- Email: [email protected]
The calculation of capital requirement using Extreme Value Theory
Economic Modelling 28 (2011) 390–395
Contents lists available at ScienceDirect
Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d
The calculation of capital requirement using Extreme Value Theory Ming-Shann Tsai a,⁎, Lien-Chuan Chen b a b
Department of Banking and Finance, National Chi Nan University, Puli, Nantou, Taiwan 54561 Department of Finance, National Central University, No. 300, Jung-da Rd., Jung-Li, Taiwan 320
a r t i c l e
i n f o
Article history: Accepted 13 August 2010 JEL classification: G21 G28 Keywords: Basel accord Extreme Value Theory Capital requirement
a b s t r a c t The Basel Committee has suggested some formulas for calculating capital requirement using the Advanced Internal Ratings-Based Approach. However, these formulas were derived under the assumption of a normal distribution. Thus, the capital requirement estimated by the Basel formula may be incorrect when the asset distributions are not normal. Using an analysis of qualifying revolving retail exposures as an example, this paper introduces a formula based on the Extreme Value Theory to calculate the capital requirement. This formula is more general and accurate than its predecessors, because it can be used with any type of distribution. Numerical examples are provided to demonstrate that the capital requirement estimated by the Basel formula is less than by our formula when the asset distribution has a heavy tail, and more than by our formula when the distribution has a short tail. Our formula is also more sensitive to risk than competing models in the context of the recent financial crisis. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Methods for managing credit risk have undergone dramatic developments in the last 20 years because of the worldwide increase in the number of bankruptcies and the dramatic growth in the use of off-balance-sheet instruments to hedge the inherent risk of default (McKinsey, 1993; Altman and Saunders, 1998). According to the theory of credit management, a bank needs to maintain a rational capital adequacy for reducing its bankrupt probability. To address this need, the Basel Committee of Banking Supervision suggested a common framework (i.e. the Basel Capital Accord) for calculating the capital adequacy. Because the Basel Capital Accord is suggested to be implemented worldwide, research on the best way to calculate capital requirement has recently become popular among both academic scholars and financial institutions. The recent version of the Basel Capital Accord is the “International Convergence of Capital Measurement and Capital Standards: A Revised Framework” (hereafter referred to as the RF). The RF suggests three methods for calculating the capital requirement of bank's credit risk: the Standardized Approach, the Foundation Internal RatingsBased Approach, and the Advanced Internal Ratings-Based Approach (hereafter referred to as the Advanced IRB Approach). In the studies related to Basel Capital Accord, the issues for accurately estimating the default probability and calculating the capital requirements have been widely discussed (Carey and Hrycay, 2001; Altman et al., 2002; Gordy, 2000; Altman, Resti and Sironi, 2003; Perli and Nayda, 2004).
⁎ Corresponding author. Tel.: + 886 49 2910960x4902; fax: +886 49 2914511. E-mail address: [email protected] (M.-S. Tsai). 0264-9993/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2010.08.010
Johnston (2009) derived a modified model for calculating the capital requirement that takes account of a bank's earnings. Repullwo and Suarez (2004) proposed a net interest income correction model for IRB Approach. Gatzert and Schmeiser (2008) extended the Basel formulas to address the issues of bank's equity investment. Most researchers calculate the credit exposure based on parameterized credit risk models. Grane and Veiga (2008) have argued that the assumption of a normal distribution is improper when estimating capital requirement. Brooks, Clareb and Persanda (2000) demonstrated that the use of GARCH-type models for the calculation of minimum capital risk requirements may lead to the production of inaccurate and therefore inefficient capital requirements. These authors give a simple modification to the model to improve the accuracy of minimum capital risk requirements estimates in both back- and out-of-sample tests. Some researchers have used nonparametric statistics to generate the credit–loss distributions needed to calculate the credit risk (Jacobson et al., 2005). The RF pays particular attention to retail credit1 because of the supposedly smaller exposure to systemic risk that it entails (Jacobson et al., 2005). In the RF, retail credit exposures are divided into three primary categories: (1) exposures secured by residential mortgages, (2) qualifying revolving retail exposures (hereafter referred to as QRRE), and (3) other non-mortgage exposures, also known as “other retail” (RF paragraphs 233). This paper uses the QRRE as an example to introduce a new formula for calculating the capital requirement. QRRE consists of unsecured revolving credits that exhibit appropriate loss characteristics. For example, the credit card is a kind of QRRE. A 1 The retail credit business in commercial banks involves consumer credit, residential mortgages, automobile loans, credit cards, and small business loans (Allen et al., 2004).
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
formula for calculating the capital requirement of QRRE is provided by the Advanced IRB approach. In so doing, bank can use a suitable distribution for estimating the probability default, because the distributions of financial data usually show heavy tails. The estimated default probability then is entered into the suggested formula for calculating the capital requirement. The RF takes into consideration the above-mentioned problems of skewed distributions in calculating the default probability. However, the calculation of the capital requirement may still be inaccurate because the formula for calculating the critical value of the default probability at a given confidence level is still derived on the assumption that the asset probability is normally distributed (Perli and Nayda, 2004). The purpose of this paper is to derive a formula for accurately calculating the capital requirement under a more general framework. The Extreme Value Theory has been widely applied in the finance and insurance literature to analyze data distributions characterized by extreme departure from normality with respect to skewness. Specific applications include portfolios, operational losses, catastrophic insurance claims, and credit losses (Embrechts et al., 1997; Fernández, 2003a,b; Coles, 2001; Reiss and Thomas, 2001; McNeil and Frey, 2000). To accurately calculate the capital requirement, we use the Extreme Value Theory to derive a formula for accurately calculating the critical value of default probability at a given confidence level. Our formula is more general and estimates capital requirement more accurately than previous models because it can be applied to any type of distribution (e.g., the distribution with heavy tails). We provide quantitative examples comparing our formula with that suggested by the RF. The results show that the RF formula can underestimate capital requirements when the sample distribution has a heavy tail and overestimate it when it has a short tail. Since 2007, the sub-prime mortgage crisis has kept US financial markets in turmoil and eventually created turbulence in financial markets worldwide. During this period, many famous banks and insurance companies, such as Lehman Brothers, Merrill Lynch, Fannie Mae, and Freddie Mac, have been overleveraged or burdened by debt, resulting in bankruptcies or takeovers. Because maintaining adequate capital is how banks avoid default, any formula that adequately estimates a bank's capital requirement must accurately reflect the huge turbulence in the financial markets. We also use a numerical example to show how our formula is more sensitive to risk when economic situations change. The remainder of this paper is organized as follows. In Section 2 we present the formula for calculating the capital requirement of QRRE as described in the New Basel Accord. We also present the formula for calculating the default probability based on the Extreme Value Theory (the RF approach). In Section 3, we use the Extreme Value Theory to derive a new formula for calculating the critical default probability at a given confidence level when the default rate distribution is nonnormal with respect to skewness. Section 4 includes quantitative examples comparing the formulas. Finally, in Section 5, we summarize our main findings and offer suggestions for future research. 2. Formulas for calculating default probabilities using the Extreme Value Theory The standard procedure for determining the QRRE can be described in terms of the following equations (defined in RF paragraph 329):
CR = VaRα ðPDÞ × LGD;
ð1Þ
VaRα ðPDÞ = PDα −PD;
ð2Þ
PDα = Φ
! pffiffiffi −1 ρΦ ðαÞ + Φ−1 ðPDÞ pffiffiffiffiffiffiffiffiffiffiffi ; and 1−ρ
Risk−weighted assets = CR × 12:5 × EAD;
391
ð3Þ
ð4Þ
where CR LGD EAD α PD
is the capital requirements, is the loss given default, is the exposure at default, is the confidence level, which is suggested to be 99.9% in the RF, is the probability of default, which can be estimated with a proper method, VaRα(PD) is the Value-at-Risk of the default probability under confidence level α, is the critical value of PD under α confidence level, PDα ρ is the correlation of consumers' assets, suggested as ρ = 0.04 in RF, Φ(⋅) is the cumulative normal distribution, and Φ − 1(⋅) is the inverse cumulative normal distribution function. RF suggests that a specific bank can use its own estimation methods to determine LGD, EAD and PD for calculating its capital requirement. In this section, we derive a formula for calculating the default probability, taking into account the skewness problems, based on the Extreme Value Theory. To begin, let Vi(T) represent the standardized asset value of debtor i at time T, and let k be a given critical asset value of bankruptcy. According to the traditional definition of bankruptcy, default occurs when Vi ðT Þbk. Therefore, the probability of default for an individual consumer can be described as follows: PDi = PrðVi ðT ÞbkÞ; i = 1; 2; :::N;
ð5Þ
where N is the number of consumers. Let a debtor's asset value be driven by a single common factor Y, such as a macroeconomic variable related to the business cycle (e.g. national income, consumer price index, stock index). We can describe the standardized asset value as follows (Perli and Nayda, 2004): Vi ðT Þ =
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ρY + 1−ρεi ;
ð6Þ
where εi is an individual variable. Assume that Y and εi are independent. Let F(⋅) be the cumulative conditional probability distribution of εi; G(⋅) be the cumulative conditional probability distribution of Y; and Q(⋅) be the cumulative conditional probability distribution of Vi(T). According to Eq. (3), Q(⋅) is a combination of the distributions of Y and εi. In the study of Perli and Nayda (2004), F(⋅), G(⋅), and Q(⋅) were assumed to be standard normal distributions for deriving the formula suggested in the RF (hereafter referred to as the RF formula). Here we relax this assumption and support a more general formula for estimating the default probability. The default probability contingent on the realization of the common factor (Y = y) can be described as follows: pffiffiffi ! k− ρy PD ≡ PDð yÞ = PrðVi ðT Þb k j Y = yÞ = Pr εi b pffiffiffiffiffiffiffiffiffiffiffi ≡ F ðsÞ; 1−ρ where s =
ð7Þ
pffiffi k− ρy pffiffiffiffiffiffiffiffi . 1−ρ
This default probability, F(s), can be estimated from the Extreme Value Theory. In practice, there are two methods for analyzing
392
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
extreme events: the block maxima with a general extreme value and the peaks-over-threshold with a general Pareto distribution (POT method). Because modeling all the block maxima is wasteful when other extreme values are available, the POT method is a more efficient and precise approach to model the behavior of extreme values above the threshold (Fernández, 2003a,b; Matthys and Beirlant, 2003). We therefore use the POT method to analyze the default probability.2 The POT method requires a sufficiently small threshold u, u N s, for analysis.3 The left-tail excess distribution is defined as follows:
Fu ðu−sÞ = Prðs≤εi ≤uj εi buÞ = 1−
F ðsÞ ; F ðuÞ
ð8Þ
where: u Fu ð⋅Þ F(u)
is the threshold value of εi that is exogenous, is the cumulative left-tail excess probability distribution of εi, and is the empirical cumulative probability of εi below the threshold u.
According to Eq. (8), F(s) = F(u)(1 − Fu(u − s)). To estimate F(s), it is necessary to first estimate F(u) and Fu(u − s). In general, F(u) is determined empirically and non-parametrically; that is F ðuÞ = Nn , where n is the number of observations that fall below the threshold u. Fu ðu−sÞ can be estimated using a theorem from the Extreme Value Theory. This theorem shows that Fu ðu−sÞ approaches a generalized Pareto distribution (Hξ, β(u − s)) given a sufficiently small u (Embrechts et al., 1997). This distribution can be described as follows:
Hξ;
8 − 1=ξ ξ > > ; if ξ≠0 < 1− 1 + ðu−sÞ ; β β ðu−sÞ = u−s > > ; if ξ = 0 : 1− exp − β
ð9Þ
where ξ is the shape parameter of Hξ; β ðu−sÞ, β is the location parameter of Hξ; β ðu−sÞ, and u − s ≥ 0 when ξ ≥ 0, and 0≤u−s≤− βξ when ξ b 0.
8 > > > > > <
1=ξ ξ for ξ≠0 1 + ðu−sÞ β
F ðsÞ = > F ðuÞ > > u−s > > : exp − β
:
ð11Þ
for ξ = 0
Finally, based on the Extreme Value Theory, the conditional default probability is derived as follows: 8 pffiffiffiffiffiffi!!− 1=ξ > n ξ k− ρy > > p ffiffiffiffiffiffiffiffiffiffi ffi for ξ≠0 1 + u− > n u k− ρy > > > p ffiffiffiffiffiffiffiffiffiffi ffi for ξ = 0 exp − + :N β β 1−ρ
ð12Þ
Eq. (12) is a formula for calculating the default probability on the basis of the Extreme Value Theory. In this equation, eight parameters (i.e., N, n, y, ρ, k, ξ, β and u) need to be determined when estimating the probability of default. The following will describe how to obtain these parameters. N, the number of consumers, is a known value. Because n is the number of the observations that fall below the threshold u, it can be determined as soon as u is determined (see later discussions). y is the current value of a macroeconomic variable (e.g., stock return) related to the business cycle. ρ is the correlation of consumers' assets. RF is assigned the value of ρ equals to 0.04, but this value can be adjusted based on an empirical evaluation of the relation between individual's wealth and the common factor. For example, pffiffiffi pffiffiffi , a bank can define ρ as the effect of the changes of because ρ = ∂V ∂Y macroeconomic variable on borrower's wealth. Because V b k refers to the default event, the value of k is critical for determining when the debtor must file for bankruptcy. In practice, we suggest that k can be an average standardized loan value in a bank. The parameters ξ and β can be estimated using the nonlinear least square method or the maximum likelihood method. For example, when using the nonlinear least square method, a bank may collect a yt, t = 1, ⋯, T, where T is a sample set of the time series of kt, ρt and pffiffiffiffi kt − ρt yt period, to obtain the series of st st = pffiffiffiffiffiffiffiffiffi . ξ and β can be 1−ρt
In theory, the general Pareto distribution can be subdivided into three types. (1) If ξ = 0, then Hξ, β(u − s) is defined as a Gumbel distribution. This specification is used when analyzing the distribution with thin tails (e.g. normal, log-normal, exponential, and gamma). (2) If ξ N 0, then Hξ, β(u − s) is defined as a Fréchet distribution, used when analyzing the distribution with heavy tails (e.g. Pareto, Cauchy, Student t, mixed models). (3) If ξ b 0, then Hξ, β(u − s) is defined as a Weibull distribution, used when analyzing the distribution with short tails. Because Fu(u − s) ≈ Hξ, β(u − s) given a sufficiently small u, we have: 8 1=ξ ξ > > for ξ≠0 < 1− 1 + ðu−sÞ ; β Fu ðu−sÞ = u−s > > for ξ = 0 : 1− exp − β
According to Eq. (8), we obtain:
ð10Þ
2 When using the POT method, an extreme value is generally defined as occurring on the right tail of the distribution (Embrechts et al., 1997). It should be noted that we define default as a debtor's assets falling below the threshold; therefore, we focus on the left-tail of the distribution. This focus should be kept in mind when deriving the formulas. 3 To maintain u N s, the range of the common factor y must be taken into account.
estimated by minimizing the sum of squared error between the data of default probability and the outcome of solving Eq. (12). The model of nonlinear least square method can be described as follows: 0 0 112 8 1=ξ > T > ξ > min @PDt −Ft ðuÞ@1 + ðu−st Þ AA for ξ ≠ 0 > > < ξ ; β t∑ β =1 ; 2 > T > > u st > >min ∑ PDt −Ft ðuÞ exp − + for ξ = 0 : β β β t =1
ð13Þ
According to Eq. (13), one can obtain the estimated parameters (i.e. ξ and β) with nonlinear estimation method if the threshold u is decided. In practice, researchers generally assume that the threshold u is an exogenous variable. Recently, some scholars have argued that the optimal value of u can be defined as the point at which the mean squared error of the estimators is minimized (Beirlant et al., 1999; Matthys and Beirlant, 2003). Based on this reasoning, we suggest that u be selected arbitrarily so long as it satisfies the initial condition u N s; this initial value is then increased until the optimal threshold (u) is reached. For example, let u = s + Δs, s + 2Δs, s + 3Δs,…, and so on, where Δs is a small interval; then, the optimal threshold is the point at which the mean squared error of the estimators is minimized. Our model is thus viable, because a bank with access to its retail credit data can assign values to all the necessary parameters. We have shown above how a default probability, PD, can be estimated using the Extreme Value Theory. This result can then be used to calculate
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
the capital requirement of QRRE, calculated on the assumption of a nonnormal asset distribution, in the RF (i.e., Eqs. (1)–(3)). However, the calculated value could be incorrect, because the RF formula shown in Eq. (3) (i.e., PDα) was derived under the assumption that the asset follows a normal distribution. In the next section, we use the Extreme Value Theory to derive a formula for accurately calculating PDα. 3. A formula for calculating the critical value of the default probability based on the Extreme Value Theory Let X be a random variable designating the default rate. The cumulative probability distribution of X is expressed as follows: ∞
∞
−∞
−∞
Mð X Þ = ∫ ðX = PDð yÞ≤qÞgð yÞdy = ∫ Ipð yÞ≤q g ð yÞdy;
ð14Þ
where M(X) q g(y)
is the cumulative probability of the default rate, is the certainty level of PD, and is the probability density of y.
Assuming ξ ≠ 0 from Eq. (12), we can describe PD(y) ≤ q as follows: pffiffiffi !!−1= ξ n ξ k− ρy ≤q: 1+ u− pffiffiffiffiffiffiffiffiffiffiffi N β 1−ρ
ð15Þ
393
Comparing Eqs. (19) and (20) with Eq. (12) we may find that to calculate the critical value of PDeα it is necessary only to replace the common factor y by y1-α. Therefore, the formula for estimating the capital requirement can be described as follows: e e CR = VaR PDα × LGD = PDα −PD × LGD;
ð21Þ
Risk-weighted assets = CR × 12.5 × EAD, and
(22)
8 !! pffiffiffi − 1=ξ > y1−α −k > > n 1 + ξ u + ρp ffiffiffiffiffiffiffiffiffiffi ffi for ξ≠0; > > > n μ k− ρy1−α > > p ffiffiffiffiffiffiffiffiffiffi ffi for ξ = 0 exp − + > : N β β 1−ρ
ð23Þ
In Eq. (23), it can be seen that if we want to calculate the Value-at-Risk of the default probability, the critical value of PDeα must be derived by assuming the worst possible situation for the common factor at confidence level 1−α. This formula seems to make more economic sense than Eq. (3). When calculating the capital requirement, we must consider not only the estimated default probability of the bank, but also the expected risk following from the common economic factors. Because the default probability PD has been estimated using the Extreme Value Theory, the critical value of PD at confidence level α must also be estimated by the same theory. Based on this reasoning, we believe that our model is more general and accurate than previous models. 4. Numerical examples
From Eq. (15), we have the following: !! pffiffiffiffiffiffiffiffiffiffiffi −ξ k β N 1−ρ −1 : y ≥ pffiffiffi q −u + pffiffiffiffiffiffiffiffiffiffiffi + n ρ ξ 1−ρ
ð16Þ
From Eqs. (14) and (16), we obtain:
Mð X Þ = 1−G
!!! pffiffiffiffiffiffiffiffiffiffiffi −ξ k β N 1−ρ −1 : ð17Þ q −u + pffiffiffiffiffiffiffiffiffiffiffi + pffiffiffi n ρ ξ 1−ρ
Combining all these results leads to a cumulative probability function for the default rate at confidence level α, based on the Valueat-Risk Theory: !!! pffiffiffiffiffiffiffiffiffiffiffi −ξ 1−ρ k β N q α = M ðXα Þ = 1−G −1 ; −u + pffiffiffiffiffiffiffiffiffiffiffi + pffiffiffi ρ n α ξ 1−ρ
In this section, we present numerical examples that compare capital requirement calculated using our formula and using the Basel formula. We also show that our formula, when applied to a bank during the recent financial crisis, is more risk-sensitive than the Basel formula. Assume the current probability of default (PD) to be 0.05. Let LGD = 0.45, ρ = 0.04, and α = 99.9%.4 Following from these assumptions and the formulas in RF (see Eqs. (1)–(4)), we have PDα = 0.1473 and CR = 0.0438. Now we demonstrate that when the distribution of asset shows heavy (short) tails, the RF formula underestimates (overestimates) the values of PDα and CF as compared with our model. For analysis purposes, we rewrite the formulas for calculating PDeα as follows (see Appendix A): 8 ! pffiffiffi > ξ PD −ξ − 1=ξ ρ > > p ffiffiffiffiffiffiffiffiffiffi ð y Þ + for ξ ≠ 0 VaR F ð u Þ − > 1−α > β 1−ρ F ðuÞ < e PDα = ! pffiffiffi > > ρ > > PD × exp 1 pffiffiffiffiffiffiffiffiffiffi > ð y Þ for ξ = 0 VaR 1−α : β 1−ρ
;
ð18Þ
ð24Þ
where qα is a critical value of probability under the confidence level α. We define y1 − α = G − 1(1 − α) as the critical value for the common factor that applies in the worst possible situation, at confidence level 1 − α. As soon as we assign a value to α (e.g. α = 99.9% in RF), we can apply the Extreme Value Theory to derive the critical value for the default probability at confidence level α (PDeα) as follows:
where VaR1 − α(y) = y − y1 − α is the Value-at-Risk of the common factor at confidence level 1 − α. Because y and y1 − α represent the current and the worst economic situations respectively, we selected the 1 year return on the S&P 500 stock index as the common factor. After generating the return distribution, we let y be the Z-score representing the current index return and y1 − α be the 0.1th percentile of the standardized return distribution. Using the data of the S&P 500 stock index from 10/4/2004 to 10/3/2007, we found that, on the latter date, y=−0.6221 and y1 − α =−4.1088. For simplicity, we let the probability that falls below the threshold u as 1.5 times the default probability, that is, F ðuÞ = 1:5 × PD.5 For comparison purposes, we use
e PDα
n ξ ≡ qα = 1+ u+ N β
!!−1 = : pffiffiffi ξ ρy1−α −k pffiffiffiffiffiffiffiffiffiffiffi 1−ρ
ð19Þ
Likewise, if ξ = 0, we obtain: ! pffiffiffi n u k− ρy1−α e pffiffiffiffiffiffiffiffiffiffiffi : PDα = exp − + N β β 1−ρ
ð20Þ
4 These values are suggested in the Foundation IRB Approach. Banks can also estimate the value when using the Advanced IRB Approach. 5 We selected this value (i.e., 1.5 times) for illustrative purpose. The main conclusion is the same as when using the other value.
394
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
Fig. 1. The relationship between the shape parameter and the critical value of the default probability. Note: The Y-axis shows the critical value of default probability, PDeα. The X-axis shows the shape of the distribution represented by ξ, which ranges from −0.6 to 0.6. The key parameters are as follows: PD = 0.05, ρ = 0.04, α = 99.9%, β = 0.6586, y = − 0.6221 and y1 − α = − 4.1088. F ðuÞ = 1:5 × PD for calculating PDαe .
the RF calculation (PDα =0.1473) to obtain the location parameter of the Pareto distribution (i.e., β). Because ξ=0 can be used for analyzing a normal distribution, we assign PDeα =0.1473 to estimate β given ξ=0. According to Eq. (19), we obtain β=0.6586 in this circumstance. We then use this value to calculate PDeα given ξ≠0. These estimates allow us to determine the relationship between PDeα and ξ. For example, if ξ=0.4 (indicating a heavy tail), PDeα =0.1572. In contrast, if ξ=−0.4 (indicating a short tail), PDeα =0.1397. According to these results, the value of PDeα is greater than 0.1473 when the asset distribution have heavy tail and less than 0.1473 when the tail is short. These results demonstrate that we need a more accurate formula for risk management than that provided by the RF. The relationship between the shape of the distribution ξ and the critical value of default probability PDeα is illustrated in Fig. 1. Let ξ range from −0.6 to 0.6. The horizontal line in the figure is equal to 0.1473, which is the calculated PDα using the suggested formulae in the RF (in the case of normal distribution). The line that stands for ξ N 0 is the calculated PDeα when asset distribution has a heavy tail. This clearly demonstrates that the estimated exposure is less using the Basel formula than using our formula when the asset distribution has this shape. In contrast, the line for ξ b 0 is the calculated PDeα if asset distribution has a short tail. In this case, it is equally clear that the estimated exposure is higher using the Basel formula than our formula. As for the capital requirement, we find that CR = 0.0482 when ξ = 0.4 and 0.0404 when ξ = − 0.4. This value is higher than the value calculated using the Basel formula when the asset distribution has a heavy tail and lower when the asset distribution has a short tail. The relationship between the shape of the distribution ξ and the capital requirement CR is shown in Fig. 2. Just as for Fig. 1, we conjecture that the estimated CR using the Basel formula is higher than that using our formula when the asset distribution has a heavy tail and lower when the distribution has a short tail. During the period of sub-prime mortgage crisis, a bank's capital requirement must accurately and fastly reflect the huge turbulence in the financial markets. Data from the S&P 500 stock index from 4/2/2007 to 4/2/2010 yield y1 − α = − 4.7367, meaning that the actual economic situation in the US is worse than in our previous example. However, the traditional Basel formula failed in this regard during the recent financial crisis. Our formula can accurately and fastly reflect the huge turbulence in the financial markets because it has considered the Value-at-Risk of the common factor. Fig. 3 shows the relationship between the critical value of common factor y1 − α and the capital requirement CR given the same parameters
Fig. 2. The relationship between the shape parameter and capital requirement. Note: The Y-axis represents the capital requirement,CR. The X-axis represents the shape of the assets distribution represented by ξ. LGD = 0.45 for the calculation of CR. For the other definitions, see notes for Fig. 1.
used in our previous example. Let y1 − α range from −5.5 to −3.5. The horizontal dotted line in Fig. 3 represents the capital requirement calculated from the traditional Basel formula. We compare this estimate to two estimates calculated from our model, one where the asset shows a heavy-tailed distribution (ξ = 0.4) and one where it shows a shorttailed distribution (ξ = − 0.4). The “+” line shows the capital requirement calculated from our formula when ξ = 0.4 The “*” line shows the capital requirement when ξ = − 0.4. As shown in Fig. 3, the capital requirement calculated from the Basel formula is 0.0438 regardless of how the economic situation changes. In our model, the capital requirement is negatively related to y1 − α in both cases. That is, if the economic situation is worse, our model indicates that a bank needs to prepare more capital for avoiding bankruptcy; if the situation is better, the bank needs less capital. Thus, our formula is more risksensitive than the Basel model when used for risk management. Turning to the example of the sub-prime mortgage crisis, we find that because y1 − α = − 4.7367 (the vertical dotted line in Fig. 3), the capital requirement calculated by our formula is 0.0503 if ξ = − 0.4 and 0.0707 if ξ = 0.4. Thus, the estimated capital requirement is greater with our model than with the Basel model. This implies that if used during the sub-prime mortgage crisis, our formula would provide a bank with greater protection against credit risk than the Basel formula. Even when the distribution for a given bank has a short tail, our formula estimates more capital requirement than the Basel formula when the economic situation is poor.
Fig. 3. The relationship between the worst economic situation and capital requirement. Note: The Y-axis represents the capital requirement, CR. The X-axis represents the critical value of common factor, y1 − α. The “+” and " " lines indicate the calculated CRs for ξ= 0.4 and ξ= − 0.4, respectively, under different economic conditions. The vertical dotted line denotes y1 − α = − 4.7367, and the horizontal dotted line denotes CR calculated using the traditional Basel formula (0.0438). For the other definitions, see notes for Fig. 2.
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
Likewise, assume ξ = 0, we have
5. Conclusion An accurate estimation of the capital requirement is important for a bank to reduce its bankruptcy probability. The RF formula for calculating the critical default probability at a given confidence level was derived under the assumption that the default rate follows a normal distribution. If that is the case of normal distribution, a bank can successfully estimate the default probability by using the Advanced IRB Approach to calculate the capital requirement. On the other hand, the capital requirement may be underestimated if the distribution has a heavy tail and overestimated if it has a short tail. Thus, in either of these cases, the RF formula is likely to inaccurately estimate the bank's exposure. In this paper, we have presented a new formula, based on the Extreme Value Theory, for calculating the critical value of the default probability at a given confidence level. This formula gives more accurate estimates than the formula associated with the Advanced IRB Approach when the tails of the financial data distribution deviate from normal in terms of skewness. We have illustrated numerical examples to show that the Basel formula tends to underestimate the capital requirement when the asset distribution has a heavy tail and underestimate it when the distribution has a short tail. Finally, we showed that our formula is more risk-sensitive than the Basel model when the economic situation changes, such as during the recent financial crisis. In future research, real data could be used to estimate parameters such as ξ and β before applying our formula to estimate the capital requirement. Although the discussion of our formula was limited to the example of QRRE, the formula can also be well applied to other risk management contexts, for example, the calculation of other kinds of exposures addressed by the Basel Accord. Appendix A This appendix shows how to obtain Eq. (24). Assume ξ ≠ 0, from Eqs. (12) and (23), we have the following: ! 0 1 pffiffiffi − 1=ξ ρy1−α −k ξ p ffiffiffiffiffiffiffiffiffiffi ffi B1 + β u + C e B C 1−ρ PDα C =B pffiffiffi ! −1 + 1C B PD ξ k− ρy @ A u− pffiffiffiffiffiffiffiffiffiffiffi 1+ β 1−ρ 0
ξ B1 + β u + B B =B @ 0
395
! " 1 pffiffiffi pffiffiffi !# ρy1−α −k ξ k− ρy − 1=ξ pffiffiffiffiffiffiffiffiffiffiffi u− pffiffiffiffiffiffiffiffiffiffiffi − 1+ C β C 1−ρ 1−ρ C + 1C pffiffiffi ! ξ k− ρy A u− pffiffiffiffiffiffiffiffiffiffiffi 1+ β 1−ρ 1
pffiffiffi − 1=ξ ρ ξ B pffiffiffiffiffiffiffiffiffiffiffi ðy1−α −yÞ C B β 1−ρ C C =B pffiffiffi ! + 1C B @ ξ k− ρy A u− pffiffiffiffiffiffiffiffiffiffiffi 1+ β 1−ρ
! pffiffiffi ρ F ðuÞ ξ PD −ξ − 1=ξ pffiffiffiffiffiffiffiffiffiffiffi ðy = −yÞ + : PD β 1−ρ 1−α F ðuÞ
pffiffi k− ρy1−α ! ffi pffiffiffi F ðuÞ exp − βu + pffiffiffiffiffiffiffi β 1−ρ 1 ρ p ffiffiffiffiffiffiffiffiffiffi ffi = exp = ðy−y1−α Þ : pffiffi PD β 1−ρ ρy pffiffiffiffiffiffiffi ffi F ðuÞ exp − βu + k−
PDeα
β
1−ρ
Based on these results, we can obtain Eq. (24).
References Allen, L., DeLong, G., Saunders, A., 2004. Issues in the credit risk modeling of retail markets. Journal of Banking & Finance 28, 727–752. Altman, E.I., Resti, A., Sironi, A., 2002. The Link Between Default and Recovery Rates: Effects on the Procyclicality of Regulatory Capital Ratios. BIS Working Papers. Altman, E.I., Saunders, A., 1998. Credit risk measurement: developments over the last 20 years. Journal of Banking & Finance 21, 1721–1742. Altman, E.I., Resti, A., Sironi, A., 2003. Default Recovery Rates in Credit Risk Modeling: A Review of the Literature and Empirical Evidence. Working paper. Beirlant, J., Dierckx, G., Goegebeur, Y., Matthys, G., 1999. Tail index estimation and an exponential regression model. Extremes 2, 177–200. Brooks, C., Clareb, A.D., Persanda, G., 2000. A word of caution on calculating marketbased minimum capital risk requirements. Journal of Banking & Finance 24, 1557–1574. Carey, M., Hrycay, M., 2001. Parameterizing credit risk models with rating data. Journal of Banking & Finance 25, 197–270. Coles, S., 2001. An Introduction to Statistical Modeling of Extreme Values. Springer series in statistics. Springer-Verlag, London. Limited. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extreme Events for Insurance and Finance. Springer-Verlag, Berlin. Fernández, V., 2003a. Extreme Value Theory and Value at Risk. Revista de Análisis Económico 18–1, 57–85. Fernández, V., 2003b. Extreme Value Theory: Value at Risk and Returns Dependence Around the World. Documentos de Trabajo 161, Centro de Economía Aplicada, Universidad de Chile. Gatzert, N., Schmeiser, H., 2008. Combining fair pricing and capital requirements for non-life insurance companies. Journal of Banking & Finance 32, 2589–2596. Gordy, M.B., 2000. A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules. Board of Governors of the Federal Reserve System, Washington D.C. Grane, A., Veiga, H., 2008. Accurate minimum capital risk requirements: a comparison of several approaches. Journal of Banking & Finance 32, 2482–2492. Jacobson, T., Linde, J., Riksbank, S., 2005. Credit risk versus capital requirements under Basel II: are SME loans and retail credit really different? Journal of Financial Services Research 28, 43–75. Johnston, M., 2009. Extending the Basel II Approach to estimate capital requirements for equity investments. Journal of Banking & Finance 33, 1177–1185. Matthys, G., Beirlant, J., 2003. Estimating the extreme value index and high quantile with exponential regression models. Statistica Sinica 13, 853–880. McKinsey, 1993. Special Report on “The New World of Financial Services.”. The Mckinsey Quarterly 2. McNeil, A., Frey, R., 2000. Estimation of Tail-Related Risk Measures for Heterocedastic Financial Times Series: an Extreme Value Approach. Journal of Empirical Finance 7, 271–300. Perli, R., Nayda, W.I., 2004. Economic and regulatory capital allocation for revolving retail exposures. Journal of Banking & Finance 28, 789–809. Reiss, R.D., Thomas, M., 2001. Statistical Analysis of Extreme Values: With Applications to Insurance, Finance, Hydrology and Other Fields. Birkhäuser Verlarg, second edition. Basel, Switzerland. Repullwo, R., Suarez, J., 2004. Loan pricing under basel capital requirements. Journal of Financial Intermediation 13, 496–521.
Contents lists available at ScienceDirect
Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d
The calculation of capital requirement using Extreme Value Theory Ming-Shann Tsai a,⁎, Lien-Chuan Chen b a b
Department of Banking and Finance, National Chi Nan University, Puli, Nantou, Taiwan 54561 Department of Finance, National Central University, No. 300, Jung-da Rd., Jung-Li, Taiwan 320
a r t i c l e
i n f o
Article history: Accepted 13 August 2010 JEL classification: G21 G28 Keywords: Basel accord Extreme Value Theory Capital requirement
a b s t r a c t The Basel Committee has suggested some formulas for calculating capital requirement using the Advanced Internal Ratings-Based Approach. However, these formulas were derived under the assumption of a normal distribution. Thus, the capital requirement estimated by the Basel formula may be incorrect when the asset distributions are not normal. Using an analysis of qualifying revolving retail exposures as an example, this paper introduces a formula based on the Extreme Value Theory to calculate the capital requirement. This formula is more general and accurate than its predecessors, because it can be used with any type of distribution. Numerical examples are provided to demonstrate that the capital requirement estimated by the Basel formula is less than by our formula when the asset distribution has a heavy tail, and more than by our formula when the distribution has a short tail. Our formula is also more sensitive to risk than competing models in the context of the recent financial crisis. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Methods for managing credit risk have undergone dramatic developments in the last 20 years because of the worldwide increase in the number of bankruptcies and the dramatic growth in the use of off-balance-sheet instruments to hedge the inherent risk of default (McKinsey, 1993; Altman and Saunders, 1998). According to the theory of credit management, a bank needs to maintain a rational capital adequacy for reducing its bankrupt probability. To address this need, the Basel Committee of Banking Supervision suggested a common framework (i.e. the Basel Capital Accord) for calculating the capital adequacy. Because the Basel Capital Accord is suggested to be implemented worldwide, research on the best way to calculate capital requirement has recently become popular among both academic scholars and financial institutions. The recent version of the Basel Capital Accord is the “International Convergence of Capital Measurement and Capital Standards: A Revised Framework” (hereafter referred to as the RF). The RF suggests three methods for calculating the capital requirement of bank's credit risk: the Standardized Approach, the Foundation Internal RatingsBased Approach, and the Advanced Internal Ratings-Based Approach (hereafter referred to as the Advanced IRB Approach). In the studies related to Basel Capital Accord, the issues for accurately estimating the default probability and calculating the capital requirements have been widely discussed (Carey and Hrycay, 2001; Altman et al., 2002; Gordy, 2000; Altman, Resti and Sironi, 2003; Perli and Nayda, 2004).
⁎ Corresponding author. Tel.: + 886 49 2910960x4902; fax: +886 49 2914511. E-mail address: [email protected] (M.-S. Tsai). 0264-9993/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2010.08.010
Johnston (2009) derived a modified model for calculating the capital requirement that takes account of a bank's earnings. Repullwo and Suarez (2004) proposed a net interest income correction model for IRB Approach. Gatzert and Schmeiser (2008) extended the Basel formulas to address the issues of bank's equity investment. Most researchers calculate the credit exposure based on parameterized credit risk models. Grane and Veiga (2008) have argued that the assumption of a normal distribution is improper when estimating capital requirement. Brooks, Clareb and Persanda (2000) demonstrated that the use of GARCH-type models for the calculation of minimum capital risk requirements may lead to the production of inaccurate and therefore inefficient capital requirements. These authors give a simple modification to the model to improve the accuracy of minimum capital risk requirements estimates in both back- and out-of-sample tests. Some researchers have used nonparametric statistics to generate the credit–loss distributions needed to calculate the credit risk (Jacobson et al., 2005). The RF pays particular attention to retail credit1 because of the supposedly smaller exposure to systemic risk that it entails (Jacobson et al., 2005). In the RF, retail credit exposures are divided into three primary categories: (1) exposures secured by residential mortgages, (2) qualifying revolving retail exposures (hereafter referred to as QRRE), and (3) other non-mortgage exposures, also known as “other retail” (RF paragraphs 233). This paper uses the QRRE as an example to introduce a new formula for calculating the capital requirement. QRRE consists of unsecured revolving credits that exhibit appropriate loss characteristics. For example, the credit card is a kind of QRRE. A 1 The retail credit business in commercial banks involves consumer credit, residential mortgages, automobile loans, credit cards, and small business loans (Allen et al., 2004).
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
formula for calculating the capital requirement of QRRE is provided by the Advanced IRB approach. In so doing, bank can use a suitable distribution for estimating the probability default, because the distributions of financial data usually show heavy tails. The estimated default probability then is entered into the suggested formula for calculating the capital requirement. The RF takes into consideration the above-mentioned problems of skewed distributions in calculating the default probability. However, the calculation of the capital requirement may still be inaccurate because the formula for calculating the critical value of the default probability at a given confidence level is still derived on the assumption that the asset probability is normally distributed (Perli and Nayda, 2004). The purpose of this paper is to derive a formula for accurately calculating the capital requirement under a more general framework. The Extreme Value Theory has been widely applied in the finance and insurance literature to analyze data distributions characterized by extreme departure from normality with respect to skewness. Specific applications include portfolios, operational losses, catastrophic insurance claims, and credit losses (Embrechts et al., 1997; Fernández, 2003a,b; Coles, 2001; Reiss and Thomas, 2001; McNeil and Frey, 2000). To accurately calculate the capital requirement, we use the Extreme Value Theory to derive a formula for accurately calculating the critical value of default probability at a given confidence level. Our formula is more general and estimates capital requirement more accurately than previous models because it can be applied to any type of distribution (e.g., the distribution with heavy tails). We provide quantitative examples comparing our formula with that suggested by the RF. The results show that the RF formula can underestimate capital requirements when the sample distribution has a heavy tail and overestimate it when it has a short tail. Since 2007, the sub-prime mortgage crisis has kept US financial markets in turmoil and eventually created turbulence in financial markets worldwide. During this period, many famous banks and insurance companies, such as Lehman Brothers, Merrill Lynch, Fannie Mae, and Freddie Mac, have been overleveraged or burdened by debt, resulting in bankruptcies or takeovers. Because maintaining adequate capital is how banks avoid default, any formula that adequately estimates a bank's capital requirement must accurately reflect the huge turbulence in the financial markets. We also use a numerical example to show how our formula is more sensitive to risk when economic situations change. The remainder of this paper is organized as follows. In Section 2 we present the formula for calculating the capital requirement of QRRE as described in the New Basel Accord. We also present the formula for calculating the default probability based on the Extreme Value Theory (the RF approach). In Section 3, we use the Extreme Value Theory to derive a new formula for calculating the critical default probability at a given confidence level when the default rate distribution is nonnormal with respect to skewness. Section 4 includes quantitative examples comparing the formulas. Finally, in Section 5, we summarize our main findings and offer suggestions for future research. 2. Formulas for calculating default probabilities using the Extreme Value Theory The standard procedure for determining the QRRE can be described in terms of the following equations (defined in RF paragraph 329):
CR = VaRα ðPDÞ × LGD;
ð1Þ
VaRα ðPDÞ = PDα −PD;
ð2Þ
PDα = Φ
! pffiffiffi −1 ρΦ ðαÞ + Φ−1 ðPDÞ pffiffiffiffiffiffiffiffiffiffiffi ; and 1−ρ
Risk−weighted assets = CR × 12:5 × EAD;
391
ð3Þ
ð4Þ
where CR LGD EAD α PD
is the capital requirements, is the loss given default, is the exposure at default, is the confidence level, which is suggested to be 99.9% in the RF, is the probability of default, which can be estimated with a proper method, VaRα(PD) is the Value-at-Risk of the default probability under confidence level α, is the critical value of PD under α confidence level, PDα ρ is the correlation of consumers' assets, suggested as ρ = 0.04 in RF, Φ(⋅) is the cumulative normal distribution, and Φ − 1(⋅) is the inverse cumulative normal distribution function. RF suggests that a specific bank can use its own estimation methods to determine LGD, EAD and PD for calculating its capital requirement. In this section, we derive a formula for calculating the default probability, taking into account the skewness problems, based on the Extreme Value Theory. To begin, let Vi(T) represent the standardized asset value of debtor i at time T, and let k be a given critical asset value of bankruptcy. According to the traditional definition of bankruptcy, default occurs when Vi ðT Þbk. Therefore, the probability of default for an individual consumer can be described as follows: PDi = PrðVi ðT ÞbkÞ; i = 1; 2; :::N;
ð5Þ
where N is the number of consumers. Let a debtor's asset value be driven by a single common factor Y, such as a macroeconomic variable related to the business cycle (e.g. national income, consumer price index, stock index). We can describe the standardized asset value as follows (Perli and Nayda, 2004): Vi ðT Þ =
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ρY + 1−ρεi ;
ð6Þ
where εi is an individual variable. Assume that Y and εi are independent. Let F(⋅) be the cumulative conditional probability distribution of εi; G(⋅) be the cumulative conditional probability distribution of Y; and Q(⋅) be the cumulative conditional probability distribution of Vi(T). According to Eq. (3), Q(⋅) is a combination of the distributions of Y and εi. In the study of Perli and Nayda (2004), F(⋅), G(⋅), and Q(⋅) were assumed to be standard normal distributions for deriving the formula suggested in the RF (hereafter referred to as the RF formula). Here we relax this assumption and support a more general formula for estimating the default probability. The default probability contingent on the realization of the common factor (Y = y) can be described as follows: pffiffiffi ! k− ρy PD ≡ PDð yÞ = PrðVi ðT Þb k j Y = yÞ = Pr εi b pffiffiffiffiffiffiffiffiffiffiffi ≡ F ðsÞ; 1−ρ where s =
ð7Þ
pffiffi k− ρy pffiffiffiffiffiffiffiffi . 1−ρ
This default probability, F(s), can be estimated from the Extreme Value Theory. In practice, there are two methods for analyzing
392
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
extreme events: the block maxima with a general extreme value and the peaks-over-threshold with a general Pareto distribution (POT method). Because modeling all the block maxima is wasteful when other extreme values are available, the POT method is a more efficient and precise approach to model the behavior of extreme values above the threshold (Fernández, 2003a,b; Matthys and Beirlant, 2003). We therefore use the POT method to analyze the default probability.2 The POT method requires a sufficiently small threshold u, u N s, for analysis.3 The left-tail excess distribution is defined as follows:
Fu ðu−sÞ = Prðs≤εi ≤uj εi buÞ = 1−
F ðsÞ ; F ðuÞ
ð8Þ
where: u Fu ð⋅Þ F(u)
is the threshold value of εi that is exogenous, is the cumulative left-tail excess probability distribution of εi, and is the empirical cumulative probability of εi below the threshold u.
According to Eq. (8), F(s) = F(u)(1 − Fu(u − s)). To estimate F(s), it is necessary to first estimate F(u) and Fu(u − s). In general, F(u) is determined empirically and non-parametrically; that is F ðuÞ = Nn , where n is the number of observations that fall below the threshold u. Fu ðu−sÞ can be estimated using a theorem from the Extreme Value Theory. This theorem shows that Fu ðu−sÞ approaches a generalized Pareto distribution (Hξ, β(u − s)) given a sufficiently small u (Embrechts et al., 1997). This distribution can be described as follows:
Hξ;
8 − 1=ξ ξ > > ; if ξ≠0 < 1− 1 + ðu−sÞ ; β β ðu−sÞ = u−s > > ; if ξ = 0 : 1− exp − β
ð9Þ
where ξ is the shape parameter of Hξ; β ðu−sÞ, β is the location parameter of Hξ; β ðu−sÞ, and u − s ≥ 0 when ξ ≥ 0, and 0≤u−s≤− βξ when ξ b 0.
8 > > > > > <
1=ξ ξ for ξ≠0 1 + ðu−sÞ β
F ðsÞ = > F ðuÞ > > u−s > > : exp − β
:
ð11Þ
for ξ = 0
Finally, based on the Extreme Value Theory, the conditional default probability is derived as follows: 8 pffiffiffiffiffiffi!!− 1=ξ > n ξ k− ρy > > p ffiffiffiffiffiffiffiffiffiffi ffi for ξ≠0 1 + u− >
ð12Þ
Eq. (12) is a formula for calculating the default probability on the basis of the Extreme Value Theory. In this equation, eight parameters (i.e., N, n, y, ρ, k, ξ, β and u) need to be determined when estimating the probability of default. The following will describe how to obtain these parameters. N, the number of consumers, is a known value. Because n is the number of the observations that fall below the threshold u, it can be determined as soon as u is determined (see later discussions). y is the current value of a macroeconomic variable (e.g., stock return) related to the business cycle. ρ is the correlation of consumers' assets. RF is assigned the value of ρ equals to 0.04, but this value can be adjusted based on an empirical evaluation of the relation between individual's wealth and the common factor. For example, pffiffiffi pffiffiffi , a bank can define ρ as the effect of the changes of because ρ = ∂V ∂Y macroeconomic variable on borrower's wealth. Because V b k refers to the default event, the value of k is critical for determining when the debtor must file for bankruptcy. In practice, we suggest that k can be an average standardized loan value in a bank. The parameters ξ and β can be estimated using the nonlinear least square method or the maximum likelihood method. For example, when using the nonlinear least square method, a bank may collect a yt, t = 1, ⋯, T, where T is a sample set of the time series of kt, ρt and pffiffiffiffi kt − ρt yt period, to obtain the series of st st = pffiffiffiffiffiffiffiffiffi . ξ and β can be 1−ρt
In theory, the general Pareto distribution can be subdivided into three types. (1) If ξ = 0, then Hξ, β(u − s) is defined as a Gumbel distribution. This specification is used when analyzing the distribution with thin tails (e.g. normal, log-normal, exponential, and gamma). (2) If ξ N 0, then Hξ, β(u − s) is defined as a Fréchet distribution, used when analyzing the distribution with heavy tails (e.g. Pareto, Cauchy, Student t, mixed models). (3) If ξ b 0, then Hξ, β(u − s) is defined as a Weibull distribution, used when analyzing the distribution with short tails. Because Fu(u − s) ≈ Hξ, β(u − s) given a sufficiently small u, we have: 8 1=ξ ξ > > for ξ≠0 < 1− 1 + ðu−sÞ ; β Fu ðu−sÞ = u−s > > for ξ = 0 : 1− exp − β
According to Eq. (8), we obtain:
ð10Þ
2 When using the POT method, an extreme value is generally defined as occurring on the right tail of the distribution (Embrechts et al., 1997). It should be noted that we define default as a debtor's assets falling below the threshold; therefore, we focus on the left-tail of the distribution. This focus should be kept in mind when deriving the formulas. 3 To maintain u N s, the range of the common factor y must be taken into account.
estimated by minimizing the sum of squared error between the data of default probability and the outcome of solving Eq. (12). The model of nonlinear least square method can be described as follows: 0 0 112 8 1=ξ > T > ξ > min @PDt −Ft ðuÞ@1 + ðu−st Þ AA for ξ ≠ 0 > > < ξ ; β t∑ β =1 ; 2 > T > > u st > >min ∑ PDt −Ft ðuÞ exp − + for ξ = 0 : β β β t =1
ð13Þ
According to Eq. (13), one can obtain the estimated parameters (i.e. ξ and β) with nonlinear estimation method if the threshold u is decided. In practice, researchers generally assume that the threshold u is an exogenous variable. Recently, some scholars have argued that the optimal value of u can be defined as the point at which the mean squared error of the estimators is minimized (Beirlant et al., 1999; Matthys and Beirlant, 2003). Based on this reasoning, we suggest that u be selected arbitrarily so long as it satisfies the initial condition u N s; this initial value is then increased until the optimal threshold (u) is reached. For example, let u = s + Δs, s + 2Δs, s + 3Δs,…, and so on, where Δs is a small interval; then, the optimal threshold is the point at which the mean squared error of the estimators is minimized. Our model is thus viable, because a bank with access to its retail credit data can assign values to all the necessary parameters. We have shown above how a default probability, PD, can be estimated using the Extreme Value Theory. This result can then be used to calculate
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
the capital requirement of QRRE, calculated on the assumption of a nonnormal asset distribution, in the RF (i.e., Eqs. (1)–(3)). However, the calculated value could be incorrect, because the RF formula shown in Eq. (3) (i.e., PDα) was derived under the assumption that the asset follows a normal distribution. In the next section, we use the Extreme Value Theory to derive a formula for accurately calculating PDα. 3. A formula for calculating the critical value of the default probability based on the Extreme Value Theory Let X be a random variable designating the default rate. The cumulative probability distribution of X is expressed as follows: ∞
∞
−∞
−∞
Mð X Þ = ∫ ðX = PDð yÞ≤qÞgð yÞdy = ∫ Ipð yÞ≤q g ð yÞdy;
ð14Þ
where M(X) q g(y)
is the cumulative probability of the default rate, is the certainty level of PD, and is the probability density of y.
Assuming ξ ≠ 0 from Eq. (12), we can describe PD(y) ≤ q as follows: pffiffiffi !!−1= ξ n ξ k− ρy ≤q: 1+ u− pffiffiffiffiffiffiffiffiffiffiffi N β 1−ρ
ð15Þ
393
Comparing Eqs. (19) and (20) with Eq. (12) we may find that to calculate the critical value of PDeα it is necessary only to replace the common factor y by y1-α. Therefore, the formula for estimating the capital requirement can be described as follows: e e CR = VaR PDα × LGD = PDα −PD × LGD;
ð21Þ
Risk-weighted assets = CR × 12.5 × EAD, and
(22)
8 !! pffiffiffi − 1=ξ > y1−α −k > > n 1 + ξ u + ρp ffiffiffiffiffiffiffiffiffiffi ffi for ξ≠0; > >
ð23Þ
In Eq. (23), it can be seen that if we want to calculate the Value-at-Risk of the default probability, the critical value of PDeα must be derived by assuming the worst possible situation for the common factor at confidence level 1−α. This formula seems to make more economic sense than Eq. (3). When calculating the capital requirement, we must consider not only the estimated default probability of the bank, but also the expected risk following from the common economic factors. Because the default probability PD has been estimated using the Extreme Value Theory, the critical value of PD at confidence level α must also be estimated by the same theory. Based on this reasoning, we believe that our model is more general and accurate than previous models. 4. Numerical examples
From Eq. (15), we have the following: !! pffiffiffiffiffiffiffiffiffiffiffi −ξ k β N 1−ρ −1 : y ≥ pffiffiffi q −u + pffiffiffiffiffiffiffiffiffiffiffi + n ρ ξ 1−ρ
ð16Þ
From Eqs. (14) and (16), we obtain:
Mð X Þ = 1−G
!!! pffiffiffiffiffiffiffiffiffiffiffi −ξ k β N 1−ρ −1 : ð17Þ q −u + pffiffiffiffiffiffiffiffiffiffiffi + pffiffiffi n ρ ξ 1−ρ
Combining all these results leads to a cumulative probability function for the default rate at confidence level α, based on the Valueat-Risk Theory: !!! pffiffiffiffiffiffiffiffiffiffiffi −ξ 1−ρ k β N q α = M ðXα Þ = 1−G −1 ; −u + pffiffiffiffiffiffiffiffiffiffiffi + pffiffiffi ρ n α ξ 1−ρ
In this section, we present numerical examples that compare capital requirement calculated using our formula and using the Basel formula. We also show that our formula, when applied to a bank during the recent financial crisis, is more risk-sensitive than the Basel formula. Assume the current probability of default (PD) to be 0.05. Let LGD = 0.45, ρ = 0.04, and α = 99.9%.4 Following from these assumptions and the formulas in RF (see Eqs. (1)–(4)), we have PDα = 0.1473 and CR = 0.0438. Now we demonstrate that when the distribution of asset shows heavy (short) tails, the RF formula underestimates (overestimates) the values of PDα and CF as compared with our model. For analysis purposes, we rewrite the formulas for calculating PDeα as follows (see Appendix A): 8 ! pffiffiffi > ξ PD −ξ − 1=ξ ρ > > p ffiffiffiffiffiffiffiffiffiffi ð y Þ + for ξ ≠ 0 VaR F ð u Þ − > 1−α > β 1−ρ F ðuÞ < e PDα = ! pffiffiffi > > ρ > > PD × exp 1 pffiffiffiffiffiffiffiffiffiffi > ð y Þ for ξ = 0 VaR 1−α : β 1−ρ
;
ð18Þ
ð24Þ
where qα is a critical value of probability under the confidence level α. We define y1 − α = G − 1(1 − α) as the critical value for the common factor that applies in the worst possible situation, at confidence level 1 − α. As soon as we assign a value to α (e.g. α = 99.9% in RF), we can apply the Extreme Value Theory to derive the critical value for the default probability at confidence level α (PDeα) as follows:
where VaR1 − α(y) = y − y1 − α is the Value-at-Risk of the common factor at confidence level 1 − α. Because y and y1 − α represent the current and the worst economic situations respectively, we selected the 1 year return on the S&P 500 stock index as the common factor. After generating the return distribution, we let y be the Z-score representing the current index return and y1 − α be the 0.1th percentile of the standardized return distribution. Using the data of the S&P 500 stock index from 10/4/2004 to 10/3/2007, we found that, on the latter date, y=−0.6221 and y1 − α =−4.1088. For simplicity, we let the probability that falls below the threshold u as 1.5 times the default probability, that is, F ðuÞ = 1:5 × PD.5 For comparison purposes, we use
e PDα
n ξ ≡ qα = 1+ u+ N β
!!−1 = : pffiffiffi ξ ρy1−α −k pffiffiffiffiffiffiffiffiffiffiffi 1−ρ
ð19Þ
Likewise, if ξ = 0, we obtain: ! pffiffiffi n u k− ρy1−α e pffiffiffiffiffiffiffiffiffiffiffi : PDα = exp − + N β β 1−ρ
ð20Þ
4 These values are suggested in the Foundation IRB Approach. Banks can also estimate the value when using the Advanced IRB Approach. 5 We selected this value (i.e., 1.5 times) for illustrative purpose. The main conclusion is the same as when using the other value.
394
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
Fig. 1. The relationship between the shape parameter and the critical value of the default probability. Note: The Y-axis shows the critical value of default probability, PDeα. The X-axis shows the shape of the distribution represented by ξ, which ranges from −0.6 to 0.6. The key parameters are as follows: PD = 0.05, ρ = 0.04, α = 99.9%, β = 0.6586, y = − 0.6221 and y1 − α = − 4.1088. F ðuÞ = 1:5 × PD for calculating PDαe .
the RF calculation (PDα =0.1473) to obtain the location parameter of the Pareto distribution (i.e., β). Because ξ=0 can be used for analyzing a normal distribution, we assign PDeα =0.1473 to estimate β given ξ=0. According to Eq. (19), we obtain β=0.6586 in this circumstance. We then use this value to calculate PDeα given ξ≠0. These estimates allow us to determine the relationship between PDeα and ξ. For example, if ξ=0.4 (indicating a heavy tail), PDeα =0.1572. In contrast, if ξ=−0.4 (indicating a short tail), PDeα =0.1397. According to these results, the value of PDeα is greater than 0.1473 when the asset distribution have heavy tail and less than 0.1473 when the tail is short. These results demonstrate that we need a more accurate formula for risk management than that provided by the RF. The relationship between the shape of the distribution ξ and the critical value of default probability PDeα is illustrated in Fig. 1. Let ξ range from −0.6 to 0.6. The horizontal line in the figure is equal to 0.1473, which is the calculated PDα using the suggested formulae in the RF (in the case of normal distribution). The line that stands for ξ N 0 is the calculated PDeα when asset distribution has a heavy tail. This clearly demonstrates that the estimated exposure is less using the Basel formula than using our formula when the asset distribution has this shape. In contrast, the line for ξ b 0 is the calculated PDeα if asset distribution has a short tail. In this case, it is equally clear that the estimated exposure is higher using the Basel formula than our formula. As for the capital requirement, we find that CR = 0.0482 when ξ = 0.4 and 0.0404 when ξ = − 0.4. This value is higher than the value calculated using the Basel formula when the asset distribution has a heavy tail and lower when the asset distribution has a short tail. The relationship between the shape of the distribution ξ and the capital requirement CR is shown in Fig. 2. Just as for Fig. 1, we conjecture that the estimated CR using the Basel formula is higher than that using our formula when the asset distribution has a heavy tail and lower when the distribution has a short tail. During the period of sub-prime mortgage crisis, a bank's capital requirement must accurately and fastly reflect the huge turbulence in the financial markets. Data from the S&P 500 stock index from 4/2/2007 to 4/2/2010 yield y1 − α = − 4.7367, meaning that the actual economic situation in the US is worse than in our previous example. However, the traditional Basel formula failed in this regard during the recent financial crisis. Our formula can accurately and fastly reflect the huge turbulence in the financial markets because it has considered the Value-at-Risk of the common factor. Fig. 3 shows the relationship between the critical value of common factor y1 − α and the capital requirement CR given the same parameters
Fig. 2. The relationship between the shape parameter and capital requirement. Note: The Y-axis represents the capital requirement,CR. The X-axis represents the shape of the assets distribution represented by ξ. LGD = 0.45 for the calculation of CR. For the other definitions, see notes for Fig. 1.
used in our previous example. Let y1 − α range from −5.5 to −3.5. The horizontal dotted line in Fig. 3 represents the capital requirement calculated from the traditional Basel formula. We compare this estimate to two estimates calculated from our model, one where the asset shows a heavy-tailed distribution (ξ = 0.4) and one where it shows a shorttailed distribution (ξ = − 0.4). The “+” line shows the capital requirement calculated from our formula when ξ = 0.4 The “*” line shows the capital requirement when ξ = − 0.4. As shown in Fig. 3, the capital requirement calculated from the Basel formula is 0.0438 regardless of how the economic situation changes. In our model, the capital requirement is negatively related to y1 − α in both cases. That is, if the economic situation is worse, our model indicates that a bank needs to prepare more capital for avoiding bankruptcy; if the situation is better, the bank needs less capital. Thus, our formula is more risksensitive than the Basel model when used for risk management. Turning to the example of the sub-prime mortgage crisis, we find that because y1 − α = − 4.7367 (the vertical dotted line in Fig. 3), the capital requirement calculated by our formula is 0.0503 if ξ = − 0.4 and 0.0707 if ξ = 0.4. Thus, the estimated capital requirement is greater with our model than with the Basel model. This implies that if used during the sub-prime mortgage crisis, our formula would provide a bank with greater protection against credit risk than the Basel formula. Even when the distribution for a given bank has a short tail, our formula estimates more capital requirement than the Basel formula when the economic situation is poor.
Fig. 3. The relationship between the worst economic situation and capital requirement. Note: The Y-axis represents the capital requirement, CR. The X-axis represents the critical value of common factor, y1 − α. The “+” and " " lines indicate the calculated CRs for ξ= 0.4 and ξ= − 0.4, respectively, under different economic conditions. The vertical dotted line denotes y1 − α = − 4.7367, and the horizontal dotted line denotes CR calculated using the traditional Basel formula (0.0438). For the other definitions, see notes for Fig. 2.
M.-S. Tsai, L.-C. Chen / Economic Modelling 28 (2011) 390–395
Likewise, assume ξ = 0, we have
5. Conclusion An accurate estimation of the capital requirement is important for a bank to reduce its bankruptcy probability. The RF formula for calculating the critical default probability at a given confidence level was derived under the assumption that the default rate follows a normal distribution. If that is the case of normal distribution, a bank can successfully estimate the default probability by using the Advanced IRB Approach to calculate the capital requirement. On the other hand, the capital requirement may be underestimated if the distribution has a heavy tail and overestimated if it has a short tail. Thus, in either of these cases, the RF formula is likely to inaccurately estimate the bank's exposure. In this paper, we have presented a new formula, based on the Extreme Value Theory, for calculating the critical value of the default probability at a given confidence level. This formula gives more accurate estimates than the formula associated with the Advanced IRB Approach when the tails of the financial data distribution deviate from normal in terms of skewness. We have illustrated numerical examples to show that the Basel formula tends to underestimate the capital requirement when the asset distribution has a heavy tail and underestimate it when the distribution has a short tail. Finally, we showed that our formula is more risk-sensitive than the Basel model when the economic situation changes, such as during the recent financial crisis. In future research, real data could be used to estimate parameters such as ξ and β before applying our formula to estimate the capital requirement. Although the discussion of our formula was limited to the example of QRRE, the formula can also be well applied to other risk management contexts, for example, the calculation of other kinds of exposures addressed by the Basel Accord. Appendix A This appendix shows how to obtain Eq. (24). Assume ξ ≠ 0, from Eqs. (12) and (23), we have the following: ! 0 1 pffiffiffi − 1=ξ ρy1−α −k ξ p ffiffiffiffiffiffiffiffiffiffi ffi B1 + β u + C e B C 1−ρ PDα C =B pffiffiffi ! −1 + 1C B PD ξ k− ρy @ A u− pffiffiffiffiffiffiffiffiffiffiffi 1+ β 1−ρ 0
ξ B1 + β u + B B =B @ 0
395
! " 1 pffiffiffi pffiffiffi !# ρy1−α −k ξ k− ρy − 1=ξ pffiffiffiffiffiffiffiffiffiffiffi u− pffiffiffiffiffiffiffiffiffiffiffi − 1+ C β C 1−ρ 1−ρ C + 1C pffiffiffi ! ξ k− ρy A u− pffiffiffiffiffiffiffiffiffiffiffi 1+ β 1−ρ 1
pffiffiffi − 1=ξ ρ ξ B pffiffiffiffiffiffiffiffiffiffiffi ðy1−α −yÞ C B β 1−ρ C C =B pffiffiffi ! + 1C B @ ξ k− ρy A u− pffiffiffiffiffiffiffiffiffiffiffi 1+ β 1−ρ
! pffiffiffi ρ F ðuÞ ξ PD −ξ − 1=ξ pffiffiffiffiffiffiffiffiffiffiffi ðy = −yÞ + : PD β 1−ρ 1−α F ðuÞ
pffiffi k− ρy1−α ! ffi pffiffiffi F ðuÞ exp − βu + pffiffiffiffiffiffiffi β 1−ρ 1 ρ p ffiffiffiffiffiffiffiffiffiffi ffi = exp = ðy−y1−α Þ : pffiffi PD β 1−ρ ρy pffiffiffiffiffiffiffi ffi F ðuÞ exp − βu + k−
PDeα
β
1−ρ
Based on these results, we can obtain Eq. (24).
References Allen, L., DeLong, G., Saunders, A., 2004. Issues in the credit risk modeling of retail markets. Journal of Banking & Finance 28, 727–752. Altman, E.I., Resti, A., Sironi, A., 2002. The Link Between Default and Recovery Rates: Effects on the Procyclicality of Regulatory Capital Ratios. BIS Working Papers. Altman, E.I., Saunders, A., 1998. Credit risk measurement: developments over the last 20 years. Journal of Banking & Finance 21, 1721–1742. Altman, E.I., Resti, A., Sironi, A., 2003. Default Recovery Rates in Credit Risk Modeling: A Review of the Literature and Empirical Evidence. Working paper. Beirlant, J., Dierckx, G., Goegebeur, Y., Matthys, G., 1999. Tail index estimation and an exponential regression model. Extremes 2, 177–200. Brooks, C., Clareb, A.D., Persanda, G., 2000. A word of caution on calculating marketbased minimum capital risk requirements. Journal of Banking & Finance 24, 1557–1574. Carey, M., Hrycay, M., 2001. Parameterizing credit risk models with rating data. Journal of Banking & Finance 25, 197–270. Coles, S., 2001. An Introduction to Statistical Modeling of Extreme Values. Springer series in statistics. Springer-Verlag, London. Limited. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extreme Events for Insurance and Finance. Springer-Verlag, Berlin. Fernández, V., 2003a. Extreme Value Theory and Value at Risk. Revista de Análisis Económico 18–1, 57–85. Fernández, V., 2003b. Extreme Value Theory: Value at Risk and Returns Dependence Around the World. Documentos de Trabajo 161, Centro de Economía Aplicada, Universidad de Chile. Gatzert, N., Schmeiser, H., 2008. Combining fair pricing and capital requirements for non-life insurance companies. Journal of Banking & Finance 32, 2589–2596. Gordy, M.B., 2000. A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules. Board of Governors of the Federal Reserve System, Washington D.C. Grane, A., Veiga, H., 2008. Accurate minimum capital risk requirements: a comparison of several approaches. Journal of Banking & Finance 32, 2482–2492. Jacobson, T., Linde, J., Riksbank, S., 2005. Credit risk versus capital requirements under Basel II: are SME loans and retail credit really different? Journal of Financial Services Research 28, 43–75. Johnston, M., 2009. Extending the Basel II Approach to estimate capital requirements for equity investments. Journal of Banking & Finance 33, 1177–1185. Matthys, G., Beirlant, J., 2003. Estimating the extreme value index and high quantile with exponential regression models. Statistica Sinica 13, 853–880. McKinsey, 1993. Special Report on “The New World of Financial Services.”. The Mckinsey Quarterly 2. McNeil, A., Frey, R., 2000. Estimation of Tail-Related Risk Measures for Heterocedastic Financial Times Series: an Extreme Value Approach. Journal of Empirical Finance 7, 271–300. Perli, R., Nayda, W.I., 2004. Economic and regulatory capital allocation for revolving retail exposures. Journal of Banking & Finance 28, 789–809. Reiss, R.D., Thomas, M., 2001. Statistical Analysis of Extreme Values: With Applications to Insurance, Finance, Hydrology and Other Fields. Birkhäuser Verlarg, second edition. Basel, Switzerland. Repullwo, R., Suarez, J., 2004. Loan pricing under basel capital requirements. Journal of Financial Intermediation 13, 496–521.