12. Concentration and Distribution Measures

International Monetary Fund. Statistics Dept.

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International Monetary Fund. Statistics Dept.

International Monetary Fund. Statistics Dept.
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Language:: English

Print Publication Date:: 27 Dec 2019

Keywords:: M&G; capital; loan; foreign currency; holding company; DT subsidiary; DT sector; parent corporation; FSI compiler; regulatory capital; capital range; Financial soundness indicators; Personal income; Loans; Real estate prices; Financial statements; Global; Western Hemisphere; Middle East and Central Asia; Sub-Saharan Africa; Europe; Caribbean; return on assets; Financial Stability Board G-20 Data Gaps Initiative; weighted quartile; IMF staff; inter-quartile range; Commercial banks; Nonperforming loans; Financial sector stability; Financial stability assessment

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Abstract

12.1 Financial soundness indicators for a sector may hide variations that could endanger the entire financial system. For example, the sector-wide capital-asset ratio for deposit takers is an average ratio for the system, but it does not reveal whether individual institutions’ capital ratios are clustered around the average value or are spread over a wide range. Moreover, data for highly capitalized deposit takers could offset the data for undercapitalized deposit takers, such that the aggregate ratio may appear robust while masking significant vulnerabilities from weak deposit takers whose failure could lead to contagion throughout the system. For this reason, FSIs need to be supplemented by concentration and distributions measures (CDMs).

I. Introduction

12.1 Financial soundness indicators for a sector may hide variations that could endanger the entire financial system. For example, the sector-wide capital-asset ratio for deposit takers is an average ratio for the system, but it does not reveal whether individual institutions’ capital ratios are clustered around the average value or are spread over a wide range. Moreover, data for highly capitalized deposit takers could offset the data for undercapitalized deposit takers, such that the aggregate ratio may appear robust while masking significant vulnerabilities from weak deposit takers whose failure could lead to contagion throughout the system. For this reason, FSIs need to be supplemented by concentration and distributions measures (CDMs).

12.2 To address this concern, in 2009, the IMF/ Financial Stability Board G-20 Data Gaps Initiative (DGI) called on the IMF “to investigate, develop, and encourage implementation of standard measures that can provide information on tail risks, concentrations, variations in distributions, and the volatility of indicators over time.”¹ The concentration and distribution measures (CDMs) for selected FSIs aim at providing critical information about vulnerabilities in the financial system, not directly captured by simple averages.

12.3 This chapter provides a brief overview of the proposed CDMs for selected FSIs and guidance on their computation. The chapter also discusses ways to overcome confidentiality concerns about CDM reporting.

II. Background

The CDM Pilot Project

12.4 As part of the calls to develop and encourage the implementation of tail risk and concentration measures for FSIs, the IMF’s Statistic Department conducted a pilot project to assess the feasibility of calculating and reporting (regularly) CDM data for selected deposit takers’ FSIs. In addition, the pilot was undertaken to ascertain (1) the effectiveness of the pilot CDMs in monitoring financial sector vulnerabilities; (2) potential confidentiality concerns over their reporting; (3) the extent of the reporting burden; and (4) the procedures and resources the Fund would need to deploy in order to gather, compile, analyze, and disseminate the CDMs along with current FSI data and metadata.

12.5 Participation in the project was broadly based and there was a strong response to the request for volunteers—with 35 participants providing data. The comprehensiveness of reporting varied across countries, indicators, and time periods. Several participants engaged IMF staff to resolve methodological issues, including the computation of quartiles.

12.6 The pilot CDMs comprised (1) minimum, maximum, and mean; (2) weighted standard deviations and skewness; and (3) quartiles and the asset share of the bottom quartile. Also, a concentration (Herfindahl) index was calculated. The CDMs were requested for a subset of six FSIs for DTs (regulatory Tier 1 capital to risk-weighted assets, NPLs to total gross loans, return on assets [ROA], return on equity [ROE], liquid assets to short-term liabilities, and capital to total assets).

12.7 The results of the pilot provided useful insights to the analytical value of CDMs.² They suggested that CDM data have analytical value that would justify efforts to compile and report them. CDMs provide important information that is not revealed by simple averages, can be used as a starting point in financial stability and performance assessments, and are useful tools for monitoring financial sector vulnerabilities. Participating countries did not report any significant resource burden associated with the compilation of CDMs.

The FSI Compilers’ Perspective

12.8 When developing the work program for Phase 2 of the DGI, the IAG³ agreed that the IMF should seek further input from FSI reporting countries and potential data users prior to taking a decision with moving ahead with the collection of FSI’s CDMs.

12.9 In April 2017, the IMF conducted a workshop on FSIs, which brought together 75 participants from 36 countries and 7 international organizations, to secure an agreement on the collection of CDMs for a selected list of FSIs. Participants, which included a significant majority of FSI compilers, agreed with the benefits of the CDM project. Most participants observed that underlying bank-by-bank supervisory data are available, thus the data required for producing CDMs are, in many cases, readily available. They acknowledged that CDMs provide valuable insights for financial stability analysis, which supports efforts by member countries to compile and report them on a regular basis to the IMF. The list of CDMs and the underlying FSIs were discussed with the workshop participants and agreed upon.

III. Compilation of CDMs

12.10 This section provides a set of concrete recommendations and guidelines on the proposed CDM measures and their potential use. This would serve to harmonize and standardize different types of statistical methodologies (e.g., interpolation, estimation or approximation techniques, and soff-ware) available to users which in turn will facilitate cross-country comparability, reproducibility, and interpretability. The chapter also discusses how to address confidentiality concerns associated with their reporting.

12.11 The technique to compute CDMs may be a function of their ease of use, type of data available, and the type of information sharing agreements for the specific institutions. When it comes to public dissemination, some CDMs may prove particularly useful, as they can shed information on the system’s vulnerabilities without revealing otherwise confidential information on individual institutions. To facilitate uniform and consistent reporting of the CDMs, the IMF is making available on its FSI website a template for the calculation of these CDMs.

12.12 To estimate concentration in the system, this Guide recommends the Herfindahl-Hirschman (thereafter Herfindahl) index. To gauge dispersion and related properties, the Guide recommends the following distribution measures: weighted standard deviation, weighted quartiles, weighted skewness, and weighted kurtosis (Table 12.1), for at least seven designated FSIs in the universe of a country’s deposit-taking institutions (Table 12.2).

^{Table 12.1}

Concentration and Distributions Measures Indicators

Source: IMF staff.

^{Table 12.1}

Concentration and Distributions Measures Indicators

Measure	Required Frequency
Sector concentration index (Herfindahl index)	Annual
Weighted quartiles	Monthly, quarterly, or annual
Weighted standard deviation	Monthly, quarterly, or annual
Weighted skewness	Monthly, quarterly, or annual
Weighted kurtosis	Monthly, quarterly, or annual

Source: IMF staff.

^{Table 12.1}

Concentration and Distributions Measures Indicators

Measure	Required Frequency
Sector concentration index (Herfindahl index)	Annual
Weighted quartiles	Monthly, quarterly, or annual
Weighted standard deviation	Monthly, quarterly, or annual
Weighted skewness	Monthly, quarterly, or annual
Weighted kurtosis	Monthly, quarterly, or annual

Source: IMF staff.

^{Table 12.2}

Subset of Financial Soundness Indicators Covered by Concentration and Distributions Measures

Source: IMF staff.

The Sample

12.13 CDMs should be computed for the same financial institutions for which the selected FSIs are reported according to the consolidation principles described in Chapter 6. The Guide also, prescribes a minimum number of DTs to ensure meaningful estimates and to preserve confidentiality, as described in section V of this chapter.

Concentration

12.14 Concentration in the banking sector has continued to increase in the recent times (e.g., BIS 2018). Some studies argued that concentration may promote financial stability (e.g., Beck et al, 2006; De Haan and Poghosyan, 2012; Evrensel, 2008), the rationale being that concentration may improve franchise value, leading to less risk taking, thus contributing to financial stability. At the same time, concentration is well-known to be linked to the moral hazard—too big-to-fail problem. The implicit assumption that large DTs in a concentrated sector will be bailed out by the government if they get into trouble provides an incentive to the DTs to acquire riskier assets and to operate with high levels of leverage, increasing the vulnerability of the financial system (e.g., Boyd and Runkle, 1993; Mishkin, 1999; O’Hara and Shaw, 1990).

12.15 Because of conflicting evidence on the concentration and stability nexus, it is important to start with a measure of the level of concentration in the financial system and then interpret the results considering additional country-specific factors for a more holistic view. There are several measures of concentration, ranging from heuristic measures (Herfindahl and Gini indices), to more sophisticated model and simulation-based approaches also known as granularity adjustments.⁴ The Guide recommends the Herfindahl Index to estimate concentration in the financial system because of its ease of use. Furthermore, empirical evidence suggests that this measure does not yield significantly different estimates than more sophisticated and computation-intensive approaches.⁵

Herfindahl concentration index

12.16 The Herfindahl Index, H, is the percentage asset share of the system. It is calculated as the sum of squares of each financial institutions asset shares (measured in percent) in a sector: $H = Σ_{i = 1}^{N} {(a_{i})}^{2}$

where

a_{i} = \frac{T o t a l a s s e s t s o f i n s t i t u t i o n i}{T o t a l a s s e s t s o f t h e e n t i r e d e p o s i t t a k i n g \sec t o r}

12.17 Values of the index range from 1/N to 1.0, with higher values indicating greater concentration. In a situation where the sector has 100 institutions each with an identical 1 percent share of the market, the index will be H = 0.01. In contrast, with perfect concentration, where one institution has a 100 percent market share, H = 1.0; that is, the contribution of the monopoly institution is 1.0 × 1.0 = 1.0. A rule of thumb sometimes used is that H below 0.1 indicates limited concentration, and H above 0.18 points to significant concentration. Table 12.3 illustrates how to compute H for a country consisting of 11 deposit takers. For this hypothetical example, the Herfindahl Index for the top-fve DTs is equal to 0.1614.

^{Table 12.3}

Example of Computing the Herfindahl Index

Source: IMF staff estimates.

^{Table 12.3}

Example of Computing the Herfindahl Index

Deposit Taker	Assets	Market Share (percent)	Market Share Squared
1	300	30	0.0900
2	200	20	0.0400
3	130	13	0.0169
4	90	9	0.0081
5	80	8	0.0064
6	50	5	0.0025
7	50	5	0.0025
8	40	4	0.0016
9	20	2	0.0004
10	20	2	0.0004
11	20	2	0.0004
Total	1,000	100	0.1692

Source: IMF staff estimates.

^{Table 12.3}

Example of Computing the Herfindahl Index

Deposit Taker	Assets	Market Share (percent)	Market Share Squared
1	300	30	0.0900
2	200	20	0.0400
3	130	13	0.0169
4	90	9	0.0081
5	80	8	0.0064
6	50	5	0.0025
7	50	5	0.0025
8	40	4	0.0016
9	20	2	0.0004
10	20	2	0.0004
11	20	2	0.0004
Total	1,000	100	0.1692

Source: IMF staff estimates.

Distribution: Dispersion Measures

12.18 FSIs include aggregated individual institution data as well as idiosyncratic elements in which the financial institutions operate. The extent to which these aggregates are good representatives of the entire sector is a function of factors such as the variability or dispersion among the underlying soundness of individual institutions. Given that large-scale disruptions to the financial stability might stem from difficulties in individual institutions, dispersion measures are needed to shed some light on such blind spots (e.g., Smaga, 2014; and Systemic Risk Centre RC, 2015).

12.19 There are various ways to measure dispersion, including variance, standard deviation, and quartiles. The Guide recommends (1) weighted standard deviation and (2) weighted quartiles as measures of dispersion, not just because of the ease of computation but also for their robustness.

Weighted standard deviation

12.20 The standard deviation (σ) estimates the variability (or spread) for an FSI among the different DTs. It indicates how close the indicators for the individual institutions are to the sectoral average. When the standard deviation is small, that is, close to zero, the values in a dataset are tightly bunched together, and consequently, the aggregate indicator is a good reflection of the overall system’s soundness. On the other hand, when the individual values vary significantly, the standard deviation will be relatively large. And this difference can have a significant economic impact. For example, an aggregate Tier 1 capital to risk-weighted assets indicator with a large standard deviation indicates that some DTs could deviate significantly from the minimum capital requirement. However, standard deviation measures are not robust because they can be greatly influenced by outliers. In addition, they do not account for sample characteristics such as the relative asset or loan size of the different DTs.

12.21 To account for the relative asset or loan size of the different DTs, the Guide recommends weighing the standard deviation by the relative share of the variable in the denominator of the relevant FSI ratio. This weighting will account for the marginal contribution of DTs with larger assets, gross loans, and so on to the relevant FSI. For instance, the weighting variable of Tier 1 capital to risk-weighted assets is the relative share of risk-weighted assets of the individual institutions. For the FSIs related to asset quality, the Guide recommends weighing the NPL indicator by the relative size of an institutions loans to total gross loans, and so on.

12.22 Specifically, the recommended weighted standard deviation is given by the positive square root of the weighted variance (σ²). Finally, the weighted variance is calculated as follows:

σ^{2} = Σ_{i = 1}^{N} {({F S I}_{i} - \bar{F S I})}^{2} \times ω_{i}

where

$\bar{F S I}$ is the sector FSI⁶ and ω is the weight used to calculate the average sector FSI.

Weighted quartiles

12.23 In addition, to understand the variability of financial soundness among the individual DTs, it is important to identify DTs’ degree of exposure. The statistical method of quartiles which divides data into quarters (with the second quartile being the median) is useful in this regard.

12.24 Standard quantiles comprise the same number of institutions, irrespective of their contribution to the FSI in question. Therefore, with this approach, the important features of the distribution may be missed if the relative contribution of each data point is not properly accounted for in the compilation of the quantile.

12.25 In addition, adopting an unweighted approach where each quantile comprises the same number of institutions may affect cross-country comparability. For example, a country where 25 percent of the DTs are very small institutions with high NPLs will exhibit a fourth quartile with low asset quality, raising concerns about the stability of the system, even though the risk to the finan-cial system as a whole may not be significant. This constitutes a major drawback of standard quantile analysis.

12.26 The Guide recommends the compilation of weighted quartiles. With weighted quartiles, the marginal contribution of the DTs to the FSI increases proportionally with their weights. Computing weighted quartiles require first mapping the DTs’ specific asset, loan, or capital characteristics to the relevant FSI and then sorting out by quartile. This technique facilitates cross-country comparability of the measures, as each quartile will display the FSI for a comparable share of, say, total assets of the deposit-taking sector.

^{Box 12.1}

Step-by-step Instructions to Compute Weighted Quartiles

The practical computation of weighted quartiles can be thought of as following the three steps outlined here.

Let FSI_k be the value of the financial soundness indicator (FSI) for deposit taker (DT) k (k = 1, . . . , N), and let A_k refer to the total assets of DT_k, with N the number of institutions.

^{Table 12.4}

Unweighted and Weighted Medians of Tier 1 Ratios for a Hypothetical Sample of 15 Deposit Takers

Source: IMF staff estimates.

^{Table 12.4}

Unweighted and Weighted Medians of Tier 1 Ratios for a Hypothetical Sample of 15 Deposit Takers

Deposit Taker	Tier 1 Ratio	Individual Deposit Taker’s Assets
1	2.1	400,000
2	3.1	300,000
3	3.3	300,000
4	4.1	400,000
5	4.1	600,000
6	6.7	300,000
7	7.1	200,000
8	8.1	500,000
9	8.2	300,000
10	9.2	400,000
11	11.2	1,500,000
12	11.3	800,000
13	13.1	1,800,000
14	13.5	2,200,000
15	13.8	2,000,000
Unweighted median: 8.1
Weighted median: 12.2

Source: IMF staff estimates.

^{Table 12.4}

Unweighted and Weighted Medians of Tier 1 Ratios for a Hypothetical Sample of 15 Deposit Takers

Deposit Taker	Tier 1 Ratio	Individual Deposit Taker’s Assets
1	2.1	400,000
2	3.1	300,000
3	3.3	300,000
4	4.1	400,000
5	4.1	600,000
6	6.7	300,000
7	7.1	200,000
8	8.1	500,000
9	8.2	300,000
10	9.2	400,000
11	11.2	1,500,000
12	11.3	800,000
13	13.1	1,800,000
14	13.5	2,200,000
15	13.8	2,000,000
Unweighted median: 8.1
Weighted median: 12.2

Source: IMF staff estimates.

^{Table 12.5}

Calculation Steps of the Unweighted and Weighted Medians of Tier 1 Ratios for a Hypothetical Sample of 15 Deposit Takers

Source: IMF staff estimates.

^{Table 12.5}

Calculation Steps of the Unweighted and Weighted Medians of Tier 1 Ratios for a Hypothetical Sample of 15 Deposit Takers

Deposit Taker	Tier 1 Ratio	Individual Deposit Taker’s Assets	Cumulative Assets	Weights
1	2.1	400,000	400,000
2	3.1	300,000	700,000
3	3.3	300,000	1,000,000
4	4.1	400,000	1,400,000
5	4.1	600,000	2,000,000
6	6.7	300,000	2,300,000
7	7.1	200,000	2,500,000
8	8.1	500,000	3,000,000
9	8.2	300,000	3,300,000	W₉
10	9.2	400,000	3,700,000
11	11.2	1,500,000	5,200,000
12	11.3	800,000	6,000,000
13	13.1	1,800,000	7,800,000	W₁₃
14	13.5	2,200,000	10,000,000	W₁₄
15	13.8	2,000,000	12,000,000
P for the quantile 0.25: 3,000,000 P for the quantile 0.5: 6,000,000 P for the quantile 0.75: 9,000,000
$Q_{0.25} = \frac{8.1 + 8.2}{2}; Q_{0.50} = \frac{13.1 + 11.3}{2}; Q_{0.75} = 13.5$
Unweighted median: 8.1
Weighted median: 12.2

Source: IMF staff estimates.

^{Table 12.5}

Calculation Steps of the Unweighted and Weighted Medians of Tier 1 Ratios for a Hypothetical Sample of 15 Deposit Takers

Deposit Taker	Tier 1 Ratio	Individual Deposit Taker’s Assets	Cumulative Assets	Weights
1	2.1	400,000	400,000
2	3.1	300,000	700,000
3	3.3	300,000	1,000,000
4	4.1	400,000	1,400,000
5	4.1	600,000	2,000,000
6	6.7	300,000	2,300,000
7	7.1	200,000	2,500,000
8	8.1	500,000	3,000,000
9	8.2	300,000	3,300,000	W₉
10	9.2	400,000	3,700,000
11	11.2	1,500,000	5,200,000
12	11.3	800,000	6,000,000
13	13.1	1,800,000	7,800,000	W₁₃
14	13.5	2,200,000	10,000,000	W₁₄
15	13.8	2,000,000	12,000,000
P for the quantile 0.25: 3,000,000 P for the quantile 0.5: 6,000,000 P for the quantile 0.75: 9,000,000
$Q_{0.25} = \frac{8.1 + 8.2}{2}; Q_{0.50} = \frac{13.1 + 11.3}{2}; Q_{0.75} = 13.5$
Unweighted median: 8.1
Weighted median: 12.2

Source: IMF staff estimates.

12.27 The difference between the unweighted and weighted quartiles may potentially be significant, as illustrated in Table 12.4 for the median of a fictitious sample of 15 deposit takers. While the unweighted median is 8.1, the weighted median is 12.2.⁸

12.28 Different approaches can be used to assign weights to each observation. The Guide recommends the weight by asset size approach, therefore providing a comparable weighing scheme across FSIs when defining the quartiles. In short, the point of this exercise is to construct cumulative weights according to the position of the DT’s assets in the sector’s distribution for the FSI and use these cumulative weights to identify the FSI quartiles. To provide concrete guidance on the compilation of weighted quartiles, Box 12.1 provides step-by step instructions for their computation.

12.29 The example in Table 12.5 illustrates the estimate differences under unweighted and weighted methodologies.

Distribution: Measures of Shape

12.30 Measures of variability and dispersion including standard deviations can help identify the presence of outlier individual institutions. However, several studies argue that these measures fail to capture fully the “true risk” of the distribution. Therefore, it is also important to identify the relative effect of the outliers.

12.31 The Guide recommends the use of (1) weighted skewness and (2) weighted kurtosis as additional distribution measures (DMs) for this purpose. Skewness and kurtosis are measures of shape of a distribution or dataset and widely applied in modern finance to study, for example, asset return risk.

Skewness

12.32 Skewness can indicate the extent to which FSIs of individual institutions are asymmetrically distributed relative to the sectoral mean. It serves a similar purpose as the standard deviation. In addition, skew, or skewness of a dataset, furthers our understanding of whether the outliers are tilted toward the low or high end of the spectrum, and whether the mass of the distribution is concentrated toward the left or right of the mean.

12.33 Specifically, the Guide recommends, the computation of weighted skewness is a function of the third moment of the distribution—with the weighting

variable being the denominator of the FSI ratio, similar to the approach recommended for the standard deviation:

μ^{3} = \frac{Σ_{i = 1}^{N} {({F S I}_{i} - \bar{F S I})}^{3} \times ω_{i}}{σ^{3}}

12.34 Value of skewness can be positive, zero, or negative. Positive skewness indicates a longer right-hand-side tail of the distribution and the mass of the distribution concentrated toward the lef of the mean as illustrated in Figure 12.1, while negative skewness indicates a longer lef tail and mass of the distribution concentrated toward the right.

^{Figure 12.1}

Example of Right Skewed Distribution: Gamma Distribution with Parameters α = 2.5 and β = 0.5

Source: IMF staff estimates.

Kurtosis

12.35 Further, to get some insights on the proportional effect of the outliers, kurtosis may be calculated. Kurtosis estimates the degree of fatness of the tails of the FSI distribution compared to a normal distribution. Put simply, the kurtosis of distributions may be used to understand if the variability in the sectoral FSI is readily attributed to a few, extreme outliers (positive kurtosis) or several, modest deviations from the mean (negative kurtosis).

12.36 The Guide recommends weighted kurtosis for this purpose—with the weights constructed using the denominator of the FSI ratio. A common measure of kurtosis is the moment coefficient of kurtosis given by

μ^{4} = \frac{Σ_{i = 1}^{N} {({F S I}_{i} - \bar{F S I})}^{4} \times ω_{i}}{σ^{4}}

12.37 As the normal distribution is often used as the standard for comparison, it is common to subtract its kurtosis from that of the distribution to estimate “excess kurtosis.” The moment coefficient of kurtosis of a normal distribution equals 3.

Positive excess kurtosis indicates that the distribution has fatter tails and sharper peak than the normal distribution. This is known as a “lep-tokurtic” distribution.
Negative excess kurtosis indicates that the tails are “leaner” than the tails of the normal distribution. Such distributions are known as “platykurtic.”
The absence of excess kurtosis indicates that the distribution does not exhibit fat tails. This is referred to as “mesokurtic” distribution. For example, the student t-distribution (Figure 12.2) exhibits leptokurtosis.

^{Figure 12.2}

Leptokurtic Distribution versus Normal Distribution

Source: IMF staff calculations.

IV. Use of CDMs

12.38 This section illustrates the use of CDMs as an early diagnostic for the assessment of financial stability with two examples.

12.39 Figure 12.3 shows the evolution of the interquartile range of the Tier 1 Capital to RWA ratio for a selected group of French banks. The gray band shows the inter-quartile range and quantifes the dispersion of this capitalization ratio for these banks. The lower and upper limits of the band represent, respectively, the lower and higher Tier 1 capitalization ratio among the banks (i.e., the 25th and 75th percentile, respectively). The wider the gray band, the higher the dispersion. The black dotted line shows the median (the 50th percentile) of this capitalization ratio for these banks.

^{Figure 12.3}

Weighted Quartiles for Selected French Banks’ Capital Adequacy Ratios (2006–2016)

Source: Fitch Connect; and IMF staff calculations.

12.40 Two things stand out from this distribution analysis. First, the black dotted line shows an increasing trend over time, indicating that major banks in France built up capital resources since the global financial crisis. The first quartile went from 7.7 percent in 2006 to 12.9 percent in 2016, exhibiting substantial improvement in Tier 1 capitalization over time. This very positive development could be, in part, the result of tighter regulation during this period. At the same time, from 2006 to 2016, we also observe a development that could merit closer analysis: the widening of the weighted interquartile capital range. As the widening of the gray band shows, this interquartile capital range went from 1.2 in 2006 to 2.2 in 2016, reflecting a wider spread across banks’ capitalization.

12.41 Figure 12.4 shows the evolution of the return on assets (ROA), over time, for a sample of 20 randomly generated bank data. The weighted mean and median show a cyclical trend around 1.50 and 1.25, respectively, in dotted gray and solid black. The mean is larger than the median, indicating that the ROA distribution is skewed to the right.

^{Figure 12.4}

Distribution Measures Analysis for the Return on Assets

Source: IMF staff calculations.

12.42 In Figure 12.4, the gray line shows the weighted standard deviation, which quantifes how spread out are the ROAs of the individual banks around the mean for the sector. The standard deviation remains broadly stable indicating a low dispersion of the ROAs among the banks during the whole period.

12.43 The symmetry of the distribution around the mean is captured with the weighted skewness, which is represented with a dotted black line in the figure. Despite the overall low dispersion of the data, the skewness is rather volatile throughout the period of analysis. From 2015 onwards, the skewness becomes negative, indicating ROA of one or more banks fall on the lef-hand tail of the distribution. Put simply, this could possibly indicate an outlier—a bank with a large negative ROA. On a closer look at the underlying data, it appears that the ROA of one of banks fell from 0.25 in 2014Q4 to –30.90 in 2015Q4 while the assets declined by close to 300 percent, indicating financial difficulties. The strong impact of this bank’s ROA on the weighted skewness of the sector warrants further investigation into the performance of the bank, the underlying causes of the stress and thus, identifies any potential risk of spillovers to the entire sector. If large interconnections exist among the banks, any adverse shock to this bank can rapidly transmit to the entire system.

V. Addressing Confidentiality Issues

12.44 As with any system that involves decomposition of aggregated data, dissemination of CDMs can be constrained by confidentiality issues. One way of addressing confidentiality issues is to establish, for each CDM, a minimum number of reporting institutions (reporting threshold) to a point where values of individual institutions cannot be derived. The Guide introduces stricter reporting thresholds for all CDMs to preserve data confidentiality (Table 12.6). In addition, the Guide recommends flexibility in reporting these measures for countries where financial systems are highly concentrated.

^{Table 12.6}

Reporting Thresholds for Concentration and Distributions Measures

Source: IMF staff estimates.

^{Table 12.6}

Reporting Thresholds for Concentration and Distributions Measures

Measure	Required Minimum Number of Deposit Takers
Herfindahl concentration index	7
Weighted quartiles (weighted by shares of assets in total assets)	28
Weighted standard deviation	7
Weighted skewness	7
Weighted kurtosis	7

Source: IMF staff estimates.

^{Table 12.6}

Reporting Thresholds for Concentration and Distributions Measures

Measure	Required Minimum Number of Deposit Takers
Herfindahl concentration index	7
Weighted quartiles (weighted by shares of assets in total assets)	28
Weighted standard deviation	7
Weighted skewness	7
Weighted kurtosis	7

Source: IMF staff estimates.

Annex 12.1: CDM Template

Annex Figure 12.1.1

Concentration and Distribution Measures Template

Source: IMF staff.

See “The Financial Crisis and Information Gaps: Report to the G-20 Finance Ministers and Central Bank Governors,” prepared by IMF Staff and the FSB Secretariat, October 29, 2009, recommendation number 3.

See Crowley et al (2016).

The IAG was established in 2008 to coordinate international statistical work following the financial crisis.

There is vast literature on granularity adjustment, which was pioneered by Gordy (2003) in the context of credit risk concentration.

See Deutsche Bundesbank (2006) and Grippa and Gornicka (2016). Equations 15 and 16 in Emmer and Tasche (2005) provide some theoretical background.

The sector FSI is a weighted average.

The new sequence is known as the sequence of order statistics.

See step 3 on the derivation of the weighted quartiles.

Step 1. Sorting	Sort the sequence {FSI_k} in ascending order. As the sorting order will be the same for all the FSIs, the interpretation of the resulting quantiles will vary depending on the FSI. For the FSIs on Tier 1 capital, provisions to nonperforming loans (NPLs), returns on assets, and returns on equity, the value of the first quartile will point to institutions with relatively higher vulnerabilities, whereas for the FSIs on NPLs to gross loans and NPLs net of provisions to capital, the data for the first quartile will point to institutions with relatively lower vulnerabilities. The sorting will result in a new sequence {FSI_j} (j = 1, . . ., N) of the FSIs.⁷ Next to it, trace the corresponding sequence of assets {A_j}. Table 12.4 presents a sequence of FSIs sorted in an ascending order with their corresponding cumulative assets.
Step 2. Threshold Identification	Let T be the sum of the assets of DTs and W_i be the cumulative assets associated with each of the DTs in the distribution: $T = Σ_{j = 1}^{N} A_{j}, W_{i} = Σ_{j = 1}^{N} A_{j}$ (the assets cumulative frequency) Let “P” be the values of the asset that identify the theoretical thresholds that would define each of the quartiles. Compute the following quantities for each quartile Q_p = 0.25, 0.50, or 0.75): P = T × p, and Find the index W_i such that W_i > P to identify the cutoff point for each of the quartile.
Step 3. Derivation of the Weighted Quartiles	Compute the weighted quartiles as follows: $Q_{P} = {\begin{matrix} \frac{{F S I}_{i = 1} + {F S I}_{i}}{2} & i f W_{i - 1} = P \\ {F S I}_{i} & O t h e r w i s e \end{matrix}$ >Thus, if the cumulative asset frequency (W_i-1) falls exactly on the cutoff point (P) for the quartile, the value for that quartile, say the first quartile, would be determined by the average of the FSI corresponding to the cumulative asset frequency W_i-1 and the next value. As a result, for the first quartile, 25 percent of the FSI values will be less than the value provided for the quartile, as intended. When W_i > P (and W_i-1 < P), then the value for the quartile is FSI_i. Table 12.5 shows the calculation steps to derive the weighted and unweighted median based on the data presented in Table 12.4.

Step 1. Sorting	Sort the sequence {FSI_k} in ascending order. As the sorting order will be the same for all the FSIs, the interpretation of the resulting quantiles will vary depending on the FSI. For the FSIs on Tier 1 capital, provisions to nonperforming loans (NPLs), returns on assets, and returns on equity, the value of the first quartile will point to institutions with relatively higher vulnerabilities, whereas for the FSIs on NPLs to gross loans and NPLs net of provisions to capital, the data for the first quartile will point to institutions with relatively lower vulnerabilities. The sorting will result in a new sequence {FSI_j} (j = 1, . . ., N) of the FSIs.⁷ Next to it, trace the corresponding sequence of assets {A_j}. Table 12.4 presents a sequence of FSIs sorted in an ascending order with their corresponding cumulative assets.
Step 2. Threshold Identification	Let T be the sum of the assets of DTs and W_i be the cumulative assets associated with each of the DTs in the distribution: $T = Σ_{j = 1}^{N} A_{j}, W_{i} = Σ_{j = 1}^{N} A_{j}$ (the assets cumulative frequency) Let “P” be the values of the asset that identify the theoretical thresholds that would define each of the quartiles. Compute the following quantities for each quartile Q_p = 0.25, 0.50, or 0.75): P = T × p, and Find the index W_i such that W_i > P to identify the cutoff point for each of the quartile.
Step 3. Derivation of the Weighted Quartiles	Compute the weighted quartiles as follows: $Q_{P} = {\begin{matrix} \frac{{F S I}_{i = 1} + {F S I}_{i}}{2} & i f W_{i - 1} = P \\ {F S I}_{i} & O t h e r w i s e \end{matrix}$ >Thus, if the cumulative asset frequency (W_i-1) falls exactly on the cutoff point (P) for the quartile, the value for that quartile, say the first quartile, would be determined by the average of the FSI corresponding to the cumulative asset frequency W_i-1 and the next value. As a result, for the first quartile, 25 percent of the FSI values will be less than the value provided for the quartile, as intended. When W_i > P (and W_i-1 < P), then the value for the quartile is FSI_i. Table 12.5 shows the calculation steps to derive the weighted and unweighted median based on the data presented in Table 12.4.

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Abstract

I. Introduction

II. Background

The CDM Pilot Project

The FSI Compilers’ Perspective

III. Compilation of CDMs

The Sample

Concentration

Herfindahl concentration index

Distribution: Dispersion Measures

Weighted standard deviation

Weighted quartiles

Distribution: Measures of Shape

Skewness

Kurtosis

IV. Use of CDMs

V. Addressing Confidentiality Issues

Annex 12.1: CDM Template

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