The information value of past losses in operational risk

Filippo Curti and Marco Migueis

Abstract

Operational risk is a substantial source of risk for US banks. Improving the performance of operational risk models allows banks’ management to make more informed risk decisions by better matching economic capital and risk appetite, and it allows regulators to enhance their understanding of banks’ operational risk. We show that past operational losses are informative of future losses, even after controlling for a wide range of financial characteristics. We propose that the information provided by past losses results from their capturing hard-to-quantify factors such as the quality of operational risk controls, the risk culture and the risk appetite of the bank.

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Introduction

1 Introduction

Operational risk is a major source of risk for financial institutions and has grown in recent years. Several institutions have been severely damaged or bankrupted due to operational loss events. Barings Bank, a 350-year old institution, failed because Nick Leeson’s rogue transactions caused a USD1.3 billion loss (Jeremy 1995). Jérôme Kerviel’s rogue trades cost Société Générale over USD7 billion after he evaded numerous layers of computer controls and audits (Clark and Jolly 2016). In the United States, large financial institutions experienced tens of billions of dollars in losses due to the improper origination, securitization and foreclosure practices in the lead up to the 2007–9 global financial crisis (see, for example, Department of Justice 2013, 2014). More recently, Wells Fargo experienced multiple costly operational failures (Wattles et al 2018). The relevance of operational risk is recognized by US regulators, as demonstrated by operational risk accounting for 29% of the Advanced Approaches’ capital requirements of the 10 large, internationally active US bank holding companies (BHCs) with approved capital models, as of the end of 2019 (more than market risk, which corresponds to 5% of the Advanced Approaches’ capital requirements, and equivalent to 47% of credit risk Advanced Approaches’ capital requirements (Afonso et al 2019)).¹¹ 1 The Advanced Approaches are model-based capital requirements in the United States to which only large, internationally active banks are subject. All US banks are also subject to standardized risk-based capital requirements, which do not currently include operational risk. Further, projected operational losses under the severely adverse scenario reached USD144 billion for BHCs participating in the 2010 Dodd–Frank Act Stress Test (DFAST), corresponding to 33% of the projected pre-provision net revenue (PPNR) losses of DFAST BHCs.²² 2 Multiple academic studies have also demonstrated the magnitude of operational risk. For example, de Fontnouvelle et al (2006) showed that operational risk capital would likely exceed market risk capital and could reach several billions of US dollars in internationally active banks. Cummins et al (2006) and Gillet et al (2010) find strong market reactions to operational loss events. Allen and Bali (2007) investigate the cyclicality in operational risk and estimate that approximately 18% of banks’ returns compensate for operational risk (a figure that increases to 39% for depository institutions). Given the magnitude of operational risk, achieving a better understanding of its drivers and enhancing its modeling is of the utmost importance for practitioners and regulators. This paper contributes to this understanding by researching whether past losses are predictive of future exposure.

We investigate whether the inclusion of past operational losses improves the performance of operational risk models, and generally find that it does even when accounting for a wide range of quantifiable controls. We do not claim that past losses cause future losses, but rather we claim that they predict future losses because they capture hard-to-quantify drivers of exposure. In particular, we believe past losses can act as a proxy for banks’ operational risk control quality, risk culture and risk appetite. Banks with worse risk controls and less risk averse cultures are likely to experience larger and more numerous operational losses. As internal controls and risk culture are unlikely to change overnight (Lazear 1995; Kreps 1996), an association between controls and culture and operational losses explains why past operational losses are predictive of future operational losses.

We analyze different operational loss event types separately and find that past losses are predictive of future losses for all event types. In addition, we investigate the degree to which losses further back in time help to predict future losses and find that past losses are informative up to three years prior to the present. Also, we show that past losses suffered by peers are informative, even after controlling for an individual bank’s past losses. This finding suggests that past losses are useful proxies not only for bank-specific risk drivers but also for systemic risk drivers. We perform quantile regressions to assess whether past losses are also predictive of tail exposure and find that they are. Lastly, we find that historical loss frequency is generally more predictive of future exposure than historical loss severity, likely because loss frequency is a more stable metric of exposure than loss severity.

We consider multiple robustness checks. First, we add firm and time fixed effects to our regressions and find that past losses remain predictive of future losses. This result implies that the exposure factors, for which past losses act as proxies, are neither immutable within firms nor fully driven by aggregate time series trends. Second, we compare the performance of models including past losses with models including other metrics of operational risk management, specifically the risk management index from Ellul and Yerramilli (2013) and the number of Federal Reserve operational risk exam findings, and find that past losses outperform these other metrics in predicting operational risk exposure. We also replicate our main regressions by assigning losses to their date of accounting (the date on which losses resulted in an impact in financial statements) instead of the date of occurrence (the date when a bank judges the loss event to have occurred) used in other regressions, and we find that the results do not materially change.

Our study contributes to the literature exploring the drivers and predictors of operational risk (Chernobai et al 2012; Cope et al 2012; Cope and Carrivick 2013; Wang and Hsu 2013; Abdymomunov and Mihov 2019; Abdymomunov et al 2020; Curti et al 2020). Previous studies have argued that operational risk is linked to poor control practices and have shown that operational risk correlates with corporate governance and executive compensation (Chernobai et al 2012), with board diversity (Wang and Hsu 2013) and with regulatory assessments of banks’ risk management practices (Abdymomunov and Mihov 2019). Before now, to the best of our knowledge, no study has examined whether past operational losses are informative of future operational losses. The persistence of operational loss experience that we find supports the view that operational risk is driven by hard-to-measure factors such as internal controls, risk culture and risk appetite. Thus, despite our difference in focus, the findings of our paper are consistent with the previous literature. In addition, our study contributes to the broader literature assessing the performance of risk models in banking. This literature includes studies on credit risk (such as Rajan et al (2015) and Shumway (2001)) and on market risk (such as Berkowitz (2002)). However, similar studies for operational risk are lacking. Our results show that past losses significantly improve models’ ability to predict future losses.

How to best model operational risk is of particular policy interest at the moment because the Basel Committee on Banking Supervision introduced a new standardized framework for calculating operational risk capital (Basel Committee on Banking Supervision 2017), which allows for two variants of the calculation, one including losses and the other not. The Basel III Accord set an expectation that the operational risk standardized approach be implemented by January 2023 and, thus, regulators around the world are having to decide which version of the calculation to adopt. Certain jurisdictions have signaled their intention to not use historical losses in the calculation of operational risk capital (eg, the European Union, Canada), while others have not yet made their intentions clear (eg, the United States). Operational risk is also a critical component of banks’ losses in stress testing exercises such as DFAST, and the Federal Reserve uses historical operational losses in the calculation of stress operational losses in its supervisory model. Our results inform policy considerations around operational risk measurement by suggesting that incorporating historical losses in the operational risk capital requirements is likely to result in a more risk-sensitive approach, and that incorporating historical losses into stress test models will likely result in a more accurate measurement of exposure.

Operational failures can significantly affect banks’ balance sheets and hamper their ability to carry out day-to-day activities. At the same time, as the size and structural complexity of financial firms have increased, so has the challenge of understanding and mitigating operational risk. Our study can help banks’ risk managers build more accurate operational risk models. Better estimates of exposure would allow banks to better match their economic capital with their risk appetite, and thus to make better business decisions.

The remainder of this paper is organized as follows. Section 2 describes the data. Section 3 presents our regression results. Section 4 states our conclusions.

2 The data sample and variable definitions

2.1 Operational loss data

This paper relies on operational loss event data, financial statements and other supervisory information on 38 publicly traded BHCs that participate in the DFAST program. Operational loss event data follows the reporting requirements of the FR Y-14Q form and is provided by financial institutions with consolidated assets of USD50 billion or more.³³ 3 More information about FR Y-14Q reporting requirements, instructions and forms can be found on the Federal Reserve website. URL: http://www.federalreserve.gov/apps/reportforms/. Loss information (including loss amounts and counts) is used from 2000 Q1, or as far back as available, and up to 2017 Q4.⁴⁴ 4 According to the FR Y-14Q reporting instructions, BHCs must report a complete history of operational losses, “starting from the point-in-time at which the institution began capturing operational loss event data in a systematic manner”. Most of the BHCs in our sample report losses for periods prior to the 2020 Dodd–Frank Act. BHCs were already collecting such loss data in a systematic manner under the Basel II supervisory framework and for internal use. These data are subject to specific data quality checks, including regular exams conducted by Federal Reserve internal audit functions. The data is highly granular and provides information such as amounts, dates, classifications and descriptions. Definitions of the variables used in the analysis are presented in Table 1.

Consistent with Basel II definitions, losses are categorized into seven event types. The event types are internal fraud (IF), external fraud (EF), employment practices and workplace safety (EPWS), clients, products and business practices (CPBP), damage to physical assets (DPA), business disruption and system failures (BDSF) and execution, delivery and process management (EDPM). Table 2 provides definitions of each event type.

Table 1: Variable definitions. [GDP growth is the year-over-year US real GDP growth rate. HPI growth is the year-over-year growth rate in the US CoreLogic House Price Index. CREPI Growth is the year-over-year growth rate in the US Commercial Real Estate Price Index. VIX is the CBOE US Market Volatility Index, converted to a quarterly frequency by using the maximum close-of-day value in any quarter. BBB-T10Yr Sprd is the spread between the US 10-year BBB rated corporate bond yield and the 10-year US Treasury bond yield (higher values denote worse conditions). CCAR denotes the Comprehensive Capital Analysis and Review.]

(a) Operational loss metrics
Variable	Definition
OpLoss	The sum of the losses resulting from operational loss events experienced by a BHC over a quarter (in millions of US dollars)
ln(OpLoss)	A natural log transformation of OpLoss, defined as $\text{ln}(\text{1}+\text{OpLoss})$
$\overline{\text{OpLoss}}$	The quarterly average of OpLoss measured over the prior four quarters
$\text{ln}(\overline{\text{OpLoss}})$	A natural log transformation of $\overline{\text{OpLoss}}$, defined as $\text{ln}(\text{1}+\overline{\text{OpLoss}})$
$\overline{\text{Frequency}}$	The quarterly average number of loss events measured over the prior four quarters
$\text{ln}(\overline{\text{Frequency}})$	A natural log transformation of $\overline{\text{Frequency}}$, defined as $\text{ln}(\text{1}+\overline{\text{Frequency}})$
$\overline{\text{Severity}}$	The average severity of loss events measured over the prior four quarters
$\text{ln}(\overline{\text{Severity}})$	A natural log transformation of $\overline{\text{Severity}}$, defined as $\text{ln}(\text{1}+\overline{\text{Severity}})$
$\overline{\text{OpLossInd}}$	The size-weighted industry average of $\overline{\text{OpLoss}}$ calculated as per (3.2)
$\text{ln}(\overline{\text{OpLossInd}})$	A natural log transformation of $\overline{\text{OpLossInd}}$, defined as $\text{ln}(\overline{\text{OpLossInd}})$
$\overline{\text{FrequencyInd}}$	The size-weighted industry average of $\overline{\text{Frequency}}$ calculated as per (3.2)
$\text{ln}(\overline{\text{FrequencyInd}})$	A natural log transformation of $\overline{\text{FrequencyInd}}$, defined as $\text{ln}(\overline{\text{FrequencyInd}})$
$\overline{\text{SeverityInd}}$	Industry average loss event severity calculated as per (3.3)
$\text{ln}(\overline{\text{SeverityInd}})$	A natural log transformation of $\overline{\text{SeverityInd}}$, defined as $\text{ln}(\overline{\text{Severity}})$
$\text{ln}(\overline{\text{OpLossInd}})$	A natural log transformation of the industry quarterly average operational loss measured over the prior year as per (3.2)
(b) Controls
Variable	Definition
Size	BHC total assets (in millions of US dollars)
ln(Size)	A natural log transformation of Size, defined as $\text{ln}(\text{Size})$
II-to-NII	The ratio of BHC net interest income to noninterest income
RoE	BHC return on equity over a four-quarter window
T1 capital	BHC Tier 1 capital
Equity vol	Annualized BHC daily equity return volatility measured over the prior four quarters
CCAR age	The number of quarters since the BHC was first subject to the CCAR (it takes a value of 0 for the quarters before the BHC become subject to CCAR)
ME	US financial and economic environment measure, defined as the first principal component of GDP growth, HPI growth, CREPI growth, VIX and BBB-T10Yr Sprd
Risk management	BHC risk management index (RMI) value developed by Ellul and Yerramilli (2013) (a higher value corresponds to a higher quality of the BHCs’ risk management function)
OpRisk MR(I)A	The number of outstanding operational risk Matters Requiring Immediate Attention and Matters Requiring Attention findings at a BHC as of a given quarter
ln(OpRisk MR(I)A)	A natural log transformation of OpRisk MR(I)A, defined as $\text{ln}(\text{1}+\text{OpRisk MR(I)A})$

Table 2: Event types.

Event type
category	Abbreviation	Description
Internal fraud	IF	Acts of a type intended to defraud, misappropriate property or circumvent regulations, which involves at least one internal party
External fraud	EF	Acts of a type intended to defraud, misappropriate property or circumvent the law, by a third party
Employment practices and workplace safety	EPWS	Acts inconsistent with employment, health or safety laws or agreements, from payment of personal injury claims or from diversity/discrimination events
Clients, products and business practices	CPBP	An unintentional or negligent failure to meet a professional obligation to specific clients or from the nature or design of a product
Damage to physical assets	DPA	Damage to physical assets from natural disasters or other events
Business disruption and system failures	BDSF	Disruption of business or system failures
Execution, delivery and process management	EDPM	Failed transaction processing or process management, from relations with trade counterparties and vendors

Figure 1: Operational losses by event type. This figure presents the allocation of operational loss amounts (the percentage of the total loss amount and the US dollar loss amounts in billions) by event type. The sample includes 300 549 operational loss events incurred by 38 large US BHCs over the period 2000 Q1–2017 Q4.

Figure 1 presents the US dollar loss amounts by event type category, as well as the share of total losses corresponding to each event type. In aggregate, DFAST-participating BHCs suffered more than USD230 billion in operational losses over the period from 2000 Q1 to 2017 Q4. The most significant event type was CPBP, which accounted for USD180 billion, or 78%, of losses. EDPM was the second largest category by contribution, representing USD31 billion or 13% of losses. The remaining five event types comprised around USD20 billion, or less than 10% of losses.

The reporting threshold for individual operational losses varies across BHCs; thus, to ensure consistency across BHCs, we discarded losses below USD20 000, the highest reporting threshold for institutions participating in the DFAST program. The final sample in our data consisted of 300 549 individual loss events from 38 BHCs over the 2000 Q1–2017 Q4 period. Our data source is substantially richer than publicly available data used in other academic studies.⁵⁵ 5 For example, Chernobai et al (2012) use 2426 operational loss events from IBM’s Algo FIRST database. Financial statement data were obtained from FR Y-9C reports when available and supplemented with data from Bloomberg otherwise.⁶⁶ 6 Some firms in our sample, such as Goldman Sachs and Morgan Stanley, did not become BHCs until late in our sample period and so were not required to fill the FR Y-9C report for the earlier portion of our sample. Also, some non-BHCs (eg, Countrywide Financial) merged with or were acquired by BHCs in our sample. For consistency between the loss data sets and the financial statement variables used for these companies, Y-9C data was augmented by Bloomberg data where appropriate.

2.2 Operational loss metrics

The dependent variable in the regressions of this paper is the natural logarithm of the total amount of operational losses suffered by a BHC in a quarter. We focus on total loss amount (instead of, for example, loss frequency as in Chernobai et al (2012)) because this is the metric of exposure that is most relevant to risk managers and regulators. We chose to take the natural logarithms (and to do the same for the explanatory variables that incorporate quantities) to mitigate the volatility of loss totals, reduce the influence of outlier observations and not give undue weight to the observations of the largest firms in our regressions. The use of logs also facilitates the inclusion of certain control variables that are not proportional to firm scale (eg, Tier 1 capital ratio or return on equity (ROE)) in regressions. The total operational loss for quarter $t$ is the sum of the loss amounts for all events having a loss amount of at least USD20 000 and an occurrence date in quarter $t$.

We consider three metrics of past loss experience to predict future losses: the natural logarithm of the average quarterly total operational loss; the natural logarithm of the average quarterly operational loss frequency; and the natural logarithm of the average operational loss severity. These three averages are calculated over one calendar year (eg, the average quarterly total operational loss for 2018 Q2 corresponds to average of the total operational loss in 2017 Q3, 2017 Q4, 2018 Q1 and 2018 Q2). Frequency and severity represent two dimensions of a bank’s operational risk exposure. The frequency reflects how often operational loss events occur in a bank, while severity reflects how damaging these events are (on average). The total operational loss combines these two dimensions into a single measurement of exposure. The average frequency is typically more stable than the average severity and average total operational loss because the latter two can experience large swings due to extreme loss events. Descriptive statistics of the operational loss metrics are presented in Table 3(a).

Table 3: Descriptive statistics. [The sample includes 1266 quarterly observations of 38 large BHCs over the period 2000 Q1–2017 Q4 for which the requisite data are available. Part (a) reports the descriptive statistics on operational risk metrics, while part (b) reports those on the other variables used in our analyses. Definitions of the variables are given in Table 1. SD, standard deviation. P25, 25th percentile. P50 50th percentile (median). P75, 25th percentile.]

(a) Operational risk measures
	Mean	SD	P25	P50	P75
${\text{OpLoss}}_{t+\text{1}}$	182.956	1 288.472	003.627	011.885	061.974
$\text{ln}({\text{OpLoss}}_{t+\text{1}})$	002.947	001.804	001.501	002.524	004.104
$\overline{\text{OpLoss}}$	187.596	745.063	005.209	015.782	082.765
$\text{ln}(\overline{\text{OpLoss}})$	003.263	001.799	001.826	002.820	004.428
$\overline{\text{Frequency}}$	240.547	398.536	031.000	067.875	228.000
$\text{ln}(\overline{\text{Frequency}})$	004.533	001.327	003.466	004.232	005.434
$\overline{\text{Severity}}$	000.490	001.107	000.124	000.203	000.453
$\text{ln}(\overline{\text{Severity}})$	000.311	000.339	000.117	000.185	000.374
$\overline{\text{OpLossInd}}$	304.940	300.462	083.911	205.400	428.166
$\text{ln}(\overline{\text{OpLossInd}})$	005.276	000.971	004.430	005.325	006.060
$\overline{\text{FrequencyInd}}$	459.855	083.472	385.320	474.340	519.059
$\text{ln}(\overline{\text{FrequencyInd}})$	006.113	000.191	005.954	006.162	006.252
$\overline{\text{SeverityInd}}$	597 326	520 255	217 709	423 060	839 678
$\text{ln}(\overline{\text{SeverityInd}})$	012.978	000.802	012.291	012.955	013.641
(b) Control variables
	Mean	SD	P25	P50	P75
Size	496 969	689 981	114 976	171 164	386 296
$\text{ln}(\text{Size})$	012.398	001.119	011.652	012.050	012.864
II-to-NII	001.806	003.356	001.007	001.770	002.657
RoE	010.267	038.456	005.785	011.046	015.719
T1 capital	011.902	003.699	010.631	012.014	013.620
Equity vol	037.288	026.966	021.068	027.933	038.728
CCAR age	007.656	008.959	000.000	004.000	014.000
ME index	109.460	011.693	101.270	105.064	112.970
Risk management	001.006	000.259	000.761	001.065	001.213
OpRisk MR(I)A	004.577	008.982	000.000	001.000	005.000
ln(OpRisk MR(I)A)	001.058	001.059	000.000	000.693	001.792

2.3 Control variables

The regressions in this paper include a variety of firm-specific control factors that previous research has found relevant in explaining operational risk exposure. The first and most important control is firm size, as measured by the natural logarithm of the total assets. Previous research has shown that asset size is positively associated with operational losses (Abdymomunov and Curti 2020; Curti et al 2020).⁷⁷ 7 Following Chernobai et al (2012), multiple studies have used the market value of equity to control for size in operational risk studies. We have chosen not to do so because meaningful market value of equity figures are not available for many of the firms in our sample, as they are US holding companies of foreign firms (eg, Deutsche Bank, Barclays), and thus using the market value of equity as a control would meaningfully restrict our sample. Also, we believe the scale of operations of a bank is better represented by its total assets than by its market value of equity. Nevertheless, we have estimated the regressions of this paper using market value of equity instead of total assets on the more restricted sample for which the market value of equity is available, and the estimates of the effect for our variables of interest remain similar. Results are available from the corresponding author upon request. The net interest income to noninterest income (net II-to-NII) ratio has been used in the banking literature as a proxy for diversification, and previous studies have shown that this ratio affects profitability and risk (Baele et al 2007; Elsas et al 2010). The design of the Basel Committee new standardized approach also indicates that firms with a greater focus on nontraditional banking activities (and thus more noninterest income) likely experience more operational risk. For these reasons, we have included the net interest income to noninterest income ratio in our regressions. We borrow three additional controls from Chernobai et al (2012): ROE, Tier 1 capital ratio and equity return volatility.⁸⁸ 8 ROE is the net income divided by equity capital. The Tier 1 capital ratio is equal to the Tier 1 capital divided by the risk-weighted assets. Equity return volatility is the annual daily volatility of returns over the previous calendar year. Chernobai et al (2012) find that operational loss frequency is negatively associated with the Tier 1 capital ratio, positively associated with equity return volatility and positively associated with ROE (albeit, in this last case, the effect is not statistically significant). We also control for how long the firm has been subject to Comprehensive Capital Analysis and Review (CCAR), as the tighter supervision associated with CCAR may have contributed to improving firms’ risk management practices and thus affected firms’ losses. Finally, we control for the macroeconomic environment by using an index proposed by Abdymomunov et al (2020). Descriptive statistics of the control variables are provided in Table 3(b).

2.4 Pairwise correlation

Table 4 presents the pairwise correlation between the variables included in this study. In the forthcoming regressions, we explain the total operational loss in a given quarter by using lagged values of the explanatory variables, and thus we include total operational loss one period ahead of the explanatory variables considered in Table 4. The log total operational loss is highly correlated with lagged log average total operational loss as well as with lagged log average operational loss frequency (78.9% and 77.3%, respectively (see part (a))). The correlation with the lagged log average loss severity is weaker (27.3%), but still statistically significant.

The log total operational loss is highly correlated with lagged log total assets (75%). Correlations with other control variables are weaker, and some of them are not statistically significant (see part (b)). Of the remaining control variables, the ones whose correlation with log total operational loss has greatest magnitude are the net II-to-NII ratio ($-18.5$%) and the Tier 1 capital ratio ($-16.9$%).

3 Empirical analysis

3.1 Hypothesis

The main hypothesis we test with this study is whether past operational loss levels are predictive of future operational loss levels. We include in our analysis a wide range of controls, drawn from the literature on the determinants of operational risk. Nevertheless, we expect past losses to be predictive of future losses because operational losses generally result from the failures of a firm’s internal control processes and correlate with the risk firm’s culture, both of which change slowly and are not directly measured by our other regression controls. In testing this hypothesis, we use the following regression equation:

$\ln ({\mathrm{OpLoss}}_{i,t+1})={\beta }_1+{\beta }_2\ln ({\overline{\mathrm{OpLossMetrics}}}_{i,t})+{\displaystyle \sum _{j=1}^k}{\beta }_j{x}_{j,i,t}+{\epsilon }_{i,t},$

(3.1)

where the total operational loss in quarter $t+1$ is denoted by ${\mathrm{OpLoss}}_{t+1}$; the metrics of operational loss experience reflecting the averages between quarter $t-3$ and quarter $t$ are denoted by ${\overline{\mathrm{OpLossMetrics}}}_{i,t}$; $x_{j,i,t}$ is a control variable as measured in quarter $t$ (control variables include the log of total assets, II-to-NII ratio, ROE, Tier 1 capital ratio, equity return volatility, the number of years the firm has been in CCAR and the macroeconomic environment index); and ${\epsilon }_{i,t}$ is the error term.

3.2 Main regression

Table 5 presents the main regression results of the paper. This table includes a regression just with the control variables (column (1)) plus two regressions that include our main explanatory variables: a regression that includes lagged average total operational loss as an explanatory variable (column (2)); and a regression that includes lagged average loss frequency and lagged average loss severity as separate explanatory variables (column (3)).

Table 4: Pairwise correlations. [This table presents pairwise correlations. The sample includes 1266 quarterly observations of 38 large BHCs over the period 2000 Q1–2017 Q4 for which the requisite data are available. Part (a) reports the pairwise correlations between different operational loss metrics, while part (b) reports pairwise correlations between the realized total operational loss in quarter $t+\text{1}$ and the control variables used in our analyses.]

(a) Operational risk measures
	$\text{𝐥𝐧}\mathbf{(}{\text{𝐎𝐩𝐋𝐨𝐬𝐬}}_{𝒕\mathbf{+}\text{𝟏}}\mathbf{)}$	$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐎𝐩𝐋𝐨𝐬𝐬}}\mathbf{)}$	$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐅𝐫𝐞𝐪𝐮𝐞𝐧𝐜𝐲}}\mathbf{)}$	$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐒𝐞𝐯𝐞𝐫𝐢𝐭𝐲}}\mathbf{)}$	$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐎𝐩𝐋𝐨𝐬𝐬𝐈𝐧𝐝}}\mathbf{)}$	$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐅𝐫𝐞𝐪𝐮𝐞𝐧𝐜𝐲𝐈𝐧𝐝}}\mathbf{)}$	$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐒𝐞𝐯𝐞𝐫𝐢𝐭𝐲𝐈𝐧𝐝}}\mathbf{)}$
$\text{𝐥𝐧}\mathbf{(}{\text{𝐎𝐩𝐋𝐨𝐬𝐬}}_{𝒕\mathbf{+}\text{𝟏}}\mathbf{)}$	1.000
$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐎𝐩𝐋𝐨𝐬𝐬}}\mathbf{)}$	0.839	1.000
	(0.000)
$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐅𝐫𝐞𝐪𝐮𝐞𝐧𝐜𝐲}}\mathbf{)}$	0.840	0.878	1.000
	(0.000)	(0.000)
$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐒𝐞𝐯𝐞𝐫𝐢𝐭𝐲}}\mathbf{)}$	0.421	0.675	0.282	1.000
	(0.000)	(0.000)	(0.000)
$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐎𝐩𝐋𝐨𝐬𝐬𝐈𝐧𝐝}}\mathbf{)}$	0.326	0.331	0.170	0.348	1.000
	(0.000)	(0.000)	(0.000)	(0.000)
$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐅𝐫𝐞𝐪𝐮𝐞𝐧𝐜𝐲𝐈𝐧𝐝}}\mathbf{)}$	0.270	0.279	0.099	0.341	0.908	1.000
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
$\text{𝐥𝐧}\mathbf{(}\overline{\text{𝐒𝐞𝐯𝐞𝐫𝐢𝐭𝐲𝐈𝐧𝐝}}\mathbf{)}$	0.331	0.334	0.183	0.340	0.995	0.861	1.000
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)

(b) Control variables
					T1	Equity	CCAR	ME	Risk	ln(OpRisk
	$\text{𝐥𝐧}\mathbf{(}{\text{𝐎𝐩𝐋𝐨𝐬𝐬}}_{𝒕\mathbf{+}\text{𝟏}}\mathbf{)}$	$\text{𝐥𝐧}\mathbf{(}\text{𝐒𝐢𝐳𝐞}\mathbf{)}$	II-to-NII	RoE	capital	vol	age	index	management	MR(I)A)
$\text{𝐥𝐧}\mathbf{(}{\text{𝐎𝐩𝐋𝐨𝐬𝐬}}_{𝒕\mathbf{+}\text{𝟏}}\mathbf{)}$	$-$1.000
$\text{𝐥𝐧}\mathbf{(}\text{𝐒𝐢𝐳𝐞}\mathbf{)}$	$-$0.770	$-$1.000
	$-$(0.000)
II-to-NII	$-$0.176	$-$0.170	$-$1.000
	$-$(0.000)	$-$(0.000)
RoE	$-$0.013	$-$0.020	$-$0.060	$-$1.000
	$-$(0.653)	$-$(0.467)	$-$(0.032)
T1 capital	$-$0.177	$-$0.037	$-$0.015	$-$0.076	$-$1.000
	$-$(0.000)	$-$(0.185)	$-$(0.600)	$-$(0.007)
Equity vol	$-$0.139	$-$0.005	$-$0.032	$-$0.061	$-$0.078	$-$1.000
	$-$(0.000)	$-$(0.861)	$-$(0.261)	$-$(0.030)	$-$(0.006)
CCAR age	$-$0.101	$-$0.216	$-$0.041	$-$0.051	$-$0.282	$-$0.376
	$-$(0.000)	$-$(0.000)	$-$(0.148)	$-$(0.071)	$-$(0.000)	$-$(0.000)
ME index	$-$0.179	$-$0.009	$-$0.081	$-$0.058	$-$0.191	$-$0.801	$-$0.411	$-$1.000
	$-$(0.000)	$-$(0.760)	$-$(0.004)	$-$(0.038)	$-$(0.000)	$-$(0.000)	$-$(0.000)
Risk management	$-$0.089	$-$0.191	$-$0.088	$-$0.141	$-$0.078	$-$0.053	$-$0.191	$-$0.010	$-$1.000
	$-$(0.041)	$-$(0.000)	$-$(0.044)	$-$(0.001)	$-$(0.074)	$-$(0.223)	$-$(0.000)	$-$(0.819)
ln(OpRisk MR(I)A)	$-$0.326	$-$0.342	$-$0.035	$-$0.058	$-$0.254	$-$0.140	$-$0.084	$-$0.160	$-$0.231	$-$1.000
	$-$(0.000)	$-$(0.000)	$-$(0.257)	$-$(0.061)	$-$(0.000)	$-$(0.000)	$-$(0.006)	$-$(0.000)	$-$(0.000)

Table 5: Operational loss metrics and current losses ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$). [This table reports coefficients from panel regressions of realized total operational loss in quarter $t+\text{1}$ on operational loss metrics and control variables measured in quarter $t$. The estimation sample comprises an unbalanced panel of 1266 quarterly observations of 38 large BHCs over the period 2000 Q1–2017 Q4 for which requisite data are available. $\text{ln}({\text{OpLoss}}_{t+\text{1}})$ is the natural log of the total operational loss (in millions of US dollars) incurred by a BHC in a given calendar quarter. In column (1), we only include control variables. In column (2) we include $\text{ln}(\overline{\text{OpLoss}})$, the natural log of BHC quarterly average total operational loss measured over the four prior quarters, in addition to the control variables. In column (3) we include $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$, the natural logs of BHC quarterly average loss frequency and quarterly average loss severity, respectively, over the four prior quarters. Standard errors are clustered by BHC and quarter. $p$-values are shown in parentheses. ${}^{*}$, ${}^{**}$ and ${}^{***}$ denote , and , respectively.]

	(1)	(2)	(3)
$\text{ln}(\overline{\text{OpLoss}})$		$-$0.510${}^{***}$
		$-$(0.000)
$\text{ln}(\overline{\text{Frequency}})$			$-$0.852${}^{***}$
			$-$(0.000)
$\text{ln}(\overline{\text{Severity}})$			$-$0.389${}^{***}$
			$-$(0.002)
$\text{ln}(\text{Size})$	$-$1.306${}^{***}$	$-$0.601${}^{***}$	$-$0.382${}^{***}$
	$-$(0.000)	$-$(0.000)	$-$(0.000)
II-to-NII	$-$0.025${}^{**}$	$-$0.014${}^{**}$	$-$0.023${}^{***}$
	$-$(0.050)	$-$(0.020)	$-$(0.000)
RoE	$-$0.002	$-$0.001	$-$0.000
	$-$(0.298)	$-$(0.279)	$-$(0.320)
T1 capital	$-$0.032${}^{**}$	$-$0.022${}^{***}$	$-$0.034${}^{**}$
	$-$(0.012)	$-$(0.009)	$-$(0.025)
Equity vol	$-$0.000	$-$0.001	$-$0.000
	$-$(0.908)	$-$(0.791)	$-$(0.833)
CCAR age	$-$0.047${}^{***}$	$-$0.024${}^{***}$	$-$0.035${}^{***}$
	$-$(0.000)	$-$(0.002)	$-$(0.000)
ME index	$-$0.009	$-$0.002	$-$0.001
	$-$(0.156)	$-$(0.694)	$-$(0.867)
$N$	1266	1266	1266
Adjusted $R^{\text{2}}$	0.677	0.738	0.778
BIC	3704.456	$-$3447.104	$-$3244.402
$\mathrm{\Delta }\text{BIC}$	0	0$-$257.352	0$-$460.054

We find that the average total operational loss is predictive of future total operational loss. A 1% increase in the average total operational loss is associated with a 0.51% increase in the expected total operational loss in the ensuing quarter. This coefficient is statistically significant, and the Bayesian information criterion (BIC) shows that the regression model including lagged average total operational loss performs meaningfully better (lower BIC) than the baseline regression with the control variables alone.⁹⁹ 9 Kass and Raftery (1995) showed that a model should be strongly preferred to another when its BIC is are more than six units larger.

The regression in column (3) includes, separately, lagged average loss frequency and lagged average loss severity. Both dimensions of operational risk exposure prove predictive of operational loss totals. Nevertheless, the estimated coefficient of the lagged average frequency is much larger than the estimated coefficient of the lagged average severity. A 1% increase in average loss frequency is associated with a 0.85% increase in the expected total operational loss in the ensuing quarter, while a 1% increase in average loss severity is associated with a 0.39% increase in the expected total operational loss in the ensuing quarter. This regression performs much better than regression (2) according to the BIC, which indicates that taking into account the frequency and severity of losses separately will likely add more information than relying solely on past loss totals.

The effects of the control variables that are statistically significant have the expected sign. Asset size is positively associated with losses, while the NI-to-NII ratio, Tier 1 capital ratio and the number of years of participation in CCAR are negatively associated with losses (a higher NI-to-NII ratio likely means less complexity, a larger Tier 1 capital ratio means a less risky bank and a longer participation time in CCAR likely means less risk). ROE, equity return volatility and the macroeconomic environment index are not statistically significant predictors of total operational loss in our regressions. Notably, the effect of asset size meaningfully decreases when past loss metrics are included in the regressions. This indicates that, while size is a good proxy for operational risk exposure, the inclusion of metrics of loss experience adds relevant information relative to models that consider size alone.¹⁰¹⁰ 10 We also considered alternative specifications where both the dependent total loss variable and the explanatory lagged losses variables were scaled by total assets, and the results regarding the significance of historical losses remain consistent (see the table in Appendix A (online)).

These regression results are consistent with operational loss exposure being persistent and partly driven by factors that cannot be easily accounted for through financial statement metrics. These factors likely include operational risk control quality, risk culture and risk appetite. Such factors influence operational loss history and are slow moving. Therefore, they likely explain why operational loss history predicts future operational losses even after controlling for various balance-sheet and business model factors. These results support the inclusion of historical operational losses in the modeling of future operational risk exposure. In addition, they support the separate consideration of frequency and severity in operational risk models.

3.3 Regressions by event type

Table 6 shows regressions by operational loss event type (ie, both the dependent variable and the lagged loss metric are calculated using only losses from a certain event type). There are seven event types under the Basel categorization (Basel Committee on Banking Supervision 2006), as listed in Section 2.1. All these regressions include the same set of controls as those in Section 3.2.¹¹¹¹ 11 Note that banks sometimes have no operational loss events in a quarter. In those cases, the log of operational loss frequency plus one and the log of operational loss severity plus one equal zero.

The lagged average total operational loss (part (a)) is predictive of future total operational loss across all event types, except for damage to physical assets. Meanwhile, the regressions where lagged average loss frequency and lagged average loss severity are accounted for separately (part (b)) show that lagged average loss frequency is always positively associated with future total loss and is statistically significant. The effect of average loss severity is positive and statistically significant for CPBP and EDPM (the event types with the greatest loss amounts in US dollars) but is negative and statistically significant for BDSF. The negative association between past BDSF loss severity and future BDSF losses may reflect firms investing in operational resilience after large disruptions, and therefore minimizing future losses. In the other event types, lagged average loss severity is not a statistically significant predictor of future total operational loss. For all event types except EPWS, the regression that accounts for frequency and severity separately performs better according to the BIC than the regression that includes lagged average total operational loss alone.

Table 6: Event types ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$). [This table reports coefficients from panel regressions of realized total operational loss in quarter $t+\text{1}$ on operational loss metrics and control variables measured in quarter $t$ by event type. $\text{ln}({\text{OpLoss}}_{t+\text{1}})$ is the natural log of the total operational loss (in millions of US dollars) of a specific event type incurred by a BHC in a given calendar quarter. $\text{ln}(\overline{\text{OpLoss}})$ is the natural log of BHC quarterly average total operational loss of a specific event type measured over the four prior quarters. $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$ are the natural logs of BHC quarterly average loss frequency and quarterly average loss severity of a specific event type, respectively, over the four prior quarters. Event types are as listed in Section 2.1. Part (a) reports results using $\text{ln}(\overline{\text{OpLoss}})$ as an explanatory variable. Part (b) reports results using $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$ as explanatory variables. Control variables ($\ln (\mathrm{Size})$, II-to-NII, RoE, T1 capital, equity vol, CCAR age, ME index) are included in all the regressions in this table, but their coefficient estimates are omitted for brevity. Standard errors are clustered by BHC and quarter. $p$-values are shown in parentheses. ${}^{*}$, ${}^{**}$ and ${}^{***}$ denote , and , respectively.]

(a) Average total operational loss as explanatory variable
	(1)	(2)	(3)	(4)	(5)	(6)	(7)
	IF	EF	EPWS	CPBP	DPA	BDSF	EPDM
$\text{ln}(\overline{\text{OpLoss}})$	0.341${}^{***}$	0.794${}^{***}$	0.681${}^{***}$	0.325${}^{***}$	0.101	0.244${}^{***}$	0.657${}^{***}$
	(0.000)	(0.000)	(0.000)	(0.000)	(0.177)	(0.006)	(0.000)
BHC controls	Yes	Yes	Yes	Yes	Yes	Yes	Yes
$N$	1266	1266	1266	1266	1266	1266	1266
Adjusted $R^{\text{2}}$	0.428	0.761	0.774	0.635	0.148	0.444	0.754
BIC	$-$1877.808	$-$2223.694	$-$1823.787	$-$4120.929	$-$1626.008	$-$1770.557	$-$2860.405
$\mathrm{\Delta }\text{BIC}$	0$-$139.9486	$-$1166.4607	0$-$494.2363	0$-$102.0532	000$-$4.1706	00$-$57.4873	0$-$559.6169
(b) Average loss frequencies and average loss severities as explanatory variables
	(1)	(2)	(3)	(4)	(5)	(6)	(7)
	IF	EF	EPWS	CPBP	DPA	BDSF	EPDM
$\text{ln}(\overline{\text{Frequency}})$	$-$0.365${}^{***}$	$-$0.533${}^{***}$	$-$0.416${}^{***}$	$-$0.534${}^{***}$	$-$0.179${}^{***}$	$-$0.376${}^{***}$	$-$0.740${}^{***}$
	$-$(0.000)	$-$(0.000)	$-$(0.000)	$-$(0.000)	$-$(0.000)	$-$(0.000)	$-$(0.000)
$\text{ln}(\overline{\text{Severity}})$	$-$0.008	$-$0.138	$-$0.088	$-$0.245${}^{***}$	$-$0.015	$-$0.138${}^{***}$	$-$0.671${}^{**}$
	$-$(0.890)	$-$(0.284)	$-$(0.464)	$-$(0.001)	$-$(0.667)	$-$(0.000)	$-$(0.017)
BHC controls	Yes	Yes	Yes	Yes	Yes	Yes	Yes
$N$	1266	1266	1266	1266	1266	1266	1266
Adjusted $R^{\text{2}}$	$-$0.458	$-$0.802	$-$0.763	$-$0.646	$-$0.202	$-$0.505	$-$0.772
BIC	$-$1814.533	$-$1995.262	$-$1890.379	$-$4086.652	$-$1548.508	$-$1629.954	$-$2772.317
$\mathrm{\Delta }\text{BIC}$	0$-$203.224	$-$1394.8931	0$-$427.6448	0$-$136.3302	00$-$81.6704	0$-$198.0902	0$-$647.7053

Increases in the lagged average total loss predict the greatest increases in EF, EPWS and EDPM losses (a 1% increase in lagged average event type total loss is associated with increases of 0.79%, 0.68% and 0.66% in the ensuing quarter’s event type total loss, respectively). While increases in lagged average event-type loss frequency predict the greatest increases in EDPM, CPBP and EF (a 1% increase in lagged average event-type loss frequency is associated with increases of 0.74%, 0.53% and 0.53% in the ensuing quarter’s event type total loss, respectively). EDPM losses also have the strongest association with lagged average loss severity (a 1% increase in lagged average event-type loss severity is associated with a 0.67% increase in the ensuing quarter’s event type total loss).

The weak relationship between DPA losses and metrics of its lagged loss experience (the lagged average total loss is not significant, and in the frequency-and-severity regression the BIC improvement obtained from introducing the historical loss metrics is the smallest for the DPA event type) is unsurprising because DPA losses are driven more by external events (eg, weather events) than by internal controls, risk culture and risk appetite. So, this weaker relationship is consistent with the mechanism that we hypothesize explains why past operational losses are predictive of future operational losses.

These event type regressions strongly support the robustness of our “top-of-the-house” results, as they show that the relevance of past operational losses in predicting future operational losses is not a feature of one kind of operational loss event, but rather is a general property of operational risk exposure across its various dimensions.

3.4 Lag structure

To further understand the dependency between operational losses over time, we performed additional regressions where we discretely added up to four years of average loss metrics. Table 7 presents the regression results. Part (a) focuses on using the lagged average total loss, while the regressions in part (b) use the lagged average loss frequency and lagged average loss severity separately. In the regression in column (1), only the average for the year previous to the quarter to be explained is used, while in column (2), the average for the year previous to that is added, and so on. To allow for fair comparisons across these regressions, we only include the observations that could be included in all regressions (ie, the first observation in these regressions refers to the start of the fifth year for which there is data for a bank because four lag years are needed, rather than just one lag year as is the case in the other regressions in this paper). This approach implies that the number of observations (1036) is less than in the other regressions in this paper (1266), and that the regression results from column (1) in both parts of Table 7 are slightly different from those in Table 5.

The regressions in part (b) generally show that the information provided by the loss frequency is almost fully captured by the frequency in the previous year (regression (4) shows that loss frequency lagged four years has a negative and statistically significant effect, but regression (4) performs worse than regression (2) or regression (3) according to the BIC). Meanwhile, the average loss severity with two-year and three-year lags appears more predictive of losses than average loss severity in the previous year. This difference between frequency and severity likely results from frequency being a stable metric, not prone to upswings and downswings year to year, and thus, once frequency increases, exposure can be expected to increase in the near future. In contrast, severity is volatile, and thus, it takes multiple years of observed higher severity to become more certain that exposure has increased.

The regressions in part (a) show that lagged average total loss is informative up to three years prior to the modeled quarter. The relevance of total losses with lags of up to three years is likely explained by the compound effect of frequency and severity, which, as discussed above, provide information about future exposure in the first lag (frequency) and in the second and third lags (severity).

3.5 Industry experience

The hard-to-measure factors driving operational risk exposure (which, we argue in this paper, can be proxied by past losses) likely have commonalities across firms in the banking industry. For example, an increase in the rate of credit card fraud across the industry is likely to also provide information about the exposure of an individual firm that has not yet observed a large increase in fraud in its own credit cards. In this subsection, we formally test whether industry-wide historical operational loss experience helps explain the operational risk exposure of individual firms. To do so, we define three metrics of the industry-wide operational loss experience based on average total loss, average loss frequency and average loss severity. Consistent with our firm-level historical metrics, we calculate industry averages over a four-quarter window. To separate the effect of a firm’s own loss experience from the effect of the industry experience, we exclude a firm’s losses from the calculations of the industry metric (thus, the industry loss metrics on a given quarter will vary slightly across the firms). Also, because the number and size of firms change over our sample period, simple averages of industry loss metrics would move even if the average riskiness of firms, controlling for their size, is unchanged. To eliminate such confounding effects from our metrics of industry historical loss experience, we calculate these metrics through the approach below.

Table 7: Lagged operational loss metrics and current losses. [This table reports coefficients from panel regressions of realized total operational loss in quarter $t+\text{1}$ on operational loss metrics and control variables measured in quarter $t$. $\text{ln}({\text{OpLoss}}_{t+\text{1}})$ is the natural log of the total operational loss (in millions of US dollars) incurred by a BHC in a given calendar quarter. $\text{ln}(\overline{\text{OpLoss}})$, $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$ are the natural logs of BHC quarterly average total operational loss, average loss frequency and average loss severity measured over the four prior quarters. $\text{ln}({\overline{\text{OpLoss}}}_{t-\text{4}})$, $\text{ln}({\overline{\text{Frequency}}}_{t-\text{4}})$ and $\text{ln}({\overline{\text{Severity}}}_{t-\text{4}})$ are the natural logs of BHC quarterly average total operational loss, average loss frequency and average loss severity measured over the quarters $t-\text{7}$ to $t-\text{4}$. $\text{ln}({\overline{\text{OpLoss}}}_{t-\text{8}})$, $\text{ln}({\overline{\text{Frequency}}}_{t-\text{8}})$ and $\text{ln}({\overline{\text{Severity}}}_{t-\text{8}})$ are the natural logs of BHC quarterly average total operational loss, average loss frequency and average loss severity measured over the quarters $t-\text{11}$ to $t-\text{8}$. $\text{ln}({\overline{\text{OpLoss}}}_{t-\text{12}})$, $\text{ln}({\overline{\text{Frequency}}}_{t-\text{12}})$ and $\text{ln}({\overline{\text{Severity}}}_{t-\text{12}})$ are the natural logs of BHC quarterly average total operational loss, average loss frequency and average loss severity measured over the quarters $t-\text{15}$ to $t-\text{12}$. Part (a) reports the regression results when the lagged quarterly average total operational loss is used as an explanatory variable. Part (b) reports the regression results when lagged quarterly average loss frequency and lagged quarterly average loss severity are used as explanatory variables. Control variables ($\text{ln}(\text{Size})$, II-to-NII, RoE, T1 capital, equity vol, CCAR age, ME index) are included, but their coefficient estimates are omitted for brevity. Standard errors are clustered by BHC and quarter. $p$-values are shown in parentheses. ${}^{*}$, ${}^{**}$ and ${}^{***}$ denote , and , respectively.]

(a) Average total operational loss as the
explanatory variable ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$)
	(1)	(2)	(3)	(4)
$\text{ln}(\overline{\text{OpLoss}})$	0.503${}^{***}$	0.327${}^{***}$	0.279${}^{***}$	0.275${}^{***}$
	(0.000)	(0.000)	(0.000)	(0.000)
$\text{ln}({\overline{\text{OpLoss}}}_{t-\text{4}})$		0.329${}^{***}$	0.264${}^{***}$	0.262${}^{***}$
		(0.000)	(0.000)	(0.000)
$\text{ln}({\overline{\text{OpLoss}}}_{t-\text{8}})$			0.178${}^{***}$	0.173${}^{***}$
			(0.000)	(0.000)
$\text{ln}({\overline{\text{OpLoss}}}_{t-\text{12}})$				0.016
				(0.617)
BHC controls	Yes	Yes	Yes	Yes
$N$	1036	1036	1036	1036
Adjusted $R^{\text{2}}$	0.741	0.760	0.766	0.766
BIC	$-$2799.761	$-$2725.988	$-$2708.425	$-$2715.157
$\mathrm{\Delta }\text{BIC}$	0$-$203.7258	0$-$277.4985	0$-$295.0614	0$-$288.3294
(b) Average loss frequency and average loss severity as
explanatory variables ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$)
	(1)	(2)	(3)	(4)
$\text{ln}(\overline{\text{Frequency}})$	0.842${}^{***}$	0.785${}^{***}$	0.795${}^{***}$	0.783${}^{***}$
	(0.000)	(0.000)	(0.000)	(0.000)
$\text{ln}({\overline{\text{Frequency}}}_{t-\text{4}})$		0.088	0.145	0.140
		(0.321)	(0.344)	(0.361)
$\text{ln}({\overline{\text{Frequency}}}_{t-\text{8}})$			$-$0.052	0.185
			(0.699)	(0.267)
$\text{ln}({\overline{\text{Frequency}}}_{t-\text{12}})$				$-$0.231${}^{**}$
				(0.021)
$\text{ln}(\overline{\text{Severity}})$	0.288${}^{**}$	0.165${}^{*}$	0.106	0.064
	(0.015)	(0.052)	(0.160)	(0.526)
$\text{ln}({\overline{\text{Severity}}}_{t-\text{4}})$		0.467${}^{***}$	0.420${}^{***}$	0.413${}^{***}$
		(0.000)	(0.000)	(0.000)
$\text{ln}({\overline{\text{Severity}}}_{t-\text{8}})$			0.382${}^{***}$	0.375${}^{***}$
			(0.000)	(0.000)
$\text{ln}({\overline{\text{Severity}}}_{t-\text{12}})$				0.110
				(0.353)
BHC controls	Yes	Yes	Yes	Yes
$N$	1036	1036	1036	1036
Adjusted $R^{\text{2}}$	0.787	0.792	0.794	0.795
BIC	$-$2601.683	$-$2593.127	$-$2592.031	$-$2600.051
$\mathrm{\Delta }\text{BIC}$	0$-$399.8257	0$-$410.3599	0$-$411.4557	0$-$403.4357

1. (1)

   To obtain the average industry-wide operational loss total and the average industry-wide operational loss frequency for each observation, we took the following steps.

   1. (a)

      We calculated the average total operational loss (average operational loss frequency) for all banks.

   2. (b)

      We summed these averages across all banks (except for the firm for which an observation refers to).

   3. (c)

      We summed the amount of assets at time $t$ for all banks (except for the firm for which the observation refers to).

   4. (d)

      We divided the sum of the averaged total losses (the sum of the averaged loss frequencies) by the total assets:

      $\overline{{\mathrm{OpLossMetricsInd}}_{i,t}}={\displaystyle \frac{{\sum }_{m=1,m\ne i}^{N_t}\overline{{\mathrm{OpLossMetrics}}_{m,t}}}{{\sum }_{m=1,m\ne i}^{N_t}{\mathrm{Size}}_{m,t}}},$

      (3.2)

      where $\overline{{\mathrm{OpLossMetricsInd}}_{i,t}}$ is the average industry-wide operational loss (loss frequency) assigned to bank $i$ in quarter $t$; $N_t$ represents the number of banks in our sample in quarter $t$; $\overline{{\mathrm{OpLossMetrics}}_{m,t}}$ is the average operational loss (loss frequency) of bank $m$ measured in quarter $t$ (calculated as described in Section 2.2); and ${\mathrm{Size}}_{m,t}$ is the total assets of bank $m$ in quarter $t$. This calculation provides the average total operational loss (average frequency of the operational losses) per US dollar of the total assets in our industry sample.

2. (2)

   To obtain the average industry-wide operational loss severity for each observation, we sum the severities of all the losses for all banks in a given four-quarter window and divide this sum by the total number of loss events (the losses experienced by the firm are excluded from the numerator and the denominator):

   $\overline{{\mathrm{SeverityInd}}_{i,t}}={\displaystyle \frac{{\sum }_{x=1,x\not\subset i}^{S_t}{\mathrm{Severity}}_x}{S_t-s_{i,t}}},$

   (3.3)

   where $\overline{{\mathrm{SeverityInd}}_{i,t}}$ is the average industry-wide operational loss severity assigned to bank $i$ in quarter $t$; $S_t$ represents the total number of operational loss events in quarters $t-3$ to $t$; ${\mathrm{Severity}}_x$ is the US dollar amount for loss event $x$; and $s_{i,t}$ represents the number of loss events at bank $i$ in quarters $t-3$ to $t$. The simple average in (3.3) provides the average severity of a loss event in the industry, where all losses are equally weighted (and thus banks with more losses have a greater influence on the average).

Table 8 presents the regression results when lagged average industry losses are included as explanatory variables together with firm lagged loss metrics (all the control variables included in previous regressions are also included). The regression in column (1) pairs the lagged average industry total loss with lagged average firm-level total loss, while the regression in column (2) separately accounts for lagged average industry loss frequency, lagged average industry loss severity, lagged average firm-level loss frequency and lagged average firm-level loss severity.

Table 8: Industry operational loss metrics and current losses ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$). [This table reports coefficients from quantile regressions of realized total operational loss in quarter $t+\text{1}$ on operational loss metrics and control variables measured in quarter $t$. $\text{ln}({\text{OpLoss}}_{t+\text{1}})$ is the natural log of the total operational loss (in millions of US dollars) incurred by a BHC in a given calendar quarter. $\text{ln}(\overline{\text{OpLoss}})$, $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$ are the natural logs of BHC quarterly average total operational loss, average loss frequency and average loss severity measured over the four prior quarters. $\text{ln}(\overline{\text{OpLossInd}})$ is the natural log of the industry quarterly average operational loss measured over the four prior quarters as in (3.2). $\text{ln}(\overline{\text{FrequencyInd}})$ is the natural log of the industry average loss frequency measured over the four prior quarters as in (3.2). $\text{ln}(\overline{\text{SeverityInd}})$ is the natural log of the industry average loss severity measured over the four prior quarters as per (3.3). Control variables ($\ln (\mathrm{Size})$, II-to-NII, RoE, T1 capital, equity vol, CCAR age, ME index) are included, but their coefficient estimates are omitted for brevity. Standard errors are clustered by BHC and quarter. $p$-values are shown in parentheses. ${}^{*}$, ${}^{**}$ and ${}^{***}$ denote , and , respectively.]

	(1)	(2)	(3)	(4)
$\text{ln}(\overline{\text{OpLossInd}})$	0.400${}^{***}$	0.230${}^{***}$
	(0.000)	(0.000)
$\text{ln}(\overline{\text{OpLoss}})$		0.480${}^{***}$
		(0.000)
$\text{ln}(\overline{\text{FrequencyInd}})$			0.050	0.589${}^{**}$
			(0.926)	(0.049)
$\text{ln}(\overline{\text{Frequency}})$				0.837${}^{***}$
				(0.000)
$\text{ln}(\overline{\text{SeverityInd}})$			0.476${}^{***}$	0.025
			(0.001)	(0.781)
$\text{ln}(\overline{\text{Severity}})$				0.374${}^{***}$
				(0.005)
BHC controls	Yes	Yes	Yes	Yes
$N$	1266	1266	1266	1266
Adjusted $R^{\text{2}}$	0.690	0.742	0.690	0.779
BIC	$-$3657.678	$-$3433.605	$-$3663.431	$-$3247.969
$\mathrm{\Delta }\text{BIC}$	00$-$46.7787	0$-$270.8512	00$-$41.0248	0$-$456.4876

The lagged average industry total losses are predictive of individual firms’ exposure. A 1% increase in the average industry total loss (excluding the firm) is associated with a 0.23% increase in the expected total loss of a firm in the ensuing quarter. The inclusion of lagged average industry total losses in the regression only slightly attenuates the coefficient of the lagged average firm-level total loss, from 0.510 (see Table 5, column (2)) to 0.480.

The regression where the effects of severity and frequency are taken separately (column (2)) shows that average industry loss frequency is a statistically significant predictor of exposure, while the average industry loss severity is not. A 1% increase in the industry average loss frequency (excluding the firm) is associated with a 0.59% increase in the expected total loss of a firm in the ensuing quarter. The coefficients of lagged average firm-level loss frequency and lagged average firm-level loss severity are only slightly reduced relative to the regression that does not include industry losses (see Table 5, column (3)). These results indicate that when losses become frequent in the industry, individual banks generally see their own loss totals increase, while the industry average severity, which is meaningfully influenced by large losses, is not a statistically significant indicator of future exposure.

The results of this subsection suggest that operational risk presents commonalities across the industry, which are reflected in banks’ loss experience. And thus, external loss data can contribute to the understanding of operational risk in individual firms. Still, these regressions also show that industry-wide factors do not explain away the association between a firm’s historical loss history and its operational risk exposure; therefore, this association is unlikely to be entirely due to systemic factors such as greater regulatory scrutiny in some periods, and is likely due in part to a firm’s idiosyncratic factors, such as its internal controls, risk culture and risk appetite.

3.6 Tail exposure

Operational risk exposure is dominated by large, often idiosyncratic, events (Nešlehová et al 2006; Cope et al 2009). Thus, as well as understanding the drivers of expected operational losses, understanding and modeling the tail regions of the operational loss distribution are critical for effective risk management. In this subsection, we examine whether historical metrics of loss experience are predictive of tail losses using quantile regression. We present the results for 95th quantile regressions, which correspond to infrequent occurrences (ie, one-in-20 year losses) but not quite to the extreme tail (eg, the 99.9th quantile used in the operational risk capital standards).¹²¹² 12 Detailed information on the US risk-based capital standards can be found in Office of the Comptroller of the Treasury, Board of Governors of the Federal Reserve System, Federal Deposit Insurance Corporation and Office of Thrift Supervision (2007). Our findings are generally robust to changes in the quantile used.¹³¹³ 13 Albeit coefficient standard errors do increase meaningfully as we move toward higher quantiles as higher quantile estimates depend more heavily on a few tail observations. Table 9 presents the quantile regression results. All regressions include the same controls as the least squares regressions discussed in previous subsections.

The quantile regressions show broadly similar results to the least squares regressions. A 1% increase in the lagged average total operational loss is associated with a 0.74% increase in the 95th quantile of the total operational loss in the ensuing quarter, while the regression that accounts for loss frequency and severity separately (column (3)) shows that a 1% increase in lagged average loss frequency is associated with a 0.89% increase in the 95th quantile of the total operational loss in the ensuing quarter, and that a 1% increase in lagged average loss severity is associated with a 0.98% increase in the 95th quantile of total operational loss in the ensuing quarter.

Loss metrics are often used by practitioners and regulators in capital and stress testing models that aim to project tail losses (Board of Governors of the Federal Reserve System 2020). These quantile regressions corroborate the usefulness of using metrics based on past losses to model tail exposure.

Table 9: Operational loss metrics and tail risk ($\text{ln}(\text{OpLoss})$). [This table reports coefficients from the 95th quantile regressions of realized total operational loss in quarter $t+\text{1}$ on operational loss metrics and control variables measured in quarter $t$. $\text{ln}({\text{OpLoss}}_{t+\text{1}})$ is the natural log of the total operational loss (in millions of US dollars) incurred by a BHC in a given calendar quarter. In column (1), we only include control variables. In column (2) we include $\text{ln}(\overline{\text{OpLoss}})$, the natural log of the BHC quarterly average total operational losses measured over the four prior quarters, in addition to the control variables. In column (3) we include $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$, the natural logs of BHC quarterly average loss frequency and quarterly average loss severity, respectively, over the four prior quarters, in addition to the control variables. The pseudo $R^{\text{2}}$ is presented to compare the performance of the regressions and is calculated following Koenker and Machado (1999). $p$-values are computed with a bootstrap (100 000 samples) and presented in parentheses. ${}^{*}$, ${}^{**}$ and ${}^{***}$ denote , and , respectively.]

	(1)	(2)	(3)
$\text{ln}(\overline{\text{OpLoss}})$		$-$0.740${}^{***}$
		$-$(0.000)
$\text{ln}(\overline{\text{Frequency}})$			$-$0.886${}^{***}$
			$-$(0.000)
$\text{ln}(\overline{\text{Severity}})$			$-$0.978${}^{*}$
			$-$(0.054)
$\text{ln}(\text{Size})$	$-$1.381${}^{***}$	$-$0.525${}^{***}$	$-$0.516${}^{***}$
	$-$(0.000)	$-$(0.000)	$-$(0.002)
II-to-NII	$-$0.096${}^{***}$	$-$0.048	$-$0.057
	$-$(0.002)	$-$(0.388)	$-$(0.146)
RoE	$-$0.013${}^{**}$	$-$0.002	$-$0.002
	$-$(0.020)	$-$(0.766)	$-$(0.330)
T1 capital	$-$0.114${}^{*}$	$-$0.029	$-$0.064${}^{**}$
	$-$(0.064)	$-$(0.156)	$-$(0.036)
Equity vol	$-$0.001	$-$0.007	$-$0.002
	$-$(0.704)	$-$(0.428)	$-$(0.814)
CCAR age	$-$0.068${}^{***}$	$-$0.044${}^{***}$	$-$0.059${}^{***}$
	$-$(0.000)	$-$(0.000)	$-$(0.000)
ME index	$-$0.013${}^{*}$	$-$0.019	$-$0.002
	$-$(0.092)	$-$(0.150)	$-$(0.836)
$N$	1266	1266	1266
Pseudo $R^{\text{2}}$	0.411	0.469	0.487

3.7 Time and bank fixed effects

Table 10(a) shows the regressions result when quarterly fixed effects are added to the regression. The statistical significance of lagged total operational loss, lagged operational loss frequency and lagged operational loss severity in explaining total operational loss remains, while the magnitude of the coefficients is only slightly reduced. These regressions show that the results in this paper are robust to time effects, as the predictive power of historical loss metrics is not an artifact of systemic effects relating to specific time periods.

In addition to quarterly fixed effects, regressions in Table 10(b) include firm fixed effects as control variables. The inclusion of firm fixed effects reduces the explanatory power of historical loss metrics. The lagged average loss severity is no longer statistically significant, and the coefficient of the lagged average total loss meaningfully decreases in magnitude. The improvement in the BIC resulting from the inclusion of historical loss metrics is also much smaller than when firm effects are not included. The diminished explanatory power of historical loss metrics once fixed effects are included is not surprising because the internal control processes and cultural characteristics of firms that originate operational risk exposure (proxied by past losses) are unlikely to change overnight, and thus are bound to be somewhat absorbed by firm fixed effects. Nevertheless, we note that, according to the BIC, the regression models that include historical operational losses are meaningfully superior to the one that does not. And the coefficient of the lagged loss frequency does not greatly decrease in magnitude or statistical significance, which suggests that the underlying risk factors for which frequency is a proxy have meaningful variation over time, and thus that loss frequency is a particularly relevant metric of future exposure.

Table 10: Operational loss metrics and current losses with time fixed effects. [This table reports coefficients from panel regressions of realized total operational loss in quarter $t+\text{1}$ on operational loss metrics and control variables measured in quarter $t$. $\text{ln}({\text{OpLoss}}_{t+\text{1}})$ is the natural log of the total operational loss (in millions of US dollars) incurred by a BHC in a given calendar quarter. In column (1), we give only the control variables. In column (2) we include $\text{ln}(\overline{\text{OpLoss}})$, the natural log of BHC quarterly average total operational loss measured over the four prior quarters. In column (3) we include $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$, the natural logs of BHC quarterly average loss frequency and quarterly average loss severity measured over the four prior quarters. Control variables ($\text{ln}(\text{Size})$, II-to-NII, RoE, T1 capital, equity vol, CCAR age, ME index) are included in all specifications, but their coefficient estimates are omitted for brevity. All specifications include time (quarterly) fixed effects. Standard errors are clustered by BHC and quarter. $p$-values are shown in parentheses. ${}^{*}$, ${}^{**}$ and ${}^{***}$ denote , and , respectively.]

(a) Time fixed effects ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$)
	(1)	(2)	(3)
$\text{ln}(\overline{\text{OpLoss}})$		0.463${}^{***}$
		(0.000)
$\text{ln}(\overline{\text{Frequency}})$			0.803${}^{***}$
			(0.000)
$\text{ln}(\overline{\text{Severity}})$			0.385${}^{***}$
			(0.003)
BHC controls	Yes	Yes	Yes
Quarterly fixed effects	Yes	Yes	Yes
$N$	1266	1266	1266
Adjusted $R^{\text{2}}$	0.708	0.754	0.788
BIC	3932.454	$-$3721.854	$-$3538.713
$\mathrm{\Delta }\text{BIC}$	0	0$-$210.6007	0$-$393.7411
(b) Bank and time fixed effects ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$)
	(1)	(2)	(3)
$\text{ln}(\overline{\text{OpLoss}})$		0.180${}^{***}$
		(0.001)
$\text{ln}(\overline{\text{Frequency}})$			0.782${}^{***}$
			(0.000)
$\text{ln}(\overline{\text{Severity}})$			0.062
			(0.607)
BHC controls	Yes	Yes	Yes
BHC fixed effects	Yes	Yes	Yes
Quarterly fixed effects	Yes	Yes	Yes
$N$	1266	1266	1266
Adjusted $R^{\text{2}}$	0.780	0.785	0.797
BIC	3796.313	$-$3776.976	$-$3708.847
$\mathrm{\Delta }\text{BIC}$	0	00$-$19.3376	00$-$87.4666

3.8 Comparison with other risk management metrics

So far we have shown that past operational losses provide information regarding operational risk exposure, even when various financial metrics and fixed effects are controlled for. To further understand whether past operational losses are useful predictors of operational risk exposure, we compare models that include historical loss metrics with models that include alternative indicators of risk management quality.

Table 11(a) compares a regression model using historical operational loss metrics with a regression model using the risk management index (RMI) from Ellul and Yerramilli (2013). Data on RMI is only available up to 2013 and only for a subset of the firms included in the rest of the analysis. Therefore, to better compare the performance of the RMI and the historical loss metrics, we restrict the sample to the period and the firms for which both are available. The bilateral correlation between the RMI and the log of future operational losses is only 5% (see Table 4(b)). In the regression analysis, once the controls used so far in this paper are accounted for, RMI is not a statistically significant predictor of operational losses, and its inclusion in the regression does not meaningfully improve the performance of the model according to the BIC. Lagged operational loss severity performs worse as a predictor of operational risk in this subsample than in our full sample and loses statistical significance; meanwhile, the coefficient of operational loss frequency does not meaningfully change with the introduction of RMI as an additional control and retains statistical significance. These results indicate that historical loss frequency is likely a more reliable proxy for the idiosyncratic factors driving the operational risk of a firm than its RMI.

Table 11: Operational loss metrics and current losses with risk management. [This table reports coefficients from panel regressions of realized total operational loss in quarter $t+\text{1}$ on operational loss metrics and control variables measured in quarter $t$. $\text{ln}({\text{OpLoss}}_{t+\text{1}})$ is the natural log of the total operational loss (in millions of US dollars) incurred by a BHC in a given calendar quarter. In column (1), we only include control variables. In column (2), in addition to the control variables, we include a metric capturing the quality of the BHCs risk management function. In part (a), the risk management metric is “Risk management”, the RMI developed by Ellul and Yerramilli (2013). In part (b), the risk management metric is $\text{ln}(\text{OpRisk MR(I)A})$, the natural log of the number of matters requiring immediate attention (MRIA) and matters requiring attention (MRA) outstanding at the beginning of the quarter of interest. In column (3) we include $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$, the natural logs of BHC quarterly average loss frequency and quarterly average loss severity measured over the four prior quarters in addition to the control variables. In column (4) we include the risk management metrics $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$. Control variables ($\text{ln}(\text{Size})$, II-to-NII, RoE, T1 capital, equity vol, CCAR age, ME index) are included in all specifications, but their coefficient estimates are omitted for brevity. All specifications in part (a) include time (quarterly) fixed effects. All specifications in part (b) include BHC and time (quarterly) fixed effects. Standard errors are clustered by BHC and quarter. $p$-values are shown in parentheses. ${}^{*}$, ${}^{**}$ and ${}^{***}$ denote , and , respectively.]

(a) Risk management index ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$)
	(1)	(2)	(3)	(4)
$\text{ln}(\overline{\text{Frequency}})$			$-$0.953${}^{***}$	$-$0.943${}^{***}$
			$-$(0.000)	$-$(0.000)
$\text{ln}(\overline{\text{Severity}})$			$-$0.105	$-$0.095
			$-$(0.411)	$-$(0.429)
Risk management		$-$0.534		$-$0.117
		$-$(0.178)		$-$(0.563)
BHC controls	Yes	Yes	Yes	Yes
$N$	530	530	530	530
Adjusted $R^{\text{2}}$	$-$0.712	$-$0.716	$-$0.785	$-$0.785
BIC	1627.363	$-$1625.897	$-$1483.200	$-$1489.002
$\mathrm{\Delta }\text{BIC}$	0	000$-$1.466	0$-$144.1628	0$-$138.3608
(b) Operational risk MR(I)A ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$)
	(1)	(2)	(3)	(4)
$\text{ln}(\overline{\text{Frequency}})$			$-$0.867${}^{***}$	$-$0.860${}^{***}$
			$-$(0.000)	$-$(0.000)
$\text{ln}(\overline{\text{Severity}})$			$-$0.368${}^{**}$	$-$0.304${}^{*}$
			$-$(0.016)	$-$(0.063)
ln(OpRisk MR(I)A)		$-$0.126		$-$0.107${}^{**}$
		$-$(0.184)		$-$(0.013)
BHC controls	Yes	Yes	Yes	Yes
$N$	1046	1046	1046	1046
Adjusted $R^{\text{2}}$	$-$0.701	$-$0.705	$-$0.796	$-$0.798
BIC	$-$2974.962	$-$2968.342	$-$2589.092	$-$2582.218
$\mathrm{\Delta }\text{BIC}$	0	000$-$6.6202	0$-$385.8703	0$-$392.7441

Table 11(b) introduces as an alternative explanatory variable the log of the number of the Federal Reserve operational risk supervisory findings that firms are subject to. Operational risk supervisory findings reflect issues where a Federal Reserve assessment finds that the firm’s operational risk processes fail to satisfy regulatory requirements. To the degree that the issues identified by the Federal Reserve are material, a relationship between supervisory findings and operational losses should be expected. The bilateral correlation between the log of the number of operational risk supervisory findings and the log of future operational losses is approximately 32%. Also, the regression results confirm this hypothesis. When included in a regression together with lagged operational loss severity and lagged operational loss frequency, the log of the number of operational risk findings is a statistically significant predictor of operational losses, and the model that includes the number of findings (column (4)) outperforms the model that does not (column (3)) according to the BIC. Meanwhile, the coefficients associated with lagged operational loss frequency and lagged operational loss severity do not change meaningfully, albeit the lagged operational loss severity loses statistical significance. These results indicate that, while the supervisory findings are a relevant indicator of a firm’s operational risk, they do not substitute for the information provided by historical operational losses.

Overall, our results show that lagged operational losses are a strong predictor of operational risk exposure and outperform other risk management metrics identified in the literature.

3.9 Using disaggregated accounting dates

The regressions in this paper (outside of this subsection) account for operational losses at their date of occurrence: the date on which a bank experiencing a loss event judges the loss event to have been triggered. Such a choice of date is justified by our focus on understanding whether past losses predict future losses, given the common factors that originate the past and future losses.

However, banks also record the accounting dates of their losses (ie, the date or dates on which a loss event results in financial impacts for the bank). The accounting date is relevant because the accounting impacts of losses are a mechanism through which operational risk can lead directly to bankruptcy. Also, accounting dates are objective, while occurrence dates, although they are useful to understand risk, can suffer from measurement error because they often have to be estimated by banks. For these reasons, recent operational risk regulations, such as the Basel III standardized approach for operational risk (Basel Committee on Banking Supervision 2017) require loss calculations to reflect accounting dates rather than occurrence dates.

To understand whether our results apply when accounting dates are used (both in constructing the dependent variable and in constructing the lagged loss metrics’ explanatory variables), we perform the regressions presented in Table 12.

Table 12: Operational loss metrics and current losses at the time of financial impact ($\text{ln}({\text{OpLoss}}_{t+\text{1}})$). [This table reports coefficients from panel regressions of realized operational losses in quarter $t+\text{1}$ on operational loss metrics and control variables measured in quarter $t$. $\text{ln}({\text{OpLoss}}_{t+\text{1}})$ is the natural log of the total operational loss (in millions of US dollars) incurred by a BHC in a given calendar quarter. In column (1), we only include control variables. In column (2) we include $\text{ln}(\overline{\text{OpLoss}})$, the natural log of BHC quarterly average total operational loss measured over the four prior quarters, in addition to the control variables. In column (3) we include $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$, the natural logs of BHC quarterly average loss frequency and quarterly average loss severity, respectively, over the four prior quarters. Standard errors are clustered by BHC and quarter. $p$-values are shown in parentheses. In this table, $\text{ln}({\text{OpLoss}}_{t+\text{1}})$, $\text{ln}(\overline{\text{OpLoss}})$, $\text{ln}(\overline{\text{Frequency}})$ and $\text{ln}(\overline{\text{Severity}})$ are measured based on the date that losses result in a financial statement impact, rather than on their occurrence date (as is done in all other tables). ${}^{*}$, ${}^{**}$ and ${}^{***}$ denote , and , respectively.]

	(1)	(2)	(3)
$\text{ln}(\overline{\text{OpLoss}})$		$-$0.529${}^{***}$
		$-$(0.000)
$\text{ln}(\overline{\text{Frequency}})$			$-$0.797${}^{***}$
			$-$(0.000)
$\text{ln}(\overline{\text{Severity}})$			$-$0.832${}^{***}$
			$-$(0.000)
$\text{ln}(\text{Size})$	$-$1.398${}^{***}$	$-$0.672${}^{***}$	$-$0.456${}^{***}$
	$-$(0.000)	$-$(0.000)	$-$(0.000)
II-to-NII	$-$0.011	$-$0.007	$-$0.010
	$-$(0.428)	$-$(0.299)	$-$(0.101)
RoE	$-$0.002	$-$0.001	$-$0.000
	$-$(0.248)	$-$(0.276)	$-$(0.858)
T1 capital	$-$0.029	$-$0.002	$-$0.007
	$-$(0.126)	$-$(0.762)	$-$(0.629)
Equity vol	$-$0.005${}^{*}$	$-$0.003	$-$0.003
	$-$(0.085)	$-$(0.205)	$-$(0.120)
CCAR age	$-$0.013	$-$0.011${}^{**}$	$-$0.011${}^{**}$
	$-$(0.136)	$-$(0.030)	$-$(0.040)
ME index	$-$0.004	$-$0.004	$-$0.007
	$-$(0.596)	$-$(0.518)	$-$(0.239)
$N$	1266	1266	1266
Adjusted $R^{\text{2}}$	0.689	0.754	0.767
BIC	$-$3746.077	$-$3452.883	$-$3393.077
$\mathrm{\Delta }\text{BIC}$	0	0$-$293.1934	0$-$352.9993

The results hold. All lagged loss metrics remain statistically significant in the regressions that include them. The magnitude of the coefficient of the lagged average loss frequency decreases only slightly. Meanwhile, the magnitude of the coefficient of historical loss severity more than doubles. This increase in the association between future exposure and lagged loss severity is likely due to the accounting measurement of severity being close to the realization of the future losses to be predicted, while in regressions that use occurrence dates some loss severity amounts have to be moved back in time several years despite the factors that influence them (such as the legal environment) potentially changing between when the loss event occurred and when the loss was accounted.

4 Conclusion

The analysis presented in this paper provides substantial evidence that historical operational losses are predictive of operational risk exposure, even after multiple drivers of operational risk that have been identified in the literature (such as size) and alternative metrics of risk management quality are accounted for. The most likely explanation for this relationship between losses across time is that operational risk exposure is partly explained by factors such as internal controls, risk culture and risk appetite, which are persistent through time (but not immutable). Historical losses are a proxy for these factors and thus provide information on exposure beyond other factors previously identified in the literature.

Among the metrics of historical loss experience, loss frequency proves the most reliable predictor of operational risk exposure. This is likely because the number of operational failures is more stable than the severity of failures and thus reflects more consistently the underlying drivers of exposure. A 1% increase in past average loss frequency is associated with a 0.85% increase in total operational loss. We also provide evidence that operational loss experience of other firms (in particular the loss frequency rate in the industry as a whole) is useful in forecasting the loss experience of a firm. Such a finding suggests that firms should expand and retain industry data sharing, as such data can help firms better understand their exposure. It also suggests that regulators should consider publishing aggregate information on operational loss trends.

The results of this paper support the use of historical losses in operational risk exposure models. Across the range of operational risk types, from fraud to legal risk, operational loss history is predictive of future exposure. Past losses are useful not only to predict expected exposure but also to predict tail exposure. Thus, industry practitioners and regulators should use historical loss experience to better understand banks’ operational risk exposure. Such experience provides a guide to areas in which firms can improve their operational risk management and controls.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper. The opinions expressed in this manuscript are those of the authors and do not represent the official positions of the Federal Reserve Bank of Richmond, the Federal Reserve Board or the Federal Reserve System.

Acknowledgements

The authors thank Andrew Hawley for his excellent research assistance and Andrew Ellul for sharing the RMI data. They also thank Brian Clark, Simon Firestone, Jeff Gerlach, Atanas Mihov, Bernardette Minton, Robert Stewart and the participants at the Federal Deposit Insurance Corporation/Journal of Financial Services Research 19th Annual Bank Research Conference, the 2019 American Bankers Association Annual Convention and the 2019 Operational Riskdata eXchange Association (ORX) Conference for helpful suggestions.

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• Keywords
• Banks
• Operational risk
• Operational risk modelling
• Regulators
• Risk management
• Original research