Validation Mythology of Maturity Adjustment Formula for Basel II Capital Requirement

Validation Mythology of Maturity Adjustment Formula for Basel II Capital Requirement Working paper Version 9..9 JRMV 8 8 6 DP.R Authors: Dmitry Petrov Lomonosov Moscow State University (Moscow, Russia) Senior analyst, EGAR Technology Inc. e-mail: dapetroff@gmail.com Michael Pomazanov PhD Associate Professor, State University Higher School of Economics (Moscow, Russia) Vice-Director of Credit Risk Management, Bank Zenit e-mail: m.pomazanov@zenit.ru

Abstract In recent years a considerable progress is observed in the development of credit risk models. Revised Framework on International Convergence of Capital Measurement and Capital Standards (4) (Basel II) raised the standards of risk management on the new level. The validation methodologies of internal rating based (IRB) systems have emerged as an important issue of the implementation of Basel II. One of the most actual and less examined problems is the theoretical investigation of maturity effects and probability of default time structure. Basel Committee recommendations include maturity adjustment for capital requirement. However the complete derivation of proposed adjustment formula rests undisclosed. In this paper authors describe a method of maturity adjustment calculation directly from open data published by rating ageneses. In addition analytical expressions revealing probability of default time structure are proposed. In order to validate Basel II recommendation the comparison of received result with Basel maturity adjustment formula is performed. The character of presented dependences is close enough, but it was discovered that for low probabilities of default (for high ratings) and maturities of -3 year there may exist considerable underestimation of risk capital. Key words: Maturity adjustment, Capital Requirement, Basel II, Probability of default, PD time structure.

The unknown credit losses the bank will suffer can be represented by two components: expected loss (EL) and unexpected loss (UL). While a bank can forecast the average level of EL and manage them, UL are peak losses that exceed expected levels. Economical capital is needed to cover the risks of such peak losses, and therefore it has a loss-absorbing function. In June 4, the Basel Committee issued the first version of Revised Framework on International Convergence of Capital Measurement and Capital Standards (or Basel II) (see Basel Committee on Banking Supervision, 4). In this document Basel II Internal Ratings- Based (IRB) approach was introduced. This approach is built on the following main parameters of credit risk: Probability of Default (PD), Loss Given Default (LGD), Exposure at Default (EAD) and effective Maturity (M). Under advanced IRB (AIRB) approach, institutions are allowed to use their own internal models for these base risk parameters as primary inputs to the capital requirement (CAR) calculation. Banks generally employ a one-year planning horizon. The majority of well known portfolio models (CreditPortfolioView, CreditRisk+, CreditPortfolioManager, Credit Metrics, etc.) as well as Basel II agree in fact, that the value of the credit portfolio is only observed with respect to a predefined time horizon (typically one year). In fact this time horizon generally does not correspond with the actual maturity of the loans in credit portfolio. It is obvious that long term credits are riskier. With respect to a three-year term loan, for example, the account of oneyear horizon could mean that more than two-thirds of the credit risk is potentially ignored. So maturity is one of the most important parameters of risk. As a consequence, it is necessary to account real risk horizon to estimate precisely multi-year probability of default, unexpected loss and consequently sufficient capital requirement. The topic of maturity effects is rather popular and widely discussed in recent literature. Number of authors worked on multi-horizon economic capital allocation on basis of Mark to Market (MTM) paradigm (see Kalkbrener and Overbeck,, Barco, 4, Grundke, 3). Under these models changes in portfolio value are caused by changes in credit spreads which in their turn strongly depend on credit rating migration. Though the Markov assumption for PD time dependence is not proved there are a lot of works on Markov chains application for maturity effects (see, for example, Jarrow et al., 997, Inamura, 6, Frydman & Schuermann, 5). Bluhm and Overbeck (7) don t rejected Markov assumption but adopts it by dropping the homogeneity assumption with Non-Homogeneity Continuous-Time Markov Chains (NHCTMCs). For models based on DM paradigm there exists few literature analyzing account of long risk horizons (Gurtler and Heithecker (5)). The Basel Risk Weight Functions used for the calculation of supervisory capital charges are based on a specific one-factor model adopted by the Basel Committee on Banking Supervision (5). The results of Merton (974) and Vasicek () lie in the bottom of this model. To account maturity effect Basel Committee proposes special maturity adjustment formula (Basel Committee on Banking Supervision, 5). However there is no available detailed explication and initial data used for its deviation. Thereby the following process of validation on basis of open data is considered:. Merton-type one-factor model of Vasicek is considered. Capital requirement formula is adjusted to account maturities longer than a year.. Cumulative default rates published by rating ageneses are analyzed. 3. Special functions are proposed to approximate continuously cumulative default rates. 4. Appropriate maturity adjustment is calculated. 5. Received maturity adjustment is compared with Basel maturity adjustment. 3

Capital Requirement Calculation We briefly consider the derivation of capital requirement formula. It should be noted that in contrast to Basel II one-time horizon is not fixed. Voluntary maturity T is considered. It is assumed that a loan defaults if the value of the borrower's assets at the loan maturity T falls below the contractual value B of its obligations payable. Let A be the value of borrower s assets, described by the Wiener process da Adt Adx. Here asset value at T can be represented as log AT log A T T T X where X is a standard normal variable. The probability of default on risk horizon T (PD T ) equals then the probability that assets fall below the level of borrower s obligations PDT P A T B P X c N c where log c B log A T T T and N( ) is a cumulative normal distribution function. The variable X is standard normal, and can therefore be represented as X Y Z where Y, Z are mutually independent standard normal variables. The variable Y can be interpreted as a common factor, such as an economic index, over the interval (, T). Then ρ represents correlation of a borrower with the state of the economy. The term Y is the company s exposure to the common factor and the term Zi represents the company s specific risk. The probability of default is evaluated as the expectation over the common factor Y. When it is fixed, the conditional probability of default is pd Y P A T B Y PX c Y P Y Z c Y cy N PDT Y N PDT Y PZ PZ N. For the worst economical scenario the common factor takes the magnitude given by N with some confidence level α (α=.999 under Basel II). Then the worst conditional probability of default is pd N PDT N N. 4

Under this worst scenario the losses will be also the most serious. The capital requirement for a loan is then given by Capital Requirement PD,,, EAD, LGD Worst Loss- Expected Loss = T = EAD LGD pd EAD LGD PD N PD T N EAD LGD N PDT EAD LGD FDaR T,, PD, N PDT N where FDaR T,, N. Thus given the probability of default on time horizon T capital requirement could be calculated on the same time horizon. Figure illustrates the dependence of capital requirement on probability of default for a maturity of one year. For simplicity EAD and LGD equals. [Figure about here] T T () Basel Maturity Adjustment Basel II capital requirement formula includes a component responsible for maturity (Basel maturity adjustment). It is noted that this adjustment follows from the regression of the output of the KMV Portfolio Manager TM.. By its sense this adjustment is a penalty for the exceeding of one year maturity. Dependence on maturity is linear, changes for risk horizon from to 5 years and has the following from: T.5bPD Basel Maturity Adjustment, ().5b PD.85.5478log PD b PD where PD is one-year probability of default. Figure a illustrates the dependence of Basel maturity adjustment on one-year probability of default. Maturity is fixed to 3 years. The maturity effect is stronger for high PDs than for low PDs. Figure b illustrates how Basel maturity adjustment formula changes with maturity. Oneyear probabilities of default equal.%, % and %. The adjustment is linear and increasing with the maturity. [Figures a and b about here] PD time structure From cumulative default rates published by major rating agencies, such as Fitch Ratings (6), Moody s (6), Standard & Poor s (7) directly follows that probability of default increases with the increase of risk horizon (Table, Figure 4). So we need to perform an adjustment in one-year PD if we want to take into account maturities longer than a year. 5

Consequently, adjustment in capital requirement is necessary when the one year time horizon is exceeded. It should be noted that similar results can be received on basis of data provided by every of rating agencies mentioned earlier. However Moody s statistical data is mainly considered in this paper (see Table ). 997): [Table and Figure 3 about here] There are some potential errors in this data (see Credit Metrics TM Technical Document, Output cumulative default likelihoods violate proper rank order. For instance, presented table shows that AAAs have defaulted more often at the -year horizon than have AAs. This is true also for B and Ba3 ratings. Limited historical observation yields granularity in estimates. For instance, the AAA row in the Table is supported by limited firm-years worth of observation. In 997 it was only,658 firm-years. This is enough to yield a resolution of.6% (i.e., only probabilities in increments of.6% or /658 are possible). This lack of resolution may erroneously suggest that some probabilities are identically zero. For instance, if there were truly a.% chance of AAA default, then we would have to watch about for another 8 years before there would be a 5% chance of tabulating a non-zero AAA default probability. In spite of these slight errors we suppose that presented statistical data reflects well the time structure of probability of default except, probably, several first ratings for the reasons mentioned earlier. Firstly Moody s cumulative probabilities are fitted with a special parametric function for every rating: PDT F PDn, a, b, T PDn - exp -T a - exp -T a - exp -T b exp b - exp -a - exp -a - exp -b b Fitting function depends on three parameters PDn, a and b different for every rating. The actual form of this function is chosen to satisfy several essential properties: For the maturity of one year parameter PDn is equivalent to one year probability of default taken in percents For zero maturity PDn equals zero The function has an asymptotic for large terms which is not equivalent to %. This property follows from the notion that with time companies either default rather fast or attain higher ratings. So with time we have some kind of stabilization. The property is satisfied when parameter a is greater than b for every rating. The function has a change in convexity (for low probabilities of default we have the concavity, for high - the salience). This property follows from the notion that companies with high rating pass several lower ratings before default. So there exist some initial period where cumulative probability of default doesn t grow very fast (concavity). Companies with low ratings can come to default rather fast so we can t observe such effect and cumulative probability of default grows immediately (salience). For the convince, numerical ratings corresponding to alphanumeric ratings are introduced. The highest rating Aaa corresponds to first numeric rating, Aa - to the second, etc. (3) 6

Of cause, proposed function is not unique, but it shows very good fitting results (see Table and Figure 4). [Figure 4 about here] In the Table the set of received data is presented for every rating: three parameters of fitting function, one-year probabilities. R-square shows that proposed function precisely takes into account particularities of used data. [Table about here] Heretofore we used probabilities of default which correspond to discrete ratings. But PD is continuous by its nature. So it is necessary to pass from discrete ratings (and corresponding oneyear PDs) to continuous default probabilities. To do that we smooth the PDn parameter, which corresponds to one-year probability of default. Generally accepted logit function is used. Linear dependence is established between numeric ratings and natural logarithm of PDn (see Figure 5). Quality of this approximation is rather high: R-square equals.974. PDA Numeric Rating exp.56 Numeric Rating 6.37, (4) where PDA is the continuous approximation for the PDn. [Figure 5 about here] To receive continuous dependency of cumulative default probabilities from one-year PD and maturity we also need to smooth other two parameter (a and b). After the analysis of dependence of parameters on PDA the following two fitting functions were proposed: f depends on two parameters ( and ): a a a f ( x) exp x, a a a f depends on three parameters (, b b b, b ): f ( x) exp x. b b b b Approximation of parameters a and b gives the following results: a PD.8 exp.639 ln( PD ) (5) b PD.78 exp.93 ln PD.938. (6) Constraint on a and b ( a have to be grater then b ) is fulfilled. From (3), (5) and (6) follows the formula which gives probability of default (PD T ) for every one-year default probability (PD) and maturity (T in years): PD F PD, a PD, b PD, T. (7) T [Figure 6 about here] Maturity Adjustment 7

Now, when the dependence of probability of default PD T for every maturity is known we can construct maturity adjustment for capital requirement in a following way: Capital Requerement PDT,,, EAD, LGD Maturity Adjustment PD, T = Capital Requerement PD,,, EAD, LGD where PD T is calculated from (7). FDaR T,, PD FDaR,, PD [Figure 7 about here] In addition, maturity adjustment dependence on model parameters was analyzed. Maturity adjustment does not change strongly with the change of coefficient (see Figure 7). This fact confirms the absence of dependence on correlation in received adjustment the same as in Basel maturity adjustment. Dependence of maturity adjustment on confidence level is rather strong, particularly for low probabilities of default (see Figure 8). The level of adjustment declines with the convergence of to. The confidence level used for Basel maturity adjustment derivation rests undisclosed as well. However under Basel Committee recommendation we work with high confidence levels (.999 or even.9999 ). [Figure 8 about here] Figure 9 illustrates received maturity adjustment and Basel maturity adjustment for several maturities, so it is possible to compare them. [Figure 9 about here] Though the form of maturity adjustments is close enough, there is a difference in Basel proposal and our results (see Figure 9). Received adjustment is higher for small probabilities of default (high ratings) and for maturities about, 3 years. It also reduces faster with the increase of one-year PD. Higher level of adjustment for high ratings is partly explainable and follows from the dependence of capital requirement on default probability (see Figure ). For small probabilities the slope of the curve is grater then for large probabilities, so the same change in probability with time gives the greater change in capital requirement. But at the moment there is no complete explanation of difference between these two adjustments. If more exact information about methodology and data used for Basel maturity adjustment would be available it seems to us to be possible to explain disagreements. T Conclusion In this article the dependence of default probability on time was continuously parameterized using data provided by Moody s. This approach gives results expressed analytically. Results correspond well with statistical data. Time structure of PD allows to calculate maturity adjustment (or penalty for excess of one year maturity) for capital requirement. Proposed approach of validation makes clearer the process of maturity adjustment calculation. 8

It was shown that the character of Basel approach maturity adjustment function can be explained rather well from open statistical data. However from received results follows that there exist possible underestimate of risk fixed by Basel maturity adjustment function. It is shown that penalty is higher for assets with good rating (investment grade) and maturities about years. The precise estimation of unexpected loss is critical for banks stability. Though Basel II recommendations are often regarded as rather conservative the possible underestimate of risk may be up to 5%. 9

Reference:. Barco M. (4), Bringing Credit Portfolio Modeling to Maturity, Risk 7(), pages 86-9.. Basel Committee on Banking Supervision (4), International Convergence of Capital Measurement and Capital Standards, Bank for International Settlements, June. 3. Basel Committee on Banking Supervision (5), An Explanatory Note on the Basel II IRB Risk Weight Functions, Bank for International Settlements, July. 4. Bluhm C., Overbeck L. (7), Calibration of PD term structures: to be Markov or not to be, Risk (), pages 98-3. 5. Credit Metrics TM Technical Document (997), J.P. Morgan & Co. Incorporated, April. 6. Fitch Ratings (6), Fitch Ratings Global Corporate Finance 99 5 Transition and Default Study, Fitch Ratings Corporate Finance Credit Market Research, August. 7. Frydman H., Schuermann T. (5), Credit Ratings Dynamics and Markov Mixture Models, Working Paper, Wharton Financial Institutions. 8. Gordy M. B. (3), A risk-factor model foundation for ratings-based bank capital rules. Journal of Financial Intermediation, 99 3. 9. Grundke P. (3), The Term Structure of Credit Spreads as a Determinant of the Maturity Effect on Credit Risk Capital, Finance Letters (6), S. 4-9.. Gurtler M., Heithecker D. (5), Multi-Period Defaults and Maturity Effects on Economic Capital in a Ratings-Based Default-Mode Model, Finanz Wirtschaft Working Paper Series FW9V/5, Braunschweig University of Technology Institute for Economics and Business Administration Department of Finance.. Inamura Y. (6), Estimating Continuous Time Transition Matrices from Discretely Observed Data, Bank of Japan Working Paper Series 6, E7, April.. Jarrow R., Lando D., Turnbull S. (997), A Marcov Model fro the Trem Structure of Credit Risk Spreads, Review of Financial Studies, pages 48-53. 3. Kalkbrener M., and Overbeck L. (), The Maturity Effect on Credit Risk Capital, Risk 4(7), pages 59-63. 4. Merton R.C. (974), On the Pricing of Corporate Debt: the Risk Structure of Interest Rates, Journal of Finance 9, pages 449-47. 5. Moody s (6), Default and Recovery Rates of Corporate Bond Issuers 9-5, Moody s Investor Service Global Credit Research, January. 6. Standard & Poor s (7), Annual 6 Global Corporate Default Study and Ratings Transitions, S&P Global Fixed Income Research, January. 7. Vasicek O. (), Loan portfolio value. RISK, December, 6 6.

Table. Example of average cumulative corporate defaults rates for several ratings/years Rating Year Year Year 3 Year 4 Year 6 Year 7 Year 8 Year 9 Year Aaa....39.8.8.8.8.8 Aa.....94.94.94.94.94 Aa...48..97.77.44.684.7 B 3.3 8.53 3.573 7.635 3.6 3.6 3.6 3.6 3.6 B 5.457.67 7.4.57 9.598 9.68 9.756 9.756 9.756 B3.46 8.653 5.49 9.887 38.964 38.964 38.985 38.985 38.985 Caa-C.98 3.74 36.5 39.5 43.36 43.36 43.36 43.36 43.36 Note: see Moody's (6), Exhibit 36

Table. Fitting results Alphanumeric Rating Numeric Rating PD oneyear, % PDn a b R Aaa..,364,44.95 Aa..,,9.899 Aa 3..7,,6.975 Aa3 4.9.,9,3.983 A 5.3.33,4,6.97 A 6.6.,,4.965 A3 7.37.63,,6.95 Baa 8.66.3,,.978 Baa 9.6.57,4,5.998 Baa3.335.538,3,584.993 Ba.753.7,84,6.996 Ba.78.47,,74.995 Ba3 3.69 4.7,6,76.986 B 4 3.3 5.98,9,864.978 B 5 5.457 7.35,97,5.99 B3 6.46.43,355,6.996 Caa-C 7.98 9.97,69,69.998

Basel capital requirement.4.35.3.5..5..5 3 4 5 6 One-year probability of default, % Figure. Dependence of Basel capital requirement on one-year probability of default. 3

Basel maturity adjustment Basel Maturity adjustment.8.7.6.5.4.3.. 5 5 One-year probability of deafult, % Figure a. Dependence of Basel maturity adjustment on one-year probability of default..5 PD =.% PD =.% PD =.%.5.5 3 4 5 Maturity, years Figure b. Dependence of Basel maturity adjustment on maturity. 4

Probability of default, % 45 4 35 Caa-C 3 B 5 5 Ba 5 Baa A Aa Aaa 5 5 Moody's Maturity, years Figure 3. Average cumulative issue-weighted corporate default rates (Moody s data). 5

Cmulative probability of deafult, % Cmulative probability of deafult, % Cmulative probability of deafult, %.5 Ratings from Aaa to A Ratings from A to Baa3 8.5 6 4.5 5 5 Maturity, year 5 5 Maturity, year Ratings from Ba to C 4 35 3 5 5 5 5 5 Maturity, year Figure 4. Results of cumulative default rates fitting (solid curves) and Moody s data (dots). 6

ln(pdn); ln(pda) 4 3 - - -3-4 -5-6 4 6 8 4 6 8 Numeric rating Figure 5. Transition from rating grades to continuous one-year probability of default. 7

Figure 6. Smoothed cumulative probabilities of default (surface) compared with Moody s data (dots). 8

Maturity Adjustment.8.6.4. PD =.% PD = % PD = %.8.6.4...4.6.8...4.6 Figure 7. Dependence of maturity adjustment on correlation coefficient ρ. 9

Maturity adjustment 3. 3.8 PD =.% PD =.% PD =.%.6.4..8.6.4.99.99.994.996.998 Confidence level, Figure 8. Dependence of maturity adjustment on confidence level α.

Maturity adjustment Maturity adjustment Maturity adjustment Maturity adjustment.5 Maturity = years 3.5 Maturity = 3 years 3.5.5.5.5.5 5 5 One-year probability of default, % 5 5 One-year probability of default, % 5 Maturity = 4 years 6 Maturity = 5 years 4 5 3 4 3 5 5 One-year probability of default, % 5 5 One-year probability of default, % Figure 9. Received maturity adjustment (solid curves) and Basel maturity adjustment (dashed curves).