Measuring systemic risk contribution with CoGVaR approach

Zhang Ping, Wang Yashi, Wu Qinyu

1. Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China；2. Department of Science and Technology， China University of Political Science and Law， Beijing 102249, China

Abstract: We propose a new conditional risk measure, conditional generalized value-at-risk (CoGVaR)， from the perspective of measuring systemic risk. The new class of risk measures is a natural generalization of the conditional quantiles including the classic CoVaR. Compared with the classic conditional value-at-risk (CoVaR) and conditional expectile (CoExpectile), it has more potential application in reality as it takes the risk attitude of the decision maker into consideration, which has not been the focus of much study to date. Using generalized quantile regression approach with state variables added, some calculation results are presented in the Dow Jones U.S. Financials Index case, and it is shown that it provides a new perspective on systemic risk contribution. In addition, the result shows that our risk measure can capture the tail risk by using more convex disutility function.

Keywords: generalized quantile; conditional risk measure; systemic risk contribution

1 Introduction

The systemic risk is a risk that could trigger severe instability or collapse of an entire industry or economy. It has attracted a lot of interests recently. Usually there exist financial links between the institutions in the financial systems. The caused failure of one institution may spread to other institutions. The contribution of a financial institution to the systemic crisis thus plays an important role in systemic risk measurement and management of the whole financial system. Measuring the contribution of each institution to overall systemic risk can help regulators identify institutions that make significant contributions to systemic risk. With strict constraints on these institutions, the tendency to generate systemic risk can be restrained. Since the constructive work of Ref.[1] with a corisk, conditional value-at-risk (CoVaR), for systemic risk measurement was proposed, many corisks has been discussed for measuring systemic risk.

CoVaR in Ref.[1] described the VaR of the financial system conditional on an institution being in financial distress (the loss of an institution being exactly its VaR). Ref.[1] defined the systemic contribution of a financial institution as the difference between CoVaR conditional on the institution being under distress and CoVaR in the median state. Girardi and Ergün[2]modified the definition of distress in Ref.[1] to no less than its VaR to consider more severe distress events. Huang and Uryasev[3]changed the systemic risk in Ref.[1] from VaR to CVaR to propose a new corisk, CoCVaR. Brownlees and Engle[4]introduced SRISK, a function of the firm’s size, leverage and risk, to measure the systemic risk contribution of a financial firm. Acharya et al.[5]proposed the systemic expected shortfall to measure financial institutions’ contribution to the systemic risk with an economic model of systemic risk presented in their paper.

πα(X,x)=α[u1((X-x)+)]+

(1-α)[u2((X-x)-)].

Here,Xrepresents the loss a financial institution face andxis the required capital.a+=max{a,0},a-=max{-a,0},α∈ (0,1) is the confidence level to balance the shortfall risk (X-x)+and over-required capital risk (X-x)-.u1andu2, two strictly increasing, convex functions on+, are disutility (loss) functions in the expected utility model, they are used to transform the two risks (X-x)+and (X-x)-respectively. Ifu1(x)=u2(x)=x, then the generalized quantile reduces to the classic quantile (VaR), and ifu1(x)=u2(x)=x2, the generalized quantile reduces to the expectile. While VaR and expectile are symmetrically viewing the two risks with same disutility functionsu1=u2, the generalized quantile can asymmetrically view them by using different disutility functions.

There is a generally believed point of view in risk management that underestimating is much more disastrous than overestimating. Thus it is natural to use more convex disutility functionu1thanu2to transform shortfall risk (X-x)+. For example, the generalized quantile will work well on such an occasion when the decision maker (regulator) is risk averse, i.e., is more concerned about upper-tail realizations of a loss random variable[9]. That is the main occasion that our paper focuses on the systemic risk measurement, to the best of our knowledge, which has not been discussed by other people.

Motivated by past work on the systemic risk, our paper propose a systemic risk measure, CoGVaR, similar to the definition of CoVaR in Ref.[1]. CoGVaR is defined as the GVaR of financial system conditional on some financial institution being under distress. We define systemic risk contribution of a financial institution asΔCoGVaR, which describes the change from its CoGVaR in its median state to its CoGVaR under distress. We consider disutility functionsu1andu2with the form of

u1(x)=xa,u2(x)=xb,a,b>1.

The rest of paper is organized as follows. In Section 2, we first recall the definition and some properties of generalized quantiles that will be used in the context. Next we give the definition of GVaR and CoGVaR and explore the stronger emphasis a GVaR with relatively large value ofamay apply on upper tail of a loss. Section 3 illustrates the approach to do estimations. Section 4 presents the Dow Jones U.S. Financials Index case study and Section 5 concludes.

2 GVaR and CoGVaR

Recall that the generalized quantile[6]of a riskXis defined as follows.

where

πα(X,x)=α[u1((X-x)+)] +

(1-α)[u2((X-x)-)]

(1)

α∈ (0,1) andu1,u2are two strictly increasing, convex functions on+withui(0)=0,ui(1)=1,i=1,2.

ProofThe proof can be seen in Proposition 1 of Ref.[6].

Throughout the paper, we only focus on disutility functionsu1andu2with the form ofu1(x)=xa,u2(x)=xb,a,b> 1. LetXibe a random variable representing the loss of a financial institution,X=(X1,X2,…,Xn) be a vector of random variables, andXsysbe the financial system loss.

Definition 2.1The generalized quantile with a confidence levelαof a financial institutionidenoted as GVaRα(Xi),

GVaRα(Xi)=

Proposition 2.2GVaR is a convex/coherent risk measure if and only ifa=b=2 andα≥1/2.

ProofWhena=b=2, it refers to expectile in Ref.[7], and by Proposition 6 (a) and (b) of Ref.[6], we can conclude that expectile is the only class of GVaRs that are convex/coherent risk measures.

Remark 2.1Disutility functionsu1andu2with the form ofu1(x)=xa,u2(x)=xb,a,b>1, become more convex functions (x≥1) asaandbincrease. Here, we want to discuss the stronger emphasis on tail risks through more convex disutility functionsu1,u2. The GVaR with confidence levelαof a riskX′ is shown as follows.

(1-α)((X′-x)-)b}.

As is often the case,X′ obeys heavy-tailed distribution in finance with much probability weight on tail. From the view of a regulator, underestimating is much more disastrous than overestimating, thus (X′-x)+is much more focused on with large value ofαand relatively large value ofa, and for fixedb, GVaRα(X′) increases in general asaincreases. SupposingX′～ Pareto (3,1) with distribution functionF(x)=1-x-3,x≥1, we calculate the corresponding results withα=0.9 in Table 1. It is clear that the larger value ofabrings more conservative outcome.

Table 1.Results of generalized quantiles with α=0.9.

Definition 2.2The CoGVaR of the financial system is defined as GVaR with confidence levelαofXsys, conditional on the event thatXis in a measurable setC,

(1-α)((Xsys-x)-)b|X∈C].

Definition 2.1 and Definition 2.2 hold becauseu1andu2are strictly convex functions and by Proposition 2.1, the minimizer is unique.

Theorem 2.1CoGVaR has the following form:

where

3 Methodology for estimation

3.1 Generalized quantile regression in static setting

Consider the following regression models:

The coefficients are estimated based on minimizing a loss function of the form

lα(x)=α((x)+)a+(1-α)((x)-)b,

that is to say we need to solve the following minimization problems

E(L)=[lα(L)],

the GVaR statistics be

S(L)=GVaR(L).

CoGVaRαsys|i=

3.2 Estimation in dynamic setting

(2)

The coefficients are estimated based on

3.3 Validation analysis

General OLS (ordinary least squares) regression models are often assessed by coefficient determination

4 Case study

In this section, we compare CoGVaR with CoVaR and CoExpectile in a case study after the estimation approach has been given before.

4.1 Data description

The data are considered in the period from January 1, 2002 to January 1, 2015 (679 weeks), which covers a recession (2007-2009) and a financial crisis (2008). We download the weekly Dow Jones U.S. Financials Index and financial institutions’ closing prices from Yahoo! Finance(1)Data are available at https://au.finance.yahoo.com/..

Table 2 lists ten publicly traded banks in the United States ranked by total assets as of December 31, 2014.

Table 2. The ten publicly traded banks in the U.S..

To estimate the time-varying GVaRα,tand CoGVaRα,t, we choose the following state variables.

(Ⅰ) The change in the three-month yield (TC): Ref.[1] found that the change in the three-month Treasury bill rate is most significant in explaining the tails in asset returns of financial institutions.

(Ⅱ) The change in the slope of the yield curve (TSC): measured by the yield spread between the ten-year Treasury rate and the three-month bill rate obtained from Federal Reserve Bank’s H.15 report(2)www.federalreserve.gov/releases/h15/..

(Ⅲ) The credit spread change (CSC): measured by the change between Baa-rated bonds and the Treasury rate (with the same ten-year maturity) from the Federal Reserve Bank’s H.15 report.

(Ⅳ) Equity volatility (VIX): The Chicago Board Options Exchange (CBOE) Volatility Index (VIX), which captures the implied volatility in the stock market reported by the CBOE.

(Ⅴ) The weekly equity market return (MER): we use the Standard & Poor’s 500 Index to calculate the equity market return.

(Ⅵ) A short-term “liquidity spread” (LS): defined as the difference between the three-month LIBOR rate and the three-month Treasury bill rate. This liquidity spread measures short-term liquidity risk. We obtain the three-month Treasury bill rate from the Federal Reserve Bank’s H.15 report. We use the three-month LIBOR rate from Wind.

Figure 1 shows the log-loss of the DJUSFN, and Figure 2 includes the log-loss of the related 10 institutions. They factually share a similar tendency in general, which coincides with the DJUSFN.

Figure 1. Log-loss of DJUSFN.

Figure 2. Log-loss of institutions.

4.2 Caculation results

According to the definition of GVaR, there is a tendency that with a fixed indexa=1(3)When a=1 or b=1, the value of GVaR is not unique according to its definition and we choose the smallest one. We take the value of a or b as 1 just for the demonstration of monotonicity tendency and comparison to VaR.andα=0.9, GVaR will decrease as indexbincreases in most cases. We only exhibit the results focusing on JPM to verify these changes here, they certainly also hold for other institutions. The plots on the left side of Figure 3 reveal the changes of a varyingb, and it is clear that whenbis relatively large enough, there is no practical significance (above 2.3 in the plots). Moreover, we discuss indexa’s variation in the plots on the right side of Figure 3. GVaR increases asaincreases in most cases. We also linearise the risk measure VaR here, as it is factually a GVaR witha=b=1. The left and right parts of Figure 4 show CoGVaR’s change in varyingband varyinga, respectively, and they share similar fluctuations with GVaR in Figure 3.

Figure 3. (a) depicts the GVaR of JPM with a=1 and varying b, and (b) depicts its GVaR of with b=1 and varying a.

Figure 4. (a) is the CoGVaR0.9,t of JPM with varying b and fixed a=1, and (b) is its CoGVaR0.9,t with varying a and fixed b=1.

Figure 5. The CoGVaR0.9,t, CoGVaR0.95,t and CoGVaR0.8,t of JPM with a=3, b=1.1.

In the following, we calculate CoGVaR witha=3 andb=1.1, which are chosen according to our calculation results for a relatively conservative corisk (compare to VaR and expectile). In the meantime, the choice reflects the regulator’s risk aversion, that is he/she concerns much more about upper tail of risks.

Table 3 presents the coefficients from model (2) for 10 institutions, which conveys the message that the state variables have different sensitivities for most institutions and some even have the opposite sign. For example, the change in the three-month yield (TC) has a positive effect on most banks, while it has a negative effect on JPM and PNC.

Table 3. Variable coefficients from regression.

In Figure 5, we calculate the corresponding CoGVaR0.9,t, CoGVaR0.95,tand CoGVaR0.8,tto explore the change generated by varyingα. CoGVaR0.95,tseems to be above the others, which is natural for GVaR’s monotonicity in confidence levelα. And in reality, large enough value ofαbrings large enough capital reservation to protect the system.

Figure 6. The time series of weekly

4.3 CoGVaR vs. CoVaR, CoExpectile

In this section, we perform a simple comparison between CoVaR, CoExpectile and CoGVaR (a=3,b=1.1) at different confidence levelsα. Note that CoExpectile is the special case of CoGVaR witha=b=2. Figure 6 shows the concrete time series of the change between the three kinds of risk measures withα=0.9. The time series plots show that for each institution, CoGVaR (a=3,b=1.1) is more conservative than CoVaR and CoExpectile, especially during the financial crisis period. As a result of the finding that the state variables have similar effects on most institutions according to Table 3, the CoRisks’ values of institutions share similar trends.

Table 4 summarizes the average results of CoVaR, CoExpectile and CoGVaR (a=3,b=1.1) withα=0.9. The Rk column is given according to the institution’s contribution to the systemic risk. We find that a high rank may not always coincide with a high value of VaR, expectile or GVaR, and obviously CoVaR, CoExpectile and CoGVaR (a=3,b=1.1) provide rather different ranks for systemic risk contributions. Tables 5 and 6 summarize the average results withα=0.95 andα=0.8. Clearly, the case withα=0.95 yields larger values of VaR, expectile and GVaR,while the case withα=0.8 yields smaller values of VaR, expectile and GVaR. The systemic risk contribution also yields divergent observations, and the ranks differ greatly. Compared to CoVaR and CoExpectile, CoGVaR (a=3,b=1.1) behaves more conservatively, especially during the financial crisis period (2008-2010). CoGVaR (a=3,b=1.1) with the regulator’s risk aversion added and more concentration on shortfall risks (X-x)+(with largerathan CoVaR) may be more suitable in such a context when the upper-tail of a risk is given more concern.

Table 4. Comparison of average results with α=0.9.

Table 5. Comparison of average results with α=0.95.

Table 6. Comparison of average results with α=0.8.

5 Conclusions

In conclusion, we explored the application of generalized quantiles to the systemic risk, inspired by Ref.[1]. It accounts for risk aversion, using two different disutility functions to transform the two risks (X-x)+and (X-x)-. We proposed the approach to estimate CoGVaR via generalized quantile regression without any distribution assumption and denoted the systemic risk contribution asΔCoGVaR. We compared CoVaR, CoExpectile and CoGVaR in our Dow Jones U.S. Financials Index case.

In the case study, we found that a high rank in terms of the systemic risk contribution may not coincide with high value of the corresponding VaR, Expectile or GVaR. Controlling an individual risk may not be sufficient to make the whole financial system safe. Our CoGVaR withα=0.9,u1(x)=x3andu2(x)=x1.1in the Dow Jones U.S. Financials Index case focuses more on the heavy upper tail of the loss and provides a new perspective on systemic risk contribution.

There is potential for more in-depth investigations. Our regression method has some dependence on subjectively selected state variables, and this dependence does not change over time. That is, on the one hand, the choice of state variables has some potential to improve the regression model for the calculation of our CoGVaR, and on the other hand, the time-invariant dependence may not work as well as time-variant dependence. These are problems to be explored in the future. Compared to CoVaR, the CoGVaR in Dow Jones U.S. Financials Index case has a different application to the systemic risk. For the decision maker who cares much about the upper tail of a risk, a generalized quantile with more convex disutility functionu1may work better.

Acknowledgments

This work is supported by Qian Duansheng Distinguished Scholar Support Program of China University of Political Science and Law (DSJCXZ180403).

Conflict of interest

The authors declare no conflict of interest.