A Guide to IMF Stress Testing
Chapter

Chapter 27. Measuring Systemic Risk-Adjusted Liquidity

Author(s):
Li Ong
Published Date:
December 2014
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Author(s)
Andreas A. JobstThis chapter is an abridged version of IMF Working Paper 12/209, which was also published as “Measuring Systemic Risk-Adjusted Liquidity (SRL)—A Model Approach,” Journal of Banking and Finance, Vol. 45, pp. 270–87 (Jobst, 2014). The model is based on previous analytical work in the context of the October 2009, October 2010, and April 2011 issues of the Global Financial Stability Report (IMF, 2009, 2010, 2011; Jobst, 2011).

The Systemic Risk-Adjusted Liquidity (SRL) model combines option pricing with market information and balance sheet data to generate a probabilistic measure of multiple entities experiencing a liquidity event. This measure links a firm’s maturity mismatch between assets and liabilities, its overall risk profile, and the stability of its funding with those characteristics of other firms that are subject to common changes in market conditions. This approach can then be used (1) to quantify an individual institution’s time-varying contribution to expected losses from systemwide liquidity shortfalls and (2) to price contingent liquidity support during times of stress within a macroprudential framework that provides incentives for liquidity managers to internalize the negative externalities from systemic risk implications of individual funding decisions. The model also can 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.

Method Summary

Method Summary
OverviewThe Systemic Risk-Adjusted Liquidity model generates a risk-adjusted, market price–based measure of multiple entities experiencing a liquidity event under stress conditions.
ApplicationThe method is appropriate in situations where prudential data are limited but market information is available; the approach relies on predefined prudential specification of liquidity risk (e.g., the net stable funding ratio, or NSFR) to assess the impact of maturity mismatches.
Nature of approachModel-based (option pricing, multivariate parametric/nonparametric estimation).
Data requirements
  • Accounting information on all constituent balance sheet elements underpinning the NSFR.

  • Market data on equity and equity option prices.

  • Various market rates (short-term sovereign rate; with maturities ranging from 3 to 12 months), long-term sovereign rates (with maturity ranging from 3 to 10 years), total equity market returns (domestic market and other equity markets), financial bond rates (investment grade, both medium- and long-term), local currency London interbank offered rate (ranging from 3 to 12 months), and the domestic short-term currency overnight indexed swap.

StrengthsThe model combines a prudential measure of liquidity risk with an advanced version of contingent claims analysis as inputs to a multivariate specification of joint liquidity risk; approach can be used to:
  • quantify an individual institution’s time-varying contribution to systemwide liquidity shortfalls; and

  • price contingent liquidity support during times of stress.

WeaknessesAssumptions are required regarding the prudential measure of liquidity risk and the specification of the option pricing model; there is no explicit treatment of the impact of solvency risk on liquidity risk; technique is complex and resource intensive.
ToolUnder development.

A defining characteristic of the recent financial crisis was the simultaneous and widespread dislocation in funding markets, which revealed that the lack of sound liquidity risk management (and suitable policy responses) can have a significant impact on financial stability. 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. 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.

Little progress has been made so far in addressing—in a comprehensive way—the externalities caused by impact of the interconnectedness within financial institutions and financial markets on systemic liquidity risk, that is, the risk that 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). Although systemic solvency risk already has entered the prudential debate in the form of additional capital rules to apply to systemically important financial institutions, proposals aimed at measuring and regulating systemic liquidity have been few and far between. This is largely owed to the rarity of systemwide liquidity risk events, the variegated interactions between institutions and funding markets within countries and across national boundaries, and the conceptual challenges in modeling liquidity conditions affecting institutions and transactions separately or jointly.

The policy objective of such efforts would be to minimize the possibility of systemic liquidity disruptions that necessitate costly public sector support. Disruptions to the flow of financial services due to an impairment of all or parts of the financial system give rise to systemic risk if there is the potential of financial instability to trigger serious negative spillovers to the real economy.1 The recent financial crisis demonstrated that higher perceived counterparty risk precipitates escalating funding constraints that further depress asset prices and eventually overwhelm individual liquidity provisions. Because every financial institution’s risk-taking affects the operation of the financial system as a whole, the magnitude of market disruptions increases with the level of asymmetric information as coordination failure accentuates the impact of common shocks. One case in point is the wholesale funding market, where these network externalities and the structural complexity across financial and nonfinancial institutions during the recent crisis has amplified aggregate vulnerabilities to shocks from counterparty risk. As the economic environment or 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. As such a behavior occurred collectively, it caused liquidation of assets under fire sale conditions, which resulted in a negative confidence-induced downward liquidity spiral, increasing funding pressures as deteriorating counterparty risk eventually weighs on solvency (Shin, 2009; Shleifer and Vishny, 2010).

A number of reforms and initiatives are under way to address shortcomings in financial institutions’ liquidity practices. They have resulted in more stringent supervisory liquidity requirements without directly targeting systemwide implications. Under the postcrisis revisions of the existing Basel Accord, known as Basel III, the Basel Committee on Banking Supervision (BCBS, 2010a, 2010b) has issued two quantitative liquidity standards to be applied at a global level and published qualitative guidance to strengthen liquidity risk management practices in banks.2 Individual banks will have to maintain higher and better-quality liquid assets as measured by the liquidity coverage ratio and the net stable funding ratio (NSFR).3 This approach assumes that sufficient liquidity would limit knock-on effects on solvency conditions in distress situations and complement the risk absorption role of capital—but without considering systemwide effects and liquidity buffers. These standard measures do not address the additional risk of such simultaneous shortfalls arising out of the interconnectedness of various institutions across a host of financial markets and jurisdictions.

In contrast, a macroprudential approach to systemic liquidity risk should offset procyclical tendencies of liquidity risk and account for changes to an institution’s risk contribution, which might not necessarily follow cyclical patterns, in order to inform countercyclical risk mitigants to joint liquidity risk. It would also motivate a risk-adjusted pricing scheme so that institutions that contribute to systemic liquidity risk through their interconnectedness or through their impact are assigned a proportionately higher charge (while the opposite holds true for firms that help absorb systemwide shocks from sudden increases in liquidity risk). In this regard, several proposals are under discussion. They include the internalization of public sector cost of liquidity risk via insurance schemes (Goodhart, 2009; Gorton and Metrick, 2009; Perotti and Suarez, 2009, 2011), capital charges (Brunnermeier and Pedersen, 2009), taxation (Acharya, Cooley, and others, 2010; Acharya, Santos, and Tanju, 2010), investment requirements (Cao and Illing, 2009; Farhi, Golosov, and Tsyvinski, 2009), as well as arrangements aimed at mitigating the systemwide effects from the fire sale liquidation of assets via collateral haircuts (Valderrama, 2010) and modifications of resolution regimes (Roe, 2009; Acharya and Oncu, 2010).

In this chapter, we propose a combined price/quantity-based mechanism—the SRL model—for the assessment and stress testing of systemic liquidity risk. This methodology complements the current Basel III liquidity framework by extending the prudential assessment of liquidity using the NSFR to a systemwide approach, which can help substantiate different types of macroprudential liquidity regulation, such as a capital surcharge, a fee, a tax, or an insurance premium for contingent liquidity access. The SRL model combines both perspectives by assessing how the size and interconnectedness of individual institutions (with varying degrees of leverage and maturity mismatches defining their risk profile) can create short-term vulnerabilities to liquidity risk on a systemwide level and under distress conditions (Davis, 2011).

The SRL model is based on an options pricing concept to gauge the general level of liquidity risk for a portfolio of institutions in a three-step process. First, using an “expected loss” notion to evaluate the level of liquidity shortfall, the model combines balance sheet information and market data (equity/equity options) in order to generate a risk-adjusted measure of stable funding. In this way, the probability of falling below the joint NSFR boundary translates into a market-implied analogue to the NSFR. Given the historical variation in the underpinnings of the NSFR and the extent to which this measure links institutions implicitly to the markets, 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 required stable funding (RSF). The joint probability function then is used to identify an individual institution’s contribution to systemic liquidity risk to price contingent liquidity support within a macroprudential framework that provides 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 and/or insurance premium.

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: (1) to measure the extent to which an institution contributes to systemic liquidity risk; and (2) to 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 toward insolvency. The systemic dimension of the model is captured by three properties:

  • Drawing on the market’s evaluation of the riskiness of a firm (including the liquidity risk that the institution will be unable to offset continuous cash outflows). That evaluation, in turn, is based on perceived riskiness 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 at varying degrees. Changes in common funding conditions (and their 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.

  • Quantifying the chance of simultaneous liquidity shortfalls via joint probability distributions. After obtaining a market-based measure of individual liquidity risk, the probability that banks will experience liquidity shortfall simultaneously—that is, the likelihood that the amount of available stable funding (ASF) for each institution falls below the amount of RSF as defined by the NSFR4—is made explicit by computing joint probability distributions (which also account for difference in the magnitude of individual 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.

1. Methodology

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 systemwide effects. So far, the CCA methodology5 been applied widely to measure and evaluate solvency risk and credit risk at financial institutions. 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 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 simultaneous liquidity shortfall, which also quantifies the marginal contribution of an institution to systemic liquidity risk.6

First, the SRL model transposes the current regulatory proposal aimed at limiting term structure transformation into a market-based measure of individual liquidity risk. In keeping with existing microprudential measures, individual liquidity risk is defined as the effective maturity mismatch between short-term liabilities, weighted by their susceptibility to market fragilities (rollover risk), and short-term assets, weighted by their market liquidity value (funding risk),7 after controlling for the maturity of off-balance-sheet hedging transactions, such as foreign exchange swaps.8 In the SRL model, the proposed Basel III liquidity standard aimed at limiting maturity transformation—the NSFR—is a regulatory benchmark in this regard and serves as the starting point for the quantification of individual liquidity risk.9 The components of the NSFR—ASF and RSF—are transposed into a time-varying measure of the NSFR at market prices, where RSF and ASF values reflect differences between the balance sheet and actual market values of total assets and liabilities of each firm (see Figure 27.1). The actual balance sheet measures of ASF and RSF values are rescaled by (1) the ratio of the book value of total liabilities to the present value of total liabilities B (which can be observed) and (2) 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, respectively.10 Doing so generates the values ARSF and BASF, which transform the NSFR ratio into a market-based measure of an institution’s liquidity risk (“market-based NSFR”).

Figure 27.1Methodology to Compute Joint Expected Losses from Systemic Liquidity Risk

Source: Author.

Given that A cannot be observed, it is estimated directly from option prices prior to the estimation of expected losses arising from the market-based NSFR. 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 two-equations-two-unknowns in the traditional Merton model, which contains empirical irregularities that can influence solving both implied asset value and asset volatility simultaneously. Using equity option prices, we can derive the risk-neutral probability distribution of the underlying equity price at the maturity date of the options. We determine the implied asset value based on the empirical SPD of the equity price by adapting the Breeden and Litzenberger (1978) method, together with a semiparametric specification of the Black-Scholes option pricing formula (Aït-Sahalia and Lo, 1998). Estimates are based on option contracts with identical time to maturity (and consistent with the residual maturity assumed in the application of the BSM model), assuming a continuum of strike prices. The SPD of the equity price includes the risk-neutral expectation of the implied asset value and removes the impact of differences of higher moments of both the equity price and the implied asset value dynamics over the chosen time horizon. Thus, the current implied asset value can be transposed from the SPD of the equity price conditional on the contemporaneous leverage of the firm (see equations (27.5) and (27.9)).

Second, the net exposure arising from individual liquidity risk is modeled as a put option in order to derive a risk-adjusted version of the market-based NSFR (see Figure 27.2). The aggregate cash flow implications of changes to liquidity risk then can be quantified as expected losses arising from insufficient stable funding of each bank. Thus, liquidity risk is viewed as 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 of observable equity option prices of a firm.11 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 provide the basis for computing a put option, which has intrinsic value (i.e., is “in-the-money”) when the discounted value of ASF falls below that of RSF over the same time horizon, constituting an expected loss owing 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 27.2Conceptual Relation between the Net Stable Funding Ratio (NSFR) at Market Prices and Expected Losses from Liquidity Risk

Source: Author.

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 ES at the 95th percentile, which is the area under the curve beyond the threshold value set by the VaR.

ES = expected shortfall.

The value of the put option increases the higher the probability of ASF falling below RSF over a one-year risk horizon so that NSFR is smaller than one. This 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 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 on the present value of ARSF as strike price in keeping with the traditional Black-Scholes-Merton (BSM) model as over time horizon Tt at the general risk-free rate r and asset growth (or drift) rate rARSF, where Φ(•) denotes the standard normal cumulative distribution function and subject to the duration of liabilities (or debt claims) B underpinning ASF, the total funding mismatch of the firm ARSF (t)/BASF under NSFR assumptions, and joint asset-liability volatility

The volatility of ASF, σARSF, draws on the general asset volatility

where E represents the equity value at time t, and σE is the equity volatility. The volatility, σARSF, is endogenous to the CCA approach and conditional on changes in leverage A(t)/B, given

where rA is the risk-free asset rate. The short-term volatility of all liabilities underpinning the ASF, σBASF, is identifiable but not readily observable for each source of funding. Thus, it is derived as a weighted average of the observed volatilities of latent factors of significant market interest rates affecting the funding components of the RSF determined by a dynamic factor model (DFM).12 The correlation ρARSF, BASF between both volatilities can be constant or estimated using the historical relation between ASF and RSF at levels over a rolling time window.

This specification of option price–based expected losses, however, does not incorporate skewness, kurtosis, and stochastic volatility, which can account for implied volatility smiles of equity prices (Backus, Foresi, and Wu, 2004). Thus, we mitigate the shortcomings of the BSoM approach and enhance equation (27.1), without altering its general analytical closed-form form, by means of a jump diffusion that follows a standard Poisson process:

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,

with average random jump size φ and volatility v of the jump size calibrated over a rolling estimation period of τ observations with periodic updating of daily log returns (which sets the time horizon for the subsequent estimation steps of the model).13 The k th term in this series corresponds to the scenario where k jumps occur over observation period. Hence, the put option value can be written as follows:14

and asset volatility of RSF with jumps σARSFkσAk=σA2+kν2/(Tt) as well as the risk-free interest rate rARSFk = rARSFλ(φ − 1) + k ln(φ)/(Tt) are updated accordingly,15 so that

and

Thus, the volatility of ASF, σARSFk, would be the same as the revised general asset volatility σAk (consistent with the above specification of a jump diffusion process),16 so that

where

given the general asset volatility with jumps σAk = σA2+kν2/(Tt) and the risk-free interest rate rAk = rAλ(φ − 1) + k ln(φ)/(Tt).

Third, individually estimated net exposures to liquidity risk are aggregated to determine the magnitude of liquidity shortfalls on a systemwide level (see Figure 27.3). 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 27.3Conceptual Relation between Expected Losses from Liquidity Risk: Two-Firm (Bivariate) Case

Source: Author.

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 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).

ES = expected shortfall.

More specifically, we adapt the Systemic CCA framework to generate a multivariate extreme value distribution that formally captures the realizations of joint liquidity shortfalls. We define a nonparametric dependence function of individual expected losses, which is combined with the marginal distributions of these individual loss estimates to generate a joint distribution that defines an aggregate measure of liquidity risk for a selected sample of firms. Based on the conditional tail expectation of this multivariate distribution, such as the conditional value at risk (VaR) (or expected shortfall [ES]), we can derive point estimates of systemic liquidity risk in times of stress at a statistical confidence level of choice.

We first specify the individual asymptotic tail behavior of individual expected losses in keeping with extreme value theory (EVT) 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. Let the vector-valued series Xmn=PNSFRk,1i(t),,PNSFRk,mi(t) denote independent and identically distributed random observations of expected losses (i.e., put option values) each estimated over the given time horizon (e.g., a daily sliding window of n = 120 days) for a total m number of j firms. Given

there exists a choice of normalizing constants βn > 0 and αn, such that the probability of each ordered n-sequence of normalized sample maxima (Xnαn)/βn converges to the nondegenerate limit distribution H(x) as n→∞ and x ∈°, so that

falls within the maximum domain of attraction (MDA) of the generalized extreme value (GEV) distribution with probability density function

which is obtained from differentiating the cumulative distribution function:

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

where 1 + ξj(xμj)/σj>0, scale parameter σj>0, location parameter μj ∈ ℝ, and shape parameter ξj≠0.17 The moments of the univariate density function are estimated via the linear combinations of ratios of spacings method (Gray and Jobst, 2010), which identifies possible limiting laws of asymptotic tail behavior of normalized extremes (Coles, Heffernan, and Tawn, 1999; Poon, Rockinger, and Tawn, 2003; Jobst, 2007).

Second, we define the comovement of expected losses as a nonparametric, multivariate dependence function. It is estimated iteratively on a unit simplex that optimizes the coincidence of multiple series of cross-classified random variables—similar to a chi-statistic that measures the statistical likelihood of observed values to differ from their expected distribution. We specify dependence function A(ω1,…,ωm−1) 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

where y^j=i=1nyi,j/n and 0 ≤ max(ω1,…,ωm−1) ≤ ϒ(ωj) ≤ 1 for all 0 ≤ ωj ≤ 1, subject to the optimization of the (m − 1)-dimensional unit simplex:

Sm establishes the degree of coincidence of multiple series of cross-classified random variables similar to a chi-statistic that measures the statistical likelihood of observed values to differ from their expected distribution. ϒ(•) represents a convex function on [0,1] with ϒ(0) = ϒ(1) = 1, that is, the upper and lower limits of ϒ(•) are obtained under complete dependence and mutual independence, respectively. This specification stands in contrast to a general copula function that links the marginal distributions using only a single (and time-invariant) dependence parameter.

Finally, we combine the marginal distributions of these individual expected losses with their dependence structure to determine a multivariate distribution (over the estimation time horizon defined earlier) by following the aggregation mechanism of the Systemic CCA approach (Gray and Jobst, 2009, 2010, 2011; Jobst and Gray, 2013).18,19 The resultant multivariate cumulative distribution function is specified as

with corresponding probability density function

at time t = τ+1.

We then obtain the general expression of joint ES (or conditional VaR) as the probability-weighted residual density beyond a prespecified statistical confidence level (say, a = 0.95) of maximum losses. ES defines the conditional average value z ∈ ℝ of aggregate potential losses in excess of the statistical confidence limit (“severity threshold”) based on observations over estimation days τ. Thus, we can obtain continuous densities from G(•) and compute ES as

where G−1(a) is the quantile corresponding to probability 0 < a < 1 and G−1(a) = G(a) with G(a) ≡ inf (x|G(xa)) and threshold quantile value

with the point estimate of joint potential losses of m firms at time t defined as follows:20

ES also can be written as a linear combination of individual ES values, ESt,τ,j, a, where the relative weights (in the weighted sum) are given by the second-order cross-partial derivatives of the inverse of the joint probability density function Gt,τ,m1() to changes in both the dependence function and the individual marginal severity of expected losses.

Thus, by rewriting ESt,τ,m,a, we obtain

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

attributable to the joint effect of both the marginal density y, t,j (•) and the change of the dependence function ϒt(•) absent institution j.

The resulting measure of the contribution of each individual firm to aggregate expected losses can be used as a basis for practical macroprudential tools designed to create incentives that help mitigate liquidity risk from both individual and systemwide perspectives.21 We use this amount (as a result of the quantity-based restriction of NSFR > 1) to develop two price-based risk mitigation mechanisms—a capital surcharge and an insurance premium—that 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.

For the capital surcharge, the method follows the current bank supervisory guidelines for market risk capital requirements (BCBS, 2009), in which the VaR is calculated each day and compared with 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 (1) the previous quarter’s ES at percentile a, ES¯t1,τ,j,a,, averaged over daily observations associated with individual liquidity risk of firm j; and (2) the average of this quarterly measure over the preceding four quarters, multiplied by an individual multiplication factor κj. This amount would be compared with the greater of (1) the last available quarterly marginal contribution, ψ¯t1,τ,j,a, and multiplied by the total ES, ES¯t,τ,a, over the same time period; and (2) 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

An alternative method is to require individual firms to pay an insurance premium for liquidity support based on their likelihood to experience expected losses from liquidity risk that is shared by other firms as well. The conditional probability of expected losses from individual shortfall risk to coincide with higher systemwide 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 dollar losses of each firm to its RSF—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 ES (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.

Assuming that this conditional probability is constant over time and can be expressed as an exponential function, the fair value of a risk-based insurance premium can be obtained as the natural logarithm of 1 minus the preceding ratio and multiplied by the negative inverse of the time period under consideration. Thus, the cost fSLR of insuring stable funding over the short term against possible liability runoffs can be calculated by multiplying the insurance premium with 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:

where r is the risk-free rate and Tt is the time horizon (i.e., residual maturity). This amount would compensate for the individual firm’s cost of future systemic liquidity support.

The SRL approach also can 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 alter significantly the net exposure to liquidity shortfall 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 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 RSF, which could result in unbounded and random liabilities.

  • Common shocks are modeled by means of stressing the historical variance—covariance matrix of latent factors of several market interest rates (see Box 27.1). In adverse conditions, we observe higher volatilities of all market interest rates but lower correlation between those rates that are more indicative of changes in short-term asset returns and those rates that better explain short-term funding costs.

2. Empirical Application: U.S. Banking Sector

The SRL model was applied to the 13 largest commercial and investment banks in the United States using firm-level data obtained from financial statements and markets. The five-year time period covers January 1, 2005, to December 14, 2010 (1,442 observations).22 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 27.1)—were used to compute the expected losses owing to liquidity shortfalls under extreme conditions.23,24

Table 27.1Factors Used in Estimations
Available Stable FundingRequired Stable Funding
Equity1.00Cash0.00
Tier 21.00Customer loans0.75
Demand deposits0.80Commercial loans0.85
Saving and term deposits0.85Advances to banks0.00
Bank deposits0.00Other commercial and retail loans0.85
Other deposits and short-term borrowing0.00Other loans1.00
Derivative liabilities0.00Derivative assets0.90
Trading liabilities0.00Trading securities0.15
Senior debt maturing after one year1.00Available for sale securities0.15
Other non-interest-bearing liabilities0.00At-equity investments in associates1.00
Other reserves1.00Other earning assets1.00
Insurance assets1.00
Residual assets1.00
Reserves for nonperforming loans1.00
Contingent funding0.05
Sources: Author; and Bankscope.Note: Categories in italics are staff judgements; others more closely approximate Basel classifications.
Sources: Author; and Bankscope.Note: Categories in italics are staff judgements; others more closely approximate Basel classifications.

Box 27.1.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 imposed mechanically 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:1

  • Estimation of latent variables from market rates affecting ASF and RSF. We derive latent variables of multiple market rates by extracting the principal component of each of the following six general categories: (a) short-term government bond yields (at maturities of 3, 5, 7, and 10 years), (b) long-term government bond yields (at maturities of 3, 6, and 12 months), (c) total equity market returns (total return of domestic stock market, MSCI Total Return Index), (d) home currency LIBOR (at maturities of 3, 6, and 12 months), (e) home currency OIS rates (at maturities of 3, 6, and 12 months), and (f) financial bond yields (1–5 years, 15+ years, and rated AAA).

  • Definition of baseline ASF volatility as a composite measure. Under baseline conditions, the volatility of ASF is defined as a weighted average volatility of these latent variables whose explanatory power on ASF—on levels—defines the relative size of these weights. The weights are derived as dynamic factor model regression coefficients of ASF on these latent variables and—once rescaled to unity—define the individual contribution of each latent variable (measured as the 120-day rolling window standard deviation) to the composite ASF volatility. The volatility of RSF is endogenous to the model specification, while the covariance between RSF and ASF is based on the 120-day rolling window correlation of their levels.

  • Definition of shocks to volatility—stressed volatility and covariance of ASF and 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 matrix2 (defined by some change to the historical correlation and volatility of all elements) in order to derive stressed series of each latent variable and its volatility under stress.3

  • 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. The stressed volatility series are multiplied by corresponding coefficient values estimated in a dynamic factor regression of the baseline composite ASF volatility on the baseline volatility 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 ASF and RSF is an explicit result of the “shocked” variance–covariance matrix.

  • Updating the individual and joint measure of liquidity risk using the stressed input values. The joint probability distribution of liquidity shortfalls is reestimated after recalculating individual expected losses from liquidity shortfall using the stressed ASF and RSF volatility measure and aggregating these results in the same way as in the baseline case.

1 Also, various liquidity stress testing methodologies could be incorporated in this approach, such as van den End (2008), Wong and Hui (2009), and Aikman and others (2009).2 The shock to the variance–covariance matrix does not need to be uniform, such as a 20 percentage point shock to volatility and/or correlation, but also can accommodate asymmetric changes based on each particular market rate, with some experiencing greater shocks than others.3 This is done technically by multiplying the inverted, lower-triangular variance–covariance matrix of historical market returns with an inverted variance–covariance matrix defined by “shocked” assumptions on correlation and individual volatility of latent variables.

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 market-based NSFR for the 13 banks stays above one and has continued to improve since the credit crisis (see Figure 27.4). 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 timeframes, 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. Because the SLR model takes into account the joint dynamics between the ASF and RSF via their covariance structure, 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.

Figure 27.4Individual Market-Based Net Stable Funding Ratio (NSFR) and Associated Expected Losses Using Option Pricing (sample median)

Sources: Author; Bankscope; and Bloomberg.

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.

Using the results for individual banks, we estimate systemwide liquidity risk under extreme stress. We derive the joint ES at the 95th percentile for all sample banks and find that accumulated expected losses associated with individual banks’ risk-adjusted NSFR would have underestimated systemwide liquidity shortfall for the period from mid-2009 to mid-2010, where the dashed line exceeds the solid line in Figure 27.5.25 It would have failed to take into account the interlinkages in banks’ funding positions and their common exposure to the risk of funding shocks—that is, the systemic component—which raises joint liquidity risk beyond the simple composition of individual net exposures if estimates are derived at a high level of statistical confidence (at percentile levels far removed from average outcomes). Conversely, during the first half of 2008, the sum of individual expected losses associated with individual banks’ risk-adjusted NSFR would have overestimated systemwide liquidity shortfall under extreme stress conditions. The results suggest that if liquidity shortfalls happen simultaneously, accounting for their interdependence is imperative for a more accurate representation of systemic liquidity risk, and contagion risk from this interdependence gets accentuated during times of extreme stress in markets.

Figure 27.5Joint Expected Losses from Systemic Liquidity Risk Using Option Pricing (expected shortfall at the 95th percentile, in billions of U.S. dollars)

Sources: Author; Bankscope; and Bloomberg.

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.

ES = expected shortfall.

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 systemwide 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 latter accounting for only the average net exposure to liquidity risk). In 2010, expected losses from the likelihood of rising systemic liquidity risk in the wake of extreme disruptions to funding markets (and their impact on asset prices and volatility) exceeded US$31 billion (see Table 27.2), which is considerable but far lower than during the peak of the financial crisis (at US$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.

Table 27.2Capital Charge for Individual Liquidity Risk and Contributions to Systemic Liquidity Risk (in billions of U.S. dollars)
Individual Liquidity RiskContribution to Systemic Liquidity RiskEconomic Significance
Stress Period: Sept. 14, 2008–Dec. 31, 2009Last quarter (2010 Q4)Average of 2010 Q1–Q4Stress Period: Sept. 14, 2008–Dec. 31, 2009Last quarter (2010 Q4)Average of 2010 Q1–Q4Capital charge (maximum of (1)–(4))Share of total capital (in percent)Share of total assets (in percent)
(1)(2)(3)(4)
Minimum0.000.000.031.550.080.270.271.570.20
Median1.460.741.186.420.662.052.054.820.73
Maximum33.328.539.8613.513.095.969.863.250.44
Sources: Author; Bankscope; and Bloomberg.Note: This exercise was run on a selected set of 13 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).
Sources: Author; Bankscope; and Bloomberg.Note: This exercise was run on a selected set of 13 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).

We also calculate both a capital surcharge and an insurance premium, which take into account the support that U.S. banks would receive in times of systemic liquidity stress. It thus represents the public cost of marginal expected losses when two or more institutions face significant liquidity shortfalls. Table 27.2 presents the distribution of the capital charges over selected U.S. banks, and Table 27.3 does so for the value of the insurance premium that would compensate for the contribution to joint expected losses caused by each bank. We find that the application of the SRL model would lead to a potential 5 percent increase of total capital to include net exposures arising mainly from the individual contribution to systemic liquidity risk. This number, however, would exceed 25 percent for the worst bank in the sample (for which individual liquidity risk dominates) and could more than triple on average for all banks if stress periods, like the recent credit crisis, were to be included in the historical estimation (see Table 27.2). Alternatively, we calculate the commensurate fair value insurance premium to compensate for the liquidity support that would be needed to bring the NSFR above one during stressful times. For most banks, the premium would amount to US$0.8 billion for most banks, or less than half the additional capital needed to offset expected losses from liquidity risk at any given point in time over a one-year risk horizon (see Table 27.3).

Table 27.3Summary Statistics of Individual Contributions to Systemic Liquidity Risk and Associated Fair Value Insurance Premium
RangePrecrisis: end-June, 2006–end-June 2007Subprime Crisis: July 1, 2007–Sept. 14, 2008Credit Crisis: Sept. 14, 2008–Dec. 31, 2009Sovereign Crisis: Jan. 1–Dec. 31, 2010
Individual contribution to systemic liquidity risk (at 95th percentile; in percent)
Minimum1.20.61.01.7
Median6.84.58.37.6
Maximum13.435.116.714.5
Total100.0100.0100.0100.0
Insurance cost based on reported exposure: Fair value insurance premium uninsured short-term liabilities (in billion U.S. dollars)
Minimum0.70.10.70.1
Median1.91.43.90.8
Maximum7.817.211.31.9
Sources: Bankscope; Bloomberg; and IMF staff calculations.Note: This exercise was run on a number of selected U.S. banks. Each bank’s percentage share reflects its contribution to total expected losses from systemic liquidity risk. 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.
Sources: Bankscope; Bloomberg; and IMF staff calculations.Note: This exercise was run on a number of selected U.S. banks. Each bank’s percentage share reflects its contribution to total expected losses from systemic liquidity risk. 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.

3. conclusion

The SRL model offers several potential benefits in areas that are largely unaddressed in the current Basel III liquidity framework. It is a price-based approach that treats liquidity risk endogenously as a dynamic exposure via the risk-adjusted value of stable funding rather than a combination of discrete accounting identities by drawing on the 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 liquidity shortfall and incurring associated expected loss (as an approximation of the economic cost of being unable to service ongoing debt payments, resulting in net cash outflows).

Overall, the SLR model provides a tractable framework for the assessment of systemwide valuation effects arising from joint liquidity risk. 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 how the size and interconnectedness of individual institutions can create short-term, systemwide vulnerabilities to cases of considerable liquidity risk shared by several entities. Thus, it seems better suited than existing prudential approaches to identify, quantify, and mitigate systemic risk because it (1) measures the marginal contribution of each institution to total systemic liquidity risk and (2) 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.

References

Impairment to the flow of financial services occurs where certain financial services are temporarily unavailable, as well as in situations where the cost of obtaining the financial services is sharply increased. It would include disruptions owing to shocks originating outside the financial system that have an impact on it, as well as shocks originating from within the financial system.

See also BCBS (2009).

The NSFR represents a structural ratio, which limits the stock of unstable funding by encouraging longer-term borrowing in order to restrict liquidity mismatches from excessive maturity transformation. Based on existing proposals, it would require banks to establish a stable funding profile over the short term, that is, the use of stable (long-term and/or stress-resilient) sources to continuously fund cash flow obligations that arise from lending and investment activities inside a one-year time horizon.

The NSFR is defined as the ratio, for a bank, of its “available amount of stable funding” divided by its “required amount of stable funding.”

The CCA is a generalization of option pricing theory pioneered by Black and Scholes (1973) and Merton (1973, 1974). It is based on three principles that are applied in this study: (1) the values of liabilities are derived from assets; (2) assets follow a stochastic process; and (3) liabilities have different priorities (senior and junior claims). Equity can be modeled as an implicit call option, while risky debt can be modeled as the default-free value of debt less an implicit put option that captures expected losses. In the SRL model, advance option pricing is applied to account for biases in the Black-Scholes-Merton specification.

This method uses publicly available information. Although the focus here is on banks given data limitations, the methodology is sufficiently flexible to be used for nonbank institutions that contribute to systemic liquidity risk. Indeed, the proposal builds on several strands of recent research that focus on the interactions between financial institutions and markets in the context of systemic liquidity risk.

This will be reflected in the liquidity of the market, i.e., the ability to trade an asset without affecting the price. We will introduce a parameter that denotes the cost of early liquidation from imperfect market depth.

This approach represents a significant improvement over the concept of systemic liquidity insurance presented in Perotti and Suarez (2009). The implementation of the model purports to be a market-driven regulation of a quantity-based restriction on liquidity transformation as opposed to a variable Pegovian tax (Perotti and Suarez, 2011).

The NSFR measures the amount of longer-term, stable sources of funding employed relative to the liquidity profiles of the assets funded and the potential for contingent calls on funding liquidity arising from off-balance-sheet commitments and obligations.

Estimations of these scaling factors, and the subsequent covariance and the joint expected losses, are computed over a rolling window of 120 working days to reflect their changing characteristics.

The NSFR reflects the impact of funding shocks as an exposure to changes in market prices in times of stress. The procedure also can be applied to other measures of an individual firm’s liquidity risk.

A DFM of the ASF is specified based on one principal component extracted from each group of observed market rates (at different maturities) as explanatory variables: (1) short-term sovereign rates (with maturities ranging from three to nine months) and long-term sovereign rates (with maturity ranging from 3 to 10 years); (2) total equity market returns (domestic market and Morgan Stanley Capital International [MSCI] index); (3) financial bond rates (investment grade, both medium- and long-term); and (4) domestic currency London Interbank Offered Rates (LIBOR, ranging from three to nine months) and the domestic currency overnight indexed swap (OIS) rates (ranging from three to nine months). In the empirical analysis of the chapter, the volatility of the ASF is calculated as the average volatility of these daily market rates weighted by the regression coefficient of each principal component, estimated over a five-year period from January 3, 2005, to December 14, 2010.

Further refinements of this option pricing model are possible, including various simulation approaches, which might come at the expense of losing analytical tractability. The ad hoc model of Dumas and others (1998) is designed to accommodate the implied volatility smile and is easy to implement, but requires a large number of market option prices. The pricing models by Heston (1993) and Heston and Nandi (2000) allow for stochastic volatility, but the parameters driving these models can be difficult to estimate. Many other models have been proposed, to incorporate stochastic volatility, jumps, and stochastic interest rates. Bakshi and others (1997), however, suggest that most of the improvement in pricing comes from introducing stochastic volatility. Introducing jumps in asset prices leads to small improvements in the accuracy of option prices. Other option pricing models include those based on copulas, Levy processes, neural networks, GARCH models, and non-parametric methods. Finally, the binomial tree proposed by Cox and others (1979) spurred the development of lattices, which are discrete-time models that can be used to price any type of option—European or American, plain-vanilla or exotic.

All parameters are calibrated over the entire sample period.

The value of derivatives with convex payoff (which includes this put option specification) increases when jumps are present (i.e., when λ > 0)—regardless of the average jump direction.

Given that the implied asset value is derived separately, this approach avoids the traditional “two-equations-two-unknowns” approach applied by Jones, Mason, and Rosenfeld (1984) to derive implied assets and asset volatility.

The upper tails of most (conventional) limit distributions (weakly) converge to this parametric specification of asymptotic behavior, irrespective of the original distribution of observed maxima (unlike parametric VaR models). The higher the absolute value of shape parameter, the larger the weight of the tail and the slower the speed at which the tail approaches its limit.

See also Gray and Malone (2010) for a more general application of this approach to the integrated balance sheets of an entire economy. An application of the Systemic CCA approach in the context of IMF stress tests and spillover analysis can be found in Chapter 26.

The analysis of dependence is completed independently from the analysis of marginal distributions and thus differs from the classical approach, where multivariate analysis is performed jointly for marginal distributions and their dependence structure by considering the complete variance–covariance matrix, such as the Multivariate Generalized AutoRegressive Conditional Heteroskedasticity approach.

ES is an improvement over VaR, which, in addition to being a pure frequency measure, is “incoherent,” that is, it violates several axioms of monotonicity, subadditivity, positive homogenity, and translation invariance found in coherent risk measures. For example, subadditivity, which is a mathematical way to say that diversification leads to less risk, is not satisfied by VaR.

Note that this approach also could be used to identify the effectiveness of closer supervisory monitoring in response to identified liquidity problems of a particular bank. That can be done if remedial actions decrease the bank’s contribution to overall systemic risk from liquidity shortfalls up to the point where it closely matches the individual liquidity risk.

The sample covers Bank of America, JP Morgan, Citigroup, Morgan Stanley, Goldman Sachs, Merrill Lynch, Wells Fargo, US Bancorp, PNC Financial Services, Bank of New York Mellon, Sun Trust, BB&T, State Street, and Regions Bank.

Extreme conditions were defined to be those that occur with a probability of 5 percent or less.

Further specifications of the model include three-month equity call prices for sample banks, put option time horizon of t = 1, risk-free rate = 0.03, and an estimation window for jump parameters over 120 days. Price changes would qualify as jumps if equity daily returns were outside an acceptable range of −10 and +10 percent and represented a deviation of more than 50 percent on a normalized scale over a 120-day rolling window (i.e., observations were in the top decile over a half-year observation period).

In Figure 27.5, the solid line represents the daily sum of individual, market-implied expected losses, and the dashed line indicates the joint tail risk of these individual expected losses. Both tail risks are measured so that the chances of such events are 5 percent or less.

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