Topics 19-20

Question 1

Three features of a good rating system

Answer 1

A good rating system will possess the following three features, which together will help entities measure the appropriateness of their internal rating systems:

• Objectivity and Homogeneity. An objective rating system will produce judgments based only on considerations tied to credit risk, while a homogeneous system implies that ratings are comparable among market segments, portfolios, and customer types.
• Specificity. A rating system is specific if it measures the distance from a default event while ignoring other financial elements that are not directly tied to potential default.
• Measurability and Verifiability. Ratings must provide correct expectations related to default probabilities which are backtested on a continuous basis.

Question 2

Key measures used to assess the risk of default: probability of default (PD), cumulative probability of default, marginal probability of default, annualized default rate (ADR)

Answer 2

Question 3

Conditional (forward) PD

Answer 3

Conditional (forward) PD = (PD_cum^t - PD_cum^t-1)/(Names - PD_cum^t-1)

Question 4

Compare agencies’ ratings to internal experts-based rating systems

Answer 4

In terms of the criteria for a good rating system, the following comparisons can be made between agencies’ ratings and internal experts-based rating systems:

• Objectivity and Homogeneity. Agencies’ ratings are 73% compliant, while internal experts-based rating systems are 30% compliant.
• Specificity. Agencies’ ratings are close to 100% compliant, while internal experts-based rating systems are 75% compliant.
• Measurability and Verifiability. Agencies’ ratings are 75% compliant, while internal experts-based rating systems are 25% compliant.

Question 5

Distinguish between structural approaches and reduced-form approaches to predicting default

Answer 5

The foundation of a structural approach (e.g., the Merton model) is the financial and economic theoretical assumptions that describe the overall path to default. Under this approach, building a model involves estimating the formal relationships that link the relevant variables of the model. In contrast, reduced form models (e.g., statistical and numerical approaches) arrive at a final solution using the set of variables that is most statistically suitable without factoring in the theoretical or conceptual causal relationships among variables.

A significant model risk in reduced form approaches results from a model’s dependency on the sample used to estimate it. To derive valid results, there must be a strong level of homogeneity between the sample and the population to which the model is applied.

Reduced form models used for credit risk can be classified into statistical and numerical-based categories.

• Statistical-based models use variables and relations that are selected and calibrated by statistical procedures.
• Numerical-based approaches use algorithms that connect actual defaults with observed variables.
• Both approaches can aggregate profiles, such as industry, sector, size, location, capitalization, and form of incorporation, into homogeneous “top-down” segment classifications. A “bottom-up” approach may also be used, which would classify variables based on case-by-case impacts. While numerical and statistical methods are primarily considered bottom-up approaches, experts-based approaches tend to be the most bottom up.

Question 6

Describe Merton model to calculate default probability and the distance to default

Answer 6

Question 7

Challenges/limitations of using the Merton model

Answer 7

There are many challenges associated with using the Merton model:

• Neither the asset value itself nor its associated volatility are observed.
• The structure of the underlying debt is typically very complex, as it involves differing maturities, covenants, guarantees, and other specifications.
• Because variables change so frequently, the model must be recalibrated continuously.
• Also, its main limitation is that it only applies to liquid, publicly traded firms.
• Using this approach for unlisted companies can be problematic due to unobservable prices and challenges with finding comparable prices.
• Finally, due to high sensitivity to market movements and underlying variables, the model tends to fall short of fully reflecting the dependence of credit risk on business and credit cycles.

Question 8

Describe linear discriminant analysis (LDA), define the Z-score and its usage

Answer 8

• Linear discriminant analysis (LDA) is one of the most popular statistical methods used for developing scoring models. The contributions (i.e., weights) of each accounting ratio to the overall score are represented by Altman’s Z-score.
• LDA categorizes firms into two groups: the first represents performing (solvent) firms and the second represents defaulting (insolvent) firms.
• A Z cut-off point is used to differentiate both groups, although it is imperfect as both solvent and insolvent firms may have similar scores. This may lead to incorrect classifications.
• Another example of LDA is the RiskCalc® model, which was developed by Moody’s. It incorporates variables that span several areas, such as financial leverage, growth, liquidity, debt coverage, profitability, size, and assets. The model is tailored to individual countries.
• With LDA, one of the main goals is to optimize variable coefficients such that Z-scores minimize the inevitable “overlapping zone” between solvent and insolvent firms. For two groups of borrowers with similar Z-scores, the overlapping zone is a risk area where firms may end up incorrectly classified, historical versions of LDA would sometimes consider a gray area allowing for three Z-score range interpretations to determine who would be granted funding: very safe borrowers, very risky borrowers, and the middle ground of borrowers that merited further investigation. In the current world, LDA incorporates the two additional objectives of measuring default probability and assigning ratings.
• Note that LDA models typically offer only two decisions: accept or reject. Modern internal rating systems, which are based on the concept of default probability, require more options for decisions.
• For Altman: a score below 1.8 means it’s likely the company is headed for bankruptcy, while companies with scores above 3 are not likely to go bankrupt.

Question 9

Calibration of LDA models

Answer 9

The process of fitting empirical data into a statistical model is called calibration.

This process implies that more work is still needed, even after the scoring function is estimated and Z-scores are obtained, before the model can be used.

• In the case of the model being used simply to accept or reject credit applications, calibration simply involves adjusting the Z-score cut-off to account for differences between sample and population default rates.
• In the case of the model being used to categorize borrowers into different ratings classes (thereby assigning default probabilities to borrowers), calibration will include a cut-off adjustment and a potential rescaling of Z-score default quantifications.

Because of the relative infrequency of actual defaults, a more accurate model can be derived by attempting to create more balanced samples with relatively equal (in size) groups of both performing and defaulting firms. However, the risk of equaling the sample group sizes is that the model applied to a real population will tend to overpredict defaults. To protect against this risk, the results obtained from the sample must be calibrated. If the model is only used to classify potential borrowers into performing versus defaulting firms, calibration will only involve adjusting the Z cut-off using Bayes’ theorem to equate the frequency of defaulting borrowers per the model to the frequency in the actual population.

Question 10

Describe the application of logistic regression model to estimate default probability

Answer 10

Logistic regression models (also known as LOGIT models), which are from the
Generalized Linear Model (GLM) family, are statistical tools that are also used to predict default.

GLMs typically have three common elements:

• A systematic component, which specifies the variables used in a linear predictor function.
• A random component, which identifies both the target variable and its associated probability function.
• A link function, which is a function of the target variable mean that the model ties to the systematic component.

Question 11

Define and interpret cluster analysis

Answer 11

Both LDA and LOGIT methodologies are considered “supervised” due to having a defined dependent variable (the default event), while independent variables are applied to determine an ex ante prediction. When the dependent variable is not explicitly defined, the statistical technique is considered “unsupervised.”

Cluster analysis looks to identify groups of similar cases in a data set. Groups represent observation subsets that exhibit homogeneity (i.e., similarities) due to variables’ profiles that allow them to be distinguished from those found in other groups.

Two approaches can be used to implement cluster analysis:

1. hierarchical/aggregative clustering and
2. divisive/partitioned clustering.

With hierarchical clustering, cluster hierarchies are created and aggregated on a case-by-case basis to form a tree structure with the clusters shown as leaves and the whole population shown as the roots. Clusters are merged together beginning at the leaves, and branches are followed until arriving at the roots. The end result of the analysis typically produces three forms:

• A small number of highly homogeneous, large clusters.
• Some small clusters with comprehensible and well-defined specificities.
• Single, very specific, nonaggregated units.

One of the key benefits of this method is the detection of anomalies. Many borrowers, such as merged (or demerged) companies, start-ups, and companies in liquidation, are unique. This analysis facilitates identifying these unique profiles and managing them separately from other observations.

Divisive clustering begins at the root and splits clusters based on algorithms that assign every observation to the specific cluster whose center (the average of all points in the cluster) is nearest. This approach serves to force the population into fewer cluster groups than what would be found under aggregative clustering. On the other side, high calculation power is needed as expanding the number of observations has an exponential impact.

As an example of applying cluster analysis, we can look to composite measures of profitability such as ROE and ROI. The task is to identify both specific aspects of a firms financial profile and latent (hidden) variables underlying the ratio system, such that the basic information from a firm’s financial statements can be extracted and used for modeling without redundant data and information.

Question 12

Define and interpret principal component analysis

Answer 12

• Principal component analysis involves transforming an original tabular data set into a second, derived tabular data set.
• The performance of a given variable (equal to variance explained divided by total original variance) is referred to as communality, and the higher the communality (the more general the component is), the more relevant its ability to summarize an original set of variables into a new composed variable.
• The starting point is the extraction of the first component that achieves maximum communality. The second extraction will focus on the residuals not explained by the first component. This process will continue until we have a new principal components set, which will be orthogonal (statistically independent) by design and explain original variance in descending order. In terms of a stopping point, potential thresholds include reaching a minimum predefined variance level or a minimum communality that assures a reasonable level of information using the new set of components.
• An eigenvalue is a measure of the communality associated with an extracted component. The ideal first component is one that corresponds to the first eigenvalue of the set of variables. The second component will ideally correspond to the first eigenvalue extracted on the residuals. All original variables once standardized contribute a value of one to the final variance.
• An eigenvalue greater (less) than one implies that this component is summarizing a component of the total variance which exceeds (is less than) the information provided by the original variable. Therefore, it is common that only principal components with eigenvalues greater than one are considered.

Question 13

Decribe factor analysis

Answer 13

Factor analysis is similar to principal component analysis, except that factor analysis is used to describe observed variables in terms of fewer unobserved variables called “factors” and can be seen as more efficient.

Factor analysis is often used as the second stage of principal component analysis. In terms of the process, step one is to standardize principal components. Then, the values of the new variables (factor loadings) should be standardized such that the mean equals zero and the standard deviation is equal to one. Even though factor loadings are not comparable (from a size and range perspective) to original variables, they are comparable to each other.

Factors will be contingent on the criteria used to conduct what is called the “rotation.” The varimax method is a rotation method used to target either small or large loadings of a particular variable associated with each factor. As a result of iteratively rotating factor pairs, the resulting solution yields results that make it feasible to identify each variable tied to a single factor. A final solution is reached once the last round provides no added benefit.

Question 14

Canonical correlation method

Answer 14

• The canonical correlation method is a technique used to address the correspondence between a set of independent variables and a set of dependent variables.
• As an example, if an analyst wanted to understand what is explaining the default rate and any changes in default rates over various time horizons, he can look at the relationship between default rate factors and financial ratio factors and understand what common dimensions existed between the tests and the degree of shared variance.
• This analysis, which is a type of factor analysis, helps us find linear combinations of the two sets that have a maximum correlation with each other. From this analysis, we can determine how many factors are embedded in the set of dependent variables and what the corresponding factors are out of the independent variables that have maximum correlations with the factors from the dependent variable set. The factors from both sets are independent of one another.
• Although this method is very powerful, the disadvantages are that it is difficult to rigorously calculate scores for factors, and measuring the borrower profiles can only be done by proxy as opposed to measuring them in new independent and dependent factors.

Question 15

Describe the use of a cash flow simulation model in assigning rating and default probability, and explain the limitations of the model

Answer 15

• A cash flow simulation model is most often used to assign ratings to companies that have non-existent or relatively meaningless track records. In an ideal situation, a given firm’s future cash flow simulation will stay in the middle between structural and reduced form models. The simulation will be based on forecasting a firm’s pro forma financial reports and studying the volatility of future performances.
• One of the biggest risks of cash flow simulation models is model risk, which stems from the fact that any model serves as a simplified version of reality. Defining default for the purposes of the model is also challenging, as it cannot always be known if and when a default will actually be filed in real-life circumstances.
• Therefore, the default threshold needs to be set such that it is not too early (the risk of having too many defaults, resulting in transactions that are deemed risky when they are not truly risky) and not too late (the risk of having not enough defaults, thereby understating the potential risk).
• Costs must also be taken into account, as models can cost a lot of money to build, maintain, and calibrate.
• Even given these issues, there are not many feasible alternatives to using the simulation model for a firm in certain conditions when historical data cannot be observed.

Question 16

Heuristic and numerical methods in predicting defaults

Answer 16

Through the application of artificial intelligence methods, other techniques have been applied to predicting default in recent years. These two primary approaches include:

• Heuristic methods. These methods are designed to mirror human decision-making processes and procedures. Trial by error is used to generate new knowledge rather than using statistical modeling. These methods are also known as “expert systems,” with a goal of reproducing high frequency standardized decisions at the highest level of quality at a low cost. The fundamental idea is to learn from both successes and errors.
• Numerical methods. The objective of these methods is to derive optimal solutions using “trained” algorithms and incorporate decisions based on relatively weak information in very complex environments. An example of this is a “neural network,” which is able to continuously update itself in order to incorporate modifications to the environment.

Question 17

Expert system

Answer 17

An expert system, which is a traditional application of artificial intelligence, is a set of software solutions designed to produce answers to problems where human experts would otherwise be needed. Expert systems will typically involve the creation of a knowledge base and will use knowledge engineering to gather and codify knowledge into a framework.

The typical components of an expert system include the working memory (short-term memory), the user interface/communication, the knowledge base (long-term memory), and the inferential engine (the heart/nervous network).

The rule base of an expert system consists of many inference rules (which are designed to resemble human behavior); these go into the knowledge base as separate rules, and the inference engine serves to bring them together to draw conclusions.

The inference engine can use either backward chaining or forward chaining.

• With backward chaining (goal driven), the starting point is a list of goals. Working backward, the expert system will look to find paths that will allow it to achieve these goals. Rules are searched until one is found which best aligns to the desired goal.
• With forward chaining (data driven), the starting point is available data. Inference rules are applied until a desired goal is achieved. Once the path is recognized as successful, it is applied to the data.

An expert system may also incorporate “fuzzy logic” applications. This logic applies “rules of thumb” based on feelings and uses approximate as opposed to precise reasoning. A fuzzy logic variable will not be confined to the extremes of zero and one; rather, they can assume any value that exists between the two extreme values.

A subset of expert systems is decision support systems (DSSs), which are applied to certain phases of the human decision-making process and involve very complex and cumbersome calculations.

Question 18

Neural networks

Answer 18

Neural networks come from biological studies and serve to simulate human brain behavior. These networks involve the interconnection of artificial neurons (software programs designed to mirror the properties of biological neurons) and have the ability to continuously learn by experience.

• One of the key benefits of the neural network method is its ability to capture nonlinear relationships. Because a network may have thousands of nodes and even more potential connections, the flexibility exists to handle highly complex, nonlinear, recursive, and independent problems. The most common structure is the “hierarchically dependent neural network.”
• In terms of limitations, there is no way to look step-by-step at neural networks to determine how results are obtained; we have to accept that the results will come from what appears like a “black box,” which makes it impossible to explain how and why we arrived at a specific result. A way around this issue is to prepare multiple data sets characterized by distinguishing profiles and then put them in the neural network to obtain results. With outputs coming from homogeneous inputs, it is possible to then deduce the critical variables and their associated weights.
• Also, these networks are highly sensitive to the quality of the inputs; as such, data sets must be carefully chosen to not have the model learn from outliers.
• In addition, continuous quantitative variables are more appropriate for neural networks than qualitative variables.
• Over-fitting is a major risk for estimating neural networks, as a network that over-fits a sample of data will not be able to produce quality results when applied to other samples, such as sectors, borrowers, economic cycle stages, and geographic areas.

Question 19

Comparison of heuristic and numerical methods

Answer 19

• An expert system is advantageous when human experts have known, clear, and well-dominated experience; this experience allows for the formalization of rules and building of effective systems.
• For the purposes of rating assignments, expert systems provide objectivity, order, and discipline to the ratings process; however, they do not provide new knowledge because they are not inferential methods or models.
• Numerical approaches, like neural networks, provide classifications, often with low granularity (like very good, pass, reject, etc.). These models are not statistical models and, therefore, do not produce outputs like probabilities of default. This limitation, along with the “black box” limitation, limits the usefulness of neural networks outside of segments such as personal loans or consumer credit. However, they can be used for potential early warnings and credit quality monitoring. Also, a neural network is very useful for processing extremely large quantities of data, adjusting quickly when a discontinuity occurs, and creating new rules when a change in the pattern of success/failure is uncovered.

Comparing heuristic approaches (i.e., expert systems and decision support systems) to numerical approaches (i.e., neural networks) across the three key features of a good ratings system discussed earlier shows the following results:

• Objectivity and Homogeneity. Both are almost entirely compliant.
• Specificity. The numerical approach is 73% compliant, while the heuristic approach is 30% compliant.
• Measurability and Verifiability. The numerical approach is 75% compliant, while the heuristic approach is 50% compliant.

Question 20

Describe the role and management of qualitative information in assessing probability of default

Answer 20

From the perspective of using judgment to ultimately determine credit approval, three categories are used to encapsulate qualitative information:

• Investment, innovation, and technology.
• Human resource management, motivation, retention of key resources, and maximizing talent.
• Effective and efficient internal processes

Categorical types of information include binary information (such as yes/no), nominal information (like locations of incorporation), and ordinal classifications with graduating levels (such as low, medium, and high).

• Binary information can be represented as dummy variables (i.e., 0 or 1).
• Ordinal information can be assigned numbers and weights differing at each level. Even with these options for quantification, the lack of historical data is a major problem with using qualitative information.

A potential mechanism for overcoming these issues is to invoke a two-stage process:

• Stage 1: Build a quantitative model along with launching a systematic qualitative data collection on new reports.
• Stage 2: Once Stage 1 has produced enough information, build a new model which includes the new qualitative information.

In spite of the challenges of incorporating qualitative data, this data set is a critical element to building powerful credit models and driving long-term value creation for banks.

Recommendations to using qualitative information in the models.

• A first recommendation is to only gather qualitative information that is not collectable in quantitative terms. For instance, growth and financial structure information can be extracted from balance sheets.
• A second recommendation regards how to manage qualitative information in quantitative models.

Question 21

Basic characteristics of Black-Scholes-Merton model (assumptions)

Answer 21

The simplest form of the model assumes the existence of a non-dividend paying firm with only one liability claim and that financial markets are perfect. That is, the model assumes away taxes, bankruptcy costs, and costs associated with enforcing contracts.

The Black-Scholes-Merton option-pricing model for European options can be modified to determine the value of equity prior to T, T — t, if additional assumptions are made, which include:

• Firm value characterized by a lognormal distribution with constant volatility, σ.
• Constant interest rate, r.
• Perfect financial market with continuous trading.

Question 22

The Value of Equity at Time t

Answer 22

Question 23

The Value of Debt at Time t

Answer 23

There are two methods for valuing risky debt in this framework. Risky debt is equal to:

• Risk-free debt minus a put option on the firm.
• Firm value minus equity value.

Figure 2 shows the general relationships between debt and equity values according to the inputs of the Merton model.

Question 24

Distance to default (DD) and Lognormal DD

Answer 24

Lognormal price-based DD = [V(t) - Default] / [sigma * V(t)]

Question 25

Explain the relationship between credit spreads, time to maturity, and interest rates, and calculate credit spread

Answer 25

A credit spread is the difference between the yield on a risky bond (e.g., corporate bond) and the yield on a risk-free bond (e.g., T-bond) given that the two instruments have the same maturity.

Question 26

Determining Firm Value and Volatility

Answer 26

• A portfolio consisting of a call option on the firm and a risk-free asset is equivalent to the value of the firm. Thus, a small change in firm value will change the value of equity by delta, Δ, times the change in firm value.
• Delta is the rate of change in the value of the call option relative to the change in the value of the underlying asset, ΔS/ΔV.
• The Merton model delta, Δ, is equal to N(d). Therefore, if we know the parameters for calculating the value of equity as a call and the value of risk-free debt, then we can determine the firm’s value and the volatility of firm value.
• Although delta is increasing as the value of the firm increases, the change in the value of equity decreases as firm value increases. This indicates that the distribution of equity values is not constant (which is sometimes referred to as a volatility smirk). The non-constant volatility of equity is a violation of the Black-Scholes-Merton model.
• The Geske compound option model is appropriate for valuing the equity call option because it assumes that the value of the firm is characterized by a lognormal distribution with a constant variance.

Question 27

Explain the differences between valuing senior and subordinated debt

Answer 27

In the event of bankruptcy, subordinate debt will receive payment only after all obligations to senior debt have been paid. Because of the uncertainty associated with financial distress, the value of subordinate debt acts more like an equity security than a debt security.

Therefore, when a firm is in financial distress, the value of subordinate debt will increase as firm volatility increases, while the value of senior debt will decline.

Subordinate debt can be valued in a portfolio as a long position in a call option on the firm with an exercise price equal to the face value of senior debt, F, and a short position on a call option on the firm with an exercise price equal to the total principal due on all debt, U + F.

Figure 5 illustrates how subordinate debt values behave like equity when the firm has low values, as during periods of financial distress, and how they behave like senior debt when the firm is not in financial distress.

Question 28

Value of subordinated debt

Answer 28

Stulz model:

V = c(V,F,T) - c(V,F+U,T)

The impact of volatility and maturity is ambiguous and tends to be different for a high-value firm (when subordinated debt act more like debt: increase in volatility tends to decrease value and decrease in maturity tends to increase price via pull to par) than for a low-value firm (when subordinated debt acts more like equity: increase in volatility tends to increase value and decrease in maturity tends to decrease value)

Question 29

Interest Rate Dynamics of Firm in Financial Distress

Answer 29

Question 30

Difficulties in application of the Merton model

Answer 30

Application of the Merton model is complicated by the complexity of firms’ capital
structures. Most firms have a variety of debt instruments that mature at different times and have many different coupon rates (i.e., not just zero-coupons). In addition to the many different types of debt issues, the Merton model does not allow the firm value to jump.

Since most defaults are surprises, the inability to have jumps in the firm value in the Merton model makes default too predictable.

The documented problems with the Merton model created the need for models to predict default more accurately (such as the KMV approach).

Question 31

Using the Merton Model to Calculate PD and LGD

Answer 31

In addition to the lack of public trading, there are four differences in measuring the risk of a debt portfolio that make estimating the probability of default and the loss due to default more challenging:

• If securities are illiquid, then the historical data is not reliable.
• The distribution of bond returns is not normal because the debtholder cannot receive more than the face amount plus the sum of the coupons.
• Debt is issued by creditors who do not have traded equity.
• Debt is not marked to market in contrast to traded securities. That is, a loss is recognized only if default occurs.

Question 32

Credit value at risk

Answer 32

Portfolio credit risk models resolve some of the difficulties of measuring a portfolio’s probability of default and the amount of loss associated with default when using the Merton model. The models also allow for the inclusion of additional securities and contracts, such as swaps. Therefore, instead of having only debtholders in the model, the model includes other obligors. Obligors include all parties who have a legal obligation to the firm.

Using various methodologies, credit risk portfolio models attempt to estimate a portfolio’s credit value at risk. Credit VaR (also called credit at risk or default VaR) is defined much the same as VaR (a.k.a. market VaR); the minimum credit loss at a given significance over a given time period (or alternatively, the maximum credit loss for a given confidence level over a given time period).

Credit VaR differs from market VaR in that it measures losses that are due specifically to default risk and credit deterioration risk. Like market VaR, credit VaR is measured over a specified time period at a specified probability.

There are two problems, however, when calculating credit VaR.

• First, calculating changes in credit quality over a 1-day period is difficult. Therefore, credit VaR is usually calculated over a year, where the potential change in credit risk is more easily estimated.
• The second problem is that changes in credit risk are highly skewed and do not follow a normal distribution. The loss distribution of changes in credit quality for investment grade bonds closely resembles a lognormal distribution.

Question 33

CreditRisk+

Answer 33

CreditRisk+ measures the credit risk of a portfolio using a set of common risk factors for each obligor. Each obligor has its own sensitivity to each of the common risk factors.

• Risk factors can only have positive values and are scaled to have a mean of one.
• The risk factors are assumed to follow a specific distribution, such as a gamma distribution.
• If an obligor has a risk factor greater than one, then the probability of default for firm i increases in proportion to the obligor’s exposure.
• After the probability of default for each obligor is calculated, the loss distribution for the portfolio can be estimated and used to assess the credit risk of the portfolio.

Question 34

CreditMetrics

Answer 34

*Correlations are important in a portfolio. If two bonds are independent, then the
probability of both bonds migrating is the product of the individual events. If the bonds are not independent, then we need to know the migration correlations. The major complexity of CreditMetrics is estimating the joint migration of bonds in a portfolio.

Question 35

Moody’s KMV Portfolio Manager

Answer 35

The KMV model, a modified Merton model, calculates the expected default frequencies (EDFs) for each obligor. This modified model allows for more complicated capital structures (e.g., short-term debt, long-term debt, convertible debt, and equity). KMV solves for firm value and volatility.

The primary advantage of the KMV model is the use of current equity values in the model. This allows for the impact of a current event to immediately affect the probability of default. Ratings changes occur with a considerable lag. The use of equity values allows for probabilities of default to change continuously as equity values change. In the CreditMetrics approach, the value of the firm can change without any impact on the probability of default.

The KMV model computes the expected return from a variation of the capital asset pricing model (CAPM), which uses a factor model to simplify the correlation structure of firm returns. This provides for a direct estimation of the loss distribution without requiring the use of simulation to estimate the credit VaR of the credit portfolio.

Question 36

CreditPortfolioView

Answer 36

CreditPortfolioView models the transition matrices using macroeconomic or economic cycle data. This is its primary distinguishing feature. Macroeconomic variables are the key drivers of default rates, and CreditPortfolioView estimates an econometric model for an index that drives the default rates of an industrial sector. The model simulates paths of the index, which produces a distribution of portfolio losses to analyze. Usually the focus is on an aggregate default rate for an entire economy.

The procedure can be summarized in four steps:

1. Measuring the autoregressive process of the macroeconomic variables.
2. Composing sector indices for the variables.
3. Estimating a default rate based on the value of that index.
4. Comparing the simulated values to historical values to determine the transition matrix to use.

Question 37

Limitations of the Credit Portfolio Models

Answer 37

Credit portfolio models have made improvements at estimating the probability of default; however, most models do not account for changes in:

• Interest rates.
• Credit spreads.
• Current economic conditions.

The state of the economy does affect probability of default for bonds.

Question 38

Assess the credit risks of derivatives, describe a credit derivative, credit default swap, and total return swap

Answer 38

A credit derivative is a contract with payoffs contingent on a specified credit event. Credit derivatives are designed as hedging instruments for credit risks. Credit derivatives are usually traded over the counter (OTC) and not on exchanges.

Credit events include:

• Bankruptcy.
• Failure to pay.
• Restructuring.
• Repudiation.
• Moratorium.
• Obligation acceleration.
• Obligation default.

One of the simplest credit derivatives is a credit default put. A credit default put pays on a loss of debt due to default at the maturity of the debt claim, T.

A more complex and popular credit derivative is the credit default swap (CDS). A CDS is similar to a typical swap in that one party makes payments to another party. The purchaser of the CDS seeks credit protection and will make fixed payments to the seller of the CDS for the life of the swap, or until a credit event occurs.

• If the terms of the swap agreement dictate settlement by physical delivery, the buyer of the CDS delivers the reference obligation to the seller of the swap and receives the par value.
• If the terms of the swap agreement are for cash delivery, dealers are surveyed a specified number of days following a credit event to determine a midpoint between bid and ask prices, called Z, which will then be used to calculate the cash payment as (100 — Z)% of the notional principal.

Total rate of return swaps (TROR) are agreements to exchange the total return of a reference asset (i.e., a risky corporate bond) for a floating rate (LIBOR) plus a specified spread. The total return of the reference asset will include both capital gains (or losses) and any flows (coupons, interest, dividends) over the life of the swap.

• If the payer owns the reference asset, a total return swap would allow the owner to transfer the credit risk of the asset to the receiver.
• If the payer does not own the reference asset, a total return swap’s cash flows would be similar to those of taking a short position in the bond. If the value of the bond declines, the payer position gains.
• If the value of the bond increases, the payer position loses.

Question 39

Derivatives With Credit Risks, vulnerable option

Answer 39

A vulnerable option is an option with default risk. An option holder receives the promised payment only if the seller of the option is able to make the payment. Without the default risk, the holder of the option at expiration receives:

Max(S - X, 0)

where:

• S = underlying asset’s price at expiration
• X = exercise price

The vulnerable option holder receives the promised payment only if the value of the counterparty firm, V, is greater than the required payment on the option. The payoff of the vulnerable option is:

Max [Min (V, S - X), 0]

The correlation between the value of the firm and the underlying asset value, ρ_(V,S), is important in the valuations of the vulnerable option. If ρ_(V,S) is strongly negative then vulnerable option has little value because firm value is low when vulnerable option payoff is to occur. If ρ_(V,S) is strongly positive then there is no credit risk because firm value is high when the value of equity is high.

An alternative approach computes the probability of default and a recovery rate estimate if default occurs. The value of the option is the weighted average of the option without default. Following this approach, the value of a vulnerable option is:

vulnerable option = [(1 — PD) x c] + (PD x RR x c)

where:

• c = value of the option without default
• PD = probability of default
• RR = recovery rate

Question 40

Explain how to account for credit risk exposure in valuing a swap

Answer 40

The credit risk in a swap can be reduced by requiring a margin or by netting the payments. Netting is a method where the payments are offset so that only one party needs to make a payment. The covenants of the swap agreement can affect the credit risk exposure.

Suppose a counterparty that is due to receive a net payment is in default. If the swap agreement has a full two-way payment covenant, then the counterparty still receives the net payment. However, if the swap has a limited two-way payment covenant, the obligations are abolished if either party is in default. Valuing a swap can be simplified by considering a swap with only one payment.

Suppose there is only one payment to be made in a swap arrangement between Market Maker, Inc. and Risky Credit, Inc., which has no liabilities at the creation of the swap. The agreement provides for Risky Credit to receive a fixed amount, F, at maturity and to pay a variable amount, S, based on some index. The index could be based on an equity value or a floating rate.

When we consider the default risk of Risky Credit, the swap’s payoff to Market Maker is:

(-)Max(F - S, 0) + Max[Min(S, V) - F, 0]

For Market Maker, the risk-free counterparty, the payoff of the swap is the same as a portfolio of a short position on a put option and a long position on a call. The put is written on an asset with a value of S and an exercise price of F. The call option is written on the lower of the variable payment, S, or the value of the risky counterparty, V, with the exercise price of the fixed payment, F. At the initiation of the swap agreement, F is selected so that the swap has no value.

The correlation between firm value, V, and the variable payment, S, is critical to the valuation of the swap. If the correlation declines, then there is no effect on the value of the put option, but the value of the option on the two risky assets declines.

Question 41

Key differences between linear and logistic regressions

Answer 41

To focus differences with the classical linear regression, consider that:

• In classical linear regression the dependent variable range is not limited and, therefore, may assume values outside the [0; 1] interval; when dealing with risk, this would be meaningless. Instead, a logarithmic relation has a dependent variable constrained between zero and one.
• The hypothesis of homoscedasticity of the classical linear model is meaningless in the case of a dichotomous dependent variable because, in this circumstance, variance is equal to π*(1-π).
• The hypothesis testing of regression parameters is based on the assumptions that errors in prediction of the dependent variables are distributed similarly to normal curves. But, when the dependent variable only assumes values equal to zero or one, this assumption does not hold. The logistic regression does not assume constant variance and the errors are not normally distributed