CHAPTER 1

Measuring Systemic Risk: Structural Approaches

Raimund M. Kovacevic

ISOR, Department of Statistics and OR, University of Vienna, Austria

Georg Ch. Pflug

ISOR and IIASA, Laxenburg

The global financial crisis of 2007–2008, often considered as the worst financial crisis since the Great Depression of the 1930s, resulted in a change of paradigms in the financial and banking sector. These crisis years saw collapses of large financial institutions, bailouts of banks by governments, and declines of stock markets. Triggered by the U.S. housing bubble, which itself was caused by giving easy access to loans for subprime borrowers, financial distress spread over the banking sector and led to failure of key businesses and to the 2008–2012 global recession. Finally, this also contributed to the European sovereign-debt crisis, with lots of aftereffects in our present times.

Uncertainties about bank solvency, declines in credit availability, and reduced investor confidence had an impact on global stock markets. Governments responded with fiscal measures and institutional bailouts, which in the long term resulted in extreme public debts and necessary tax increases.

This negative experience demonstrates that the economy as a whole, but especially the financial sector is subject to risks, which are grounded in the interdependencies between the different economic actors and not in the performance of individual actors. This type of risk is generally called systemic risk. While aspects of systemic risk (e.g., bank run and contagion) were always an issue in discussions about the financial system, the recent crises have increased the interest in the topic, not only in academic circles, but also among regulators and central banks.

SYSTEMIC RISK: DEFINITIONS

If one aims at measuring—and in a further step managing and mitigating—systemic risk, it is important to start with a definition. However, despite the consent that systemic risk is an important topic, which is reflected by an increasing number of related papers and technical reports, there is still not a single generally accepted definition.

As a first step, one should distinguish between systemic and systematic risk. Systematic risks are aggregate (macroeconomic) risks that cannot be reduced by hedging and diversification. Systemic risk, on the other hand, is a different notion. It refers to the risk of breakdown or at least major dysfunction of financial markets. The Group of Ten (2001) gave the following, often cited definition:

This formulation describes many aspects related to systemic risk but can hardly be called a definition in the technical sense, as it is very broad and hard to quantify. In addition, it seems to confuse cause (confidence) and consequence (breakdown).

As an alternative, Kaufmann and Scott (2003) introduced the following definition:

In similar manner, but naming the cause and again considering larger consequences, the European Central Bank (2004) defines systemic risk as follows:

All discussed definitions focus on the banking or financial system as a whole, and relate systemic risk to the interconnectedness within the system. Often, they stress the risk of spillovers from the financial sector to the real economy and the associated related costs. This effect is emphasized even more after the financial crisis as described in the following definition of systemic risk from Adrian and Brunnermeier (2009):

A similar definition can be found in Acharya et al. 2009.

Given the described diversity of definitions, which are similar but also different with respect to their focus, it is hard to develop universally accepted measures for systemic risk. Different definitions refer to different important nuances of systemic risk, which means that on the operational level a robust framework for monitoring and managing systemic risk should involve a variety of risk measures related to these different aspects. See Hansen (2012) for a deeper discussion of the basic difficulties in defining and identifying systemic risk.

We will focus on the first part of the definition by Kaufmann and Scott (2003), which summarizes the most important aspect of systematic risk in financial systems, without addressing more general economic aspects. Such an approach could be seen as “systemic risk in the narrow sense” and we state it (slightly modified) as follows:

Systemic risk is the risk of breakdowns in an entire system, as opposed to breakdowns in individual parts or components.

Three issues have to be substantiated, if one wants to apply such a definition in concrete situations: system, breakdowns, and risk.

System

In financial applications, the focus lies on parts of the financial system (like the banking system, insurance, hedge funds) or the financial system as a whole. Any analysis has to start with describing the agents (e.g., banks in the banking system) within the analyzed system. This involves their assets and liabilities and the main risk factors related to profit and loss.

For a systemic view, it is important that the agents are not isolated entities at all. Systematic risk can be modeled by joint risk factors, influencing all profit and losses. Systemic risk in financial systems usually comes by mutual debt between the entities and the related leverage.

Breakdowns

In single-period models, breakdown is related to bankruptcy in a technical sense—that is, that the asset value of an agent at the end of the period does not reach a certain level (e.g., is not sufficient to pay back the agents debt). A lower boundary than debt can be used to reflect the fact that confidence into a bank might fade away even before bankruptcy, which severely reduces confidence between banks. In a systemic view, it is not sufficient to look at breakdowns of individual agents: Relevant are events that lead to the breakdown of more than one agent.

Risk

Risk is the danger that unwanted events (here, breakdowns) may happen or that developments go in an unintended direction. Quantifiable risk is described by distributions arising from risk. For financial systems this may involve the probability of breakdowns or the distribution of payments necessary to bring back asset values to an acceptable level. Risk measures summarize favorable or unfavorable properties of such distributions.

It should be mentioned that such an approach assumes that a good distributional model for the relevant risk factors can be formulated and estimated. During this chapter, we will stick to exactly this assumption. However, it is clear that in practice it is often difficult to come up with good models, and data availability might be severely restricted. Additional risk (model risk) is related to the quality of the used models and estimations; see Hansen (2012) for a deeper discussion of this point.

FROM STRUCTURAL MODELS TO SYSTEMIC RISK

Structural models for default go back to Merton (2009) and build on the idea that default of a firm happens if the firm's assets are insufficient to cover contractual obligations (liabilities). Simple models such as Merton (2009) start by modeling a single firm in the framework of the Black–Scholes option pricing model, whereas more complex models extend the framework to multivariate formulations, usually based on correlations between the individual asset values. A famous example is Vasicek's asymptotic single factor model (see Vasicek 1987; 1991; and 2002), which is very stylized but leads to a closed-form solution.

In most structural default models, it is not possible to calculate the portfolio loss explicitly; hence, Monte Carlo simulation is an important tool for default calculations. Even then, the models usually make simplifying assumptions.

Consider a system consisting of k economic entities (e.g., banks), and let A₁(t), A₂(t), …, A_k(t) denote the asset processes—that is, the asset values at time t for the individual entities. Furthermore, for each entity i a limit D_i, the distress barrier, defines default in the following sense: default occurs if the asset value of entity i falls below the distress barrier:

The relation between asset value and distress barrier is usually closely related to leverage, the ratio between debt and equity.

Finally, let X₁(t), X₂(t), …, X_k(t) with

denote the distance to default of the individual entities. Note that alternatively the distance to default can also be defined in terms of X_i(t) as a percentage of asset value, divided by the asset volatility (see e.g., Crosbie and Bohn 2003).

In a one period setup—as used throughout this chapter—one is interested at values A_i(T), X_i(T) at time T, the end of the planning horizon. Analyzing systemic risk then means analyzing the joint distribution of the distances to default X_i(t), in particular their negative parts X_i(T)⁻ = max{−X_i(T), 0}, and the underlying random risk factors are described by the joint distribution of asset values A_i(T).

Many approaches for modeling the asset values exist in literature. In a classical finance setup, one would use correlated geometric Brownian motions resulting in correlated log-normal distributions for the asset values at the end of the planning horizon. Segoviano Basurto proposes a Bayesian approach (Segoviano Basurto 2006); for applications, see also Jin and Nadal de Simone (2013). In this chapter, we will use copula-based models, as discussed later.

The second component of the approach, the distress barrier, is in the simplest case (Merton 2009), modeled just by the face value of overall debt for each entity. Other approaches distinguish between short-term and long-term debt (longer than the planning horizon). Usually, this is done by adding some reasonable fraction of long-term debt to the full amount of short term debt; see, for example, Servigny and Renault (2007).

Still, such classical credit default models (see, e.g., Guerra et al. 2013), although classified as systemic risk models, neglect an important aspect: Economic entities like banks are mutually indebted, and each amount of debt is shown as a liability for one entity but also as an asset for another entity. Default of one entity (a reduction in liabilities) may trigger subsequent defaults of other entities by reducing their asset values. We call such models systemic models in the strict sense.

Such approaches with mutual debt have been proposed, such as in Chan-Lau et al. (2009a; 2009b). Models neglecting this aspect are systemic models in a broad sense; in fact, they are restricted to the effects of systematic risk related to asset values.

The basic setup of systemic models in the strict sense can be described as follows: Let denote the amount of debt between entities i and j —that is, the amount of money borrowed by entity i from entity j. We also include debt to the nonbank sector, denoted by H_i for each entity i and credit C_i to the nonbanking sector, both repayable (including interest) at the end of the planning horizon, time T. Furthermore, S_i(T) is the value at time T of other financial assets held by entity i. Then the asset value of entity i at the end of the planning horizon is given by

the distress barrier (in the simplest case) is

and the distance to default can be written as

The random factors are the values S_i(T) of financial assets, and (in an extended model) the credits C_i from outside the system, payable back at time.

Again, one could stop at this point and analyze the distances to default , respectively the sum of all individual distances to default in the framework of classical default models. Systemic models in the strict sense, however, go farther.

Consider now all entities in distress (defaulted banks), that is, . Each of these entities is closed down and the related debt has to be adjusted, because entity i cannot fully pay back its debts. In a simple setup, this can be done by reducing all debts to other entities as follows:

Here, the factor

is an estimate for the loss given default of entity i.

It is now possible to calculate new asset values, new distress barriers, and new distances to default, after the default of all entities in . For this purpose, we replace in (1.3) to (1.5) all occurrences of by and all occurrences of by . This first default triggers further ones and starts a loss cascade: It may happen that after the first adjustment step new defaults can be observed, which results in a new set of bankrupt entities after the second round. In addition, bankruptcy of additional entities may reduce even further the insolvent assets of entities that already defaulted in the first round.

This process can be continued, leading to new values , , , , and an augmented set of defaulted entities after each iteration k. The loss cascade terminates, when no additional entity is sent to bankruptcy in step k, that is, .

The sequences [] and [] of debt are nonincreasing in k and, furthermore, are bounded from below by zero values for all components, which implies convergence of debt. At this point we have

This system describes the relation between the positive and negative parts of the distances to default X_i(T) = X_i(T)⁺ − X_i(T)⁻ for all entities i. It holds with probability 1 for all entities. Note that previous literature, such as Chan-Lau et al. (2009a; 2009b), uses fixed numbers instead of the estimated loss given defaults in (1.9).

In fact, the system (1.9) is ambiguous, and we search for the smallest solution, the optimization problem

has to be solved in order to obtain the correct estimates for X_i(T)⁺ and X_i(T)⁻.

This basic setup can be easily extended to deal with different definitions of the distress barrier, involving early warning barriers, or accounting for different types of debt (e.g., short-term and long-term, as indicated earlier).

MEASURING SYSTEMIC RISK

The distances to default, derived from structural models, in particular from systemic models in the strict sense, can be used to measure systemic risk. In principle, the joint distribution of distances to default for all involved entities contains (together with the definition of distress barriers) all the relevant information. We assume that the joint distribution is continuous and let p(x) = p(x₁, x₂, …, x_k) denote the joint density of the distances to default X₁(T), X₂(T), …, X_k(T) for all entities.

Note that the risk measures discussed in the following are often defined in terms of asset value, which is fully appropriate for systemic models in the broader sense. In view of the previous discussion of systemic models in the strict sense, we instead prefer to use the distances to default or loss variables derived from the distance to default.

The first group of risk measures is based directly on unconditional and conditional default probabilities. See Guerra et al. (2013) for an overview of such measures. The simplest approach considers the individual distress probabilities

The term in squared brackets is the marginal density of X_i(T), which means that it is not necessary to estimate the joint density for this measure. In similar manner, one can consider joint distributions for any subset I ⊆ {1, …, k} of entities by using the related (joint) marginal density p_I(x), which can be obtained by integrating the joint density p(x) over all other entities, that is, j ∉ I.

Joint probabilities of distress for a subset I can be achieved by

where the set I contains the elements i₁, i₂, …, i_kI. Of special interest are the default probabilities of pairs of entities (see, e.g., Guerra, et al., 2013). Joint probabilities of distress describe tail risk within the chosen set I. If I represents the whole system (i.e., it contains all the entities), then the joint probability of distress can be considered as a tail risk measure for systemic risk (see, e.g., Segoviano & Goodhart, 2009).

Closely related are conditional probabilities of distress, that is, the probability that entity j is in distress, given that entity j is in distress, which can be written as

These conditional probabilities can be presented by a matrix with as its ijth matrix element, the distress dependency matrix.

While conditional distress probabilities contain important information, it should be noted that they only reflect the two-dimensional marginal distributions. Conditional probabilities are often used for analyzing the interlinkage of the system and the likelihood of contagion. However, such arguments should not be carried to extremes. Finally, conditional probabilities do not contain any information about causality.

Another systemic measure related to probabilities is the probability of at least one distressed entity; see Segoviano and Goodhart (2009) for an application to a small system of four entities. It can be calculated as

Guerra et al. (2013) propose an asset-value-weighted average of individual probabilities of distress as an upper bound for the probability of at least one distressed entity. Probabilities of exactly one, two, or another number of distressed entities are hard to calculate for large systems because of the large number of combinatorial possibilities.

An important measure based on probabilities is the banking stability index, measuring the expected number of entities in distress, given that at least one entity is in distress. This measure can be written as

Other systemic risk measures based directly on the distribution of distances to default. Adrian and Brunnermeier (2009) propose a measure called conditional value at risk,¹ CoVaR⁼. It is closely related to value at risk, which is the main risk measure for banks under the Basel accord.

CoVaR⁼ is based on conditional versions of the quantile at level β for an entity j given that entity i reaches the α-quantile. In terms of distances to default, this reads

where

The contribution of entity i to the risk of entity j then is calculated as

That is, the conditional value at risk at level α is compared to the conditional value at risk at the median level. From all the ΔCoVaR_α⁼(j|i) values, it is possible to construct another kind of dependency matrix.

This idea can also be applied to the system as a whole: If X_j(T) is replaced by , the distance to default of the whole system, (1.16) to (1.18), leads to a quantity ΔCoVaR_α⁼(j | i) that measures the impact of entity i on the system. In this way one is able to analyze notions like “too big to fail” or “too interconnected to fail.”

In contrast to probability-based measures, CoVaR emphasizes the role of potential monetary losses. This approach can be carried forward, leading to the idea that systemic risk should be related to the losses arising from adverse events. Given a model for the distances to default X_i(T), the overall loss of the system can be written as

L^tot covers all credit losses in the whole system, both from interbank credits and from credits to the public.

From the viewpoint of a state, this notion of total loss may be seen as too extensive. One may argue that only losses guaranteed by the state are really relevant. Definition (1.19) therefore depicts a situation in which a state guarantees all debt in the system, which can be considered as unrealistic. However, in most developed countries, the state guarantees saving deposits to a high extend, and anyhow society as a whole will have to bear the consequences of lost debt from outside the banking system. Therefore, a further notion of loss is given by

which describes the amount of lost nonbanking debt For the structural model, which has been described in the previous section, loss given default can be calculated using (1.8) and (1.9).

In general, the notion of loss depends on the exact viewpoint (loss to whom). We will therefore use the symbol L to represent any kind of loss variable in the following discussion of systemic risk measures.

An obvious measure is expected loss—that is, the (discounted) expectation of the risk variable L. For simple structural models like (1.2), this measure can be calculated from the marginal distribution of asset values, respectively, of distances to default. Modeling the joint distributions is not necessary. Note that this is different for the strict systemic model (1.9).

The expectation can be calculated with respect to an observed (estimated) model, or with respect to a risk-neutral (martingale) model. Using observed probabilities may account insufficiently for risk, which contradicts the aim of systemic risk measurement. Using risk-neutral valuation seems reasonable from a finance point of view and has been used, for example, in Gray and Jobst (2010) or Gray et al. (2010). However, it should be kept in mind that the usual assumptions underlying contingent claims analysis—in particular, that the acting investor is a price taker—are not valid if the investor has to hedge the whole financial system, which clearly would be the case when hedging the losses related to systemic risk.

Using expectation and the concept of loss cascades, Cont et al. (2010) define a contagion index as follows: They define first the total loss of a loss cascade triggered by a default of entity i and the contagion index of entity i as the expected total loss conditioned on all scenarios that trigger the default of entity i.

Clearly, the expectation does not fully account for risk. An obvious idea is to augment expectation by some risk measure ρ, which, with weight a, leads to

Typical choices of ρ are dispersion measures like the variance or the standard deviation. Such measures are examples of classical premium calculation principles in insurance. Further, more general premium calculation principles are for example, the distortion principle or the Esscher premium principle. For an overview on insurance pricing, see Furmann and Zitikis (2008). In the context of systemic risk, the idea to use insurance premiums was proposed in Huang et al. (2009). In this chapter, empirical methods were used for extracting an insurance premium from high-frequency credit default swap data. Even more generally, it should be noted that any monetary risk measure—in particular, coherent measures of risk—can be applied to the overall loss in a system. See Kovacevic and Pflug (2014) for an overview and references.

In this broad framework, an important class of risk measures is given by the quantiles of the loss variable L:

With probability α, the loss will not be higher than the related quantile.

Quantiles are closely related to the value at risk (VaR), which measures quantiles for the deviation of the loss from the expected loss. Note the slight difference between (1.22) and (1.17), because (1.17) is stated in terms of distance to default and (1.12) in terms of loss.

Q_α(L) can also be interpreted in an economic way, as follows. Assume that a fund is built up in order to cover systemic losses in the banking system. If we ask how large the fund should be, such that it is not exhausted, with probability α over the planning period, then the answer will be Q_L(α). This idea can also be reversed. Assume now that a fund of size q has been accumulated to deal with systemic losses. Then the probability that the fund is not exhausted,

is a reasonable systemic risk measure. Clearly, F_L(q) is the distribution function of the loss, and q is the quantile at level F_L(q).

Unfortunately, quantiles do not contain any information about those (1 − α)100 percent cases, in which the loss lies above the quantile. Two different distributions, which are equal in their negative tails, but very different in the positive tails, are treated equally.

The average value at risk (AVaR) avoids some drawbacks of quantiles. It is defined for a parameter α, which again is called level. The AVaR averages the bad scenario,

The latter formula justifies the alternative name conditional value at risk (CVaR), which is frequently used in finance. In insurance, the AVaR is known as conditional tail expectation or expected tail loss.

The effect of individual banks can be analyzed in obvious manner by defining conditional versions of the quantile or AVaR loss measures—that is, by conditioning the overall loss on the distance to default of an individual bank in the style of CoVaR; see (1.16).

SYSTEMIC RISK AND COPULA MODELS

The distinction between risk factors that are related to individual performances and risk factors that are a consequence of the interrelations of the economic agents has its parallel in a similar distinction for probability distributions or stochastic processes:

Suppose that X₁(t), …, X_k(t) describe the performance processes of k economic agents. The individual (marginal) processes are assumed to follow certain stochastic models as discrete Markov processes, diffusion models, or jump-diffusion models. The joint distribution, however, depends on the copula process, which links the marginal processes.

To simplify, suppose only a single-period model is considered and that the performance after one period is X₁,…, X_k. If this vector has marginal cumulative distribution functions F₁,…, F_k (meaning that P(X_i ≤ u) = F_i(u)), then the joint distribution of the whole vector can be represented by

where C is called the copula function. Typical families of copula functions are the normal copula, the Clayton copula, the Gumbel copula or—more generally—the group of Archimedean copulas.

While the marginal distributions describe the individual performances, the copula function models the interrelations between them and can thus be seen as representing the systemic component. In particular, the relation between underperformance of agent i and agent j can be described on the basis of the copula. To this end, we use the notion of conditional value at risk (CoVaR), as already described. Following Mainink and Schaaning (2014) we use the notations

for the notion of CoVAR introduced by Adrian and Brunnermeier (2009) and

for the variant introduced by Girardi and Ergün (2012). Keep in mind that we work here with profit and loss variables and not with pure loss variables. The latter variant can be expressed in terms of the conditional copula

Its inverse

and the marginal distribution of X can be used to write the CoVaR in the following way:

For the , the conditional copula

is needed. With

one gets

Notice that both notions of CoVaR depend only on the copula and the marginal distribution of X.

If underperformance of an agent means that its performance falls below an α-quantile, then indicates the probability that the other agent also underperforms. CoVaR_α,α(X_i|X_j) gives the necessary risk reserve for agent i to survive a possible default of agent j. For a system of k agents, the notion of CoVaR_α,β can be generalized in a straightforward manner to k components.

Example 1. Consider a financial institution A, which faces a gamma-distributed loss with mean 10 and variance 20. Then A's unconditional 99% VaR is 23.8.

If A's performance related to B's performance with a normal copula with correlation ρ, then A's conditional VaR (the CoVaR) increases with increasing ρ, see Table 1.1.

Example 2. Consider a system of seven banks, where the performances X_i, i = 1, …, 7 are related by a normal copula stemming from a correlation matrix with all off-diagonal elements ρ (the diagonal elements are 1). Suppose that the first bank defaults if its performance drops below the 5 percent quantile. Given the copula, one may determine the number of other banks that also fall below the 5 percent quantile (i.e., default as a consequence of the first bank's default). Figures 1.1 and 1.2 show the distribution of these numbers for the choice of ρ = 0, ρ = 0.2, ρ = 0.5 and ρ = 0.8. One may observe that in the independent case (ρ = 0) the other banks are practically not affected by the default of one bank, while for higher correlated cases a contagion effect to other banks can be easily seen.

A very interdependent banking system carries a high systemic risk. It has therefore been proposed to limit the dependencies by creating quite independent subsystems. Example 3 gives evidence for this argument.

Example 3. Here, we consider seven banks, each of which has a performance given by a negative gamma distribution with mean 100 and variance 200, but shifted such that with probability 5 percent a negative performance happens, which means bankruptcy. The total losses of the system are calculated on the basis of a normal copula linking the individual losses. By assuming that the government (or the taxpayer) takes responsibility for covering total losses up to the 99 percent quantile, this quantile (the 99 percent VaR) can be seen as a quantization of the systemic risk.

In Figures 1.3 to 1.6, we show in the upper half a visualization of the correlations (which determine the normal copula) by the thickness of the arcs connecting the seven nodes representing the banks. The lower half shows the distribution of the total systemic losses, where also the 99 percent VaR is indicated. As one can see, the higher correlation increases the systemic risk. If the system is divided into independent subsystems, the systemic risk decreases.

CONCLUSIONS

Systemic financial risk is an important issue in view of the distress the banking systems all over the world have experienced in the recent years of crises. Even if breakdowns are prevented by the government, the related societal costs are extremely high.

We described the measurement of systemic risk, based on the structural approach originating from structural credit risk models. In particular, the cascading effects that are caused by mutual debt between the individual banks in the system were analyzed in detail. Furthermore, we related the notion of systemic risk to the copula structure, modeling dependency between the performances of the individual banks. The effects of different levels of dependency on the total systemic risk in terms of the value at risk of total losses were demonstrated by examples.

REFERENCES

Acharya, V., L. Pedersen, T. Phillipon, and M. Richardson. 2009. Regulating systemic risk. In Restoring Financial Stability: How to Repair a Failed System. Hoboken, NJ: John Wiley and Sons.

Adrian, T., and M. K. Brunnermeier. 2009. CoVar. In: Staff Report 348: Federal Reserve Bank of New York.

Chan-Lau, J., J. M. Espinosa-Vega, and J. Sole. 2009a. On the use of network analysis to assess systemic financial linkages. Washington, D.C.: International Monetary Fund, IMF.

Chan-Lau, J., M. A. Espinosa-Vega, K. Giesecke, and J. Sole. 2009b. A. Assessing the systemic implications of financial linkages. In: Global Financial Stability Report. Washington, D.C.: International Monetary Fund, IMF.

Cont, R., A. Moussa, and E.e.S. Bastos. 2010. Network structure and systemic risk in banking systems, s.l.: Preprint, electronic copy available at http://ssrn.com/abstract=1733528.

Crosbie, P. and J. Bohn. 2003. Modeling default risk: Moody's KMV.

European Central Bank. 2004. Annual Report, Frankfurt, available at http://www.ecb.europa.eu/pub/pdf/annrep/ar2004en.pdf.

Furmann, E., and R. Zitikis. 2008. Weighted premium calculation principles. Insurance, Mathematics and Economics, 459–465.

Girardi, G., and T. Ergün. 2012. Systemic risk measurement: Multivariate GARCH estimation of CoVaR. available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1783958.

Gray, D. F., A. A. Jobst, and S. W. Malone. 2010. Quantifying systemic risk and reconceptualizing the role of finance for economic growth. Journal of Investment Management 8(2).

Gray, D., and A. A. Jobst. 2010. New directions in financial sector and sovereign risk management. Journal of Investment Management 8(1).

Group of Ten. 2001. The G10 Report on Consolidation in the Financial Sector, Chap. 3, http://www.imf.org/external/np/g10/2001/01/Eng/pdf/file3.pdf

Guerra, S. M., B. M. Tabak, R. A. Penaloza, and R. C. de Castro. 2013. Systemic Risk Measures. Working paper 321, Banco do Brasil, http://www.bcb.gov.br/pec/wps/ingl/wps321.pdf [Online].

Hansen, L. P. 2012. Challenges in identifying and measuring systemic risk, s.l.: National Bureau of Economic Research.

Huang, X., H. Zhou, and H. Zhu. 2009. A framework for assessing the systemic risk of major financial institutions. Journal of Banking and Finance 33: 2036–2049.

Jin, X., and F. Nadal de Simone. 2013. Banking Systemic Vulnerabilities: A Tail-Risk Dynamic CIMDO Approach. Banque centrale de Luxembourg.

Kaufmann, G. G., and K. E. Scott. 2003. What is systemic risk, and do bank regulators retard or contribute to it? Independent Review 7: 371–391.

Kovacevic, R., and G. Ch. Pflug. 2014. Measuring and Managing Risk. Chapter 2 In: Investment Risk Management, edited by K. Baker and G. Filbeck, Oxford University Press, Oxford, UK.

Mainink, G., and E. Schaaning. 2014. On dependence consistency of CoVaR and some other systemic risk measures. Statistics and Risk Modeling.

Merton, R. C. 2009. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. In: Continuous Time Finance (first published 1992). Wiley-Blackwell Publishing, New York, 388–412.

Segoviano Basurto, M. A. 2006. Consistent information multivariate density optimizing methodology, s.l.: London School of Economics, Discussion Paper 557.

Segoviano, M. A., and C. Goodhart. 2009. Banking Stability Measures. Washington, D.C.: International Monetary Fund.

Servigny, O. D., and O. Renault. 2007. Measuring and Managing Credit Risk. New York: McGraw-Hill.

Vasicek, O. 1987. Probability of loss on loan portfolios. K.M.V. Corporation 12(6).

_________. 1991. Limiting loan loss distribution. K.M.V. Corporation.

_________. 2002. Loan portfolio value. Risk, available at www.risk.net. December: 160–162.