Abstract
Consider a credit portfolio with the investments in various sectors and exposed to an external stochastic environment. The portfolio loss due to defaults is of critical importance for social and economic security particularly in times of financial distress. We argue that the dependences among obligors within sectors (intradependence) and across sectors (interdependence) may coexist and influence the portfolio loss. To quantify the portfolio loss, we develop a multi-sector structural model in which a multivariate regular variation structure is employed to model the intradependence within sectors, and the interdependence across sectors is implied in the arbitrarily dependent macroeconomic factors, although, given them, obligors in different sectors are conditionally independent. We establish some sharp asymptotic formulas for the tail probability and the (tail) distortion risk measures of the portfolio loss. Our results show that the portfolio loss is mainly driven by the latent variables and the recovery rate function, and is also potentially affected by the macroeconomic factors and the intradependence within sectors. Moreover, we implement intensive numerical studies to examine the accuracy of the obtained approximations and conduct some sensitivity analysis.
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Notes
For example, during the worst three-year period of the Great Depression, the annual default rates totalled to \(12.88\%\) of the total par value of the corporate bond market; and even worse, in the the railroad crisis of 1873–1875, the total defaults amounted to \(35.90\%\), see Giesecke et al. (2011).
See page 19 of the Global Financial Stability Report of IMF, available at https://www.imf.org/en/Publications/GFSR/Issues/2022/10/11/global-financial-stability-report-october-2022.
The Annual Report 2021 of AIG reported that its corporate debt securities are mainly invested in the sectors including finance, utilities, communications, and energy, see page 122, available at https://www.aig.com/content/dam/aig/america-canada/us/documents/investor-relations/2022/aig-annual-report-2021.pdf.
High-level quantiles and related risk measures of the portfolio LGD are of considerable interest in economic capital assessment. To maintain its superior credit quality, the Bank for International Settlements (BIS) sets economic capital measured at the 99.99% confidence level assuming a one-year holding period. See the Annual Report 2020/2021 of BIS, available at https://www.bis.org/about/areport/areport2021.pdf.
For two random vectors \(\varvec{\xi }=(\xi _1,\dots , \xi _d)\) and \(\varvec{\eta }=(\eta _1,\dots , \eta _d)\), \(\varvec{\xi }\) is stochastically dominated by \(\varvec{\eta }\), written as \(\varvec{\xi }\le _{st} \varvec{\eta }\), if \( P(\varvec{\xi }\in \Delta )\le P(\varvec{\eta }\in \Delta ) \) holds for every increasing set \(\Delta \subset \mathbb R^d\) (namely, a set whose indicator function is component-wise increasing). Particularly, in the univariate case, \(\xi \le _{st}\eta \) if and only if \(P(\xi>x)\le P(\eta >x)\) for every \(x\in \mathbb R\). See Section 17.A of Marshall et al. (2011) for this concept and related discussions.
The dataset collects the returns data during the period from April 2, 1992 to February 27, 2004, rendering 3,106 return observations per bank.
The dataset consists of the daily stock returns data of top 20 technology firms from 2 April 1992 to 31 December 2019.
VaR is popular in the banking industry and is used for both external and internal reporting purposes.
CTE has already replaced VaR in regulatory requirements of, e.g., Canada, Israel, and Switzerland. On the insurance side, CTE has been regarded as the key risk measure, since the adoption of the C-3 Phase II revision to the regulatory risk-based capital model for variable annuities in 2005, and the implementation of the analogous principles-based reserving methodology in 2009 by the National Association of Insurance Commissioners, as well as the analogous reserve and capital methodology for life insurance products in 2014 by the Life Reserves Work Group and the Life Capital Work Group.
Practically, the regulatory requirement in Canada, Israel, and Switzerland is \(q = 0.99\) over a one-year time horizon, and the Solvency II Accord designed by the EU Commission sets \(q = 0.995\) over a one-year time horizon. See, e.g., Asimit et al. (2011).
The Basel Accord banking book risk measure under certain conditions is asymptotically equivalent to the \(99.9\%\) VaR, see Gordy (2003). The Basel II and Basel III risk measures for trading books are both special cases of VaR, see Basel Committee on Banking Supervision (2006); Basel Committee on Banking Supervision (2011).
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Acknowledgements
Yang Yang acknowledges the financial support by the National Social Science Fund of China (No. 22BTJ060), the Humanities and Social Sciences Foundation of the Ministry of Education of China (No. 20YJA910006), Natural Science Foundation of Jiangsu Province of China (No. BK20201396), and Natural Science Foundation of the Jiangsu Higher Education Institutions (No. 23KJA110002). Zhimin Zhang acknowledges the financial support by the the National Natural Science Foundation of China (Nos. 11871121, 12271066, 12171405).
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Appendix A: Proofs
Appendix A: Proofs
For convenience, we abbreviate the conditional probability measure \(P(\cdot |{\varvec{S}})\) to \(P^{\varvec{S}}(\cdot )\), and for each \(i=1,\dots , k\), construct an associated random vector \(\varvec{Y}_i=(Y_{i 1},\ldots ,Y_{i d_i})\) with
\(j=1,\dots , d_i\). The following lemma can be found in Lemma 4.3 of Tang and Yuan (2013), which is crucial in the proof of Theorem 3.1.
Lemma A.1
If Assumption 3.1(ii) is satisfied and \(R\in \textrm{RV}_{-\beta }\) for some \(0<\beta \le \infty \), then it holds that for each \(i=1,\dots , k\), almost surely,
on \([0,\infty ]^{d_i}\setminus \{\varvec{0}\}\), where \(\tilde{\nu }^{\varvec{S}}_{i}\) is defined by (3.3).
Proof of Theorem 3.1
We start with a claim that, almost surely,
Recalling \(\varvec{Y}_{i}, i=1,\dots , k\), in (A.1) and \(\sum _{i=1}^k\sum _{j=1}^{d_i} e_{i j} =1\), we rewrite the conditional probability \(P^{\varvec{S}}\left( L>1-\frac{1}{x} \right) \) as
If we can prove that, almost surely, for any \(e_{i j}>0\), \(j=1,\dots , d_i\); \(i=1,\dots , k\),
then (A.2) holds. For (A.3), we proceed by induction on k. Trivially, (A.3) holds for \(k=1\) from Lemma A.1. Now we assume by induction that it holds for all \(k\le k_0\) and some \(k_0\in \mathbb N\), and we are going to prove it for \(k=k_0 +1\). First consider the lower bound of (A.3) with \(k=k_0 +1\). For each \(i=1,\dots , k_0+ 1\), by (3.4) it is easy to see that \({\varvec{y}}_1 \in A_i\) implies \({\varvec{y}}_2 \in A_i\) for any \({\varvec{y}}_2 \ge {\varvec{y}}_1\), and thus \(t (\partial A_i)\cap \partial A_i = \emptyset \) for every \(t>1\). Then, by Lemma 7.1 of Shi et al. (2017), we have \(\tilde{\nu }^{\varvec{S}}_i( \partial A_i )=0\). For any large but fixed \(n\in \mathbb N\), almost surely,
where we used (A.3) for all \(k\le k_0\) and (2.4) in the last step. Letting \(n\rightarrow \infty \), the sum in (A.4) converges to \(\int _0^1 (1-y)^{\frac{\alpha _{k_0+1}}{\beta }} \textrm{d} \left( y^{\frac{1}{\beta }\sum _{i=1}^{k_0}\alpha _i}\right) ={\Gamma \left( 1+\frac{1}{\beta }\sum _{i=1}^{k_0}\alpha _i\right) \Gamma \left( 1+ \frac{\alpha _{k_0 +1}}{\beta }\right) }/{\Gamma \left( 1+\frac{1}{\beta } \sum _{i=1}^{k_0 +1}\alpha _i\right) }\). For the upper bound of (A.3) with \(k=k_0+1\), we do the split
Then, a similar discussion to (A.4) can be carried out to derive the upper bound. Thus, the desired relation (A.3) holds for \(k=k_0+1\).
Therefore, applying the dominated convergence theorem and by (A.2) we can obtain
In the derivation above, the applicability of the dominated convergence theorem can be justified as follows. For any fixed \(x\ge 1\), denote by
\(i=1,\dots , k\). Clearly, it can be verified that, for any fixed \(x\ge 1\), the sets \(\Delta _i(x), i=1,\dots , k\), are all increasing, since \(R(\cdot )\) is a non-increasing function. By Assumption 3.1(i), (iii), for any fixed \(\epsilon >0\), there exists a large constant \(M>0\), such that for all sufficiently large \(x\ge 1\), almost surely,
which is integrable with respect to \(P(\varvec{S}\in \textrm{d}\varvec{s})\). Here we used (2.5) in the fourth step, and (2.2) in the last step. This completes the proof of Theorem 3.1.\(\square \)
Proof of Theorem 3.2
Combined with Corollary 3.1, for relation (3.6), it suffices to prove that
Note that for any \(0<q<1\),
Since the distortion function g is left continuous, so is \(g_q\). Then, by Theorem 6 of Dhaene et al. (2012), we have
We first consider the convergence of the integrand in (A.7) as \(q\uparrow 1\). By \(\overline{F_i}\in \textrm{RV}_{-\alpha _i}, i=1,\dots ,k\), \(R\in \textrm{RV}_{-\beta }\), and \(E \left[ \prod _{i=1}^k \tilde{\nu }^S_{i}(A_{i})\right] >0\), Theorem 3.1 implies that \(1/\overline{F_{\frac{1}{1-L}}}\in \textrm{RV}_{\frac{1}{\beta }\sum _{i=1}^k\alpha _i}\). Then, applying Proposition 0.8(v) of Resnick (1987) gives that for any fixed \(u\in (0,1]\) and \(0\le \frac{1}{\beta }\sum _{i=1}^k\alpha _i\le \infty \),
Therefore, the desired relation (A.6) can be derived by applying the dominated convergence theorem. Indeed, for all \(u\in (0,1]\) and \(q\in (0,1)\),
which is integrable with respect to \(g(\textrm{d}u)\) in \(u\in [0,1]\). It ends the proof of Theorem 3.2.\(\Box \)
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Chen, S., Yang, Y. & Zhang, Z. Asymptotics for credit portfolio losses due to defaults in a multi-sector model. Ann Oper Res 337, 23–44 (2024). https://doi.org/10.1007/s10479-024-05934-5
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DOI: https://doi.org/10.1007/s10479-024-05934-5
Keywords
- Credit portfolio loss due to defaults
- Multi-sector model
- Sharp asymptotics
- Macroeconomic factors
- Multivariate regular variation