ‘seniority’ of the bank's claim on the borrower's assets. Additionally, ‘recovery’ could also entail ‘financial cost’ (such as interest income lost due to time taken for recovery or the cost of ‘workout’ as discussed in a previous chapter). In reality, therefore, the LGD is modelled as the ‘expected severity’ of the loss in the event of default.
 According to Schuermann,^{4} there are three broad methods to measure LGD.
 Market LGD, which can be computed directly from market prices or trades of defaulted bonds or loans;
 Workout LGD that depends on the magnitude and timing of the cash flows (including costs) from the workout process. LGD can be computed under this method as EAD less recovery plus workout costs incurred expressed as a percentage of EAD. Since the recovery process can be long winded, the cash flows should be expressed in present value (PV) terms. The question then arises as to what the appropriate discount rate should be. According to Schuermann, the discount rate could be that for an asset of similar risk, or the bank's own hurdle rate. In practice, workout LGD is the most popular, especially with banks with prior experience in such defaults; and
 Implied Market LGD that is derived from prices of fixed income and credit derivatives, using a theoretical asset pricing model.
 Approaches to modelling LGD are evolving rapidly. The initial models identified the factors driving the LGD values, including the correlation between PD and LGD. Schuermann's paper quoted above belongs to the ‘first generation’ models, as does Altman (et al)'s 2005 paper.^{5} The ‘second wave’ of models developed and empirically applied frameworks to quantify the correlation between PD and LGD. The current stream of models derives concepts to stress LGDs in economic downturns, as in the events happening since 2007. There is a need to now calculate a ‘downturn LGD’ (BLGD). The Basel committee on Banking Supervision has proposed such a model in its 2006 comprehensive version of the revised framework,^{6} and the Federal Reserve uses a simple model to arrive at the BLGD^{7}:
BLGD = 0.08 + 0.92 ELGD
where ELGD is the ‘expected LGD’.
The above linear relationship between BLGD and ELGD, however, does not take into account the degree to which the risk segments are exposed to ‘systematic risks’.
Why Do We Need Credit Risk Models?
There are important differences between market risk^{8} and credit risk. While market risk is the risk of adverse price movements in equity, foreign currency, bonds, etc., credit risk is the risk of adverse price movements due to changes in credit events, such as borrower or counterparty defaults or credit rating downgrades. These risks are also called ‘market VaR’ and ‘credit VaR’, the term VaR denoting ‘Value at Risk’, a technique to measure risk.
First, the portfolio distribution for credit VaR hardly conforms to the normal distribution. Credit returns tend to be highly skewed—while lenders’ benefits are limited when there is an upgrade in credit quality, their losses are substantial in the case of downgrade or default, as shown in Figure 9.3.
FIGURE 9.3 COMPARISON OF DISTRIBUTION OF CREDIT RETURNS AND MARKET RETURNS
Second, market VaR can be directly calculated based on daily/periodic price fluctuations. However, measuring portfolio effect due to credit diversification is more complex than for market risk. To measure this effect, estimates of correlations in credit quality changes among all pairs of borrowers/counterparties have to be obtained. However, these correlations are not readily observable, and several assumptions are required for such correlation estimates.
Third, though the default probability of a borrower or counterparty can be likened to ‘asset volatility’ in market risk, credit risk is more complicated. Take the instance of currency risk, an important market risk. One of the ways of arriving at the rate volatility is to observe the fluctuations in the currency over a period of time, and compute a reasonable estimate of the currency's volatility. However, a borrower's history may indicate nothing about the probability of future default; in fact, in many cases, the very fact that the lender stays exposed to the borrower signifies no prior default!
Credit risk models are valuable since they provide users and decision makers with insights that are not otherwise available or can be gathered only at a prohibitive cost. A credit risk model should be able to guide the decisionmaking manager on allocating scarce capital to various loan assets, so that the bank's riskreturn trade off is optimized. Optimization, of course, does not imply ‘risk elimination’—it means achieving the targeted or maximum return at minimum risk.
Banks typically have a diversified portfolio of assets. Most credit risk models compare riskreturn characteristics between individual assets or businesses in order to quantify the diversification of risks and ensure that the bank is well diversified.
A good credit risk management model is expected to generate the outputs as given in Figure 9.4.
FIGURE 9.4 OUTPUTS OF A CREDIT RISK MODEL
 Credit risk of portfolio in Figure 9.4 indicates the output defining the probability distribution of losses due to credit risk.
 The marginal credit risk of the portfolio in Figure 9.4 depicts how the riskreturn trade off would change if an incremental (marginal) asset is added to the portfolio. This output helps bank managers to decide on new investment options for the bank and the price that can be paid for the opportunity.
 The optimal portfolio is the optimal mix of assets that the bank should have at that point in time.
Every credit risk model is built on certain assumptions and, therefore, may not be able to generate all the outputs as above. Further, they cannot be used mechanically for predictions, since they are constructed mostly out of historical data. The outputs of the models are useful tools for decision making, but cannot substitute judgment and experience of bank managers.
Credit Risk Models—Best Practice Industry Models
Credit risk has been recognized as the largest source of risk to banks. The increasing focus of measuring credit risk through models—internal to banks and industrysponsored—can be attributed to the following:
 Substantial research has been carried out and advanced analytical methods have been evolved for formulating and implementing credit risk models;
 The ‘economic capital’ of a bank is aligned closely to the bank's credit risk. Hence, efficient allocation of capital within a bank presupposes accurate quantification of credit risk, and better understanding of the impact of credit concentration and diversification on the bank's asset portfolio on the bank's capital requirements.^{9}
 The other incentives flowing from credit risk measurement include better pricing of credit due to better valuation of financial contracts, and better management of funds, due to accurate assessment of risk and diversification benefits.
 Regulatory developments, such as the Basel II Capital Accord^{10} have necessitated use of risk models.
The previous chapter lists some industrysponsored credit risk models, based on approaches, such as the ‘credit migration approach’, ‘option pricing approach’, (these two classes of models are called ‘structural models’), the ‘actuarial approach’ and the ‘reducedform approach’.
The above models can also be classified as given in Figure 9.5. This classification is based on the approach adopted by the model for credit risk measurement.
FIGURE 9.5 CLASSIFICATION OF SOME INDUSTRYSPONSORED BEST PRACTICE CREDIT RISK MODELS
In the following sections, we will provide an overview of each of the above approaches.
SECTION II
SELECT APPROACHES AND MODELS—THE CREDIT MIGRATION APPROACH
The Credit Migration Approach (Used by Credit Metrics™^{11})
We have seen that credit quality can vary over time. ‘Credit migration’ is analysed as the probability of moving from one credit quality to another, including default, within a specified time horizon, usually a year.
‘CreditMetrics™’, the most wellknown credit migration model, builds upon a broad body of research^{12} that applies migration analysis to credit risk measurement. The model computes the full (1 year) forward distribution of values for a loan portfolio, where the changes in values are assumed as due to credit migration alone, while interest rates are assumed to evolve in a deterministic manner. ‘Credit VaR’^{13} is then derived as a percentile of the distribution corresponding to the desired confidence level.
Though we have earlier drawn the distinction between ‘market’ and ‘credit’ risks in order to isolate them and deal with them, the real world picture is different. For example, credit risk could arise due to volatility in currency movements, in which case, we have to capture market risk components as well in the credit risk models. The credit risk models are being increasingly refined to account for the impact of market risks.
However, as a first step, it would help to understand how the basic credit migration approach works for measuring credit risk.
Step 1: Assessing Real World Probabilities of Default—The Transition Matrix Credit rating agencies periodically construct, from historical data, a ‘transition matrix’, which shows the probability of an existing (rated) loan getting upgraded or downgraded or defaulting during a specified period in time (typically 1 year). The matrix is usually presented in the way as shown in Table 9.1. (in the case of S&P, CRISIL^{14}):
TABLE 9.1 CRISIL'S AVERAGE 1YEAR TRANSITION RATES
Source : CRISIL Ratings
Each cell in the above matrix contains a probability. The rating categories in the columns of the matrix signify the expected credit rating 1 year in the future. The rating categories in the rows of the matrix show current credit ratings.
For example, the probability in the cell AAAAA will signify the chances of a loan, currently assigned an AA rating, upgrading to AAA over the ensuing year. Similarly, the probability in the cell ABBB will signify the chances that a loan, currently rated A, downgrades to BBB over the next 1 year. The highlighted cells AAAAAA, AAAA and so on, therefore, signify the probability that the loan stays in the same category. These probabilities are ‘real world probabilities’^{15} as they have been calculated from historical data. For example, the transition matrix given by CRISIL in the reference provided has been constructed out of data over the period 1992–2007.^{16}
Box 9.1 explains the distinction between ‘real world’ and ‘risk neutral’ probabilities, and their usage in credit risk analysis.
BOX 9.1 REAL WORLD VERSUS RISK NEUTRAL PROBABILITIES
Table 9.2 shows the average cumulative default rates estimated by CRISIL^{17} for rated credit instruments.
TABLE 9.2 CRISIL AVERAGE CUMULATIVE DEFAULT RATES (WITHDRAWALADJUSTED)
Source: CRISIL ratings
From Table 9.2, we can infer that the probability of an Arated loan defaulting in 1 year is.42 per cent, in 2 years is 1.03 per cent and in 3 years is 1.97 per cent. These are ‘real world’ probabilities, estimated from historical data—in the above case, data from 2000 to 2007.
Now let us assume that an Arated loan is priced 100 bps (fixed) over the bank's Prime lending rate (the rate at which it lends to ‘risk free’ borrowers). Assuming a zero RR from the loan in case of default, the PD would be given by the equation 1 – [1/(1 + r)^{n}], where r denotes the premium over the riskfree rate and n the number of periods. Using this equation, the estimated probabilities of default would be 1 per cent, 1.97 per cent and 2.94 per cent for 1, 2 and 3 years, respectively.
Compare these probabilities with the historical probabilities of default in Table 9.2. Are the results inconsistent? They are not.
To understand the concept, we should note that the value of a riskfree loan is higher than the value of an Arated loan that carries some element of risk. In other words, (1) the expected cash flow from the Arated loan at the end of 3 years would be 2.94 per cent less than the expected cash flow from a riskfree loan at the end of 3 years, and (2) both cash flows would be discounted at the same rate in a risk neutral world, i.e., at the prime lending rate or riskfree rate.
The above is analogous to stating that (1) the expected cash flow from the Arated loan at the end of 3 years is 1.97 per cent less than the expected cash flow from a riskfree loan at the end of 3 years, and (2) the appropriate discount rate for the Arated loan's expected cash flow is about 0.32 per cent higher than the discount rate applicable to the riskfree loan's cash flow—because a 0.32 per cent increase in the discount rate leads to the Arated loan's value being reduced by the difference between the risk neutral and real world PD, i.e., 1.97 per cent and 2.94 per cent, which is 0.97 per cent over 3 years or 0.32 per cent per year.
It, therefore, follows that 1.97 per cent is a correct estimate of the real world PD, if the correct discount rate to be used for the debt cash flow in the real world is 0.32 per cent higher than in the risk neutral world. The increase in the discount rate seems reasonable as Arated loans have been perceived to have higher systematic risks—when the economy does badly, they are more likely to default.^{18}
The question, therefore, is—which PD is appropriate for credit risk analysis—the real world or the risk neutral one?
Typically, analysts use risk neutral default probabilities while pricing credit derivatives or estimating the impact of credit risk on the pricing of instruments such as in the calculation of the PV of the cost of default (since risk neutral valuation is commonly used in the analysis). Real world probabilities are commonly used in scenario analysis as for estimating Credit VaR (to estimate potential future losses in the event of default).
The information contained in transition matrices provided by renowned credit rating agencies is considered robust and useful and is being used as the starting point for modelling credit risk. However, there are certain criticisms, the most obvious of them being that these transition probabilities only reflect averages over several years, and lack the predictive ability to determine how serious or benign the forthcoming year's credit transitions would be. To address this shortcoming, smaller historical periods are sometimes chosen to reflect the most likely current scenario, and transition matrices constructed. Another method used to improve the predictive ability is to model the relationship between transition of credit defaults and select macroeconomic variables, say, industrial production.
However, it has to be recognized here that predicting the PD or downgrade/upgrade in credit quality is a difficult task in practice. Hence, it would be prudent to stress test the portfolio under a variety of transition assumptions.
In many cases, banks develop their own transition matrices based on their borrowers’ profile, since the probabilities provided by rating agencies are average statistics obtained from a heterogeneous sample of firms, with varying business cycles.
Step 2: Specifying the Credit Risk Horizon The risk horizon is typically taken to be 1 year, as assumed by the transition matrix. This is a convenient assumption, but can be an arbitrary one.^{19} It should be noted here that the transition matrix should be estimated for the same time interval over which the credit risk is being assessed. For example, a semi annual risk horizon would use a semi annual transition matrix.^{20}
Step 3: Revaluing the Loan In this step, we calculate the value of the loan at the end of the risk horizon. However, since there are eight possible ratings in the transition matrix, we have to estimate eight values of the same loan. This estimation has to be done within two broad, mutually exclusive assumptions.
 In the worst case, if the loan defaults over the horizon, some recovery would be possible based on the available collaterals.
 Alternatively, the loan can simply move between the ratings—an ‘upgrade’ or a ‘downgrade’. In the second case, we need to revalue the loan based on the rating to which it migrates.
Assume that credit risk is being assessed for one large, longterm loan.
a. In the worst case of default, we require two important inputs—the ‘seniority’ of the loan and the ‘recovery’ in case of default. The likely value of the loan will depend on how much can be recovered, say, by enforcing securities offered for the loan, which, in turn will depend on the ‘seniority’ of the loan, the mean recovery rate and the volatility (standard deviation) of the RR. The CreditMetrics technical document provides the Table 9.3^{21} :
TABLE 9.3 RECOVERY RATES BY SENIORITY CLASSS (PER CENT OF FACE VALUE—‘PAR’)
Seniority class  Mean (%)  Standard deviation (%) 

Senior secured  53.80 
26.86 
Senior unsecured  51.13 
25.45 
Senior subordinated  38.52 
23.81 
Subordinated  32.74 
20.18 
Junior subordinated  17.09 
10.90 
Source: Carty & Liberman [96a]—Moody's Investors Service
Table 9.3 can be interpreted easily. If the loan or bond is ‘senior secured’, its mean recovery in default would be 53.8 per cent of its face value, i.e., for a loan of say Rs. 100 crore, the average recovery would be Rs. 53.80 crore, with a volatility (standard deviation) of 26.86 per cent.
See Box 9.2 for further discussion on the relationship between the PD and recovery rates.
BOX 9.2 RELATIONSHIP BETWEEN PROBABILITY OF DEFAULT AND RECOVERY RATE
From our discussion earlier in this chapter, we know that EL under credit risk is calculated as the product of three factors—PD, LGD and EAD. RR is 1 – LGD.
While credit risk literature abounds with work on estimating PD, much less attention has been devoted to estimation of RR and its relationship with PD^{22}. However, empirical evidence seems to suggest that (1) recovery rates can be volatile and (2) they tend to decrease (or LGDs increase) when PD increases, say, due to economic downturns.
However, many credit risk models continue to work with simplifying assumptions based on static losses for a given type of debt as in Table 9.3. It should also be noted that credit VaR models such as the one being discussed, treat RR and PD as two independent variables.
b. In case of an expected upgrade or downgrade in credit rating, i.e., if a BBBrated loan moves up to A or moves down to BB over the risk horizon, the ‘value’ of the loan would also change. The magnitude of this change can be assessed by estimating the forward zero curves^{23} for each rating category, stated as of the risk horizon, going up to the maturity of the loan. Table 9.4 shows a sample of 1year forward zero curves by rating category.^{24}
We can now revalue the loan using forward rates as above, and the promised cash flows from the loan over the risk horizon for the appropriate rating category. The result of this exercise would be a distribution of loan values over all the rating categories.
TABLE 9.4 EXAMPLE 1YEAR FORWARD ZERO CURVES BY CREDIT RATING CATEGORY (PER CENT)
Source: CreditMetrics technical document
Step 4: Estimating Credit Risk The probabilities of migration of a singlerated loan to other categories from Step 1, and the result of the calculation in Step 3 would enable us calculate the expected value (mean) of the loan over the risk horizon, as well as the volatility (standard deviation). This standard deviation is the measure of ‘credit risk’ of the loan.
Another way to estimate the credit risk is to use ‘percentile levels’. For example, if we determine the 5th percentile level for the loan portfolio, it denotes the level below which the portfolio value will fall with probability 5 per cent. For large portfolios, percentile levels are more meaningful. This is also called the Credit VaR.
Illustration 9.1 clarifies the methodology through a numerical example.
ILLUSTRATION 9.1
Estimating the credit risk of a single loan exposure
The problem
Using the Credit Migration approach, calculate the Credit Risk (Credit VaR) of a senior secured loan of Rs. 100 crore, to be repaid in 5 years, at an annual interest rate of 10 per cent.
Key inputs and assumptions
Risk horizon is 1 year
CRISIL's credit transition matrix given in Table 9.1 will be used.
The current credit rating for this senior secured loan is ‘A’
The recovery rates and the 1year forward zero curves as given in step 3 above would be used in the calculations
For calculation convenience, it can be assumed here that the principal is paid at the end of 5 years, while the annual interest of Rs. 10 crore is paid every year.
Step 1  Using the transition matrix, recovery rates and the forward curves given above, we will estimate the credit risk of the loan over the risk horizon. 
We first determine the cash flows over the loan period—Rs. 10 crore every year at the end of next 4 years as interest, and Rs. 110 crore at the end of the 5th year, including principal payment. This is similar to a straightforward bond valuation. The CreditMetrics document mentions that, among others, the methodology is applicable to bank loans as for bonds.
Using the forward rates applicable to an Arated loan, we determine the value of the loan over the risk horizon. The discount rates are taken from the earlier 1year forward zero curve as applicable to an Arated instrument, and calculated as follows:
+ 10/(1 + 4.93%)^{3} + 110/(1 + 5.32%)^{4}
That is, loan value = Rs. 126.89 crore over the next 1 year
Now let us calculate the value of the loan, assuming it downgrades to BBB. The loan value will be calculated as follows:
+ 10/(1 + 5.25%)^{3} + 110/(1 + 5.63%)^{4}
That is loan value now changes to Rs. 125.67 crore, showing erosion in value due to the downgrade.
Continuing this exercise for all rating categories, we arrive at the possible 1 year forward values of the loan for all rating categories as in Table 9.5.
TABLE 9.5 ONEYEAR FORWARD VALUES OF THE LOAN FOR ALL RATING CATEGORIES
Yearend rating  Possible loan value 

AAA  127.66 
AA  127.46 
A  126.89 
BBB  125.67 
BB  119.66 
B  115.41 
CCC  99.232 
Default  53.8 
The default value is the mean value in default for a senior secured loan of Rs. 100 crore (53.8 per cent of Rs. 100 crore) as given in Table 9.3 containing recovery rates. Also from the same table, the standard deviation of the RR is 26.86 per cent.
Step 2  With the above inputs, we can calculate the volatility in loan value due to credit quality changes, using Table 9.6. Note that the probabilities of rating migrations have been obtained from CRISIL's transition matrix in Table 9.1. 
TABLE 9.6 VOLATILITY IN LOAN VALUE DUE TO CREDIT RATING CHANGES
We have now obtained one measure of credit risk—the standard deviation. The mean is calculated as an expected loan value using the probabilities of migration, and the standard deviation measures the dispersion of the likely loan values from the mean. Rs. 7.38 crore is, therefore, one measure of the absolute amount of the credit risk in the loan of Rs. 100 crore.
However, this value of credit risk has not taken into consideration the volatility (standard deviation) in the recovery rate. The RR in case of default was assumed to be the mean value of Rs. 53.80 crore, but this amount of recovery is uncertain since the volatility was estimated at 26.86 per cent. This uncertainty has to also find a place in our estimate of credit risk. The Credit metrics technical document uses the following formula to include the volatility in RR in case of default^{25}:
Accordingly, the standard deviation of the default state is included in the calculation. In the other cases of downgrade and upgrade, this volatility is assumed to be 0. This implies that in Table 9.6, the last term under ‘D’, 46.44, should be multiplied with the product of the PD (.009) and the square of the standard deviation of default, that is, 26.86^{2}. This yields a higher standard deviation of Rs. 7.81 crore, amounting to a 5.7 per cent increase in volatility.
The revised Table 9.7 is shown as follows:
TABLE 9.7 REVISED VOLATILITY
We should note here that we have considered the volatility of only the RR in case of default to arrive at a measure of the credit risk. This implies that we are assuming that there is zero volatility or uncertainty in the upgrade and downgrade states. However, this cannot be true in a practical situation—the volatility in each rating would be determined by the credit ‘spreads’ within each rating category. (The CreditMetrics technical document is aware of this shortcoming, arising out of nonavailability of data to determine what portion of the credit spread volatility is due to ‘systematic’ factors, and what portion is diversifiable.)
Calculating credit risk using the percentile level—the credit VaR
If we want to ascertain the level below which our loan value will fall with a probability of say, 1 per cent, we can use the values generated by Table 9.7. Let us move upwards in the probability column (generated by the transition matrix), stopping at the point where the probability becomes 1 per cent. In our transition matrix, we see that for an Arated loan, the PD is 0.9 per cent, and the probability that it moves to CCC is 0.7 per cent. The joint probability of being in default or in the CCC state is, therefore, 1.6 per cent, which is more than 1 per cent. Hence, we read off the value from the CCC row—99,33. This is the first percentile level value, which is Rs. 26.40 crore lower than the expected (mean) loan value.
Calculating credit VaR
The changes in the forward value of the Arated loan is shown in Table 9.8.
TABLE 9.8 CHANGES IN FORWARD VALUE OF THE LOAN
We can see that this distribution of changes in loan value under various rating changes shows long downside tails, the variance and standard deviation of the distribution remaining the same. In the earlier paragraph, we calculated the first percentile at 99.23. The corresponding first percentile of the distribution of change in loan value is – 27.66, which is the credit VaR.
(However, if we assume a normal distribution for the change in loan value distribution, the credit VaR at the first percentile would be much lower at –18.46)^{26}
Calculation of Portfolio Risk^{27}
The methodology in the Illustration 9.1 can be extended and employed to assess the credit risk in a portfolio of loans with the bank. The approach is described in detail in the CreditMetrics technical document.
In brief, the methodology for a two loan portfolio would be as follows:
The above procedure, as well as the Monte Carlo simulation that is used in practice for large portfolios, is described in detail in the CreditMetrics technical document.
The Credit Migration Approach (Used by CreditPortfolioView)
CreditPortfolioView (CPV) was developed by McKinsey and Company based on two papers by Thomas C Wilson^{29}, then Principal consultant at McKinsey and company.
In summary, CPV is a ratingsbased portfolio model used to define the relationship between macroeconomic cycles and credit risk in a bank's portfolio. It is based on the observation that default and migration probabilities downgrade when the economy worsens (i.e., defaults increase), and the contrary happens when the economy strengthens. The model simulates joint conditional distribution of default and migration probabilities for noninvestment grade (nonIG) borrowers (whose default probabilities are more sensitive to credit cycles, which are assumed to follow business cycles closely, than those of highly rated borrowers) in different industries and for each country, conditional on the value of macroeconomic factors, such as follows:
 Unemployment rate
 GDP growth rate
 Level of longterm interest rates
 Foreign exchange rates
 Government expenditure
 Aggregate savings rate
CPV calls these ‘conditional migration probabilities’ for a particular year, since it is constructed conditional on the economic conditions of that year. It should be noted here that the previous model, CreditMetrics, used ‘unconditional migration probabilities’, which is the average of conditional probabilities sampled over several years.
Thomas C Wilson^{30} points out that this approach differs from others in that it models the actual, discrete loss distribution (as opposed to using normal distributions or mean variance approximations); the losses or gains are measured on a default/no default basis for illiquid credit exposures as well as liquid secondary market positions, and retail lending, such as mortgages and overdrafts; the loss distributions are driven by the state of the economy, since most of the systematic risk in a portfolio has been found to arise from economic cycles; and the approach is based on a multifactor systematic risk that is closer to reality, while other models capture default correlations based on a single systematic risk factor.
The model follows three essential steps:
Step 1  Determine the state of the economy. 
Step 2  Estimate the PD of customers/customer segments. 
Step 3  Arrive at the loss distribution. 
Illustration^{31} 9.2 would help in understanding the methodology through a simple numerical example.
ILLUSTRATION 9.2
Credit portfolio view basic methodology
(Based on Thomas Wilson's paperportfolio credit risk—October 1998)
Step 1 The states of the economy
State  GDP  PD% 

Growth  1 
33.33 
Normal  0 
33.33 
Recession  –1 
33.33 
Step 2 Assume two customers/customer segments, X and Y, for simplicity
Customers have different risk profiles
State  Customer X—Medium risk—PD  Customer Y—High risk—PD 

Growth  2.5 
0.75 
Normal  3 
3.5 
Recession  4.5 
5 
Note that we assume that the high risk customer has lower
likelihood of defaulting when the economy is growing.
Step 3 Arrive at the loss distribution
There are four possibilities in each state of the economy—X defaults, Y defaults, both X and Y default, or there is no default Assume loss in case of default as Rs. 100 in the case of both X and Y
Here, we have assumed that X and Y are independent of each other, that is, correlation = 0
FIGURE 9.6 PROBABILITY OF LOSS ACROSS ALL STATES OF THE ECONOMY
It can be seen from the above simple example that the ‘conditional’ probability of, say, a Rs. 200 loss in the growth state of the economy is 0.01 per cent, while, from the above graph the ‘unconditional’ loss of Rs. 200 across all states in the economy has a probability of 0.12 per cent.
Thus, ignoring the effect of the economy (systematic risk), it appears that X and Y are correlated. However, if the economic state is considered, the PD of X and Y as well as the joint conditional PD for X and Y is seen to be significantly higher during times of recession than during economic growth. This would imply more correlated defaults during recessionary periods, rather than an overall correlation between defaults of X and Y arising from the presence of both systematic and nonsystematic risks.^{32}
Therefore, the basic premise of the CPV is that credit migration probabilities show random fluctuations due to the volatility in economic cycles. It also assumes that different ‘risk segments’ into which borrowers are classified (in the above example, ‘X’ is a ‘risk segment’ and so is ‘Y’, and a bank can classify its borrowers into varying risk segments that could correspond with the ‘ratings’ given in the migration matrix in the previous example on CreditMetrics) react differently to economic conditions.
If we assume X and Y to be large ‘risk segments’ of borrowers whose nonsystematic risks have been diversified away, then the systematic risks alone have to be considered to arrive at joint default correlations. Though this assumption is made implicitly by other models, this model extends the standard single factor approach to a multifactor approach to capture country and industryspecific shocks.
For each ‘risk segment’, CPV simulates a ‘conditional migration matrix’ as done in the simple numerical example above. Hence, in the three different ‘states of the economy’ envisaged in the ‘risk horizon’ (say, 1 year), the migration matrix will exhibit different characteristics, as in the simple two borrower example above: (1) in the ‘growth state’, it is likely that there are a lower number of probable downgrades and higher upgrade probabilities; (2) in the ‘normal state’, the conditional migration matrix is likely to be similar to the ‘unconditional migration matrix’ derived from historical observations (i.e., similar to the migration matrix shown in the CreditMetrics example); and (3) in the ‘recession state’, the downgrades are more likely than upgrades. This concept of ‘segmentspecific risk’ for each macro economic scenario gives the CPV flexibility to simulate a ‘systematic risk model’ with a large number of such scenarios and migration possibilities.
In practice, CPV uses simulation tools, such as ‘Monte Carlo’ methods to generate a systematic risk model and simulates the conditional default probabilities for each risk segment, using two different methods of ‘calibration’^{33}—CPV Macro and CPV Direct.
CPV Macro uses a ‘macroeconomic regression model’ using time series of historical data on macroeconomic factors, as described by Wilson in his papers quoted above.
CPV Direct was developed later to make the calibration easier, and derives default probabilities from a ‘gamma distribution’.^{34}
Finally, all conditional loss distributions are aggregated to arrive at the unconditional loss distribution for the portfolio. The technical documentation for CPV is not available freely, and will have to be obtained from McKinsey and company.
SECTION III
SELECT APPROACHES AND MODELS—THE OPTION PRICING APPROACH
The KMV^{35} Model
The models discussed above have made three critical assumptions: (1) all borrowers included in a specified rating category carry the same default risk, (2) the actual default rate within a rating category would be the same as the historical default rate that has been observed over the years and (3) all borrowers within each rating category carry the same PD or migration. These assumptions may not be totally valid in real world situations.
The KMV model strongly refutes these assumptions on the following grounds: (1) default rates are continuous, while ratings are adjusted in a discrete fashion by rating agencies; (2) rating agencies take time to upgrade or downgrade borrowers/counterparties, and such credit quality changes would be carried out at some point in time after the default risk is observed.
Further, KMV has demonstrated through simulation exercises that (a) the historical average default rate and transition probabilities can deviate significantly from the actual rates, (b) that substantial differences in default rates may exist within the same rating class and (c) the overlap in default probability ranges may be quite large. In some cases, the historical default rate could, in reality, overstate the actual PD of a specific borrower. Such inconsistencies may lead to banks overcharging their better borrowers (since the loan pricing depends on the PD), which may endanger their banking relationship, or worse still, benefit riskier borrowers.
The KMV model relates the default probability of a firm to three factors:
The Market Value of the Firm's Assets The Market value is computed as the PV of the future stream of free cash flows that is expected to be generated by the firm. An appropriate ‘discount rate’^{36} is used to discount the future cash flows and arrive at the PV.
The Risk of the Assets The value computed above is scenariobased and can be impacted by business and other risks in future. The asset values can fluctuate in future, and could be higher or lower than expected. This ‘volatility’ is asset risk.
Leverage The firm would have to repay all its contracted liabilities. The relative measure of outstanding liabilities (this would typically be taken at book value) to the market value of assets is an indication of ‘leverage’.^{37}
Why are these three factors important? See Box 9.3 for a simplified explanation.
BOX 9.3 THE MAIN DETERMINANTS OF PROBABILITY OF DEFAULT
Assume a firm with only one asset on its balance sheet—1 crore shares (book value Rs. 100 crore) of a blue chip company—financed by Rs. 80 crore of debt due at the end of 1 year, and Rs. 20 crore of equity.
Over the next 1 year, analysts forecast that one of the two things could happen with equal probability—the market could do well, in which case the value of the investment could rise to Rs. 150 crore, or the market would do badly, in which case the value of the investment falls by 50 per cent to Rs. 50 crore. If the first event happens, the firm can pay off its debt of Rs. 80 crore (ignore interest for the time being) and add Rs. 70 crore to its equity value. If the market does badly, however, the firm will not have sufficient money to pay off its debt (equity + asset value will be Rs. 70 crore, while the debt will be Rs. 80 crore), and will default.
In a second scenario, assume the same firm with only Rs. 40 crore of debt on its balance sheet, Rs. 60 crore being equity. In this case, the firm would not default even when the investment value falls to Rs. 50 crore, since the debt can still be paid off. However, only Rs. 10 crore would be added to equity value.
In a third scenario, if other things remain as in the first scenario, but Rs. 80 crore of debt can be repaid after 2 years, the firm would not default at the end of the first year even when the market does badly.
We can infer from the above that
 Equity derives its ‘value’ from the firm's cash flows (determined by the market value of the firm's assets)
 The firm's cash flows (or market value) could fluctuate depending on various factors—this is called ‘volatility’
 The lower the book value of liabilities as compared to market value of assets (leverage), the lower the likelihood of default by the firm
In other words, the lenders to the firm essentially ‘own’ the firm until the debt is fully paid off by the equity holders. Conversely, equity holders can ‘buy’ the firm (its assets) from the lenders as and when the debt is repaid. That is, equity holders have the ‘right’, but not the ‘obligation’ to pay off the lenders and take over the remaining assets of the firm.
Thus, simply stated, equity is a ‘call option’^{38} on the firm's assets with a ‘strike price’ equal to the book value of the firm's debt (liabilities). The value of the equity depends on, among other things, the market value of the firm's assets, their volatility and the payment terms of the liabilities. Implicit in the value of the option is an estimate of the probability of the option being ‘exercised’—i.e., for equity, it is the probability of not defaulting on the firm's liability. It is also obvious that the PD would increase if (1) the firm's market value of assets decreases, (2) if the volatility of the assets’ market value increases or (3) the firm's liabilities increase. These three variables are therefore the main determinants of a firm's PD.
The approach in Box 9.3 follows one of the major pillars of modern finance—the Merton Model used in quantification of credit risk.
In 1973, Black and Scholes^{39} proposed that equity of a firm could be viewed as a call option. This paved the way for a coherent framework in the objective measurement of credit risk. The approach, subsequently developed further by Merton in 1974,^{40} Black and Cox in 1976,^{41} and others after them, has come to be called ‘the Merton model’.
Box 9.4 presents an overview of the ‘Merton model’.
BOX 9.4 THE MERTON MODEL
The rationale behind Merton's ‘Pricing of corporate debt’ is as follows:
We know that a firm's cash flows are paid out to either debt holders (lenders) or equity holders. The lenders have priority over the cash flows till their debt to the firm (along with interest and other fees) is completely serviced. The remaining cash flows are paid to the equity holders.
Assume that the debt L of the firm has to be repaid only at the end of the term T, and there are no intermediate payments. If V is the value of the firm at T, the cash flow D to lenders can be expressed as
The remaining cash flows E accrue to equity holders who will receive at T
Hence, the firm value at T would be
The above implies that equity holders have the option to purchase the firm from lenders by paying the face value of debt L at time T—that is, they can ‘exercise’ the option to ‘call’ the firm away from the lenders at the ‘strike price’ L. In other words, given the firm value at a point in time, we can price its equity using an options pricing method, such as the Black–Scholes formula, in which E depends on the following factors: V the firm value, L the debt value, T – t the time to expiration, r the riskfree rate of interest, and σ the volatility of the underlying asset (the firm).
What is the option that the lenders have? The lenders have taken a risk in lending to the firm. In the event that the firm value falls below the debt value at T, the lenders have a claim on all the assets of the firm. In other words, the lenders have given the firm the ‘option’ of ‘buying’ away the assets by repaying the debt, that is, the lenders are ‘selling’ the assets to the firm. This implies that the cash flow to the lenders is similar to those from a riskfree asset less the ‘credit risk’ of the firm. This credit risk is similar to a ‘put option’,^{42} and, therefore, can be valued using the option pricing formula.
The classic Merton model (1974) makes use of the Black and Scholes option pricing model (1973—referred above) to value corporate liabilities. In doing so, it makes the following simplifying assumptions:
The (publicly traded) firm has a single debt liability and equity. It has no other obligations. Its balance sheet at time T appears as follows:
Liabilities Assets Debt L Market value of Assets A Equity E Total firm value V Total firm value V (=A) Thus, we would have the equation V = L + E at time T.
The firm's debt matures at time T and is due to the lender in a single payment. That is, the debt is ‘zero coupon’ debt.
The firm's assets are tradable and their market value evolves as a lognormal process.^{43}
The market is ‘perfect’—that is, there are no coupon or dividend payments, no taxes, no ban on short sales (selling the assets without owning them); the market is fully liquid, and investors can buy or sell any quantity of assets at market prices; trading in assets takes place continuously in time; borrowing and lending are at the same rate of interest; the conditions under which the Modigliani–Miller theorem^{44} of firm value being independent of capital structure are present; the ‘term structure’^{45} is flat and known with certainty; and importantly, the dynamics for the value of the firm through time can be described through a diffusiontype stochastic process.^{46} The last assumption requires that asset price movements are continuous and the asset returns are serially independent.^{47}
Merton modelled his firm's asset value as a lognormal process and stated that the firm would default if the market value of the firm's assets, V, fell below a certain default boundary, X. The default could happen only at a specified point in time, T. The equity of the firm is a European call option^{48} on the assets of the firm with a strike price equal to the face value of debt. The equity value of the firm at time T would therefore be the higher of the following scenarios: the positive difference between the market value of assets at T and the value of debt, or, if the difference is negative, zero (since the firm would default—see Box 9.3 titled “Main determinants of PD”).
We describe Merton's option pricing model through the following equations
The current equity price is
and
and
σ _{v}^{2} being the implied variance of asset values (asset volatility)
The above equation enables estimation of the firm value V (= market value of assets) and its volatility from the equity value and its volatility.
The ‘distance to default’ (DD) is given by the value of (d_{2})
The PD is 1–the probability of the equity option being exercised. This is given by N(–d_{2}) or 1 – N(d_{2}). From the lender's point of view, the model throws up the ‘value’ of debt for every unit change in firm value as 1 – N(d_{1}).
Oldrich Vasicek and Stephen Kealhofer extended the Black–Scholes–Merton framework to produce a model of default probability known as the Vasicek–Kealhofer (VK) model.^{49} This model assumes the firm's equity is a perpetual option on the underlying assets of the firm, and accommodates five different types of liabilities—shortterm and longterm liabilities, convertible debt, preferred and common equity.
Moody's KMV (MKMV) uses the option pricing framework in the VK model to obtain the market value of a firm's assets and the related asset volatility. The default point^{50} term structure (for various risk horizons) is calculated empirically.
What is the ‘default point’?
The Merton model assumption that a firm defaults when the market value of assets falls below the book value of its liabilities is modified by Crosbie and Bohn (2002) in their paper—‘While some firms certainly default at this point, many continue to trade and service their debts. The longterm nature of some of their liabilities provides these firms with some breathing space. We have found that the default point, the asset value at which the firm will default, generally lies somewhere between total liabilities and current, or shortterm, liabilities’.^{51} Further, the ‘market net worth’ is assumed as the ‘market value of assets less the default point’, and a firm is said to default when the market net worth reaches zero.
How Merton's basic options framework is used in the VK Model of KMV is shown in Figure 9.7.
FIGURE 9.7 RELATING MERTON'S MODEL WITH KMV
How the Black–Scholes–Merton model differs from the VK model is shown in Table 9.9^{52}:
TABLE 9.9 KEY DIFFERENCES BETWEEN THE BSM AND VK MODELS
BlackScholesMerton  VasicekKealhofer EDF model 
Two classes of liabilities: Short Term Liabilities and Common Stock.  Five classes of liabilities: Short Term and Long Term Liabilities, Common Stock, Preferred Stock, and Convertible Stock. 
No cash payouts  Cash payouts: Coupons and Dividends (Common and Preferred). 
Default occurs only at horizon.  Default can occur at or before horizon. 
Default barrier is total debt.  Default barrier is empirically determined. 
Equity is a call option on Assets, expiring at the Maturity of the debt.  Equity is a perpetual call option on Assets. 
Gaussian relationship between probability of default (PD) and distance to default (DD).  DDtoEDF mapping empirically determined from calibration to historical data. 
Source: Moody's KMV documents
MKMV combines market value of assets, asset volatility and default point term structure to calculate a DD term structure, which is then translated into a credit measure termed ‘expected default frequency’ (EDF). The EDF is the PD for the risk horizon (1 year or more for publicly traded firms)—and ‘default’ is defined by MKMV as the nonpayment of any scheduled payment, interest or principal.
MKMV's EDF credit measures can be viewed and analysed within the context of a software product called CreditMonitor™ (CM) on Moodys KMV Web site. CM calculates EDF values for years 1 through 5 allowing the user to see a term structure of EDF values.
The procedure adopted by MKMV^{53} for calculating the PD of a public firm can be described in the following three steps:
 Estimate asset value and volatility from the market value and the volatility of equity and the book value of liabilities. This is done using the option nature of equity, as given in the Merton model in Box 9.4 The Market value of equity is observed and the market value of the firm (assets) is derived from it. The model solves the following two relationships^{54} simultaneously:
The model solves for ‘asset value’ and ‘asset volatility’ based on the observed inputs of equity value and volatility and the capital structure. The ‘interest rate’ used in this model would be the ‘riskfree’ rate (option pricing theory).
 Calculate the DD as = (market value of assets – default point)/(market value of assets × asset volatility) The numerator reinforces the fact that if the market value of assets falls below the default point, the firm defaults. Hence, the PD is the probability that this event happens.
If the future distribution of the DD were known (over the relevant risk horizon), the default probability—called the expected default frequency (EDF)—would be the likelihood of the final asset value falling below the default point.
In practice, however, it is difficult to accurately measure the DD distribution. MKMV uses the fact that defaults occur either due to large adverse changes in market value of assets or changes in the firm's leverage, or a correlation of these two factors, and measures DD as the number of standard deviations the asset value is away from default, and then uses empirical data to determine the corresponding PD.
This process is pictorially depicted in MKMV documents^{55} in figure 9.8.
 The current asset value.
 The distribution of the asset value at time H.
 The volatility of the future assets value at time H.
 The level of the default point, the book value of the liabilities.
FIGURE 9.8 PROCEDURE ADOPTED BY MKMV FOR CALCULATING THE PD OF A FIRM
Source: Moody's KMV documents
 The expected rate of growth in the asset value over the horizon.
 The length of the horizon, H.
 Calculate the default probability
As stated in the earlier paragraph, default probability is empirically determined from data on historical default and bankruptcy frequencies.
Summarising, the calculations for EDF are done in Table 9.10 using the relevant variables shown in step 2:
TABLE 9.10 CALCULATIONS FOR EDF USING THE RELEVANT VARIABLES
S. No.  Variable description  How calculated 

a.  Market value of equity  Current value = current share price × no. of shares outstanding. 
b.  Book value of liabilities  From the firm's balance sheet 
c.  Default point  Liabilities payable within the risk horizon, say 1 year. KMV also computes a ‘critical threshold’ set as the aggregate of the firm's shortterm debt and half of longterm debt value 
d.  Market value of assets  Using the option pricing model 
e.  Asset volatility  Using the option pricing model 
f.  DD  [(d) – (c)]/(d × e) 
Note: The asset growth rate has been ignored in this summary
Source: MKMV documents
The final step is to map the DD to actual probabilities of default over the determined risk horizon. These are the EDFs, which are calculated from a large sample of firms including those that defaulted. Based on this historical information, the model estimates, for each time horizon, the proportion of firms with say DD = 4 that actually defaulted after, say, 1 year. If this proportion were, say, 0.33 per cent,^{56} it is taken to be the EDF of 33 bps. This PD is then assigned an implied rating by the model.
What do the EDF measures indicate?
 EDF measures are not credit scores. They are actual probabilities.
 If a firm has a current EDF credit measure of 3 per cent, it implies that there is a 3 per cent probability of the firm defaulting over the next 12 months.
 That is, out of 100 firms with an EDF of 3 per cent, we can expect, on an average, three firms to default over the next 12 months
 Further, a firm with 3 per cent EDF measure is 10 times more likely to default than a firm with 0.3 per cent EDF measure
For a technical note on calculating EDF measures using the VK model, and calculating longterm EDF, the MKMV document referenced in this section can be accessed at www.moodyskmv.com.
As in the credit migration approach described above, an empirical credit migration matrix is derived using the EDF measures.
The above model is suitable for a publicly listed firm, where market value of equity would be readily available to estimate asset prices. If the firm is a private firm, whose market value of equity is not readily available, KMV's private firm model requires the following additional steps to precede estimation of the firm's DD:
Step 1  Calculate the Earnings before interest, taxes, depreciation and amortization (EBITDA) for the private firm P in industry I 
Step 2  Divide the industry average market value of equity by the average industry EBITDA. This yields the average ‘equity multiple’ for the industry. 
Step 3  Multiply the average industry equity multiple from Step 2 by the private firm's EBITDA. This gives an estimate of the private firm's market value of equity. 
Step 4  The private firm's asset value can now be calculated as the market value of equity (Step 3) + the firm's book value of debt. 
From this point onwards, the calculation of EDF can proceed as for a public firm.
For a credit portfolio, as in the earlier credit migration approach, a global factor model has been developed using asset correlations. This is called the GCorr Model, and it derives asset return correlations from a structural model that links correlations to fundamental factors—systematic and idiosyncratic or firmspecific factors. The variables that define the GCorr correlation structure are the following:
 The relation between the firm and its systematic risk—this is given by R^{2}
 The relation between the firm's industry and country composition.
 The other factors—global, regional and industrial—that impact the correlations between industry and country combinations.
The GCorr model's factor structure^{57} is presented in Figure 9.9.
FIGURE 9.9 MKMV'S GCORR MODEL
Source: MKMV Web site^{58}
SECTION IV
SELECT APPROACHES AND MODELS—THE ACTUARIAL APPROACH
CreditRisk+™ Model
This model follows the typical insurance mathematics approach, which is why it is called the ‘actuarial model’.
The insurance industry widely applies mathematical techniques to model ‘sudden’ events of default by the insured. This approach contrasts with the prevalent mathematical techniques used in financial modelling, which is typically concerned with modelling continuous price changes rather than ‘sudden events’.
How does the actuarial model differ from other models outlined above—say, the CreditMetrics model, based on the credit migration approach?
In the credit migration approach, credit events (such as default or downgrade) are driven by movements in unobserved ‘latent variables’, which in turn give rise to ‘risk factors’. In a credit portfolio, correlations in credit events occur since different borrowers/counterparties depend on the same risk factors. In the actuarial approach, on the other hand, it is assumed that each borrower has a ‘default probability’, but no latent variables. This PD changes in response to macroeconomic factors. If two borrowers are sensitive to the same set of macroeconomic factors, their default probabilities change together, which in turn, give rise to ‘default correlations’.
CreditRisk+ has been developed based on the actuarial approach (insurance modelling techniques) by Credit Suisse Financial Products (CSFP).^{59} In this model, only default risk is modelled, not downgrade risk. The model makes no assumptions about the causes of default. Each borrower/counterparty can assume only one of two ‘states’ at the end of the risk horizon, say, 1 year—‘default’ (0) or ‘no default (1). The basic model setting is outlined in Box 9.5.
BOX 9.5 THE BASIC MODEL SETTING OF THE ACTUARIAL APPROACH IN CREDITRISK+
Assume a portfolio of N borrowers where n = 1…N, where the exposure to borrower n is E_{n} and the PD of borrower n is p_{n}.
Default of borrower n is represented by a ‘Bernoulli’^{60} random variable D_{n} such that at the end of the risk horizon, say, 1 year, the PD P(D_{n} = 1) = p_{n} and the probability of ‘survival’ (no default) P(D_{n} = 0) = 1—p_{n}.
If E_{n} is the exposure of the bank to borrower n, the total loss of the portfolio at the end of the year is given by
The objective is to determine the probability distribution of L under various sets of assumptions. Knowing the probability distribution of L will enable computing VaR and other risk measures for this portfolio.
There are two assumptions made under this model:
 For a loan or borrower or counterparty, the PD during 1 year (the risk horizon) is the same as the PD for any other year; and
 Given a large number of borrowers/counterparties, the PD of any one borrower is small, and the number of defalts that occur during any 1 year is independent of the number of defaults that occur during (N) any other year.
The probability distribution for the number of defaults during 1 year is then modelled as a Poisson distribution.^{61} The probability of realising n default events in the portfolio in 1 year is given by the expression^{62}
where n = 0,1,2,… is a stochastic variable with mean μ and standard deviation
μ = average [expected] number of defaults per year. For example, if μ = 4, over the next one 1 year,
It can be noted from the above equation that the distribution has only one parameter, μ. The distribution does not depend on the number of exposures in the portfolio, or the individual probabilities of default, provided these probabilities are uniformly small.
CreditRisk+ models credit risk in two stages as shown in Figure 9.10. This approach is typically used in evaluating losses originated in insurance portfolios. The cumulative average EL is arrived at using the loss frequency average and the loss severity average. However, for a dynamic analysis of risks, it is necessary to quantify loss variability by computing the distribution of cumulative losses that would combine the effects of frequency and severity of losses.
Applied to a credit portfolio, the concepts are understood as follows:
 Frequency—the number of defaults in a portfolio over the risk horizon (typically 1 year), and
 Severity—the amount, in currency units, of each individual default, that is, the LGD.
FIGURE 9.10 CREDITRISK+ STAGES IN MODELLING CREDIT RISK
We will describe these processes briefly.
Stage 1 

FIGURE 9.11 LOSS DISTRIBUTIONS GENERATED BY CREDITRISK+ ^{63}
Box 9.6 summarizes the process and notations used by the model in arriving at the loss distributions.
BOX 9. 6 PROCESS AND NOTATIONS USED IN CREDITRISK+
Scenario 1
Default losses with fixed default rates
Step 1 Slotting exposures into independent ‘bands’
Reference  Symbol 

Obligor/borrower  A 
Exposure  L_{A} 
PD  P_{A} 
Expected loss  λ_{A} 
L is chosen as the base currency unit (e.g., Rs. in lakh, $1,00,000, etc.), such that for every borrower,
Where v_{A} and ε_{A} are the exposure and EL, respectively, of the borrower.
Each exposure v_{A} is then rounded off to the nearest whole number.
Thus, the portfolio gets divided into ‘m’ exposure bands, with 1 ≤ j ≤ m.
Step 2 Calculating the expected number of defaults in each band
The following additional definitions are made:
Reference  Symbol 

Portfolio exposure in band j in units of L  vj 
EL in band j in units of L  ε_{j} 
Expected number of defaults (mean) in band j  μ_{j} 
The above implies that
If μ is the total expected number of default events in a portfolio in 1 year (the risk horizon), then μ should be the sum of expected number of default events in each exposure band. That is,
Step 3  To derive the loss distribution, calculate the Probability Generating function^{64} (PGF) for each band, and then for the portfolio. 
Step 3A  Calculate the PGF for each band, j. This is done using equation (2) of the CreditRisk+ technical document (Appendix A, page 34), where the PGF is defined in terms of an auxiliary variable z by^{65} 
Since the probability of n defaults is assumed to be a Poisson distribution, we can substitute this with the expression in item 6, Box 9.6. Hence,
Step 3B  Calculate the PGF for the entire portfolio. Since each band j is assumed to be a portfolio of exposures, independent from the other bands, the PGF of the entire portfolio is just a product of the PGFs of each band. 
Step 3C  Calculate the Loss distribution of the entire portfolio. The CreditRisk+ document (equation 25, page 38) derives the following formula: 
, where A_{n} is the probability of loss of ‘n’ units of the portfolio. It should be noted here that the calculation depends only on knowledge of ε and v.
The above initial model, in which loss distribution has been calculated over a 1 year horizon, can be extended to multiyear periods by building a term structure of default for the portfolio. To achieve this, marginal rates of default must be specified for each future year in the portfolio, for each borrower. It can be seen from the CreditRisk+ document (equations 36, 37, page 40) that the PGF and the loss distribution have the same form as in the single year model described above.
Scenario 2
This scenario assumes variable default rates. The variability in default rates is assumed to result from ‘background factors’, each representing a sector of activity. The sectors are assumed to be independent. To arrive at the loss distribution in such cases, a ‘Mixed Poisson’ process is used.
Simply stated, the ‘Mixed Poisson’ is a two stage process—in the first stage, the external factor controlling the default is drawn from a ‘gamma’ distribution function, and in the second stage, the severity of this external factor is obtained from a conditional distribution that is assumed to be Poisson distributed. The ‘Mixed Poisson’ process is also called a ‘Negative Binomial’ distribution with μ and σ as its mean and standard deviation, respectively.
The ‘gamma distribution’ is a skew distribution that approximates to the Normal distribution when the mean is large. This continuous probability distribution that has two parameters—a scale and a shape parameter—called α and β. It is fully described by its mean μ and standard deviation σ such that μ = αβ and σ^{2} = αβ^{2}. Hence for every sector ‘k ’, the parameters of the related gamma distribution are given by
In this scenario, the portfolio is divided into n sectors with annual mean default rates following a gamma distribution defined by α and β as defined above. The PGF for each sector is given by the equation
This can be summed up to the PGF for n sectors, as in equation 59 (page 46) of the CreditRisk+ document.
The loss distribution is then obtained as described in the CreditRisk+ technical document, Section A10.
Source: Creditrisk+ technical document, 1997
A simple numerical example in Illustration 9.3 explains the calculation of loss distribution under Scenario 1.
ILLUSTRATION 9.3 CALCULATING THE LOSS DISTRIBUTION WITH FIXED DEFAULT RATES
A bank holds a portfolio of loans with 1000 different borrowers.
For simplicity, we will assume that the exposures fall between Rs. 50,000 and Rs. 10 lakh
Scenario 1
Default Losses with Fixed Default Rates
Step 1 Slotting Loans into Independent Bands
We will consider below the first six borrowers
Step 2 Calculating the Expected Number of Defaults in Each Band
Borrowers in each band are aggregated
In CreditRisk+ each band is viewed as an independent portfolio of loans
Consider the hypothetical grouping of borrowers in various bands
The loss distributions are then modelled using steps 3A, 3B and 3C outlined above.
Appendix B of the CreditRisk+ document (1997) provides an illustrative spreadsheetbased example.
SECTION V
SELECT APPROACHES AND MODELS—THE REDUCED FORM APPROACH
We have seen in the foregoing sections that two distinct classes of credit models exist—structural and reduced form.
The structural models discussed in the earlier sections focus on the borrower's/firm's fundamental financial variables—assets and liabilities—and link these to the PD. Since we assume complete knowledge of information related to the financial variables (from the firm's balance sheet), given the firm's capital structure, and having specified the stochastic process that drives the firm's asset value (assuming that the market value of the firm is the fundamental source of uncertainty that drives credit risk), the PD for any time horizon (1 year, 2 years, etc.) can be calculated. This implies that a firm's time of default is predictable. Further, Merton's model predicts default by allowing a gradual diffusion of asset values to the ‘default point’—the level of debt (see Section III—the KMV model). Such a process would imply that the PD becomes almost zero as the debt approaches maturity—an assumption that may not be true in the real world.
In contrast, ‘reducedform’ models assume ‘default’ to be an unexpected, sudden event (hence, unpredictable or inaccessible), and assume no explicit relationship between default and firm value. The models basically use the market prices of the firm's defaultprone instruments, such as bonds or credit default swaps (CDS) to extract default probabilities. Thus they rely solely on market information to determine the firm's credit risk structure, which is much less detailed or complete (compared to the information available from the firm's financial statements). Since there is no observable link between the firm's credit risk and its financial situation, the firm's default time becomes ‘inaccessible’.
The key difference therefore, between structural and reducedform models, is in the ‘information’ available to model credit risk. While ‘defaults’ are specified ‘endogenously’ (the credit quality being determined by the assets and liabilities of the firm) in structural models, defaults are modelled exogenously from market data in the reducedform approach. Another difference is in the treatment of RRs—structural models specify RRs based on the value of assets and liabilities within the firm, while reducedform models look to the market to specify RRs.
Reducedform models are also called ‘intensity’ models since the input for the models is the intensity of the random time of default within the risk horizon being considered. In practice, the ‘intensity’ of default can be inferred through observed prices of bonds. ‘Default time’ is assumed such that default occurs in a time interval (0, T), where T is the risk horizon (as assumed in the structural models in the previous sections). The default time is driven by an intensity process that is stochastic in nature. These intensity processes define default events, or more generally, credit transitions (as in the structural models), but are directly calibrated to market prices. Though the terms ‘reduced form’ and ‘intensity’ models are used interchangeably, they are in fact two approaches based on the type of information required.
Reducedform models go back to Artzner and Delbaen (1995),^{66} Jarrow and Turnbull (1995)^{67} and Duffie and Singleton (1999).^{68} The earliest models dealt with just two credit states—default and nodefault. Subsequently, various researchers extended the approach to the case of multiple, discrete credit states or ratings, leading to ‘ratingsbased’ reducedform models. The model was further extended to allow for stochastic credit ‘spreads’ (the indicator of credit risk), which were assumed to arise when the underlying default probabilities (intensities) and/or the RRs were stochastic. The theory underlying intensity models is mathematically complex, and will not be explained in this section^{69}
However, Box 9.7 sets out the basic mathematical premises on which the reducedform approaches work.
BOX 9.7 BASIC MATHEMATICAL PREMISES OF REDUCEDFORM MODELS
There is no model definition of default in the reducedform approach. If t, the random default time lies in the time interval (0, T), where T is the risk horizon, in the simplest case, the default process This is called a ‘point process’ with one ‘jump’ of size ‘1’ at default. (Contrast this with the Merton structural model where there is a gradual fall—‘diffusion’—in asset values to the ‘default point’—the debt level).
Typically, reducedform models characterize default as a Poisson process stopping at the ‘first jump’. The corresponding ‘default time’—called ‘stopping time’—is usually totally unpredictable (‘inaccessible’) implying nonzero credit ‘spreads’ for short maturity debt. Defaults are assumed to occur randomly with a probability determined by the ‘intensity’ or ‘hazard’ function. From this viewpoint, the intensity model can be considered a time continuous extension of the actuarial framework embodied by CreditRisk+ introduced in the previous section.
The model works by decomposing observed ‘credit spreads’ on ‘risky’ debt (debt prone to default) to ascertain the PD and the LGD. This PD is a cumulative, conditional probability—conditional on there being no default before time t. Hence the observed credit spread can be expressed as follows:
where  CS is the credit spread (calculated as the risky debt yield minus the riskfree rate), 
PD is the probability of default, and  
LGD is the Loss given default equal to 1—RR. 
Intensity models, therefore, are basically empirical, since they use observable prices of risky debt (and therefore credit spreads).
The differences in the various reducedform models arise from the assumptions made for calculating PD and LGD. In one of the earliest forms of the approach, Jarrow and Turnbull (1995) infer that the EL over a time period t would be the product of the conditional PD and the RR under the equivalent martingale (the risk neutral measure).^{70} They assumed that the RR was a known fraction of the value at maturity of the debt. The other well known paper by Duffie and Singleton (1998) assumed that the RR was a known fraction of the value of debt just prior to default, while in their 1999 paper they modelled both PD and LGD as a function of economic state variables. Madan and Unal (1998)^{71} and Unal et al. (2001) model different RRs on senior and junior debt.
The model developed by Jarrow, Lando and Turnbull in 1997^{72} incorporates probabilities of rating transitions into the process. This can be viewed as a special case of the credit migration approach described earlier.
Kamakura Risk Manager Version 7.0 and Kamakura Public Firm Models Version 4.1^{73}
Founded in 1990, with Dr. Robert Jarrow (who pioneered the reducedform approach) as its Managing Director of Research, Kamakura Corporation (headquartered in Honolulu, Hawaii), provides daily default probabilities and default correlations for listed companies, as well as pricing services for collateralized debt obligations (CDO). The service is called KRIS—Kamakura Risk Information Services.
Kamakura's Public Firm Models currently offer four different quantitative approaches to modelling default probabilities: two versions of the Jarrow–Chava Model^{74} (current version 4.1), the Merton Structural Model and the Jarrow—Merton Hybrid Model.^{75}
Which Model is Better—Structured or Reduced Form?
Specific inputs required for structural models
 Firm's asset value process
 Firm's capital structure
 LGD
 Terms and conditions of debt issue
Specific inputs required for reducedform models
 Firm's default process (bankruptcy process)
 LGD specified as a stochastic process
Common inputs required for both models
 Riskfree interest rate process
 Correlation between riskfree interest rates and asset prices
Thus it can be seen that the solutions/inferences from the models depend on the assumptions, restrictions and different techniques for calibration and solution.
It has, therefore, been claimed by Jarrow and Protter (2004)^{76} that these models are not disconnected or disjoint model types as is typically assumed. The two types of models, in fact, are really the same model containing different informational assumptions. If information available to the modeller/bank is partial or incomplete, a structural model with default being a predictable ‘stopping time’ becomes a reducedform model with default time being unpredictable.
To explain the concept further, we know from the Merton's model that default occurs when the value of the firm falls below a threshold level L. Once the firm defaults, it is assumed that it cannot recover. If we assume that the default barrier is constant (though it can be stochastic), then default occurs at time t, the stopping time, which is predictable. However, in reality, the firm value cannot be monitored continuously, nor is the value observable. Hence, information is not ‘complete’. Even assuming we are able to observe asset values at discrete intervals, there is a likelihood of default occurring between two observations, quite unexpectedly. This implies that the predictable default time t has now become unpredictable. Therefore, the structural model, due to lack of partial information, has taken the form of an intensitybased, reducedform model.
Hence, the purpose for which the model is to be used determines which model is more appropriate. Jarrow and Protter sum up thus: ‘If one is using the model for risk management purposes—pricing and hedging—then the reducedform perspective is the correct one to take.
Prices are determined by the market, and the market equilibrates based on the information that it has available to make its decisions. In marktomarket (MTM) or judging market risk, reducedform models are the preferred modelling methodology. Instead, if one represents the management within a firm, judging its own firm's default risk for capital considerations, then a structural model may be preferred. However, this is not the approach one wants to take for pricing a firm's risky debt or related credit derivatives’.^{77}
SECTION VI
PRICING CREDIT DERIVATIVES
The previous chapter introduced some types of credit derivatives. Though there are several types, credit derivatives typically fall into three discernible classes, and structuring more complex derivatives can be achieved through a combination of the basic features.
FIGURE 9.12 MAJOR CLASSES OF CREDIT DERIVATIVES
However, we have seen that the most popular is the CDS. We will therefore understand the pricing of CDS, an ‘unfunded’ credit derivative that works more like an insurance contract.
Pricing Credit Default Swaps—Understanding the Cash Flows
Understanding the valuation of a single entity (plain vanilla) CDS would provide insights into valuing variations of the CDS, such as binary CDS (where the payoff in the event of default is a specific amount), basket CDS (where more than one reference asset is specified, and the payoff happens when the first reference asset defaults), contingent CDS (where the payoff specifies a credit event as well as an additional trigger) or dynamic CDS (where the notional payoff is linked to a MTM value of a swap portfolio).
It is also important to understand CDS since it is a ‘pure’ credit risk derivative—default probabilities can be ‘extracted’ directly from default swap prices (credit spreads), which can then be used in pricing other credit riskbearing instruments. Hence, CDS can be traded securities^{78} as well as risk management tools. In our context, we are interested in valuing CDS as a credit risk management methodology.
Several formulas, using different approaches, have been developed for pricing or valuing CDS. However, the default probabilities that they yield are quite comparable, and the various pricing formulas are actually equivalent.^{79}
Let us recap the basics of a single entity CDS.
A CDS is a contract that provides a lender or bank with ‘insurance’ against the risk of default by a specific borrower, firm or counterparty. The borrower is called the ‘reference entity’. The default is the ‘credit event’.^{80} If the credit event occurs, the bank, called the ‘protection buyer’, has the ‘right’ to sell the loan against whose default risk the insurance has been bought. The specific loan whose default risk is being protected against is called the ‘reference obligation’ or ‘reference asset’, and the total value of the loan for which the insurance contract has been created is called the ‘notional principal’.
The cash flows in a CDS occur in two ‘legs’. The ‘protection seller’ provides insurance against default risk. In return, the protection buyer makes a stream of periodical payments (usually quarterly, though semiannual and annual payments can also happen), until the end of the CDS contract or till a credit event occurs. This is called the ‘premium leg’ (Figure 9.13). The size of these periodic payments are determined from quoted ‘default swap spreads’ to be paid on the notional principal on which default protection is sought.
Default swap spreads, (often simply called ‘spreads’, or sometimes referred to as ‘credit swap premium’,) are quoted in ‘basis points per annum’^{81} (bpa or bps) on the contracts’ notional principal value. Another point to note that these ‘spreads’ are not the same concept as ‘yield spread’ of corporate bonds to government bonds, and not based on any risk free or benchmark interest rate.^{82} Several large banks and other institutions are market makers in the CDS market. For example, when quoting on a 5year CDS on firm F, a bank may ‘bid’ 250 bps and ‘offer’ 260 bps. This is called the ‘bidoffer’ spread and is the typical manner in which quotes are made. This means the bank is willing to ‘buy’ protection on F by paying 250 bps per year (or 2.5 per cent of the notional principal) and ‘sell’ protection at 260 bps per year (2.6 per cent of notional principal). This implies that if Bank A wanted to buy protection against exposure of Rs. 1 crore to firm F, it would have to pay 62.5 bps [(250/4) bps] per quarter on Rs. 1 crore or Rs. 62,500.
If the credit event does not occur till the maturity of the contract, the protection seller does not pay anything. However, if the credit event occurs before the maturity of the contract, the protection seller has to pay as specified in the contract—the swap is then settled either through cash or physical delivery. This is called the ‘protection leg’ and is shown in Figure 9.13.
FIGURE 9.13 CASH FLOWS IN A SINGLE ENTITY CDS
In a CDS settled through cash, the protection seller is required to compensate the loss suffered by the protection buyer by paying the amount of difference between the notional principal and the ‘market price’ of the defaulted bank credit (this approximates to the reduced recovery value). In the case of physical settlement, which is the prevalent market practice, the protection seller buys the defaulted loan from the protection buyer at par. Here the loan purchased by the protection seller is called the ‘deliverable obligation’. The deliverable obligation may be the reference asset itself or one of a broad class of assets meeting certain specifications, usually in respect of seniority and maturity.^{83} After a credit event, the value of the obligation may be different. Therefore, the protection buyer will deliver the ‘cheapest’ of all deliverable obligations. Hence, this is called the ‘delivery option’. Typically, physical settlement takes place within 30 days of the credit event. Box 9.8 elucidates the ‘cheapesttodeliver’ concept.
BOX 9.8 THE CHEAPEST–TO–DELIVER CONCEPT
The cheapest–to–deliver bond
The RR of a bond is defined as the value of a bond immediately after default as a percentage of face value. If the CDS is used to hedge the investment in a default prone bond, the payoff from the CDS in the event of default would be (1 – R) × notional principal.
Typically, a CDS specifies that a number of different bonds can be delivered in case of default. These bonds may have the same seniority, but may not be selling at the same proportion of face value immediately after default. This happens because the claim made on default is usually equal to the bond's face value plus the accrued interest up to default. If a bond has higher accrued interest at the time of default, it tends to have a higher price just after default. Another reason is because the market might feel that there would be differential treatment of bondholders in the event of the firm issuing the bonds being reorganized or restructured. Thus, in the event of default, the protection buyer would tend to choose for delivery the bond that can be purchased most cheaply in the market.
The cheapest–to–deliver loan
How does the above concept apply to bank loans? Bank A has made a senior, secured loan of Rs. 100 to borrower firm X and has bought protection through a long CDS position from Bank B. If the credit quality of the loan deteriorates and a credit event is triggered, then the buyer, under physical settlement, delivers the loan now valued at Rs. 70, down from Rs. 100 par, to the CDS seller. The seller pays Rs. 100 to the CDS buyer. Thus, the CDS buyer has made no loss (or profit) on the transaction. On the other hand, the buyer could instead sell the loan at Rs. 70 (making a loss of Rs. 30), and buy a junior ranking bond at, say, Rs. 50, to deliver to the seller (provided the deliverable obligation permits it). The CDS buyer would still receive Rs. 100 in settlement, and also make a gain of Rs. 20 (Rs. 50 gain on CDS less Rs. 30 loss on loan). Such a practice is useful especially where restructuring is deemed a credit event, since Bank A may not want to sell the asset under restructuring.
In case of cash settlement the protection seller pays the protection buyer an amount equal to the notional principal × (1 – R). Cash settlement is less common since it is difficult to obtain market quotes for the distressed asset. In practice, cash settlement occurs within 5 days after the credit event, and the price will be determined through the Calculation Agent by reference to a poll of price quotations obtained from dealers for the reference obligation on the valuation date.^{84} The assumption here is that defaulted loans will trade at a price reflecting the market's estimate of recovery value, irrespective of maturity or interest. In some cases, the cash settlement would be a sum predetermined by the counterparties, which is termed ‘binary settlement’.
Though they seem different, the net value of asset transfer under physical settlement and the net transfer under cash settlement is the same. Under physical settlement, the protection buyer delivers the market value of the deliverable obligation (R × notional principal) and receives in return 100 per cent of the notional principal in cash. The total effect is that of being compensated for the notional principal. Under cash settlement, the asset is with the protection seller and can be realized at R × notional principal. The protection seller merely compensates the loss by paying (1 – R) × notional principal.
Illustration 9.4 clarifies through a numerical example.
Pricing Credit Default Swaps—Grasping the Basics
According to Duffie (1999),^{85} pricing a CDS involves two issues.
The first issue is estimating the credit swap spread. We have seen in Figure 9.13 that, at the beginning of the contract, the standard credit swap involves no exchange of cash flows and, therefore, has a market value of zero. For convenience, we ignore dealer margins and transaction costs here. The market premium rate, therefore, is that for which the market value of the credit swap is zero.
ILLUSTRATION 9.4 UNDERSTANDING CDS CASH FLOWS
Basic inputs
Protection buyer, Bank A, wants protection against default risk of firm XYZ.
The notional principal (face value) on which protection sought for is Rs. 10 crore. (Bank A may have other exposures to firm XYZ but it wants protection only for Rs. 10 crore that it considers most vulnerable).
The protection is required for 2 years (T in Figure 9.13). (Again, the credit limit to firm XYZ may have a maturity period of 5 years, but Bank A needs protection only for the first 2 years).
The default spread is 300bps over the riskfree rate (prime rate).
Bank A has to make quarterly payments to Bank B, the protection seller.
Cash flows
The quarterly payment would amount to 10 crore × 0.03 × 0.25 = Rs. 7,50,000.
Assume that at point τ, (τ < T), the loan to firm XYZ suffers a credit event. At this point, the ‘cheapest to deliver’ (CTD) asset of the firm XYZ has a recovery price of Rs. 60 per Rs. 100 of face value.
The cash flows are as follows:
 Bank A, the protection buyer, pays Bank B Rs. 7.5 lakh per quarter up to the time of the credit event, τ. If the credit event happens, say 1 month after the completion of a quarter, then a proportionate amount (Rs. 7.5 lakh/3 months) of Rs. 2.5 lakh would be paid.
 Bank B, the protection seller, compensates Bank A for the loss on the face value of the asset, by paying Rs. 10 crore × (100 per cent – 60 per cent) = Rs. 4 crore.
The second issue is that, with passage of time, interest rates or credit quality of the reference asset may change, and would impact the market value of the swap. Hence, for a given CDS, with defined periodic payments determined by the credit spread, the current market value would have to be ascertained. This market value would not be zero.
It should be noted that the default probabilities used to value a CDS are ‘risk neutral’ probabilities (and not ‘real world probabilities’).^{86}
The simplest situation is where there is no counterparty default risk.^{87} The other simplifying assumption is that default probabilities, interest rates and RRs are independent. Box 9.9 outlines the basic equations to determine the single entity, CDS credit spread.
BOX 9.9 THE BASIC EQUATIONS TO VALUE A SINGLE ENTITY CDS
Assume Bank A has made a new loan L to firm X, an existing borrower. Bank A wants to transfer the risk of the new loan N (the Notional Principal) to Bank B through a CDS. Bank A is the protection buyer and Bank B, the protection seller. The protection against credit risk is sought for T years (the swap maturity, which need not be the same as the loan maturity). Bank A accepts to pay a fixed fee f for the protection provided by B. For understanding the basics, we will ignore, for the time being, accrued fee amount, if any, at the time of the credit event, and also assume that Bank B has zero PD on its contract with A.
At the time of the credit event (loosely, default), assume the value of the contract to be RN, where R is the already known RR. This means that Bank A would lose (1 – R) × N if it did not buy protection in case of default on the new loan by X. In other words, (1 – R) × N is the payoff of the CDS contract in the event of default. Assuming R as known, the fee f to be paid periodically by Bank A would depend on the market view of the PD by firm X.
Under the contract, Bank A pays f at predetermined periods (e.g. every quarter) until the contract ends at T or until a credit event happens before T. Therefore, the PV of the cash payments by Bank A is the discounted value of each fee payment multiplied by the probability that X does not default up to the date of payment.
Assume that CDS premiums (fee payments) are paid on a set of predetermined dates represented by Ґ = t_{1}, t_{2},…t_{n}. In other words, in a CDS of T maturity starting time 0 (t_{0}) and having quarterly payments, the payment dates would be t_{1} = t_{0} + 3 (months), t_{2} = t_{0} + 3 × 2 months, and so on, and t_{n} = T. If p is the PD of X, then p(t_{1}) would be the probability (as seen at time t_{0}) that X will default at time t_{1}. That is, 1 – p(t_{1}) [call it q(t_{1})] is the probability of no default by X at time t_{1} If d(t_{1}) is the discount factor pertaining to the prevailing riskfree rate, the expected PV of fee payments by Bank A can be obtained as follows:
The PV of cash flow for Bank B, the protection seller, will be the discounted value of the recovered fraction of N multiplied by the PD at each fee payment date. This implies that for a default occurring at time t_{(i + 1)}, it is necessary that the firm X has not defaulted up to time t_{i}, and that it defaults exactly at t_{(i + 1)}. Hence, the weight to be used for calculating expected payment at time t_{(i + 1)} is the product of the probability of no default up to t_{i} and the conditional PD between t_{i} and t_{(i + 1)} The PV of expected cash flow for Bank B can be represented by the following equation:
However, since default does not occur on a predetermined date, and can occur at any time between t_{i} and t_{(i + 1)}, we can rewrite the above equation as follows:
Here, the discount factor is the average of the discount factors of the periods within which default may occur, and P_{ti}, _{ti + 1} is the conditional PD between times i and i + 1, given that default did not occur up to time i.
This implies that the ‘value’ of the CDS to the protection buyer is the difference between the PV of the expected payoff and the PV of payments made by the buyer.
Since we operate under the ‘no arbitrage’^{88} assumption, the two PVs should be equal at the beginning of the contract. Setting PV_{f}, equal to PV_{recovery}, we can find the CDS rate or fee:
Here, f indicates the CDS ‘spread’. This is an annualized spread.
In practice, CDS spreads are available in the market for different maturities based on the reference asset.
We can use the market CDS spreads to arrive at a term structure of default probabilities using the equation above. Assuming we have 1 year, 2 year,…, n year CDS spreads available from the market, we can value the PV of a CDS contract at any point in time using the above equation. For example, if we know that the midmarket CDS spread for a newly issued 5 year CDS is 100 bps, we can work backwards using the above equation and estimate the implied default probability.
One parameter required for valuing a CDS, but cannot be directly observed from the market, is the expected RR. So far, we have assumed the same RR for estimating PD densities, and for calculating the payoff, which means that the estimated CDS spread will not be very sensitive to the RR. This is because the implied probabilities of default are proportional to (1/1–R) and the payoffs from a CDS are proportional to (1–R).
However, the above will not be true in the case of a binary CDS. The payoff in a binary CDS is a fixed amount, regardless of the RR. Since implied probabilities are still proportional to (1/1–R), but the payoff is independent of R, if we have the CDS spread of both a plain (vanilla) CDS and a binary CDS, we can estimate both the RR and the default probability.
The theory implicitly assumes that the reference asset is a floating rate asset, trading at par. However, where such floating rate assets are not available, the market often tries to estimate the CDS spread from a fixed rate asset, trading away from par, using an ‘asset swap’. Asset swaps break down the asset's yield into a yield (say LIBOR) curve plus a spread. This spread is usually taken as an estimate for the CDS spread.^{89}
Another complication is when the protection seller is prone to default risk.^{90} In such cases, the CDS spreads reflect the default risk of the seller, or the protection buyers try to get additional/backup protection from the market.
There are other methodologies used to price CDS spreads. The reducedform models (described in section V) are used extensively for this purpose. The Jarrow–Turnbull (1995)^{91} model assumes recovery to be a fixed amount in the event of default, and is, therefore, called a ‘fixed recovery model’. In a ‘market standard’ model based on the Jarrow–Turnbull model, O'Kane and Turnbull (2003)^{92} propose that the PD occurring within the time interval (t, t + dt), conditional on ‘no default’ up to time t, is proportional to a timedependent function λ (called the ‘hazard rate’), and the length of the time interval dt. Hence, the model uses a simple ‘binomial tree’^{93} where the probability of no default is 1 – λdt, and if there is default the recovery value R is received with probability λdt. This model can be extended to multiple time periods and continuous time probabilities (as compared with discrete time probabilities used in Box 9.9) of default and no default can be arrived at.
The Duffie–Singleton (1999)^{94} model assumes the amount of recovery is restricted and is, therefore, referred to as the ‘fractional recovery model’.
The CDS market is largely overthecounter (OTC). This means that the protection buyer and seller negotiate each CDS trade, at the time it is required—hence, each CDS will have a separate reference asset, notional amount, maturity, credit event definition, CDS spread, etc. Once the terms are agreed to by the counterparties, the trade is finalised through proper documentation. Default swaps are looked upon as a good source of expected return for protection sellers.
Credit derivatives are used by different groups of investors for a variety of reasons. Banks are understandably major protection buyers and sellers, in order to mitigate the credit risk in their large loan portfolios. Insurance firms are major protection sellers, who also provide protection against counterparty risk for protection buyers.
We will illustrate estimation of CDS spread with a simple numerical example (Illustration 9.5).
ILLUSTRATION 9.5 CALCULATION OF CDS SPREAD—A SIMPLE EXAMPLE
Assumptions:
The notional principal is 1 (monetary value in any currency, e.g. Rs. 1 or $1, etc.)
The PD p of a reference entity is assumed to be 2 per cent at the end of year 1, provided there is no earlier default. This is the conditional PD.^{95}
The above implies that the probability of no default q at the end of year 1 is 98 per cent.
The protection buyer seeks protection for 5 years
Step 1  Construct Table 9.11 of unconditional probabilities of default over 5 years (i.e., PD viewed at time ‘0’) 
TABLE 9.11 UNCONDITIONAL PROBABILITIES OF DEFAULT OVER FIVE YEARS
Time  p  q 

1 
0.0200 
0.9800 
2 
0.0196 
0.9604 
3 
0.0192 
0.9412 
4 
0.0188 
0.9224 
5 
0.0184 
0.9039 
How did we obtain the above probabilities? For year 1, we know that p = 2 per cent and q = 98 per cent. For year 2, p = q of year 1 × p = 0.9800 × 0.02 = 0.0196 and q (year 2) = probability of no default in both years = 0.9800 × 0.9800 = 0.9604. For year 3, p = probability of no default for 2 years (q of year 2) × p = 0.9604 × 0.02 = 0.0192 and q = probability of no default for 2 years × q = 0.9604 × 0.9800 = 0.9412 and so on.
Step 2  Calculate the cash flows of the protection buyer, assuming the riskfree rate^{96} to be 6 per cent per annum. Since payments will be made by the protection buyer only in the case of no default, we take q over 5 years to estimate the expected fee payment. Let the fee to be paid be f. 
The total expected cash flow of the protection seller over the 5 year CDS would be 3.9756f
Step 3  Cash flow of protection seller in case of credit event 
We know that p is the probability of credit event (default) over the life of the CDS. If we assume 50 per cent RR, the protection seller will have to settle 100 – R or 100 – 50 in case of a credit event (see Figure 9.13). We also assume that the credit event is likely to occur only at the end of the year.
The expected payoff is calculated as p × (1 – RR) for each year. The PV of the expected payoff for the protection seller is seen to be 0.0406.
Step 4  Equate the two PVs in steps 2 and 3 and solve for f. 
3.9756f = 0.0406
Hence, f = 0.0102 or 102 bps per year credit spread.
Note that the simple example is given to illustrate the calculation methodology. In practice, there are more complications, such as fee payments by the protection seller may be made quarterly or half yearly, or the credit event may occur at any time during the year. We have also assumed the protection seller will not default in settling the CDS.
Let us now turn to the second issue raised by Duffie (1999) outlined in an earlier paragraph.
At the time it is priced, the CDS, like most other swaps,^{97} is worth almost nothing. Assume that the protection buyer in the Illustration 9.5 wants to value its position after 1 year. On the date of such revaluation, CDS spreads quoted in the market are 125 bps for a 4year CDS (the 4year spread is relevant since the protection was sought for 5 years in the example, and 1 year has since elapsed). What is the current market value of the position?
The MTM value = current market value of remaining 4year protection less the expected PV of 4year premium leg at the earlier calculated spread of 102 bps. It is to be noted here that the CDS contract has increased in value since the protection buyer is paying only 102 bps for something the market is now willing to pay 125 bps. Since the MTM of a new CDS is zero, the current market value of remaining 4year protection becomes equal to the expected PV of the premium leg at 125 bps. It therefore follows that the MTM value to the protection buyer is the difference between the expected PV of 4year premium leg at 125 bps and the expected PV of 4year premium leg at 102 bps.
In Illustration 9.5, the expected PV of the protection buyer's fee payments would change to 3.9756 × 0.0125 = 0.0497. The PV of the payoff (assuming RR is the same) would remain at 0.0406. The market value of the swap would now be 0.0497 – 0.0406 = 0.0091 times the notional principal. In other words, the MTM value of the CDS to the protection buyer would now be – 0.0091 times the notional principal.
In case of a binary default swap, the numbers in Illustration 9.5 would change as follows:
In 2009, ISDA introduced the ISDA CDS standard model, the source code for CDS calculations, as a free download from its Web site, which can be accessed at www.cdsmodel.com
Pricing Collateralized Debt Obligations—The Basics
The CDS discussed above transfers the credit risk of a single entity from the protection buyer to the protection seller for a contracted period. Where protection buyers seek to securitize large portfolios of default risk prone loans, or even CDSs, the collateralized debt obligations (CDO) is used.
How is a loan portfolio different from a single loan? We have seen above that the individual firms (in a loan portfolio) could default, leading to losses in the portfolio value. This risk can be assessed by estimating the PD for each firm and the LGD (1 – RR). Additionally, the degree of dependence in the portfolio between firms’ default probability, known as ‘default correlation’, has an impact on the timing of firms’ default, and, therefore, on the portfolio loss.
Understanding the CDO We have seen in the previous chapter that a CDO, like securitization, is a way of creating securities with differing risk characteristics from a portfolio of debt instruments.
However in a CDO, the portfolio comprises of heterogeneous instruments, such as senior secured bank loans, CDSs or high yield bonds (in securitization, it is typically a homogeneous pool of, say, home loans, auto loans, etc.). The basic structure of a CDO is as given in Figure 9.14.
Bank A, owner of a loan portfolio, wants to protect itself against the possible losses in the portfolio, but it does not want to ‘sell’ the portfolio. One way is to buy CDS for each borrower/firm in the loan portfolio, but that would be quite expensive. Further, individual CDS contracts do not protect against correlated defaults.
Bank A decides to buy protection through a CDO. In a typical CDO, the portfolio's risk is ‘sliced’ into ‘tranches’ (‘tranche’ in French, means ‘slice’ and, in simple terms, every tranche is akin to tradable commercial paper) of increasing seniority and default risk of the underlying pool of loans (or CDSs).
Thus, each tranche in a CDO shows the priority of receiving payments from the underlying pool of assets, and also bearing the default risk. Each tranche is sold separately to investors according to their risk preferences, for a fee.
Cash flows are the critical link between the asset and liability sides of the CDO transaction. The asset side of the CDO is the underlying portfolio of reference assets—loans (or CDSs). The liabilities side comprises the securities issued by an issuer, which is typically a special purpose vehicle (SPV) that is a separate company or trust specially created for the transaction by the owner of the pool of assets. The SPV is created to insulate potential investors from the risk of failure of the CDO originator itself, say a bank.
Bank A, the originator of the CDO, sells the selected portfolio of assets whose credit risk is to be transferred, to the SPV, which in turn issues structured notes in the market backed by the portfolio of assets.
Let us now understand Figure 9.14. The first tranche (called the ‘equity tranche’) holds 5 per cent of the total notional loan principal, and absorbs all credit losses from the portfolio (during the life of the CDO) until the losses aggregate to 5 per cent of the notional principal. Hence the equity tranche is also called the ‘first loss tranche’, if the losses exceed 5 percent; the second tranche, which has 10 per cent of the loan principal, absorbs losses up to a maximum of 15 per cent (cumulatively). The third tranche holds 15 per cent of the principal and absorbs losses over 15 per cent up to a maximum of 30 per cent of the principal. The fourth tranche holds the bulk of the loan principal, 70 per cent, and absorbs all losses exceeding 30 per cent of the principal.
FIGURE 9.14 THE BASIC CDO STRUCTURE
Now look at the yields (rate of interest paid) to the respective tranche holders. These rates are paid on the notional principal remaining in each tranche after the losses have been paid. Assume that tranche 1 suffers a loss of 1 per cent. The tranche holders, who were paid 40 per cent in case of no default, would now lose 20 per cent (1/5) of their investment, and would earn only on the remaining 80 per cent of their investment. Since they assume the first loss, taking the maximum risk, they are called the ‘equity tranche holders’. If the default loss were to rise to 3 per cent, this tranche would lose up to 60 per cent of the notional principal. Compare this with tranche 4. Defaults on the portfolio must exceed 30 per cent if this tranche is to suffer losses. Hence, this tranche is usually given the top rating (say AAA) by rating agencies. Since this is relatively safe investment, the tranche earns less for its holders. It is usual for Bank A, the originator of the CDO, to retain the equity tranche. The remaining tranches are sold in the market.
The transaction in the above example is termed ‘cash CDO’.
Typically, four broad types of CDOs are recognized—cash flow CDOs, market value CDOs, managed CDOs and static CDOs. The attributes of a specific CDO, such as MTM, rules related to trading of the underlying securities, etc., are determined by the type of CDO. For example, in a cash flow CDO, payments to investors are in the form of interest earnings and principal repayments from the underlying assets. Erosion in the underlying asset values would not usually have an impact on the cash flows to the investors in the CDO, unless the erosion is accompanied by a credit event. In this case, the interest/principal payments cease to flow from the asset originators to the SPV, leading to lower cash flow for the CDO investors. In the case of the market value CDO, changes in the MTM value are passed on to the CDO investors through the SPV.^{98} In a managed CDO, the CDO manager actively trades the securities in the pool, with performance being monitored by the investors. In contrast, the static CDO's portfolio is determined upfront and does not change over time.
If the portfolio of loans in the example is replaced by a portfolio of CDSs, the CDO is termed ‘synthetic CDO’.
Cash inflows to the synthetic CDO are the underlying CDS premiums, and cash outflows are losses consequent to credit events related to the underlying assets. Each tranche in a synthetic CDO has a specified upper and lower ‘detachment’ (U) and ‘attachment’ (L) point. Defaults affect the tranche according to the seniority of the tranche in the capital structure. The buyer of the tranche with attachment point L and detachment point U will bear all losses in the portfolio value in excess of L and up to U per cent of initial value of the portfolio. Table 9.12 presents the typical tranching done in the iTraxx index (Please see Box 9.10 for an introduction to credit indices).
TABLE 9.12 A SAMPLE CDO TRANCHED STRUCTURE, ITRAXX^{99}
Source: www.markit.com
Assume that the above CDO experiences a 12 per cent loss of its initial value. This implies that the equity tranche holders will bear up to 10 per cent of the loss while the junior mezzanine holders would bear the remaining 2 per cent. The holders of more senior tranches (senior mezzanine, junior senior, super senior) will not suffer any loss. The super senior suffers only if the total collateral portfolio loss exceeds 35 per cent of its notional value.
Hence, it can be seen that CDO tranching enables holders of the respective tranches to limit their loss exposure to (U – L) per cent of the initial portfolio value.
For example,^{100} assume a hypothetical CDS index comprises of 100 names with equal weight (1/100) for each name. The tranches are indicated in the market as follows: 0 – 5, 5 – 10, 10 – 15, 15 – 22, 22 – 100. In other words, the 10–15 tranche has an attachment point L of 10 and a detachment point U of 15 and so on. We assume a specific investor has bought protection on the 0–5 per cent tranche with a notional principal of Rs. 10 crore. If one name out of the 100 defaults, and recovery is set at 65 per cent, the LGD is 35 per cent. In such a case the payout from the protection seller would be calculated using the formula—(notional principal × LGD × weighting)/tranche size, i.e., a payout of Rs. 70,000. Once the payout happens, the 0–5 tranche is recalculated to reflect the reduced notional principal. Hence 4.65 per cent (5 – 0.35 LGD) of the notional principal remains, which implies that the new detachment point should reflect that only 99 names now remain in the CDO. The new detachment point has to be now adjusted for the remaining names in the Index. It is to be noted that the original principal of the other tranches are not impacted due to this change, but have less protection against further losses.
The same concept can be used for determining the number of credit events at which the equity tranche stops covering losses, and the next tranche starts losing. The loss level in any given tranche depends on two random variables—the number of credit events in the entire portfolio and the related LGD. Assuming the LGD of 35 per cent in the previous paragraph applies to all defaulted firms in the portfolio (in reality LGD for each asset may be uncertain), the scenario under which the equity holders lose their entire notional principal (i.e., when the equity tranche gets fully written down to zero) will be given by the equation n number of defaults × Rs. 10 lakh per name notional × 35 per cent loss rate = 5 per cent equity tranche width × Rs. 10 crore million total portfolio notional.
Solving for n yields 14—the number of defaults for the equity tranche to be written down to zero. This means that the portfolio would have lost Rs. 50 lakh, at which point the equity tranche would be fully written down to zero. The next tranche, say, the junior mezzanine, would start losing from the 15th default onwards, and so on.
Pricing the CDO The concept of ‘default correlation’ is important to determine the pricing of a tranched CDO, such as tranched iTraxx.
For any two given firms, their default correlation approximates to a measure of the correlation between the times at which they default. In other words, two reference assets or firms which are highly correlated (correlation close to 1) are likely to behave similarly—either defaulting or not defaulting, together. This implies that a portfolio with several reference assets, who bear a high default correlation with one another, will have higher probabilities of both higher and lower loss rates. Consequently, the emanating loss distribution is likely to have fatter tails.
As default correlation increases in a portfolio, it is easy to understand why the cash flows from the underlying asset pool of loans or CDS would exhibit higher volatility. As volatility increases, the market value of the equity tranche (the highest risk bearing tranche) increases (and the spread decreases). On the other hand, since increased volatility (risk) is not favoured by the senior most tranche, the market value declines with an increase in default correlation, leading to higher spreads. The mezzanine and other tranches in between these two can have exposures to correlation in either direction, depending on the degree of volatility.
The basic methodology is to average the discounted cash flows to a particular tranche over several independently simulated scenarios. This implies that cash flows for each underlying single name CDS in the CDO have to be simulated, with correlation in the default times of the various underlying firms as well. The widely used industry methodology for the correlated default time simulation is the ‘Gaussian copula’ (in Statistics, ‘copula’ is used to couple the behaviour of two or more variables—however, a discussion on this is outside the scope of this book).^{101}
Box 9.10 introduces credit indices developed in credit derivative markets.
BOX 9.10 CREDIT INDICES
It is obvious from the discussion this far that CDOs were created and customized to meet specific needs of investors and credit risk transferring institutions. However, customization also means higher cost—creating exclusive SPVs, legal documentation and approvals, managers to design and oversee trade in the CDO and so on. The primary advantages of indices over customized CDOs are (1) liquidity, since issue sizes are large with varying credit spreads, (2) transparency (easier to obtain price quotes and information on pricing and spreads—rules, constituents, fixed coupon and daily prices are publicly available), (3) cost (lower transaction costs due to possibility of trading in portions of the market and standardization), (4) tradability, (5) operational efficiency (standardized terms, legal documentation and electronic processing) and (6) industry support (most major dealer banks and other players use and support the indices).
The standardization also contributed to better hedging patterns. For example, before the introduction of indices, issuers would hedge unbalanced positions of customized CDOs through even more complex, multitranche structures. This made the transaction even less transparent.
The evolution
2001 saw the launch of synthetic credit indices by J P Morgan (JECI and Hydi), and Morgan Stanley (Synthetic TRACERS). Subsequently, in 2003, these indices were merged under the name Tracx. At around the same time, credit derivative indices were introduced by iBoxx. Further consolidation took place in 2004 when Tracx and iBoxx combined to form the CDX in North America, and the iTraxx in Europe and Asia. Since November 2007, Markit (www.markit.com), which had earlier been the administrator for CDX and the calculation agent for iTraxx, owns both families of credit indices. Markit owns the iTraxx, CDX, LevX and LCDX indices for derivatives and iBoxx indices for cash bonds.
The role of Delphi's bankruptcy in the evolution of new protocols
As CDS markets evolved, CDS indices were fast becoming the centre of trading activity. Simultaneously, the notional value of contracts written on insurers who found a place in these indices also increazed rapidly. At times the notional value of CDS contracts began to exceed the bonds outstanding. This trend was more pronounced when CDS contracts were written on issuers who found a place in the major CDS indices. The result was a scramble by protection buyers to find CTD bonds when credit events happened.
When Delphi, a major supplier of auto parts to General Motors, filed for bankruptcy (Chapter 11 in the United States) in October 2005, it was reported to have $28 billion notional value of CDS contracts against $5.2 billion in outstanding bonds. The rush to find bonds to deliver to the CDS contracts led to a rise in the bond prices after the bankruptcy filing. This was followed by a steep fall in the bond prices.
The Delphi episode was one of the primary triggers for the ISDA, the trade organization for derivatives, to step in and set up protocols for settling derivative trades, as also determining, through auctions, the value of the obligations of the defaulted reference asset. The option to settle CDS contracts in cash (as opposed to physical settlement that led to complications as described) was popularized around this time.
The auctions happen in two stages. In the first stage, about 15 dealers submit bid and offer prices for the bonds of the reference asset, with a bidoffer spread of maximum 2 points. In the second stage, these bids and offers are bifurcated. Where the bids are higher than offers by other bidders, they are excluded from the calculation as ‘tradeable’. The objective of this process is to prevent manipulation or misuse of the auction mechanism. The remaining bids are again bifurcated, and the best half of the two is selected. The average of the bids and offers in this best half is the auction settlement price or the ‘inside market midpoint’. This process establishes the price of the instruments.
Market participants
Apart from Markit, which owns and operates the indices, the other participants include all major banks dealing in derivatives, institutional investors, ISDA (responsible for globally approved legal documentation) and other parties who use and trade in the indices.
Trading practices
Markit categorizes its tradable credit indices into three broad categories
Structured finance
Synthetic fixed income
Cash fixed income
The indices under ‘structured finance’ are the ABS, CMBX and the TABX, traded exclusively in the United States. ‘Synthetic fixed income’ indices form the largest category with the following subcategories
FIGURE 9.15 CATEGORIES OF SYNTHETIC FIXED INCOME INDICES
Each index has its unique features and notations. Indices roll every 6 months, and a new series is created with updated names. However, the previous series will continue trading.
The CDX indices, e.g., are broken out between investment grade (IG), high yield (HY), high volatility (HVOC), crossover (XO) and emerging market (EM). The CDX.NA.HY is the notation to indicate an index based on a basket of North American (NA) singlename HY CDSs. The XO index includes names that are splitrated, meaning they are rated IG by one agency, and ‘below IG by another. The CDX index rolls over every 6 months. There are 125 names in the IG index, with maturities of 1, 2, 3, 5, 7 and 10 years, and 100 names in the HY index, with a single maturity of 5 years. These names enter and leave the index as appropriate. For example, if one of the names is upgraded, it will move, say, from the HY index to the IG index when the rebalance occurs. The tranches for each index are standardized. For example, for the CDX, NA.IG, the tranches are 0–3, 3–7, 7–10, 10–15, 15–30 and 30–100. The significance of these tranches is as explained earlier using the hypothetical iTraxx example.
The LCDX basket is made up of 100 singlename, senior secured loans in the United States. Prices are quoted for maturities of 3 and 5 years, rolled over every 6 months.
The iTraxx Europe has 125 names, while the iTraxx Asia (Japan) has 50 names. Almost all the indices are rolled over every 6 months, but the maturities for which prices are quoted differ from index to index.
Payments from the protection buyer to the protection seller are made quarterly—the 20th of March, June, September and December—except in the case of CDX.EM, where payments are semiannual. The payments accrue on an Actual/360 basis.
The ‘cashfixed income’ category comprises of bonds. The index is iBoxx and is traded in Europe, Asia, United States and the emerging markets (EM).
More on trading rules can be found on Markit's Web site, www.markit.com. (May 2009)
Credit event fixings
The Credit Event Fixings have been developed by Creditex and Markit along with ISDA and major dealers in credit derivatives. The objective of credit fixings is to ensure a fair, efficient and transparent process for settlement of credit derivative trades following a corporate default. They are an integral part of ISDA's CDS Index protocols. Creditex and Markit have jointly acted as administrators of the Credit Event Fixings since their inception in June 2005.
Tradable credit fixings are determined based on a welldefined methodology for iTraxx Europe—iTraxx 5 year Europe, HiVol and XO indices. A North American version is shortly expected to be in the market. Credit fixings take place weekly on Fridays. More details can be accessed at www.creditfixings.com.
During a Credit Event Fixing, dealers place executable orders on the Creditex platform for the reference asset of a particular company in respect of which a credit event, such as filing for bankruptcy, has occurred. A market standard methodology is used to simultaneously execute these orders and generate a final cash settlement price for eligible credit derivative contracts in respect of this reference asset. Markit verifies the integrity of the process and calculates the final price which it publishes on the Credit Fixings Web site.
Source: www.markit.com, www.isda.org, www.creditfixings.com
SECTION VII
CREDIT RISK MEASUREMENT AFTER THE FINANCIAL CRISIS
The Financial Crisis—An Overview and Analysis^{102}
When some of the world's largest banks folded up quietly overnight in 2007–2008, afflicted by the ‘subprime crisis’, shaking up powerful economies, leaving stock markets and investors in a state of panic, taking centre stage in the crisis were the ‘villains’—structured products and the credit derivative.
The subprime crisis that began in mid 2007 is widely regarded as the first financial crisis in the age of mass securitization. It has unleashed a flurry of daunting questions on the very foundations of securitized finance and structured products. Just about everything related to the credit market is being questioned—is it right to originate and distribute? Is Basel II (to be discussed in a later chapter) adequate to control credit risks and ensure adequacy of bank capital? With its reliance on internal models and external credit rating agencies, can the credit risk be wished away by merely dispensing with complex derivatives? Is ‘financial engineering’ a dirty word?—and the debate goes on.
Funds and banks around the world purchased bonds or the risk related to bonds, backed by home loans, often bundled into CDOs. These CDOs were backed by pools of mortgages or other incomeproducing assets and considered essentially ‘bond like’ in that they offered investors a steady stream of returns. Investors found these securities attractive as they offered higher returns at a time when traditional fixed income or debtrelated products were yielding lower returns. Investors also viewed these structured products to be quite ‘risk free’, when they were certified to be so by the top rating agencies of the world. The ability of ‘structured finance’ to repackage risks and create ‘safe’, tradable assets from risky, illiquid collateral was the reason for the meteoric increase in the issue of structured securities.
However, as low interest rates in many parts of the world fuelled a lending boom to less than creditworthy borrowers, banks looked for new avenues to package and sell these loans, so that they could get liquidity to lend more. By selling off risky loans, banks could maintain less regulatory capital commensurate with the risk. Hence banks and other financial institutions pooled these assetbacked securities into new pools, dividing them up and issuing securities against them, thus creating CDOs. The concept caught on pretty fast, with new combinations that were further and further removed from the original underlying asset. Such innovations included CDOs of CDOs or CDOsquared and even CDOcubed.
According to J. P. Morgan, there are about $1.5 trillion in CDO, referring to those made up of bonds backed by subprime mortgages, slightly safer mortgages and commercial mortgagebacked securities.
Then what went wrong?
At the core of the crisis has been the housing boom in the United States since the beginning of the decade. Low interest rates meant home buyers could take larger loans, giving rise to a housing bubble marked by unrealistic optimism and a failure to consider the downside, since it was assumed that all foreseeable downside was taken care of while designing the securitized products. The cracks began showing in 2006, when subprime borrowers increasingly defaulted on monthly payments due to annual interest rate resets on their floating rate loans. Fears of recession caused bloated housing prices to dip alarmingly, and in 2007, prices of securities based on subprime loans were in free fall as investors feared that they would not get the promised payments from the structured securities. Lenders became wary and showed reluctance to lend.
With the underlying assets—the subprime mortgages—fast losing value, dangerous levels of leverage were revealed in the packaged securities, leading to the discovery that the ‘safe’ CDOs and other securities were actually far riskier than originally envisaged. As the complexity of these products increased, so did their opaqueness. The extent of losses that they have generated has surprised not only investors, but also the funds and bankers themselves. This was when weaknesses in the system were laid bare, including ratings that did not accurately reflect risk and faulty assumptions on how diversified pools would act on multiple layers of leverage.
The innovation in structured products was assisted by the rise of credit derivatives—especially CDSs. CDSs became staggeringly popular over the last few years throughout the developed world. CDSs resemble an insurance policy to the extent that they can be used by debt owners to hedge, or insure against a default on a debt. However, because there is no requirement to actually hold any asset or suffer a loss, CDSs could also be used for speculative purposes.
Thus, credit derivatives allowed banks to hedge their exposure to the subprime loans they had made, and also freed up capital—since they did not have to reserve capital for potential losses. The banks partnered with hedge funds—lightly regulated pools of capital with high fees—looking for better returns. Insurance companies and pension funds also sought the higher yields as interest rates hit historically low rates.
Annexure I presents a brief chronology of events leading to the crisis that hit the United States and Europe. Are structured products and credit derivatives really the villains of the high drama in the financial world? Who is responsible for the mammoth fiasco—Basel II, the rating agencies, the regulators or the banks themselves?
One possible inference is that the benefits of financial structuring and risk management are small compared to the huge costs they impose on financial stability and the reputation of banks. An alternative, therefore, is to return to ‘good, oldfashioned banking’ where banks create loans and hold them on their balance sheets and do not synthetically pool, package and distribute them.
Another argument advanced against credit risk transfer mechanisms is that they create the problem of ‘moral hazard’. The originators of loans do not assess the credit quality with the same rigor that they would exercise were they to hold them on their balance sheets. This allegation seems to be true in the light of the findings of the 2008 and earlier surveys of credit underwriting practices in the United States.^{103} The findings show that credit appraisal and delivery standards had in fact declined over the previous periods and banks have begun exercising more prudence in granting credit post the credit crisis. Table 9.13 summarizes the changes.
TABLE 9.13 TRENDS IN CREDIT APPRAISAL AND DELIVERY STANDARDS
Source: US OCC documents
Basel II has also been squarely blamed for the incentive for banks to take risky assets off their balance sheet, or simply transfer the risk to willing sellers of protection—to comply with capital requirements. Theoretically, originators of loans can assume that assets or risks are off their balance sheet when they distribute them. Hence, the need to maintain capital on these assets is obviated. But for reputation, the originators may feel compelled to repurchase securities sold earlier. Then the assets come right back on the balance sheet—and when this would happen cannot be determined. For instance, Citigroup was seen to act responsibly when it included put options on CDOs backed by subprime mortgages that it sold to customers. The ‘puts’ gave the buyers the right to sell the securities back to the originator in case of financing problems. This contingency was not accounted for in the bank's balance sheet.
Sophisticated financial models of asset portfolios have been built and used by the banking system and these have now come under the scanner. Were the financial models flawed or was there too much reliance on the results that they threw up? For example, Northern Rock, one of the biggest casualties of the UK, had reportedly carried out extensive stress testing as stipulated by the UK Financial Services Authority only in the first half of 2007 and the results sounded no alarm. Evidently, the probability that all the bank's funding sources could dry up simultaneously was not one of the scenarios tested—since this looked highly improbable at that time!
The leading rating agencies of the world—S&P, Moody's, Fitch—have all drawn huge flak for their ‘failure’ to distinguish the riskiness of different securities. They have been accused of being too generous with their AAA ratings, and worse still, overlooking the potential downfall of the market as the underlying mortgage assets deteriorated in value. The worst cut of all—they reacted with wholesale downgrades when the market collapsed. The possible conflicts of interest arising from rating agencies being paid by the issuers and the same agencies offering advisory services to the issuers are also areas drawing criticism.
And there could be other reasons too—Was it the structured investment vehicles (SIVs, similar to special purpose vehicles/entities (SPVs) or (SPEs) and other mechanisms that used shortterm bank funding to invest in longterm derivatives, or was it their opaqueness, or was it the lack of proper regulation?
The Way Forward There are two alternatives before banks.
They return to ‘good, oldfashioned banking’, where banks originate and hold the assets till they are liquidated. But this option ignores economic realities. Deregulation has already come to stay, and structured products and risk management are inexorably linked with the broader deregulation of the financial markets and financial technology, backed by the dramatic leap forward of information technology and communication. Even if we turn the clock back and launch the era of strict and restrictive regulation, the advances made in communication and technology would make it possible for banks to move risks and assets offshore, and securitization would take a different form.
The second alternative is to recognize the real benefits of securitization and financial innovation—after all, these innovations have shown us methods of packaging and redistributing risk, transforming illiquid assets on bank balance sheets to tradable ones with attractive income streams; they have shown banks how to reduce holding costly capital to absorb the credit risk; they have given liquidity to the system and lowered funding costs.
Innovation has its costs too—as the present crisis has shown. But this is a necessary though unwelcome cost of learning. Pioneering efforts in any industry have always run into rough weather for lack of prior experience. The global turbulence is a heavy price to pay for innovation, but central banks and governments have acted swiftly to soften the blow to the banking system and markets.
The present crisis can therefore be considered a maturity crisis of the credit market after the development of credit derivatives. At first glance, it looks like a massive failure of most used quantitative models of credit derivative pricing. However, we would have to remember here that there is a basic risk involved in lending money, and that this risk comes at a price, and credit derivatives have found a way to merely transfer this risk, but not to wish risk away! Probably, armed with a deeper understanding of the causality of the credit default process, deductive models with better and better predictive ability can take care of the changing environment.
The signals are now quite clear. The OCC Survey sums it up neatly. ‘While the competitive environment will inevitably cause changes in credit underwriting standards, banks need to have risk management and control processes to signal when standards veer away from safe and sound banking practices. Banks should underwrite credit based upon an expectation that the borrower can repay the loan, regardless of whether the loan is intended for portfolio or for distribution. As recent events have clearly shown, liquidity conditions in credit markets can change abruptly. Banks originating credit for distribution should maintain underwriting standards reasonably consistent with the standards for their own portfolio holdings.’^{105}
Current Developments and Regulatory Changes
In a bid to ensure the stability of the financial system, several regulatory and marketrelated changes are being introduced. One of the focus areas of these changes is credit derivatives. Even though public interest has turned towards the credit derivatives market only recently, the derivatives market has been focused on by regulators over the years. Regulatory attention is now most pronounced in the United States and Europe, where the crisis has taken its severest toll.
In the United States, far reaching changes are being brought about in both regulation of credit derivatives as well as market infrastructure for derivatives trading. Some of the important measures instituted include
 A comprehensive regulatory reform of the financial regulatory system, called the ‘Treasury framework’ (March 2009). This framework will cover four broad areas—systemic risk, consumer and investor protection, eliminating gaps in the US regulatory structure and international coordination—embedded in the ‘core principles’ of the framework.^{106}
 One of the key elements of the above framework is the ‘Derivatives framework’, whose main components would be the oversight, protection and disclosures in the OTC derivatives market.
 The improvements in market infrastructure, as discussed by industry participants and the Federal Reserve of New York (April 2009)^{107} would be consistent with the priorities in the President's Working Group (PWG) on Financial Markets and the Treasury framework mentioned above.
 The PWG's March 2008 ‘Policy Statement on financial markets’ has been one of the most influential in deciding on regulatory issues. In this statement, the PWG outlines its diagnosis of the causes of the financial crisis, pointing out loosening lending standards, inadequate risk management systems, residential mortgagebacked securities (RMBS) and the lacunae in the functioning of the rating agencies as the most significant. The document also makes some recommendation for the OTC derivatives markets. The PWG consists of the Secretary of the Treasury (as Chairman), the Chairman of the Board of Governors of the Federal Reserve System, the Chairman of the Securities and Exchange Commission and the Chairman of the Commodity Futures Trading Commission.
 The important steps taken to improve market infrastructure include (a) establishing CDS central counterparties (CCPs) to mitigate counterparty credit and operational risks, and (b) increasing market transparency, by expanding information publicly available on the credit derivative markets (such as release of periodical data on trading and prices of derivatives by the Depository Trust and Clearing Corporation (DTCC), dealers and the CCPs).
 The CDS contract itself and the trading of credit derivatives would soon undergo substantial changes. The changes to the CDS contract will be global changes. However, the convention changes would initially only apply to North American markets. In what is known as the CDS ‘Big Bang’, both contract and convention changes are to be introduced simultaneously.^{108}
 The global changes to the CDS contract would be as follows:
 The contract will be changed to ‘hardwire’ the auction mechanism for CDS following a credit event. The standard CDS document will incorporate this requirement in accordance with ISDA requirements. The current CDS contract deals only with physical settlement of trades. In 2005, an auction process was initiated to determine the final RR of a defaulted reference asset. The practice was for participants to sign up for a separate protocol (a legal document amending all previous trades) for each auction. Under this process, every time there was a credit event, the participants had to be tracked down to determine if they wanted to adhere to the protocol or not. This was a complex and time consuming process. Hardwiring the auction mechanism language into the protocol would make the process more efficient and reliable.^{109}
 Determination committees will be formed to make binding decisions on the terms of the auction as well as whether credit events have occurred.
 The effective date for all CDS contracts will be current day less 60 days for credit events. The standardization of the effective date is to eliminate residual risk between two trades for participants.
 The contract and convention changes proposed are intended to support the CCP initiative by bringing in more standardization, and hence mitigate systemic risk.
CHAPTER SUMMARY
 Traditional methods such as the Altman's Z score and other credit scoring models try to estimate the PD, rather than potential losses in the event of default (LGD). The traditional methods define a firm's credit risk in the context of its ‘failure’—bankruptcy, liquidation or default. They ignore the possibility that the ‘credit quality’ of a loan or portfolio of loans could undergo a mere ‘upgrade’ or ‘downgrade’
 The credit risk of a single borrower/client is the basis of all risk modelling. In addition, credit risk models should also capture the ‘concentration risk’ arising out of portfolio diversification and correlations between assets in the portfolio. Typically, credit risk models are expected to generate (a) loss distributions for the default risk of a single borrower and (b) portfolio value distributions for migration (upgrades and downgrades of a borrower's creditworthiness) and default risks.
 Credit risk models have a wide range of applications. They are prevalently used for the following:
 Assessing the ‘EL’ of a single borrower.
 Measuring the ‘economic capital’ of a financial institution
 Estimating credit concentration risk
 Optimising the bank's asset portfolio
 Pricing debt instruments based on the risk profile
Credit risk models are valuable since they provide users and decision makers with insights that are not otherwise available or can be gathered only at a prohibitive cost.
 ‘Creditmetrics™’, the most wellknown industrysponsored credit migration model applies migration analysis to credit risk measurement. The model computes the full (1 year) forward distribution of values for a loan portfolio, where the changes in values are assumed to be due to credit migration alone, while interest rates are assumed to evolve in a deterministic manner. ‘Credit VaR’ is then derived as a percentile of the distribution corresponding to the desired confidence level.
 CPV is a ratingsbased portfolio model used to define the relationship between macroeconomic cycles and credit risk in a bank's portfolio. It is based on the observation that default and migration probabilities downgrade when the economy worsens, i.e., defaults increase, and the contrary happens when the economy strengthens. The model simulates joint conditional distribution of default and migration probabilities for nonIG borrowers whose default probabilities are more sensitive to credit cycles, which are assumed to follow business cycles closely, than those of highly rated borrowers in different industries and for each country, conditional on the value of macroeconomic factors.
 MKMV uses the option pricing framework in the VK model to obtain the market value of a firm's assets and the related asset volatility. The default point term structure for various risk horizons is calculated empirically. MKMV combines market value of assets, asset volatility and default point term structure to calculate a DD term structure, which is then translated into a credit measure termed (EDF). The EDF is the PD for the risk horizon (1 year or more for publicly traded firms).
 Creditrisk+ has been developed based on the actuarial approach by Credit Suisse financial products (CSFP). In this model, only default risk is modelled, not downgrade risk. The model makes no assumptions about the causes of default. Each borrower/counterparty can assume only one of two ‘states’ at the end of the risk horizon,—‘default’ (0) or ‘no default’ (1). All the above models are commonly known as ‘structural models’.
 The other class of popular models is ‘reducedform’ models. The key difference between structural and reducedform models is in the ‘information’ available to model credit risk. While ‘defaults’ are specified ‘endogenously’ (the credit quality being determined by the assets and liabilities of the firm) in structural models, defaults are modelled exogenously from market data in the reducedform approach. Another difference is in the treatment of RRs—structural models specify RRs based on the value of assets and liabilities within the firm, while reducedform models look to the market to specify RRs. Reducedform models are also called ‘intensity’ models. If information available to the modeller/bank is partial or incomplete, a structural model with default being a predictable ‘stopping time’ becomes a reducedform model with default time being unpredictable.
 There are several methodologies used to price CDS spreads. The reducedform models are also used for this purpose. The concept of ‘default correlation’ is important to determining the pricing of a tranched CDO, such as tranched iTraxx.
 2001 saw the launch of synthetic credit indices by J P Morgan (JECI and Hydi), and Morgan Stanley (Synthetic Tracers). In 2003, these indices were merged under the name Tracx. At around the same time, credit derivative indices were introduced by iBoxx. Further consolidation took place in 2004 when Tracx and iBoxx combined to form the CDX in North America, and the iTraxx in Europe and Asia. Since November 2007, Markit owns the iTraxx, CDX, LevX and LCDX indices for derivatives, and iBoxx indices for cash bonds.
 The Credit Event Fixings have been developed by Creditex and Markit along with ISDA and major dealers in credit derivatives. The objective of credit fixings is to ensure a fair, efficient and transparent process for settlement of credit derivative trades following a corporate default. They are an integral part of ISDAs CDS Index protocols.
 The subprime crisis that began in mid 2007 is widely regarded as the first financial crisis in the age of mass ‘securitisation’. It has unleashed a flurry of daunting questions on the very foundations of securitized finance and structured products, as well as credit risk models and rating agencies.
 In a bid to ensure the stability of the financial system, several regulatory and marketrelated changes are being introduced. One of the focus areas of these changes is credit derivatives. Regulatory attention is now most pronounced in the United States and Europe, where the crisis has taken its severest toll.
TEST YOUR UNDERSTANDING
 Which of the following loans is most risky? Assume that other things are equal—the borrowers are rated the same, are from the same industry and have the same risk profile.
 Rs. 30 crore loan with 50 per cent LGD
 Rs. 10 crore loan with no collateral
 Rs. 40 crore loan with 40 per cent RR
 Rs. 40 crore loan with 40 per cent LGD
 Which of the following loans is likely to have the highest PD, other things being equal?
 The loan with the longest maturity
 The loan with the lowest LGD
 The loan whose value has the lowest volatility
 The loan whose borrower is assessed most creditworthy
 Which of the following is NOT used to estimate the PD of a firm in the KMV model?
 Book value of equity
 Market value of equity
 Book value of debt
 Price volatility
 Time to maturity of loan to the firm
 Using the KMV credit risk model, calculate the DD of a firm whose assets stand at Rs. 50 crore at current value, and whose liabilities are at Rs. 30 crore. Also assume the asset return volatility is 10, and the conditions of the Merton model are met.
 Which of the following is NOT TRUE of MKMV's EDF?
 EDF measures are actual probabilities
 EDF measures are credit scores
 If a firm has a current EDF credit measure of 2 per cent, it implies that there is a 2 per cent probability of the firm defaulting over the next 1 year
 If there are 100 firms with an EDF of 4 per cent, we can expect, on an average, four firms to default over the next year
 A firm with 4 per cent EDF measure is 10 times more likely to default than a firm with 0.4 per cent EDF measure
 For a portfolio of risky assets, CreditMetrics uses the following to estimate default correlations.
 Correlation of changes in corporate bond yields
 No default correlations are assumed
 Correlation of changes in corporate bond defaults
 Correlation of equity returns
 In Illustration 9.3 (CreditRisk+), if in step 2, for the same bands and number of borrowers in each band, the expected number of defaults in each band doubles, what will be the effect on the EL in each band?
 A 5year CDS requires semi annual payment at the rate of 60 bps per annum. The notional principal is Rs. 30 crore. A credit event occurs after 4 years and 5 months. If the RR is 40 per cent, and the CDS is settled in cash, list the cash flows and their timing for (a) the protection buyer and (b) the protection seller.
 What is the difference between (a) a plain vanilla CDS and a binary CDS, and (b) a plain vanilla CDS and a basket default swap?
 How would the CDS fee (spread) change in Illustration 9.5 if (other things remaining constant)
 The fee payments were made quarterly
 The fee payments were made semi annually
 The default occurred mid way during the year
 The RR was more than 50 per cent
 The RR was less than 50 per cent
 It is a binary CDS.
 Denote conditional default probability for a firm X as p per year, and the RR as R. The riskfree rate is 5 per cent per year. Default always occurs at the end of the year. The spread for a 5year plain vanilla CDS is 120 bps and the spread for a similar 5year binary CDS is 160 bps. What is the value of p and R?
 How would p and R in the above example change if defaults always occurred mid way through every year? Would you be able to draw any conclusion about the relationship between the spreads of a similar plain vanilla CDS and binary CDS?
TOPICS FOR FURTHER DISCUSSION
 For a hypothetical loan/loan portfolio, apply the credit risk models and compare the results.
 MKMV publishes on its Web site, case studies on calculating EDF for several firms. Take a sample of these firms and trace their credit ratings as awarded by the top credit rating companies to these firms. Are the results consistent?
 Should credit derivatives be continued as credit risk transfer mechanisms, in the light of their role in the current global financial crisis?
 What are CDOsquared and CDOcubed and how are they valued?
 What are the various types of exotic CDOs and how are they priced?
SELECT REFERENCES
 Bluhm, Christian, Ludger Overbeck and Christoph Wagner (2003) An introduction to credit risk modelling. CRC Press LLC, USA, ISBN 158488326X.
 Crouhy, M, Dan Galai and Robert Mark, (2000). ‘A Comparative Analysis of Current Credit Risk Models’, Journal of Banking and finance 24: 59–117.
 Elizalde, Abel. (2005a) ‘Credit Risk Models I: Default Correlation in Intensity Models’, accessed at www.abelelizalde.com.
 Elizalde, Abel. (2005b) ‘Credit Risk Models II: Structural Models’, accessed at www.abelelizalde.com.
 Elizalde, Abel (2006), ‘Credit Risk Models III: Reconciliation ReducedStructural Models’, CEMFI working paper 0607, accessed at www.cemfi.es.
 Gordy, Michael (2000), ‘A Comparative Anatomy of Credit Risk Models’, Journal of Banking and Finance 24: 119–149.
 Hull, John, and Alan White (2008), ‘Dynamic Models of Portfolio Credit Risk: A Simplified Approach’, Journal of Derivatives 15, 4 (Summer): 9–28.
 Jarrow, Robert A. and Stuart M. Turnbull (2000), ‘The Intersection of Market and Credit Risk’, Journal of Banking and Finance 24: 271–299.
 Merton, Robert C. (1974), ‘On the Pricing of Corporate Debt: The Risk Structure of Interest Rates’, Journal of Finance 39: 449–470
 Uwe, Wehrspohn. (2002), ‘Credit Risk Evaluation: Modelling–Analysis–Management’, accepted as a doctoral thesis at the faculty of economics at Heidelberg University, Germany, available in ebookformat at http://www.riskandevaluation.com.
 http://www.defaultrisk.com
 http://www.bis.org
ANNEXURE I
THE GLOBAL CREDIT CRISIS—A BRIEF CHRONOLOGY OF EVENTS IN 2007–08
Sources: Gathered from various Web sites and published material.
ABBREVIATIONS EXPANDED
1. TPG—Texas Pacific Group; FDIC—The Federal Deposit Insurance Corporation; TSLF—Term Securities Lending Facility; PDCF—Primary Dealer Credit Facility; AIG—American International Group; TARP—Troubled Asset Relief Program; TAF—Term Auction Facility.