Modeling Correlated Systemic Liquidity and Solvency Risks in a Financial Environment with Incomplete Information
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

Contributor Notes

Authors’ E-Mail Addresses: Barnhill@gwu.edu, and LSchumacher@imf.org

This paper proposes and demonstrates a methodology for modeling correlated systemic solvency and liquidity risks for a banking system. Using a forward looking simulation of many risk factors applied to detailed balance sheets for a 10 bank stylized United States banking system, we analyze correlated market and credit risk and estimate the probability that multiple banks will fail or experience liquidity runs simultaneously. Significant systemic risk factors are shown to include financial and economic environment regime shifts to stressful conditions, poor initial loan credit quality, loan portfolio sector and regional concentrations, bank creditors' sensitivity to and uncertainties regarding solvency risk, and inadequate capital. Systemic banking system solvency risk is driven by the correlated defaults of many borrowers, other market risks, and inter-bank defaults. Liquidity runs are modeled as a response to elevated solvency risk and uncertainties and are shown to increase correlated bank failures. Potential bank funding outflows and contractions in lending with significant real economic impacts are estimated. Increases in equity capital levels needed to reduce bank solvency and liquidity risk levels to a target confidence level are also estimated to range from 3 percent to 20 percent of assets. For a future environment that replicates the 1987-2006 volatilities and correlations, we find only a small risk of U.S. bank failures focused on thinly capitalized and regionally concentrated smaller banks. For the 2007-2010 financial environment calibration we find substantially elevated solvency and liquidity risks for all banks and the banking system.

Abstract

This paper proposes and demonstrates a methodology for modeling correlated systemic solvency and liquidity risks for a banking system. Using a forward looking simulation of many risk factors applied to detailed balance sheets for a 10 bank stylized United States banking system, we analyze correlated market and credit risk and estimate the probability that multiple banks will fail or experience liquidity runs simultaneously. Significant systemic risk factors are shown to include financial and economic environment regime shifts to stressful conditions, poor initial loan credit quality, loan portfolio sector and regional concentrations, bank creditors' sensitivity to and uncertainties regarding solvency risk, and inadequate capital. Systemic banking system solvency risk is driven by the correlated defaults of many borrowers, other market risks, and inter-bank defaults. Liquidity runs are modeled as a response to elevated solvency risk and uncertainties and are shown to increase correlated bank failures. Potential bank funding outflows and contractions in lending with significant real economic impacts are estimated. Increases in equity capital levels needed to reduce bank solvency and liquidity risk levels to a target confidence level are also estimated to range from 3 percent to 20 percent of assets. For a future environment that replicates the 1987-2006 volatilities and correlations, we find only a small risk of U.S. bank failures focused on thinly capitalized and regionally concentrated smaller banks. For the 2007-2010 financial environment calibration we find substantially elevated solvency and liquidity risks for all banks and the banking system.

I. Introduction1,2

This paper proposes and demonstrates a methodology for modeling correlated systemic solvency and liquidity risks for a banking system.3 Correlated financial and economic shocks impact various sectors of an economy and regions of a country in different ways. For example, real estate prices (sector equity returns) may fall much more sharply in some regions (sectors) than others. All entities (individuals, businesses, financial institutions, regulators, and governments, among others) existing at a particular point in time will be impacted simultaneously by adverse financial and economic environment events which may produce correlated defaults by many bank borrowers in various sectors and regions. Correlated recovery rates on loans may also decline in adverse periods. Correlated loan portfolio losses can be expected to produce correlated solvency and potentially liquidity risks for many banks with similar asset and liability structures. In our view most risk assessment methodologies do not adequately model the interaction of the four main drivers of bank solvency risk including financial and economic environment volatility, bank loan sector and region concentration levels, bank loan credit quality, and bank capital levels. The principal contributions of this paper are to model these risk factors in significant detail for 10 banks simultaneously, estimate the probability of banking system systemic solvency and liquidity risks, and evaluate measures that may be adopted in advance to moderate the magnitude of such systemic risks and their potential impacts.

Our view is in line with the literature relating bank runs to extreme episodes of market discipline and with the empirical evidence on the causes of the 2008–2009 global crises. Recent research has made it clear that the global financial crisis has not been a pure liquidity shock but was triggered instead by concerns about the value of bank assets—subprime mortgages and structured products affected by the fall in house prices (e.g., Gorton and Metrick, 2009, and Afonso, Cover, and Schoar, 2010). Our approach is also related to current supervisory approaches for stress testing in which a systemic liquidity shock is triggered by solvency concerns, such as those developed by the Bank of England (Aikman et al., 2009, Wong and Hui, 2009, and van den End and Tabbae, 2009).

Given the interaction between solvency risk and systemic liquidity risk, our framework will jointly model both. Comprehensive macro stress testing is a useful instrument for central banks and supervisors to assess the consequences of severe market disruptions, to understand the different dimensions of systemic risk and estimate the contribution made by different institutions and transactions to potential systemic losses.

The stress tests proposed in this paper were applied to a stylized set of U.S. banks. Using Call Report and other publicly available data, we constructed detailed balance sheets for 10 aggregate banks in four categories: two large banks that aggregate the asset and liabilities of all U.S. banks with assets above $500 billion (excluding Morgan Stanley and Goldman Sachs); three large banks that aggregate the assets and liabilities of all U.S. banks with assets between $100–500 billion; three medium-size banks that aggregate banks with assets between $10-100 billion and two small banks that aggregate banks with assets below 10 billion.

Section II defines systemic liquidity risk. Section III presents the methodology and modeling steps, and data requirements. Section IV calibrates the model for the stylized U.S. banking system. Section V reports the results. Section VI concludes.

II. Systemic Liquidity

A systemic liquidity shock is an aggregate shortage of liquidity, i.e. a situation in which many institutions face liquidity shortages simultaneously, as opposed to one institution suffering a liquidity shortage. Systemic liquidity risk is the probability that this situation takes place. A liquidity shortage can manifest as an inability for institutions to roll over funding (funding liquidity risk), the inability to trade assets at normal bid/ask spreads (market liquidity risk), or very frequently, both.

The stress test approach developed in this paper takes the view that absent an aggregate preference shock (i.e. a sudden shift of preferences in favor of higher present consumption) or infrastructure malfunctioning, a systemic liquidity shock—i.e. many institutions suffering a liquidity shortage—is more likely to happen in the presence of a shock to fundamentals that depresses asset values and makes the market reluctant to fund these (suddenly) lower quality assets or the institutions that hold them.4 In the presence of incomplete and asymmetric information on the values of assets and the financial condition of banks, this reluctance can also be extended to good assets and solvent institutions. Our approach is consistent with the stress testing literature in which liquidity withdrawals are linked to banks’ solvency risk (Table 1).

Table 1.

Selected Liquidity Stress Testing (ST) Frameworks

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Note: Bank of England reflects the ST framework proposed by Aikmen and others (2009); De Nederlandsche Bank reflects the ST framework proposed by van den End (2008); and the Hong Kong Monetary Authority reflects the ST framework proposed by Wong and Hui (2009).Source: Global Financial Stability Report, October 2011

We propose a stress test of systemic liquidity in which systemic liquidity shocks are modeled as a reaction to shocks to asset values resulting from borrower defaults and other factors. In our approach a liquidity shock (or a “run”) is an extreme episode of “market discipline” by which those providing funding (depositors, wholesale investors, and other banks, among others) attempt to sort among ex-ante “good” (solvent) and ex-ante “bad” (insolvent) users of funds in a world of asymmetric information regarding asset values. While the exact timing of a systemic liquidity shock is difficult to forecast, we postulate that they are highly correlated with solvency concerns and contractions in bank lending.

A systemic liquidity shock is closely associated with the notion of bank panics. Traditionally, there have been two leading alternative views to explain the triggers of panics in the more traditional setting of depositors’ behavior: the random withdrawals theory and the information-based theory. The random withdrawal approach, as developed in Diamond and Dybvig (1983) postulates that a panic is the realization of a bad equilibrium due to the fulfillment of depositors’ self-expectations concerning the behavior of other depositors (a pure liquidity shock). On the other hand, the information-based approach, as reflected in Allen and Gale (1998), claims that a panic is an episode of market discipline during which depositors attempt to sort among ex-ante “good” (solvent) and ex-ante “bad” (insolvent) banks in a world of asymmetric information regarding bank asset values. In this context, bank panics can be a normal outcome of business cycles: an economic downturn will reduce the value of bank assets raising the possibility that banks cannot meet their commitments. Gorton (1988) undertook an empirical study to differentiate between the “sunspot” view and the business-cycle view of banking panics. He found evidence consistent with the view that banking panics are related to the business cycle.

There is also consensus that the global financial crisis has not been a pure liquidity shock but was triggered instead by concerns about the value of bank assets—subprime mortgages and structured products affected by the fall in house prices. Gorton and Metrick (2009) characterized the global crisis as system-wide “run” in the securitized banking system-- more precisely a “run on the repo market”—similar to the banking panics of the 19th century. Both episodes, in their view, were triggered by insolvency problems. They find that during 2007– 2008, changes in the LIBOR-OIS spread, a proxy for counterparty risk in the interbank market, was strongly correlated with changes in credit spreads and repo rates for securitized bonds. These changes implied higher uncertainty about bank solvency and lower values for repo collateral. They conclude that the market slowly became aware of the risks associated with the subprime market, which then led to doubts about repo collateral and bank solvency. At some point—August 2007—a critical mass of such fears led to the first run on repo, with lenders no longer willing to provide short-term finance at historical spreads and haircuts.

Afonso, Cover, and Schoar examined the connections between solvency and liquidity over the global crisis. They test two hypothesis by which shocks to individual banks can lead to market wide reductions in liquidity: (i) an increase in counterparty risk leading to a drying up in liquidity; and (ii) liquidity hoarding, i.e. banks not willing to lend even to high quality counterparties in order to keep liquidity for precautionary reasons. Their findings suggest that concerns about counterparty risks played a much larger role than liquidity hoarding. Moreover, in the days after Lehman’s bankruptcy, loan amounts and spreads became more sensitive to borrower’s characteristics: they observe that large borrowers accessed the fed funds market less after Lehman’s bankruptcy and from fewer counterparties. Furthermore, it was the worst performing large banks (the “bad” banks) that accessed the market least. They do not observe the complete cessation of lending predicted by some theoretical models that focus on liquidity hoarding.

The October 2008 GFSR showed that systemic (joint) default risk has been the dominant factor in the explanation of interest rate spreads and that systemic (joint) default risk has influenced the spreads since July 2007. It also showed that the repo spread began to presents signs of stress in 2005 when the U.S. housing market began its downturn. It then concluded that broadening access to emergency liquidity alone would not resolve bank funding stresses until broader policy measures, including those aimed at the underlying counterparty credit concerns, were implemented.

This paper also highlights the importance of relating the policy response to the diagnosis of the shock. Systemic liquidity shocks may all look similar, despite their origin (aggregate liquidity preference shock infrastructure malfunctioning or solvency concerns). However, the different origin is important to inform the policy response, both preemptively, to minimize the probability of a systemic liquidity crisis, and to manage the crisis, once it happens.5 For our application to the U.S. banks, we develop a capital surcharge aimed at minimizing the probability that any given bank would experience a destabilizing run. For crisis management, we propose recapitalizing or closing insolvent banks and disclosing enough information to eliminate uncertainties about bank solvency. Liquidity injections by a central bank—that can solve the problem in the case of a change in intertemporal preferences for consumption— would likely not be effective if there is reluctance to provide funding for suddenly poor quality assets.6 Balance between supply and demand of liquidity would only be achieved by deleveraging, restoring asset quality and confidence, and by providing enough information to avoid contagion problems for solvent institutions.

III. Modeling Steps and Data Requirements

Our approach starts with a detailed solvency stress test for multiple banks and then adds, as an innovation, a systemic liquidity component. It can be used to measure correlated systemic solvency and liquidity risk, assess a bank’s vulnerability to a liquidity shortfall, and develop a capital surcharge aimed at minimizing the probability that any given bank would experience a destabilizing run.

The ST framework assumes that systemic liquidity stress is caused by rising solvency concerns and uncertainty about asset values. The ST approach models three channels for a systemic liquidity event:

  • a stressed macro and financial environment leading to a reduction in funding from the unsecured funding markets due to a heightened perception of counterparty and default risk;

  • a fire sale of assets as stressed banks seek to meet their cash flow obligations. Lower asset prices affect asset valuations and margin requirements for all banks in the system, and these in turn affect funding costs, profitability, and generate systemic solvency concerns; and

  • lower funding liquidity because increased uncertainty over counterparty risk and lower asset valuations induce banks and investors to hoard liquidity, leading to systemic liquidity shortfalls.

The approach proceeds in four stages as illustrated in Figure 1: (i) modeling the financial and economic environment; (ii) modeling correlated borrower credit risk; (iii) modeling systemic banking system solvency risk; and (iv) modeling correlated systemic liquidity risk. First, thousands of Monte-Carlo simulations are used to simulate correlated changes in many asset prices (foreign exchange rates, interest rates, real estate prices, and market equity indexes), as well as macroeconomic factors that drive bank clients’ defaults (equity values, bank clients’ leverage) between the current time (T0) and a future time (T1). These simulated prices and macroeconomic factors are used to revalue banks’ balance sheets.

Figure 1.
Figure 1.

Modeling Steps

Citation: IMF Working Papers 2011, 263; 10.5089/9781463924614.001.A001

Source: GFSR April 2011

A large shock to these prices and macroeconomic factors affect the quality of bank assets directly (higher credit and market risk) and also indirectly through a network of interbank claims. Our model estimates the correlated value of banks’ economic capital to asset ratios, the number of bank solvency defaults at T1, and the probability of future solvency defaults at T2 (as measured at T1).

In simulations with higher bank probabilities of default, bank creditors react by showing reluctance to fund bank assets. Confronted with increasing difficulties to roll over their liabilities, banks need to fire sell assets at distressed prices. This in turn aggravates their economic capital to asset ratios. At the end of the simulation, the model generates a final distribution of economic capital to asset values as well as a distribution of cash flows. Banks are modeled as failing when their capital-to-asset ratios reach a critical threshold value (2 percent) or in the presence of liquidity shortages. Periods with multiple bank failures are likely to also have multiple banks in weakened financial conditions. This is just the time when losses on interbank credit defaults can lead to correlated banking system solvency and liquidity crises.

Data Requirements

The ST approach has the following intensive data requirements. In some cases, it may be possible to substitute expert opinion for data that may not be available.

  • Time series related to the financial and economic environment in which banks operate. These series need to be of sufficient length to allow trends, volatilities, and correlations to be estimated during both “normal” and “stress” periods. The following data are of interest:

    • short-term domestic and foreign interest rates and their term structures;

    • interest rate spreads for loans of various credit qualities (securities);

    • foreign exchange rates (as relevant);

    • economic indicators (Gross Domestic Product (GDP), consumer price index; unemployment, and so on);

    • commodity prices (oil, gold, and so on);

    • sector equity indices, and

    • regional real estate prices.

  • Information on banks’ assets, liabilities, and, ideally, off-balance-sheet transactions, including hedges, such as:

    • various categories of loans, including information about their credit quality, maturity structure, and currencies of denomination;

    • currency and maturity structure of the other assets and liabilities;

    • capital as well as operating expenses and tax rates;

    • clients’ leverage ratios and recovery rates, to be able to calibrate credit risk models, and

    • interbank exposures, including bilateral credit exposures among the various banks.

  • Information to enable calibration of behavioral relationships, such as:

    • between banks’ default probabilities and a reduction in funding due to bank creditors’ concerns about solvency

    • between asset fire sales and asset values (including haircuts), which in turn affect liquidity and solvency ratios.

IV. Model Calibration to the U.S. Financial Environment and the U.S. Banking System

A. The U.S. Financial and Economic Environment

For the characterization of the U.S. financial and economic environment we utilize a set of variables including interest rates, interest rate spreads, foreign exchange rates, U.S. economic indicators, global equity indices, 14 S&P sector equity returns, and 20 Case-Shiller regional real estate price returns. The Financial and Economic Environment Model is calibrated for two different “regimes.” The first calibration is based on monthly data from the twenty year period 1987 to 2006. The second calibration is based on data from the period 2007 to 2010.

Table 2 gives data on the trends and volatilities of a number of the financial and economic variables used in the risk analysis. Clearly 2007–2010 was a much more adverse period than the previous 20 years. For example average sector equity returns fell from approximately 13 percent to approximately 2 percent per year. Average sector equity return volatility increased from approximately 18 percent per year to 24 percent per year. Average regional real estate price changes fell from approximately 6 percent to -9 percent. It is also important to note the variation in real estate prices by region (e.g., Nevada and Florida suffered much larger real estate price declines). Average regional real estate index volatility increase from approximately 2 percent to 5 percent. The financial environment calibration also included estimations of Hull and White term structure models for domestic and foreign interest rates, volatilities for interest rate spreads, and correlations among the various risk variables.

Table 2.

U.S. Financial and Economic Calibrations (1987–2006 and 2007–2010)

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We view the 2007–2011 banking crises in the United States as having substantial similarities to the one that occurred in Japan during the 1990s. In both cases, the bursting of an asset price bubble resulted in large correlated defaults and losses on bank loans and the failure of many institutions. Table 3 gives a comparison of the percentage changes in real estate prices by state7 (Percent _Change_Real_Estate_Prices) and the percentage failure rates of U.S. banks by state8 (Percent_Bank_Failure_Rate) for the period 2007–2011. This data demonstrates that real estate price changes and bank failure rates vary greatly by state and appear to be highly correlated, particularly for large declines in real estate prices (e.g., declines greater than 20 percent). A regression of real estate price changes on bank failure rates finds:

Percent_Bank_Failure_Rate = -.021 – 0.387 Percent_Change_Real_Esate_Prices

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Table 3.

Percentage Bank Failure Rates and Percentage Changes in Real Estate Prices by State 2007–2011

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Sources: FDIC, Freddie Mac.

This regression supports the proposition that regional real estate prices are an important and statistically significant factor explaining bank failures. We explicitly model regional real estate price changes and how such changes impact banks with more or less concentrated real estate loan portfolios.9

Table 4 provides information on the asset size distribution of 342 U.S. bank failures in the January 1, 2007 to February 25, 2011 period as reported by the FDIC.10 Some 280 of these failed banks had assets of less than $1 billion; an additional 54 banks had assets of between $1 and $10 billion. However, larger institutions also failed, including six with assets in the $10 to $25 billion range. Also failing were IndyMac ($32 billion), Washington Mutual ($307 billion), Lehman Brothers ($639 billion), Wachovia ($780 billion), Freddie Mac ($850 billion of assets, plus approximately $4 trillion of guarantees), and Fannie Mae ($912 billion of assets, plus approximately $6 trillion in guarantees). A large majority of the failed banks were smaller in size and we believe regionally oriented with large concentrated positions in real estate loans. However, a number of medium sized and larger institutions having concentrated exposure to real estate price risk also failed in this same period.

Table 4.

Distribution of Asset Sizes for U.S. bank failures January 2007 to February 2011

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B. Modeling Banks’ Assets, Liabilities and Income

Using Call Report Data,11 we constructed 10 stylized U.S. banks in four categories as shown in Tables 58: Two mega banks that aggregate the asset and liabilities of two groups of banks, with higher and lower equity capital to asset ratios, and assets above $500 billion; three large banks that aggregate the assets and liabilities of groups of banks with assets between $100–500 billion; three medium size banks that aggregate groups of banks with assets between $10–100 billion and two small banks that aggregate groups of banks with assets below 10 billion. The various banks are sized so that they have an appropriate weighting relative to the overall U.S. banking system. For example, the mega banks have approximately 62 percent of the total assets for the model banking system. A larger or smaller number of banks could be modeled.

Table 5.

Small Banks Balance Sheet (percent assets).

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Source: SNL Financial, staff estimates.
Table 6.

Medium Banks Balance Sheet (percent assets).

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Source: SNL Financial, Staff estimates.
Table 7.

Large Banks Balance Sheet (percent assets)

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Source: SNL Financial, Staff estimates
Table 8.

Mega Banks Balance Sheet (percent assets).

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Source: SNL Financial, Staff estimates.

Sector and regional concentrations of bank loan portfolios are also a significant risk factor. The smaller banks are modeled as making mortgage loans in one or two states (e.g. California, or Florida and Georgia) and three sectors of the economy (industrial, retail, and services). Medium sized banks are modeled as lending in larger regions (West coast, Mid-America, or East Coast) and four sectors of the economy. Large and mega banks are modeled as lending nationally in 20 regions and 14 sectors of the economy.

Solvency risk in our model depends on bank exposure to borrower creditworthiness (credit risk), including credit concentration, as well as correlated market prices (market risk). The risk assessment horizon was set at one year. The model is flexible to accommodate other risk modeling time steps.

Credit Risk Modeling

Business and mortgage loan credit risk assessments are based on simulations of business debt to value ratios and property loan to value ratios using a contingent claims type model.12 The future values of companies are systematically related to simulated sector equity returns plus a company specific random return. The credit rating of the loans are assumed to change when business debt to value ratios cross-critical boundaries. At identified high debt to value ratios the loans are assumed to default.13 In this study we use the U.S. business credit risk model estimated by Barnhill and Maxwell (2002). Correlated variations in recovery rates on business loans are also an important systematic risk factor. In our analysis recovery rates on business loans are modeled as increasing (decreasing) as stock market returns increase (decrease).14

Given that we did not have information on the credit quality of corporate borrowers for each bank, we assumed that initially, the set of business loans in U.S. bank portfolios have the same credit quality distribution for all banks and this distribution is the one described in the Shared National Credits Review issued annually by the Board of Governors of the Federal Reserve System, Federal Deposit Insurance Corporation, Office of the Comptroller of the Currency and Office of Thrift Supervision (see Table 9).15 Foreign corporate loans are modeled following the same credit risk analysis procedures used for domestic loans. However we do account for foreign exchange rate risk. For those assets and liabilities where credit risk is not modeled, valuation is based on a present value approach where the cash flows are discounted using the simulated interest rates of a selected term structure and the simulated values for the correlated exchange rates, in the case of securities denominated in foreign currency.

Table 9.

Credit Quality of Committed and Outstanding Commercial and Industrial Loans (In Billions of Dollars per Year)

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Note: Sources Shared National Credit Report 2010. Figures may not add to totals due to rounding.

Due to lack of an alternative model, loans to individuals were modeled entirely as a portfolio of mortgage loans. This approach has obvious limitations but does capture any correlations among the default rates on other loans to individual and mortgage loans resulting from unemployment rates, and low property prices, among others. The initial loan to value ratios for mortgage loans were estimated from data given in Fannie Mae’s and Freddie Mac’s annual report plus estimates of the likely distributions of mortgage portfolio loan to value (LTV) ratios based on assumed initial LTV’s and trends in national real estate prices (see Table 10).

Table 10.

Assumed Distribution of Initial Mortgage Loan to Value Ratios

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In our model the future values of properties are systematically related to regional real estate returns plus a property specific random term. The probability of a mortgage loan defaulting is modeled as being related to its LTV ratio.16 For LTV’s between 1.2 and 1.4, the default rate is set at 20 percent. For LTV’s between 1.4 and 1.6, the default rate is set at 40 percent. For LTV’s between 1.6 and 1.8, the default rate is set at 60 percent. For LTV’s over 1.8 the default rate is set at 80 percent. Recovery rates on mortgage loans are correlated with real estate prices and are assumed to be the value to loan ratio less a 30 percent liquidation cost.

Correlated changes in the values of real estate assets by region and business’s by sector are driven by the correlated returns on regional real estate indices and sector equity indices in the financial and economic environment. Correlated default rates on mortgage loans in various regions and business loans in various sectors are driven by the assumed initial loan to value ratios and correlated changes in the values of the real estate and business assets securing the bank loans. Such correlated defaults on bank loan portfolios produce correlated banking system systemic solvency and liquidity risks.

Loan Portfolio Concentration Modeling

The concentration of bank loans in various sectors (e.g. energy), regions (e.g. Florida), and security types (e.g. mortgage loans) are particularly significant bank risk factors that are often not modeled adequately. To account for loan portfolio concentration risk we model the correlated market and credit risk on 200 business loans distributed across up to 20 sectors of an economy and 200 mortgage loans distributed across up to 20 regions of a country. We find this to be an adequate number of loans, sectors, and regions to statistically distinguish between more concentrated and more diversified portfolios. More sectors, regions, and loans could be modeled. We also model correlated market risk for approximately 100 other bank assets and liabilities.

Systemic Solvency Risk Modeling

One of the outcomes of the risk assessments of the financial and economic environment and bank portfolios after many simulation runs are joint distributions of each of the 10 banks’ market value of equity capital at T1.

MVEt=i=1nAi,ti=1nLi,t,

where MVEt is the simulated market value of the bank’s equity at time t, Ai,t is the simulated market value of the i’th asset at time t which reflects the simulated financial environment variables (e.g., interest rates, exchange rates, equity prices, real estate prices, and etc.) and where appropriate, the simulated credit rating of the borrower, Li,t is the simulated market value of the i’th liability at time t which reflects the simulated financial environment variables (e.g., interest rates, exchange rates, etc.). The bank’s asset and liability levels are also adjusted to reflect bank net interest income, fee income plus other income less operating expenses, and taxes over the simulation period.17

After many simulation runs joint distributions of the various banks’ capital to asset ratios are estimated and used to assess bank defaults and systemic banking system solvency risks at T1.

Capital_Ratiot=MVEt/i=1nAi,t

For each run of the simulation the simulated capital ratio is also used to estimate each bank’s correlated probability of defaulting at T2. These future default probabilities are derived under the assumption that the distribution of changes in capital ratios between T1 and T2 is identical to the distribution of changes in capital ratios between T0 and T1.

During times of economic stress, it is likely that default losses on loans will increase, and many banks will either fail or be weakened significantly, particularly if they have similar asset and liability structures. This is just the time when the failure of several banks could, through interbank credit defaults, precipitate a number of simultaneous bank failures. Interbank credit risk is modeled using a network methodology. Since we do not have precise information on inter-bank borrowers/lenders identities, we assumed that the amount of interbank loans made between each bank is proportional to their total inter-bank borrowing and lending.

In the current study, and consistent with current U.S. regulations, we model a bank as failing when its ratio of equity capital to assets falls below 2 percent.18 In this case, the bank becomes incapable of honoring its interbank obligations and defaults on them. The recovery rate on defaulted interbank obligations is assumed to be 40 percent. Such losses could affect counterparty banks’ capital ratios and potentially lead to additional bank failures. A network methodology is applied repeatedly until no additional banks fail, after which the probability of multiple simultaneous bank failures (that is, systemic solvency risk) can be computed. The outcome of this step is again equity to asset ratios and bank failures for T1—that includes losses due to defaults on interbank claims—and a probability of default for each bank at T2 (estimated as discussed in the previous paragraph) for each run of the simulation.

C. Modeling Correlated Systemic Liquidity Risk

A primary contribution of the model to stress testing is the addition of correlated liquidity runs on banks, driven by heightened risks, or uncertainties, regarding future bank solvency.19 Changes in bank liabilities observed over the period from 2007 to the first quarter of 2010 were used to develop an estimated relationship between a bank’s probability of default and the rate of withdrawal of total liabilities over the period T1 to T2.

Because of incomplete information on particular banks, we assume that bank creditors are also aware of and react to developments in the overall banking system. We thus model system wide weighted average banking system default probabilities and assume that they have some impact on liquidity runs. In particular liquidity runs for a particular bank are modeled as being driven by the probability of failure for that bank at T2 plus a factor equal to ten percent of the system wide weighted average default probability (i.e. the adjusted probability of failure).

Bank liquidity outflows are estimated under two cases. In case 1, total liability withdrawal rates match those experienced by bank holding companies (BHC) with elevated default probabilities during the 2007–2010 period.20,21 In case 2, at the highest default probabilities withdrawal rates match those experienced by investment banks; since investment banks have a very low level of insured deposits, this case provides a way to calibrate a more stressed scenario where funding sources may dry up very quickly. In case 2 for lower default probabilities we modeled potential reductions in specific liability accounts (e.g., demand deposits, time deposits, jumbo time deposits, Fed Funds, and repos, among others), which reflect actual liability structures of the stylized BHC’s. Table 11 summarizes assumptions on total liability withdrawal rates associated with different default probability ranges for each case.

Table 11.

Withdrawal Rate Assumptions for Decline in Total Liabilities (percent)

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Sources: SNL Financial; and author estimates.

When multiple banks fail, it is highly likely that the risk of future insolvency for the remaining banks is elevated. At the end of each run of the simulation (for example, at T1), future (T2) solvency risks for each bank are computed as previously discussed. These estimated T2 probabilities of default drive assumed bank liquidity flows as shown in Table 11.

Banks that face a liquidity run are assumed to follow one of two strategies. In the first strategy banks stop lending in the interbank and repo markets, liquidate interest bearing bank deposits, sell government securities, and sell other securities. If these steps do not produce adequate liquidity, they ultimately default on their obligations.22 In the second strategy banks sell their liquid securities and reduce their loan portfolios in proportions similar to that observed in U.S. bank holding companies having elevated failure probabilities.23

Banks pay a high cost when they are forced to sell assets during periods of extreme financial market stress. We model bank losses resulting from the fire sale of assets. This cost is given by a selling price with an embedded high liquidity premium and consequently well below its fundamental price. Developments in bid-ask spreads in several securities markets during the 2000–09 period24 were used as a proxy for fire sale prices. At the peak of the crisis (September 2008), the size of the bid-ask spread was in the 5-10 percent range across different asset qualities, suggesting a discount factor of 3–5 percent to represent the loss suffered by the bank under distress when forced to liquidate assets (see Figure 2).

Figure 2.
Figure 2.

Average Corporate Bond Bid-Ask Spread (in basis points)

Citation: IMF Working Papers 2011, 263; 10.5089/9781463924614.001.A001

Source: Bloomberg, staff estimates.

These values are in line with Coval and Stafford (2007), Aikman and others (2009), and Duffie and others (2006).

  • Coval and Stafford (2007) examine fire sales in equity markets using market prices of mutual fund transactions caused by capital flows from 1980 to 2003. They find significantly negative abnormal returns in stock prices around widespread forced sales. In a situation where around at least 15 percent of the owners are distressed sellers of the same stock, average abnormal stock return is -10.1 percent for the first quarter, and less than 2 percent for months 4–12.

  • Aikman et al (2008), following Duffie et al (2006) conclude that the relation between prices and the magnitude of fire sales is concave and specifically, for asset j is equivalent to the following expression
    Pji=max{0,Pj(2exp(θSijMj+εj))}

    The price of asset j following the fire sale, Pji, is the maximum of zero and the price before the fire sale, Pj, multiplied by a discount term. The discount term is a function of value of assets sold by bank i in the fire sale, Sij, divided by the depth of the market in normal times Mj, and scaled by a parameter θ that reflects frictions, such as search problems, that cause markets to be less than perfectly liquid. Market depth can also be shocked by a term εj to capture fluctuations in the depth of markets as macroeconomic conditions vary. For the U.K. banks, they calibrate this relation for the case in which the U.K. bank with the largest holdings of an asset class in its trading portfolio and AFS assets sells all these assets, it generate price falls of 2 percent for equities, 4 percent for corporate debt and 5 percent for mortgage-back securities.

Both liquidity failures of counterparty banks and the fire sale of assets may produce further losses for banks that adversely affect their solvency. Again, these can be modeled with a network methodology applied repeatedly until no additional banks fail. In this way the probability of multiple simultaneous bank failures (that is, correlated systemic solvency and liquidity risk) can be assessed.

The stress test assesses whether banks faced with these withdrawal rates can deleverage in an orderly manner. Initially banks that suffer a run are assumed to stop lending in the interbank market and sell government securities and other liquid assets. Banks may pay a high cost if they are forced to sell potentially less liquid assets, in particular if those assets are associated with a high liquidity premium. In this way, the model captures the interaction between funding and market liquidity and the second round feedback between solvency and liquidity risks. It is important to notice that even if banks can deleverage in an orderly manner, this does not eliminate systemic liquidity risks: banks may avoid liquidity failures and potential defaults, but at the expense of lower credit provided to the economy.

V. Results

U.S. Systemic Solvency Risk- with no Inter-bank Defaults

Table 12 below shows the distributional analysis for the simulated equity capital ratios for the stylized 10 bank system using the 1987–2006 financial environment calibration and assuming a one year risk assessment time step. In this analysis potential inter-bank defaults are not modeled. We assume that simulated capital ratios below 0.02 (2 percent) result in bank failure. We also show information on the distribution of simultaneous bank failures. We find only a small risk of bank failures focused on thinly capitalized and regionally concentrated smaller banks. We find no likelihood of systemic solvency or systemic liquidity risks. These results are generally consistent with U.S. bank risk in the period prior to 2007.

Table 12.

Simulated Capital Ratios for Banks using the 1987–2006 Financial Environment with No Inter-Bank Default Losses

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Source: ValueCalc Estimates.

For the 2007–2010 calibration of the financial and economic environment model, and a one year time step, we find substantially elevated solvency risks for all banks and the banking system. Figure 3 shows the emergence of fatter tails as the mass of the distribution of equity capital ratios (and estimated default probabilities) shifted in a negative direction. Table 13 shows that some of the small more regionally concentrated banks have high failure probabilities.25 Eight of the ten banks, including the two mega banks, have a risk of failure in the 0.5 percent range. There is a 1 percent joint probability of four banks failing simultaneously.

Figure 3.
Figure 3.

Capital Ratios, 1987–2006 and 2007–2010; Before Interbank Failures

Citation: IMF Working Papers 2011, 263; 10.5089/9781463924614.001.A001

Source: ValueCalc estimates.
Table 13.

Simulated Capital Ratios for Banks using the 2007–2010 Financial Environment Calibration with No Inter-Bank Default Losses (percent)

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Source: ValueCalc Estimates.

The only difference between the analyses presented in Tables 12 and 13 are changes in the trends, volatilities, and correlations of the financial environment variables (e.g., sector equity returns, regional real estate prices, and credit spreads, among others) for the two model calibrations. Thus within our model financial and economic regime shifts to more adverse conditions (as occurred during the 2007–2010 period) is clearly a significant systemic risk factor.

U.S. Systemic Solvency Risk with Inter-bank Defaults

Adverse financial and economic regime shifts can be expected to cause an increase in correlated defaults on loan portfolios for all banks resulting in the correlated failures of some banks and the weakening of others. When analyzing correlated inter-bank default risk we apply a network methodology repeatedly until no additional banks fail, after which the probability of multiple simultaneous bank failures (i.e., systemic solvency risk) can be computed.

Table 14 presents the distributional analysis for the simulated equity capital ratios for the stylized 10 bank system using the 2007–2010 financial environment calibration including potential inter-bank defaults. When potential inter-bank default losses are modeled there is a 1 percent joint probability of six banks failing simultaneously. Thus in our model inter-bank default losses have the potential to significantly increase the number of correlated bank failures (e.g., from four to six at a 1 percent probability).

Table 14.

Simulated Capital Ratios for Banks using the 2007–2010 Financial Environment Calibration with First and Second Round of Inter-bank Default Losses

(percent)

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Source: ValueCalc Estimates.

Table 15 shows an analysis of the correlations among the initial defaults for individual banks and incremental defaults resulting from losses in the inter-bank credit market. As shown in the last row of Table 15 failures by the mega banks have the highest correlations with subsequent bank failures. This occurs in our simulations due to large inter-bank credit losses imposed by the mega banks. Such mega banks can thus be considered to be systemically important.

Table 15.

Correlations among Incremental Bank Failures Due to Inter-Bank Default losses and Initial Bank Failures.

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Source: ValueCalc Estimates

U.S. Correlated Systemic Solvency and Liquidity Risks

Our goal is to ultimately model correlated systemic solvency and liquidity risks. As discussed earlier on each run of the simulation we estimate whether each of the ten banks will fail at T1. We also estimate the probability of each of the remaining solvent banks failing at T2. We then model correlated liquidity runs as a response to elevated probabilities of bank failure at T2. By design the model is thus structured to estimate correlated solvency and liquidity risks. For example a correlation analysis of simulation results finds a 0.55 correlation between the simulated weighted average probabilities of solvent banks failing at T2 versus the simulated percentage of banking system assets held by banks failing at T1.26

Table 16 below gives a distributional analysis on the banks’ probabilities of failure at T2 as measured at T1. To these probabilities of default we add a factor equal to 10 percent of the banking system’s weighted average probability of default to account for system wide stress impacts. These adjusted default probabilities are used to estimate potential bank liquidity runs as discussed previously. This methodology is simply illustrative of one possible method of accounting for the impact of system wide stress levels on potential bank runs. Clearly more research on how to measure and model these relationships would be useful.

Table 16.

Distributional Analysis of Bank Probabilities of Default at T2 as Measured at T1

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Source: ValueCalc Estimates