Measuring Systemic Risk-Adjusted Liquidity (SRL)
A Model Approach
Author: Andreas Jobst

Contributor Notes

Authors Email Address: ajobst@imf.org

Little progress has been made so far in addressing—in a comprehensive way—the externalities caused by impact of the interconnectedness within institutions and markets on funding and market liquidity risk within financial systems. The Systemic Risk-adjusted Liquidity (SRL) model combines option pricing with market information and balance sheet data to generate a probabilistic measure of the frequency and severity of multiple entities experiencing a joint liquidity event. It links a firm’s maturity mismatch between assets and liabilities impacting the stability of its funding with those characteristics of other firms, subject to individual changes in risk profiles and common changes in market conditions. This approach can then be used (i) to quantify an individual institution’s time-varying contribution to system-wide liquidity shortfalls and (ii) to price liquidity risk within a macroprudential framework that, if used to motivate a capital charge or insurance premia, provides incentives for liquidity managers to internalize the systemic risk of their decisions. The model can also accommodate a stress testing approach for institution-specific and/or general funding shocks that generate estimates of systemic liquidity risk (and associated charges) under adverse scenarios.

Abstract

Little progress has been made so far in addressing—in a comprehensive way—the externalities caused by impact of the interconnectedness within institutions and markets on funding and market liquidity risk within financial systems. The Systemic Risk-adjusted Liquidity (SRL) model combines option pricing with market information and balance sheet data to generate a probabilistic measure of the frequency and severity of multiple entities experiencing a joint liquidity event. It links a firm’s maturity mismatch between assets and liabilities impacting the stability of its funding with those characteristics of other firms, subject to individual changes in risk profiles and common changes in market conditions. This approach can then be used (i) to quantify an individual institution’s time-varying contribution to system-wide liquidity shortfalls and (ii) to price liquidity risk within a macroprudential framework that, if used to motivate a capital charge or insurance premia, provides incentives for liquidity managers to internalize the systemic risk of their decisions. The model can also accommodate a stress testing approach for institution-specific and/or general funding shocks that generate estimates of systemic liquidity risk (and associated charges) under adverse scenarios.

I. Introduction

A defining characteristic of the recent financial crisis was the simultaneous and widespread dislocation in funding markets, which can adversely affect financial stability in absence of suitable liquidity risk management and policy responses. In particular, banks’ common asset exposures and their increased reliance on short-term wholesale funding in tandem with high leverage levels helped propagate rising counterparty risk due to greater interdependence within the financial system. The implications from liquidity risk management decisions made by some institutions spilled over to other markets and other institutions, contributing to others’ losses, amplifying solvency concerns, and exacerbating overall liquidity stress as a result of these negative dynamics. Thus, private sector liquidity (as opposed to monetary liquidity), which is created largely through banks and other financial institutions via bilateral arrangements and organized trading venues, is invariably influenced by common channels of market pricing that can amplify cyclical movements in system-wide financial conditions with the potential of negative externalities resulting from individual actions (CGFS, 2011).

The opportunity cost of holding liquidity is invariably cyclical, resulting in a notorious underpricing of liquidity risk, which tends to perpetuate a disregard for the potential inability of markets to sustain sufficient liquidity transformation under stress. Banks have an incentive to minimize liquidity (and mitigate the opportunity cost of holding excess liquidity in lieu of return-generating assets) in anticipation that central banks will almost certainly intervene in times of stress as lenders-of-last-resort. Even without central bank support, liquidity risk is most expensive when it is needed most while generating little if any additional return in good times. While central banks can halt a deterioration of funding conditions in order to maintain the efficient operation of funding markets (see Figure 1), prevent financial firms from failing, and, thus, limit the impact of liquidity shortfalls on the real economy, their implicit subsidization of bank funding accentuates the magnitude of liquidity risks under stress. Central bank measures during the credit crisis have further reinforced this perception of contingent liquidity support, giving financial institutions an incentive to hold less liquidity than needed (IMF, 2010a).

Figure 1.
Figure 1.

Conceptualization of Liquidity Risk.

Citation: IMF Working Papers 2012, 209; 10.5089/9781475505597.001.A001

Current systemic risk analysis—as a fundamental pillar of macroprudential surveillance and policy—is mostly focused on solvency conditions. Disruptions to the flow of financial services become systemic if there is the potential of financial instability to trigger serious negative spillovers to the real economy.2 Macroprudential policy aims to limit, mitigate or reduce systemic risk, thereby minimizing the incidence and impact of disruptions in the provision of key financial services that can have adverse consequences for the real economy (and broader implications for economic growth).3 Substantial work is underway to develop enhanced analytical tools that can help to identify and measure systemic risk in a forward-looking way, and, thus, support improved policy judgments. While systemic solvency risk has already entered the prudential debate in the form of additional capital rules that apply to systemically important financial institutions (SIFIs),4 little progress has been made so far in addressing systemic liquidity risk.

In contrast, proposals aimed at measuring and regulating systemic liquidity risk caused by the interconnectedness across financial institutions and financial markets have been few and far between. Systemic liquidity risk is associated with the possibility that maturity transformation in systemically important institutions and markets is disrupted by common shocks that overwhelm the capacity to fulfill all planned payment obligations as and when they come due. For instance, multiple institutions may face simultaneous difficulties in rolling over their short-term debts or in obtaining new short-term funding (much less long-term funding). However, progress in developing a systemic liquidity risk framework have been hampered by the rarity of system-wide liquidity risk events, the multiplicity of interactions between institutions and funding markets, and the conceptual challenges in modeling liquidity conditions affecting institutions and transactions separately or jointly.5

The policy objective of such efforts would be to minimize the possibility of systemic risk from liquidity disruptions that necessitate costly public sector support. While a financial institution’s failure can cause an impairment of all or parts of the financial system, firms are not charged for the possibility that their risk-taking affects the operation of the financial system as a whole. In fact, individual actions might cause losses elsewhere in the system through direct credit exposures and financial guarantees, forced asset sales, and greater uncertainty regarding mutual exposures (possibly in combination with greater risk aversion of investors), which increases the cost of funding for all financial institutions. These “negative externalities” impose costs to the system, which increases the greater the importance of a single institution to the system (“too-important-to-fail”) and the higher the level of asymmetric information as coordination failures accentuate the impact of common shocks.6 Thus, more stringent prudential liquidity requirements, much like higher capital levels, might be beneficial ex ante by creating incentives of shareholders to limit excessive risk-taking, which would otherwise increase the potential loss in case of failure (Jensen and Meckling, 1976; Holmstrom and Tirole, 1997). However, certain liquidity standards might also encourage greater concentrations in assets that receive a more favorable regulatory treatment based on their liquidity characteristics during normal times (which remains to be tested during times of stress).

A number of prudential reforms and initiatives are underway to address shortcomings in financial institutions’ liquidity practices, which have resulted in more stringent supervisory liquidity requirements. Under the post-crisis revisions of the existing Basel Accord, known as Basel III, the Basel Committee on Banking Supervision (BCBS, 2010a, 2010b and 2009) has proposed two quantitative liquidity standards to be applied at a global level and published a qualitative guidance to strengthen liquidity risk management practices in banks. Under this proposal, individual banks are expected to maintain a stable funding structure, reduce maturity transformation, and hold a sufficient stock of assets that should be available to meet its funding needs in times of stress—as measured by two standardized ratios:

  • Liquidity Coverage Ratio (LCR). This ratio is intended to promote short-term resilience to potential liquidity disruptions by requiring banks to hold sufficient high-quality liquid assets to withstand the run-off of liabilities over a stressed 30-day scenario specified by supervisors. More specifically, “the LCR numerator consists of a stock of unencumbered, high-quality liquid assets that must be available to cover any net [cash] outflow, while the denominator is comprised of cash outflows less cash inflows (subject to a cap at 75 [percent] of total outflows) that are expected to occur in a severe stress scenario (BCBS, 2011 and 2012b).”7

  • Net Stable Funding Ratio (NSFR). This structural ratio limits the stock of unstable funding by encouraging longer term borrowing in order to restrict liquidity mismatches from excessive maturity transformation. It requires banks to establish a stable funding profile over the short term (i.e., the use of stable (long-term and/or stress-resilient) sources to continuously fund short-term cash flow obligations that arise from lending and investment activities). The NSFR reflects the proportion of long-term assets that are funded by stable sources of funding with maturities of more than one year (except deposits), which includes customer deposits, long-term wholesale funding, and equity (but excludes short-term funding).8 A value of this ratio of less than 100 percent indicates a shortfall in stable funding based on “the difference between balance sheet positions after the application of available stable funding factors and the application of required stable funding factors for banks where the former is less than the latter (BCBS, 2011 and 2012b).”9

However, these prudential measures do not directly targeting system-wide implications. The current approach assumes that sufficient institutional liquidity would reduce the likelihood of knock-on effects on solvency conditions in distress situations and complement the risk absorption role of capital—but without considering system-wide effects.10 Larger liquidity buffers at each bank should lower the risk that multiple institutions will simultaneously face liquidity shortfalls, which would ensure that central banks are asked to perform only as lenders of last resort—and not as lenders of first resort. However, this rationale underpinning the Basel liquidity standards ignores the impact of the interconnectedness of various institutions and their diverse funding structures across a host of financial markets and jurisdictions on the probability of such simultaneous shortfalls.11Moreover, in light of the protracted adoption of both the LCR and the NSFR (whose implementation is envisaged in 2015 and 2018, respectively) and the associated risk of undermining timely adjustment of industry standards, Perotti (2012) argues for strong transitional tools in the form of “prudential risk surcharges.” These would be imposed on the gap between current liquidity positions of banks and the envisaged minimum liquid standards at a level high enough to compensate for and discourage the creation of systemic risk in order to ensure early adoption of safer standards while offering sufficient flexibility of banks to chart their own path towards compliance.

An effective macroprudential approach that targets systemic liquidity risk presupposes the use of objective and meaningful measures that can be applied in a consistent and transparent fashion (and the attendant design of appropriate policy instruments). Ideally, any such methodology would need to allow for extensive back-testing and should benefit from straightforward application (and avoid complex modeling (or stress-testing)). While it should not be too data intensive to compute and implement, enough data would need to be collected to ensure the greatest possible coverage of financial intermediaries in order to accommodate different financial sector characteristics and supervisory regimes across national boundaries. In addition, the underlying measure of systemic risk should be time-varying, and, if possible, it should offset the procyclical tendencies of liquidity risk and account for changes to an institution’s risk contribution, which might not necessarily follow cyclical patterns. Finally, it would also motivate a risk-adjusted pricing scheme so that institutions that contribute to systemic liquidity risk are assigned a proportionately higher charge (while the opposite would hold true for firms that help absorb system-wide shocks from sudden increases in liquidity risk).

In this regard, several proposals are currently under discussion (see Table 1), including the internalization of public sector cost of liquidity risk via insurance schemes (Goodhart, 2009; Gorton and Metrick, 2009; Perotti and Suarez, 2009 and 2011), capital charges (Brunnermeier and Pedersen, 2009), taxation (Acharya and others, 2010a and 2010b), investment requirements (Cao and Illing, 2009; Farhi and others, 2009), as well as arrangements aimed at mitigating the system-wide effects from the fire sale liquidation of assets in via collateral haircuts (Valderrama, 2010) and modifications of resolution regimes (Roe, 2009; Acharya and Oncu, 2010). In particular, Gorton (2009) advocates a systemic liquidity risk insurance guarantee fee that explicitly recognizes the public sector cost of supporting secured funding markets if fragility were to materialize. Roe (2009) argues that the internalization of such cost would ideally be achieved by exposing the lenders to credit risk of the counterparty (and not just that of the collateral) by disallowing unrestricted access to collateral even in case of default of the counterparty. In this way, lenders would exercise greater effort in discriminating ex ante between safer and riskier borrowers.12 Such incentives could be supported by time-varying increase in liquidity requirements, which also curb credit expansion fueled by short-term and volatile wholesale funding and reduce dangerous reliance on such funding (Jácome and Nier, 2012).

Table 1.

Selected Regulatory Proposals for Managing Systemic Liquidity Risk.

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Source: IMF (2011b).

In this paper, we propose a structural approachthe systemic risk-adjusted liquidity (SRL) model—for the structural assessment and stress testing of systemic liquidity risk. Although macroprudential surveillance relies primarily on prudential regulation and supervision, calibrated and used to limit systemic risk, additional measures and instruments are needed to directly address systemic liquidity risk. This paper underscores why more needs to be done to develop macroprudential techniques to measure and mitigate such risks arising from individual or collective financial arrangements—both institutional and market-based—that could either lead directly to system-wide distress of institutions and/or significantly amplify its consequences. The SRL model complements the current Basel III liquidity framework by extending the prudential assessment of stable funding (based on the NSFR) to a system-wide approach, which can help substantiate different types of macroprudential tools, such as a capital surcharge, a fee, a tax, or an insurance premium that can be used to price contingent liquidity access.13

The SRL model quantifies how the size and interconnectedness of individual institutions (with varying degrees of leverage and maturity mismatches defining their risk profile) can create short-term liquidity risk on a system-wide level and under distress conditions.14 The model combines quantity-based indicators of institution-specific funding liquidity (conditional on maturity mismatches and leverage), while adverse shocks to various market rates are used to alter the price-based measures of monetary and funding liquidity that, in turn, form the stress scenarios for systemic liquidity risk within the model (see Table 2 and Box 2). In this way, the SRL model fosters a better understanding of institutional vulnerabilities to the liquidity cycle and related build-ups of risks based on market rates that are available at high frequencies and which lend themselves to the identification of periods of heightened systemic liquidity risk (CGFS, 2011).

Table 2.

Overview of Liquidity Indicators.

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Sources: CGFS (2011), IMF (2011b), and author.Note: The solid line around the quantity-based liquidity indicators reflects the model focus on funding liquidity; however, in the stress testing application (see Box 2), price-based measures of monetary and funding liquidity are taken into account by way of various market rates.

This approach forms the basis for a possible capital charge or an insurance premium—a pre-payment for the contingent (official) liquidity support that financial institutions eventually receive in times of joint distress—by identifying and measuring ways in which they contribute to aggregate risk over the short-term.15 Such a liquidity charge should reflect the marginal contribution of short-term funding decisions by institutions to the generation of systemic risk from the simultaneous realization of liquidity shortfalls. Proper pricing of the opportunity cost of holding insufficient liquidity—especially for very adverse funding situations—would help lower the scale of contingent liquidity support from the public sector (or collective burden sharing mechanisms). The charge needs to be risk-based, should be increasing in a common maturity mismatch of assets and liabilities, and would be applicable to all institutions with access to safety net guarantees. Since liquidity runs are present in the escalating phase of all systemic crises, our focus is on short-term wholesale liabilities, properly weighted by the bank’s maturity mismatch.

The remainder of the paper is organized as follows. After presenting some macroprudential considerations affecting the conceptualization of systemic liquidity risk in the next section, we provide an overview of the SRL model and its contribution to the systemic risk measurement and modeling literature. We then present the technical specification and its application for stress testing in Section III. Section IV illustrates the empirical case of measuring systemic liquidity risk of the largest U.S. commercial and investment banks. Section V concludes the paper.

II. Macroprudential Considerations of Systemic Liquidity

The recent financial crisis demonstrated the devastating impact of systemic liquidity events. Under normal circumstances, regulation and supervision (both capital adequacy and appropriate liquidity risk management) ensure, as far as possible, that the maturity transformation in the banking sector in conducted safely with the necessary access to central bank lending facilities and depositor protection preventing sudden run-offs of liabilities that could deplete the availability of sufficient funding under stress. However, higher perceived counterparty risk can result in funding constraints and can depress asset prices to a point where they eventually overwhelm individual liquidity buffers despite these mitigating factors. In addition, market failure derived from ill-designed mechanisms for coordinating financial intermediaries and investors in different funding markets (see Figure 1) can exacerbate these problems (Franke and Krahnen, 2008).

During the recent crisis, several aspects contributed to systemic liquidity risk that escaped microprudential oversight:

  • Banks were allowed to enter improvident off-balance sheet commitments (such as liquidity back-up facilities for special investment vehicles (SIVs)) without appropriate liquidity management, which appeared efficient and profitable as liquidity risk was not priced in (Haldane, 2010).

  • Microprudential supervision failed to recognize vulnerabilities from currency-specific maturity transformation in offshore financial centers. In particular, European banks used short-term wholesale U.S. dollar markets to fund long-term U.S. dollar-denominated assets. When funding was no longer available, this double mismatch required liquidity support via mutually agreed U.S. dollar swap agreements between the U.S. Federal Reserve and other central banks, illustrating the significant size and systemic nature of offshore dollar markets.16

  • Non-banks engaged in maturity transformation, creating a “shadow banking sector” without safeguards imposed—at least in principle—on the banking sector. A number of short-term money market investments, such as mutual funds, asset-backed commercial paper (ABCP), and repo placements, were used to fund longer-term assets while being treated by cash providers as if they were deposits (i.e., they expected to be able to redeem them at par on demand) while in fact they were not.

These shortcomings in the liquidity regulation of depository institutions increased aggregate vulnerabilities of funding arrangements to shocks from counterparty risk. One case in point is the wholesale funding market, where network externalities and the structural complexity across financial and non-financial institutions amplified these vulnerabilities. As the economic environment and asset allocation behavior changed, lenders were more likely to increase haircuts on repo financing, limit eligibility of collateral, or stop rolling over short-term funding altogether in order to offset an asset shock by means of deleveraging their balance sheets (Shin, 2009; Shleifer and Vishny, 2010).17 As such a behavior occurred collectively, it caused liquidation of assets under fire sale conditions (Coval and Stafford, 2007), which resulted in a negative confidence-induced downward liquidity spiral, increasing funding pressures as deteriorating counterparty risk eventually weighed on solvency and required official liquidity support (CGFS, 2011).

The proper identification, monitoring and mitigating of systemic liquidity risk requires a profoundly macroprudential view.18 The traditional approach to financial stability analysis concentrates analytical efforts on the identification of vulnerabilities prior to stress from individual failures, assuming that the financial system is in equilibrium and adjusts when it experiences a shock. From a more technical perspective on liquidity risk, functional discussions have tended to focus on market liquidity, which can be measured by spreads, turnover ratios, or other price impact measures, while the supervisory focus was on funding liquidity of institutions based on liquidity ratios. As opposed to this conventional approach, the potential build-up of systemic vulnerabilities warrants comprehensive monitoring of ongoing developments in areas where the impact of disruptions to financial stability is deemed most severe and wide-spread—and especially in areas of economic significance to both the financial sector and the real economy. Although the crisis has shifted attention of supervisors from market liquidity to funding risk of individual institutions (see Table 2), microprudential liquidity requirements still remain largely devoid of systemic risk considerations.19

Macroprudential regulation would discourage the underpricing of liquidity risk while limiting system-wide vulnerabilities of funding sources to common asset price shocks. Any macroprudential approach would need to encourage the fair pricing of liquidity risk in good times while acknowledging the adverse impact of market stress on funding conditions, which, in turn, can affect perceived solvency. Such conjunctural assessment of liquidity risk conditional on both time and the cross-sectional variability of funding would be motivated by the belief that systemic liquidity risk results from (i) contingent claims on other institutions and markets (via off-balance sheet activities), (ii) the cyclicality of volatility-based margin requirements and haircuts in wholesale funding markets that amplify exposures to common asset price shocks, and (iii) the negative correlation between asset returns and funding costs during times of stress.

Aligning private incentives to mitigate liquidity risk beyond individual institutions requires methodological and empirical approaches aimed at the identification and measurement of systemic risk. While there is still no consistent theory of regulating systemically important activity in the financial sector, existing approaches can be broadly distinguished based on their conceptual underpinnings regarding several core principles used to determine systemic relevance. In general, there are two general approaches: (i) a particular activity causes a firm to fail, whose importance to the system imposes marginal distress on the system (“contribution approach”),20 or (ii) a firm experiences losses from a single (or multiple) large shock(s) due a significant exposure to the commonly affected sector, country and/or currency (“concentration of activity”), which are large relative to overall losses (“participation approach”). In the case of the former, the contribution to systemic risk arises from the initial effect of direct and indirect exposures to the failing institution (e.g., defaults on liabilities and/or asset fire sales), which escalates to spillover effects to previously unrelated institutions and markets as a result of greater uncertainty or the reassessment of financial risk (i.e., changes in risk appetite and/or the market price of risk). In contrast, the participation in systemic risk occurs via the an institution’s credit and market risk exposure to other financial institutions and market risks, which result in expected losses that exceed the loss bearing capacity of bank creditors. Table 3 below shows the distinguishing features of both approaches.

Table 3.

General Systemic Risk Measurement Approaches.

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Sources: Drehmann and Tarashev (2011), FSB (2011), Weistroffer (2011), and author.Note: The policy objectives and different indicators to measure systemic risk under both contribution and participation approaches are not exclusive to each concept. Moreover, the availability of certain types of balance sheet information and/or market data underpinning the various risk indicators varies between different groups of financial institutions, which requires a certain degree of customization of the measurement approach to the distinct characteristics of a particular group of financial institutions, such as insurance companies.

The distinction of measurement approaches also reflects varying channels of risk transmission, with the contribution of banks via common exposures and asset liquidation causing most systemic liquidity risk. In general, banks are prone to contribute to systemic risk from individual failures that propagate material financial distress or activities via intra-and inter-sectoral linkages to other institutions and markets (“contribution approach”, see Table 2). Thus, material financial distress at such a bank, or the nature, scope, size, scale, concentration, or connectedness with other financial institutions, including through either their reliance on the same providers of funding and large common exposures to similar types of assets or their substitutability as providers of liquidity in critical payment and settlement systems could pose a threat to financial stability in the absence of close substitutes (FSOC, 2011). Among the main channels that facilitate the transmission of the negative effects caused by individual distress—substitutability, complexity of operations, connectedness via common exposures, and asset liquidation—especially the latter two are of greatest relevance from a technical perspective in this paper:

  • Exposures within the sector and/or financial system. Claims by creditors, counterparties, investors, or other market participants, as well as common exposures to certain asset classes, industry sectors, and markets establish relationships that can affect both the probability but also the magnitude of systemic risk if these exposures are significant enough to cause either material impairment of other significant financial institutions (by threatening their financial condition and/or competitive position) or disruptions to critical functions of the sector and/or financial system.

  • Timing of payments and asset liquidation. Liquidity risk and maturity mismatches are important criteria for assessing the potential of material financial distress to pose systemic risk. The sudden disposal of large asset positions of an institution in distress could significantly disrupt trading and/or cause significant losses for other firms with similar holdings due to increases in asset and funding liquidity risk (which could perpetuate and possibly accentuate the impact of individual failure on financial stability.

III. Methodology

A. Overview

The SRL model is based on an options pricing concept to gauge the general level of liquidity risk for a portfolio of institutions based on the current regulatory proposal aimed at limiting term structure transformation—the Net Stable Funding Ratio (NSFR). The NSFR defines individual liquidity risk as the effective maturity mismatch between short-term liabilities (“available stable funding”, or ASF), weighted by their susceptibility to market fragilities (rollover risk), and short-term assets (“required stable funding”, or RSF), weighted by their market liquidity value (funding risk), after controlling for the maturity of off-balance sheet hedging transactions, such as FX swaps.21,22 Using an “expected loss” notion to evaluate any level of liquidity shortfall to meet the regulatory minimum standard of stable funding, the model combines balance sheet information and market data (equity and equity options) in order to generate a stochastic measure of the NSFR. In this way, the probability of falling below the lower boundary of this structural ratio translates into a risk-adjusted analog of stable funding. Given the historical variation of the market data underpinning the components of the NSFR, and the extent to which this measure links institutions implicitly to common changes in market prices, it is then possible to derive a joint distribution of expected losses and determine when firms simultaneously fall below a funding threshold defined by the RSF. The joint probability function is then used to identify an individual institution’s contribution to systemic liquidity risk to price liquidity risk within a macroprudential framework that can provide incentives for liquidity managers to internalize the systemic risk of their decisions. The contribution to systemic liquidity risk depends on an institution’s funding and asset structure and its interconnectedness with other institutions, which informs the calculation of a firm-specific capital charge or insurance premium. Capital could be assigned based on own-liquidity risk plus some marginal contribution to joint liquidity risk, if larger. The sensitivity of these estimates can then be examined within a stress testing approach using shocks to asset values and the sources of funding.

The innovation of the SRL approach is its use of contingent claims analysis (CCA) to measure individual liquidity risk consistent with proposed prudential standards in order to quantify its system-wide effects. So far, the CCA methodology has been widely applied to measure and evaluate solvency risk and credit risk at financial institutions (see Appendix 1).23 In the SRL model, however, CCA is used to derive a forward-looking measure of liquidity risk that helps determine the probability of an individual institution experiencing a liquidity shortfall and the associated expected loss when the shortfall indeed occurs. For a sample of financial institutions, these individual estimates are aggregated to a joint probability of expected losses from simultaneous liquidity shortfall, which also quantifies the marginal contribution of an institution to systemic liquidity risk.24

The SRL model accounts for changes in common factors determining individual funding conditions, their implications for the market-implied linkages between financial institutions, and the resulting impact on systemic liquidity risk. Thus, it accomplishes two essential goals of risk measures in this area: (i) measure the extent to which an institution contributes to systemic liquidity risk, and (ii) use this to indirectly price the liquidity assistance that an institution would need to receive in cases of severe funding problems in order to head off further escalation towards insolvency. The systemic dimension of the model is captured by the following three properties (see Table 4):

  • drawing on the market’s evaluation of a firm’s risk profile (including the liquidity risk that the institution will be unable to offset continuous cash outflows conditional on the regulatory expectation of stable funding). That evaluation, in turn, is based on perceived riskiness of liquidity shortfall—i.e., the likelihood that the amount of available stable funding for each institution falls below the amount of required stable funding as defined by the NSFR—as implied by the institution’s equity and equity options in the context of the economic and financial environment present at the time of measurement;

  • controlling for the firm’s sources of stable funding. The sources are modeled as being sensitive to the same markets as the funding sources of every other institution but by varying degrees. Changes in common funding conditions (and its impact on the perceived risk profile of each firm) establish market-induced linkages among institutions. The proposed framework thus combines market prices and balance sheet information to inform a risk-adjusted measure of systemic liquidity risk. That measure links institutions implicitly to the markets in which they obtain equity capital and funding; and

  • quantifying the chance of simultaneous liquidity shortfalls via joint probability distributions. After obtaining a market-based measure of individual liquidity risk, the probability that two or more banks will experience liquidity shortfall simultaneously is made explicit by computing joint probability distributions (which also accounts for differences in the magnitude of an individual firm’s liquidity shortfall). Hence, the liquidity risk resulting from a particular funding configuration is assessed not only for individual institutions but for all firms within a system in order to generate estimates of systemic risk. Because the SLR model takes into account the joint dynamics between the ASF and RSF via their covariance structure (and the impact of potential stresses on it), it provides a far deeper analysis of the liquidity risk to which a firm is exposed than does looking at them separately or with only accounting data.

Table 4.

Main Features of the SRL Model.

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B. Model Specification and Estimation Steps

The SRL model follows a three-step estimation process. First, the components of the NSFR are valued at market prices in order to generate a time-varying measure of funding risk in keeping with the proposed prudential liquidity standards aimed at limiting maturity transformation. Second, the aggregate cash flow implications of changes to liquidity risk causing a bank to fail the market-based NSFR are modelled as a put option in order to estimate expected losses arising from insufficient stable funding. Finally, these individually estimated net exposures from liquidity risk are aggregated via a multivariate distribution that determines the probabilistic measure of joint liquidity shortfalls on a system-wide level.

  1. Step 1 – Calculating the market-based measure of stable funding (“market-based NSFR”)

First, the SRL model transposes the proposed Basel III liquidity standard aimed at limiting maturity transformation—the Net Stable Funding Ratio (NSFR)into a market-based measure of individual liquidity risk. This regulatory benchmark serves as the starting point for the quantification of individual liquidity risk.25 The components of the NSFR—the ASF in the numerator and the RSF in the denominator—are transposed into a time-varying measure of the NSFR at market prices, where the RSF and ASF values reflect differences between the balance sheet and actual market values of total assets and liabilities of each firm (see Figure 2).26 The actual balance sheet measures of the ASF and RSF values are re-scaled by (i) the ratio of the book value of total liabilities to the present value of total liabilities B (which can be observed) and (ii) the ratio of the book value of total assets to market-implied value of total assets A (which is obtained as a risk-neutral density from equity option prices with maturities between three and 12 months, see below), respectively.27 Doing so generates the values ARSF and BASF, which transform the NSFR ratio into a marked-based measure of an institution’s liquidity risk (“market-implied NSFR”).

Figure 2.
Figure 2.

Methodology to Compute Systemic Liquidity under the Systemic Risk-adjusted Liquidity (SRL) Model.

Citation: IMF Working Papers 2012, 209; 10.5089/9781475505597.001.A001

Since the asset value A cannot be observed, it is estimated directly from option prices. More specifically, the state-price density (SPD) of the implied total asset value underpinning ARSF is estimated from equity option prices without any assumptions on the underlying diffusion process. This avoids the calibration error of using the “two-equations-two unknowns” approach in the traditional Merton model, which contains empirical irregularities that can influence solving both implied asset value and asset volatility simultaneously.28 Using equity option prices, we can derive the risk-neutral probability distribution of the underlying asset price at the maturity date of the options. We determine the implied asset value as the expectation over the empirical SPD by adapting the Breeden and Litzenberger (1978) method (see Appendix 2), together with a semi-parametric specification of the Black-Scholes option pricing formula (Aït-Sahalia and Lo, 1998). More specifically, this approach uses the second derivative of the call pricing function (on European options) with respect to the strike price (rather than option prices as identifying conditions). Estimates are based on option contracts with identical time to maturity, assuming a continuum of strike prices. Since available strike prices are always discretely spaced on a finite range around the actual price of the underlying asset, interpolation of the call pricing function inside this range and extrapolation outside this range are performed by means nonparametric (local polynomial) regression of the implied volatility surface (Rookley, 1997).

  • 2. Step 2 – Calculating expected losses from funding liquidity risk (“risk-adjusted NSFR”)

Second, the net exposure arising from individual liquidity risk is modelled as a put option in order to derive a risk-adjusted version of the market-based NSFR (see Figure 3). The aggregate cash flow implications of changes to liquidity risk can then be quantified by viewing liquidity risk as if it were a put option written on the NSFR, where the present value of the RSF represents the “strike price,” with the short-term volatility of all assets underpinning the RSF determined by the implied volatility derived from equity options prices.29 The value of the ASF is assumed to follow a random walk with intermittent jumps that create sudden and large changes in the valuation of the liabilities. The volatility of these liabilities included in the ASF is computed as a weighted average of the observed volatilities of latent factors derived from a set of market funding rates deemed relevant for banks. These two time-varying elements, the adjusted RSF and ASF values, provide the basis for computing a put option, which has intrinsic value (i.e., is “in-the-money”) when the discounted value of the ASF falls below that of the RSF over the same time horizon, constituting an expected loss due to liquidity shortfall. The value of this derived put option can be shown to result in significant hypothetical cash losses for an individual firm as the risk-adjusted NSFR declines.

Figure 3.
Figure 3.

Conceptual Relation between the Net Stable Funding Ratio at Market Prices and Expected Losses from Liquidity Risk.

Citation: IMF Working Papers 2012, 209; 10.5089/9781475505597.001.A001

Note: This figure illustrates the relation between the net exposure from liquidity risk and the NSFR at market prices distribution functions (based on multiple observations of each over a certain period of time). Expected losses, which are modeled as a put option that approximates the cash flow profile over a given risk horizon, arise once there is some probability that the NSFR drops below the regulatory requirement to be greater than one. The greater the potential for funding distress projected by a declining NSFR, the greater are these losses. The tail risk of individual expected losses from a liquidity shortfall is represented by the expected shortfall (ES) at the 95th percentile, which is the area under the curve beyond the threshold value set by the Value-at-Risk (VaR).

The value of the put option increases the higher the probability of the ASF falling below the RSF over a one-year risk horizon(so that the NSFR is smaller than one, which would breach the lower boundary that banks will be mandated to maintain under the current Basel III proposal). Such probability is influenced by changes to the firm’s funding pattern, risk profile, and market perceptions of risk, which can be derived from changes to the implied asset value and volatility reflected in the institution’s equity option prices and from its asset and liability structure. Thus, the present value of market-implied expected losses associated with the current liquidity position of a single institution can be valued as a modified implicit put option of BASF based on the present value of ARSF as the strike price in keeping with the traditional Black-Scholes-Merton (BSM) model (see Appendix 1):

PNSFR(t)=ARSFer(Tt)Φ(dNSFRσARSF,BASFTt)BASFΦ(dNSFR),(1)

over time horizon T–t at the general risk-free rate, r, and asset growth rate, rARSF, where

dNSFR=In(BASFARSF(t))+((rARSF+σARSF,BASF22)(Tt))/σARSF,BASFTt.(2)

Note the dependence on the duration of liabilities (or debt claims) B underpinning the ASF, the total funding mismatch of the firm ARSF (t)/BASF using the definition underpinning the NSFR assumptions, and the joint asset-liability volatility given by

σARSF,BASF=σARSF2+σBASF22ρARSF,BASFσARSFσBASF(3)

Φ() in equation (1) denotes the standard normal distribution. The volatility of the RSF, σARSF, draws on the general asset volatility (which is “embedded” in the equity option formula)

σARSFσA=E(t)A(t)Φ(d)σE=(1Ber(Tt)Φ(dσATt)A(t)Φ(d))σE,(4)

similar to that employed in the CCA approach (see Appendix 1) and conditional on changes in leverage A(t)/B, where

d=In(A(t)B)+((rA+σA22)(Tt))/σATt(5)

E(t) represents the equity value at time t, σE is the equity volatility, and rA is the risk-free asset rate.30 The volatility of short-term liabilities in the ASF, σBASF, is derived as a weighted average of the observed volatilities of latent factors of significant market interest rates identified by a dynamic factor model (DFM)31 (see Box 2). The correlation ρAASF, BASF between both volatilities can be constant or estimated using the historical relation between the ASF and the RSF (at levels) over a rolling time window.

This specification of option price-based expected losses, however, does not incorporate skewness and kurtosis, and stochastic volatility, which can account for implied volatility smiles of equity prices (Backus and others, 2004). Thus, we mitigate the shortcomings of the BSM approach and enhance equation (1), without altering its general analytical form, by means of a jump diffusion that follows a standard Poisson process

Pr(N(t)=k)=(λt)kk!eλt,(6)

where λ is the average number of jumps per unit time (i.e., the number of jump events up to time t). The jump size follows a log-normal distribution

Jφexp[v22+vΦ(0,1)](7)

with average jump size φ and volatility v of the jump size calibrated over an estimation time period.32 The kth term in this series corresponds to the scenario where k jumps occur over a specific time horizon. By conditioning the present value of ARSF on the expected jump process over a rolling window of τ observations with periodic updating, the put option value can be written as

PNSFRk(t)=Σk=0exp(φλt)(φλt)kk!ARSFer(Tt)Φ(dNSFRkσARSF,BASFt)BASFΦ(dNSFRk).(8)

The asset volatility of the RSF with jumps σARSFk=σARSF2+kv2/t as well as the risk-free interest rate rARSFk = rARSF – λ(φ–1)+k ln(φ)/t are updated accordingly,33 so that equations (2) and (3) can be re-written as

dNSFRk =ln(BASFARSF(t))+((rARSFk+σARSFk,BASF22)(Tt))/σARSFk,BASFTt(9)

and

σARSFk,BASF=σARSFk2+σBASF22ρARSFk ,BASFσARSFkσBASF.(10)

Analogous to equation (4) under the BSM model, the volatility of the RSF, σARSFk, is set equal to the revised general asset volatility σAk (consistent with the above specification of a jump diffusion process),34 which is derived by solving

σARSFkσAk=(E(t)A(t)Φ(dk)σE)2+kv2t,(11)

where

dk=ln(A(t)B)+((rAk+σAk22)(Tt))/σAkTt,(12)

with general asset volatility with jumps σAk=σA2+kv2/t and the risk-free interest rate rAk = rA – λ(φ–1) + k ln(φ)/t.

Derivation of the Implied Asset Volatility Using the Moody’s KMV Model.

Default risk in the global Moody’s KMV model (Crosbie and Bohn, 2003) is defined using a Merton-style option pricing definition (see Appendix 1). But under the physical measure, the asset volatility under risk-neutral valuation can also be derived explicitly in absence of information about equity volatility—in lieu of equation (4) above.

We can recover a measure of the adjusted asset volatility (see Appendix 4)

σA* =(Φ1(EDFAMKMV)+ρAMKMV,MμMrσM)±(Φ1(EDFAMKMV)+ρAMKMV,MμMrσM)2+2ln(AMKMV(t)BMKMV)+rAMKMV

to account for the market price of risk, given the modified estimated default frequency (EDF) after adjustment for risk-neutrality based on

EDFAMKMV=Φ(dMKMVσA*Tt)=Φ((ln(AMKMV(t)BMKMV)+(rAMKMV+ρAMKMV,MσA*μMrσM(σA*)22)(Tt))/σA*Tt),

which satisfies equation (4) above, and uses the correlation between the asset and market return ρAMKMV,M, asset return rAMKMV, the “Market Sharpe Ratio” (μM – r)/σM (which changes daily but is assumed to be the same for all firms and financial institutions, consisting of the general risk-free rate r, average market return μM, and market volatility σM).

Thus, the actual (but unobservable) asset volatility, σA, as risk-neutral model input, is derived by solving the optimization constraint

1τΣt=1τ(σA*A(t)E(t)γ(t)η*)=1τΣt=1τσA*

over a rolling time window of τ number of estimation days (consistent with the maximum-likelihood estimation of the multivariate distribution below) on η* subject to the intertemporal error correction term

γ(t)=σA(t)σA(t)MKMVσA(t+τ)MKMVσA(t+τ)

over the same estimation window (with σA(t)MKMV and σA(t + τ)MKMV denoting the asset volatility at time t and t + τ days underpinning the observed “real” default probability EDFAMKMV reported by Moody’s KMV for an individual entity). Thus, the general asset volatility under risk-neutrality can be written as

σA=σA*γ(t)η*A(t)E(t).
  • 3. Step 3 – Estimating the joint expected losses from liquidity risk (“joint, risk-adjusted NSFR”)

Finally, the individually estimated net exposures to liquidity risk are aggregated to determine the magnitude of liquidity shortfalls on a system-wide level (see Figure 4). The expected losses arising from the variation of each individual firm’s risk-adjusted NSFR over time are treated as a portfolio for which we calculate the joint probability of all firms experiencing a liquidity shortfall simultaneously. In this way, each firm’s maturity mismatch between assets and liabilities, its implication on the market-based assessment of its risk profile, and the stability of its funding are linked with those characteristics at other firms that are subject to common changes in market conditions.

Figure 4.
Figure 4.

Conceptual Relation between Expected Losses from Liquidity Risk: Two-Firm (Bivariate) Case.

Citation: IMF Working Papers 2012, 209; 10.5089/9781475505597.001.A001

Note: This figure illustrates the bivariate case of aggregating expected losses in order to determine the joint probability of two sample firms experiencing a liquidity shortfall at the same time, using the estimation results for individual institutions. The left panel of Figure 4 shows the kernel density function of two firms (Bank A and Bank B). The probability of systemic liquidity risk is captured by combining the individual bank estimates (depicted by the green and blue panels), which generates the joint expected losses at a defined level of statistical confidence, such as the 95th percentile (red cube). The left panel can also be shown in two-dimensions as a so-called “contour plot” (see right panel of Figure 4).

We adapt the Systemic CCA framework (Gray and Jobst, 2011a, 2011b, and forthcoming) to formally capture the realizations of joint liquidity shortfalls by controlling for the realization of tail events.35 As part of a four-step process, we define a non-parametric dependence function of individual expected losses, which is combined with the marginal distributions of these individual loss estimates, in order to generate a joint distribution that defines an aggregate measure of liquidity risk. Using estimates of the joint tail risk of this multivariate distribution, such as the conditional Value-at-Risk (VaR) (or expected shortfall (ES)), we can gauge systemic liquidity risk in times of stress at a statistical confidence level of choice.

(i) Estimating the marginal distributions of individual expected losses from liquidity risk

We first specify the individual asymptotic tail behavior of individual expected losses in keeping with extreme value theory (EVT). EVT is as a general statistical concept of deriving a limit law for sample maxima, where the Fisher-Tippett-Gnedenko theorem (Fisher and Tippett, 1928; Gnedenko, 1943) defines the attribution of a given distribution of normalized maxima (or minima) to be of extremal type (assuming that the underlying function is continuous on a closed interval). Let the vector-valued series

Xjn=PNSFRk,1n(t),,PNSFRk,mn(t)(13)

denote i.i.d. random observations of expected losses (i.e., a total of n-number of daily put option values PNSFRk,mn(t) up to time t), each estimated over a rolling window of τ observations with periodic updating for jm firms in the sample. Given X=max(PNSFRk,j1(t),,PNSFRk,jn(t)), there exists a choice of normalizing constants βjn>0 and αjn, such that the probability of each ordered n-sequence of normalized sample maxima (X – αn) / βn converges to the non-degenerate limit distribution G(x) as n → ∞ and x ∈, so that

Fn[βnx+αn](x)=limn Pr(Xαnβny)=[F(βn y+αn)]nG(x),(14)

falls within the maximum domain of attraction (MDA) of the generalized extreme value (GEV) distribution and conforms to one of three distinct types of extremal behavior, Gumbel (EV0), Fréchet (EV1) or negative Weibull (EV2)) as limiting distributions of maxima of dependent random variables:

EV0:G0(x)=exp(exp(x))ifx0,ξ=0,(15)
EV1:G1,ξ(x)=exp(x1/ξ)ifx[μσ/ξ,),ξ>0,and(16)
EV2:G2,ξ(x)=exp((x)1/ξ)ifx(,μσ/ξ],ξ<0.36(17)

The limit distributions above are combined into a unified parametric family of the GEV probability distribution function with the shape parameter determined by the type of sub model (EV0, EV1 or EV2), so that

gμ,σ,ξ(x)={1σ((1+ξ(xμ)σ)1/ξ)ξ+1exp((1+ξ(xμ)σ)1/ξ)if1+ξ(xμ)σ>0,ξ01σ((1+ξ(xμ)σ)1/ξ)ξ+1exp(exp(xμσ))ifx(,+),ξ=0(18)

or simply

gμ,σ,ξ(x)=1σ((1+ξ(xμ)σ)1/ξ)ξ+1exp((1+ξ(xμ)σ)+1/ξ),(19)

and the cumulative distribution function

Gμ,σ,ξ(x)={exp((1+ξ(xμ)σ)1/ξ)if1+ξ(xμ)σ0exp(exp(x))ifx,ξ=0.(20)

Thus, in the context of multiple series, the jth univariate marginal density function based on GEV is defined as

yi(x)=(1+ξ^j(xμ^j)σ^j)+1/ξ^j(forj= 1,…,m)(21)

where 1 + ξj (x – μj)/σj > 0, the scale parameter is σj > 0, location parameter is μj, and shape parameter is ξj ≠ 0.37,38 The moments of the univariate density function in equation (21) above are estimated via the linear combinations of ratios of spacings (LRS) method (see Appendix 3), which determines how quickly the probability of extreme observations converges to zero), which identifies possible limiting laws of asymptotic tail behavior, i.e., the likelihood of even larger extremes as the level of statistical confidence approaches certainty (Coles and others, 1999; Poon and others, 2003; Jobst, 2007).

(ii) Estimating the dependence structure of individual expected losses from liquidity risk

Second, we define the co-movement of expected losses as a non-parametric, multivariate dependence function by expanding the bivariate logistic method proposed by Pickands (1981) to the multivariate case and adjusting the margins according to Hall and Tajvidi (2000) so that

ϒ(ω)=min{1,max{n(Σi=1nΛj=1myi,j/y^jωj)1,ω,1ω}},(22)

where y^j=Σi=1nyi,j/n reflects the average marginal density of all i ∈ n put option values 0 ≤ max (ω1,…, ωm–1) ≤ ϒ(ωj) ≤ 1 for all 0 ≤ ωj ≤ 1.39 ϒ() represents a convex function on [0,1] with ϒ (0) = ϒ (1) = 1, i.e., the upper and lower limits of ϒ() are obtained under complete dependence and mutual independence, respectively. It is estimated iteratively (and over a rolling window of τ observations with periodic updating (e.g., a daily sliding window of 120 days)) subject to the optimization of the (m-1)-dimensional unit simplex

Sm={(ω1,,ωm1)+m:ωj0,1jm1;Σj=1m1ωj1andωm=1Σj=1m1ωj},(23)

which establishes the degree of coincidence of multiple series of cross-classified random variables similar to a χ-statistic that measures the statistical likelihood that observed values differ from their expected distribution (Jobst and Kamil, 2008). This specification stands in contrast to a general copula function that links the marginal distributions using only a single (and time-invariant) dependence parameter.

(iii) Estimating the joint distribution of expected losses from liquidity risk

We then combine the marginal distributions of these individual expected losses with their dependence structure to generate a multivariate extreme value distribution (MGEV) over the a rolling window of τ number of days with daily updating by following the aggregation mechanism of the Systemic CCA approach.40 The resultant multivariate cumulative distribution function is specified as

Gt,m(x)=exp{(Σj=1myt,j)ϒt(ω)}(24)

with corresponding probability density function

gt,m(x)=σ^t,m1((Σj=1myt,j)ϒt(ω))ξ^t,m+1exp{(Σj=1myt,j)ϒt(ω)}(25)

at time t = τ + 1 by maximizing the likelihood Πj=1mgt,m(x|θ) over all three parameters θ = (μ, σ, ξ) simultaneously. Equation (24) represents the functional form of the daily series of expected losses from liquidity shortfall for all sample firms based on the empirical panel of observations Xjn in equation (13) above. Since the logarithm is a continuously increasing function over the range of likelihood, the parameter values that maximize the likelihood will also maximize the logarithm as a global maximum. Thus, we can write the lognormal likelihood function as Σj=1m In g(x|θ) so that the maximum likelihood estimate (MLE) of the true values θ0 is

θ^MLE=argθΘmax^(θ|x)θ0(26)

(iv) Estimating a tail risk measure of joint expected losses from liquidity risk

Finally, we obtain the general expression of the joint expected shortfall (ES) (or conditional Value-at-Risk (VaR)) as the probability-weighted residual density beyond a pre-specified statistical confidence level (say, a=0.95) of maximum losses. ES defines the conditional average value z of aggregate potential losses in excess of the statistical confidence limit (“severity threshold”) based on all observations over estimation days τ. Thus, we can obtain continuous densities from G() and compute ES as

ESt,τ,m,a=1a0a Gt,τ,m(x)dx=1a0a Gt,τ,m1(x)dx=E[zτ|zτGt,τ.m1(a)=VaRt,a],(27)

where G-1 (a) is the quantile corresponding to probability a and G-1 (a) = G (a) with G (a) ≡ inf (x | G(xa)) and threshold quantile value

VaRt,a=sup{Gt,τ,m1()|Pr[zτ>Gt,τ,m1()]a=0.95},(28)

with the point estimate of joint potential losses of m firms at time t defined as41

Gt,τ,m1(a)=μ^t,m+σ^t,m/ξ^t,m((ln(a)ϒt(ω))ξ^t,m1)(29)

C. Pricing the cost of liquidity support

The aggregate measure of expected losses can then be used to base practical macroprudential tools that help mitigate liquidity risk from both an individual and system-wide perspective. We extract the time-varying contribution of each individual institution as the discounted capital loss resulting from a possible funding shortfall or an approximation of a fair value risk premium for systemic liquidity risk affecting the likelihood of individual funding shortfall.

The contribution of each individual firm is determined by calculating the cross-partial derivative of the joint distribution of expected losses.42 The joint expected shortfall (ES) can also be written as a linear combination of individual ES values, ESt, τ, j, a, where the relative weights ψτ, m, a (in the weighted sum) are given by the second order cross-partial derivative of the inverse of the joint probability density function Gt,τ,m1() to changes in both the dependence function ϒt() and the individual marginal severity yt,j of expected losses. Thus, by re-writing ESt,τ,m,a in equation (27) above, we obtain

ESt,τ,m,a=Σjmψτ,m,aE[Zτ,j|Zτ,mGt,τ,m1(a)=VaRt,a],(30)

where the relative weight of institution j at statistical confidence level a is defined as the marginal contribution

Ψτ,m,a=2Gt,τ,m1(a)yt,jϒt(ω)s.t.ΣjmΨτ,j,a=1andΨτ,j,aZτ,jZτ,m,(31)

attributable to the joint effect of both the marginal density and the change of the dependence function due to the presence of institution jm in the sample.

We can use this amount to develop two price-based risk mitigation mechanisms, a capital surcharge and an insurance premium, which take into account the contingent support that banks would receive from a central bank in times of systemic liquidity stress, and, thus, represent the potential public sector cost arising from two or more institutions experiencing a significant liquidity shortfall (see Appendix 5):

  • (i) a capital surcharge would need to offset expected losses from liquidity risk at any given point in time during the envisaged risk horizon and be based on an institution’s own liquidity risk (reflected in the expected losses derived from the risk-based and market-implied NSFR) or on its marginal contribution to joint liquidity risk, whichever of the two is higher.

  • (ii) an insurance premium would reflect the cost of covering all expected losses from liquidity risk subject to the chance that an institution, in concert with other banks, fails to meet required funding needs and falls below the minimum required market-implied NSFR of one.

The capital charge would be the present value of the sum of money (in billions and as a percent of total capital), which would be needed by each firm to offset expected losses from liquidity shortfalls when the marked-implied NSFR of one is breached either individually or jointly with a predetermined probability consistent with prudential standards for market risk. Basing the capital surcharge on the higher of two indicators—the maximum capital that offsets the amount of individual expected losses or the contribution of an institution to overall expected losses) under both normal and stressed conditions—is motivated by the fact that sometimes the individual component is higher and sometimes the contribution to the systemic risk is higher depending on the state of distress of the system, which varies in its impact on firms, which, in turn, also changes their susceptibility to funding shocks. Such a macroprudential measure, if adopted, could encourage de-leveraging assuming all else equal (e.g., the potential of the offsetting effect of higher returns from holding less liquid assets is ignored).

By contrast, the fair value insurance premium would compensate for the liquidity support that would be needed to maintain a market-implied NSFR above one even during stressful times (occurring with a likelihood of five percent, for instance). The fair value insurance premium is derived as the actuarial value associated with the ASF exceeding the present value of the RSF over a risk horizon of one year, which is modeled based on the hazard rate of expected losses from system-wide liquidity shortfall given system-wide funding needs. This premium is multiplied by all short-term uninsured liabilities, i.e., the portion of deposits that is not covered by the insurance scheme. This reflects the cost of insuring against the downside risk that no cash inflows are available to cover debt service obligations in times of stress. As with any insurance premium, the primary objective would not be the collection of fees for the ex post funding of such a guarantee but the change in behavior it would hope to elicit to internalize an institution’s own liquidity risks and/or the impact of its own funding choices on system-wide liquidity risk. Given that the pricing would be based on the aggregate incidence of liquidity shortfall, the risk is “pooled” with others in the sample that made up the joint probability—making it less expensive.

D. Stress testing

The SRL approach can also be used within a stress testing framework to examine the vulnerabilities of individual institutions—and the system as a whole—to shocks affecting the valuation of assets and liabilities that underpin the NSFR. Volatility shocks to both asset returns and funding costs as well as the joint dynamics (i.e., dependence structure) between them can significantly alter the net exposure to liquidity shortfalls reflected in a market-based measure of liquidity risk. Thus, stress scenarios can be determined based on the historical calibration of market factors affecting the valuation of both the ASF and RSF, the constituent components of NSFR—as an approximation of net cash flows in stress situations. The stresses can be based on firm-specific shocks, common shocks or both:

  • Firm-specific shocks are modeled by modifying the jump diffusion process (with the frequency, average size, and volatility of jumps calibrated to past stress scenarios) of assets underpinning the RSF, which could result in unbounded and random liabilities (Eberlein and Madan, 2010).

  • Common shocks are modelled by means of stressing the historical variance-covariance matrix of latent factors of several market interest rates that impact the valuation of both the ASF and RSF (see Box 2). In adverse conditions, we observe higher volatilities of all market interest rates but lower correlations between those rates that are more indicative of changes in short-term asset returns and those rates that better explain short-term funding costs.

Stress Testing Within the SRL Model Framework.

In adverse conditions, shocks to the volatilities of market funding rates and the correlation between asset returns and funding rates can be mechanically imposed in the model to better examine short-term funding vulnerabilities. The following five-step process offers a useful way to apply market-informed common shocks to the liquidity position of sample firms:43

1. Estimation of latent variables from market rates affecting the ASF and RSF. We derive latent variables of multiple market rates by extracting the principal component of each of the following six general categories: (i) short-term government bond yields (at maturities of 3, 6 and 12 months), (ii) long-term government bond yields (at maturities of 3, 5, 7 and 10 years), (iii) total equity market returns (total return of domestic stock market, MSCI total return index), (iv) home currency LIBOR rates (at maturities of 3, 6 and 12 months), (v) home currency OIS rates (at maturities of 3, 6 and 12 months), and (vi) financial bond yields (1-5 years, 15+ years, and rated “AAA”).

2. Definition of baseline ASF volatility as a composite measure. Under baseline conditions, the volatility of the ASF is defined as a weighted average volatility of these latent variables whose explanatory power on the ASF—on levels—defines the relative size of these weights. The weights are derived as dynamic factor model (DFM) regression coefficients of the ASF on these latent variables, and—once re-scaled to unity—define the individual contribution of each latent variable volatility (measured as the standard deviation over a rolling window of τ observations with periodic updating (e.g., a daily sliding window of 120 days)) to the composite ASF volatility. Note that the volatility of the RSF, in contrast, is endogenous to the model specification, while the covariance between the RSF and the ASF is based on the correlation of their levels over the same rolling window time period.

3. Definition of shocks to volatilitystressed volatility and covariance of the ASF and the RSF. The historical return series of each latent variable are multiplied by the product of the empirical variance-covariance matrix and a “shocked” variance-covariance matrix44 (defined by an ad hoc change to the pairwise correlation and volatility of all elements according to a historical or synthetic scenario) in order to derive stressed series of all latent variables and their corresponding volatilities.45

4. Estimation of stressed ASF and RSF volatility. Like in the baseline case, the stressed ASF volatility is a composite measure derived as a weighted average of the stressed individual volatility series of each latent variable. But this time, the stressed volatility series are multiplied by corresponding coefficient values estimated in a dynamic factor regression of the baseline composite ASF volatility (see above) defined by the baseline volatilities of these latent variables. The same process is repeated for the stressed RSF volatility based on the endogenously generated RSF volatility and the “shocked” variance-covariance matrix of the same latent variables. The stressed covariance between the ASF and RSF is an explicit result of the “shocked” variance-covariance matrix.

5. Update the individual and joint measure of liquidity risk using the stressed input values. The joint probability distribution of liquidity shortfalls is re-estimated after re-calculating individual expected losses from the liquidity shortfall using the stressed ASF and RSF volatility measure and aggregating these results in the same way as in the baseline case.

IV. Empirical Application – U.S. Banking Sector

The SRL model was applied to 14 largest commercial and investment banks in the United States based on firm-level data obtained from financial statements and markets covering a five-year time period from January 1, 2005 to December 14, 2010 (1,442 observations).46,47 The variations in the components of the NSFR—that is, in the ASF and RSF as weighted sums of their constituent liabilities and asset positions (see Table 5)—were used to compute the expected losses due to liquidity shortfalls under extreme conditions.48,49

Table 5.

Weighting Factors for the Calculation of Available and Required Stable Funding.

article image
Source: Bankscope.Note: The figures in italics are based on assumptions published in IMF (2011b), which reflect the availability of relevant prudential data available in Bankscope.

The results suggest that the current liquidity standards in Basel III (whether as an accounting measure or a risk-adjusted measure) are not able to capture the potential liquidity shortfall under stressed conditions. The median of the marked-based NSFR for the 14 banks stays above one and has continued to improve since the credit crisis (see Figure 5). In contrast, the median expected losses generated by the SRL model would suggest that banks have become more vulnerable to extreme liquidity shocks and that their expected losses arising from potential liquidity shortfalls were higher during some time frames, namely in the run-up to the March 14, 2008, Bear Stearns rescue and around year-end 2008. Those results apply especially to firms dependent on funding sources that are more susceptible to short-term (and more volatile) market interest rates; that dependency, in combination with their relatively higher exposure to maturity mismatches, accentuates their vulnerability to liquidity risk.

Figure 5.
Figure 5.

Individual Market-based NSFR and Associated Expected Losses Using Option Pricing (sample median)

Citation: IMF Working Papers 2012, 209; 10.5089/9781475505597.001.A001

Sources: Bloomberg and Bankscope.Note: Dates of vertical lines are as follows: (1) March 1, 2008—Bear Stearns Rescue; (2) September 14, 2008—Lehman Brothers failure; and (3) April 27, 2010—Greek debt crisis. Expected losses are at the 95percent confidence level.

Using the results for individual banks, we estimate system-wide liquidity risk under extreme stress. We derive the joint expected shortfall (ES) at the 95th percentile for all sample banks and find that interlinkages in banks’ funding positions and their common exposure to the risk of funding shocks raise joint liquidity risk beyond the simple sum of individual net exposures if liquidity shortfalls happen simultaneously. The contagion risk from this interdependence gets accentuated during times of extreme stress if estimates are derived at a high level of statistical confidence (at percentile levels far removed from average outcomes). Failing to take into account this systemic component by only adding up expected losses associated with individual banks’ risk-adjusted NSFR would have resulted in an underestimation of system-wide liquidity shortfall for the period between mid-2009 and mid-2010, where the green line exceeds the red line in Figure 6.50,51 Conversely, the sum of individual expected losses would have overestimated system-wide liquidity shortfall under extreme stress conditions during the first half of 2008 when the probability of several banks experiencing a liquidity shortfall was low; however, this changes in the run-up to the collapse of Lehman Brothers in September 2008 when a general increase of expected losses and their sensitivity to extreme shocks generated higher systemic liquidity risk.

Figure 6.
Figure 6.

Joint Expected Losses from Systemic Liquidity Risk Using Option Pricing

(expected shortfall at the 95th percentile, In billions of U.S. dollars).

Citation: IMF Working Papers 2012, 209; 10.5089/9781475505597.001.A001

Sources: Bloomberg and Bankscope.Note: Dates of vertical lines are as follows: (1) March 1, 2008—Bear Stearns Rescue; (2) September 14, 2008—Lehman Brothers failure; and (3) April 27, 2010—Greek debt crisis. Expected losses are at the 95 percent confidence level. 1/ expected shortfall at the 95th percentile level based on the multivariate distribution of individual expected losses. 2/ sum of individual estimates of expected shortfall at the 95th percentile level over 30-day rolling window.

The results suggest that, accounting for their interdependence is imperative for a more accurate representation of systemic liquidity risk. The joint expected losses may be easier to discern by looking at averages over specified periods (see Table 6). During the crisis period from late 2008 to 2009, the joint expected losses were largest, as one would surmise. Thus, the joint tail risk of the expected shortfalls takes into account nonlinear dependence and the probability of extreme changes in funding costs affecting the valuation of firms.

Table 6.

Joint Expected Losses from Systemic Liquidity Risk

(expected shortfall at the 95th percentile, In billions of U.S. dollars).

article image
Sources: Bloomberg and Bankscope.Note: This exercise was run on a set of 14 large U.S. commercial and investment banks. The expected shortfall measure of systemic liquidity risk was estimated at the 95th percentile level.

The results imply that some banks contribute to systemic liquidity risk beyond their individual exposure to liquidity shortfalls in times of distress, which underscores the usefulness of a system-wide assessment of liquidity risk. During the height of the credit crisis, the average contribution of the largest U.S. banks to the tail risk from simultaneous liquidity shortfalls was higher than their individual liquidity risk (with the later accounting only for the average net exposure to liquidity risk) (see Table 3, first half). During 2010, expected losses from the likelihood of rising systemic liquidity risk as measured by the 95 percent ES (based on the multivariate distribution of expected losses) exceeded $31 billion on average (see Table 6), which is considerable, but far lower than during the peak of the financial crisis (at $150 billion). Appreciable differences among banks indicate varying sensitivity to common changes in funding conditions and the extent to which these translate into externalities affecting general liquidity risk in the overall sample.

We also calculate both a capital surcharge and an insurance premium, which take into account the support that U.S. banks would likely to receive in times of systemic liquidity stress, and, thus, represent the potential public cost of marginal expected losses when two or more institutions face significant liquidity shortfalls. Table 7 presents the distribution of the capital charges over selected U.S. banks, and Table 8 does so for the value of the fair value insurance premium that would compensate for the contribution to joint expected losses caused by each bank. Both capital surcharges and the cost of insurance have been calculated based on the methodology presented in Appendix 5. After application of the SRL model we find that:

  • most banks would need to increase total capital by at least 4.82 percent to offset expected losses of $2.05 billion (median) arising from their individual contributions to systemic liquidity risk (as of end-Q4 2010). This number, however, would reach $9.86 billion for the worst bank in the sample (if only individual liquidity risk were considered given the bank’s lower contribution to systemic liquidity risk of a maximum of $5.96 billion). After considering stress periods, like the recent credit crisis, in the assessment methodology, the additional capital need to mitigate systemic liquidity risk could more than triple on average to 15.09 percent of total capital (see Table 7).

  • alternatively, the commensurate insurance premium would amount to $0.8 billion, or less than half the additional capital needed to offset expected losses from liquidity risk and compensates for the liquidity support that would be needed to maintain the NSFR above one over a one-year risk horizon (see Table 8).

Table 7.

Capital Charge for Individual Liquidity Risk and Contributions to Systemic Liquidity Risk

(In billions of U.S. dollars).

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Sources: Bloomberg and Bankscope.Note: This exercise was run on a selected set of 14 large U.S. commercial and investment banks. The last column matches the distributions of the individual capital charges and reported total capital of all sample institutions. In this case, the maximum capital charge for the worst bank in 2010 coincides with a disproportionately higher total capital amount, which reduces the percentage share of the capital add-on for systemic liquidity from 4.82 percent (median) to 3.25 percent (maximum). The expected losses during the stress period (covering the height of the recent credit crisis) are listed here for illustrative purposes (indicating a potential extension of the presented approach consistent with the revised market risk estimation under the new regulatory framework for banks (BCBS, 2010b). For details about the calculation of the capital charge, see Appendix 5.
Table 8.

Summary Statistics of Banks’ Contributions to Systemic Liquidity Risk and Associated Fair Value Insurance Premium

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Sources: Bloomberg and Bankscope.Note: This exercise was run on a set of 14 large U.S. commercial and investment banks. Insured deposits here are defined as 10 percent of demand deposits reported by sample banks. Note that the share of deposits covered by guarantees varies by country and could include time and savings deposits. Robustness checks reveal that reducing the amount of uninsured short-term liabilities does not materially affect the median and maximum. For details of the calculation see Appendix 5.

Each bank’s percentage share reflects its contribution to total expected losses from systemic liquidity risk.

V. Conclusion

In the wake of the global financial crisis, there has been a concerted effort to establish a regulatory framework to address systemic risk, which has resulted in a multi-faceted approach comprising complementary measures in areas of regulatory policies, supervisory scope, and resolution arrangements as part of a sustainable solution towards a more resilient financial sector while avoiding impairment to efficient activities that do not cause and/or amplify stress in any meaningful manner. However, systemic liquidity risk from a macroprudential perspective remains largely unaddressed.

The SRL model seems better suited than existing prudential approaches to identify, quantify and mitigate systemic liquidity risk since it (i) measures the marginal contribution of each institution to total systemic liquidity risk, and (ii) and can be used to construct a supervisory charge for the institution’s contribution to systemic liquidity risk that provides incentives for the internalization of the cost of contingent liquidity support in times of stress. In particular, it offers several potential benefits by generating an objective and meaningful measure of systemic risk in areas that are absent in the current Basel III liquidity framework:

  • Liquidity risk is treated as a dynamic exposure via a risk-adjusted value of the net stable funding ratio (rather than a combination of discrete accounting identities as in the current Basel III framework) by drawing on a market-based evaluation of the riskiness of a firm. This approach generates a forward-looking measure of liquidity risk (subject to different degrees of leverage and maturity mismatches defining the risk profile of institutions), which helps determine the probability of an individual institution experiencing a liquidity shortfall and incurring an associated expected loss (as an approximation of the economic cost of being unable to service on-going debt payments, resulting in net cash outflows); and

  • The impact of a particular funding configuration is not assessed individually but in concert with all banks to generate estimates of systemic liquidity risk. As such, it takes the systemic components of liquidity risk into account by estimating the joint sensitivity of assets and liabilities (and their volatilities) of multiple entities to common changes in market prices. In this way, the SRL model helps assess (and quantify consistently) how the size and interconnectedness of individual institutions can create short-term, system-wide vulnerabilities to cases of considerable liquidity risk shared by several entities.

However, there are also distinct drawbacks to this approach, which are worthwhile bearing in mind. These constitute key limitations that need to be acknowledged and reflected appropriately as caveats in the discussion of findings. The presented findings are derived from valuation models that are subject to varying degrees of estimation uncertainty and assumptions, which need to be taken into account when drawing policy conclusions. For instance, the chosen option pricing model might fail to capture some relevant economics, especially during times when the validity of valuation models driving market prices might be undermined by rare and non-recurring events, and not by repeated realizations of predictable outcomes generated by a process of random events that exhibits stochastic stability (in violation of its steady state assumptions). Since the statistical apparatus underlying conventional asset pricing theory fails to capture sudden and unexpected realizations beyond historical precedent, exploring different alternative approaches, such as cash flow models, can help assess any pricing distortions aberrant price dynamics might impose on the option pricing method underpinning the SLR model.52

Going forward, several technical improvements to the SLR model could be considered. For instance, it could be usefully extended to include a dynamic configuration of weighting factors of both the ASF and RSF (possibly informed by empirical calibration), which would provide insights about the sensitivity of prudential measures of liquidity risk to exogenous assumptions.

Overall, the SLR model provides a tractable framework for the assessment of system-wide valuation effects arising from joint liquidity risk. However, much like any other efforts aimed at the design of macroprudential tools for managing systemic liquidity risk, its effectiveness and expedient use are contingent on data availability given the increasing sophistication of financial products and transactions, the diversity of financial institutions, as well as the growing interdependence of capital markets and their impact on liquidity risk management of financial institutions. New sources of systemic liquidity risk are likely to be found in areas where financial activities are loosely organized and governed by incentive structures that encourage greater-risk taking in a benign economic environment but entail more adverse consequences when stress occurs.

Appendix 1—Standard Definition of Contingent Claims Analysis (CCA)

CCA is used to construct risk-adjusted balance sheets, based on three principles: (i) the values of liabilities (equity and debt) are derived from assets; (ii) liabilities have different priority (i.e., senior and junior claims); and (iii) assets follow a stochastic process. Assets (present value of income flows, proceeds from assets sales, etc.) are stochastic and over a horizon period may be above or below promised payments on debt which constitute a default barrier. Uncertain changes in future asset value, relative to the default barrier, are the driver of default risk which occurs when assets decline below the barrier. When there is a chance of default, the repayment of debt is considered “risky,” to the extent that it is not guaranteed in the event of default (risky debt = risk-free debt minus guarantee against default). The guarantee can be held by the debt holder, in which case it can be thought of as the expected loss from possible default or by a third party guarantor, such as the government.

In the first structural specification, commonly referred to as the Black-Scholes-Merton (BSM) framework (or short “Merton model”) of capital structure-based option pricing theory (OPT), the total value of firm’s assets follows a stochastic process and may fall below the value of outstanding liabilities.53 Thus, the asset value A(t) at time t describes a continuous asset process so that the physical probability distribution of the end-of-period value is:

A(Tt)~A(t)exp{(rA+σA2/2)(Tt)+σATtz},(A1.1)

for time to maturity T-t. More specifically, A(t) is equal to the sum of its equity market value, E(t), and its risky debt, D(t), so that A(t) = E(t) + D(t). Default occurs if A(t) is insufficient to meet the amount of debt owed to creditors at maturity, which constitute the bankruptcy level (“default threshold” or “distress barrier”). The equity value E(t) is the value of an implicit call option on the assets, with an exercise price equal to default barrier. It can be computed as the value of a call option

E(t)=A(t)Φ(d)Ber(Tt)Φ(dσATt),(A1.2)

with

d=ln(A(t)B)+(r+οA22)(Tt)σATt,(A1.3)

asset return volatility σA, and the cumulative probability Φ() of the standard normal density function. Both the asset, A(t), and asset volatility, σA are valued after the dividend payouts. The value of risky debt is equal to default-free debt minus the present value of expected loss due to default,

D(t)=Ber(Tt)PE(t).(A1.4)

Thus, the present value of market-implied expected losses associated with outstanding liabilities can be valued as an implicit put option, which is calculated with the default threshold B as strike price on the asset value A(t) of each institution. Thus, the present value of market-implied expected loss can be computed as:

PE(t)=Ber(Tt)Φ(dσATt)A(t)Φ(d),(A1.5)

over time horizon T–t at risk-free discount rate r, subject to the duration of debt claims, the leverage of the firm, and asset volatility. Note that the above option pricing method for PE(t) does not incorporate skewness, kurtosis, and stochastic volatility, which can account for implied volatility smiles of equity prices.

Since the implicit put option PE(t) can be decomposed,

PE=Φ(dσATt)(1Φ(d)Φ(dσATt)A(t)BerT)Ber(Tt)=PD × LGD,(A1.6)

there is no need to introduce a potential inaccuracy of assuming a certain LGD. As a consequence of the assumptions on the underlying asset price process, the risk-neutral probability distribution (or state price density, SPD) of A(t) is a log-normal density

ft*(A(Tt))=ert,TtTt2E(t)B2|B=A(Tt)=1A(Tt)2πσA2(Tt) exp[[ln(A(Tt)/A(t))(rt,TtσA2/2)(Tt)]22σA2(Tt)](A1.7)

with mean (rσA2/2)(Tt) and variance σA2(Tt) for ln(A(Tt)/A((t)) where rt,T –t and f*() denote the risk-free interest rate and the risk-neutral probability density function (or SPD) at time t, with risk measures:

Δ=defE(t)A(t)=Φ(d) andΓ=def2E(t)A(t)2=Φ(d)A(t)σATt.(A1.8)

Appendix 2—Estimation of the Empirical State Price Density (SPD)

Breeden and Litzenberger (1978) show Arrow-Debreu prices can be replicated via the concept of the butterfly spread on European call options as basis for extracting the state price density (SPD) from observable derivatives prices. This spread entails selling two call options at strike price K and buying two call options with adjacent strike prices K = K – ΔK and K+ = K + ΔK respectively, with the stepsize ΔK between the two call strikes. If the terminal underlying asset value A(T) = K then the payoff Z() of 1/ΔK of such butterfly spreads at time T–τ (with time to maturity τ and maturity term T) is defined as

Z(A(T),K;ΔK)=Price(A(Tτ),τ,K;ΔK)|τ=0=u1u2ΔK|A(T)=K,τ=0=1,(A2.1)

with

u1=C(A(Tτ),τ,K+ΔK)C(A(Tτ),τ,K),(A2.2)

and

u2=C(A(Tτ),τ,K)C(A(Tτ),τ,KΔK).(A2.3)

C(A, τ, K) denotes the price of a European call option with an underlying asset price A, time to maturity τ and strike price K. As ΔK →0, Price(A(Tt),τ,KK) of the position value of the butterfly spread becomes an Arrow-Debreu security paying one if A(T) = K and zero in other states. If A(T) ∈+ is continuous, we can obtain a security price

limΔK0(Price(A(t),τ,K;ΔK)ΔK)|K=A(T)=f*(A(T))ert,τ,(A2.4)

where rt,τ and f*() denote the risk-free rate and the risk-neutral probability density function (or SPD) at time t. On a continuum of states K at infinitely small ΔK a complete state pricing function can be defined. Moreover, as ΔK →0, this price

limΔK0(Price(A(t),τ,K;ΔK)ΔK)=limΔK0u1u2(ΔK)2=2Ct()K2(A2.5)

will tend to the second derivative of the call pricing function with respect to the strike price evaluated at K, provided that C() is twice differentiable. Thus, we can write

2Ct()K2|K=A(T)=ft*(A(T))ert,ττ,(A2.6)

across all states, which yields the SPD

ft*(A(T))=ert,ττ2Ct()K2|K=A(T),(A2.7)

under no-arbitrage conditions and without assumptions on the underlying asset dynamics. Preferences are not restricted since no-arbitrage conditions only assume risk-neutrality with respect to the underlying asset. The only requirements for this method are that markets are perfect, i.e., there are no transactions costs or restrictions on sales, and agents are able to borrow and lend at the risk-free interest rate.

Appendix 3—Moments of the GEV Distribution and Estimation of the Shape Parameter Using the Linear Combination of Ratios of Spacings (LRS) Method

Since all raw moments of G() are defined contingent on the tail shape, the natural estimator of ξj^ is derived by means of the LRS method using the linear combination

ξ^=(n/4)1Σi=1(n/4)(log(ν^i)/log(c))(A3.1)

for n observations, where

ν^i=(χn(1a):nχnac:n)/(χnac:nχna:n)(A3.2)

and

c=log(1a)/log(a)(A3.3)

for quantile a = i/n. Since χna:n=Gξ^,μ^,σ^1(a) at statistical significance a, the approximation

ν^iGξ1(1a)Gξ1(ac)Gξ1(ac)Gξ1(a)=c1+ξ(A3.4)

holds. The simple statistics are defined as

mean:μσξ+σξg1 given {μ+σ(Γ(1ξ)1)/ξifξ0,ξ<1μ+σγifξ=0ifξ1
variance:σ2(g2g12)ξ2 given {σ2(g1g12)/ξ2ifξ0,ξ<1/2σ2π2/6ifξ=0ifξ1/2
skewness:g33g1g2+2g13(g2g12) given {g33g1g2+2g13(g2g12)3/2ifξ0126ζ(3)π3ifξ=0
kurtosis:g44g1g3+6g2g123g14(g2g12)23 given {g44g1g3+6g2g122g14(g2g12)2ifξ0125ifξ=0,

where gp = Γ(1–) for p = 1, …, 4, Euler’s constant γ (Sondow, 1998) and Riemann zeta function ζ(t) (Borwein and others, 2000) and gamma probability density function Γ(t).

Appendix 4—Derivation of the Asset Volatility Under Risk-neutrality Using the Moody’s KMV Model-based EDF Value

EDFA(t)MKMV=Φ(dMKMVσAMKMVTt)(A4.1)
EDFA(t)MKMV=Φ(ln(AMKMV(t)BKMV)+(rAMKMV+ρAMKMV,MσA*μMrσM(σA*)22)(Tt)σA*Tt)(A4.2)
Φ1(EDFA(t)MKMV)=ln(AMKMV(t)BKMV)+(rAMKMV+ρAMKMV,MσA*μMrσM(σA*)22)(Tt)σA*Tt(A4.3)

Using T-t=1, and rearranging gives

ln(AMKMV(t)MKMV)+rAMKMV+ρAMKMV,MσA*μMrσMΦ1(EDFA(t)MKMV)σA*(σA*)22=0,(A4.4)

so the quadratic formula can be used to solve for

σA*=b±b24ac2a=(ρAMKMV,MμMrσMΦ1(EDFA(t)MKMV))±(ρAMKMV,MμMrσMΦ1(EDFA(t)MKMV))24(ln(AMKMV(t)BKMV)+rAMKAV)(12)2(12)(A4.5)
σA*=b±b24ac2a=(ρAMKMV,MμMrσMΦ1(EDFA(t)MKMV))±(ρAMKMV,MμMrσMΦ1(EDFA(t)MKMV))24(ln(AMKMV(t)BKMV)+rAMKAV)(12)2(12)(A4.6)=(ρAMKMV,MμMrσM+Φ1(EDFA(t)MKMV))±(ρAMKMV,MμMrσMΦ1(EDFA(t)MKMV))2+2(ln(AMKMV(t)BKMV)+rAMKAV)

Appendix 5—Technical Description of Price-based Macroprudential Measures for Systemic Liquidity using the Systemic Risk-Adjusted Liquidity (SRL) Model

The SRL model, if applied to a banking system, can be used to calibrate two price-based measures, a capital surcharge and an insurance premium, either of which could be used as a macroprudential tool to help mitigate systemic liquidity risk:

  • a capital surcharge would need to offset expected losses from liquidity risk at any given point in time based on a firm’s own liquidity risk (highest risk-based NSFR over some pre-specified period, such as one quarter) or its marginal contribution to joint liquidity risk, whichever is higher; and

  • an insurance premium would be an actuarial fee imposed on firms, which would compensate them for expected losses from liquidity risk in a systemic event when they fall below the minimum required NSFR of one in concert with other institutions.

Implicitly, these two measures proxy for the amount of contingent support that banks would receive from a central bank in times of systemic liquidity stress. Numerical examples of these two approaches are in the main text of the chapter (see Tables 6 and 7), and their calculations are explained below. Unlike the capital surcharge, which is meant to absorb losses at any point in time, the insurance premium is measured over time (in this case, one year ahead) and, thus, spreads out the probability of the firm’s experiencing a liquidity shortfall over a risk horizon.

For the capital surcharge, the method follows the current bank supervisory guidelines for market risk capital requirements (BCBS, 2009), in which the Value-at-Risk (VaR) is calculated each day and compared to three times the average quarterly VaRs over the last four quarters. The maximum of these two numbers becomes the required amount of regulatory capital for market risk. In a similar way, each firm would need to meet an additional capital requirement, cSLR (in currency units), at time t, to offset the net exposure from either its individual liquidity risk or its contribution to systemic liquidity risk at a statistical confidence level of a=0.95. First, we choose the higher of (i) the previous quarter’s expected shortfall at percentile a, ES¯t1,τ,j,a, averaged over daily observations associated with individual liquidity risk of firm j, and (ii) the average of this quarterly measure over the preceding four quarters, multiplied by an individual multiplication factor κj. This amount would be compared to greater of (i) the last available quarterly marginal contribution, Ψ¯t1,τ,j,a multiplied by the total expected shortfall, ES¯t,τ,a, over the same time period, and (ii) the average of this quarterly measure over the preceding four quarters, multiplied by a general multiplication factor κ. The higher of the two maximums would then be the surcharge. Therefore, based on an estimation window of τ days in each quarter t, the capital surcharge cSLR would be

cSLR=max{max{ES¯t1,τ,j,a;κj × 14Σt=40ES¯t,τ,j,a};max{ψ¯t1,τ,j,a × ES¯t1,τ,a;κ × 14Σt=40(Ψ¯t,τ,a×ES¯t,τ,a)}}(A5.1)

The comparison of the two maximums is motivated by Figure 6, whereby a firm’s individual liquidity risk is not commensurate to its contribution to systemic liquidity risk depending on existing interlinkages between firms and how they influence the probability of joint liquidity shortfalls. Note that the amount of capital to be withheld under this measure is exactly the present value of funds needed to offset the expected losses that would be incurred when the requirement of NSFR greater than one is violated either jointly or individually for a given level of statistical confidence.

An alternative method is to require individual firms to pay an insurance premium for liquidity support based on their likelihood of experiencing expected loss from liquidity risk that is shared by other firms as well. The conditional probability of expected losses from individual shortfall that raises system-wide liquidity risk, i.e., the marginal contribution of an institution to system-wide liquidity risk, can be used to calculate a fair value price for the necessary insurance coverage specific to each firm. To illustrate this, we calculate the ratio of the potential systemically-based expected losses of each institution to the total amount of required stable funding—the probabilistic proportion of underfunding (relative to the existing funding level), akin to a probability of distress for a certain risk horizon. More specifically, we estimate the average marginal contribution of each firm to the expected shortfall (with statistical significance a) as a measure of systemic liquidity, averaged over the previous four quarters, which is then divided by the average of the discounted present value of the asset value underpinning RSF over the previous four quarters.

The fair value of a risk-based insurance premium can be obtained as the natural logarithm of 1 minus the ratio between the sum of the marginal contribution to the joint expected shortfall at percentile a (over the preceding four quarters) and the sum of the present values of the quarterly RSFs of all sample firms, multiplied by the negative inverse of the time period under consideration (see equation (A5.4) below). This assumes that the conditional probability of expected losses from liquidity shortfall under the risk-neutral measure is constant over time and can be expressed as an exponential function, given a survival probability

exp(0th(u)du)=exp(ht)(A5.2)

at time t with a constant hazard rate

hΣt=40(ψ¯t,τ,j,a × ES¯t,τ,a)ΣjmΣt=40RSFt,j × exp(r(Tt))(A5.3)

over one period t=1. Thus, the cost fSLR of insuring stable funding over the short-term against possible shortfalls can be calculated by multiplying the insurance premium (in basis points) by the value of average uncovered short-term liabilities L¯t,jST (i.e., excluding secured deposits and investments) over the previous four quarters as a nominal base.

fSLR=1Tln(1h)insurancepremium × 10,000 × 14Σt=40L¯t,jST,(A5.4)

where r is the risk-free rate and T-t is the time horizon (i.e., residual maturity). This amount would compensate for the individual firm’s cost of future systemic liquidity support.

Because they take into account a single firm’s contribution to systemic liquidity risk, either the capital surcharge or the insurance premium could be used as price-based macroprudential tool to instill incentives for more resilient and diversified funding structures. Based on estimates during times of stress, both measures could be refined to avoid procyclical tendencies. For instance, in the context of the capital surcharge, the multiplication factor κj could be calibrated on data obtained during times of stress and set such that minimum prudential levels of capital charges are maintained “through the cycle” with suitable latency of time-varying adjustment. Equally, the maximization of the capital surcharge (and the determination of the conditional probability of individual underfunding for the insurance premium calculation) could be directly conditioned on an estimate of the marginal contribution to systemic liquidity risk based on data obtained during times of stress.

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