Unbiased Estimation of Risk

Pitera, Marcin; Schmidt, Thorsten

doi:10.2139/ssrn.2890034

Cited by 11 publications

(28 citation statements)

References 33 publications

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“…As a matter of fact, while not all general law invariant risk measures are concave in µ, this is often the case 1 ; see Acciaio and Svindland [1]. In particular, the above discussion also applies to general risk measures presented in the next section, and we refer to Pitera and Schmidt [40] for further discussion on the issue of biasedness and some empirical evidence.…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

“…These include Pal [37,38] who analyzes hedging under risk measures which can be written as the finite maxima of expectations. Let us further refer to [47,10,40] for other (asymptotic) estimation results, mostly for the average value at risk and under some assumptions on the distribution µ; see e.g. Hong, Hu and Liu [28] for a review.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Non-asymptotic convergence rates for the plug-in estimation of risk measures

Bartl¹,

Tangpi²

2020

Preprint

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Consider the problem of computing the riskiness ρ(F (S)) of a financial position F written on the underlying S with respect to a general law invariant risk measure ρ; for instance, ρ can be the average value at risk. In practice the true distribution of S is typically unknown and one needs to resort to historical data for the computation. In this article we investigate rates of convergence of ρ(F (S N )) to ρ(F (S)), where S N is distributed as the empirical measure of S with N observations. We provide (sharp) non-asymptotic rates for both the deviation probability and the expectation of the estimation error. Our framework further allows for hedging, and the convergence rates we obtain depend neither on the dimension of the underlying stocks nor on the number of options available for trading.Let us first consider the case of plain risk measures, without trading or optimization issues. Denote the underlying by S and by µ its distribution, that is, µ is a probability measure on some measurable space X and S is a random variable distributed according to µ. Given a financial position F : X → R written on S, the task is to compute ρ µ (F ) := ρ(F (S)) where ρ is a law invariant convex risk measure. In practice however, the true distribution µ is unavailable, and one often resort to historical data. This means that instead of ρ µ (F ) one computes the (plug-in) estimator ρ µN (F ), where µ N is the empirical measure built from N i.i.d. historical observations of the underlying S. As we will soon observe, while this estimator is consistent, it typically underestimates the true risk ρ µ (F ). Thus an essential question for risk managers is:How far is ρ µN (F ) from ρ µ (F ) for a fixed sample size N ?

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Section: Introduction and Main Resultsmentioning

confidence: 96%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Non-asymptotic convergence rates for the plug-in estimation of risk measures

Bartl¹,

Tangpi²

2020

Preprint

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“…When evaluating non-parametric ES estimators in such environments, we first have to estimate the parameters of the time series model (via quasi maximum likelihood) based on a simulated sample, then apply the ES formulas discussed above to the (mean 0 and variance 1) model residuals and finally use the time-varying mean and variance predictions of the time series model to scale the obtained ES estimate to its proper level (Gao and Song, 2008) [2]. Consequently, the simulated estimation error of an ES estimator has two components, namely, the non-negligible estimation error related to the time series model (Mancini and Trojani, 2011; Kellner and Rösch, 2016; Pitera and Schmidt, 2018) and what we call the actual or pure error of the ES formula. Because we are interested in the pure error of our ES estimators, the design of our simulation study is close to Peracchi and Tanase (2008) and Yu et al (2010), who simulate iid returns from normal, student t , as well as normal and t mixture distributions.…”

Section: Simulation Setupmentioning

confidence: 99%

A Monte Carlo evaluation of non-parametric estimators of expected shortfall

Mehlitz

Auer

2020

JRF

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Purpose Motivated by the growing importance of the expected shortfall in banking and finance, this study aims to compare the performance of popular non-parametric estimators of the expected shortfall (i.e. different variants of historical, outlier-adjusted and kernel methods) to each other, selected parametric benchmarks and estimates based on the idea of forecast combination. Design/methodology/approach Within a multidimensional simulation setup (spanned by different distributional settings, sample sizes and confidence levels), the authors rank the estimators based on classic error measures, as well as an innovative performance profile technique, which the authors adapt from the mathematical programming literature. Findings The rich set of results supports academics and practitioners in the search for an answer to the question of which estimators are preferable under which circumstances. This is because no estimator or combination of estimators ranks first in all considered settings. Originality/value To the best of their knowledge, the authors are the first to provide a structured simulation-based comparison of non-parametric expected shortfall estimators, study the effects of estimator averaging and apply the mentioned profiling technique in risk management.

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“…Definition 1.1 has been formalized and appropriate methods modelling a risk capital requirement under parameter uncertainty have been proposed (see e.g. [2,3,6,7,15,14]). For practical applications we search for a integrated risk capital model which is simultaneously appropriate in the sense of Definition 1.1 to model each subrisk X j , j = 1, .…”

Section: Introductionmentioning

confidence: 99%

Parameter uncertainty for integrated risk capital calculations based on normally distributed subrisks

Fröhlich,

Weng

2017

Preprint

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In this contribution we consider the overall risk given as the sum of random subrisks X j in the context of value-at-risk (VaR) based risk calculations. If we assume that the undertaking knows the parametric distribution family subrisk X j = X j (θ j ), but does not know the true parameter vectors θ j , the undertaking faces parameter uncertainty. To assess the appropriateness of methods to model parameter uncertainty for risk capital calculation we consider a criterion introduced in the recent literature. According to this criterion, we demonstrate that, in general, appropriateness of a risk capital model for each subrisk does not imply appropriateness of the model on the aggregate level of the overall risk. For the case where the overall risk is given by the sum of normally distributed subrisks we prove a theoretical result leading to an appropriate integrated risk capital model taking parameter uncertainty into account. Based on the theorem we develop a method improving the approximation of the required confidence level simultaneously for both -on the level of each subrisk as well as for the overall risk.

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Unbiased Estimation of Risk

Cited by 11 publications

References 33 publications

Non-asymptotic convergence rates for the plug-in estimation of risk measures

Non-asymptotic convergence rates for the plug-in estimation of risk measures

A Monte Carlo evaluation of non-parametric estimators of expected shortfall

Parameter uncertainty for integrated risk capital calculations based on normally distributed subrisks

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