A Guide to IMF Stress Testing

Chapter 3. Stress Tester: A Toolkit for Bank-by-Bank Analysis with Accounting Data

Li Ong
Published Date:
December 2014
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Martin ČihákThis chapter is an abridged and updated version of IMF Working Paper 07/59 (Čihák, 2007a). The author would like to thank R. Sean Craig, Dale Gray, Plamen Iossifov, Peter Chunnan Liao, Thomas Lutton, Christiane Nickel, Nada Oulidi, Richard Podpiera, Leah Sahely, Graham Slack, and participants in a regional conference on financial stability issues at Sinaia, Romania, and in seminars at the IMF, the World Bank, the Central Bank of Russia, and the Central Bank of Trinidad and Tobago for helpful comments on the working paper; Matthew Jones and Miguel Segoviano for insights on stress testing methodologies; and Roland Straub for help with the overview of stress tests.

Stress testing is a useful and increasingly popular, yet sometimes misunderstood, method of analyzing the resilience of financial systems to adverse events. This chapter aims to help demystify stress tests and illustrate their strengths and weaknesses. Using an Excel-based exercise with institution-by-institution data, readers are walked through stress testing for credit risk, interest rate and exchange rate risks, liquidity risk, and contagion risk and are guided in the design of stress testing scenarios. The chapter also describes the links between stress testing and other analytical tools, such as financial soundness indicators and supervisory early-warning systems. Furthermore, it includes surveys of stress testing practices in central banks and the IMF.

Method Summary

Method Summary
OverviewThe Stress Tester 3.0 is a basic yet flexible model for implementing bank-by-bank stress tests for a set of risks. It covers basic solvency and liquidity risks, contagion, reverse stress testing, and links between stress tests and early-warning systems.
ApplicationThe method can be applied to a wide variety of banking systems but is particularly appropriate for small, noncomplex banking systems.
Nature of approachBasic balance sheet shocks, combined with basic contagion analysis.
Data requirementsAccounting information on capital, loans, and risk-weighted assets. Supervisory data on classified loans and provisions.
StrengthsEasy to use, flexible (institutions, shocks, assumptions can be easily added), transparent (all assumptions are in one sheet), and covers a relatively wide range of risks.
WeaknessesNumerous assumptions are required. The mechanism for generating the shocks (e.g., historical scenario and macroeconomic model) is left unspecified.

Reliability and comparability of results depend on quality of input data (which includes, e.g., accounting data on nonperforming loans and provisions).
ToolThe Excel spreadsheet macro (Stress Tester 3.0) is available in the toolkit, which is on the companion CD and at www.elibrary.imf.org/stress-test-toolkit.

The subject of this chapter and the accompanying Excel file are system-oriented stress tests carried out on bank-by-bank data.1 The chapter and the accompanying file aim to illustrate strengths and weaknesses of stress testing, using concrete examples. Readers will familiarize themselves with how common types of stress tests can be implemented in practice, in a small and noncomplex banking system. They should gain an understanding of how the various potential shocks can be fitted together and how stress testing complements other analytical tools, such as financial soundness indicators (FSIs) and supervisory early-warning systems. They should also learn how to interpret the results of stress tests.

As the title indicates, this chapter is about applying stress tests to actual data. It devotes relatively little space to general discussions of what stress testing is. There is already a plethora of studies that provide a general introduction to stress testing, discussing its nature and purpose. For example, Blaschke and others (2001), Jones, Hilbers, and Slack (2004), Čihák (2004, 2005), and IMF and World Bank (2005) provide a general introduction to stress testing. In contrast, relatively little is available in terms of practical technical guidance on how to actually implement stress tests for financial systems, using concrete data as examples. This study and the accompanying Excel file are an attempt to fill that gap.

As the title also suggests, this is an introduction, not a comprehensive “stress testing cookbook.” The chapter covers basic versions of the most common stress tests. Depending on the sophistication of the financial system and the type of its exposures, it may be necessary to elaborate on these basic versions of tests (e.g., by estimating econometrically some relationships that are only assumed in this file) and perhaps include also tests for other risks (e.g., asset price risks or commodity risks) if financial institutions have material exposures to those risks. The accompanying file can be developed in a modular fashion, with additional modules capturing additional risks or elaborating on the existing ones. The final part of this chapter provides an overview of the main extensions that could be considered.

One of the key messages of this chapter is that assumptions matter and that they particularly matter in stress testing. The chapter calls for transparency in presenting stress testing assumptions and for assessing robustness of results to the assumptions. To highlight the importance of assumptions in stress testing, this chapter uses boldface for references to assumptions used in the accompanying Excel file. To achieve transparency in the Excel file itself, assumptions are highlighted (in blue and green) and grouped in one worksheet.

The remainder of this chapter is structured as follows. Section 1 provides an overview of the general issues one needs to address when carrying out stress tests and describes the design of the accompanying Excel file. It also introduces the general setting of the fictional economy described by the Excel file. Section 2 discusses the input data. Sections 37 discuss the stress tests for the individual risk factors, namely, credit risk (Section 3), interest rate risk (Section 4), foreign exchange risk (Section 5), interbank contagion risk (Section 6), and liquidity risk (Section 7). Section 8 shows how to create scenarios from the individual risk factors, how to present the results, and how to link the results to supervisory early-warning systems. Section 9 concludes and discusses possible extensions.

The accompanying Excel file, “Stress Tester 3.0.xls,” constitutes an essential part of this chapter. For each of the concepts introduced in the subsequent sections, we include specific references to the file. To distinguish references to tables in the Excel file from those in this chapter, the Excel table names start with a capital letter A–H (denoting the order of the spreadsheet) followed by a number (denoting the order of the table within the spreadsheet). For instance, Table A2 denotes the second table in the first Excel spreadsheet.

1. Overview of the File and of the Stress Testing Process

Stress testing can be thought of as a process that includes (1) identifying specific vulnerabilities or areas of concern; (2) constructing a scenario; (3) mapping the outputs of the scenario into a form that is usable for an analysis of financial institutions’ balance sheets and income statements; (4) performing the numerical analysis; (5) considering any second-round effects; and (6) summarizing and interpreting the results (Jones, Hilbers, and Slack, 2004; IMF and World Bank, 2005). The aim of this exercise is to illustrate the stages of this process. It will also illustrate that these stages are not necessarily sequential, as some modification or review of each component of the process may be needed as work progresses.

The stress testing exercise is performed on the banking system in a fictional country named Bankistan. Given that confidentiality restrictions do not allow the IMF to pass on individual bank data, we refer to a fictional country rather than an actual one. The exercise is modeled on stress tests conducted in a number of Financial Sector Assessment Program (FSAP) missions, and the input data were created to make the exercise realistic. However, compared with typical FSAP stress tests, the exercise was simplified significantly to make it suitable for a short workshop that would give an overview of the FSAP stress tests (see IMF and World Bank 2003). It is a version that could be used for a noncomplex banking system. For larger systems characterized by complex financial institutions and markets, more elaborate tests may be necessary.

To understand how to design stress testing shocks and scenarios, it is important to have a good understanding of the structure of the financial system and the overall environment in which the system operates. Box 3.1 therefore provides short briefing information on the macroeconomic and macroprudential situation in Bankistan.

A. How to operate Stress Tester 3.0: A quick guide to the accompanying file

This section introduces the accompanying Excel file, “Stress Tester 3.0.xls.” Working with the file requires relatively little prior knowledge. Users should be proficient in operating standard Excel files. Knowledge of intermediate macroeconomics is useful for understanding the linkages between the financial sector and the broader macroeconomic framework.

The file can be viewed as a module belonging to a broader stress testing framework (Figure 3.1). Such a framework would typically include a model characterizing linkages among key macroeconomic variables, such as GDP, interest rates, the exchange rate, and other variables. Medium-scale macroeconomic models (e.g., those used by a central bank for macroeconomic forecasts) including dozens of estimated or calibrated relationships are often used for this purpose (if such models are not available, vector autoregression or vector error-correction Given that such models generally do not include financial sector variables, the stress testing framework can also include a “satellite model” that maps (a subset of) the macroeconomic variables into financial sector variables, in particular asset quality. Such a satellite model can be built on data on individual banks over a period of time: using panel data techniques, asset quality in individual banks can be explained as a function of individual bank variables and system-level variables. Together with the macroeconomic model, the satellite model can be used to map assumed external shocks (e.g., a slowdown in world GDP) into bank-by-bank asset quality shocks.

Figure 3.1Stress Testing Framework

Source: Author.

Box 3.1Background Information on Bankistan’s Economy and Banking Sector

The economic environment in which banks are operating in Bankistan is challenging, with increasing macroeconomic imbalances, inappropriate macroeconomic policies, and deep uncertainty fueled by political tensions. Real activity is sharply contracting, and inflation has almost doubled to 65 percent.

Unsustainable fiscal imbalances and loose monetary conditions were key to the deteriorating situation in Bankistan. The government deficit more than doubled in 2005, and a sharp increase in central bank financing of the government has significantly accelerated money growth.

The policy response to the deteriorating situation has been inappropriate. Expansionary monetary policy measures (e.g., a lowering of reserve requirements) have induced a further easing of liquidity conditions. The ensuing excess liquidity induced a drop in treasury bill rates from 60 percent to below 15 percent. This means that together with an inflation rate of 65 percent, real interest rates are sharply negative. (Note: This is used for assessing interest rate risk.)

The official exchange rate of the Bankistan currency, Bankistan dollar (B$), is fixed at 55 B$/US$. However, the black market exchange rate has depreciated in recent months from about 60 B$/US$ to about 85 B$/US$. (Note: This is important information for assessing foreign exchange risk.)

The deteriorating macroeconomic environment has put considerable strain on the financial condition of the banking system. Even though the system has proved so far to be remarkably resilient, some banks have been weakened considerably and are prone to further deterioration in light of the significant risks. Reported high-capital adequacy ratios were found to be overstated because of insufficient provisioning. (Note: This information is used for the assessment of asset quality.) In addition, asset quality has deteriorated. The ratio of gross nonperforming loans (NPLs) to total loans has increased from 15 percent at end-2009 to 20 percent at end-2010. (Note: This information will be used for assessing credit risk.)

The banking system of Bankistan consists of 12 banks. Three of them are state owned (with code names SB1 to SB3), 5 are domestic privately owned banks (DB1 to DB5), and 4 are foreign owned (FB1 to FB4). The banking system, and particularly the state-owned banks, have been plagued by a large stock of NPLs and weak provisioning practices. Data on the structure and performance of the 12 banks are provided in the “Data” sheet of the accompanying Excel file. An assessment of Bankistan’s compliance with the Basel Core Principles for Banking Supervision (BCP) suggests that even though existing loan classification and provisioning rules in Bankistan are broadly adequate, they are not well implemented in practice, and banks are underprovisioned.

We focus on calculating the bank-by-bank impacts resulting from external shocks and expressing the impacts in terms of variables such as capital adequacy or capital injection as a percentage of GDP. We spend relatively little time discussing the broader macroeconomic framework or possible feedback effects to the broader economy. The cells that contain sizes of shocks and numerical assumptions in this file can be thought of as interfaces between this module and the other modules of the stress testing framework. The cells that contain results (e.g., capital injections as a percentage of GDP) can be viewed as interfaces to a module analyzing the feedback effects.

The modular design has the advantage that as a user becomes more experienced in stress testing or as more data become available for analysis, additional modules can be added or developed. Incorporating, for example, the underlying econometric calculations in Excel would make the resulting file too big and unwieldy (it also may not be possible because the econometric tools in Excel are more limited than in other packages).

All data in the file relate to end-2010, unless indicated otherwise, and are expressed in millions of Bankistan dollars (B$), except for ratios (shown in percent).

The file contains the following eleven worksheets: Read Me, Data, Assumptions, Credit Risk, Interest Risk, Foreign Exchange (FX) Risk, Interbank, Liquidity, Scenarios, Reverse, and Bottom Up. Table 3.1 contains a description of the individual worksheets.

Table 3.1Stress Tester 3.0: Description of the Worksheets
Read MeBasic information, acknowledgments, and explanation of the workbook.
DataSix tables. Input data as compiled by the National Bank of Bankistan (NBB). The data were collected in March 2011 and generally relate to end-December 2010, unless noted otherwise. Table A1 contains basic financial statement data. Table A2 contains other relevant data, including more detailed breakdowns, results of bottom-up stress tests reported by individual banks, and GDP number (for calculating ratios). The next two tables include key ratios based on the input data. Specifically, Table A3 contains the financial soundness indicators, and Table A4 characterizes the structure of the banking sector. The following two tables show how the financial soundness indicators can be combined into institution-by-institution rankings, using a simple early warning system calibrated by the NBB (see the Assumptions sheet). Specifically, Table A5 provides the rankings, and Table A6 converts them into probabilities of default.
AssumptionsOne table. Table B puts together all the assumptions. This worksheet also contains several charts allowing the user to see how changes in the assumptions affect the results.
Credit RiskTwo tables. Table C1 summarizes the reported data on asset quality. Table C2 shows the credit risk stress test. It consists of four components: (i) a correction for underprovisioning of NPLs; (ii) an aggregate NPL shock; (iii) a sectoral shock, allowing different shocks to different sectors; and (iv) a shock for credit concentration risk (large exposures).
Interest RiskTwo tables. Table D1 sorts assets and liabilities into three time-to-repricing buckets, using the input data provided by the NBB. Table D2 shows the corresponding interest rate stress test. The test itself consists of two components: (i) flow impact from a gap between interest-sensitive assets and liabilities; and (ii) stock impact resulting from the repricing of bonds.
FX RiskTwo tables. Table E1 contains information on the foreign exchange exposure of the banks and the direct exchange rate risk shock. Table E2 shows a basic calculation of the indirect foreign exchange shock (using FX loans to approximate impact on credit quality).
InterbankThree tables. Table F1 is a matrix of net interbank exposures. Table F2 uses the interbank exposure data to show “pure” interbank contagion, i.e., to illustrate what happens to the other banks when one bank fails to repay its obligations in the interbank market. Table F3 shows a “macro” contagion exercise, in which banks’ failures to repay obligations in the interbank market are not assumed, but rather a result of the “macro” shocks modeled in the sheet “Scenarios.”

To differentiate the various types of cells (input data, numerical assumptions, and formulas), different colors are used in the file. The following color coding is used:

  • Yellow denotes data reported by the National Bank of Bankistan (NBB). The yellow cells are found only in the Data worksheet. When a new set of data arrives from the NBB, the contents of the yellow cells should be replaced with the new data, and all the results are recalculated automatically.

  • Blue denotes the assumed sizes of the shocks to risk factors, for example, an increase in interest rates. The blue cells are found only in the Assumptions worksheet. The users can change the values of these factors in the Assumptions worksheet and observe the impact of these changes on the stress test results.

  • Green denotes numerical assumptions (parameters) of the stress test. Like the blue cells, the green cells are found only in the Assumptions worksheet. As with the blue cells, the users can change the values of these factors in the Assumptions worksheet and observe the impact of these changes on the stress test results.

  • No background. Cells that have no background and generally normal black font contain formulas linked to the yellow, green, and blue cells. If the values of the blue or green calls are changed (or if new input data are entered in the yellow cells), the results of the stress tests are recalculated automatically.

  • Yellow stripes indicate consistency checks. These cells contain sums or other functions of the input data. Those would normally also come from the authorities as hard numbers but are calculated in this file to avoid inconsistencies.

  • Green/white stripes denote numerical assumptions imported from the Assumptions sheet. Under normal circumstances, it is expected that the user will leave these cells (each of which contains a link to the Assumptions sheet) unchanged and will carry out the changes in the corresponding green cells in the Assumptions sheet. However, if users want to see the impact of changes in an assumption directly in the corresponding sheet (e.g., for credit risk assumptions in the Credit Risk worksheet), they can do so by changing the value of the green/white cell rather than going back to the Assumptions sheet. Of course, users need to be aware that if they save these changes, it will result in overwriting the original links in the file and some of the links between the Assumptions sheet and the results in the Scenarios sheet may be broken. However, if they do not save those changes and afterward return to the original template, they will be able to use the file again in its original form.

  • Blue/white stripes denote numerical assumptions imported from the Assumptions sheet. Like the green/white stripe cells, these cells allow the user to change the values of assumed shocks directly in the individual worksheets without going back to the Assumptions sheet. However, these changes should be used with caution to avoid breaking the links in the file; changes in the shock sizes should primarily be done in the Assumptions sheet.

In addition to explanations in the Read Me sheet, many of the cells in the stress testing file contain comments explaining the calculations carried out in these cells.

All assumptions and shock parameters are in the Assumptions sheet. This is the sheet that a regular user would use the most, changing the assumptions (in green) and shock sizes (in blue) and observing the results. Given that a summary presentation of the stress test results is provided in the Assumptions sheet in charts, the user can change the assumptions and directly see the impacts in a graphical form. If the user wants to examine the overall stress test results in a tabular form, the results are available in the Scenarios and other sheets.

Expert users who have become familiar with the file are invited to suggest improvements in the file or to develop the file further themselves. Such developments can include new types of risks, making the modeling of the existing risks more realistic or including more institutions in the system.2 Not all the developments need to take place in the same file: users can think about some of the blue or green cells as interfaces between this tool (module) and other tools (modules), such as macroeconomic models that provide scenarios. Those can provide inputs that feed into this stress testing tool. The main advantage of Excel-based tools, such as this one, is the relative ease with which they can be adapted and extended. For longer-term usage, it may be useful to develop the file into a program, for example, in Microsoft Access. This may reduce the flexibility for regular users, but it may, among other things, allow development of the file from the current one-period snapshot to a multiperiod framework.

B. Top-down or bottom-up?

There are two main approaches to translating macroeconomic shocks and scenarios into financial sector variables: the “bottom-up” approach, where the impact is estimated using data on individual portfolios, and the “top-down” approach, where the impact is estimated using aggregated data.3 Among central banks’ Financial Stability Reports (FSRs), reports by the Bank of England and Norges Bank can be used as examples of FSRs that rely more on top-down approaches to stress testing, whereas reports by the Austrian National Bank and Czech National Bank are examples of stress tests using more bottom-up approaches, even though in all these cases, the reports in fact combine elements from both approaches.

The disadvantage of a top-down approach is that applying the tests only to aggregated data could overlook the concentration of exposures at the level of individual institutions and linkages among the institutions. This approach may therefore overlook the risk that failures in a few weak institutions can spread to the rest of the system. The bottom-up approach should be able to capture the concentration of risks and contagion and therefore should lead generally to more precise results, but it may be hampered by insufficient data and by calculation complexities. Having detailed information on exposures of individual banks to individual borrowers should in principle lead to more accurate results than using more aggregated data, but, especially for large and complex financial systems, it may lead to insurmountable computational problems. Most macroprudential stress tests therefore try to combine the advantages and minimize the disadvantages of the bottom-up and top-down approaches.

This chapter and the accompanying Excel file focus on the bottom-up implementation of stress tests in a relatively small, noncomplex banking system. The spreadsheet illustrates why using institution-by-institution data is important: a relatively minor change in the distribution of risks among banks can result in substantial changes in the overall impacts. The spreadsheet also illustrates how the bottom-up approach can be complemented by a top-down approach. For example, a model estimated on aggregate data can be used to identify how a combination of shocks to macroeconomic variables can translate into an increase in nonperforming loans. Stress Tester 3.0 can then be used to calculate how this aggregate impact influences individual banks and the system as a whole.

Although much of the Excel file illustrates a “centralized” approach to bank-by-bank stress testing, where all the calculations are done in one center (e.g., at the central bank or a supervisory agency or by an IMF expert), the Bottom Up worksheet illustrates an alternative, more “decentralized” approach. The approach often used in advanced-economy FSAPs is to involve banks themselves in carrying out the stress testing calculations. The advantage of the “decentralized” approach is that it can provide a richer, more detailed modeling, using a wider set of data and leveraging on the expertise and calculation capacity of banks’ risk management.4 The disadvantage is that the “decentralized” calculations may not sufficiently reflect contagion effects among banks, so just adding up the results for individual banks may not be a good proxy for the systemic impacts. Also, if the calculations are complex and done by many institutions, it may be a major challenge to ensure that all banks implement the assumed shocks or scenarios in a consistent fashion. In comparison, the “centralized” stress tests discussed in this chapter are less refined (for reasons of computational complexity and data availability), but they (1) are more focused on linkages to macroeconomic factors; (2) can better integrate credit and market risks; (3) are implemented consistently across institutions; (4) can better analyze correlation across institutions (interportfolio correlation); and (5) can analyze network effects (contagion). Therefore, even if the “decentralized” calculations are carried out, it is important to complement them with the “centralized” calculations along the lines discussed in this chapter.

C. Presenting stress test results: What variables can be stressed?

For a variable to be used to measure the impacts of the stress tests, it should have two key properties: (1) it should be possible to interpret the variable as a measure of financial soundness of the system in question; and (2) it can be credibly linked to the risk factors. Of the various variables that have been used in the literature so far, each has advantages and disadvantages, which should be clear to the reader and user. Here is a list of the commonly used variables:

  • Capital. Using capital as a measure of impact has a clear motivation. If a risk has a material impact on solvency, it has an impact on capital. Also, commercial banks’ capital is part of the Other Items Net in monetary surveys, so expressing impacts in terms of capital could be used to directly link stress tests to other parts of the financial programming framework (even though it is only a small part of the possible feedback effects from the financial sector to the macroeconomy, as discussed in Section 8.C). The disadvantage of using impact in terms of capital is that it is just a number in B$ that needs to be compared with something else to give the reader an idea about the impact on soundness (e.g., dividing it by risk-weighted assets) and on the macroeconomic framework (e.g., dividing it by GDP). Nonetheless, it is a key measure, and the accompanying Excel file illustrates the presentation of stress testing impact in terms of capital.

  • Capitalization. The advantage of capitalization measures (capital or equity to assets or capital to risk-weighted assets) is that capital adequacy is a commonly recognized soundness indicator. Compared with capital, this measure is scaled, so it allows comparison among institutions of different size. For this reason, the accompanying Excel file uses capitalization as a key indicator of impact. The disadvantage of this measure is that a change in capitalization does not by itself indicate macroeconomic relevance of the calculated impacts. It therefore needs to be accompanied by other measures.

  • Capital injection needed (e.g., as a percentage of GDP). This indicator provides a direct link to the macroeconomy. It provides an upper bound on the potential fiscal costs of bank failures associated with the assumed stressful scenario. The accompanying Excel file illustrates the use of capital injection.

  • Profits. In a normal, nonstressful situation (“baseline scenario”), banks would typically create profits. When carrying out stress tests, it is important to bear in mind that we are evaluating impacts against such a baseline, as banks would normally use profits as the first line of defense before dipping into capital. Expressing shocks only in terms of capital may result in overestimating the actual impacts if banks were profitable in the baseline. The accompanying Excel file allows profits to be taken into account. More specifically, it indicates the “profit buffer” that banks would have available in the baseline. The profit buffer is based on the average annual profits over the last 10 years, but the file also allows the user to run a separate “test” for the impact of an autonomous shock affecting profits (one can think about autonomous shocks to net interest income, e.g., related to an increase in competition from abroad; alternatively, this can be viewed as a measure of the risks not reflected in the other shocks specified in the stress testing scenario). However, to reflect the views of some observers that it is better to be prudent and disregard profits, it shows them as a separate item rather than directly deducting the impacts from profits

  • Profitability (return on equity, assets, or risk—weighted assets). Compared with profits, these measures are scaled by bank size, thus allowing comparison among banks of different size. The accompanying Excel file shows the profit buffers as ratios to the risk-weighted assets, to make them comparable with the impact shown in terms of risk-weighted assets.

  • Net interest income and other components of profits. Sometimes it can be useful to stress test separately individual components of profits. For example, net interest income is likely to have a more direct relationship to interest rates and may therefore be more amenable to econometric analysis. However, such an approach provides only a partial picture of the economic value of a bank and its resilience to adverse events.

  • z-scores. The z-score has become a popular measure of bank soundness (e.g., Boyd and Runkle, 1993; or Hesse and Čihák, 2007). Its popularity stems from the fact that it is related to the probability of a bank’s insolvency, that is, the probability that the value of its assets becomes lower than the value of the debt. The z-score can be summarized as z=(k+µ)/σ, where k is equity capital as percentage of assets, µ is average aftertax return as percentage on assets, and σ is standard deviation of the aftertax return on assets, as a proxy for return volatility. The z-score measures the number of standard deviations a return realization has to fall in order to deplete equity, assuming normality of returns. A higher z-score therefore implies a lower probability of insolvency risk. The accompanying Excel file illustrates this presentation.5

  • Loan losses. Stress tests presented by staff from Norges Bank (e.g., Evjen and others, 2005) and the Bank of England (e.g., Bunn, Cunningham, and Drehmann, 2005) are just two examples of presenting stress testing results in terms of loan losses. Although this approach has its advantages (in particular, it is easier to implement in top-down calculations than measures calculating losses to capital), its drawback is that it does not take into account banks’ buffers (profits and capital) against those losses. It may underestimate the overall impact if losses are concentrated in weak institutions.

  • Liquidity indicators. For liquidity stress tests, the impacts have to be measured differently than for solvency tests, namely, in terms of liquidity indicators. The accompanying Excel file illustrates this presentation.

  • Ratings and probabilities of default (PDs). Ratings and PDs provide a useful way of combining solvency and liquidity risks. By definition, ratings try to combine various solvency and liquidity risks into a single measure. We can use the system designed for ratings and see how changes in the various variables translate into changes in ratings. If we have a model linking ratings and probabilities of default, we can also calculate how a stressful scenario influences PDs. The accompanying Excel file illustrates this presentation.

This list is far from complete. It is possible to calculate and present stress test impacts in terms of other variables that capture soundness of financial institutions and can be credibly linked to the development of risk factors. For example, instead of the accounting-based data discussed earlier, it is possible to present the impact of a stressful scenario in terms of market-based indicators of financial sector soundness, such as relative prices of securities issued by financial securities, the distance to default for banks’ stocks,6 or credit default swap premia (for a review, see, e.g., Čihák, 2007b). One of the advantages of the market-based indicators is that they are usually available on a much more frequent basis than accounting data. However, one of their major disadvantages is the absence in many countries of sufficiently deep markets from which such indicators can be derived (e.g., bank stocks are not traded, or the market for such stocks is illiquid). There have been some attempts to link market-based indicators, such as distance to default, to macroeconomic variables (see, e.g., IMF, 2005), but work on using such relationships for stress tests has been limited so far.

D. How are the results presented in Stress Tester 3.0?

Each stress test conducted in this exercise aims to address two main questions: (1) Which banks could withstand the assumed shocks and which ones would fail? (2) What are the associated potential costs for the government given the failure of banks in times of stress?

A commonly used approach to assessing question (1), as mentioned in the previous section, is to look at banks’ capital adequacy ratio (CAR). According to the original Basel Agreement, a bank has to hold a minimum CAR, defined as total regulatory capital to risk-weighted assets (RWA), of 8 percent. In our example, given that Bankistan as an emerging market country faces more risks than an industrial country, it is appropriate to require from banks a higher CAR. On these considerations, Bankistan’s supervisors use a minimum CAR of 10 percent. Whenever the CAR of a bank in Bankistan falls below 10 percent, its owners are obliged to inject capital in order to stay in business.7 If they fail to do so, the bank will be closed and its banking license withdrawn. A CAR below 0 means that a bank has negative capital and is insolvent.8

Stress Tester 3.0 allows profits to be taken into account, which—as explained earlier—is important because shocks always take place over time and therefore need to be compared with a “baseline scenario.” The Scenarios sheet indicates the profit buffer that banks would have available in the baseline scenario. We refer to annual profits, which is consistent with the fact that we evaluate the shocks in a horizon of one year (see the interest rate shock). The value of the profit buffer is based on the average annual profits over the last 10 years, but the file also allows the user to run a separate “test” for the impact of an autonomous shock affecting profits or net interest income. Take, for instance, autonomous shocks to net interest income, for example, related to an increase in competition from abroad; alternatively, this can be viewed as a measure of the risks not reflected in the other shocks specified in the stress testing scenario. However, to reflect the views of some papers that it is more prudent to disregard profits and measure shocks directly against capital (e.g., Blaschke and others, 2001), the file shows the profit buffers as a separate item rather than directly deducting the impacts from profits. For some banks, the profit “buffer” is nonexistent or negative (they have been creating losses).

An assessment of question (2) requires consideration of the following question: if bank owners fail to inject new capital, how much capital would the government need to inject in order to bring the CAR up to 10 percent again? For state-owned banks, it is obvious that the government would have to inject capital to keep banks operating. For private-owned banks, asking this question assumes that the government has an implicit or explicit guarantee for the banking sector, which may or may not be the case. If it is, an answer to this question can be found from the following accounting relationship:

where C is the bank’s existing total regulatory capital, RWA are its existing risk-weighted assets, I is the capital injection, q is the percentage of the capital injection that is immediately used to increase risk-weighted assets, and ρ is the regulatory minimum CAR (ρ =10 percent in the case of Bankistan).

From the above equation, we can express the necessary capital injection as follows:

If q=0, that is, the capital injection is not used for an increase in RWA (at least immediately), and if we substitute for ρ the 10 percent value used in Bankistan, we can calculate the capital injection as I= 0.1*RWA – C. The values of the parameters ρ and q are assumed. The stress test file enables us to change the values of the two parameters in the appropriate green cells (B71 and B72 in the worksheet Assumptions). We can see that if ρ is lower than 10 percent, the necessary capital injection is lower, and vice versa. If RWA increase as a result of the increase in capital (i.e., if q>0), the necessary capital injection is higher (but the impact of changes in q is generally rather small).

2. Understanding and Analyzing the Input Data

The Data sheet summarizes the input data (Tables A1 and A2), as reported by the NBB. It also shows key ratios (Tables A3 and A4) and illustrates how these ratios can be used in an off-site supervisory assessment system (Tables A5 and A6).

A. Coverage of stress tests

In the Excel file, the stress tests cover all 12 commercial banks in the country. An overview of FSAP stress tests suggests that only a minority of FSAPs have covered all banks in a country. Most FSAPs have covered only a subsample of large banks, accounting for a substantial majority (generally 70–80 percent) of the banking system’s total assets. Including all banks rather than a subsample has the obvious advantage of being more comprehensive. For this reason, this approach is also often favored by supervisors, who are expected to supervise all institutions, not only the larger ones. However, for someone interested primarily in macroprudential issues (e.g., a central bank not involved in microprudential supervision), it may be sufficient to include only the systemically important institutions (that is why some authors, such as Jones, Hilbers, and Slack [2004] refer to this type of stress testing as “system-oriented stress testing” rather than “systemwide stress testing”). Excluding the other institutions may be practical for reasons of computational complexity.9

The Stress Tester illustrates a basic peer group analysis. To do that, the 12 banks in Bankistan are grouped into three peer groups according to their ownership: state-owned banks (SB1 to SB3), domestic privately owned banks (DB1 to DB5), and foreign-owned banks (FB1 to FB4). This is just one possible grouping. Depending on the analytical purpose, the banks in Bankistan can be grouped into other groups, for example, by their size (e.g., large, medium, and small) or financial performance (e.g., strong and weak).

In the Bankistan example, foreign banks are present only through locally incorporated subsidiaries. In practice, foreign banks may also be present through branches. The difference from a stress testing perspective is that branches typically do not have their own capital against which the impact of shocks could be shown (and comparing the impacts with the parent institution’s capital would overlook risks faced by the same institution in other countries). However, if there are separately available data on assets, liabilities, incomes, expenses, and other input data indicated in the Data worksheet, most of the standard tests can be performed on the branches of foreign banks as well. The main practical difference, then, is that the impacts have to be expressed in terms of a different variable than capital or capital adequacy (e.g., in terms of profits).

Mirroring the practice of FSAPs and central banks’ FSRs, this chapter focuses on stress tests for banks and banking systems. In most countries, banks tend to dominate the financial system and are key to assessing systemic risk. Some FSAPs and FSRs contained explicit stress tests of insurance companies and pension funds. Stress testing pension funds and insurance companies themselves can be a rather complex task. Some of the insurance sector risks, such as market risk, liquidity risk, or credit risk, can be modeled similarly to the banking stress tests. Modeling of other risks, especially some of those stemming from the liabilities side (e.g., catastrophe risk), is beyond the scope of this study.

The impact of failures in nonbank financial institutions can be assessed as part of the credit risk. The accompanying Excel file allows for two ways of incorporating such an analysis. First, it can be incorporated as part of the sectoral credit risk, with nonbank financial institutions as one of the sectors. We can then run a basic stress test for what would happen if a certain percentage of loans to the nonbank financial sector became nonperforming. Second, it can be incorporated as part of the large exposures tests. If we have data on the largest exposures of banks to nonbank financial institutions, we can run a test on what would happen to banks’ solvency should their largest counterparties in the nonbank financial sector fail.

B. Balance sheets, income statements, and other input data

Data availability is a key determinant of the quality of a stress test. The purpose of Tables A1 and A2 in the Data worksheet is to illustrate what data are typically needed for carrying out a set of stress tests. These are not the minimum requirements—it is possible to do very rudimentary stress tests with even less data, as discussed in Box 3.2. However, these are the types of data that one usually looks for when doing a basic stress test.

Throughout the sheet, as well as the file, the first data column shows aggregated data for the whole banking system, the next 3 columns show the aggregated data for the three peer groups of banks (state banks, private domestic banks, and foreign banks), and the remaining 12 columns show the data for the individual banks.

The input data presented in the top part of the Data worksheet (Table A1) form a set of balance sheets and income statements. To keep the exercise straightforward and transparent, the aggregated data and the peer group data can be calculated as sums of the bank-by-bank data. This means that we disregard interbank exposures for the time being. This assumption is relaxed later on, in the analysis of interbank contagion risk (Section 6).

Table A2 in the Data worksheet lists—again for the system in aggregate, for the peer groups, and for individual banks—other key input data that are used for the stress test calculations. Such data are on the following: (1) regulatory capital and risk-weighted assets (to be able to express the stress test impacts in terms of capital adequacy); (2) asset quality and structure of lending by sector and by size of borrower; (3) provisioning and collateral (for the credit risk calculation); (4) structure of assets, liabilities, and off-balance-sheet items by time to repricing; (5) the structure of the bond portfolio (for the interest rate risk calculation); (6) net open positions in foreign exchange and lending in foreign currency (for the foreign exchange solvency risk calculation); (7) average profits and standard deviation of profits over time (to have a measure of “baseline” profitability); (8) liquidity structure of assets and liabilities (for liquidity risk calculation); and (9) bank-to-bank uncollateralized exposures, presented in matrix form (for the interbank solvency contagion risk).

Box 3.2Stress Testing When Some Input Data Are Unavailable

It may happen that some of the input data in Stress Tester 3.0—such as those on nonperforming loans, net open positions, and time to repricing—are not available, for example, owing to weak reporting systems or legal restrictions on data sharing. A rudimentary version of the stress tests can still be performed with the basic financial statements, provided that they are available for a number of periods. Those tests can be based on the observed relationships among the risk factors, the various items of the income statement, and the balance sheet. For example, even when no data are available on repricing buckets of assets and liabilities and bond portfolios in banks, a rudimentary stress test for interest rate risk could be based on the net interest income on banks’ income statements. In particular, past data on individual banks’ net interest income over time can be regressed on interest rates and other potential variables to estimate how banks’ net interest income responded to changes in interest rates, and the estimated slope coefficient(s) can be used to translate a change in interest rates into the impact in terms of profits (and potentially capital). Similarly, provisions for loan losses from the income statement can be regressed on the risk factors and other explanatory variables to analyze the impact on banks’ profitability. If longtime series are not available to carry out such regressions, the slope coefficients can be calibrated based on expert information or experience from other countries.

Even if the individual items of the financial statements are not available, “reduced-form” stress tests can still be carried out if there are reliable time series data. For example, one needs just capital, asset, and return data on individual banks over time to calculate the z-score, as a proxy for individual bank soundness, discussed in Section 2.C. The z-scores for individual banks can be regressed on a range of macroeconomic variables (e.g., real GDP growth rate, interest rate, and exchange rate) and bank-level variables (e.g., asset size or loan-to-asset ratio). The slope coefficients from this regression can then be used to map a macroeconomic scenario into the z-scores, to approximate the impact of macroeconomic stress on individual bank soundness. (A similar approach can also be used with distance to default data or other market-based indicators of individual institution soundness.) The main challenge in this type of approach is how to aggregate the bank-by-bank soundness data into a systemwide indicator (an issue that is discussed in more detail in Čihák, 2007b).

Most of the data in Table A2 usually are available to bank supervisors through standard regulatory returns. However, several issues need to be mentioned:

  • Stress tests analyze the economic position (net worth) of banks. In principle, this should be aligned with the reported data on capital, but in practice, there may be important differences between the calculated economic net worth of a bank and the reported regulatory data on capital. This may be the case, for example, when some assets are overvalued in banks’ balance sheets or when regulators accept as capital some liabilities that, in fact, are not capital (e.g., some long-term loans). The person carrying out stress tests should first try to adjust the input data for such biases. In the file, we show one example of such adjustments, namely, when banks underprovision their nonperforming loans. Another example (not shown in the file) might be when banks do not mark to market some bonds that they are holding in their portfolios.

  • The input data should reflect not only assets and liabilities but also off-balance-sheet positions. For example, the net open positions in foreign currency should reflect the delta equivalents of foreign exchange options.

  • Data on bank-to-bank exposures may be difficult to collect in many countries. In those cases, approximate calculations using less data (e.g., only data on each bank’s exposure to the rest of the system as a whole) can be used to at least broadly assess the associated risks, even though such methods may result in overlooking some exposures in the system.

  • For interest rate risk calculations, data on time to repricing are crucial. For example, from an interest rate risk perspective, a 20-year mortgage loan with an interest rate that can change every 6 months should be treated the same as a 6-month fixed-rate loan, not as a 20-year loan. However, getting data on time to repricing may be difficult to obtain in some cases. In many countries, banks report instead a breakdown of assets by maturity or by residual maturity. Although data on maturity or residual maturity are important for analyzing liquidity, using them as proxy for time to repricing may lead to misleading results (typically overstating the interest rate risk).

Additional data may be needed for stress tests for other risks not covered in this file. For example, to carry out stress tests for equity price risk and commodity price risk, data on net open positions in equities and in commodities would be needed (the mechanics of the test would be similar to the direct foreign exchange solvency test). Also, breakdowns of assets and liabilities by residual maturity/time to repricing and currency would be needed to perform stress tests separately for foreign currency.

C. Indicators of financial sector soundness and structure

Table A3 contains a set of core FSIs and other important ratios characterizing Bankistan’s banking sector and its components. These ratios can be used to provide a summary picture of the soundness of the financial sector and its components (peer groups and individual banks) in Bankistan. For more on the definitions and compilation of FSIs, see IMF (2004).

Table A3 also includes, at the bottom, individual banks’ z-scores. The z-score is defined as z=(k+µ)/σ, where k is equity capital as a percentage of assets, µ is average aftertax return as percentage on assets, and σ is standard deviation of the aftertax return on assets, as a proxy for return volatility. As mentioned earlier, z-scores have become popular as measures of bank soundness, because they are directly linked to the probability of a bank’s insolvency. The table also shows the z-scores for the peer groups of banks (for a discussion on the calculation of z-scores for groups of banks, see, e.g., Čihák, 2007b).

Table A4 characterizes the structure of the banking sector in Bankistan, showing the shares of the peer groups and the individual banks in total assets, loans, deposits, and capital. It shows that the share of foreign-owned banks is relatively high—about 55 percent in terms of assets and 83 percent in terms of capital (reflecting their higher capitalization). It also shows the total assets as a ratio to GDP, to indicate the size of the banking system relative to the Bankistan economy. The GDP figure is used again later, to put into macroeconomic perspective the capital injection needed to get all banks to comply with the minimum capital adequacy requirement.

D. Ratings and probabilities of default

Tables A5 and A6 show how banking sector ratios can be combined into institution-by-institution rankings, using a supervisory early-warning system.10 Such systems (e.g., Sahajwala and Van den Bergh, 2000) are very common in supervisory agencies, and they are used typically for assessing soundness of banks in “baseline” conditions; we will show in this exercise that they can be also used for assessing soundness in stressful conditions. One caveat needs to be borne in mind, however, namely, that these early-warning systems typically treat each bank separately and do not look at contagion among banks—an issue that will be analyzed as part of the stress test exercise presented here.

Table A5 provides the rankings, based on the off-site supervisory ranking system of the NBB, characterized in rows 3–22 of the Assumptions sheet. The system has three thresholds (columns B, C, and D in the Assumptions sheet) for each indicator, determining a numerical ranking for each of the indicators (with 1 indicating the best ranking and 4 the worst ranking). The rankings for the individual variables are weighted (using weights established by the NBB and provided in column E of the Assumptions worksheet) to derive an overall ranking for a bank. Similar off-site ranking systems are used by supervisors to identify banks that deserve increased attention (Sahajwala and Van den Bergh, 2000).

Underlying each ranking system is a more or less explicit link to probability of default (or probability of technical insolvency, that is, probability that the capital adequacy ratio declines below the regulatory minimum). Table A6 illustrates this by converting the rankings into probabilities of default, using a “step function” illustrated in Figure 3.2 and described in the case of Bankistan rows 6 and 22 of the Assumptions worksheet: according to NBB’s estimates, a bank with a rating of 1 has a 0.1 percent probability of default in a given year; a bank rated 2 fails with a 1 percent probability; a bank rated 3 has a 5 percent probability of default, and a bank rated 4 has a 30 percent probability of default in the coming year.

Figure 3.2Step Function (Example)

Source: Author, based on the default settings in the Assumptions sheet.

How are the parameters of such step functions derived? In some cases, they are based on expert estimates. In other cases, central banks or supervisory agencies attempt to “back-test” such systems to see whether they actually identify institutions that fail (or institutions that need interventions). When the step function is reestimated, the new parameters should be entered in row 22.

Figure 3.3 provides an illustration of such “back-testing” in the case of two variables (capital adequacy and gross nonperforming loans to total loans) and one threshold per variable. The dots represent observations of banks, and the two bigger boxes indicate two banks that have actually failed. The supervisory ranking system attempts to single out the failed banks (i.e., the bigger boxes), minimizing the signal-to-noise ratio for the estimate. If we want to capture all failed banks in Figure 3.3 (i.e., eliminate Type I errors), the early-warning system characterized by the CAR threshold and the NPL threshold shown in the figure allows a decrease in the percentage of banks misclassified as failures (i.e., Type II errors) from 88 percent (15/17, if we do not have any prior information) to 33 percent (1/3, for the “northwest” subset identified by the two thresholds). That is a major improvement in forecast precision.

Figure 3.3Back-Testing a Supervisory Early-Warning System (example)

Source: Author.

Box 3.3Linking Credit Risk and Macroeconomic Models

A number of papers have attempted to link credit risk to macroeconomic variables using econometric models. For example, Pesola (2005) presents an econometric study of macroeconomic determinants of credit risk and other sources of banking fragility and distress in the Nordic countries, Belgium, Germany, Greece, Spain, and the United Kingdom from the early 1980s to 2002. An even broader cross-country analysis is presented in IMF (2003). For Austria, Boss (2002) and Boss and others (2004) provide estimates of the relationship between macroeconomic variables and credit risk. For Finland, Virolainen (2004) develops a macroeconomic credit risk model, estimating the probability of default in various industries as a function of a range of macroeconomic variables. For Norway, the Norges Bank has single-equation models for household debt and house prices and a model of corporate bankruptcies based on annual accounts for all Norwegian enterprises (Eklund, Larsen, and Berhardsen, 2003). For Hong Kong SAR, several studies are available on the topic, employing both single-equation aggregate estimates and panels using bank-by-bank data (Shu, 2002; Peng and others, 2003; and Gerlach, Peng, and Shu, 2004). For the Czech Republic, Babouček and Jančar (2005) estimate a vector autoregression model with nonperforming loans and a set of macroeconomic variables.

Similar models are also common in FSAP missions. For example, the technical note from the Spain FSAP includes an estimate of a regression explaining nonperforming loans on an aggregate level with financial sector indicators and a set of macroeconomic indicators (IMF, 2006).

Several issues need to be considered when interpreting the macroeconomic models of credit risk. In particular, the literature is dominated by linear statistical models. The linear approximation may be reasonable when shocks are small, but nonlinearities are likely to be important for large shocks: doubling the size of the shock may more than double its impact. Indeed, microlevel credit risk models often find a nonlinear relationship between the scale of shocks and the likelihood of default; for macroeconomic shocks, Drehmann (2005) also reports a nonlinear link to credit risk. Moreover, the models are subject to the Lucas critique (Lucas, 1976), because their parameters or functional forms may become unstable, especially if exposed to a major stress. As an extreme example, when considering a scenario that involves depegging in a country with a currency board regime, models estimated on past data are likely to say very little about the impact of the exchange rate change on credit risk. In such a situation, other approaches, such as calibration that uses parameters based on experience from other countries, may be more appropriate.

3. Credit Risk

Lending is the core of the traditional banking business. In most banking systems, credit risk is the key type of risk. At the same time, it is the type of risk where existing models are most in need of strengthening.

There are three basic groups of approaches to modeling credit risk as part of stress tests. First, there are mechanical approaches (typically used if there are insufficient data or if shocks are different from past ones). Second, there are approaches based on loan performance data (e.g., probabilities of default, losses given default, nonperforming loans, and provisions) and regressions (e.g., single equation, structural, and vector auto regression). Third, there are approaches based on corporate sector data (e.g., leverage or interest coverage) and possibly on household sector data (even though such data are typically much more difficult to collect than corporate sector data).

The exposition in this section and in the accompanying file starts with the basic mechanical approaches. We then present how these can be extended into more realistic approaches. In particular, Box 3.3 discusses the links between credit risk and macroeconomic risk.

The Credit Risk worksheet contains calculations relating to the credit risk of banks, that is, the risk that banks’ borrowers will default on their contractual obligations. Table C1 in the worksheet Credit Risk summarizes the reported data for asset quality. Table C2 is used for the credit risk stress tests. It includes four different types of credit shocks, labeled 1, 2, 3, and 4.

A. Credit Shock 1 (“adjustment for underprovisioning”)

The purpose of the first part of the credit risk calculation is to make the point that stress testing should focus on the underlying economic value (net worth) of the bank. The economic value may differ in general from the bank’s reported regulatory capital. For example, as part of reported capital, some banks may include items that, in fact, are not capital and should be treated rather as something else (e.g., long-term loan). Or they can overstate some assets, resulting in overstated capital. Because all the stress testing calculations relate to the economic value of the bank, the stress testing analyst needs to first adjust the reported data to get a better picture of the starting (“baseline”) economic situation of the bank. Credit Shock 1 shows an example of such an adjustment.11

In Credit Shock 1, we assess what would happen if banks corrected their currently insufficient provisioning to fully meet the existing provisioning requirements. As mentioned earlier, a BCP assessment suggested that the current loan classification and provisioning standards in Bankistan are broadly adequate but not well enforced. The regulations prescribe the following loan provisioning rules: 1 percent general provision for pass loans, 3 percent general provision for special mention loans, 20 percent specific provision for substandard loans, 50 percent specific provision for doubtful loans, and 100 percent for loss loans. Reflecting the fact that it is very difficult to foreclose collateral in Bankistan, the test assumes that the actual value of the collateral is only 25 percent of the reported values (i.e., we assume a 75 percent “haircut” on the value of the collateral).

B. Credit Shock 2 (“increase in NPLs”)

If Credit Shock 1 is a “starting point adjustment,” then Credit Shock 2 can be seen as the first “real” stress test. It models a general decline in asset quality, affecting all banks proportionately. It is assumed that NPLs increase by a certain percentage, the default values being 25 percent of the existing stock of NPLs. This means that a bank would have to undertake additional provisioning by 25 percent for each of the three groups that constitute NPLs (substandard, doubtful, and loss loans). The increased provisioning requirements will reduce the value of the RWA as well as the capital. As regards the impact on the RWA, a common assumption is that the full increase in NPLs is subtracted from RWA. However, the impact on RWA may be smaller if the affected assets have a weight of less than one in the RWA. Typically, precise information of the distribution of NPLs across risk categories is not available. Nonetheless, a green cell (B40) in the Assumptions worksheet allows the user to change the assumed weight from 100 percent to a smaller number (say, 80 percent) and observe the impact of this assumption on the results.

The initial assumption in Credit Shock 2 is that the increase in NPLs in individual banks is proportional to the existing NPLs in these banks. In other words, banks that had more NPLs in the past are assumed to have more new NPLs as a result of the shock. This is the most straightforward calculation, but there are alternative approaches. For example, the new NPLs can be proportional to the overall stock of loans or to the stock of new performing loans. The green cells in this part of the worksheet allow the relaxation of this assumption and choose the weight of the existing NPLs and performing loans in determining the bank-by-bank increases in NPLs.

Whether to use existing NPLs or existing performing loans as a basis for assessing future credit risk is an open question. It is an empirical question that—if there are sufficient empirical data—can and should be decided empirically, by testing for the factors explaining bank-by-bank changes in NPLs (see Box 3.3 for a discussion of this issue). In the absence of reliable or sufficiently detailed empirical data, however, it is often necessary to resort to simplifying assumptions. As a rule of thumb, using the existing NPLs is correct if the existing bad loans are a good proxy for the quality of a bank’s risk management and therefore of the risk faced by the bank going forward. Conversely, using the existing performing loans can be justified by the fact that performing loans are those loans that may “go bad,” that is, be shifted into the NPL category, and therefore indicate the potential for credit risk. Using performing loans as a basis may be warranted if there has been a structural change in the economy (and therefore the past NPL ratios are of limited guidance in assessing future credit risks). For example, in a number of emerging markets in Central and Eastern Europe in the early 2000s, there was a marked shift from corporate lending to household lending, with very different qualities and parameters. Using past NPLs as a basis for the bank-by-bank increases in such a situation might lead to misleading results.

The importance of the bank-by-bank distribution of NPL increases can be illustrated in the Credit Risk sheet. For example, in the default settings, NPLs increase by 25 percent in each bank (i.e., cell B46 in the Credit Risk sheet has a value of 25, and cells B48 and B49 have the values of 1 and 0, respectively).12 The total volume of additional NPLs in the system (cell B50) is B$2,206 million and the state-owned banks’ postshock capital is B$ −437 million (cell C53). If we instead make the increase in NPLs proportional to the existing performing loans (i.e., we make cells B48 and B49 equal 0 and 1, respectively) and make the overall volume of additional NPLs the same (which we can achieve by using the Tools/Goal Seek function, setting the value of B50 back to B$2,206 million by changing the values of B46), the increase in NPLs has to be about 4.3 percent of performing loans, and the state-owned banks’ postshock capital is B$ −296 billion (cell C53), that is, smaller than if the increase in NPLs is proportional to the stock of NPLs. So, if we are concerned about the fiscal costs of the state-owned banks, the distribution of credit risks and buffers is important.

C. Credit Shock 3 (“sectoral shocks”)

As an illustration of how bank-by-bank credit risk can be modeled in a more realistic fashion, the stress testing file also includes sectoral shocks (see the bottom part of Table C2).

This exercise allows the user to select different shocks to economic sectors and observe how each bank would be affected, depending on the relative sizes of the banks’ credit exposures to these sectors. The case shown in the file as a starting example models a “terrorist attack” scenario, increasing credit risk in the tourism and trade sectors. The calibration of the sectoral shocks can be based on a historical scenario (e.g., a concrete example of a terrorist attack in Bankistan or in a neighboring country) or on empirical models explaining, based on past data, default rates in different sectors as a function of macroeconomic and other explanatory variables. The B43–B49 cells in the Assumptions worksheet can then be seen as an interface between the econometric model (or historical scenario) and the balance sheet implementation calculated in Stress Tester 3.0.

The definitions of sectors may be adapted depending on the country and the analyzed topic. For example, the “other” sector may be broken down into a number of subsectors. Alternatively, instead of using the main economic activity of the counterparty as the defining feature, sectors can be defined by the nature of the counterparty. For example, the “sectors” can be households (residents/nonresidents), nonfinancial enterprises (residents/nonresidents), nonbank financial institutions, and government (domestic/foreign).

The increase in NPLs is assumed to be proportional to the particular bank’s credit exposure to a sector, approximated by the bank’s total loans to that sector. Along the lines of the discussion in the previous subsection, it is possible to envisage that the increases reflect the bank’s existing NPLs or total loans to each sector and relax this assumption accordingly.

D. Credit Shock 4 (“concentration risk”)

Stress Tester 3.0 incorporates a block at the end of Table C2 that allows testing for the failure of the largest counterparties of individual banks. The user can change the assumed number of failures per institution and the assumed provisioning rate for those failures. If supervisors have data on banks’ exposures to nonbank financial institutions (NBFIs), this type of test can also be used to model the credit impact on banks of failures of the largest NBFIs.

4. Interest Rate Risk

The interest rate shock in the Interest Risk worksheet tests for direct interest rate risk. Direct interest rate risk is the risk incurred by a financial institution when the interest rate sensitivities of its assets and liabilities are mismatched. In addition, the financial institution is also exposed to indirect interest rate risk, resulting from the impact of interest rate changes on borrowers’ creditworthiness and ability to repay. The indirect interest rate risk is a part of credit risk. We discuss how to account for the interest rate–related credit risk in the section devoted to designing scenarios involving several shocks (Section 8).

A. Direct interest rate risk

The direct interest rate risk calculation in Stress Tester 3.0 consists of two parts, reflecting, respectively, flow and stock impacts of interest rate changes. The upper part, in Table D1, works with the repricing gap information from the Data worksheet. It calculates the changes in interest income and interest expenses resulting from the “gap” between the flow of interest on the holdings of assets and liabilities in each bucket. The “gap” in each time band or time-to-repricing bucket shows how net interest income will be affected by a given change in interest rates. It sorts assets and liabilities into three time-to-repricing buckets (due in less than 3 months, due in 3 to 6 months, due in 6 to 12 months).13

The bottom part of the calculation, in Table D2, shows the impact of interest rate changes on the value of bonds held by the commercial banks. The calculations assume that the bonds are “marked-to-market,” that is, changes in their market value have a direct impact on the capitalization of the banks. The impact of an interest rate change on the market value is approximated using the duration of the bonds held by the banks. The data on duration are given to us by the NBB, which calculated it using more detailed data on the parameters of the bonds and the structure of their holdings by banks—see the formula for duration in the Financial Soundness Indicators Compilation Guide (IMF, 2004).

The direct impact of higher nominal interest rates on capital and capital adequacy is typically negative, resulting from the fact that financial institutions operate with a duration gap between their assets and liabilities. Duration of assets (liabilities), DA (DL), is defined as the weighted average, term-to-maturity of an asset’s (liability’s) cash flow, the weights being the present value of each future cash flow as a percentage of the asset’s (liability’s) full price.14 Duration approximates the elasticity of the market values of assets and liabilities to the respective rates of return:

where A(rA) and L(rL) are market values of assets and liabilities of the financial system, and rA and rL are annual interest rates of assets and liabilities.15 Differentiating the capital adequacy ratio with respect to the interest rate on assets and substituting from equation (3.1), we obtain

Assuming that the risk-weighted assets move proportionately to total assets, that is, ΔARW/ARW=ΔA/A, equation (3.2) can be simplified into

where GAPD is the duration gap, defined as:16

Most financial institutions, and banks in particular, operate by transforming short-term, low-interest-rate liabilities into long-term, higher-interest-rate assets. This means that DA>DL, rA>rL, and GAPD>0. Thus, an increase in interest rates has a negative impact on the institutions’ net worth and capitalization, leading to increased financial sector vulnerability.

In the Bankistan example illustrated in Stress Tester 3.0, only two bonds are available to banks. Both of them are government-issued bonds. The Assumptions worksheet shows the key parameters of these bonds (in rows 57–59) and calculates their duration, using the Duration function in Excel (see cells H58 and H59).17

B. Indirect interest rate risk

The indirect effects, related to the interest-risk nexus, work in the same direction. An increase in nominal interest rates—to the extent that it increases real interest rates and makes it more difficult for borrowers to repay their debts and to obtain new credit—is likely to have a negative effect on the credit risk of the financial institutions’ borrowers. Other things being equal, higher risk eventually translates into higher losses and a decline in the financial institutions’ net worth. The exact impact depends on factors such as the borrowers’ earnings in relation to interest and principal expenses, loan loss provisions, and the degree of collateralization of the loans. Country case studies find a positive relationship between higher interest rates and nonperforming loans or loan losses.

The basic calculation does not cover the impact of nominal interest rate changes on real interest rates and thereby on borrowers’ creditworthiness and ability to repay. The impact size depends mostly on the corporate sector’s leverage and exposure to the real estate market (which is also likely to be influenced by changes in interest rates). In order to assess this type of risk, one would usually need to estimate the impact of changes in interest rates on nonperforming loans, using a regression model. In our example, the joint impact of changes in interest rates and in credit quality can be simulated using the worksheet Scenarios.

5. Foreign Exchange Risk

The foreign exchange risk is the risk that exchange rate changes affect the local currency value of financial institutions’ assets, liabilities, and off-balance-sheet items. The foreign exchange risk is composed of three types: the direct solvency risk (resulting from banks’ net open positions in foreign currency and those in local currency that are indexed to exchange rates); the indirect solvency risk (resulting from the impact of foreign exchange positions taken by borrowers on their creditworthiness and ability to repay and thereby on financial institutions); and the foreign exchange liquidity risk (resulting from liquidity mismatches in foreign currency). In this section, we will focus on the direct solvency risk, and we will discuss implementation of the indirect solvency risk; we relegate the foreign exchange liquidity risk to the section dealing with the liquidity stress test.18

The foreign exchange shock in the FX Risk worksheet is composed of two parts. The first part, shown in Table E1, tests direct foreign exchange rate risk. The second part, shown in Table E2, shows the impact of the change in the nominal exchange rate on banks through changes in the credit risk.

A. Direct foreign exchange risk

Table E1 in the FX Risk worksheet tests direct foreign exchange rate risk based on the net open position in foreign exchange at end-2010. This figure is calculated by the NBB, using the methodology described in the Financial Soundness Indicators Compilation Guide (IMF, 2004), and is copied to Table E1 using a link to the Data worksheet.

The direct exchange rate risk can be assessed using the net open position in foreign exchange, one of the “core FSIs,” defined in IMF (2004). The direct exchange rate risk is arguably the easiest part of stress tests to implement. To illustrate this test, let F denote the net open position in foreign exchange, C the capital, ARW the risk-weighted assets (all in domestic currency units), and e the exchange rate in units of foreign currency per unit of domestic currency. A depreciation (decline) in the exchange rate leads to a proportional decline in the domestic currency value of the net open position, that is, Δe/eF/F (for F≠0). Let us assume that this translates directly into a decline in capital, that is, ΔCF=1.19 The impact of the exchange rate shock on the ratio of capital to risk-weighted assets would then be

which uses the fact that ΔCeFe=F/e. The symbol “≅” means that the equation is only approximate for larger than infinitesimal changes. Equation (3.1) can be rewritten as

The term ΔARWC can have values from 0 to 1, reflecting the degree of co-movement of capital and the risk-weighted assets.20 In the special case of ΔARWC=0, that is, if the risk-weighted assets do not change, the change in the capital adequacy ratio equals the exchange rate shock times the exposure, measured as a product of the net open position to capital (F/C) and capital adequacy (C/ARW), both of which are “core FSIs” as defined by IMF (2004). This is sometimes used as a shorthand calculation of the direct exchange rate stress test. It should be noted that equation (3.6) holds only as a linear approximation, which works well in nonsophisticated financial systems. However, if financial institutions have large positions in foreign exchange options, the relationship between the exchange rate change and the impact on capital can become highly nonlinear. In such cases, stress tests based on detailed decomposition of financial institutions’ open positions are a superior analytical tool.21

In Stress Tester 3.0, a depreciation of the Bankistan dollar against the U.S. dollar from the official exchange rate of 55 B$/US$ to the currently prevailing parallel market rate of 85 B$/US$ is assumed (at constant cross-exchange rates to the U.S. dollar). Information on the banks’ foreign exchange exposure is provided in Table E1. A depreciation will benefit banks that have a long (positive) open position in foreign currency and hurt banks that have a short (negative) position in foreign currency.

Only a very limited number of banks have short positions; therefore, the direct depreciation effects are very small—some banks would even gain from a depreciation. Given that most central banks impose limits on foreign exchange positions to capital, this result is not unusual. For most banking systems, the direct foreign exchange solvency risk is rather small. Banks’ net open positions in foreign exchange are typically under close scrutiny from banks’ risk managers and supervisors. Banks in some countries have explicit limits on these positions as a percentage of the bank’s capital (the ceilings typically being in the range of 10–20 percent of capital). In other countries, this is addressed by including the net open positions in the capital adequacy calculation. In general, the open positions tend to be rather small, and consequently the direct impact of an exchange rate depreciation (or appreciation) tends to be rather small.

B. Indirect foreign exchange risk

Besides direct depreciation effects, a change in the exchange rate would also influence the creditworthiness and ability to repay of the corporate sector. A change in the exchange rate influences the corporate sector in two main ways: first, it changes its competitiveness relative to the foreign corporate sector; second, it influences the corporate balance sheets directly via firms’ net open positions in foreign currencies (for instance, companies can borrow massively in foreign currencies).

The indirect foreign exchange risk seems to be very important. FSAP missions generally have not been able to collect comprehensive data on the corporate sector’s foreign exchange exposure, but those FSAP missions that analyzed the corporate sector in detail generally found that the banking sector’s indirect exchange rate risk was more important than its direct one. The indirect foreign exchange rate risk appears to be particularly substantial in countries with closely managed exchange rate pegs.

To illustrate the significance of the indirect risk in overall banking sector risk, let us denote the corporate sector’s debt, equity, and open foreign exchange position as Dc(e), Ec(e), and Fc(e), respectively.22 Let us assume that, similar to the case of banks’ net open position, a percentage change in the exchange rate will translate into the same percentage change in the domestic currency value of the net open position, which will in turn lead to an equivalent change in the corporate sector’s equity, that is, ΔEce=ΔFce=F/e. The impact of the exchange rate on the corporate leverage (Dc/Ec) is then given by

Thus, if the corporate sector is short in foreign exchange, a depreciation (decline) in the exchange rate would lead to an increase in its leverage. Corporate leverage typically is positively correlated with the share of banks’ nonperforming loans in total loans (denoted as NPL/TL), that is, Δ(NPL/TL)/Δ(Dc/Ec)=a>0.23 The impact of a change in the exchange rate on the NPL/TL ratio can then be expressed as

In the special case when ΔDcEc=0, the change in the NPL/TL ratio would equal the exchange rate change times the respective FSI (the net open position), times the parameter a, which can be estimated empirically, as shown, for example, in IMF (2003) or Boss and others (2004). To find the impact on capital adequacy, we can assume, as has been done in some FSAP missions, that the credit shock moves some of the previously performing loans into the nonperforming category. By differentiating C/ARW with respect to NPL/TL, and substituting for NPL/TL from equation (3.8), we obtain

where we assume (as several FSAP missions have done) that provisions are expressed as a fixed percentage (π) of nonperforming loans and that they are deducted directly from capital.

The incorporation of the indirect effect makes the analysis of foreign exchange rate risk more complex and dependent on additional assumptions or regression analysis. One of the reasons adding to the complexity of the indirect exchange rate stress test is the fact that it includes the effects on stocks as well as on flows. The calculation of the indirect effect as per equation (3.9) would need to reflect the impact of exchange rate changes on the net present value of the corporate sector, which means to take into account changes in the net present value of future earnings. For example, in export-oriented companies, a depreciation could be expected generally to increase their future earnings. In terms of the net present value, the effect would be essentially equivalent to the impact of a long position in foreign currency. However, it may be more practical to calculate the impact on flows, by estimating the elasticity of earnings to interest and principal expenses (an encouraged FSI) with respect to the exchange rate and then to estimate the relationship between this FSI and the NPL/TL ratio. Alternatively, it would be useful to compile an indicator measuring the corporate sector’s flow exposure, for example, a ratio of foreign exchange earnings to total earnings or (ideally) a ratio of earnings in foreign exchange to interest and principal expenses in foreign exchange.

Table E2 in the FX Risk worksheet shows the impact of the change in the nominal exchange rate on banks through changes in the credit risk. The impact is approximated here by assuming that the change in the NPLs is proportional to the volume of foreign exchange loans in a bank. The idea behind this assumption is that a depreciation would increase the domestic currency value of these loans, which would make it more difficult for the borrowers to repay—for example, because some of the loans are extended to borrowers with limited access to foreign currency.

6. Interbank (Solvency) Contagion Risk

We have so far assumed that there is no contagion among banks in the event of a failure. This assumption is relaxed in the Interbank worksheet, which presents a basic calculation of the interbank contagion risk.24

In this section, we focus on contagion through insolvencies. There is also, potentially very important, the risk of liquidity contagion through bank runs triggered by a run on another bank. A basic model of liquidity contagion is shown as part of the liquidity risk (Section 7). The principle of having a matrix of bank-to-bank “exposures” is the same in all the contagion tests, but the specification of the matrix is different for a liquidity test.

We focus here on contagion within the (domestic) banking system. The Bankistan banking system includes foreign-owned banks, but we do not study cross-border exposures and cross-border contagion. In principle, the same framework would have to be applied, but the definition of “system” would have to be wider to include the foreign banks. For simplicity (and because obtaining good bank-by-bank data on cross-border exposures is not trivial in practice), we focus here on interbank exposures in the domestic market. We have to be at least aware, however, that in addition to the domestic exposures, there may also be important cross-border exposures and the related risk of contagion.

The upper panel in the Interbank worksheet (Table F1) derives a matrix of interbank exposures for the banks in Bankistan. The first panel shows the matrix of net interbank credits, in which each cell shows, for the bank in the column (i.e., for the bank listed in the same column in row 4), its net credit to the bank in the row (i.e., the bank listed in the same row in column A). For example, the value of 70 in cell 111 means that the bank in column I (i.e., DB1) has an outstanding net credit to the bank in row 11 (i.e., SB2) in the amount of B$70 million. A corresponding entry of −70 in cell G13 confirms this by showing that the bank in column G (i.e., SB2) has an outstanding negative net credit of B$70 million to the bank in row 13 (i.e., DB1), that is, it has a net borrowing of the same amount. The matrix is obtained by netting out the gross figures in the interbank lending table reported in the Data worksheet, that is, reported by the NBB. In this matrix, positive figures mean that the bank in the column is a net creditor of the bank in the row, whereas negative figures mean that the bank in the column is a net borrower of the bank in the row. The diagonal cells are left empty, because the focus of this exercise is on exposures between each bank and the other banks.

To facilitate further calculation, the matrix of net interbank credit is converted into the matrix of net interbank exposures. This is done by “stripping” the matrix of net creditors by focusing on the positive numbers (because those are the banks exposed to interbank credit risk). The other cells in the table are left empty. For example, the value of 70 in cell I26 means that the bank in column I (i.e., DB1) has an outstanding net exposure to the bank in row 26 (i.e., SB2) in the amount of B$70 million. The corresponding cell, G28, is empty: the bank in column G (i.e., SB2) has no net exposure to the bank in row 28 (i.e., DB1), because SB2 is a net borrower with respect to DB1 and therefore has no direct credit exposure to DB1.

A. “Pure” interbank contagion

The middle panel in the Interbank worksheet (Table F2) provides a calculation of the “pure” interbank contagion exercise. The exercise shows what would happen with the capital of the bank in the column if the bank in the row failed and defaulted on all its interbank borrowing. This is actually a series of 12 separate stress tests, one in each row, showing for each bank what would be the direct impact of its failure on the capital of each of the other banks. The stress test is run in several iterations, as the contagion-induced failures (“first iteration”) can induce failures in other banks (“second iteration”), which can lead to further failures (“third iteration”), and so on.

The first part of Table F2 shows, in each row, the post-shock capital of the individual banks in the column after the assumed failure of the bank in the row. For example, row 49 shows in columns F to Q what would be the postshock capital of banks SB1 to FB4 if bank SB3 failed. The results in row 49 suggest that none of the banks would have a negative postshock capital as a direct result of SB3’s failure.

The next part of Table F2 shows which banks fail as a result of the first iteration (“1” denotes a newly failed bank; all others have zeros). The table shows that two banks would fail as a result of failures in other banks, namely, that DB1 can fail as a result of failures in SB2 or FB3 (see cells I62 and I71) and DB2 can fail as a result of failures in SB1, SB2, FB1, and FB3 (see cells J61, J62, J69, and J71).

We are assuming for simplicity that if a bank’s capital stays positive after an iteration, the bank does not fail and remains able to repay all its interbank obligations; if its capital becomes negative, it fails and does not repay its obligations. The calculation can be made more realistic by estimating a more complex mapping between the capital adequacy ratio and the bank’s probability of failure (rows 6 and 22 in the Assumptions worksheet contain an example of such a mapping). Such mapping is likely to indicate a wider scope for interbank contagion than the introductory calculation presented here; however, the calculations would be based on similar techniques.

To keep the calculations straightforward, we assume here that the impact of shocks is deducted directly from capital. It is not complicated, however, to extend the file to take into account banks’ profits.

The second iteration needs to be calculated only for the failures in SB1, SB2, FB1, and FB3, because these are the only banks whose failures lead to failures in other banks (namely, DB1 or DB2). The failures in DB1 and DB2 could lead, in the second iteration, to failures in other banks if the other banks had substantial net credit outstanding with respect to DB1 or DB2. However, as illustrated in rows 73–113, this is not the case: the banks that are creditors to SB3 and DB1 have exposures to SB3 and DB1 that are well below their capital.

Rows 73–85 start the second iteration by inverting the table of the failed banks from the first iteration. This is to illustrate that in the second iteration, the banks that were exposed to contagion in the first iteration become the source of further contagion.25 Rows 88–99 show the capital after the second iteration, which generally equals the capital after the first iteration, but for failures in SB1, SB2, FB1, and FB3 (i.e., in rows 88, 89, 96, and 98), the second iteration capital of the bank in the column is lowered by the amount of exposure of this bank to the bank that failed in the first round. This is highlighted in rows 101–112, which show the change in capital between the first and second iterations: capital is lower for stress tests in rows 101, 102, 109, and 111, because the corresponding banks (SB1, SB2, FB1, and FB3) cause additional bank failures in the second iteration.

Rows 115–126 show that the third iteration is not needed in this case, as none of the banks are affected enough to fail in the second round. If such failure occurred, the third round (and any subsequent round) could be implemented in the same way as the second round.

To present the results in terms of capital adequacy, the worksheet uses an assumption on the risk weight of the interbank loans that become unpaid. Reflecting the typical Basel weight for interbank loans, this assumption is set at 20 percent but can be changed by changing the value of the B77 cell in the Assumptions worksheet.

The “pure” interbank contagion test could be interpreted as a measure of systemic importance of individual banks: the bigger the decline in the system’s capital (or capital adequacy ratio), the more systemically important is the bank whose default is assumed. In our example, FB2 is the systemically most important bank, using this criterion, because assuming its failure yields the lowest postshock systemic capital (B$3,570 million in the cell B138) and postcontagion CAR (9.7 percent in the cell B152). This test (or rather set of tests) is useful, but it does not take into account the different likelihood of failures in different banks, an issue that is addressed by the “macro” interbank contagion test.26

B. “Macro” interbank contagion

The lower table in the Interbank worksheet (Table F3) shows a “macro” interbank contagion test. In this case, we model the case when bank failures are triggered by macroeconomic developments, in particular by the scenario that is already modeled in the Scenario sheet (Figure 3.4). The starting point for the “macro” interbank contagion is therefore the postshock values of capital and risk-weighted assets for each bank from the Scenarios sheet.27 For the failed banks (SB2 and SB3 for the default settings in the spreadsheet), we run an interbank contagion exercise using the matrix of net interbank exposures. Then we search for banks that fail in this first iteration. If there were no new failures, the contagion exercise would stop here. In our example, however, there are three new failures: SB1, DB1, and DB2. We therefore run a second iteration, looking at the impact of these additional failures on other banks. We find that they lead to one additional failure, namely, in DB4. If this failure led to other new failures, we would need to run a fourth iteration. However, in this particular case, the contagion-induced failure of DB4 does not lead to other failures, and the process stops at the third iteration.

Figure 3.4“Macro” Interbank Contagion

Source: Author.

What is the key difference between the “pure” and the “macro” contagion tests? The “pure” contagion test assumes that a failure occurs in a single bank, for example, for some internal reason (e.g., because of a large fraud in the bank); it does not distinguish the relative likelihood of the failure of various banks. This is what the “macro” contagion test does. It analyzes situations when all banks are weakened at the same time by a common external (typically macroeconomic) shock, which affects each bank differently depending on its exposures to the various risk factors and makes some of the banks (perhaps more than one) fail. For the default settings for Bankistan, for example, the first iteration of the “macro” contagion test involves a simultaneous failure of two banks (SB2 and SB3), resulting in a second-iteration failure of three other banks (SB1, DB1, and DB2), leading to a third-iteration failure of DB4. The process stops at the third iteration, which does not lead to additional failures.

7. Liquidity Tests and Liquidity Contagion

Testing for liquidity risks is less common than testing for risks to solvency in central banks’ stability reports and in IMF work. This mostly reflects the fact that modeling liquidity risks is more complex. First, to properly model liquidity fluctuations in banks, one needs to have very detailed, high-frequency data, such that are typically used by commercial banks themselves in their liquidity management models. Second, to model the impact of large liquidity shocks, one needs to consider the broader liquidity management framework, in particular the lender-of-last-resort function of many central banks.

At the same time, testing for liquidity risks is important. In the last two decades, much of the attention in risk management and prudential supervision was on capital, partly in relation to the efforts to standardize capital adequacy requirements across countries. In the process, relatively less attention has been paid to cash flows and analysis of liquidity (e.g., Goodhart, 2006). Analyzing the response of liquidity to stress is an important undertaking, because liquidity is how a stressful situation often manifests itself in the short run.

The presentation of the stress test impact is different from the solvency tests discussed so far. The impact is shown for each bank in terms of the number of days it would be able to survive a liquidity drain without resorting to liquidity from outside (i.e., from other banks or the central bank).28 This is a relatively narrow approach to liquidity stress testing, but it is one that allows for an introductory exposition without going into details.

The Stress Tester 3.0 file contains two basic examples of liquidity tests. The Liquidity worksheet contains these two examples in two tables. Figure 3.5 shows the results of these two tests for the default values contained in the spreadsheet.

Figure 3.5Results of Liquidity Stress Tests

Source: Author’s calculations, using Stress Tester 3.0.

Table G1 models a liquidity drain that affects all banks in the system proportionally to their volumes of demand and time deposits. The worksheet allows the user to change assumptions on the percentage of demand deposits and time deposits that get withdrawn each day and on the percentage of liquid assets and other assets that banks can convert to cash each day.29

Table G2 models “liquidity contagion,” where the liquidity drain starts in the smallest or weakest banks and proceeds to the larger or stronger banks. The test allows for three possible measures of “bank safety”: (1) total assets; (2) total assets, with a premium for state ownership; and (3) preshock rating. In the first case, depositors perceive bank safety as linked to the size of the bank, approximated by total assets. In the second case, they also perceive state-owned banks as being safer than privately owned banks (because of an explicit or implicit government guarantee in the former case). In the third case, depositors’ perceptions of bank safety are correlated with the banks’ recent financial performance.30 Table G2 also combines the liquidity impact of government default with a bank run. It allows users to change the assumption on the percentage of the government bonds that are in default.

8. Scenarios

The Scenarios worksheet illustrates, in Tables H1–H4, how shocks to the various risk factors can be combined into a single scenario. The main reason for using scenarios rather than single-factor shocks is that in the macroeconomic context, changes in several risk factors are typically interrelated. For example, a large increase in nominal interest rates can lead to an increase in real interest rates, which can (perhaps with a lag) contribute to an increase in NPLs. If this is the case, banks will be hit not only by the direct impact of the nominal increase in interest rates but also by the indirect impact through credit risk.

The purpose of this worksheet is for the users to see how different combinations of the shocks affect the capital adequacy of the system. For simplicity of presentation, the worksheet contains only formulas linked to the other worksheets, namely, Credit Risk for credit risk, Interest Risk for the interest rate risk, FX Risk for the direct and indirect foreign exchange solvency risk, Interbank for interbank contagion, and Liquidity for liquidity risk. When a number of different specifications of a stress test are available, Stress Tester 3.0 allows the user to specify which of the alternative specifications is being considered in the overall scenario. These choices are assumptions that can be made in cells B69–B75 of the Assumptions worksheet (a summary of the chosen scenario is then shown on top of Table H in the Scenarios worksheet).

The Scenarios worksheet adds up the impacts of the selected shocks to arrive at an aggregate impact. Two issues are important in considering whether the impacts can be added up. First, the calculations need to take into account concentration of risks in institutions. Simply adding up aggregate losses caused by individual shocks could overlook situations when risks are concentrated in an institution or a group of institutions. Stress Tester 3.0 addresses this issue by calculating the impacts bank by bank. This allows us to see whether some banks are hit by the selected combination of shocks much harder than others (indeed, they are). Second, it is not trivial to combine solvency and liquidity risks. Stress Tester 3.0 illustrates how this can be done. It uses the NBB’s supervisory early-warning system to combine the changes in solvency and liquidity (and other measures) to identify the change in the supervisory rating and the implied change in probability of default. There are some limitations to this approach (in particular, the PDs cannot be easily aggregated for the system as a whole), but it provides a useful illustration.

As a basic presentational approach, the Scenarios worksheet shows the impacts of the scenarios in terms of the capital adequacy and decomposes the overall impact into the individual risk factors (in percentage points of the CAR ratio). The charts in the Assumptions worksheet allow the user to immediately and graphically see the results of the various assumptions on the outcome in terms of changes in capital adequacy ratios. For illustration, we reproduce these charts here as Figure 3.6. The values shown in these charts represent the default sizes of shocks and starting assumptions in the accompanying file. Dark blue indicates the baseline (preshock) values of the capital adequacy ratio, medium blue indicates the corresponding postshock values, and light blue indicates values after the shocks and after taking into account the subsequent contagion among banks. The preshock (baseline) CAR values are taken from the Data worksheet, whereas the values corresponding to the stressful scenario are taken from the Scenarios worksheet.

Figure 3.6Impact of Stress on Capital Adequacy Ratios

Source: Stress Tester 3.0.xls, Assumptions worksheet, based on default values of shocks and assumptions.

The worksheet also compares the overall impact with the banks’ profits. Although we have so far assumed that the impacts are deducted directly from capital, in reality banks could use profits as their first line of defense. Table H in the Scenarios worksheet shows for comparison, for each bank, what its profits were in the past. It also allows the user to assume (in the appropriate green cell) an autonomous shock to net interest income.

A. Designing consistent scenarios

How can one design consistent scenarios?31 In general, there are two ways of asking questions about exposures in the financial system. The first way is to ask, for a given level of plausibility, which scenario has the worst impact on the system (“the worst case approach”). The second way is to ask, for a given impact on the system, what is the most plausible combination of shocks that would need to occur to have that impact (“threshold approach”). In Stress Tester, the “threshold approach,” also called “reverse stress testing,” is illustrated in the Reverse worksheet.

Figure 3.7 shows the process of scenario selection under the worst case approach and the threshold approach, for a simplified case when there are only two risk factors (e.g., changes in interest rates and exchange rates). Each ellipse depicts the set of combinations of the two risk factors with the same probability of occurrence. The shape of the ellipse represents the correlation between the two factors, and its size represents the level of plausibility (the larger the ellipse, the smaller the plausibility). The diagonal lines depict combinations of the risk factors leading to the same overall impact, measured here by a change in the system’s CAR. The impact increases with the size of the shocks to the risk factors, so the CAR decreases in the northeast direction. The diagonal lines do not have to be straight; they are depicted here as such only for simplicity. Figure 3.7 illustrates that the worst case approach and the threshold approach are two essentially equivalent ways of analyzing the same problem.32

Figure 3.7“Worst Case Approach” versus “Threshold Approach”

Source: Author.

The worst case approach starts with selecting a level of plausibility (e.g., 1 percent) and searching for the combination of shocks with this level of plausibility that have the worst impact on the portfolio. This means searching for the point on the largest ellipse that lies as far northeast as possible. In Figure 3.7, this is point A.

The threshold approach starts with selecting the threshold, that is, the diagonal line; it then searches for the most plausible (i.e., smallest) shocks reaching this threshold. This is straightforward if there is only one risk factor; if there are two risk factors, one needs to take into account the correlation between the risk factors. For the specific correlation pattern in Figure 3.7, selecting a threshold of zero capital adequacy would lead again to the combination of shocks corresponding to point A.

Establishing the plausibility level of a scenario can be difficult in practice, given that the scenario should be a low-probability “tail” event. For risk factors with good time series of historical data (in particular, for market risks), the natural starting point is to base the scenarios on the past volatility and covariance patterns. Calibrating the shocks is particularly straightforward for single-factor stress tests: an exchange rate shock can be based on 3 standard deviations of past exchange rate changes (corresponding roughly to a 1 percent confidence level). With multiple risk factors, one needs also to look at the covariance statistics of the variables or use stochastic simulations based on macroeconomic models. Such calculations are subject to a number of caveats. In particular, models can break down for large shocks. Nonetheless, the models, if used cautiously, can help to find a first-cut approximation of stress test scenarios (see Box 3.4 for an additional discussion of choosing the “right” scenario).

Box 3.4Picking the “Right” Scenario

In discussions on designing stress tests, too much is often made of establishing the “right” scenario. Of course, it is important, at least in theory, for scenarios to be internally consistent, as highlighted, for example, by Jones, Hilbers, and Slack (2004) or by Figure 3.7 in this chapter. In practice, assessing such consistency is tricky, because the scenarios are also supposed to be exceptional (but plausible).

How to address this challenge? One approach is to choose a concrete extreme historical scenario (e.g., the East Asian crisis of 1997) and calculate what would be the impact of repeating such a scenario (or an adaptation of such scenario) in the present situation of the banking system. The main advantage is that historical scenarios are easy to communicate and to implement. Also, they are plausible, because such a situation actually happened. Their main disadvantage is that past crises may not be good models for future crises. Also, the probability level of the past historical scenario may be unclear.

Another approach, used by some FSAPs and central bank FSRs, is to use an existing macroeconomic model (e.g., a model used by the central bank for macroeconomic forecasts and policy analysis) as a basis for stochastic simulations showing the distribution of the key risk factors in the case of shocks to the model’s exogenous variables. The challenge of this approach often arises from the fact that such macroeconomic models typically do not include a measure of credit risk (e.g., nonperforming loans to total loans or another measure of asset quality). Therefore, this approach usually involves estimating a “satellite” model that links a measure of credit risk to the variables from the macroeconomic model. Unlike the macroeconomic model, the satellite model can be estimated (and generally should be, if adequate data are available) on individual bank (and even individual borrower) data. The estimates from the satellite model can then be used for the balance sheet implementation.

Yet another, and perhaps more direct, approach can be to plot the existing observations of the various risk factors (in a similar fashion as shown, for two risk factors, in Figure 3.7) against a measure of soundness (e.g., capital adequacy ratio) and use this to identify the most stressful combinations of risk factors (in terms of Figure 3.7, this would mean identifying the points lying the most toward the northeast).

In sum, there is a range of methods, each with its advantages and disadvantages. Picking a scenario that is stressful and tells an interesting and consistent story is important. However, in most cases, identifying “the right scenario” is close to impossible. Much more important than fine-tuning scenarios is (1) being transparent about the underlying assumptions of the scenarios; (2) being transparent about the sensitivity of the results to those assumptions; and (3) showing how results with the same assumptions change over time. Showing results over time allows making judgments about the developments in the overall pool of risks and in the structure of risks faced by a financial system.

B. Linking stress tests to rankings and probabilities of default

The Scenarios sheet also illustrates the links between the stress tests and the supervisory early-warning system. It follows the same approach as the illustration shown in the Data worksheet, but it is derived from the bank-by-bank data after the shocks rather than those before the shocks. Specifically, Table H2 provides the postshock FSIs and other ratios for individual banks. These can be compared with the corresponding preshock ratios, shown in Table A3 (in the Data worksheet).

Table H3 converts these ratios into postshock ratings of the individual banks, using the same “step functions” that were applied in Table A5 to the preshock ratios. In both cases, the “step function” reflects the off-site supervisory assessment model and is specified in the Assumptions worksheet (rows 3–22). Table H3 also provides averages (weighted by banks’ total assets) for the three peer groups and for the banking system as a whole.

Table H4 converts the postshock ratings into postshock probabilities of default for each bank. These can be compared with the preshock probabilities of default, shown in Table A6 in the Data worksheet.

The charts in the Assumptions worksheet, reproduced in this chapter as Figure 3.8 and Figure 3.9, illustrate such comparisons. Dark blue indicates the baseline values of the indicators (ratings in Figure 3.8 and probabilities of default in Figure 3.9), and light blue indicates the corresponding values in situations of stress. The values of the baseline are taken from the Data worksheet, whereas the values corresponding to the stressful scenario are taken from the Scenarios worksheet. The values shown in the charts reproduced here reflect the default sizes of shocks and starting assumptions in the accompanying file.

Figure 3.8Impact of Stress on Supervisory Ratings

Source: Stress Tester 3.0.xls, Assumptions worksheet, based on default values of shocks and assumptions

Figure 3.9Impact of Stress on Banks’ Probabilities of Default

Source: Stress Tester 3.0.xls, Assumptions worksheet, based on default values of shocks and assumptions.

Figure 3.10 shows another form of presentation of stress testing results. It captures the impact of stress on the banks’ z-scores. As indicated earlier, the z-score has become a popular measure of bank soundness, because it is directly related to the probability of a bank’s insolvency. Figure 3.10 illustrates how the banks’ z-scores decline as a result of the assumed impacts, generally mirroring the decline in banks’ probabilities of default, shown in Figure 3.9 (one needs to bear in mind that higher z-scores correspond to lower probabilities of default). If the user changes the key sizes of shocks and assumptions in the accompanying Excel file, the charts in the file will change automatically.

Figure 3.10Impact of Stress on Banks’ Z-Scores

Source: Stress Tester 3.0.xls, Assumptions worksheet, based on default values of shocks and assumptions.

C. Modeling the feedback effects

The stress test calculations presented here focus on the impacts of shocks arising from the macroeconomic environment and affecting the financial sector. From a macroeconomic perspective, an important issue is whether the shocks in the financial sector can have feedback effects affecting the macroeconomic environment.

The practical problem in modeling the feedback effects is that there are too many. One direct effect that can be incorporated easily into the financial programming framework employed by the IMF is an impact on capital on “Other Items Net” in the monetary survey and thereby on other macroeconomic variables.

However, this is just one of many potential impacts. In some cases, the effects depend on the behavior of the institutions in situations of stress. For example, if banks attempt to sell off certain types of assets (e.g., real estate) in situations of stress, they may bring down the asset prices, with repercussion effects for other sectors (e.g., household consumption). Also, bank failures triggered by stress may result in a credit crunch.

In the accompanying Excel example, we approximate the potential macroeconomic impacts by the capital injection needed to bring all banks to the minimum required capital adequacy ratio. This indicator does not capture all the potential macroeconomic effects, but it is a useful broad indicator of potential fiscal costs associated with averting failures in the banking system. It is an upper-bound estimate of such costs. The public sector will most likely inject capital into state-owned banks, but it is less clear whether (and if so, how much) it will inject into the other banks, which are privately owned (it is generally more likely to do so for larger banks that are considered “too big to fail”). For this reason, the charts and results in the file show a breakdown of the necessary capital injection by ownership. The charts are included in the Assumptions worksheet and are reproduced here as Figure 3.11. The values shown in the figure reflect the default sizes of shocks and default assumptions in the accompanying file.

Figure 3.11Capital Injections Needed to Bring Banks to Minimum Capital Adequacy

Source: Stress Tester 3.0.xls, Assumptions worksheet, based on default values of shocks and assumptions.

9. Conclusion

It is difficult to really understand stress testing without going through actual stress testing calculations. This chapter enables readers to do just that: working with the accompanying Excel file, they can change the assumptions and observe changes in results.

Stress tests are complementary to other tools for financial stability analysis, and the exercise illustrates this complementarity. In particular, it illustrates that stress tests are complementary to the FSIs, which allow for “benchmarking,” that is, for a baseline assessment of the financial system under no stress. FSIs can also be used to describe the impact of stress on a system. Stress tests are also complementary to the supervisory early-warning system, an example of which was shown as well. The early-warning system traditionally is used to calculate ratings and probabilities of default in the “baseline” scenario, but as shown in Stress Tester 3.0, it can also be used to produce ratings and probabilities of default in a stressful scenario. Stress tests are complementary also to other tools, such as assessments of compliance with regulatory standards and codes, and assessment of the broader financial stability framework.

The exercise highlights several challenges. The basic challenge is that stress testing is data intensive. It deals with low-probability events, implying that there will always be a lack of data and a need for simplifying assumptions. The main thing an analyst should do is to be transparent about the nature of the assumptions. To make things more complicated, nonlinearities are likely to kick in for the large shocks that are being contemplated in stress tests. Also, macroeconomic and other models tend to break down in crises. Past crises may not be a good guide for the future. For example, a change in borrower characteristics may leave credit more vulnerable to interest rate risk. Another challenge is that the impact of shocks is distributed over time. It takes time for asset quality to deteriorate and for that deterioration to have an impact. A crisis evolves over a period of time, sometimes several years. Modeling stressful scenarios therefore has to take the time dimension into account, and it is important to clarify what the benchmark scenario is against which the stressful one is being compared. Finally, mitigating measures can be taken by participants and authorities, especially when looking at longer time periods. These measures and the feedback effects play an important role over time, and they make the stress testing calculations more complex.

Two main steps can be taken to address these challenges in stress tests. The first is to keep assumptions transparent, and be clear on the sensitivity of the results to the assumptions. The accompanying stress testing file is trying to do that. All assumptions are brought into one place and highlighted. Users can experiment with the assumptions to see the impact of the changes on the results. The second step is to present stress test results over time. Presenting results over time helps to say whether the overall pool of risks has changed or whether the structure of risks has changed. Most financial stability reports still do not provide results over time, which makes interpretation of the presented results more difficult for readers (Čihák, 2006). The Stress Tester 3.0 exercise is based on a one-period snapshot, but the idea behind the exercise is that it is run repeatedly, replacing the input data (in yellow) by data from other periods.

The idea behind the accompanying file is that it can be developed in a modular fashion, with additional modules capturing additional risks or elaborating on the existing ones. One can think about Stress Tester 3.0 as a module in a broader stress testing tool kit and about the cells with assumptions as interfaces between this module and other modules. The following extensions are particularly worth considering:

Credit risk–macro nexus. Credit risk is the main source of risk in most financial systems. At the same time, it is the part where this exercise is perhaps the most simplified. Ideally, we would need to have more detailed data on loan exposures and loan performance by economic sector and also data on the financial soundness of the corporate and household sectors. Time series of historical data ideally should be used to establish linkages between macroeconomic variables and loan performance (or, in the absence of reliable time series, estimates from other countries could be used). The calculations presented in this exercise try to convey the gist of the credit risk stress test, without going into technical details.

Credit VaR models. Developed country commercial banks and some banks in medium-income countries are using value-at-risk (VaR) models as a basis for stress testing for credit risk. There is a range of these models, all of which share the purpose of determining the probability distribution of the losses on a portfolio of loans and other debt instruments. Being able to compute the loss distribution of a portfolio allows the determination of the economic capital required by credit operations. Implementing VaR tests for credit risk on the macroprudential level is more complex than for market risk VaR models, because (1) fewer banks use them; and (2) the risk factors and their parametrization are likely to differ across banks, making comparability and aggregation more difficult. Nonetheless, implementing credit risk models on a macro-prudential level is possible and can provide a useful benchmark for credit risk evaluation, as illustrated in Avesani and others (2006) on the CreditRisk+ model. Indeed, some recent FSAPs have made use of these approaches.

Stress testing based on factor models. This includes (1) portfolio risk management models using structural credit risk models of obligors’ assets and risky debt based on domestic and international factor models, such as the portfolio manager models, CreditMetrics, and the default and conditional probability of default models (e.g., Segoviano and Padilla, 2006); and (2) Merton-type structural models of banks, which calibrate risk-adjusted balance sheets and implied assets of banks, which are then linked to domestic and international factor models. Stress testing can then analyze how changes in key international and domestic factors drive individual bank risk and systemic risk).

Other risk factors. Depending on the sophistication of the financial system and the type of its exposures, it may be necessary to perform stress tests for other risk factors. These might include asset price shocks (including, e.g., shocks to real estate prices) and shocks to commodity prices (especially in developing economies with significant exposures to commodities).


We concentrate on banks, even though the impact of risks in nonbank financial institutions is also discussed.

Including more institutions requires adding columns (and some rows in the “Interbank” worksheet), copying the relevant links in other worksheets, and checking summation formulas for the peer groups and the system.

For a longer discussion of this distinction, see, e.g., Čihák (2004), Jones, Hilbers, and Slack (2004), and IMF and World Bank (2005).

Čihák and Heřmánek (2005) provide a more detailed discussion of such “decentralized” stress tests. Hoggarth, Logan, and Zicchino (2005) discuss how such stress tests were carried out in the case of the United Kingdom.

The file also shows the z-scores for the peer groups of banks, using a similar methodology as used by some authors to translate bank-by-bank distance to default measures to a “portfolio distance to default” (e.g., IMF, 2005a). These calculations need to be treated with a degree of caution, given that they overlook issues related to contagion (see, e.g., Čihák, 2007b).

Distance to default is in effect an implementation of the z-score, for banks with stocks listed in liquid equity markets. Distance to default uses stock price data to estimate the volatility in the economic capital of the bank (see, e.g., Danmarks Nationalbank, 2004).

The 10 percent minimum capital adequacy requirement is just an assumption in Stress Tester 3.0, contained in the B71 cell of the Assumptions worksheet. The reader can change it, for example, to 8 percent and see in the relevant chart in the Assumptions worksheet how the capital injection (in percentage of GDP) declines.

Total capital may differ in general from regulatory capital; in this workbook, we use for simplicity the same numbers, but the file is set up in a way that allows for differences between equity and regulatory capital.

A practical complication with the system-oriented stress tests is the fact that the systemic relevance of an institution is established only after, not before, the stress tests. As a practical shortcut, many FSAPs and other authors use a measure of size (e.g., total assets) as a first approximation for systemic importance. It is a good proxy in most cases, but in some it can miss institutions that are small but have large potential for impact on other institutions, for example, through exposures in the interbank market. In Section VI, we provide a measure of systemic importance that takes the contagion effects into account.

The early-warning system does not have to comprise only ratios. Inclusion of variables in the early-warning system should reflect their power to identify weak banks. One variable that in some cases has good discriminatory power is deposit rates: high deposit rates in a bank can indicate that it has difficulty retaining depositors, a potential sign of problems (Kraft and Galac, 2007).

In a strict sense, therefore, this is just a starting point adjustment, not a part of stress tests themselves. In a broader sense, however, it is a “stress test” that shows how the reported data would change if the reporting system changed to one that reflected more closely the economic value of a bank. Of course, if the reporting system already reflects the economic value, there is no adjustment, and one can proceed directly to Credit Shock 2 and beyond.

These values originate in cells B35, B37, and B38 of the Assumptions worksheet.

This is a crude breakdown; for more precise calculations, more maturity brackets are used, especially for the very short periods but also for longer periods.

See the Financial Soundness Indicators Compilation Guide (IMF, 2004, paragraph 3.52) for a formula. In practice, the calculation of the duration of total assets and total liabilities of a financial system is a difficult computational task, and various simplifications are used (e.g., duration is computed for groups of assets and liabilities with common features and aggregated across such groups; or duration is replaced by residual maturity and time to repricing).

See, for example, Bierwag (1987). The results are first-order approximations; for large changes in interest rates, second derivative terms need to be included to account for convexity of portfolios. Alternatively, the elasticity of bond prices to interest rate changes can be empirically estimated using past data.

If the interest rates for assets and liabilities move simultaneously, the duration gap can be approximated as a difference of the two durations, DADL.

Excel also has a Price function, which can also be used to implement this calculation.

The direct and indirect solvency risks in foreign exchange are often also referred to as “exchange rate risks.”

An alternative, and arguably more realistic, approach is to deduct the impact first from profits (if any) and then from capital. See Section 1 for a more detailed discussion of this issue.

Empirically, ΔARWC could be estimated by a regression.

As a general point, stress tests should include all relevant off-balance-sheet items.

Given the practical difficulties involved in obtaining empirical data on open positions in the household sector, we refer here for simplicity only to the corporate sector, even though the theoretical analysis would be essentially the same even if we included the household sector.

IMF (2003) shows that for a panel of 47 countries, a 10 percentage point rise in the corporate leverage was associated with a 1.1 percentage point rise in the ratio of NPLs to total loans after a one-year lag.

In a real stress testing exercise, we would have to start from the very beginning with balance sheets that include the interbank exposures. This would make the aggregation of the bank-by-bank data more cumbersome (we would have to net out the bank-to-bank exposures). For simplicity, this exercise starts without the interbank exposures, which are added only later.

The calculation can be implemented without the inversion, but inverting makes calculations easier, because one can use the SUMPRODUCT function. Also, the inversion highlights the nature of the calculation: in the second iteration, we look at the failed creditors from the previous iteration and analyze who are their creditors. Given that the exposure table has creditors in columns and borrowers in rows, going from the first iteration to the second one requires inverting either the matrix of exposures or the table with results from the first iteration. In this implementation, we chose the latter approach.

“Pure” interbank contagion tests are more common in the literature. For example, Sveriges Riksbank presents its results regularly in its Financial Stability Report (for the methodology, see Blåvarg and Niemander 2002). “Macro” interbank contagion tests are less common but are presented for example by Elsinger, Lehar, and Summer (2003) for Austria and by Čihák, Heřmánek, and Hlaváček (2007) for the Czech Republic.

The calculations in the Scenarios sheet disregarded these contagion effects, so the calculation in Table F3 of the Interbank sheet can be seen as an extension of the calculation in the Scenarios sheet.

As a rule of thumb, supervisors often see five days as an important threshold for a bank’s ability to withstand a liquidity run. After five days or less, banks are likely to close for a weekend or a holiday, providing some “breathing time” for bank management and supervisors to regroup, assess the situation, and decide on measures and public announcements to make. This rule of thumb has been partly diluted with the growth of direct banking.

Another useful distinction in many cases is between domestic and foreign exchange deposits (and assets). It is possible to extend the file to assume different withdrawal rates by currency of denomination.

Liquidity contagion can also be thought of in terms of a similar “exposure” matrix as used in the solvency contagion, except that instead of “net uncollateralized interbank exposures,” the value of the cell in the matrix for liquidity contagion would be the difference between bank i’s and bank j’s measure of “bank safety.”

For more details, see Čihák (2004, 2005).

To some readers, these two approaches may resemble the dual tasks of microeconomics.

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