On stability of operational risk estimates by LDA: From causes to approaches

The stability of estimates is critical when applying advanced measurement approaches (AMA) such as loss distribution approach (LDA) for operational risk capital modeling. Recent studies have identified issues associated with capital estimates by applying the maximum likelihood estimation (MLE) method for truncated distributions: significant upward mean-bias, considerable uncertainty about the estimates, and non-robustness to both small and large losses. Although alternative estimation approaches have been proposed, there has not been any comprehensive study of how alternative approaches perform compared to the MLE method. This paper is the first comprehensive study on the performance of various potentially promising alternative approaches (including minimum distance approach, quantile distance approach, scaling-based bias correction, upward scaling of lower quantiles, and right-truncated distributions) as compared to MLE with regards to accuracy, precision and robustness. More importantly, based on the properties of each estimator, we propose a right-truncation with probability weighted least squares method, by combining the right-truncated distribution and minimizing a probability weighted distance (i.e., the quadratic upper-tail Anderson–Darling distance), and we find it significantly reduces the bias and volatility of capital estimates and improves the robustness of capital estimates to small losses near the threshold or moving the threshold, demonstrated by both simulation results and real data application.     

Estimation of multiple period expected shortfall and median shortfall for risk management

With the regulatory requirements for risk management, Value at Risk (VaR) has become an essential tool in determining capital reserves to protect the risk induced by adverse market movements. The fact that VaR is not coherent has motivated the industry to explore alternative risk measures such as expected shortfall. The first objective of this paper is to propose statistical methods for estimating multiple-period expected shortfall under GARCH models. In addition to the expected shortfall, we investigate a new tool called median shortfall to measure risk. The second objective of this paper is to develop backtesting methods for assessing the performance of expected shortfall and median shortfall estimators from statistical and financial perspectives. By applying our expected shortfall estimators and other existing approaches to seven international markets, we demonstrate the superiority of our methods with respect to statistical and practical evaluations. Our expected shortfall estimators likely provide an unbiased reference for setting the minimum capital required for safeguarding against expected loss.     

Bank Capital Regulation of Trading Portfolios: An Assessment of the Basel Framework

In setting minimum capital requirements for trading portfolios, the Basel Committee on Banking Supervision (1996, 2011a, 2013) initially used Value‐at‐Risk (VaR), then both VaR and stressed VaR (SVaR), and most recently, stressed Conditional VaR (SCVaR). Accordingly, we examine the use of SCVaR to measure risk and set these requirements. Assuming elliptically distributed asset returns, we show that portfolios on the mean‐SCVaR frontier generally lie away from the mean‐variance (M‐V) frontier. In a plausible numerical example, we find that such portfolios tend to have considerably higher ratios of risk (measured by, e.g., standard deviation) to minimum capital requirement than those of portfolios on the M‐V frontier. Also, we find that requirements based on SCVaR are smaller than those based on both VaR and SVaR but exceed those based on just VaR. Finally, we find that requirements based on SCVaR are less procyclical than those based on either VaR or both VaR and SVaR. Overall, our paper suggests that the use of SCVaR to measure risk and set requirements is not a panacea.     

Solvency Ⅱ与C-ROSS非寿险保险风险最低资本要求比较研究——基于中国60家财产保险公司的经验证据

本文以非寿险业务保险风险最低资本要求为考察对象,研究了欧盟Solvency Ⅱ与中国C-ROSS的差异,并利用中国保险市场60家财险公司的经验数据,对两者之间的差异进行了实证和模拟分析。研究结果表明,Solvency Ⅱ和C-ROSS对中国财险公司保险风险最低资本要求存在差异。对于拥有传统业务结构的财险公司,Solvency Ⅱ对保险风险最低资本要求更高,但是这种差距随着公司业务规模的缩小而减弱;对于以经营某些专业险种为主的财险公司,主营业务险种对两者差异具有决定性影响。本文的研究结论详细解释了Solvency Ⅱ与C-ROSS在非寿险保险风险最低资本计算上的异同,对C-ROSS下一步的修订工作提供了一定的支持与参考。     

Estimating cost of capital in firm valuations with arithmetic or geometric mean – or better use the Cooper estimator?

We test the extent and determinants of bias effects of the arithmetic as well as the geometric mean estimator and the estimator of Cooper [1996. Arithmetic versus geometric mean estimators: Setting discount rates for capital budgeting. European Financial Management 2 (July): 157–67] regarding discount rate estimation for firm valuation by way of a bootstrap approach for 13 different countries. The Cooper estimator is superior to both the geometric and the (conventional) arithmetic mean estimator. However, a ‘truncated’ version of the arithmetic mean estimator leads generally to better estimation outcomes than the Cooper estimator. This means that, in order to reduce problems of upward-biased firm value estimates, expected cash flows beyond a certain time horizon are completely neglected in terminal value estimation. Such an approach seems particularly reasonable for the valuation of young growth companies as well as for companies from quickly developing countries such as Brazil, China, or Thailand, because the bias in terminal value estimation is increasing in the growth rate of future expected cash flows.     

Minimizing Basel III Capital Requirements with Unconditional Coverage Constraint

The new Basel III framework increases the banks’ market risk capital requirements. In this paper, we introduce a new risk management approach based on the unconditional coverage test to minimize the regulatory capital requirements. Portfolios optimized with our new minimum capital constraint successfully reduce the Basel III market risk capital requirements. In general, portfolios with value-at-risk and conditional-value-at-risk objective functions and underlying empirical distribution yield better portfolio risk profiles and have lower capital requirements. For the optimization we use the threshold-accepting heuristic and the common trust-region search method.     

Private equity and regulatory capital

Regulatory capital requirements for European banks have been put forward in the Basel II Capital Framework and subsequently in the capital requirements directive (CRD) of the EU. We provide a detailed discussion of the capital requirements for private equity investments under different approaches. For the internal model approach we present a structural model that we calibrate to a proprietary dataset. We modify the standard Merton structural model to make it applicable in practice and to capture stylized facts of private equity investments. We also implement the early default feature with a fast simulation algorithm. Our results support capital requirements lower than in Basel II, but not as low as in CRD, thereby giving adverse incentives to banks for using advanced risk models. A sensitivity analysis shows that this finding is robust to parameter uncertainty and stress scenarios.     

Deposit Insurance, Capital Adequacy Requirements and Interest Rate Dynamics

This paper examines the long run interaction among deposit insurance, bank deposit rates and capital adequacy requirements. Using analysis similar to the price discrimination model of Lott and Roberts (1991) we find that a competitive environment among banks would link the spread between insured and uninsured deposit rates to the size of the insurance premium. We also find that banks that choose to operate at the regulatory minimum capital level, would increase asset risk with increased capital requirements if (1) the implicit interest paid to insured and uninsured depositors is equally sensitive to changes in risk and capital adequacy and (2) the insurance premium is independent of the level of risk and capital adequacy. Under the present risk-based premium structure, asset risk has the potential to decline when the regulatory agency raises capital requirements. Finally, we examine the time series behavior of insured and uninsured interest rates to see if it is consistent with our theoretical model. We find that insured and uninsured rates, along with deposit insurance premiums, are cointegrated series as suggested by our model.     

Portfolio Optimization under Solvency Constraints: A Dynamical Approach

We develop portfolio optimization problems for a nonlife insurance company seeking to find the minimum capital required that simultaneously satisfies solvency and portfolio performance constraints. Motivated by standard insurance regulations, we consider solvency capital requirements based on three criteria: ruin probability, conditional Value-at-Risk, and expected policyholder deficit ratio. We propose a novel semiparametric formulation for each problem and explore the advantages of implementing this methodology over other potential approaches. When liabilities follow a Lognormal distribution, we provide sufficient conditions for convexity for each problem. Using different expected return on capital target levels, we construct efficient frontiers when portfolio assets are modeled with a special class of multivariate GARCH models. We find that the correlation between asset returns plays an important role in the behavior of the optimal capital required and the portfolio structure. The stability and out-of-sample performance of our optimal solutions are empirically tested with respect to both the solvency requirement and portfolio performance, through a double rolling window estimation exercise.     

Empirical performance of Gaussian affine dynamic term structure models in the presence of autocorrelation misspecification bias

Recently, several authors have documented the presence of estimation bias in Gaussian affine dynamic term structure models (GADTSM). However, only a few applications involving its impact on the empirical performance of GADTSM exist in the extant literature, and these studies focus solely on discrete-time vector autoregressive (VAR) based GADTSM and concentrate on issues of small-sample bias and persistence. In this paper, we provide a comprehensive investigation of this issue that includes the estimation of both discrete-time VAR based GADTSM and continuous-time GADTSM at multiple data frequencies through a unique empirical design and two Monte Carlo simulation experiments, within which we construct estimation bias from the serial correlation in yield pricing errors. Our findings show that, although, empirical performance of all studied GADTSM are severely impacted by estimation bias, discrete-time GADTSM are more severely impacted by estimation bias than continuous-time GADTSM. Building on theoretical arguments developed in previous works, we attribute this finding to the strong dependence of discrete-time VAR based GADTSM on the ordinary least squares econometric technique relative to the continuous-time GADTSM for which general maximum likelihood estimation is more suitable.     

Optimal Reinsurance Arrangements Under Tail Risk Measures

Regulatory authorities demand insurance companies control their risk exposure by imposing stringent risk management policies. This article investigates the optimal risk management strategy of an insurance company subject to regulatory constraints. We provide optimal reinsurance contracts under different tail risk measures and analyze the impact of regulators' requirements on risk sharing in the reinsurance market. Our results underpin adverse incentives for the insurer when compulsory Value-at-Risk risk management requirements are imposed. But economic effects may vary when regulatory constraints involve other risk measures. Finally, we compare the obtained optimal designs to existing reinsurance contracts and alternative risk transfer mechanisms on the capital market.     

Incorporating Systemic Influences Into Risk Measurements: A Survey of the Literature

Procyclicality has emerged as a potential drawback to adoption of risk-sensitive bank capital requirements. Systematic risk factors may result in increases (decreases) in bank capital requirements when the economy is depressed (overheated), thereby decreasing (increasing) bank lending capacity and exacerbating business cycle fluctuations. Procyclicality may result from systematic risk emanating from common macroeconomic influences or from interdependencies across firms as financial markets and institutions consolidate internationally. We survey the literature on cyclical effects on operational risk, credit risk and market risk measures.     

Bank regulation and international financial stability: A case against the 2006 Basel framework for controlling tail risk in trading books

In attempting to promote international financial stability, the Basel Committee on Banking Supervision (2006) provided a framework that sought to control the amount of tail risk that large banks around the world would take in their trading books relative to their corresponding minimum capital requirements. However, many of these banks suffered significant trading losses during the recent financial crisis. Our paper examines whether the Basel framework allowed banks to take substantive tail risk in their trading books without a capital requirement penalty. We find that it allowed banks to do so and that its minimum capital requirements can be notably procyclical. Hence, focusing on the way the Basel framework sought to control the amount of tail risk in trading books relative to their corresponding minimum capital requirements, our paper supports the view that it was not properly designed to promote financial stability. We also discuss alternative regulatory frameworks that would potentially be more effective than the Basel framework in preventing banks from taking substantive tail risk in their trading books without a capital requirement penalty.     

Authors’ Reply: A Direct Approach to the Discounted Penalty Function - Discussion by Volkmar Lautscham; Yi Lu; Eric C. K. Cheung

The stochastic modeling and determination of reserves and risk capitals for variable annuity guarantee products are relatively new developments in the insurance industry. The current market practice is largely based on Monte Carlo simulations, which have great engineering flexibility, but the demand for heavy computational power can be prohibitive in many cases. In this article we distinguish and compare two types of risk models to determine the commonly used risk measures for reserving and capital calculations. Using an example of the guaranteed minimum maturity benefit, we investigate alternative numerical methods that require less computational resources and yet achieve high accuracy and efficiency.     

An analysis of VaR-based capital requirements

We study the behavior of a financial institution subject to capital requirements based on self-reported VaR measures, as in the Basel Committee's Internal Models Approach. We view these capital requirements and the associated backtesting procedure as a mechanism designed to induce financial institutions to reveal the risk of their investments and to support this risk with adequate levels of capital. Accordingly, we consider the simultaneous choice of an optimal dynamic reporting and investment strategy. Overall, we find that VaR-based capital requirements can be very effective not only in curbing portfolio risk but also in inducing revelation of this risk.     

Fair Lending Analysis of Mortgage Pricing: Does Underwriting Matter?

This paper evaluates the impact of underwriting decision on fair lending risk assessments of loan pricing. Using data from one national bank that contain a rich set of decisioning variables, we compare the estimation results from the single equation model with those from the sample selection type of models. Then we conduct three simulation studies to evaluate the sample selection bias and omitted variable bias under various scenarios. We demonstrate that the single equation approach could potentially generate biased estimates of pricing disparities when it fails to consider the impact of the underwriting decision.     

Are non-risk based capital requirements for insurance companies binding?

We investigate whether insurers base their solvency margins on risk factors even when operating under a supervisory regime where minimum solvency requirements do not fully take such risk factors into account. To do this, we use a dataset of about 350 Dutch insurers from all major lines of business during the pre-Solvency II period 1995–2005. We find that the levels of insurers’ actual solvency margins are related to their risk characteristics and not to regulatory solvency requirements. Consequently, the vast majority of insurers hold much more capital than required, i.e. non-risk based capital requirements generally are not binding. Requirements are found to affect solvency adjustment behaviour, though. More specifically, below-target capital ratios are raised most rapidly by those insurers whose targets are relatively close to the regulatory minimum. One implication from our results is that, because insurers already follow a risk-based approach, the transition to the new European regulatory framework, Solvency II, is likely to be smooth.     

Hierarchical structures in the aggregation of premium risk for insurance underwriting

In the valuation of the Solvency II capital requirement, the correct appraisal of risk dependencies acquires particular relevance. These dependencies refer to the recognition of risk diversification in the aggregation process and there are different levels of aggregation and hence different types of diversification. For instance, for a non-life company at the first level the risk components of each single line of business (e.g. premium, reserve, and CAT risks) need to be combined in the overall portfolio, the second level regards the aggregation of different kind of risks as, for example, market and underwriting risk, and finally various solo legal entities could be joined together in a group.

Solvency II allows companies to capture these diversification effects in capital requirement assessment, but the identification of a proper methodology can represent a delicate issue. Indeed, while internal models by simulation approaches permit usually to obtain the portfolio multivariate distribution only in the independence case, generally the use of copula functions can consent to have the multivariate distribution under dependence assumptions too.

However, the choice of the copula and the parameter estimation could be very problematic when only few data are available. So it could be useful to find a closed formula based on Internal Models independence results with the aim to obtain the capital requirement under dependence assumption.

A simple technique, to measure the diversification effect in capital requirement assessment, is the formula, proposed by Solvency II quantitative impact studies, focused on the aggregation of capital charges, the latter equal to percentile minus average of total claims amount distribution of single line of business (LoB), using a linear correlation matrix.

On the other hand, this formula produces the correct result only for a restricted class of distributions, while it may underestimate the diversification effect.

In this paper we present an alternative method, based on the idea to adjust that formula with proper calibration factors (proposed by Sandström (2007)) and appropriately extended with the aim to consider very skewed distribution too.

In the last part considering different non-life multi-line insurers, we compare the capital requirements obtained, for only premium risk, applying the aggregation formula to the results derived by elliptical copulas and hierarchical Archimedean copulas.     

Seasoned equity offerings: Quality of accounting information and expected flotation costs

Flotation costs represent a significant loss of capital to firms and are positively related to information asymmetry between managers and outside investors. We measure a firm's information asymmetry by its accounting information quality based on two extensions of the Dechow and Dichev [2002. The quality of accruals and earnings: the role of accrual estimation errors. Accounting Review 77, 35–59] earnings accruals model, which is a more direct approach to assessing the information available to outside investors than the more commonly used proxies. Our main hypothesis is that poor accounting information quality raises uncertainty about a firm's financial condition for outside investors, though not necessarily for insiders. This accounting effect lowers demand for a firm's new equity, thereby raising underwriting costs and risk. Using a large sample of seasoned equity offerings (SEOs), we show that poor accounting information quality is associated with higher flotation costs in terms of larger underwriting fees, larger negative SEO announcement effects, and a higher probability of SEO withdrawals. These results are robust to joint determination of offer size and flotation cost components and to adjustments for sample selection bias.     

The market's perception of the riskiness of large U.S. Bank commercial letters of credit

This article improves upon the market discipline studies of commercial letters of credit (CLC) by employing two new capital market tests, in addition to traditional beta and equity risk tests. Option pricing models have been used to calculate implied asset risk. Two implied asset risk measures have been used to examine the riskiness of commercial letters of credit. The results indicate that stockholders view CLCs as reducing bank risk, and debt holders are indifferent about CLCs. A policy conclusion of these findings is that additional capital requirements of CLCs for large commercial banks may be inappropriate.