GENERAL FRAMEWORK

In this section we recall the factor model for portfolio credit risk, which forms a foundation for regulatory and economic capital methodologies. Subsequently, we present these frameworks and highlight the difference between them.

Risk-factor model for portfolio credit risk

We consider a credit portfolio of n exposures, where the ith exposure has a principal amount A_iover a specific time horizon (eg, one year).

All amounts A_i are assumed to be known and non-random. The weight w_i of exposure i in the portfolio is given by a ratio of its principal to the total principal of the portfolio

w_i= '_nAi i=1Ai

For simplicity, we assume that every obligor has only a single liability and, in the case of the obligor’s default, the fraction of principal amount which is lost (loss given default, LGD) is deterministic and equals LGDi. We consider a book-value approach to risk, with losses arising from a default event only. Obligor i defaults if its asset value V_i falls below some threshold c_i. The total portfolio loss fraction (hereafter, portfolio loss) L is expressed as

L=

n i=1

wiLGDi1_{Vi
ci} (6.1) Asset value Viis modelled as a random variable, which depends on a vector X of d systemic risk factors and on random variable ε_i, which represents the idiosyncratic risk

V_i= ρiX+ σiε_i, (6.2)

where a d-dimensional vector of factor loadings ρi ∈ R^d with 0 <ρi < 1 defines the dependency of Vion systemic risk factors.

To stay within the Gaussian framework, we assume that systemic risk X is a d-dimensional standard normal vector with a correlation matrix Ω, and idiosyncratic risk factors ε_iare independent, identi-cally distributed standard normal random variables independent of X. By setting the factor loading σ_iof the idiosyncratic risk factor to be equal to

ε_i=

1− ρ^T_iρ_i

we ensure that Viare also standard normal distributed.

Under the independency assumption (εi is independent of X), the conditional probability p_i(x)that obligor i defaults for a given realisation x of X is given by

pi(x)= Pr[Vi<ci| X = x] where Φ denotes the cumulative distribution function of standard normal distribution.

Regulatory capital

The regulatory capital requirements are calculated according to the advanced Internal Rating Based Approach (IRBA) for corporate loans as suggested by the Basel Committee on Banking Supervi-sion (2004). The IRBA assumes a single risk factor for systemic risk (d= 1) and infinitely fine-grained credit portfolio, which reduces the general multi-factor risk model outlined in the previous section to the asymptotic single risk factor (ASRF) model. Gordy (2003) shows that, in an ASRF framework, portfolio loss L conditional on X con-verges in probability to its conditional expectation E[L | X = x], ie

L[X= x] almost surely

−

−−−−−−−→ E[L | X = x] =

n i=1

wiLGDipi(x)

with p_i(x) specified by Equation 6.3 We define the mapping l :R → (0, 1) as

l(x)= E[L | X = x] =

n i=1

w_iLGD_ip_i(x)

Loss function l(x) is strictly decreasing and continuous, and thus we express a distribution of L by

Pr[L x] = 1 − Φ(l⁻¹(x)) with x∈ (0, 1) (6.4) The α-quantile of loss distribution L, which corresponds to the value-at-risk (VaR) is given by

VaRα(L)= qα(L)= inf(

In accordance with results of Tasche (1999), the contribution of the ith exposure VaR_i,αto portfolio VaRα(L) can be attained by partial differentiation

where threshold ci is determined by the unconditional default probability pdias ci= Φ⁻¹(pdi).

The supervisory capital charges K_ifor unexpected losses are mea-sured as a difference between VaR at a confidence level of 99.9%

and expected losses EL_i(where EL_i= LGDi× pdi) amended by the maturity adjustment function MA(pd_i, M_i). They are given by

Ki= (VaRi,99.9%− ELi)MA(pdi, Mi) (6.6) The Basel maturity adjustment function MA(pd, M) addresses the higher risk of loans with longer maturities, which is ignored by the VaR in a one-period risk-factor framework. It is modelled as an increasing function of maturity M, taking a value of 1 for one-year maturities and decreasing with respect to pd.³Figure 6.1 shows the behaviour of the IRBA maturity adjustment function.

To make Equation 6.6 operative and comparable among differ-ent institutions, the Basel Committee sets factor loadings ρi to be modelled as an exponentially decreasing function of pdi, bounded between 12% and 24%⁴

ρ_i= ρ(pdi)= 24%(1 − ai)+ 12%ai with a_i= 1− e^−50pdi

1− e⁻⁵⁰ (6.7)

Figure 6.1 IRBA maturity adjustment as function of pd and maturity

We calculate economic capital with a multi-factor CreditMetrics-type portfolio model for a one-period time horizon and default-only mode. The loss distribution of the portfolio is generated by Monte Carlo simulations of idiosyncratic and systemic risk factors. In addi-tion to the portfolio described in the previous secaddi-tion, there is only one input parameter to be specified: the correlation matrix of sys-temic risk factors Ω. We derive this correlation from daily returns of corresponding MSCI equity indexes for the period of one year from June 2008 to June 2009.

To establish comparability with the IRBA framework, the eco-nomic capital of the portfolio is measured as VaR at a confidence level of 99.9%. The allocation of capital to sub-portfolios in a multi-factor framework is a more subtle matter than in the ASRF framework:

VaR is not subadditive and as such is not a coherent risk measure in the sense of Artzner et al (1999). With expected shortfall (ES), Acerbi and Tasche (2001) introduced a coherent alternative to VaR. Expected shortfall is a risk measure that describes an average loss in the worst cases. It can be calculated for smooth loss distribution L as

ESα(L)= 1 1− α

₁

αVaR_u(L) du (6.8) To allocate the total portfolio VaR VaRα(L) we choose a confidence level β, resulting in the following relation

ESβ(L)= VaRα(L) (6.9)

Finally, we allocate

ESβ(P)=

i∈P

ESβ(L_i)

Table 6.1 Asset correlation in IRBA and multi-factor models Regulatory Economic approach (%) approach (%)

Maximum 23.8 65

Average 21.2 10.6

Weighted average^† 21.3 15.3

Minimum 12.0 ₋10.9

†Netto-exposure weighted average correlation.

units of capital (adjusted for expected losses) to a sub-portfolio P. For the benchmark portfolio P used in this study, we found numerically that β equals 99.7%, satisfying Equation 6.9 for the regulatory fixed confidence level α of 99.9%.⁵

Differences between both frameworks

The most striking difference between economic and regulatory frameworks, multi-factor versus single-factor modelling, appears to be an “innocent” one. Thus, Tarashev and Zhu (2007) showed that a single risk-factor model performs sufficiently well by a proper parameter calibration.

We examined other input parameters: maturity adjustment and correlation. In the default mode of the CreditMetrics-type model, longer maturities play hardly any role, as the model is only a one-period model. The second parameter, correlation, is defined as a cor-relation of systemic risk factors and is independent of the obligor’s default probabilities.

Recent studies on credit risk (eg, Tarashev and Zhu (2007) and Rosenow et al (2007)) have shown that calibration of correlation parameter has a far-reaching impact on portfolio risk.⁶ Table 6.1 illustrates the properties of asset correlations in a portfolio calculated within regulatory and economic frameworks.