Why Simple, When it Can Be Difficult?

Bank, Matthias; Lawrenz, Jochen Article Why Simple, When it Can Be Difficult? Kredit und Kapital Provided in Cooperation with: Duncker & Humblot, Berlin Suggested Citation: Bank, Matthias; Lawrenz, Jochen (2003) : Why Simple, When it Can Be Difficult?, Kredit und Kapital, ISSN 1865-5734, Duncker & Humblot, Berlin, Vol. 36, Iss. 4, pp. 534-556, https://doi.org/10.3790/ccm.36.4.534 This Version is available at: https://hdl.handle.net/10419/293492 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Kredit und Kapital, 36. Jahrgang, Heft 4 Seiten 534-556 Why Simple, When it Can Be Difficult? Some Remarks on the Basel IRB Approach Matthias Bank and Jochen Lawrenz, Innsbruck I. Introduction In the focus of the current discussion about the New Basel Capital Accord is the calibration of the risk weight function for the credit risk measurement. From the Quantitative Impact Study it turned out that the initial proposed risk weight function will result in an average capital requirement well above the current 8%, this caused a kind of a storm of protest among market participants and especially among politicians. The committee reacted by proposing potential modifications to the risk weight function, intended to mitigate the average capital burden, which are now part of the recently published third Consultative Document. But, while altering the parameters, the approach in general has been left unchanged. The IRB approach and in particular the IRB risk weight function can be considered as one of the major innovations in Basel II. The calculation of risk weights depends directly on the probability of default of a given obligor, and the connection between PD and corresponding risk weights is given by a continuous function. It is obvious, that this risk weight function is one of the crucial building blocks in the IRB approach and in the overall Basel II proposal.1 The shape, i.e. the graph of the risk weight function is thereby the decisive factor that determines the economy-wide level of regulatory capital that will be required in future. Not at least for this reason, the weighting factors have been vividly disputed in recent discussions.2 The parties involved not 1 Recall that it is intended, to induce the majority of banks to use the IRB approach. 2 See for example the compilation of comments on www.bis.org/bcbs/cacom ments.htm. Kredit und Kapital 4/2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 Why Simple, When it Can Be Difficult? 535 only included academics and bankers, but even politicians felt that they had to comment on the proposals. In our opinion, it is a necessary prerequisite for a sound discussion to get a clear picture of the basic underlying interrelations, and the aim of the following section is to provide a detailed treatment of the IRB risk weight function and to disclose its basic assumptions. Further on, we will propose a much easier approach to model the risk weight function. The paper is organized as follows: In the next section the methodical shift in capital rules will be described. Section III. will provide a detailed derivation of the IRB risk weight function. Section IV. is going to discuss the basic assumptions, and section V. will describe the process as being basically determined by political goals. Section VI. is going to propose a radically simpler approach. We will finish with a conclusion that sums up the main points. n. Risk-bucketing Approach and Portfolio Models As has been already mentioned, the IRB approach can be considered as a conceptually new approach to regulatory capital rules. The current rules (Basel I, 1988), as well as the proposed standard approach, follow the so-called risk-bucketing approach, whereby the assets of a bank with similar risk characteristics are bundled together in risk-buckets, and are charged with a uniform capital requirement ratio. It is assumed that the single assets in a bucket have the same risk, that is risk-buckets are considered as being homogeneous. The decisive characteristic of the risk-bucketing approach is the socalled portfolio-invariance. The level of the marginal requirement for single assets does only depend on its specific characteristics and not on the characteristics of the portfolio in which it is embedded. This implies in particular that no diversification effects are taken into account. It need not be stressed that this is a major drawback for modelling credit risk. Nevertheless, the approach has some important advantages: The administrative burden is relatively low, since the capital requirement of the overall portfolio is simply the weighted sum of the marginal requirements of the single assets. Related to this, risk-bucketing systems do not impose complex reporting requirements. In particular for small and medium sized banks, the simplicity of a risk-bucketing approach is the major argument. Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 536 Matthias Bank and Jochen Lawrenz On the other hand, sophisticated credit risk models are available3, which are not only able to measure risk more accurately, but can account explicitly for portfolio effects. The committee did recognize the need to apply portfolio models, but also realized that full-blown portfolio models are currently too complex, given the limitations on the part of the majority of banks. A solution for this dilemma is provided by the results due to Gordy.4 Gordy showed that under certain assumptions, portfolio models behave like risk-bucketing models. That is, the marginal requirements for single assets do only depend on their specific characteristics. Therefore, it is possible to retain on the one hand the practical simplicity of the risk-bucketing approach, while on the other hand diversification effects can be captured. From a regulatory point of view, Gordy seems to have cut the Gordian knot. But there is a price you have to pay for it: The result is subject to mainly two assumptions. These are: 1. There is only a single systematic factor driving correlations across obligors. 2. Portfolios are asymptotically fine-grained, i.e. no single exposure accounts for a substantial share of total exposure. We will give an assessment of the assumptions in a later section, but first, we will turn to an analysis of this simplified portfolio-model approach. III. Derivation of the Risk Weight Function In the following, we will restrain ourselves to the risk weight function for corporate exposures. The risk weight function (RWCd3) according to the third Consultative Document5 has the following structure: (1) RWCD3=LGD N(^-^^-+• MA • 12.5 where N(-) denotes the cumulative distribution function of the standard normal distribution and iV_1( ) denotes its inverse. LGD denotes the loss 3 E.g. CreditMetrics, CreditRisk+, CreditPortfolioView, KMV. 4 Gordy (2000), Gordy (2001). 5 Basel (2003). Kredit und Kapital 4/2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 Why Simple, When it Can Be Difficult? 0.241 0.22 537 pc(pd) 0.17 - 0 0.025 0.05 0.075 0.1 0.12 0.15 pd Fig. 1: Correlation for corporate exposure given default, MA stands for a maturity adjustment factor to account for different effective maturities (m) and is determined by the equation: (2) MA(m PD) = 1 + {m ~ 2 5) ' (Q Q8451 ~ 0 05898 • lQë(PD))2 U ^ ; 1 - 1.5 (0.08451 - 0.05898 log(PD))2 and p is a correlation parameter, determined by the equation: (1 — e~50 'PD) \ (3) / 1 _ e-50 PD\ / p(PD) = 0.12^ l e_5Q j+0.24^1p is specified as a decreasing function of PD and ranges from a minimum of 0.12 to a maximum of 0.24.6 Note by the way, that e"50 = 1.9.10-22, so that, for all practical purposes, the denominator in (3) equals 1, and you can simply write: p(PD) = 0.12 + 0.12 • e~™PD. Figure 1 shows the graph of the correlation as a function of PD. The formula (RWCD3) differs from the formula put forward in the second Consultative Document in some aspects, but from a conceptual point of view, both have the same structure. The CD2-formula reads as follows: (4) RWCD2 = 0.02 LGD - N(l.11S N^iPD) + 1.288) M976.5 6 Or more precisely: 0.12 < p < 0.2382, since min(PD) = 0.0003 is given. Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 538 Matthias Bank and Jochen Lawrenz In comparison to the CD2-formula, the fixed coefficients of 1.118 and 1.288 have been replaced by a term, involving the correlation parameter PThe second important modification concerns the scaling factor of 976.5, which does not appear in the CD3-formula. Instead, the implicit level of confidence has increased from 99.5% to 99.9%. (The confidence level is reflected in the term N'1 (0.999) in the CD3-formula.) The LGD factor remains unchanged and is still assumed to be 0.45 in the IRB basic approach. The factor 12.5 in the CD3-formula only serves as to transform the capital requirements into risk weights. If RWCD3 is multiplied by the solvency ratio of 8 % this factor cancels out. So far, those are the main formulas, as they have been presented in the Basel II proposals. But besides some comments made in the supporting document7, most of the theoretical background, from which the formulas have been derived, remains unexplained. One may argue that the majority of readers might have a more userorientated approach and are not very much interested in the theoretical foundations of the risk weight function. But, we think that more transparency would increase the acceptance of the approach and would enable a well-founded discussion. 1. Portfolio Model Underlying the IRB Approach The portfolio model, which underlies the committees proposal, is mainly based on results due to Gordy8 and has the following basic assumption: A risk index for a given obligor i is assumed, which is composed of two components. On the one hand, it is driven by the idiosyncratic risk of the corresponding obligor and on the other hand, it is driven by a systematic risk factor, which underlies every obligor. Let Z* denote the risk index for obligor i, X the systematic risk, and e* the idiosyncratic risk, then Zi is specified according to: (5) Zi = Wi • X + yjI - w\ • €i i = 1, - • ,n 7 Basel 2001b, section 172. 8 See e.g. Gordy (2000), (2001), see also Finger (1999, 2001) and Vasicek (1991) for related work. Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 Why Simple, When it Can Be Difficult? 539 where Wi can be thought of as the weight by which the risk index of obligor i is driven by the systematic risk factor X. It can be assumed that every obligor has the same weight, i.e. Wi = Wj, so that we can drop the subscript in what follows. Further, it is assumed that both X as well as e* are standard normally distributed random variables and that they are mutually independent, i.e.: It is also important to understand why X and e* are weighted with Wi and yj 1 — wf respectively, since these terms eventually show up in the risk weight function. The expectation in the present context is zero, while Var(wX) = w2Var(X) = to2 and Var(V 1 - w2 • e*) = 1 - w2 and thus, we have: (7) E(Zi) = 0 Var(Zi) = 1 which means that the weights are chosen such that Z; is again standard normally distributed. If we further assume that the portfolio consists of N loans, each having an exposure of 1/N, a loss given default of LGD = 1 and each having a probability of default of p, then obligor i defaults, if (8) Zi<a where a = N~1(p). For a given realization of X = x (i. e. for the conditional probability p( |X)), the condition of default is given by the inequality: Since €i is by definition standard normally distributed, the conditional probability that obligor i will default is given by: (6) E(X,ei) = 0, E(euej) = 0, a - w • x €i< VT^ (10) Kredit und Kapital 4/2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 540 Matthias Bank and Jochen Lawrenz The crucial point is that once we have conditioned on the common factor X, the individual obligor defaults are driven only by the idiosyncratic terms and are therefore independent. This independence in turn is eventually the justification to apply risk bucketing rules. Next, we can ask who accounts for the variance in the portfolio? Is it largely driven by the variance stemming from the market, or is it driven by idiosyncratic variance. In the appendix it is shown, that for an asymptotically fine-grained portfolio the variance of the portfolio is completely determined by the variance of the market. Or speaking more technically, we have9 An important implication of this asymptotical behavior of the portfolio is, that it enables us to compute the qth quantile of the portfolio distribution. Given the fact, that the portfolio value of this asymptotic portfolio depends solely on the value of X, it is possible to map the quantile of the X-distribution to the portfolio quantile. And if we impose the additional restriction, that the individual conditional expected loss functions are non-decreasing, we can find the relationship: In words: The qth quantile of the conditional expected loss function equals the expected loss function conditional on the qth quantile of the distribution of the systematic factor X. We only have to take care of the fact that X enters the formula with a negative sign, so that the qth quantile of the loss distribution equals the (1 - q)th quantile of the distribution of X. Substituting into (10) gives: (11) lim Var(PFn) - Var(E(PFn\X)) = 0 n — » oo (12) aq(E(PFn\X)) = E(PFn\aq(X)) (13) Finally, substituting N 1 (PD) for a, we get: (14) 9 See Gordy (2001), p. 7. Kredit und Kapital 4/2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 Why Simple, When it Can Be Difficult? 541 Now, we are close to the formula, we wanted to arrive at. The last step is merely to replace w2 by p. This is justified, since in this special setup, we can replace the weighting parameter w10 by yfp, which stands for the asset-correlation between pairwise assets. This follows from the fact that we have assumed the idiosyncratic factors to be independent and identically distributed random variables. So evaluating the correlation of the risk index of two obligers i and j, we get (15) p = corr{Zi, Zj) = E{Z{ • Zj) = E({wX)2) = w2 where we have repeatedly used the fact that X and €j are standard normally distributed. So, finally we arrive at the conditional portfolio loss function for the qth quantile as a function of the probability of default (PD) and the asset-correlation parameter (p): We can think of this formula as being the marginal contribution of a given asset to the portfolio capital requirements. And since in the context of this model, marginal requirements are portfolio-invariant, i.e. they do not depend on how they affect diversification, they can simply be summed up to arrive at the capital requirement for the overall portfolio. Going back to (1), we see that the above conditional portfolio loss function (choosing 1 — q = 0.999) is the key ingredient of the IRB-risk weight function. What is added in (1) are merely scaling parameters, the loss given default (LGD), and the maturity adjustment factor {MA). With the above function it gets clear, where the coefficients 1.118 and 1.288 in the initial (CD2) formula stem from. They have been calculated by assuming a fixed correlation of p = 0.2 (or a weight of w = 0.447) and a confidence level of 1 - q = 0.995.11 !0 Recall that w indicates the weight, by which the risk index of an individual obligor is driven by the common systematic factor, ii See Basel 2001b, section 172. (16) p(PFn\Xq)=N^ Kredit und Kapital 4/2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 548 Matthias Bank and Jochen Lawrenz This again shows, the quite arbitrary nature of p. VI. Proposed Simple Risk Weight Function Given the model-inherent difficulties concerning the underlying assumptions and input data and - more importantly - the "trial and error" approach of the committee in fitting the model to the targeted 8% on average, we think that we can arrive at the same goal - notably providing a risk-sensitive capital rule - with much less model-theoretical burden.27 As it has been already pointed out, the committee has stated some basic target values for the risk weights. For example, in the initial proposal a PD of 0,7% should generate a risk weight of 100%. In the potential modification paper the benchmark risk weight of 100% was shifted to a PD of 1%. Another benchmark seems to be given by a PD of 20%, for which a risk weight of 375% is chosen, or the minimum PD of 0.03%, for which a risk weight of roughly 20% is chosen. The point we want to make is the following: Given the fact, that the determination of these benchmarks is for the most part a political decision and only partly based on model results, why not simply fix these values and construct an "appropriate" risk weight function around them. Given these anchor values, it is easy to find a corresponding function through regression techniques, which fits well to the data. For example, by using power regression and taking the above benchmarks as input data, we arrive at the following relationship: (19) RWreg = 0.645 • PD°556 + 0.0 34 Figure 4 compares this risk weight function with the current Basel IRB-risk weight function (see (1)). Obviously, and not surprisingly, the RWREG conforms well with the IRBrisk weight function (RWNOV), yielding comparable risk weights. A more meaningful test how the regressed function performs, may be done by calculating the average capital requirements. 27 Since at the time when the following calculations have been performed, the third Consultative Document has not yet been published, the following section uses the risk weight function (RWNov) as put forward in the Potential Modification paper (Basel (2001c)). But since RWQDZ and RWNov do not differ significantly, the argument remains valid. Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 Why Simple, When it Can Be Difficult? 549 374.59 400 RWreg(pd) 200 RwNov(pd) 18.108 300 100 0 0.05 0.1 Pd 0.15 0.2 0.2 3x10" v-4 Fig. 4: Risk weight function from regression In order to do this, we rely on the data provided in the study by Schwaiger, Lawrenz (2002), where the average capital requirement was calculated based upon a hypothetical (unsecured) credit portfolio of roughly 2.5 million obligors, representing the overall German economy.28 For this credit portfolio the authors arrived at an average capital requirement of 9.41% using the modified IRB-risk weight function. We took the same data base and calculated correspondingly the average capital requirement using our regression-risk weight function and arrived at a figure of 9.88%, showing that results are comparable.29 The proposed risk weight function not only fits well in the IRB risk weight function, but is also easily adapted to the committees goals - notably achieving a 8 % average. Using the distribution of corporate exposures, according to Schwaiger, Lawrenz (2002) and applying a numerical approximation algorithm, we arrive at the adjusted risk weight function: (20) RW*reg = 0.645 • PD°641 + 0.034 28 See Schwaiger; Lawrenz (2002); The data is due to Creditreform. 29 For a complete assessment of the result, one has to take account of the assumptions underlying the calculation. For a full discussion see Lawrenz, Schwaiger (2002). Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 550 Matthias Bank and Jochen Lawrenz Fig. 5: Approximated risk weight function where only the exponent has changed. The graph of the function, together with the IRB risk weight function is shown, in figure 5. The average regulatory capital requirement, assuming the same credit portfolio as previously described and the risk weight function (RW*reg) derived form the approximation, now turns out to be 8,08% and thus pretty close to the desired level. The careful reader will certainly have noticed, that the proposed regression risk weight function lies above the IRB risk weight function for very small PD-values (below 25 basispoints). We admit that this is the case, but we consider this not to be a major drawback of the proposal, since firstly a function that fits closely this lower part can easily be found, and then we might simply put the two regression risk weight functions together. The resulting, composed function then might look like: _ f 0.38 • PD° 283 - 0.026 if PD < 0.01 (21) I 0.645 PD°556 + 0.034 if PD > 0.01 The resulting graph in the range of 0.0003 to 0.02 is shown in figure 6, and one recognizes that this function fits the lower part very closely. Calculations show that the overall capital requirement average is practically not affected. With the same data as above, the average capital requirement changes from 8,08% to 8,07 %. Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 Why Simple, When it Can Be Difficult? 551 Fig. 6: Risk weight function from regression for low PDs VII. Conclusion A careful analysis of the IRB risk weight function, as it has been provided in the first part of the present paper, shows that the theoretical background is quite complex, relying on formal results which may not be obvious to all practitioners and supervisors. This fact alone is certainly not an argument against using theoretical models in regulatory capital rules, but on the other hand, we want to stress that the application of models should neither be a goal per se. This is particularly true for the use in a regulatory context. However, the acting of the committee gives the impression, that it tries to maintain a model-based approach only for the model's sake. The portfolio model underlying the IRB risk weight function is based on four crucial assumptions, namely (i) one single systematic factor, (ii) asymptotically fine-grained portfolios, (iii) non-decreasing conditional loss functions and (iv) normally distributed variables. We have pointed out, that each assumption can be challenged with reasonable arguments, implying that the model generates at best rough approximations of real conditions. Especially, the single systematic factor assumption is difficult to justify, given the fact that the Basel Capital Accord is intended to apply for large international banks. The problem of homogeneity has been encountered by introducing the granularity adjustment, but since the adjustment was cancelled from the third consultative paper, these assumptions become critical again. Kredit und Kapital 4/2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 552 Matthias Bank and Jochen Lawrenz We argued, that this model-inherent problems come along with the vagueness of input data quality. Taken together, these problems imply, that the model may only be able to generate quite rough results. Therefore, the committee seems to have followed a rather "trail and error" approach, whereby the paramount aim was to achieve an economy-wide average of 8%. Given this political goal, we think that it is neither necessary nor desirable to maintain a model-driven approach, since the model insinuates a theoretically founded accurateness of results that is not justified. Instead, we consider it more appropriate to admit that the determination of the risk weight function is mainly a political question. Then, given the politically determined benchmarks, it is easy to construct a function that fulfills the same task, notably providing a risk-sensitive capital rule and is easily implemented and understood. What is more, the function can be scaled to any given target level by simple approximation algorithms. If the committees aim is to provide risk-sensitive capital rules, together with an given average level of capital requirement, this can be achieved more easily. VIII. Appendix To see who accounts for the portfolio variance, we can use the standard decomposition of the variance of any random variable that is: (22) Var(il>) = Var(E(i/>\Q) + E(Var(^\0) In words: The variance of a random variable can be decomposed in the sum of the variance of the conditional expectation and the expectation of the conditional variance.30 Applying this to our example, we find for the conditional expectation of the portfolio value: 1 - p(PFn\X).31 Taking the variance of this is just the variance of p(PFn\X). (23) Var(p(PFn\X)) = E(p(PFn\X)2) - E(p(PFn\X))2 = E{p(PFn\X)2)-p2 30 See e.g. Mood, Graybill, Boes (1974) p. 159. 31 Note that the conditional expectation on a single obligor is (l/iV)(l -p(i|X)) and we have N such obligors. Given the conditional independence, we are allowed to sum over p(i\X) and we are left with 1 - p(PFn\X). Kredit und Kapital 4/2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 Why Simple, When it Can Be Difficult? 553 Evaluating the first term in (23) means finding the expected value of a random variable in a conditional distribution and results in the present context in a bivariate normal cumulative distribution function32, denoted by N2(-, p), with correlation of w2. / oo n(x) p(PFn\x)2 dx = N2(a,a,w2) •oo where n(x) denotes the density function of the normal distribution. Putting the result together, the variance of the conditional expectation of the portfolio value equals: N2{a, a, w2) - p2. To evaluate the expectation of the conditional variance of the portfolio value, we first note that the conditional variance of the value of individual loans is p(i\X) • (1 -p(i\X))/N2. Since the loan values are conditionally independent, the conditional portfolio variance is the sum of the individual conditional variances: p(PFn\X) • (1 - p(PFn\X))/N. Taking the expectation and using the previous result for E(p(PFn\X)2\ we get for the second term: (p - iV2(a, a,w2))/N. Putting both parts together, we obtain for the portfolio variance: VarPF= (N2(a,a,w2) — p2) + (p - N2(a,a,w2))/N) (25) " ' * * ' market variance idiosyncratic variance Note, that since the first term has been calculated as the variance of the conditional expectation, we can think of this term as the part of the portfolio variance, that is due to market movements. Equivalently, the second term, represented as the expectation of conditional variance can be thought of as the part of portfolio variance, that is attributable to idiosyncratic variance. Now, the following result is obvious. If N, the number of loans in the portfolio is very large, the second term - which is the only one depending on N - gets very small and vanishes eventually, so that the variance of the portfolio is completely determined by the variance of the market. 32 See Finger (1999). Kredit und Kapital 4/2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 554 Matthias Bank and Jochen Lawrenz References Altman, Edward I. 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A./ Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 Why Simple, When it Can Be Difficult? 555 Lawrenz, Jochen (2002): Standard - versus IRB-Ansatz: Auswirkungen auf die Bank Deutschland. Zeitschrift für das gesamte Kreditwesen, 55 (3-4), p. 147. - Vasicek, Oldrich (1987): Probability of loss on loan portfolio, working paper, KMV Corporation. - Vasicek, Oldrich (1991): Limiting loan loss probability distribution, working paper, KMV Corporation. - Schönbucher, Philipp J. (2000): Factor models for portfolio credit risk, working paper, Department of statistics, University of Bonn. - Wilkens, Marco/Entrop, Oliver/Scholz, Hendrik (2002): Eigenkapitalanforderungen für Kreditrisiken - Analyse des modifizierten IRB-Ansatzes, Zeitschrift für das gesamte Kreditwesen, 55 (3-4), p. 141. - Zhou, Chunsheng (1997): Default Correlation: An Analytical Result, Finance and Economics Discussion Series Papers, Board of Governors of the Federal Reserve System. Summary Why Simple, When it Can Be Difficult? Some Remarks on the Basel IRB Approach One of the major innovations in the New Basel Capital Accord (Basel II) is represented by the Internal Ratings Based (IRB) Approach. It can be considered as a conceptually new approach to capital rules, since the IRB risk weight function is derived from a simple portfolio model. A careful analysis of the underlying model reveals, that it is based on a quite complex theoretical background and depends on some critical assumptions. Beside this inherent vagueness of model-based results, the committee shows a politically determined will to calibrate the parameters of the model so as to obtain an economy-wide average of 8 % - implying an unchanged level of regulatory capital requirements on average. Taken together, this suggests that the model-driven approach is more like a pretext to disguise politically determined decisions, and pretend, an accurateness, that is not given. We argue, that if it is the committees aim to provide risk-sensitive capital rules, together with some target level of average capital requirements, this can be achieved much easier. (JEL G21, G28) Zusammenfassung Warum einfach, wenn's auch umständlich geht? Anmerkungen zum Basler IRB-Ansatz Eine der wesentlichen Neuerungen im Konsultationspapier zur Neuen Basler Eigenkapitalverordnung (Basel II) stellt der auf internen Ratings basierte Ansatz (IRB-Ansatz) dar. Er kann als eine konzeptionelle Innovation betrachtet werden, da die Risikogewichtungsfunktion aus einem einfachen Portfoliomodell abgeleitet ist. Eine sorgfältige Analyse des zugrunde liegenden Modells zeigt jedoch, dass die Ergebnisse auf nicht ganz „harmlosen" Annahmen beruhen und eine Genauigkeit vorspiegeln, die in dieser Form nicht gerechtfertigt ist. Andererseits war und ist es das erklärte Ziel des Basler Ausschusses für Bankenaufsicht, die Eigenmittelquote Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34 556 Matthias Bank and Jochen Lawrenz im volkswirtschaftlichen Durchschnitt unverändert bei 8 % zu belassen. Insgesamt drängt sich damit der Eindruck auf, dass hier der Ursache-Wirkungs-Zusammenhang umgekehrt wird: Statt mit dem zugrunde gelegten Modell die Höhe der Eigenmittelanforderungen zu bestimmen, wird ein politisch bestimmtes Niveau vorgegeben und das Modell entsprechend diesen Anforderungen „kalibriert". Damit drängt sich allerdings die Frage auf, ob dann ein modelltheoretischer Hintergrund überhaupt noch notwendig und sinnvoll ist. Im vorliegenden Artikel wird argumentiert, dass das Ziel, risikosensitive Eigenkapitalvorschriften zu schaffen und dabei ein gewünschtes durchschnittliches Eigenmittelniveau zu erreichen, auch wesentlich einfacher erreicht werden könnte. Résumé Pourquoi simple quand cela peut être difficile? Quelques remarques sur rapproche IRB du Nouvel Accord de Bâle sur l'adéquation des capitaux L'approche IRB représente une des innovations majeures du Nouvel Accord de Bâle sur l'adéquation des capitaux (Bâle II). Elle peut être considérée comme une nouvelle approche conceptuelle de la régulation des capitaux puisque la fonction de risque IRB est dérivée d'un simple modèle de portefeuille. Une analyse minutieuse du modèle sous-jacent révèle que l'approche est basée sur une théorie assez complexe et dépend de certaines hypothèses douteuses. A ce manque de précision inhérent des résultats basés sur le modèle s'ajoute la volonté politique du Comité d'étalonner les paramètres du modèle de sorte à obtenir une moyenne économique de 8 % - impliquant un niveau inchangé des exigences régulatrices de capitaux. Ceci suggère que l'approche dérivée du modèle ressemble plus à un prétexte pour camoufler des décisions politiques et qu'elle prétend à une exactitude qui n'est pas donnée. Nous affirmons que, si le but du Comité est de fournir des régulations des capitaux sensibles aux risques, avec certains niveaux d'exigences de capitaux moyens, ceci peut être obtenu beaucoup plus facilement. Kredit und Kapital 4 / 2003 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.36.4.534 | Generated on 2023-01-16 13:20:34