Bayesian estimation of truncated data with applications to operational risk measurement

Data insufficiency and reporting threshold are two main issues in operational risk modelling. When these conditions are present, maximum likelihood estimation (MLE) may produce very poor parameter estimates. In this study, we first investigate four methods to estimate the parameters of truncated distributions for small samples—MLE, expectation-maximization algorithm, penalized likelihood estimators, and Bayesian methods. Without any proper prior information, Jeffreys’ prior for truncated distributions is used. Based on a simulation study for the log-normal distribution, we find that the Bayesian method gives much more credible and reliable estimates than the MLE method. Finally, an application to the operational loss severity estimation using real data is conducted using the truncated log-normal and log-gamma distributions. With the Bayesian method, the loss distribution parameters and value-at-risk measure for every cell with loss data can be estimated separately for internal and external data. Moreover, confidence intervals for the Bayesian estimates are obtained via a bootstrap method.     

Robust estimation with flexible parametric distributions: estimation of utility stock betas

The distributions of stock returns and capital asset pricing model (CAPM) regression residuals are typically characterized by skewness and kurtosis. We apply four flexible probability density functions (pdfs) to model possible skewness and kurtosis in estimating the parameters of the CAPM and compare the corresponding estimates with ordinary least squares (OLS) and other symmetric distribution estimates. Estimation using the flexible pdfs provides more efficient results than OLS when the errors are non-normal and similar results when the errors are normal. Large estimation differences correspond to clear departures from normality. Our results show that OLS is not the best estimator of betas using this type of data. Our results suggest that the use of OLS CAPM betas may lead to erroneous estimates of the cost of capital for public utility stocks.     

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.     

Credibility Estimation of Distribution Functions with Applications to Experience Rating in General Insurance

This article presents a new credibility estimation of the probability distributions of risks under Bayes settings in a completely nonparametric framework. In contrast to the Ferguson's Bayesian nonparametric method, it does not need to specify a mathematical form of the prior distribution (such as a Dirichlet process). We then show the applications of the method in general insurance premium pricing, a procedure commonly known as experience rating, which utilizes the insured's claim experience to calculate a proper premium under a given premium principle (referred to as a risk measure). As this method estimates the probability distributions of losses, not just the means and variances, it provides a unified nonparametric framework to experience rating for arbitrary premium principles. This encompasses the advantages of the well-known Bühlmann's and Ferguson's approaches, while it overcomes their drawbacks. We first establish a linear Bayes method and prove its strong consistency in nonparametric settings that require only knowledge of the first two moments of the loss distributions considered as a stochastic process. Then an empirical Bayes method is developed for the more general situation where a portfolio of risks is observed but no knowledge is available or assumed on their loss and prior distributions, including their moments. It is shown to be asymptotically optimal. The performance of our estimates in comparison with traditional methods is also evaluated through theoretical analysis and numerical studies, which show that our approach produces premium estimates close to the optima.     

Overconfidence and optimism: The effect of national culture on capital structure

This study assesses whether variations in capital structure across countries can be explained by cultural traits. We analyze capital structure choices of firms in 42 countries and provide evidence that these decisions are affected by the degree of individualism of the country where the firm is located. We assert that managers in countries with high level of individualism exhibit strong optimism and overconfidence which cause an upward bias in perception of supportable debt ratios. Our results are robust to controlling for other firm- and country specific determinants of capital structure choices and to using alternative model specifications and estimation techniques.     

Optimal Capital Allocation Principles

This article develops a unifying framework for allocating the aggregate capital of a financial firm to its business units. The approach relies on an optimization argument, requiring that the weighted sum of measures for the deviations of the business unit's losses from their respective allocated capitals be minimized. The approach is fair insofar as it requires capital to be close to the risk that necessitates holding it. The approach is additionally very flexible in the sense that different forms of the objective function can reflect alternative definitions of corporate risk tolerance. Owing to this flexibility, the general framework reproduces several capital allocation methods that appear in the literature and allows for alternative interpretations and possible extensions.     

Gibbs posterior inference on value-at-risk

ABSTRACT

Accurate estimation of value-at-risk (VaR) and assessment of associated uncertainty is crucial for both insurers and regulators, particularly in Europe. Existing approaches link data and VaR indirectly by first linking data to the parameter of a probability model, and then expressing VaR as a function of that parameter. This indirect approach exposes the insurer to model misspecification bias or estimation inefficiency, depending on whether the parameter is finite- or infinite-dimensional. In this paper, we link data and VaR directly via what we call a discrepancy function, and this leads naturally to a Gibbs posterior distribution for VaR that does not suffer from the aforementioned biases and inefficiencies. Asymptotic consistency and root-n concentration rate of the Gibbs posterior are established, and simulations highlight its superior finite-sample performance compared to other approaches.     

Accounting‐based Risk Measurement: An Alternative to Capital Asset Pricing Model Derived Discount Factors

Discount factors have a long tradition of being computed using capital market inputs for the estimation of systematic risk. They are of increasing importance in financial accounting, including the valuation of goodwill and other intangibles. In view of the volatility of stock market returns and their inaccuracy and disjunction from the underlying cash flows of the firm, this paper proposes an alternative accounting‐based approach: accountingbased risk measurement. Alternatives to beta are computed from planning and budgeting metrics at firm level to produce consistent risk estimates factoring patterns of revenue and cost behaviour weighted according to their impact on the accounting rate of return. This approach is contrasted with the analysis and interpretation of asset betas in the corporate finance literature.     

Outliers,GARCH-type models and risk measures: A comparison of several approaches

In this paper we focus on the impact of additive outliers (level and volatility) on the calculation of risk measures, such as minimum capital risk requirements. Through simulation and empirical studies, we compare six alternative proposals that are used in the literature to reduce the effects of outliers in the estimation of risk measures when using GARCH-type models. The methods are based on [1] correcting for significant outliers, [2] accommodating outliers using complex (e.g. fat-tail) distributions and [3] accounting for outlier effects by robust estimation. The main conclusions of the simulation study are that the presence of outliers bias these risk measures, being the proposal by Grané and Veiga (2010) that providing the highest bias reduction. From the out-of-sample results for four international stock market indexes we found weak evidence that more complex models (specification and error distribution) perform better in estimating the minimum capital risk requirements during the last global financial crisis.     

Confidence regions for the mean-variance efficient set: An alternative approach to estimation risk

The mean-variance efficient set is used extensively in portfolio analysis and in addition underlies many of the models and tests of asset pricing. Despite this vital role, little direct attention has been paid to efficient frontier estimation. This article illustrates that an estimator for the efficient set hyperbola is composed of three mutually independent statistics whose distributions are known. This result is used to develop a confidence region for the efficient set hyperbola in (σ, μ) space. Two alternative approaches are used to define a confidence region. The first approach can be used to obtain an expression for a confidence region for σ given μ or for μ given σ. The second approach defines a confidence region in (σ, μ) space that contains the true hyperbola with a specified probability. In addition, anF test for mean-variance efficiency is used to generate a sample acceptance region. The sample acceptance region and the two confidence regions are compared graphically. A simulation experiment is used to examine the properties of the various procedures.     

(Non-)robustness of maximum likelihood estimators for operational risk severity distributions

The quality of operational risk data sets suffers from missing or contaminated data points. This may lead to implausible characteristics of the estimates. Outliers, especially, can make a modeler's task difficult and can result in arbitrarily large capital charges. Robust statistics provides ways to deal with these problems as well as measures for the reliability of estimators. We show that using maximum likelihood estimation can be misleading and unreliable assuming typical operational risk severity distributions. The robustness of the estimators for the Generalized Pareto distribution, and the Weibull and Lognormal distributions is measured considering both global and local reliability, which are represented by the breakdown point and the influence function of the estimate.     

Gradient-based simulated maximum likelihood estimation for stochastic volatility models using characteristic functions

Parameter estimation and statistical inference are challenging problems for stochastic volatility (SV) models, especially those driven by pure jump Lévy processes. Maximum likelihood estimation (MLE) is usually preferred when a parametric statistical model is correctly specified, but traditional MLE implementation for SV models is computationally infeasible due to high dimensionality of the integral involved. To overcome this difficulty, we propose a gradient-based simulated MLE method under the hidden Markov structure for SV models, which covers those driven by pure jump Lévy processes. Gradient estimation using characteristic functions and sequential Monte Carlo in the simulation of the hidden states are implemented. Numerical experiments illustrate the efficiency of the proposed method.     

The Robustness of the APT to Alternative Estimators

We test the robustness of the APT to two alternative estimation procedures: the Fama and MacBeth (1973) two-step methodology; and the one-step procedure due to Burmeister and McElroy (1988). We find that the APT is indeed sensitive to the chosen estimator and assumptions about the factor structure of stock returns. We believe that our findings have implications for the estimation of asset pricing models in general.     

Conventional and downside CAPM: The case of London stock exchange

Many studies on asset pricing have highlighted the importance of downside risk, in line with the actual losses of investors. In addition, the capital asset pricing model (CAPM), although presented as a universal theory, may provide significantly different rates of return in bull and bear markets. Using the CAPM under different conditions could be regarded as an alternative measurement and valuation approach to downside risk. This paper investigates conventional and downside approaches to risk taking into account different measures of downside beta coefficients. A further contribution of this research is the development of an alternative approach to testing the CAPM relationship. For this purpose, conditional relationships of the CAPM are proposed in which risk premiums are set separately in bull and bear periods. Using equity data and portfolios from the United Kingdom, we obtained positive and statistically significant downside risk premiums. We observed a slight advantage of downside measures over conventional beta measures. Conditional models provide evidence of a positive risk premium in rising markets and a negative risk premium in falling markets. The robustness analysis in subperiods indicates that these findings are largely unchanged for downside beta coefficients, which is not fulfilled by the model in a variance approach.     

Loss expectation and income shifting

This study examines the effects of expected losses on the income-shifting strategies of multinational corporations (MNCs). Using a set of worldwide MNC affiliates, this study first finds empirical evidence for ex ante adjustments of income-shifting strategies according to reverse incentives for potential losses. The results also support the existence of limited flexibility introduced by Hopland et al. (2018, 2021). Second, the estimates of income shifting measured using expected tax rate differences reveal that the traditional methodology of using statutory tax rate differences is subject to an estimation bias. This estimation bias varies depending on the loss expectation status and tax-rate levels of the affiliates.     

An analysis of earnings management by European private firms

This paper addresses the questions of whether private firms in eight European countries engage in earnings management, and if so, whether tax incentives affect such practices. To measure earnings management, we analyze the earnings distributions of private firms and compare these distributions with those of public firms in the same countries. The empirical evidence suggests that in absence of capital market pressures, firms still have incentives to manage earnings, as we find that private firms avoid reporting small losses. We further find that private firms in some countries where tax regulation strongly influences financial accounting do not avoid reporting small losses. We attribute this finding to tax incentives reducing firms’ benefits of (upward) earnings management. Finally, our results suggest that some types of earnings management are due to capital market pressures and are specific to public firms since we do not find evidence that private firms avoid earnings decreases.     

Effect of Analysts' Optimism on Estimates of the Expected Rate of Return Implied by Earnings Forecasts

Recent literature has used analysts' earnings forecasts, which are known to be optimistic, to estimate implied expected rates of return, yielding upwardly biased estimates. We estimate that the bias, computed as the difference between the estimates of the implied expected rate of return based on analysts' earnings forecasts and estimates based on current earnings realizations, is 2.84%. The importance of this bias is illustrated by the fact that several extant studies estimate an equity premium in the vicinity of 3%, which would be eliminated by the removal of the bias. We illustrate the point that cross‐sample differences in the bias may lead to the erroneous conclusion that cost of capital differs across these samples by showing that analysts' optimism, and hence, bias in the implied estimates of the expected rate of return, differs with firm size and with analysts' recommendation. As an important aside, we show that the bias in a value‐weighted estimate of the implied equity premium is 1.60% and that the unbiased value‐weighted estimate of this premium is 4.43%.     

The Interaction of the IFRS 9 Expected Loss Approach with Supervisory Rules and Implications for Financial Stability

This paper examines the interaction of the International Financial Reporting Standard (IFRS) 9 expected credit loss (ECL) model with supervisory rules and discusses potential implications for financial stability in the European Union. Compared to the incurred loss approach of IAS 39, the IFRS 9 ECL model incorporates earlier and larger impairment allowances and is more closely aligned with regulatory expected loss. The earlier recognition of credit losses will reduce the build-up of loss overhangs and the overstatement of regulatory capital. In addition, extended disclosure requirements are likely to contribute to more effective market discipline. Through these channels IFRS 9 might enhance financial stability. However, due to the reliance on point-in-time estimates of the main input parameters (probability of default and loss given default) IFRS 9 ECLs will increase the volatility of regulatory capital for some banks. Furthermore, the ECL model provides significant room for managerial discretion. Bank supervisors might play an important role in the implementation of IFRS 9, but too much supervisory intervention bears the risk of introducing a prudential bias into loan loss accounting that compromises the integrity of financial reporting. Overall, the potential benefits of the standard will crucially depend on its proper and consistent application across jurisdictions.     

The Tail Probability of Discounted Sums of Pareto-like Losses in Insurance

In an insurance context, the discounted sum of losses within a finite or infinite time period can be described as a randomly weighted sum of a sequence of independent random variables. These independent random variables represent the amounts of losses in successive development years, while the weights represent the stochastic discount factors. In this paper, we investigate the problem of approximating the tail probability of this weighted sum in the case when the losses have Pareto-like distributions and the discount factors are mutually dependent. We also give some simulation results.     

Capital structure in an emerging stock market: The case of India

This paper applies two alternative methods of estimation, viz., fully modified OLS (FMOLS) and generalized method of moments (GMM), to analyse the determinants of the capital structure of Indian firms using a panel of 1169 non-financial firms listed in either the Bombay Stock Exchange or the National Stock Exchange over the period 1995-2008. The results thus obtained are robust across the estimation methods. Among the three alternative theories of capital structure, the pecking order theory and the static trade-off theory both seem to explain Indian firms’ decisions. However, there is little evidence to support the agency cost theory.