3.7 MEASURING CREDIT RISK FOR CREDIT PORTFOLIOS
3.7.1 Economic Capital Allocation .1 Probability Density Function of Credit Losses
Sophisticated financial organizations use an analytical setup that relates the overall required economic capital of the institute to its credit activities.
The required economic capital is linked to the portfolio’s credit risk with the probability density function(PDF) of credit losses,a key result from the credit risk model.10An important element is the probability that credit losses will exceed a given amount, which is represented by the area under the density function to the right of the given limit (the upper limit covering expected credit losses). A portfolio with a higher risk of credit loss is one whose den- sity under the function has a relatively long and fat tail (see Figure 3-6).
The expected credit lossis defined as the total amount of credit losses the bank would expect from its credit portfolio over a specific time hori- zon (leftmost vertical line in Figure 3-6). The expected loss is one of the costs of transacting business that gives rise to credit risk. For marketing and other reasons, financial institutes prefer to express the risk of the credit portfolio by the unexpected credit loss,i.e., the amount by which the actual losses exceed the expected loss. The unexpected loss is a measure of the uncertainty or variability of actual losses versus expected losses. The estimated economic capital needed to support the institution’s credit port- folio activities is generally referred to as its required economic capital for credit risk. A cushion of economic capital is required for loss absorption,
Allocated economic capital Expected credit loss
99% confidence level 1% loss
Extreme credit losses — tail events
Probability
Covered by adequate pricing and loss provisions
Covered by loss provisions and/or proprietary capital
Covered by loss provisions and/or proprietary capital quantified using scenario analysis and controlled with concentration limits
F I G U R E 3-6
Probability Density Function (PDF) of Credit Losses.
because the actual level of credit losses suffered in any one period could be significantly higher than the expected level. The procedure for calcu- lating this amount is similar to the VaR methods used in allocating eco- nomic capital for market risks. The understanding is that economic capital for credit risk is determined in such a way that the estimated probability of unexpected credit loss exhausting economic capital is less than some target insolvency rate.
The capital allocation systems generally assume that it is the role of reserve funding to cover expected credit losses, while it is the role of eco- nomic capital to cover unexpected credit losses. The required economic capital represents the additional amount of capital necessary to achieve the target insolvency rate, over and above that needed for coverage of ex- pected losses (the distance between the two lines in Figure 3-6).
3.7.1.2 Measuring Credit Loss
Loss in a credit portfolio is defined by three variables: the difference be- tween the portfolio’s current valueand its future valueat the end of a certain time horizon. Because many functions depend on the time dimension in one way or another, the time horizon is a key input variable for the defini- tion of risk. There are basically two ways of selecting the time horizon over which credit risk is monitored:
• The first approach is the application of a standardized time period across all contracts (instruments). Most financial institutions adopt a one-year time horizon across all asset classes (not only credit instruments).
• The second is the liquidation periodapproach, in which each credit contract (instrument) is associated with a unique time period, coinciding with the instrument’s maturity or with the time needed for liquidation. Some vendor systems allow specific time periods for asset classes or portfolios (for structured credit portfolios). The hold-to-maturityapproach seems to be applicable if the exposures were intended to be held to maturity and/or liquid markets for trading these instruments are limited.
The factors influencing the fixing of the period are as follows:
• Availability or publication of default information
• Availability of borrower information
• Internal control rhythms and renewal processes
• Capital planning
• Account statement preparations; changes to the capital structure (capital increase or reduction).
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3.7.1.3 Default-Mode Paradigm
The default-mode (DM) paradigm states that a credit loss occurs only if a borrower defaults on a repayment obligation within the planning horizon.
As long as the default event does not become a fact, no credit loss would exist. In the event that the borrower defaults on obligations, the credit loss is calculated as the difference between the bank’s credit exposure (the amount due to the bank at the moment of default) and the present value of future net recoveries (discounted cash flows from the borrower less workout expenses). The current and the future values of the credit are de- fined in the default-mode paradigm based on the underlying two-state (default versus nondefault) notion of the credit loss. The current valueis typically calculated as the bank’s credit exposure (e.g., book value). The future valueof the loan is uncertain. It would depend on whether the bor- rower defaults during the defined time horizon. In the case of nondefault, the credit’s future value is calculated as the bank’s credit engagement at the end of the time horizon (adjusted so as to add back any principal pay- ments made over the time horizon).
In the case of a default, the future value of the credit is calculated as the credit minus loss rate given default(LGD). The higher the recovery rate following default, the lower the LGD. Using a credit risk model, the cur- rent value of the credit instrument is assumed to be known, but the future value is uncertain. Applying a default-mode credit risk model for each in- dividual credit contract (e.g., loan versus commitment versus counter- party risk), the financial institution must define or estimate the joint probability distribution between all credit contracts with respect to three types of random variables:
• The bank’s associated credit exposure
• A binary zero/one indicator denoting whether the credit contract defaults during the defined time horizon
• In the event of default, the associated LGD 3.7.1.4 Mark-to-Market Paradigm
The mark-to-market (MTM) paradigm treats all credit contracts under the assumption that a credit loss can arise over time, deteriorating the asset’s credit quality before the end of the planned time horizon. The mark-to- market paradigm treats all credit contracts as instruments of a portfolio being marked to market (or, more accurately, marked to model) at the be- ginning and end of the defined time horizon. The credit loss reflects the difference of the valuation at the beginning and at the end of the time hori- zon. This approach considers changes in the asset’s creditworthiness, re- flecting events that occur before the end of the time horizon. These models must incorporate the probabilities of credit rating migrations (through the rating transition matrix), reflecting the changes in creditworthiness.
For each credit position in the portfolio, migration paths have to be calculated, using the rating transition matrix and Monte Carlo simulation.
For all positions, the simulated migration (including the risk premium as- sociated with the contract’s rating grade at the end of the rating period) is used to mark the position to market (or, more accurately, to model) as of the end of the time horizon. For the purpose of estimating the current and future values of the credit contracts, two approaches can be used:
• The discounted contractual cash flow (DCCF) approach
• The risk-neutral valuation (RNV) approach
3.7.1.5 Discounted Contractual Cash Flow Approach The discounted contractual cash flow(DCCF) approach is commonly associ- ated with J. P. Morgan’s CreditMetrics methodology. The current value of a nondefaulted loan is calculated as the present discounted value of its fu- ture contractual cash flows. For a credit contract with an assigned risk rat- ing (i.e., BBB), the credit spreads used for the calculation of the discounted cash flows of the credit contract equal the market-determined term struc- ture of credit spreads corresponding to similar corporate bonds (maturity, coupon, sinking-fund provision, etc.) with that same rating. The current value is treated as known, because the future value of the credit depends on the (uncertain) end-of-period rating and the market-determined term structure of credit spreads associated with the specific rating. The future value of the credit is subject to changes in migration (creditworthiness) or in the credit spreads according to the market-determined term structure.
In the event of default, the future value of a credit would be given by its recovery value, calculated as the credit minus loss rate given default (a similar approach to that used in the default-mode approach).
The DCCF approach is a practical approach, but it is not completely in line with modern finance theory. For all contracts with the same rating, the same discount rates are assigned. Thus, for all contracts not defaulted within the defined time horizon, the future value does not depend on the expected loss rate given default, as they are not defaulted. Modern finance theory holds that the value of an asset (borrower’s asset) depends on the correlation of its return with that of the market. Borrowers in different market segments, exposed to different business cycles and other risk factors, will still be as- signed to the same risk grade according to the DCCF approach.
3.7.1.6 Risk-Neutral Valuation Approach
To avoid the pitfalls of the discounted contractual cash flow approach, Robert Merton developed a structural approach imposing a model of firm value and bankruptcy.11A company defaults when the value of its under- lying assets falls beneath the level required to serve its debt. The risk- neutral valuation (RNV) approach discounts contingent payments instead
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of discounting contractual payments. A credit can be considered as a set of derivative contracts on the underlying value of the borrower’s assets. If a payment (interest rate, amortization, etc.) is contractually due at date t,the payment actually received by the lender will be the contractual amount only if the firm has not defaulted by date t.The lender receives a portion of the credit’s face amount equal to the credit minus loss rate given default (an approach similar to that used in the default-mode method) if the bor- rower defaults at date t,and the lender receives nothing at date tif the bor- rower has defaulted prior to date t.The value of the credit equals the sum of the present values of these derivative contracts (each payment obliga- tion at time tis regarded as an option). The difference from the discount rates used for the discounted contractual cash flow approach is that the dis- count rate applied to the contracts’ contingent cash flows is determined using a risk-free term structure of interest rates and the risk-neutral pricing measure. Similar to the option-adjusted spread approach, the risk-neutral pricing measure can be regarded as an adjustment to the probabilities of borrower default at each horizon t,which incorporates the market risk pre- mium associated with the borrower’s default risk. The magnitude of the adjustment depends on the expected return and volatility of the bor- rower’s asset value. Returns modeled consistently with the capital asset pricing model (CAPM) can be expressed in terms of the market expected return and the firm’s correlation beta (β) with the market. This approach combines pricing of the credits with the respective credit losses:
• The expected default frequency and the loss rate given default by the borrower
• The correlation between the borrower’s risk and the systematic (market) risk
This is consistent with modern finance theory.
The interpretation of default is key in understanding the risk-neutral valuation approach. A credit is considered to be in default once it migrates to a predefined limit. This worst-case scenario is not clearly defined; it varies according to the institution’s risk appetite and risk capacity, thus af- fecting measures of default, migration, credit loss, and the probability density function.