An equilibrium-based measure of systemic risk

Ivanov, Katerina; Schulte, James; Tian, Weidong; Tseng, Kevin Article An equilibrium-based measure of systemic risk Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Ivanov, Katerina; Schulte, James; Tian, Weidong; Tseng, Kevin (2021) : An equilibrium-based measure of systemic risk, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 14, Iss. 9, pp. 1-24, https://doi.org/10.3390/jrfm14090414 This Version is available at: https://hdl.handle.net/10419/258518 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article An Equilibrium-Based Measure of Systemic Risk Katerina Ivanov 1, James Schulte 2, Weidong Tian 3,* and Kevin Tseng 4   Citation: Ivanov, Katerina, James Schulte, Weidong Tian, and Kevin Tseng. 2021. An Equilibrium-Based Measure of Systemic Risk. Journal of Risk and Financial Management 14: 414. https://doi.org/10.3390/jrfm14090414 Academic Editor: Shigeyuki Hamori Received: 19 July 2021 Accepted: 24 August 2021 Published: 2 September 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 McColl School of Business, Queens University of Charlotte, Charlotte, NC 28274, USA; [email protected] 2Department of Economics, Florida State University, Tallahassee, FL 32306, USA; [email protected] 3Belk College of Business, University of North Carolina at Charlotte, Charlotte, NC 28223, USA 4College of Management, National Taiwan University, Taipei 10617, Taiwan; [email protected] *Correspondence: [email protected] Abstract: This paper develops and implements an equilibrium model of systemic risk. The model derives a systemic risk measure, loss beta, in characterizing all too-big-to-fail banks using a capital insurance equilibrium. By constructing each bank’s loss portfolio with a recent accounting approach, we perform a comprehensive empirical study of this loss beta measure and document all TBTF banks from 2002 to 2019. Our empirical findings suggest a significant number of too-big-to-fail banks in 2018–2019. Keywords: systemic risk; too big to fail; capital insurance; loss beta JEL Classification: G11; G12; G13 1. Introduction The financial crisis of 2007–2009 generates a significant amount of interest in measuring systemic risk and ensuring the health of the financial system. The most vivid response is the passing of the Dodd-Frank Act. Measuring systemic risk has been at the center of academic research since 2008. For example, Acharya (2009), Acharya et al. (2012), and Brownless and Engle (2016) show that time-varying correlation structure plays a crucial role in their systemic risk measurements and develop an expected shortfall measure approach. Adrian and Brunnermeier (2016) develop a CoVaR approach conditional on financial institutions being in a state of financial distress. Acemoglu et al. (2015), and Elliott et al. (2014) develop a network approach for systemic risk. 1 While these quantitative and statistical metrics are straightforward to apply, the precise economic channel to which the systemic risk enters the metrics is somewhat lacking. Moreover, Benoit et al. (2019) identify several shortcomings in the systemic-risk scoring methodology. From an insurance equilibrium perspective, Panttser and Tian (2013) and Ivanov (2017) present a capital insurance approach to address systemic risk and identify too-big-to-fail (TBTF) banks. Developing a rational expectation equilibrium model of the capital insurance market to address systemic risk was initially proposed in Kashyap et al. (2008). In the equilibrium model of capital insurance, a central player (regulator or insurer) sells an insurance product, capital insurance, on the systemic risk of all banks. The insurer injects the guaranteed capital contingent upon a stressed period, while each bank pays insurance premium upfront in exchange for an implicit guarantee subsidy. In equilibrium, each bank predicts the optimal insured amount and the insurer determines the optimal pricing structure. As a result, those banks purchasing capital insurance are characterized as TBTF banks under this approach, so TBTF banks are identified endogenously. In characterizing the equilibrium, this approach introduces a new equilibrium-based systemic risk measure, loss beta. Compared with numerous systemic risk measures in previous literature, the equilibriumbased systemic risk measure, such as loss beta, is unique since it captures the perspective from all banks together as well as an issuer about each bank’s systemic risk. Moreover, J. Risk Financial Manag. 2021,14, 414. https://doi.org/10.3390/jrfm14090414 https://www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2021,14, 414 2 of 24 both the implicit guarantee subsidy and TBTF banks are determined endogenously and simultaneously. 2 Since the framework and its pricing structure are motivated by a classical insurance setting, the capital insurance equilibrium is similar to classical insurance equilibrium models such as Borch (1962), Arrow (1971), and Raviv (1979) for a new type of insurance product on the systemic risk. Nevertheless, the strong law of large numbers is no longer satisfied for capital insurance because of the particular correlated feature of systemic risk among banks (Acharya et al. (2012) and Adrian and Brunnermeier (2016)). This paper extends the equilibrium analysis in Panttser and Tian (2013) and Ivanov (2017) in several theoretical and empirical aspects. First, this paper characterizes the equilibrium for a general class of capital insurance. We focus on identifying TBTF banks, whereas Panttser and Tian (2013) study the welfare analysis of capital insurance as a financial innovation. Second, and more importantly, we introduce a new methodology to construct an individual bank’s loss portfolio with systemic risk components consistent with Basel documentations. 3 Building on the deposit insurance model developed in Diamond and Dybvig (1983) and Peck and Shell (2003), we construct an appropriate loss portfolio with systemic risk exposure by modifying a recent accounting model in Atkeson et al. (2019) for commercial banks. Using deposit, loan, equity, and debt together, we construct a loss portfolio in every period and the time series of loss beta for each bank. In contrast, an assetequity leverage ratio is used in Ivanov (2017) to construct the asset loss portfolio, but other specific information for commercial banks is not incorporated. Finally, we implement the proposed approach for U.S. banks and provide a comparison to the previously developed systemic risk measures. Specifically, there are three significant contributions in this paper. First, we characterize the equilibrium of general capital insurance for the entire banking sector. This equilibrium model helps us identify TBTF and how much premium TBTF banks should pay upfront to hedge the systemic risk. Since our model is building on the insurance principle, it complements the equilibrium asset-pricing model of systemic risk in Allen and Gale (2000). At the same time, since our model focuses on the systemic risk only, we can construct an equilibrium-based systemic risk measure and identify TBTF simultaneously, which is not addressed in Allen and Gale (2000). Second, we propose a new methodology to construct the loss portfolio and use this loss portfolio to develop an equilibrium systemic risk measure—bank loss beta. Given each bank’s loss portfolio, a bank’s loss beta is a ratio of the covariance between a bank’s loss portfolio with the aggregate loss portfolio in the entire banking sector to the variance of the aggregate loss portfolio. The innovation of this methodology is to use all deposit, loan, and equity and debt information that is important to analyze a commercial bank’s systemic risk exposure. Third, we implement the equilibrium approach for U.S. banks. 4 Using the “Call Report" of all U.S. banks with assets over $50 billion in each quarter from 2002 to 2019, we identify TBTF banks in each quarter. Our empirical findings suggest highly concentrated TBTF banks in the financial crises period and a significant number of TBTF banks starting from 2019Q1. We also calculate CoVaR simultaneously and find a positive and significant relationship between CoVaR and loss beta for a comparative purpose. As a comparison, we further implement the asset loss beta approach in Ivanov (2017) for all institutions with a Standard Industrial Classification (SIC) code between 6000 and 6499 and assets over $50 billion from quarter 1 of 2002 to quarter 2 of 2019. In our empirical findings, both approaches to constructing the loss portfolio generate a reasonably consistent group of TBTF banks during the same period, but the accounting approach covers a much larger group of commercial banks. Overall, the empirical implementation and comparison between these two approaches support the idea of using capital insurance to identify TBTF banks. The paper contributes to the literature on TBTF banks and identifies such banks through a new equilibrium approach. For instance, in the network approach to systemic risk of Acemoglu et al. (2015) and Elliott et al. (2014), the connectedness amongst the banks J. Risk Financial Manag. 2021,14, 414 3 of 24 plays a key role. In other words, only the most relevant banks affect a bank’s systemic risk. By contrast, whether one bank is TBTF in our approach relies on all loss betas of financial institutions in the market. Therefore, the capital insurance approach highlights the relative weight of loss beta in the whole financial system. In Acharya et al. (2012) and Adrian and Brunnermeier (2016), specific distribution-based measures are calculated and sorted, so all banks’ systemic risks are compared with each other. In our approach, we use the loss beta as a criterion to measure and sort the systemic risk, so this approach provides an intuitive and robust way to measure the systemic risk in the spirit of CAPM. Eisenberg and Noe (2001) consider each bank’s cash flow and outflow and study the clearing payment to ensure each system member clears the obligation (not default). We also use the deposit, loan, debt, and equity information in our approach, but this information is used to construct a loss portfolio with systemic risk exposure. Instead of examining the clearing payment vector (with a topology method) in Eisenberg and Noe (2001), we characterize the equilibrium of systemic risk based on economic and finance principles. Since we study the appropriate amount of guarantee subsidy in exchange for a government bailout (injection), this paper also complements a recent paper by Berndt et al. (2021) on TBTF banks. Statistically speaking, it is impossible to estimate the government bailout probability since Lehman Brothers might be the only large bank that has not been bailed out. Therefore, Berndt et al. (2021) develop a structural model to compute a government bailout’s market-implied (risk-neutral) probabilities, and these authors find a significant post-Lehman reduction in market-implied probabilities of government bailout for both globally systemically important banks (GSIBs) in the U.S. and other large banks that are not large enough to be classified as GSIBs. 5 Kelly et al. (2016) also demonstrate the collective government guarantee for the financial sector using options data on the financial sector index and individual banks. Because Kelly et al. (2016) use the options data, the government guarantee is also interpreted under the risk-neutral probability measure. By contrast, our approach builds on the insurance principle to compute the insurance premium under the real-world expectation of the loss portfolio. In our approach, we do not estimate the real-world bailout probability directly. However, since the firm should pay a premium to buy insurance from the government in exchange for the government bailout, the capital insurance premium is essentially the expected bailout amount, a product of the bailout exposure and the bailout probability under the real-world probability measure. Like Berndt et al. (2021), we find that the number of TBTF banks gradually decreases post-Lehman until 2019.6 The paper proceeds as follows. Section 2introduces a simple accounting model to construct each bank’s loss portfolio with a systemic risk component. In Section 3, we characterize the capital insurance equilibrium and derive a general property about the loss beta measure. Finally, we report our empirical results in Section 4, and Section 5concludes. Appendix Apresents several properties of general capital insurance. 2. Bank’s Loss Portfolio In this subsection, we explain bank’s loss portfolio in a Diamond-Dybvig framework (Diamond and Dybvig (1983) and Peck and Shell (2003)). We start with a motivated example. Then we present its formal description. 2.1. A Motivated Example Consider a bank with equity investment $100 at time zero. There are two future times t= 1 and t= 2. There are identical households who are ex-ante identical and endowed with $1 at time t= 0. We assume that there are in total 800 such households to deposit in the bank, thus the bank has $800 of short-term debt (deposit) at time t= 0. In addition, the bank issues a subordinated debt of face value $100 at maturing time t= 2 and the coupon rate is 5%. 7 On the other hand, the bank loans $1000 to long-term borrowers with random gross return ˜ R1 in the first time period (if liquid at t= 1) and random gross return ˜ R2 at time t= 2. Depositors have no risk because of deposit insurance issued by the Federal J. Risk Financial Manag. 2021,14, 414 4 of 24 Deposit Insurance Corporation (FDIC). For simplicity, we assume that the deposit cost (including the deposit insurance premium and operational cost) for the bank is zero, and the deposit interest rate is also zero. There are two types of depositors in the presence of idiosyncratic uncertainty. The first type of depositor needs to consume at time t= 1 while the second type of depositor only withdraws and consumes at t= 2. The depositor’s type is revealed in t= 1 in equilibrium. Since we focus on systemic risk in the banking system, we assume the withdraw probability is λ= 0.4 at time t= 1 in a good (Diamond-Dybvig) equilibrium, and there is no bank run equilibrium in the presence of deposit insurance. At time t= 1, $320 (= λ× 800) are withdrawn and a $5 coupon payment for the subordinated debt is paid. We assume that the prepaid probability is µ and no loan defaults; thus, the cash inflow is 1000µ˜ R1from the loan (project). Let Y1=1000µ˜ R1+E1−325, where E1 is the market price of the equity at time t= 1. Y1 presents a “profit and loss (P & L) portfolio” including the liquid common equity asset. If Y1> 0, there exists sufficient cash inflow to cover the cash outflow, thus no systemic risk to the economy. If, however, 1000 µ˜ R1+E1≤ 325 at time t= 1, the bank might sell the long-term illiquid asset (and could be subject to asset fire-sales) to avoid the default. In this situation, the total sum of 1000 µ˜ R1 and the fair value (at time t= 1) of the future payment 1000(1−µ)˜ R2 is the “fair value” of the long-term loan. Similarly, the subordinated debt obligation is represented by the market value of the debt. Therefore, we replace the P & L portfolio Y1by Z1=FL1+E1−FD1−B1 where •FL1is the fair value of the long-term loan at time t=1; •FD1is the fair value of the deposit at time t=1; •B1is the market price of the subordinated debt. If Z1 is positive, the bank does not generate systemic risk to the economy since the total asset value is greater than the total obligation to both short-term and long-term debt holders. On the other hand, if Z1 is negative, since the asset value is not sufficient to meet its obligation to both depositor (short-term debt holder) and long-term debt holder, the bank not only defaults but also generates systemic risk to the economy. Therefore, the bank’s loss portfolio is characterized as L1=max(−Z1, 0), the maximum between negative Z1and zero. Similarly, let Y2=1000(1−µ)˜ R2+E2−580 −(100 +5), where E2 is the market price of the equity at time t= 2. Conditional on no-default in the previous time period, we notice that Y2=Z2=FL2+E2−FD2−B2 where FL2= 1000 ( 1 −µ)˜ R2 is the fair value of the loan, FD2=$ 580 is the fair value of the deposit and B2=$ 105 is the market price of the subordinated debt at time t= 2. Hence, the banks’ loss portfolio at time 2 is L2=max(−Z2, 0). J. Risk Financial Manag. 2021,14, 414 5 of 24 2.2. Construction of Bank’s Loss Portfolio In this subsection we define the bank’s loss portfolio by modifying a recent accounting model in Atkeson et al. (2019). The construction extends the idea in the last motivated example. For simplicity, we assume the Treasury interest rate is a constant r. Specifically, D denotes the total face value (or the book value) of the deposit on the bank’s balance sheet. In each time period, every dollar of deposits costs the bank cD , including the interest paid on deposit, the serving cost and the deposit insurance premium paid to FDIC. The deposit is withdrawn with probability, µD , which depends on the depositor’s type in Diamond-Dybvig’s framework. Since a depositor’s withdrawal decision relies on an idiosyncratic shock, his type is revealed upon new information, so the probability µD is a random variable at each time period. The fair value of the debt is denoted by FDtat time t. Following Atkeson et al. (2019), the fair value FDtis given by8 FDt=cD+EQ t[µD,t+1] r+EQ t[µD,t+1]D(1) where EQ t[·] denotes the conditional expectation operator under a risk-neutral probability measure. On the other hand, L denotes the total face value of the loan. In every time period, every dollar of loan pays a coupon cL , net of serving cost. The fair value of the loan depends on the prepaid probability µL , and the loan default probability δL on the face value. By a similar derivation, the fair value of the loan at time tis 9 FLt=cL+EQ t[µL] r+EQ t[µL] + EQ t[δL]L. (2) The bank issues a subordinated debt as well as the equity. At each time t , the common equity’s market value is denoted by Et , and the market value of the subordinated debt is written as Bt . By the discussion in the motivated example, the bank’s P & L portfolio at time tis Zt=FLt+Et−FDt−Bt, (3) and the bank’s loss portfolio at time tis Lt=max{FDt+Bt−FLt−Et, 0}. (4) According to its definition, Lt represents the bank’s loss exposure to systemic risk. To see it, the loss exposure is positive if any only if FDt+Bt>FLt+Et. (5) There are two terms on the left side of the last formula, denoting the obligation to short-term debt holders (depositor) and long-term debt holders, while the right side is a sum of the (liquid asset) common equity and (illiquid asset) loan. Regardless depositors or debt holders, as long as the bank’s obligation is greater than the total asset value, the bank endures a loss to the economy; thus a positive loss exposure Lt. We notice that Atkeson et al. (2019) consider a representative bank with an additional loan-making arm and the deposit-taking arm. Then the government guarantee is one component in the bank market-to-book ratio. In contrast, we consider a group of heterogenous banks and we argue that the bank’s systemic risk and the government (implicit) guarantee are driven by the “correlated structure” of all banks’ loss portfolios, in a theory of capital insurance below. J. Risk Financial Manag. 2021,14, 414 6 of 24 3. Loss Beta Measure In this section, we introduce an equilibrium-based loss beta measure of a bank’s systemic risk. We present an equilibrium model of a capital insurance market and characterize TBTF banks in this equilibrium approach. Our classification of TBTF relies on equilibrium-based loss beta measures of all banks simultaneously in the banking system. We further discuss several examples of the loss beta measures. 3.1. An Equilibrium Model Following Panttser and Tian (2013) and Ivanov (2017), we present a model of capital insurance market. There are N financial institutions, namely banks, indexed by i= 1, · · · , N , in a financial sector. Each bank is endowed with a loss portfolio (or exposure, and we do not distinguish between these two concepts in this paper), L1 , · · · , LN , respectively. The aggregate loss portfolio is L=L1+· · · +LN. (6) While our empirical results are based on the bank loss portfolio introduced in the last section, we highlight that the loss portfolios are merely inputs in the presented equilibrium model. To hedge the potential loss, each bank decides whether or not to purchase a capital insurance contract from an insurance company, which determines the insurance premium. The insurance contract is written on the aggregate loss portfolio L instead of individual loss to each bank. Specifically, the payoff of one unit of the insurance payoff is written as I(L) , where I(·) is a continuous monotonic increasing function. 10 Thus, the insurance company and all banks are the market participants in the capital insurance market, in which the equilibrium is obtained if all participants achieve their highest expected utilities, respectively. Following the standard insurance equilibrium literature (Arrow (1971) and Raviv (1979)), the insurance premium for each unit is P= ( 1 +ρ)E[I(L)] , where ρ is a load factor that is determined by the issuer. Given the premium structure, each bank i decides how much insurance to purchase, namely, aiI(L) , where ai≥ 0, for a premium aiP accordingly. A bank decides the percentage coefficient ai to optimize its expected utility. At the same time, the insurance company determines the load factor, i.e., the premium structure, by the market demand aiI(L) from each bank i . Therefore, {ρ , ai , i= 1, · · · , N} are determined in this equilibrium. Specifically, we assume that each bank is risk-averse, and its risk preference is represented entirely by the mean and the variance of the wealth with the reciprocal of risk aversion parameter A> 0. 11 Given a premium structure ρ , bank i solves an optimal portfolio problem by choosing the best coinsurance coefficient: max {ai≥0}E[˜ Wi]−1 2AVar(˜ Wi), (7) where ˜ Wi=Wi 0−Li+aiI(L)−( 1 +ρ)E[aiI(L)] is the ex post terminal wealth for the bank i after purchasing the capital insurance and Wi 0 is the initial wealth of bank i . Similarly, Wi=Wi 0−Lirepresents the ex ante wealth of bank ibefore buying capital insurance. By the first-order condition in (7), the optimal coinsurance coefficient for bank i is given by ai(ρ) = maxCov(Li,I(L)) −ρE[I(L)]A Var(I(L)) , 0. (8) The issuer is assumed to be risk-neutral. 12 Therefore, the expected terminal wealth of the issuer is Wr= N ∑ i=1 (1+ρ)E[aiI(L)]− N ∑ i=1 aiI(L)− N ∑ i=1 c(aiI(L)), (9) J. Risk Financial Manag. 2021,14, 414 7 of 24 where c(·) denotes the issuance cost, which can be a fixed cost, a constant percentage of the indemnity or a general function of the indemnity. To focus on the equilibrium analysis of TBTF, we assume that the regulatory cost is a constant for each bank. Given the optimal demand ai(ρ) for each bank with a load factor ρ in (8), the issuer maximizes the expected welfare E[Wr] . By plugging Equation (8) into Equation (9), the issuer’s optimal load factor is derived from the following optimization problem: max {ρ>0}ρ N ∑ i=1 maxCov(Li,I(L)) −ρAE[I(L)] Var(I(L)) , 0(10) and the optimal coinsurance coefficient for each bank i= 1, · · · , N is given by ai(ρ∗) , where ρ∗ is the optimal load factor in (10). As a result, the capital insurance’s payoff for each bank i , ai(ρ∗)I(L) , relies on both demand (from all banks) and supply (from the regulator) in a rational expectation equilibrium. Both the optimal ai(ρ∗) and ρ∗ are determined endogenously. Bank iis TBTF under the capital insurance I(L)if and only if ai(ρ∗)>0. Equation (10) is first discussed and solved for particular capital insurance contract in Panttser and Tian (2013) and Ivanov (2017). We next discuss in detail how to determine TBTF banks in this equilibrium approach for general capital insurance. 3.2. Characterizing TBTF Banks In this subsection, we characterize the equilibrium for general capital insurance. Our new result is to demonstrate that this approach captures each bank’s systemic risk to some extent (Proposition 1) for a general capital insurance contract. We first assume that all banks are TBTF under this approach. Then, Equation (10) becomes max {ρ>0}ρ N ∑ i=1 Cov(Li,I(L)) −ρAE[I(L)] Var(I(L)) , yielding the optimal load factor13 ˆ ρ≡∑N i=1Cov(Li,I(L)) 2ANE[I(L)] =Cov(L,I(L)) 2ANE[I(L)] >0. Since ai(ˆ ρ) must be positive, for each i , the covariance between the loss Li with I(L)satisfies Cov(Li,I(L)) >Cov(L,I(L)) 2N. Clearly, if all covariances Cov(Li , I(L)) are sufficiently close so that each of them is greater than the half of their average, the last condition is satisfied, and the optimal load factor is ˆ ρ. In particular, if Cov(Li,I(L)) = Cov(Lj,I(L)),∀i6=j, all banks are TBTF. From the above analysis, we see that the covariance Cov(Li , I(L)) provides some insights about the bank’s systemic risk. More importantly, the TBTF concept depends on all covariances Cov(Li , I(L)) , instead of on individual bank’s loss portfolio. Indeed, by virtue of Equation (8), bank iis too-big-to-fail as long as Cov(Li,I(L)) >ρ∗AE[I(L)]. (11) But the optimal load factor ρ∗ in (11) is determined endogenously. The optimal load factor is solved by (10), and it depends on all loss portfolio information. Even the load factor ˆ ρ depends on all covariance terms Cov(Li , I(L)) . Therefore, an individual bank’s loss beta is not sufficient yet to recognize whether it is too big to fail or not; rather, we have to study the entire financial sector as a whole to identify all TBTF banks simultaneously. Briefly speaking, a bank is TBTF only when its covariance with the aggregate loss portfolio J. Risk Financial Manag. 2021,14, 414 8 of 24 is relatively large compared with other banks’ corresponding covariances in the same financial sector. Therefore, we introduce the loss beta of bank iwith a capital insurance I(L), βi=Cov(Li,I(L)) Var(I(L)) , (12) in analyzing systemic risk. We reorder {β1 , · · · , βN} such that β1≥ · · · ≥ βN> 0. We omit those banks with negative or zero loss betas. Clearly, N ∑ i=1 βi=Cov(L,I(L)) Var(I(L)) . (13) The total beta of all banks is the beta (regression) coefficient of the aggregate loss portfolio to the indemnity of the capital insurance. Appendix Apresents an algorithm to characterize all TBTF banks. In particular, a bank with high loss beta is TBTF. Conversely, the next result shows that all TBTF banks’s loss betas must be bounded below by 1 2N Cov(L,I(L)) Var(I(L)) , regardless of the distribution of loss exposure of each bank. Therefore, a TBTF must have a large loss beta, and vice versa to a certain extent. Proposition 1thus justifies our systemic risk measurement in terms of loss betas. Proposition 1. For any capital insurance I(L) , the loss beta of a TBTF bank must be greater than or equal to 1 2N Cov(L,I(L)) Var(I(L)) . Proof. See Appendix A. We present two examples below to illustrate the equilibrium approach to identify TBTF banks. Example 1. We assume that N= 15. The loss portfolios follow a one-factor model, Li=αiY+ei ; E[Y] = 1billion and Var(Y) = 5%, ei has standard deviation moving from 40% to 12% equally from i= 1to i= 15, while αi moves from 4% to 6.8% equally. We also assume that A= 1and I(L) = L. Table 1represents loss betas, ρ∗ and the optimal coinsurance ai(ρ∗) . We note that when i= 13, 14, 15, Cov(Li , L)> ∑i j=1Cov(Lj,L) 2i for i= 13, 14, 15. Moreover, the first 12 banks are TBTF. The optimal load factor is ρ∗= 8.1%. The expected total aggregate loss is ∑12 i=1aiE[Y] = 0.81 billion. The coinsurance coefficient ai(ρ∗)increases with respect to the loss beta. Example 2. Assume that N= 5, and any two different banks’ loss portfolios have the same correlation coefficient ρ=0.10. Var(Li) = k2iσ2for 0<k<1. Table 2demonstrates that TBTF banks must have large betas, but the inverse statement is not completely valid. In fact, only the last bank has a small beta, but the first three banks are TBTF banks. Both examples provide a crucial insight into the equilibrium approach. Even though some banks contribute positively to systemic risk and banks are heavily correlated, those banks might still not be TBTF banks, as shown by the last example. The intuition is as follows. By insuring the bank with the most significant systemic risk exposure, other banks’ systemic risks can be insured to some extent. As a consequence, other banks are not TBTF anymore. J. Risk Financial Manag. 2021,14, 414 15 of 24 Table 4. Summary Statistics. This table reports summary statistics of bank assets, loss portfolios, loss beta, TBTF, and CoVaR. Bank assets are from Call Reports. Loss portfolio and loss beta following the models in Section 2using Call Reports. TBTF is a dummy variable the equals one if the bank is identified as a TBTF bank based on the loss betas constructed. CoVaR is constructed following Adrian and Brunnermeier (2016). The sample is from the first quarter of 2002 to the second quarter of 2019 with U.S.banks with assets over $50 billion. Observations Mean SD Min Max Assets 2824 211.62 383.42 0.58 2354.81 Loss Portfolio 2824 129.74 237.00 0.00 1388.28 Loss Beta 2824 0.03 0.05 0.00 0.35 TBTF 2824 0.16 0.33 0.00 1.00 CoVaR 2018 0.01 0.01 −0.02 0.05 Table 5. Loss Beta and CoVaR. This table reports coefficients and standard errors of panel regressions of loss beta on CoVaR. Loss beta following the models in Section 2using Call Reports. CoVaR is constructed following Adrian and Brunnermeier (2016). Controls include bank asset obtained from Call Report. The sample is from the first quarter of 2002 to the second quarter of 2019 with U.S.banks with assets over $50 billions. Standard errors are reported in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively, based on a t-test. (1) (2) (3) (4) CoVaR CoVaR CoVaR CoVaR Loss Beta 0.494 *** 0.438 *** 0.330 * 0.396 ** (0.127) (0.126) (0.187) (0.185) Controls Yes Yes Quarter FE Yes Yes Observations 2018 2018 2018 2018 R-squared 0.007 0.075 0.008 0.075 4.4. Alternative Approach: Asset Loss Beta In this subsection, we present the result of loss beta using an alternative approach. Specifically, we calculate the loss beta using the total asset from Compustat and total equity from CRSP, in the spirit of Adrian and Brunnermeier (2016). This approach was implemented in Ivanov (2017) for 14 big financial institutions between 2004 to 2012. 17 For a comparison purpose, we extend Ivanov (2017) to include all institutions with a Standard Industrial Classification (SIC) code between 6000 and 6499 and assets over $50 billion from quarter 1 of 2002 to quarter 2 of 2019. This sample has significant overlap from our banking loss portfolio, in particular, the publicly traded Large Banking Organization. It includes financial institutions that are not banks, such as insurance companies and securities brokers and dealers. 18 To distinguish, we name the loss beta in this approach as “asset loss beta”, whereas the first one is “bank loss beta”. Our purpose is to identify TBTF banks in this period with the asset loss beta and present the effect of loss portfolio input to the identification of TBTF banks. In particular, we explain that the bank loss beta might be more appropriate for commercial banks. In this alternative approach, we estimate the key input parameters in this approach as follows. For each financial institution i. Let •li t : the leverage ratio of institution i at time t , the ratio of total asset value over the total equity value; •Mi t: the market capitalization of institution iat time t; •Yi t: the profit and loss of institution iat time t, that is, Yi t≡li t·Mi t−li t−1·Mi t−1; •Li t: the loss portfolio of institution iat time t, that is, Li t≡max{−Yi t, 0}. J. Risk Financial Manag. 2021,14, 414 16 of 24 Table 6shows the TBTF financial institutions identified using this alternative approach. Among the TBTF are traditional large commercial banks, such as JP Morgan Chase, Bank of America, Citigroup, and Wells Fargo, large investment banks such as Morgan Stanley and Goldman Sachs, and large insurance firms like American International (AIG) and Metlife. Interestingly, AIG, Wachovia, and Countrywide Financial were identified as TBTF financial institutions before they were bankrupt or acquired. Specifically, Table 6 identifies Countrywide Financial as TBTF in the third quarter of 2007, but this firm was not considered in Ivanov (2017). We also notice that American Express, which is missing in Ivanov (2017), is identified as a TBTF in 2001Q1–2003Q2 and 2004Q1–2005Q4. Table 6 provides additional empirical evidence on the validity of our measure. Table 6. List of TBTF Financial Institutions based on Asset Loss Beta. This table reports TBTF financial institutions from 2002Q1 to 2019Q2 from the publicly listed firms with assets over $50 billion by using asset loss portfolio. The asset loss portfolio is estimated based on the total asset from Compustat and total equity from CRSP in the spirit of Adrian and Brunnermeier (2016). This table extends Ivanov (2017) from 14 financial institutions between 2004 to 2012 to a more extensive set of institutions with a Standard Industrial Classification (SIC) code between 6000 and 6499 and assets over $50 billion from quarter 1 of 2002 to quarter 2 of 2019. Name TBTF, Quarters American Express 2002Q1–2003Q2, 2004Q1–2005Q4 Bank of America 2002Q1–2004Q2, 2008Q2–2019Q2 Bank One 2002Q1–2002Q2 Citigroup 2002Q1–2019Q2 JP Morgan Chase 2002Q1–2007Q3, 2008Q4–2019Q2 Morgan Stanley 2002Q1–2005Q3, 2006Q1–2006Q2, 2007Q3–2013Q3, 2014Q1–2019Q2 American International Group 2002Q3–2007Q3, 2008Q2–2008Q3 Goldman Sachs 2004Q2–2005Q3, 2007Q3, 2008Q1–2013Q3, 2014Q1–2019Q2 Countrywide Financial 2007Q3 Wachovia 2008Q3 Wells Fargo 2009Q1–2019Q2 Metlife 2014Q1–2018Q1 Compared with Ivanov (2017), the identified TBTF banks are relatively similar even though our dataset contains significantly more financial institutions than Ivanov (2017). We also find that the Big Four commercial banks—Bank of America, Citigroup, JP Morgan Chase, and Wells Fargo—are identified as TBTF since 2008. In addition, Morgan Stanley and Goldman Sachs are also consistently identified as TBTF after 2014. A comparison of bank loss beta with asset loss beta yields interesting results. Asset loss beta can identify non-bank financial firms that are TBTF, such as AIG and Metlfie, or the large investment banks before they reorganized in 2008. However, asset loss beta fails to take into account any firms that are not publicly traded. Since bank loss beta relies on only banks’ data, it cannot identify non-bank financial institutions as TBTF. However, bank loss beta has the benefit of identifying TBTF regardless of the public status of the firm. It also identifies more banks as TBTF than the asset loss beta measure. Moreover, it is convenient from a policy perspective as banks fall under the same regulatory structure, unlike other non-bank financial institutions. Table 3identifies TBTF institutions using bank loss beta, and Table 6uses asset loss beta. The findings are slightly different even when only looking at banks. Both approaches are fairly consistent in identifying the largest commercial banks, such as Bank of America, J. Risk Financial Manag. 2021,14, 414 17 of 24 JP Morgan Chase, Citigroup, Wachovia, and Wells Fargo, as TBTF. However, the bank loss beta, by not taking into consideration non-bank financial institutions, is more informative within the banking sector as it much more consistently and frequently identifies relatively smaller banks as TBTF, such as Bank One, Fleet National, US Bank, and Suntrust as TBTF before the 2008 Financial Crisis, and BNY Mellon and State Street after the crisis, and even Capital One, Charles Schwab, Ally, and TD Bank in the most recent quarters of the data sample. Notably, in the most recent quarter 2019Q2, using asset loss beta, only 6 institutions, Bank of America, Citigroup, JP Morgan Chase, Morgan Stanley, Goldman Sachs, and Wells Fargo, were TBTF. However, using bank loss beta, 11 institutions are TBTF, the six under asset loss beta, and Capital One, Charles Schwab, Ally, TD Bank, and American Express. Overall, the bank loss beta provides a more consistent and frequent signal that a commercial bank should be considered TBTF. 5. Conclusions In this paper, we provide an equilibrium approach based on economic theories to measure the systemic risk of financial institutions and identify TBTF financial intermediaries such as banks, insurance companies, or other types of intermediaries. This paper sheds light on why and how financial crises such as the 2007–2009 form and grow in magnitude. The recent episode of Covid-19 also sparks considerable interests on the systemic risk due to unexpected shocks. By constructing each financial intermediary’s expected loss portfolio in a recent accounting model, we suggest a capital-based insurance approach to address the systemic risk that is crucial to the health of the entire financial market. We derive a systemic risk measure, termed loss beta, in a characterization of the capital-based insurance equilibrium. We perform a comprehensive empirical implication and document U.S. TBTF financial intermediaries from 2002 to 2019. We demonstrate a time-series pattern of U.S. TBTF financial intermediaries, and a significant number of TBTF banks starting from 2019. Compared with previous literature on systemic risk, our approach relies on an insurance equilibrium in which all banks with systemic risk exposures and a regulator with implicit guarantee obligation maximize expected utility separably. Previous important studies such as Acharya et al. (2012), Billio et al. (2012) and Adrian and Brunnermeier (2016) introduce some statistical measures that are easy to apply, but these measures are mostly constructed exogenously. On the other hand, other measures that provide specific channels such as liquidity mismatching or consumer/real property risk require future data, which is impossible to implement ex-ante. Our proposed approach advances the literature by providing a forward-looking systemic risk measure with an insurance equilibrium approach to determine the TBTF financial intermediaries endogenously. Moreover, since we develop this approach in an insurance framework, this approach can also be helpful to estimate the expected bailout and bailout probability under the real-world probability measure. Our results have policy implications and practical appeals. First, the implementation of this approach is straightforward. By using a loss portfolio for each bank (provided by each bank or constructed by the regulator), the regulator can calculate each bank’s loss beta and identify TBTF banks with a simple algorithm in this conceptual framework. Second, the regulator is allowed to choose capital insurance contracts in the framework. In this way, various loss betas can be used to check the robustness of a bank’s systemic risk. Third, the regulator can use the capital insurance premium to assess the implied guarantee subsidy or another type of capital for TBTF banks. Finally, through the analysis of the loss portfolio, the bank can reduce the systemic risk exposure with this approach. According to our theoretical and empirical study, a loss portfolio is a crucial input in our approach and affects the identification of TBTF banks. However, the construction of a loss portfolio is far from unique. While we suggest some accounting-based or asset-based methods, constructing the most appropriate loss portfolio with systemic risk exposure is not resolved yet. In other words, we need to understand which bank factors/characteristics contribute to the systemic risk before constricting the loss portfolio, and there are many studies in the literature to identify those factors. This paper only compares the loss beta J. Risk Financial Manag. 2021,14, 414 18 of 24 approach with CoVaR, but it is also essential to compare it with numerous other systemic risk measures to understand the loss beta measure better. More importantly, we do not consider the availability of this approach to other regions, particularly European banks. We leave these topics to a future study. Author Contributions: Writing—original draft, K.I., J.S., W.T. and K.T. All authors share first authorship. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data is available upon request. Acknowledgments: We are grateful to Azamat Abdymomunov, Carole Bernard, Ethan Chiang, Atanas Mihov, Jeffrey Gerlach, Scott Frame, Christopher Kirby, Steven Zhu, and seminar participants at Federal Reserve Bank of Richmond, Shanghai Jiaotong University, Xiamen University, and University of North Carolina at Charlotte for very insightful and helpful comments. We thank Markus Brunnermeier for providing the codes to estimate CoVaR. The authors thank the financial support from the University of North Carolina at Charlotte, and Queens University of Charlotte. Both James Schulte and Kevin Tseng were working at the Federal Reserve Bank of Richmond while the project was developed. However, the views expressed in this paper are those of the authors and do not necessarily represent those of the Federal Reserve Bank of Richmond or the Federal Reserve System. The authors would like to thank the editor and anonymous referees for their constructive comments and insightful suggestions. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. Properties of Capital Insurance In this appendix, we present several properties of a general capital insurance, extending Panttser and Tian (2013) and Ivanov (2017). Let Z=I(L),b=AE[Z] Var(Z)>0, and f(ρ) = N ∑ i=1 maxβiρ−ρ2b, 0. (A1) Without loss of generality, we assume that β1>β2>· · · >βN> 0. For each m=1, · · · ,N, we define gm(ρ) = m ∑ i=1 ρ(βi−ρb), (A2) and    Im=hβm+1 b,βm bi,m=1, · · · ,N−1, IN=h0, βN bi.(A3) It can be shown that (see Ivanov (2017), Appendix A) the insurer’s optimization problem is reduced to max ρ>0f(ρ) = max m=1,··· ,Nmax ρ∈Im gm(ρ). (A4) Let Bm≡maxρ∈Imgm(ρ),m=1, · · · ,N, and τm=β1+···+βm 2mb ,m=1, · · · ,N. Case 1. m=1. In this case, A1=g1(max(τ1 , β2 b)) . The function g1(·) is decreasing over I1 if and only if β2 b≥τ1, that is, β2≥β1 2. Case 2. m=N. In this case, AN=gN(min(τN , βN b)) . The function gN(·) is increasing over IN if and only if βN b≤τN ; that is, βN≤β1+···+βN 2N . Otherwise, if βN>β1+···+βN 2N , then AN=gN(τN) = (β1+···+βN)2 4Nb . J. Risk Financial Manag. 2021,14, 414 19 of 24 Case 3. m=2, · · · ,N−1. 3.a. If βm+1 b<τm<βm b, then Am=gm(τm) = (β1+···+βm)2 4mb . 3.b. If τm≤βm+1 b, then gm(·)is decreasing over Im. 3.c. If τm≥βm b, then gm(·)is increasing over Im. Characterization of TBTF banks and ρ∗: Choose m∗ as argmax1≤m≤NBm , and the least one if there are multiple solutions (see Panttser and Tian (2013)). Then, • Bank 1, · · · ,m∗−1 are TBTF; • Bank m∗+1, · · · ,Nare NOT TBTF; • Bank m∗is TBTF if and only if τm∗<βm∗, that is, βm∗>β1+···+βm∗ 2m∗. Moreover, the optimal load factor is ρ∗=β1+· · · +βm∗ 2m∗. (A5) The next three results follow from the above property of each function gm(ρ) over Im, 1 ≤m≤N. Proposition A1. Bank 1 is always a TBTF bank. If β1 is sufficiently large such that following conditions hold, β1+· · · +βm 2m≥βm,m=2, · · · ,N then Bank 1 is the only one TBTF bank. Proof. The first part follows from the above characterization of TBTF banks; see also Panttser and Tian (2013). By the above discussion, under the proposed condition, the function gm(ρ) is increasing over Im , m= 2, . . . , N . Therefore maxρf(ρ) = g1(max(τ1 , β2 b)) , m∗=1. Since max(τ1,β2 b)<β1 b, Bank 1 is the only TBTF bank. Proposition A2. Assume that for each m =1, · · · ,N−1 βm+1≥β1+· · · +βm 2m, and βN>β1+· · · +βN 2N, then all banks are TBTF. Proof. Under the first condition, the function gm(ρ) is decreasing over Im , m= 1, · · · , N− 1. Then m∗= 1. If βN>β1+···+βN 2N , then all banks are TBTF. If βN is not large enough such that βN≤β1+···+βN 2N, then bank Nis not TBTF, but all other banks are TBTF. Proposition A3. Assume 2≤m≤N−1and the following conditions holds, βi≤β1+· · · +βi 2i,i=m+1, · · · ,N, and βj≤β1+· · · +βj 2j,j=1, · · · ,m−1. J. Risk Financial Manag. 2021,14, 414 20 of 24 1. If β1+···+βm 2m<βm, then all TBTF banks are banks 1, · · · ,m. 2. If β1+···+βm 2m≥βm, then all TBTF banks are banks 1, · · · ,m−1. Proof. Given the condition we see that the function gi(·) is increasing over Ii for i=m+ 1, · · · , N , and the function gj(·) is decreasing over Ij for j= 1, · · · , m− 1. Therefore m∗=m , maxρf(ρ) = maxρ∈Imgm(ρ) . If τm<βm b , then maxρf(ρ) = gm(max(τm , βm+1 b)) . Then all TBTF banks are 1, · · · , m . Otherwise, τm≥βm b implies that maxρf(ρ) = gm(βm b) . Then all TBTF banks are 1, · · · ,m−1. To prove Proposition 1, we need the following lemma that is shown in Ivanov (2017), Appendix A. Lemma A1. Given N positive numbers such that b1≥b2≥ · · · ≥ bN and ∑N i=1bi= 1. If there exists an integer i such that bi ∑i k=1bi >1 2i, (A6) then bi>1 2N. Moreover, if “>" is replaced by ≥in (A6), then bi≥1 2N. Proof of Proposition 1. The proof is given for aggregate capital insurance in Ivanov (2017). We extend its proof as follows. Case 1. Assume m∗=mand τm<βm b. In this case, βm>β1+···+βm 2m . Then by Lemma A1 for bm=βmVar(Z) Cov(L,Z) we obtain bm>1 2N . That is, βm>1 2N Cov(L,Z) Var(Z) . Here, all TBTF banks are 1, · · · , m . Hence the loss betas of all TBTF banks are greater than 1 2N Cov(L,Z) Var(Z). Case 2. Assume m∗=mand τm≥βm b. In this case, maxρf(ρ) = maxρ∈Img(ρ) = gm(βm b) . Therefore, the function gm−1(·) must be decreasing over Im−1. Hence, βm−1 b>βm b≥τm−1. It can be written as βm−1>β1+· · · +βm−1 2(m−1). Then by Lemma A1 again and similar to Case 1, we obtain βm−1>1 2N Cov(L,Z) Var(Z) . In this case, all TBTF banks are 1, · · · , m− 1. Hence, the loss betas of all TBTF banks are greater than 1 2N Cov(L,Z) Var(Z). Finally, Ivanov (2017) Proposition 3 shows that the capital insurance market improves the market participants’ utilities. See also Panttser and Tian (2013) for more welfare analysis of capital insurance and its application to classical insurance market. J. Risk Financial Manag. 2021,14, 414 21 of 24 0 500 1000 1500 Frequency 0 200 400 600 800 1000 1200 1400 Loss Portfolio in Billions Distribution of Loss Portfolio by Bank−Quarter Figure A1. Histogram of Loss Portfolio. This graph displays the histogram bank’s loss portfolios by quarter. Loss portfolios are constructed following the model in Section 2using Call Reports. The sample contains 2821 quarterly observations from the first quarter of 2002 to the second quarter of 2019 with 67 unique commercial U.S.banks with assets over $50 billions. Figure A2. Loss Beta. This graph displays the mean and the standard deviation (SD) of all loss betas in each quarter. Loss betas are constructed following the models in Section 2using Call Reports. The sample contains 2821 quarterly observations from the first quarter of 2002 to the second quarter of 2019 with 67 unique commercial U.S.banks with assets over $50 billions. 2 4 6 8 10 12 # of TBTF Banks 2000q1 2005q1 2010q1 2015q1 2020q1 date Number of TBTF Banks by Quarter Figure A3. TBTF banks. This graph displays the number of TBTF time series by each quarter. J. Risk Financial Manag. 2021,14, 414 22 of 24 Figure A4. CoVaR. This graph displays the CoVaR time series in each quarter from the first quarter of 2002 to the second quarter of 2019. CoVaR is estimated following Adrian and Brunnermeier (2016) and extended to the second quarter of 2019. The sample contains 2018 bank-quarter observations. Notes 1 Other important approaches include the default probability approach of the whole financial system developed by Shin (2008), and the CDS premium approach in Zhou et al. (2009). See also Battiston et al. (2012), Billio et al. (2012), Eisenberg and Noe (2001), Hanson et al. (2011), Hellwig (2009), Lehar (2005) and Van Oordt and Zhou (2016). Hansen (2012) explains the challenge in measuring systemic risk. A comprehensive survey of systemic risk measures is presented by Bisias et al. (2012). 2 In previous studies, systemic risk is primarily measured exogenously, and the implicit guarantee subsidy or capital surcharge is not addressed directly. See IMF Report (2014), Greens/EFA Report (2014), and BCBS (2013) for assessment methodology. 3 For a comprehensive discussion about the new Basel capital requirement, we refer to Tian (2017). See also Allen and Carletti (2013) about the classification of systemic risk. 4 The construction of a loss portfolio for European banks could be different from U.S. banks. Moreover, we also need to modify the capital insurance approach for European banks since there are at least two layers of regulators for European banks—the European central bank and the regulator in each country. Based on our available dataset, we consider U.S. banks over the period 2002 to 2019 in this paper. Therefore, our approach here only applies to U.S. commercial banks. 5These GSIBs U.S. banks are Bank of America, Citigroup, Goldman Sachs, JP Morgan Chase, Morgan Stanley, and Wells Fargo. 6 We notice that our result has no implication to European TBTF banks and implicit government intervention (guarantee). See Steinruecke (2018) about European TBTF banks from the regulatory reform of expected bailout perspective. 7 Under these particular numbers, the bank meets Tier 1 capital requirement (at least 6 percent) of risk-weighted asset and total capital (Tier 1 and Tier 2) requirement (10.5%) of risk-weighted asset in Basel III, assuming risk-weight 50% of the subordinated debt. 8 The derivation is as follows. Let vD,t denote the ratio of the fair to book value of deposits at time t . Then in that time period [t,t+1], vD,t=1 1+rEQ t[cD+µD,t+1+ (1−µD,t+1)vD,t+1]. Assume that µD,t+1is independent from vD,t+1and vD,t=EQ t[vD,t+1], we obtain vD,t=1 1+rcD+µD,t+ (1−µD,t)vD,t where µD,t=EQ t[µD,t+1] is the conditional deposit withdrawn under risk-neutral probability at time t . Solving vD,t we obtain the expression of FDt. 9It follows from the same derivation as in the last footnote, or Atkeson et al. (2019), Section 3.1. 10 The monotonicity condition is consistent with the classical revelation principle and reduces the moral hazard issue. Similarly, the continuity condition helps to resolve some implementation issues. It is possible to obtain optimal discontinuous and non-monotonic insurance contracts. See Bernard and Tian (2009) and Huberman et al. (1983). J. Risk Financial Manag. 2021,14, 414 23 of 24 11 In other words, each bank is distinguished from the other by the loss portfolio instead of risk attitude. It is straightforward to extend to different risk-aversion parameters in this setting. 12 The issuer of capital insurance contracts can be a private-sector, reinsurance company, a central bank or a government entity such as Financial Stability Oversight Council (FSOC) in Dodd-Frank Act, which is universally named as an issuer. 13 It is well known that Cov(L,I(L)) >0 for any increasing function I(·)and a general random variable L. 14 We only need the finite covariance Cov(Li,I(L)) and finite variance Var(I(L)) in the framework. 15 Here, we use CoVaR as an example of other numerous systemic risk measures to compare with our approach for the following reasons. (1) The CoVaR approach is intuitive and also based on aggregate loss, (2) Similar to our approach, the CoVaR approach is robust and also simple to be implemented, and (3) the CoVaR approach also generates consistent empirical insights with other measures. See discussions in Adrian and Brunnermeier (2016). 16 We would like to thank Markus Brunnermeier for providing the codes to estimate CoVaR. 17 They are: Freddie Mac, Fannie Mae, American International Group, Merrill Lynch, Bank of America, Bear Sterns, Citigroup. Goldman Sachs, JP Morgan Chase, Lehman Brother, Metlife, Morgan Stanley, Wachovia, and Wells Fargo. We notice that Merril Lynch, Bear Sterns, Lehman brother, and Wachovia have not existed since 2008. 18 For these reasons, Metlife, American International Group, and Countrywide Financial are included in this approach, while the first approach concentrates on commercial banks. References Acemoglu, Daron, Asuman Ozdaglar, and Alireza Tahbaz-Salehi. 2015. Systemic Risk and Stability in Financial Networks. American Economic Review 105: 564–608. [CrossRef] Acharya, Viral V. 2009. A Theory of Systemic Risk and Design of Prudential Bank Regulation. Journal of Financial Stability 5: 224–55. [CrossRef] Acharya, Viral, Robert Engle, and Matthew Richardson. 2012. Capital Shortfall: A New Approach to Ranking and Regulating Systemic Risks. American Economic Review 102: 59–64. [CrossRef] Adrian, Tobias, and Markus K. Brunnermeier. 2016. CoVaR. American Economic Review 106: 1705–41. [CrossRef] Allen, Franklin, and Elena Carletti. 2013. What is Systemic Risk? Journal of Money, Credit and Banking 45: 121–27. [CrossRef] Allen, Franklin, and Douglas Gale. 2000. Financial Contagion. Journal of Political Economy 108: 10–33. [CrossRef] Arrow, Kenneth J. 1971. Essays in the Theory of Risk Bearing. Chicago: Markham Publishing Co. Atkeson, Andrew G., Adrien d’Avernas, Andrea L. Eisfeldt, and Pierre-Olivier Weill. 2019. Government Guarantees and the Valuation of American Banks. NBER Macroeconomics Annual 33: 81–145. [CrossRef] Bawa, Vijay S., and Eric B. Lindenberg. 1977. Capital Market Equilibrium in A Mean-lower Partial Moment Framework. Journal of Financial Economics 5: 189–200. [CrossRef] Battiston, Stefano, Domenico Delli Gatti, Mauro Gallegati, Bruce Greenwald, and Joseph Stiglitz. 2012. Liaisons Dangereuses: Increasing Connectivity, Risk Sharing, and Systemic Risk. Journal of Economic Dynamics and Control 36: 1121–41. [CrossRef] BCBS. 2013. Global Systemically Important Banks: Updated Assessment Methodology and the Higher Loss Absorbency Requirement. In Basel Committee on Banking Supervision Consultative Document. Chicago: BCBS. Benoit, Sylvain, Christophe Hurlin, and Christophe Perignon. 2019. Pitfalls in Systemic-Risk Scoring. Journal of Financial Intermediation 38: 19–44. [CrossRef] Bernard, Carole, and Weidong Tian. 2009. Optimal Reinsurance Arrangements Under Tail Risk Measure. Journal of Risk Insurance 76: 253–80. [CrossRef] Berndt, Antje, Darrell Duffie, and Yichao Zhu. 2021. The Decline of Too Big to Fail. Working Paper. Available online: https: //papers.ssrn.com/sol3/papers.cfm?abstract_id=3497897 (accessed on 18 July 2020). Billio, Monica, Mila Getmansky, Andrew W. Lo, and Loriana Pelizzon. 2012. Econometric Measures of Connectedness and Systemic Risk in the Finance and Insurance Sectors. Journal of Financial Economics 104: 535–59. [CrossRef] Bisias, Dimitrios, Mark Flood, Andrew W. Lo, and Stavros Valavanis. 2012. A Survey of Systemic Risk Analytics. Annual Review of Financial Economics 4: 255–96. [CrossRef] Borch, Karl. 1962. Equilibrium in a Reinsurance Market. Econometrica 30: 424–44. [CrossRef] Brownless, Christian, and Robert F. Engle. 2016. SRISK: A Conditional Capital Shortfall Measure of Systemic Risk. Review of Financial Studies 30: 48–79. [CrossRef] Diamond, Douglas W., and Philip H. Dybvig. 1983. Bank Runs, Deposit Insurance and Liquidity. Journal of Political Economy 91: 401–19. [CrossRef] Elliott, Matthew, Benjamin Golub, and Matthew O. Jackson. 2014. Financial Networks and Contagion. American Economic Review 104: 3112–53. [CrossRef] Eisenberg, Larry, and Thomas H. Noe. 2001. Systemic Risk in Financial Systems. Management Science 47: 236–49. [CrossRef] Greens/EFA Report. 2014. Implicit Subsidy in the EU Banking Sector. January. Available online: https://www.greens-efa.edu (accessed on 18 July 2020). Hansen, Lars Peter. 2012. Challenges in Identifying and Measuring Systemic Risk. In Risk Topography: Systemic Risk and Macro Modeling. Edited by Markus Brunnermeier and Arvind Krishnamurthy. Chicago: University of Chicago Press, Chapter 1. J. Risk Financial Manag. 2021,14, 414 24 of 24 Hanson, Samuel G., Anil K. Kashyap, and Jeremy C. Stein. 2011. A Macroprudential Approach to Financial Regulation. Journal of Economic Perspectives 25: 3–28. [CrossRef] Hellwig, Martin F. 2009. Systemic Risk in the Financial Sector: An Analysis of the Subprime-Mortgage Financial Crisis. De Economist 157: 129–207. [CrossRef] Hogan, William W., and James M. Warren. 1974. Toward the Development of an Equilibrium Capital-Market Model Based on Semivariance. Journal of Financial and Quantitative Analysis 9: 1–11. [CrossRef] Huberman, Gur, David Mayers, and Clifford W. Smith, Jr. 1983. Optimal Insurance Policy Indemnity Schedules. Bell Journal of Economics 14: 415–26. [CrossRef] IMF Report. 2014. How Big is the Implicit Subsidy for Banks Considered Too Important To Fail? April. Available online: https: //papers.ssrn.com/sol3/papers.cfm?abstract_id=2419118 (accessed on 18 July 2020). Ivanov, Katerina. 2017. Identify Too Big To Fail Banks and Capital Insurance: An Equilibrium Approach. Risk Governance and Control: Financial Market & Institutions 7: 62–87. Kashyap, Anil K., Raghuram G. Rajan, and Jeremy C. Stein. 2008. Rethinking Capital Regulation. In Maintaining Stability in a Changing Financial System. Kansas City: Federal Reserve Bank of Kansas City, pp. 431–71. Kelly, Bryan, Hanno Lustig, and Stijn Van Nieuwerburgh. 2016. Too-Systemic-To-Fail: What Option Markets Imply about Sector-Wide Government Guarantees. American Economic Review 106: 1278–319. [CrossRef] Lehar, Alfred. 2005. Measuring Systemic Risk: A Risk Management Approach. Journal of Banking and Finance 29: 2577–603. [CrossRef] Panttser, Ekaterina, and Weidong Tian. 2013. A Welfare Analysis of Capital Insurance. Risks 1: 57–80. [CrossRef] Peck, James, and Karl Shell. 2003. Equilibrium Bank Runs. Journal of Political Economy 111: 103–23. [CrossRef] Raviv, Artur. 1979. The Design of an Optimal Insurance Policy. American Economic Review 69: 84–96. Shin, Hyun Song. 2008. Risk and Liquidity in a System Context. Journal of Financial Intermediation 17: 315–29. [CrossRef] Steinruecke, Lea. 2018. Are European Banks Still Too-Big-To-Fail? The Impact of Government Interventions and Regulatory Reform, on Bailout Expectations in the EU. Working Paper. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3098296 (accessed on 18 July 2020). Tian, Weidong. 2017. Commercial Banking Risk Management: Regulation in the Wake of the Financial Crisis, 1st ed. New York: Palgrave Macmillan. van Oordt, Maarten R. C., and Chen Zhou. 2016. Systemic Tail Risk. Journal of Financial and Quantitative Analysis 51: 685–705. [CrossRef] Zhou, Hao, Huang Xin, and Haibin Zhu. 2009. A Framework for Assessing systemic risk of Major Financial Institutions. Journal of Banking and Finance 33: 2036–49.