The Relevance of Expected Shortfall Models in Different Time Window Sizes

Abstract

Risk management has become increasingly important in the financial world. Considering its importance, it is necessary to measure these risks. The financial market uses two risk measures: Value at Risk (VaR) and Expected Shortfall (ES). After the subprime crisis, the market began to emphasize ES instead of VaR. The hypothesis of this paper to be tested is that longer periods provide better information than shorter, more recent periods for measuring ES volatility to hedge trades. The ES can be adopted using parametric, semi-parametric, and non-parametric methods, and the analyses of the log return indicators started on 3 January 2000 and ended on 5 May 2023. The analyses carried out to evaluate these log return indicators covered the period from 6 May 2023 to 1 August 2025, where it was found that the exchange rate volatility of the Brazilian Real exceeded the VaR limits and even reached the Expected Shortfall risk zone. Then, a different analysis was performed, starting on 11 March 2020 and ending on 5 May 2023. This second analysis, as the first analysis, was carried out to evaluate these log return indicators that covered the period from 6 May 2023 to 1 August 2025. In this latest period analysis, the exchange rate volatility of the Brazilian Real reached the Exchange Shortfall risk zone in a different way compared to the first way. All three types of methods—parametric, non-parametric, and semi-parametric—show distinct behaviors depending on the period evaluated. The hypothesis was rejected, but the hedging strategies should account for asset volatility. The software used to calculate the estimators was Microsoft Excel 365 and Stata 14.2.

1. Introduction

Financial institutions use Value at Risk (VaR) as an important indicator of financial risk, which tells them, considering a confidence level and time interval, the worst expected outcome of a given portfolio. However, along with this indicator, VaR involves some conceptual issues. This indicator of the worst possible loss is only valid within the confidence interval where the analysis has been delimited. If this confidence interval were extrapolated, VaR would not be able to predict the worst possible loss in the portfolio. Finally, to be coherent, VaR must be sub-additive, i.e., the VaR of a portfolio of assets must be less risky than the sum of the VaR of these assets individually (Yamai & Yoshiba, 2004).

According to Boudt et al. (2008), when the return of a portfolio is disparate, VaR may not be acceptable as a risk measure, since the portfolio is not sub-additive. The uncertainty of the investments cannot be greater than the uncertainties of the positions added together, despite the effects of portfolio diversification in the case of VaR. However, since VaR is not a convex function that assigns weights to the portfolio, optimizing the average VaR portfolio is more difficult. In contrast, the Expected Shortfall (ES) is a coherent risk measure and a convex function that assigns weights to the portfolio. It is also widely used in portfolios with abnormal returns. ES addresses the risk when VaR limits are exceeded, analyzing the extreme risks that VaR is unable to reach.

VaR and ES are used to set benchmarks for the Basel Committee’s capital requirements, to measure the risk assumed by financial agents, and to allocate available capital in risk management policies. The main procedure of said indicators is to guarantee that the risk by financial institutions stays within their risk appetite parameters. This assessment must take place in a timely manner so that the necessary corrective action can be taken if necessary (Scaillet, 2004).

Financial institutions calculate potential future risk, which is very important today. Besides complying with the Basel Committee on Banking Supervision regulations, risk management helps to better allocate capital. In practice, ES provides a more reasonable risk measure in the context of risk magnitude, which is also higher than the value provided by VaR. These measures are used by financial institutions to determine the minimum capital required to withstand any terrible market event. This can be useful for planning and putting into practice appropriate risk management policies against uncertain events (Kumar & Maheswaran, 2016).

Risk management evaluates the marginal impacts of the positions of risk measures and regulatory capital, referenced to justify that knowing sensitivity helps optimizing the computational time spent processing large portfolios, as they need to reprocess the risk measures every time the portfolio breakdown is slightly modified (Fermanian & Scaillet, 2004). Furthermore, risk management helps break down the risk of the overall portfolio, component by component, and identify the greatest risk factors.

The literature presents a large number of ways to estimate the asset’s ES (Mehlitz & Auer, 2020). The risk analysis based on the Expected Shortfall is poorly explored in the literature, and there is no simulation study that provides a structured ES indicator comparison for hedging purposes, according to a previous bibliometric analysis and systematic review of the literature by Fukui and Basso (2022). We resolved this issue by comparing the ES estimators with large-time-window log returns of exchange rates from 3 January 2000 to 5 May 2023 against a smaller and most recent time window from 11 March 2020 to 5 May 2023. These log return windows are compared by analysing their effectiveness over the time window from 6 May 2023 to 1 August 2025.

The purpose of this paper is to study the properties of the estimation methods for hedge and identify, based on parametric, non-parametric, and semi-parametric methods, better options for Expected Shortfall volatility analysis of the exchange rates of the Brazilian Real (BRL) versus Dollar (USD), Brazilian Real versus Euro (EUR), Brazilian Real Versus Yene (JPY), and Brazilian Real versus Yuan (CNY). The main research hypothesis is that, after comparing their volatilities across different time windows, long-term analysis is better than short-term analysis. The findings revealed that short-term analysis tends to be the most advantageous choice, but hedging strategies should be tailored to the volatility of the assets and the estimation method employed, aiming for better protection against extreme losses.

2. Literature Review

Our research carried out a comparative analysis using the ES as an extreme risk indicator between the exchange rates of five major currencies in the world economic scenario: USD, EUR, JPY, CNY, and BRL.

Tests were carried out for the exchange rates between BRL and USD, BRL and EUR, BRL and JPY, and BRL and CNY. These exchange rates were chosen to be able to compare the volatility of the exchange rate of an emerging country (Brazil) against USD, as well as the volatility of the exchange rate of countries that are more representative of the international scenario (Europe, Japan, and China) against BRL. The returns are calculated based on log returns to achieve distribution smoothing. The period defined for calculating the log return indicators started on 3 January 2000 and ran until 5 May 2023, and a second period started on 11 March 2020 and ran until 5 May 2023. To analyze the effectiveness of the log return indicators, data was collected from 6 May 2023 to 1 August 2025. The period was selected because it is a period that begins on the day that the World Health Organization (WHO) declared COVID-19 a pandemic, and it ends on the day that WHO declared the end of COVID-19 as a public health emergency.

According to Yamai and Yoshiba (2004), the ES has both advantages and disadvantages. When there is a fat-tailed distribution, the ES error estimators are much higher than the VaR, i.e., it is much more costly than the VaR. However, on the other hand, ES works very well as a complement to VaR for risk management purposes.

2.1. Financial Regulation

The risk measure on its own does not consider the distribution from which this risk measure is formed or its confidence level. Therefore, the risk measure partially captures the risk exposure and loses part of the risk that the institution is subject to. In the case of the subprime crisis, systemic risk jeopardized the economy as a whole (Guegan & Hassani, 2017).

Financial regulation focuses on ensuring the individual security of financial organizations with the stated purpose of protecting holders against the organization’s liabilities. This is the focus of this regulation, known as micro prudential regulation, but following the subprime crisis in 2008, regulation also started, including macroprudential objectives. Macroprudential regulation aims to guarantee financial stability and the resilience of the financial system. Therefore, regulators had several tools to help financial institutions focus on success, including determining capital adequacy rules, which remained one of the central pillars of prudent regulation. To comply with these requirements, two measurement risk tools have been used by financial institutions: Value at Risk (VaR) and Expected Shortfall (ES). In the financial industry, until the subprime crisis in 2009, VaR was the primary indicator for regulating capital adequacy at financial institutions. However, after this crisis, the need for the third Basel accord arose, which, among other measures, stipulated the ES for assessing market risk (Koch-Medina & Munari, 2015).

According to Guegan and Hassani (2017), it was mandatory for each section of the financial institution to assess the risks associated with the institution’s different activities. For each risk factor X, several pieces of information could be defined that match various values of X collected in previous periods. Therefore, a series of information I = (${\mathit{X}}_{\mathbf{1}},{\mathit{X}}_{\mathbf{2}}$, ${\mathit{X}}_{\mathbf{3}}$, … ${\mathit{X}}_{\mathit{n}}$) was obtained for risk factor X. Said values represented the results of the past risk factor evolution and were, a priori, unknown. Because of this ambivalence, vulnerability X was an indeterminate parameter and, for each number, i = 1, …, n, a chance could be connected. The expression representing the alternatives of an aleatory variable and their connected likelihoods was known as a probability density function. In this perspective, the risk factor X was an aleatory variable defined on a probability space, whose outcomes were numbers in R, and the probability density function F was constant, picking any number within a given range or set of ranges, as defined by its density function. It is possible to define the cumulative distributive function (c.d.f.): R ≥ [0, 1] of the aleatory variable X after establishing the continuous and strictly monotonic distribution function F. Then, the quantile function Q returned to a limit value x, below which was drawn at random from the provided c.d.f., and which would occur p percent of the time. As a statistical distribution, the quantile function Q returned the value x such that

F_x(x): = Pr (X ≤ x) = p

(1)

and

Q (p) = inf {x ϵ R: p ≤ F(x)},

(2)

for a probability 0 ≤ p ≤ 1.

VaR and ES are risk measurements used by authorities to assess the regulatory capital requirement to ensure against a major loss. Analyzing from an economic perspective, the weighted risk of assets was impacted by these capital requirements. In other words, the institutions’ assets or liability exposures that do not appear on the balance sheet have been balanced according to risk. The capital weighting uses the weighted risk to reflect capital exposures at financial institutions. These relationships were monitored by authorities to guarantee that financial establishments manage to take in a large part of the losses and that they evolved as the regulations of the Basel meetings took place. Non-compliance with these standards regarding capital suitability ratios would require financial institutions to increase capital at such a cost that it would block them from creating ways to generate new earnings streams or new business and thus increase their profitability (Guegan & Hassani, 2017).

Puccetti and Rüschendorf (2014) stated that financial institutions should set aside an amount of regulatory capital. The capital risk used at Value at Risk, for operational risk, with a high level of confidence for a random variable L of aggregate loss, is summarized as

$$\mathrm{L} = {\sum }_{\mathit{i}=\mathbf{1}}^{\mathit{d}}{\mathit{L}}_{\mathit{i}},$$

(3)

where ${\mathit{L}}_{\mathbf{1}}$, ${\mathit{L}}_{\mathbf{2}}$, …, ${\mathit{L}}_{\mathit{d}}$ indicate random loss factors for different lines of business or types of risk over a certain period. The VaR of total losses L, at a confidence index α ϵ (0.1), was the α-quantile of the distribution

$${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\alpha }}\left(\mathrm{L}\right) = {\mathit{F}}_{\mathit{L}}^{-\mathbf{1}}\left(\mathrm{\alpha }\right) = \inf \{\mathrm{x} \mathrm{ϵ} \mathbb{R}: {\mathit{F}}_{\mathit{L}}(\mathrm{x})\ge \mathrm{\alpha }\},$$

(4)

where ${\mathit{F}}_{\mathit{L}}$(x) = P(L ≤ x) is the frequency function of L. Typical values in operational risk practice within Basel II are α = 0.99, α = 0.995, α = 0.999, and T = 1 year.

The ES has features that are familiar to mathematicians, but they lead to anomalies for financiers, who seek to meet both microprudential and macroprudential objectives. The way the ES treated the tail values explains these anomalies. The averages were poor indicators for this purpose. Such anomalies have not diminished the importance of ES as a risk measure but have provided arguments against its indiscriminate use (Koch-Medina & Munari, 2015).

Although VaR became the most popular risk measure, shortcomings were identified. VaR cannot be considered a coherent risk measure because it does not meet the subadditivity criterion, and it was unable to capture the impact of events that exceeded its confidence index. The subprime crisis raised the question of the applicability of VaR as a metric risk parameter, and the Basel Committee placed ES as the natural alternative for quantifying this risk (Puccetti & Rüschendorf, 2014).

2.2. Value at Risk and Expected Shortfall

VaR has become the risk metric adopted by the financial market to define the exposure to market risk of a financial position (Consigli, 2002). VaR defines the worst-case loss, given a certain level of confidence and a certain time interval, of a chosen portfolio.

There are three approaches to calculating VaR: non-parametric, semi-parametric, and parametric. Any such approach can be validated by the number of exceptions in comparison with the errors of the estimated VaR, which can be reported as the projection of the tail probability (Kumar & Maheswaran, 2016).

Considering that Y represents a financial position expressed as a random variable of a real number. Thus, VaR describes the upper threshold of the loss interval [−∞, ${\mathit{z}}_{\mathit{\alpha }}$], which occurs variably within the likelihood of the probability function for the returns of Y, within the limit of α ϵ (0.1) (J. M. Chen, 2018):

$$\mathrm{\alpha } ={\int }_{-\mathbf{\infty }}^{{\mathit{z}}_{\mathit{\alpha }}}{\mathit{f}}_{\mathit{Y}}\left(\mathit{x}\right)\mathrm{dx}$$

(5)

This means that the VaR, mathematically, at the 100(1 − α)% level of confidence is defined as the upper 100α percentile of the distribution of adverse outcomes. If X is the aleatory variable addressing loss in the portfolio, VaR can be defined at a confidence level of 100(1 − α)% as

VaRα(X) = sup{x|P[X ≥ x] > α},

(6)

where sup{x|A} is the upper limit of x given event A, and where sup{x|P[X ≥ x] > α} indicates the upper 100α percentile of the loss distribution. This definition applies to both continuous and discrete loss distributions (Yamai & Yoshiba, 2004).

Both VaR and ES are law-invariant risk measures, as long as both risk measures rely solely on loss distributions. Law-invariant measures attract special interest in financial regulation since their values only depend on the loss distribution, and estimation does not need additional information beyond the stress scenarios (J. M. Chen, 2018).

ES is an alternative to smooth out the problems generated by VaR (Yamai & Yoshiba, 2004). Also known as Conditional VaR, ES is the conditional expectation of loss that exceeds the VaR limits. The ES is given by

ESα(X) = E[X | X ≥ VaRα(X)]

(7)

The ES indicates the average loss when it exceeds the VaR limits. In other words, Mehlitz and Auer (2020) state that VaR only considers the probability but not the impact of large losses. Furthermore, VaR does not consider risk diversification of the portfolio since it does not comply with the main properties of subactivity and convexity (Mehlitz & Auer, 2020). On the other hand, ES does not suffer from these deficiencies, since it provides the average value of losses that exceed the parameters defined by VaR. Thus, the regulators of the Basel III Committee suggest the use of ES in place of VaR in determining the result of bank funding resources. The use of ES in the valuation of stocks, bonds, commodities, currencies, and even bitcoin is becoming increasingly popular (Mehlitz & Auer, 2020).

According to Boudt et al. (2008), when it comes to estimating the portfolio’s VaR and ES, the VaR and ES equations are specified as a function of the ordering of the investment returns, and said frequency functions are undefined for real time series. VaR and ES are often calculated under the assumption of normality. The estimators calculated, the Gaussian VaR and the Gaussian ES do not respect the evidence that the returns of various assets do not follow a normal distribution. Methods that do not consider the normality of the distribution of real returns will produce more accurate estimates of downside risks than the risks provided by Gaussian VaR and Gaussian ES.

The most popular methodologies for estimating distribution functions as normal distributions plus correction terms that consider the asymmetry and excessive kurtosis observed in the data are the Cornish–Fisher and Edgeworth expansions. The results of said methodologies are the modified VaR and the modified ES. The modified VaR is one of the most popular estimators for assessing the hedge risk of funds and other assets with non-normal distributions. It has been commonly used as a portfolio selection criterion. The modified ES was obtained based on Cornish–Fisher and Edgeworth density expansions and quantile functions and estimates risk and decline more assertively, even in the presence of non-normal returns (Boudt et al., 2008).

Kumar and Maheswaran (2016) noted that volatility is the main ingredient in the calculation and design of VaR and ES. Accordingly, accurate volatility projections are fundamental to generating more accurate VaR and ES projections. The estimate and projection of VaR and ES focus on the volatility of generalized autoregressive conditional heteroscedasticity (GARCH) models, with specifications of error terms drawn from several distributions. The popularity of GARCH volatility models allows them to capture several stylized facts, such as group volatility, fat tails and mean reversion volatility.

According to Yamai and Yoshiba (2002), VaR and ES have tail risk when VaR or ES fail to show the results of relative portfolio choices based on the underestimation of tail risk and its fat tail properties, and even its potential for large losses.

Risk managers have a wide choice of ES estimation methods and generally use two approaches: regressions or simulations. Regressions face the difficulty of having the results linked to the characteristics of the empirical data set analyzed. Moreover, more complex ES estimation instructions are limited to specific categories of estimators. On the other hand, simulation configurations are more dynamic because they allow for the distribution of several configurations to see which is the best estimator, or even which is the best comparatively, considering a given scenario (Mehlitz & Auer, 2020).

2.3. VaR and ES Tail Risk in Normal Distributions

In cases of normal gain and loss distributions, VaR and ES provide very similar information. VaR fails when it presents tail risk and thus cannot summarize the relative risk of portfolios due to misinterpretation of the risk of investments and their fat tail properties, i.e., the elevated probability of large losses. The risk of fat tails from VaR arises from its analysis, which is limited to excluding the coverage area that exceeds VaR values from its parameters. This can lead to looking for assets with a high capability for significant deficits and lower risks compared to assets with a low possibility for significant exposure (Yamai & Yoshiba, 2004).

When the distribution of the losses is the Gauss distribution, the ES is calculated as follows:

$${\mathit{E}\mathit{S}}_{\mathit{\alpha }}\left(\mathrm{X}\right) - \mathrm{E}[\mathrm{X} | \mathrm{X} \ge {\mathit{V}\mathit{A}\mathit{R}}_{\mathit{\alpha }}(\mathrm{X})] = {\displaystyle \frac{\mathbf{1}}{\mathit{\alpha }{\mathit{\sigma }}_{\mathit{x}}\sqrt{\mathbf{2}\mathit{\pi }}}}{\int }_{{\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\alpha }}\left(\mathit{X}\right)}^{\mathbf{\infty }}\mathit{t}. {\mathit{e}}^{{\displaystyle \frac{-{\mathit{t}}^{\mathbf{2}}}{\mathbf{2}{\mathit{\alpha }}_{\mathit{X}}^{\mathbf{2}}}}}\mathrm{dt} = {\displaystyle \frac{{\mathit{e}}^{\frac{{-\mathit{q}}_{\mathit{\alpha }}^{\mathbf{2}}}{\mathbf{2}}}}{\mathit{\alpha }\sqrt{\mathbf{2}\mathit{\pi }}}}{\mathit{\sigma }}_{\mathit{X}}$$

(8)

where ${\mathit{q}}_{\mathit{\alpha }}$ is greater than the 100α percentile of the standard normal distribution.

Yamai and Yoshiba (2004) also showed that VaR and ES have no tail risk when the distribution of gains and losses is normal. The non-normality of the distribution of gains and losses is a result of the irregularity of the investment’s position or asset prices. To illustrate, if you use ES at a 99% confidence level in this equation, this is the same as multiplying the standard error by 2.67, which is the same level as VaR at a 99.6% confidence level.

VaR and ES are often calculated based on the normality assumption, and, actually, it is a stylized fact that the returns of many financial assets do not follow a normal distribution. The methods that consider the non-normality of the return distribution produce, in cases of non-normal distributions, more accurate downside risk estimates than the Gaussian VaR and ES (Boudt et al., 2008).

Estimating the tail index is one of the first objectives in Extreme Value Theory. For distributions with heavy tails, the Hill estimator is the most common way of estimating this parameter (Németh & Zempléni, 2020).

2.4. Non-Parametric ES Estimation

S. Chen (2008) considers ${\left\{{\mathit{X}}_{\mathit{t}}\right\}}_{\mathit{t}=\mathbf{1}}^{\mathit{n}}$ the time series of market values and considers ${\mathit{Y}}_{\mathit{t}}$ = −log(${\mathit{X}}_{\mathit{i}\mathit{t}}$| ${\mathit{X}}_{\mathit{i}\mathit{t}-\mathbf{1}}$) the negative log return over period t. Then, suppose that ${\left\{{\mathit{Y}}_{\mathit{t}}\right\}}_{\mathit{t}=\mathbf{1}}^{\mathit{n}}$ is a stationary process with a stationary distribution function F. Given a positive p-value close to zero, the VaR with a confidence level γ is

v_p = inf{u: F (u) ≥ γ}

(9)

where γ is the increasing position of the lack frequency quantile F. VaR defines a index of extreme losses, such that the chance of a loss greater than is lower than p. A major discrepancy of VaR, added to the fact that it is not a coherent risk measure, is that it does not provide information on the magnitude of losses, nor does it specify a level that defines the thereshold. On the other hand, ES is a coherent risk measure that is informative in relation to losses greater than ${\mathit{\nu }}_{\mathit{p}}$.

Mehlitz and Auer (2020) addressed some formal ES notations and definitions. Considering that (${\mathit{X}}_{\mathit{t}}$) is a time series of loss on investments in which a sequence of independent variables is assumed to be independent and identical, for each confidence level γ, ${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\gamma }}$ is considered the 1 − p quantile of the probability distribution function of ${\mathit{X}}_{\mathit{t}}$. Given the probabilistic loss probability distribution function, the ES is defined by

$${\mathit{E}\mathit{S}}_{\mathbf{1}-\mathit{p}}= {\displaystyle \frac{\mathbf{1}}{\mathbf{1}-\mathit{\gamma }}}{\int }_{{\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\gamma }}}^{\mathbf{\infty }}\mathit{x}\mathit{f}(\mathrm{x})\mathrm{dx} = {\displaystyle \frac{\mathbf{1}}{\mathbf{1}-\mathit{\gamma }}}{\int }_{\mathit{\gamma }}^{\mathbf{\infty }}{\mathit{V}\mathit{a}\mathit{R}}_{\mathit{v}}\mathrm{dv}$$

(10)

S. Chen (2008) further stated that the ES associated with a confidence level γ, which can be considered as ${\mathit{\mu }}_{\mathit{p}}$, is the conditional expectation of loss if the loss is greater than ${\mathit{\nu }}_{\mathit{p}}$, and then one can find that

$${\mathit{\mu }}_{\mathit{p}}= \mathrm{E}({\mathit{Y}}_{\mathit{t}}|{\mathit{Y}}_{\mathit{t}}> {\mathit{\nu }}_{\mathit{p}})$$

(11)

In the words of S. Chen (2008), the ES is estimated by providing a parametric loss distribution, which is the most common procedure explored in studies on the subject. However, the non-parametric method has the advantage of being a free model, making it robust and avoiding bias caused by incorrect distributional assumptions about the expected loss. Risk management focuses on the characteristics of a loss distribution concentrated in the tail, which makes it difficult to adapt the model to a parametric form. It should also be noted that data is usually sparse in the tail. The non-parametric approach also has the advantage of handling large amounts of data, which makes it favorable for financial loss purposes.

The preference for using ES over VaR in risk measurement due to its superior properties (Scaillet, 2004). The ES, also called the VaR tail, is sub-additive for continuous risk distributions, unlike the VaR. This property of sub-additivity is a necessary part of being considered a coherent risk measure. This expresses the idea that the total portfolio risk cannot be larger than the sum of the individual risks. Moreover, unlike ES, which tells us the size of the loss when it exceeds the limit determined in VaR, VaR does not tell us anything about the potential loss when that limit is exceeded.

The ES can be estimated by providing a parametric loss distribution. A binomial model is relevant for estimating ES and VaR for a large, balanced portfolio (S. Chen, 2008). Regarding portfolio allocation, Scaillet (2004) discusses a non-parametric kernel estimator for sensitivity analysis. A kernel density estimate is a non-parametric way of estimating the probability density function. It is a data-smoothing problem where population inferences are made based on a finite population.

2.5. Tail Risk Example

Embrechts et al. (1997) considered an investment portfolio with a certain number of assets, which have their respective values in their respective periods. As every asset has a distribution of profits and losses, it can be represented by a probability distribution that shows the changes in values. By estimating the covariances of the portfolios, the portfolio manager estimates the overall profit and loss distribution of the portfolio. Managers and regulators calculate the VaR, and after estimating the VaR, it is important to estimate the probability of the value reached exceeding the limits defined by the regulator.

Yamai and Yoshiba (2002) noted that the VaR distribution for extreme events fails to indicate the best portfolio choice, potentially leading to an underestimation of the uncertainty of investments with heavy-tailed distributions and significant exposure. The danger of tail risk from VaR comes to light because it measures the distribution of gains and losses and does not account for losses that exceed the VaR limit. Portfolio A of 10 million, with a VaR confidence level of 99 percent, compared with portfolio B of 15 million with the same VaR confidence level, could lead to the conclusion that portfolio B is riskier than portfolio A. However, the investor may not know how much they stand to lose when they reach a point outside the confidence level. When the maximum loss of portfolio A is greater than the maximum loss of portfolio B, portfolio A should be considered riskier than portfolio B in the worst-case scenario.

Regarding the tail risk of ES, as well as the tail risk of VaR, Yamai and Yoshiba (2002) stated that the tail risk of ES happens when it fails to indicate the best choice among portfolios, which can result in the choice underestimating the risk of portfolios with fat tail properties and a high potential for large losses. If portfolio A never reaches a 50 percent loss point and portfolio B can reach a 75 percent loss, then portfolio B is riskier than portfolio A in terms of extreme losses.

2.6. Expected Shortfall Definition

Scaillet (2004) considered a precisely stationary process {${\mathit{Y}}_{\mathit{t}}$, t ϵ ℤ} considering the values in ${\mathit{R}}^{\mathit{n}}$ and assumed that the data consist of the recognition of {${\mathit{Y}}_{\mathit{t}}$; t = 1, …, T}. The vector (${\mathit{Y}}_{\mathit{t}}$ = ${\mathit{Y}}_{\mathbf{1},\mathit{t}},\dots ,{\mathit{Y}}_{\mathit{n},\mathit{t}}$) corresponds to risk frequency over a given period. The data are parametric simulations (VARMA, multivariate GARCH, or diffusion processes), possibly within another data set. Simulations are necessary when the structure of financial assets is very complex, such as for some derivatives. Thus, by controlling the sample size T, it can be increased to obtain satisfactory estimates.

The ES linked to the investment distribution is defined by

m(a, p) = E[−a′Y > |−a′Y > VaR (a, p)],

(12)

where VaR (a, p) is implicitly defined by

P[−a′Y > VaR(a, p)] = p

(13)

According to Puccetti and Rüschendorf (2014), different from VaR, ES takes tail risk into account more comprehensively, taking into account the size of the probability of excess losses above the limit defined by VaR. ES is also a more pessimistic measure, since

$${\mathit{E}\mathit{S}}_{\mathit{\alpha }}\left(\mathit{L}\right)\ge {\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\alpha }}\left(\mathrm{L}\right), \mathrm{for} \mathrm{all} \mathrm{ϵ} (0,1)$$

(14)

Also in the words of Puccetti and Rüschendorf (2014), both the calculation of ${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\alpha }}$(L) and of ${\mathit{E}\mathit{S}}_{\mathit{\alpha }}$(L) require understanding of the joint probability function of risk and portfolio (${\mathit{L}}_{\mathbf{1}}$, …, ${\mathit{L}}_{\mathit{d}}$). Accordingly, one usually needs a series of d-variable data for deficits, which can be difficult to obtain. Normally, only the additional probability functions of the types of variables can be estimated scientifically. Then, it is normal to demand for a traditional estimate of ${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\alpha }}$(L) and ${\mathit{E}\mathit{S}}_{\mathit{\alpha }}$(L) when the marginal probability functions of the various risks ${\mathit{L}}_{\mathit{i}}$ are provided, but without knowing any dependency information about the portfolio risk pool (${\mathit{L}}_{\mathbf{1}}$, …, ${\mathit{L}}_{\mathit{d}}$). For a α ϵ (0, 1), and a series of marginal distributions ${\mathit{F}}_{\mathbf{1}}$, …, ${\mathit{F}}_{\mathit{d}}$, the poorest VaR and the poorest ES for aggregate position L are defined as

$${\acute{\mathit{V}\mathit{a}\mathit{R}}}_{\mathit{\alpha }} \left(\mathrm{L}\right) = \sup \left\{{\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\alpha }}\right({\mathit{L}}_{\mathbf{1}}+ \dots +{\mathit{L}}_{\mathit{d}}); {\mathit{L}}_{\mathit{i}}~{\mathit{F}}_{\mathit{i}},1\le \mathrm{i}\le \mathrm{d}\},$$

(15)

$${\acute{\mathit{E}\mathit{S}}}_{\mathit{\alpha }} \left(\mathrm{L}\right)=\sup \left\{{\mathit{E}\mathit{S}}_{\mathit{\alpha }}\right({\mathit{L}}_{\mathbf{1}}+\dots +{\mathit{L}}_{\mathit{d}}); {\mathit{L}}_{\mathit{i}}~{\mathit{F}}_{\mathit{i}},1\le \mathrm{i}\le \mathrm{d}\},$$

(16)

${\acute{\mathit{V}\mathit{a}\mathit{R}}}_{\mathit{\alpha }}$ (L) and ${\acute{\mathit{E}\mathit{S}}}_{\mathit{\alpha }}$(L) correspond to the largest estimates of ${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\alpha }}$(L) and ${\mathit{E}\mathit{S}}_{\mathit{\alpha }}$(L), accordingly, when the distributions of the aleatory variables ${\mathit{L}}_{\mathbf{1}}$, …, ${\mathit{L}}_{\mathit{d}}$ are known.

2.7. Expected Shortfall Sensitivity

To characterize the ES’s sensitivity to local changes in the portfolio structure, we use the first derivative (a, p) = ∂m(a, p)/∂a′ of the ES in relation to the portfolio allocation.

The first-order derivative of ES in relation to portfolio allocation is

m⁽¹⁾(a, p) = E [−Y|−a′Y > VaR (a, p)]

(17)

The ES and its first-order derivative are directly related through m(a, p) = a′(a, p). It is due to the grade one similarity of the ES. The quantity ${\mathit{a}}_{\mathit{i}}$∂m(a, p)/∂ ${\mathit{a}}_{\mathit{i}}$ is called the incremental ES of asset i by analogy with existing VaR terminology. In fact, the homogeneity property is also shared by VaR. Asset positions are ordered by incremental ES and incremental VaR in relation to their contribution to the total portfolio risk measured by ES and VaR (Scaillet, 2004).

2.8. Static Hedging and Multivariate Derivatives

In a static hedge, if the call is redundant, there is a portfolio of traded assets that have negative correlations without the need to continually rebalance this correlation. When it comes to derivatives, we can talk about put–call parity, which is a very simple example when you apply this equation to hedge a put portfolio with a call (or put). This approach works in the case of a binomial tree, but static coverage has no extra cost, apart from the cost of the initial purchase (Pellizzari, 2004).

It is possible to change the VaR to an arbitrary level by trading derivatives of the assets in question. Given the desired VaR level, it is possible, for example, to write a put with a set exercise price just below the VaR of the lower quantile of the profit and loss distribution and buy a put with a set strike price just above the desired VaR. This strategy increases the risk of a major loss. In other words, in this situation, the VaR tail risk can be significant when trading derivatives. However, VaR fails to consider losses that exceed its limits (Yamai & Yoshiba, 2002).

In cases of partial hedging, which involves hedging part of the asset (Pellizzari, 2004), where the volume for a perfect hedge is not available, agents can set up a hedge that protects part of the total value of the asset. If the hedging strategy fails to hedge the assets, the agent can take steps to remedy the failure. A European call option, for example, will only hedge the asset if the strike price is not below the option’s strike price. This is the least expensive, though least secure, way of hedging against risk. Thus, as showing how to optimize a hedge, minimize a series of requests under a budget restriction, and weight the ES by a deficit, show how to optimize a hedge, minimizing a series of requests under a budget restriction, weighting the ES by a deficit is contingent on the risk taken by the agent (Pellizzari, 2004).

2.9. Exemplifying the Hedging Strategy

To illustrate the hedging strategy, Pellizzari (2004) assumed a basket of two log-normal risk stocks, whose dynamics are described by

$$\mathrm{D}{\mathit{S}}_{\mathit{i}}= \mathrm{r}{\mathit{S}}_{\mathit{i}}\mathrm{dt} + \mathrm{\sigma }{\mathit{S}}_{\mathit{i}}\mathrm{d}{\mathit{Z}}_{\mathit{i}},\mathrm{i}=1,2,$$

(18)

where r, ${\mathit{\sigma }}_{\mathbf{1}}$, ${\mathit{\sigma }}_{\mathbf{2}}$ are constants, and ${\mathit{Z}}_{\mathbf{1}}$ and ${\mathit{Z}}_{\mathbf{2}}$ are standard Brownian movements. The asset values at time t are highlighted by ${\mathit{S}}_{\mathbf{1}\mathit{T}}$, ${\mathit{S}}_{\mathbf{2}\mathit{T}}$ for 0 ≤ t ≤ T. The result of the maturity is

$$ƒ({\mathit{S}}_{\mathbf{1}\mathit{T}}, {\mathit{S}}_{\mathbf{2}\mathit{T}}) = \max ({\mathit{S}}_{\mathbf{1}\mathit{T}}+ {\mathit{S}}_{\mathbf{2}\mathit{T}}-\mathrm{k},0)$$

(19)

where starting prices ${\mathit{S}}_{\mathbf{10}}$ = ${\mathit{S}}_{\mathbf{20}}$ = 100, volatilities ${\mathit{\sigma }}_{\mathbf{1}}$ = 0.3 and ${\mathit{\sigma }}_{\mathbf{2}}$ = 0.2, correlations between the returns p = 0.5, maturity T = 1 year, strike price k = 190, and risk-free rate r = 0.1.

Applying a Monte Carlo simulation using 100 repetitions results in a price of ${\mathit{C}}_{\mathit{M}\mathit{C}}$ = 3.89 with a standard deviation of ${\mathit{\sigma }}_{\mathit{M}\mathit{C}}$ = 3.26.

Thus, Pellizzari (2004), to reduce the size of the standard error, used another more refined method to improve the average Monte Carlo price ${\mathit{C}}_{\mathit{M}\mathit{M}\mathit{C}}$ = 33.34 with a standard error of ${\mathit{\sigma }}_{\mathit{M}\mathit{M}\mathit{C}}$ = 0.40 for 100 simulations. In this way, the standard error was reduced to 88%. The technique also shows that an investment in call options can be used to hedge the results of a basket of options. Considering t = 0 in a long position, with 0.96 call options on the first asset and 1.01 call options on the second asset, the strike prices are 94.88 and 89.71, respectively. To obtain this portfolio, at t = 0, an additional 5.59 must be taken to be paid with yield to maturity. This portfolio provides a result that, in sum between its assets, can hedge against the random results of the basket of options. In fact, the joint difference between the results of the index options and the value of the hedge portfolio is the null mean, with the standard error exp(rT)${\mathit{\sigma }}_{\mathit{M}\mathit{M}\mathit{C}}\sqrt{\mathbf{100}}$ = 4.4. The standard error of the options result is around 36, and so, operating the hedge portfolio introduced above, the risk was reduced to a factor of around 8.

Once a purchase option is established, the price is guaranteed. This gives us two possibilities: no hedging or delta hedging, which is a strategy for partially or totally hedging the operation. On the one hand, it is possible to accept the fact that one has to bear a large risk, with an average final payout of zero and a standard deviation of 36, as shown above. The other possibility, delta hedging, is inaccurate and expensive in practice. There is also a third possibility, derived from the portfolio already reported, which partially hedges the results of the option with a risk of 4.4, as shown in Table 1 (Pellizzari, 2004).

Table 1. Hedge performance according to different methods.

Method	Hedge	Risk (σ)	Risk Reduction (%)	Cost	Portfolio Hedge
MC	None	36	0	None	None
Static	Partial	4.4	88	Low	Static, 2 call options
Dynamic	Total	0	100	High	Dynamic, delta base

Source: Pellizzari (2004, p. 510).

The ES is obtained by averaging the ordering of the errors when they reach the maturity value, given the confidence interval. The confidence intervals adopted by the market vary between 95% and 99%. Using these parameters, we consider a portfolio of European options for two risky assets, ${\mathit{S}}_{\mathbf{1}}$ and ${\mathit{S}}_{\mathbf{2}}$, which results in max(${\mathit{S}}_{\mathbf{1}\mathit{T}}$ + ${\mathit{S}}_{\mathbf{2}\mathit{T}}$ − k, 0) and selects the parameters of the hedging strategy described above. Pellizzari (2004) simulated 1000 strategies with assets with approximate rebalancing of delta-based hedge portfolios. As there is no formula for calculating the deltas of an options portfolio, they are best calculated using Monte Carlo simulations. The parameters of the static hedge are calculated in Table 2 below:

Table 2. Parameters used for a static hedge portfolio (standard deviations shown in parentheses).

	Quantity	Value
Call option $S_1$	1.00	98.06
	(0.010)	(0.38)
Call option $S_2$	1.01	89.17
	(0.006)	(0.36)
Cash	−3.87
	(0.27)

Source: Pellizzari (2004, p. 514).

Table 2 shows that the optimal static hedge portfolio consists of one call option, with a value of 98.06, and 1.01 call options, with a value of 89.17. Both options expire in one year and use 3.87 at t = 0. The constant ${\mathit{b}}_{\mathbf{0}}$ is used in the regression to calculate the weights, already discounted at time t = 0 (Pellizzari, 2004). Thus, one can see that this amount is a small part of around 10% of the price, which is around 33.34. It should also be noted that the Greeks of the options portfolio, ${\mathbf{\Delta }}_{\mathbf{1}}$~0.8 and ${\mathbf{\Delta }}_{\mathbf{2}}$~0.77, show that the loan needed to start a rebalancing portfolio strategy is formidable, comparing a value above 120 against the previous 3.27. This is common to all deltas based on hedges, almost regardless of the results profiles and the volume of risk resources. One must take a significant quantity or use another cash resource at the start of the rebalancing portfolio strategy. The most interesting practical aspect of static hedging may lie in this small loan requirement.

As strategies for VaR can be used for ES, Yamai and Yoshiba (2002) stated that VaR can be reduced to an arbitrary level by buying and selling derivatives of the asset in question. Assuming that the desired VaR level is ${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{D}}$, the establishment of this arbitrariness lies in setting up a put option with an exercise price just below ${\mathit{V}\mathit{a}\mathit{R}}_{\mathbf{0}}$, which is the first quantile of the profit and loss frequency, and buying a put option with a strike price just above ${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{D}}$. This trading strategy shows that it is possible to handle VaR by buying and selling options on assets. However, because of this manipulation, the potential for major losses is high. This strategy can be used in ES as a risk estimator for disaster prevention.

3. Methodology

The methodology used is that of Mehlitz and Auer (2020), because they also work with a wide range of calculations to estimate the Expected Shortfall of an asset. Next, risk managers have to choose the best estimator option. Therefore, these estimators are different ways of calculating ES according to the distribution of the population being analyzed and comparing them. Furthermore, the comparative estimators can inform an average of them, yielding a new ES estimator. Within a multidimensional simulation setup, this study ranks innovative performance profile techniques and compares them in different time windows to test the hypothesis that long-period time windows generate better information than smaller and more recent time windows. Then, the results can be compared to identify the best time windows to use the ES as a risk estimator.

3.1. Normal Method

One of the techniques for estimating ES is to assume that losses are normally distributed, with the mean µ and standard deviation σ (Mehlitz & Auer, 2020):

$${\mathit{E}\mathit{S}}_{\mathbf{1}\mathbf{-}\mathit{p}}^{\mathit{N}\mathit{D}} = \mu + {\displaystyle \frac{\mathit{\sigma }}{\mathbf{1}\mathbf{-}\mathit{\gamma }}}Ø \left[{\mathsf{\Phi }}^{\mathbf{-}\mathbf{1}}\mathrm{\gamma }\right],$$

(20)

where Ø is the standard normal probability density function, and ${\mathsf{\Phi }}^{\mathbf{-}\mathbf{1}}$ is the inverse standard normal cumulative distributive function. Therefore, to obtain an empirical estimate, µ and σ must be estimated using their sample of counterparts, and the estimated values must be entered into Equation (20).

3.2. Peak-over-Threshold Method

As ES focuses on collapses, Extreme Value Theory is an interesting tool for deriving new ES estimators. McNeil (1999) says that Peaks-Over-Threshold (POT) class models can be semi-parametric models, which are based on Hill’s estimator, and parametric models based on the Generalized Pareto Distribution or GPD.

The Peak-Over-Threshold method serves as the basis for the theorem, which expresses that, for the loss distribution, the distribution of excesses, ${\mathit{Y}}_{\mathit{t}}$ = ${\mathit{X}}_{\mathit{t}}$ − u, over a large parameter u is well approximated using GPD. This is important because this result models the tail of the loss distribution without the need to specify the shape of the loss distribution (Mehlitz & Auer, 2020):

$$\begin{array}{l}\mathrm{G}\left(\mathrm{y}\right)=\left\{\begin{array}{ll}1-{(\mathbf{1}+{\displaystyle \frac{\mathbf{\xi }}{\mathit{\sigma }}})}^{{\displaystyle \frac{-1}{\mathbf{\xi }}}}& \mathrm{if} \mathrm{\xi } \ne 0\\ 1-{\mathit{e}}^{{\displaystyle \frac{-\mathit{y}}{\mathit{\sigma }}}}& \mathrm{if} \mathrm{\xi } = 0\end{array}\right.\end{array}$$

(21)

where ξ is the shape parameter, and σ > 0 is the scale parameter (Mehlitz & Auer, 2020). The support of this function is y ≥ 0 when ξ ≥ 0 and 0 ≤ y ≤ −${\displaystyle \frac{\alpha }{\xi }}$ when ξ < 0.

$$\begin{array}{l}\mathrm{F}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}1-\mathrm{q}{(\mathbf{1}+{\displaystyle \frac{\mathit{\xi }(\mathit{x}-\mathit{u})}{\mathit{\sigma }}})}^{{\displaystyle \frac{-\mathbf{1}}{\xi }}}& \mathrm{if} \mathrm{\xi } \ne 0\\ 1-\mathbf{q}{\mathbf{e}}^{{\displaystyle \frac{\mathbf{x}-\mathbf{u}}{\mathbf{\sigma }}}}& \mathrm{if} \mathrm{\xi } = 0\end{array}\right.\end{array},$$

(22)

where q > 1 − y is the percentage of losses that exceeds u. Then, VaR can be obtained by inverting Equation (23):

$$\begin{array}{l}\mathit{V}\mathit{a}{\mathit{R}}_{\mathbf{1}-\mathit{p}}^{\mathit{P}\mathit{O}\mathit{T}}=\left\{\begin{array}{ll}\mathrm{u}+{\displaystyle \frac{\mathit{\sigma }}{\xi }}\left({\left({\displaystyle \frac{\mathbf{1}-\mathit{\gamma }}{\mathit{q}}}\right)}^{-\xi }-1\right)& \mathrm{if} \mathrm{\xi } \ne 0\\ \mathrm{u}-\sigma ln\left({\displaystyle \frac{\mathbf{1}-\mathit{\gamma }}{\mathit{q}}}\right)& \mathrm{if} \mathrm{\xi } = 0\end{array}\right.\end{array}$$

(23)

Then we calculate ES using the calculation in Equation (24) (Mehlitz & Auer, 2020):

$${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{P}\mathit{O}\mathit{T}}= {\displaystyle \frac{{\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\gamma }}^{\mathit{P}\mathit{O}\mathit{T}}-\mathit{\xi }\mathit{u}+\mathit{\sigma }}{\mathbf{1}-\mathit{\xi }}} \mathrm{for}\mathrm{\xi }<1$$

(24)

3.3. Non-Parametric Estimators

Historical estimators do not need to follow theoretical distributions for empirical data, and the properties of the underlying sequence (${\mathit{X}}_{\mathit{t}}$) are represented by a given data sample (Mehlitz & Auer, 2020).

It is used to calculate the classic estimator of historical ES as

$${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}}= \mathrm{E}({\mathit{X}}_{\mathit{t}}|{\mathit{X}}_{\mathit{t}}\ge {\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\gamma }}^{\mathit{H}}) = {\displaystyle \frac{{\mathit{\Sigma }}_{\mathit{t}=\mathbf{1}}^{\mathit{n}}{\mathit{X}}_{\mathit{t}}\mathit{I}\left({\mathit{X}}_{\mathit{t}}\ge {\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\gamma }}^{\mathit{H}}\right)}{{\mathit{\Sigma }}_{\mathit{t}=\mathbf{1}}^{\mathit{n}}\mathit{I}\left({\mathit{X}}_{\mathit{t}}\ge {\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\gamma }}^{\mathit{H}}\right)}},$$

(25)

where ${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\gamma }}^{\mathit{H}}$ = ${\mathit{x}}_{\lceil \mathit{n}\mathit{\gamma }\rceil }$, and I(.) is the binary function where, if its premise is true, shows 1, and 0 if it is false (Mehlitz & Auer, 2020). Thus, ${\mathit{X}}_{\mathit{i}}$ shows that the i-nth smallest value of (${\mathit{X}}_{\mathit{t}}$), and ⌈.⌉ is the least-integer function. Because $\mathit{n} \mathit{\gamma }$ is close, even if $\mathit{n} \mathit{\gamma }$ has a small decimal number, the ceiling function can put in place a significant error in contrast to the floor function. Because of this fact, some modifications to Equation (25) have been proposed. Considering it as natural, these variables make the classic historical method more streamlined, then we have the first modification (Mehlitz & Auer, 2020):

$${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}\mathbf{1}}= {\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}}+ \left(1-{\displaystyle \frac{⌊\mathit{n}(\mathbf{1}-\mathit{\gamma })⌋}{\mathit{n}(\mathbf{1}-\mathit{\gamma })}}\right)\hspace{0.33em}{\mathit{X}}_{\left(\lfloor \mathit{n}\mathit{\gamma }\rfloor \right)},$$

(26)

where an adjustment term has been included in classic historical estimation, and the rounding-down function rounds the value down if it is not a whole number. The adjustment focuses on the loss, when sorted in ascending order (${\mathit{X}}_{\mathit{t}})$, that would find the point before ${\mathit{V}\mathit{a}\mathit{R}}_{\mathit{\gamma }}^{\mathit{H}}$. Then, it will be explained as the VaR that would be used if the risk is underestimated. ${\mathit{X}}_{\left(\lfloor \mathit{n}\mathit{\gamma }\rfloor \right)}$ is balanced by the element $\left(1-{\displaystyle \frac{⌊\mathit{n}(\mathbf{1}-\mathit{\gamma })⌋}{\mathit{n}(\mathbf{1}-\mathit{\gamma })}}\right)\hspace{0.33em}$.

Then, we have a second modification (Mehlitz & Auer, 2020):

$${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}\mathbf{2}}= \mathrm{\gamma } {\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}}+ (1 - \mathrm{\gamma }) \mathrm{E}({\mathit{X}}_{\mathit{t}}| \ge {\mathit{X}}_{\left(\lfloor \mathit{n}\mathit{\gamma }\rfloor \right)})$$

(27)

The purpose is to add the weightings of ${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}}$ and ES, whose outcomes for danger were underestimated. The proportions are γ and 1 − γ, respectively. Finally, it was used γ and 1 − γ weighting in Equation (27) (Mehlitz & Auer, 2020); then it was substituted by 1 − (⌈nγ⌉ − nγ), which is the place of decimals of nγ, and ⌈nγ⌉ − nγ, accordingly. Thus,

$${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}\mathbf{3}}= (1 - \lceil \mathrm{n}\mathrm{\gamma }\rceil + \mathrm{n}\mathrm{\gamma }) + (\lceil \mathrm{n}\mathrm{\gamma }\rceil - \mathrm{n}\mathrm{\gamma }) \mathrm{E}({\mathit{X}}_{\mathit{t}}|{\mathit{X}}_{\mathit{t}}\ge {\mathit{X}}_{\left(\lfloor \mathit{n}\mathit{\gamma }\rfloor \right)}),$$

(28)

where the estimated ES gains more weight, which corresponds to a lower VaR, if nγ has an insignificant decimal place.

As already discussed, historical estimators address the biggest losses and are therefore sensitive to outliers. Then we eliminate outlier data by stating a constant a ϵ [0, 0.1] (Mehlitz & Auer, 2020), opted by the user to declare the vulnerability of having anomalies by the following algebraic expression:

$$\mathrm{k}(\mathrm{t}) = (\mathrm{n} + 1) \left(1-\mathrm{\gamma }-{\displaystyle \frac{\mathit{t}(\mathbf{1}-\mathit{\gamma })}{⌊\mathit{n}(\mathbf{1}-\mathit{\gamma })⌋+1}}\right)\hspace{0.33em}\mathrm{for}\mathrm{t} \mathrm{ϵ} \mathbb{R}$$

(29)

Thus, the alternative estimator is generated as follows:

$${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{J}\mathbf{1}}= {\displaystyle \frac{\mathbf{1}}{\lfloor \mathit{n}{\left(\mathbf{1}-\mathit{\gamma }\right)}^{\mathbf{1}+\mathit{a}}\rfloor +\mathbf{2}}}{\sum }_{\mathit{t}=\mathbf{0}}^{\lfloor \mathit{n}{\left(\mathbf{1}-\mathit{\gamma }\right)}^{\mathbf{1}+\mathit{a}}\rfloor +\mathbf{1}}{\mathit{X}}_{\mathit{n}-\lfloor \mathit{k}\left(\mathit{t}\right)\rfloor }$$

(30)

The classical historic estimator can be decreased by ${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{J}\mathbf{1}}$ when a is small enough. Under other conditions, k excludes the greatest damages in ES estimation. Another version of the estimator is

$${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{J}\mathbf{2}}= {\displaystyle \frac{\mathbf{1}}{\lfloor \mathit{n}{\left(\mathbf{1}-\mathit{\gamma }\right)}^{\mathbf{1}+\mathit{a}}\rfloor +\mathbf{2}}}{\sum }_{\mathit{t}=0}^{\lfloor \mathit{n}{\left(\mathbf{1}-\mathit{\gamma }\right)}^{\mathbf{1}+\mathit{a}}\rfloor +\mathbf{1}}\left(\mathbf{1}-\mathit{w}\left(\mathit{t}\right)\right){\mathit{X}}_{\left(\mathit{n}-\lfloor \mathit{k}\left(\mathit{t}\right)\rfloor \right)}+ \mathrm{w}\left(\mathrm{t}\right){\mathit{X}}_{\left(\mathit{n}-\mathbf{1}-\lfloor \mathit{k}\left(\mathit{t}\right)\rfloor \right)}$$

(31)

The weighted sum of ${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{J}\mathbf{1}}$ and the ES, which is smaller and a result of the average of the smallest values of (${\mathit{X}}_{\mathit{t}}$), is considered.

3.4. Combined Estimations

Studies across various fields have shown that mixing different forecasting models produces more effective results than individual analyses (Mehlitz & Auer, 2020). They argue that when combinations are formed by averaging the weights, equal weights are higher than the estimated ideal proportion according to specific standards, but these are potentially wrong (Mehlitz & Auer, 2020).

Based on these results, this ES series is used to calculate a new one. In other words, a simple average is calculated using these previously calculated estimators as a basis:

$${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{M}\mathit{V}}= {\displaystyle \frac{{\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{N}\mathit{D}}+{\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{P}\mathit{O}\mathit{T}}+{\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}}+{\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}\mathbf{1}}+{\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}\mathbf{2}}+{\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{H}\mathbf{3}}+{\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{J}\mathbf{1}}+{\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{J}\mathbf{2}}}{\mathbf{8}}}$$

(32)

4. Discussion and Results

When comparing VaR with ES, it is important to focus on tail risk or where the problem lies when VaR ignores losses, given the VaR level. These facts can cause serious problems in the real world. ES can protect investors against this type of problem, as long as the method considers losses that exceed the VaR limits. On the other hand, ES has disadvantages, such as a fat-tailed distribution, which leads to much higher error estimators in ES than in VaR. To reduce this estimation error, it is necessary to increase the simulation sample size. Thus, the ES is more costly when it needs to avoid tail risk, as a fat-tailed distribution can lead to higher losses. Considering ES as a risk measurer, it is possible to set up hedging strategies to avoid unpleasant surprises or even minimize these negative results.

The hypothesis to be tested is that larger time windows, which contain a large volume of data, produce more accurate answers, based on Expected Shortfall, than smaller time windows, which concentrate on more recent data.

The data was taken from the Central Bank of Brazil website. The daily foreign exchange parities between BRL and USD, between BRL and EUR, between BRL and JPY, and between BRL and CNY were collected. The analyzed time window started on 3 January 2000 and ended on 5 May 2023. Then, a second analysis was done with a smaller time window, starting on 11 March 2020 and ending on 5 May 2023, to compare the results. The analysis was carried out to evaluate whether these log return indicators covered the period starting on 6 May 2023 and ending on 1 August 2025. This period was selected because, as a tail distribution, the COVID crisis provided a perfect time window to study and analyze the Expected Shortfall estimators as risk estimators. The confidence level is 0.95. The Hill estimator is the average of the tail of the distribution. To define the tail size, the EXQUANTILE function was used in Stata 14.2. The software used to calculate the estimators were Microsoft Excel 365 and Stata 14.2.

Assessment of the Results

The calculations made in the empirical analyses used the exchange rates between the USD X BRL, EUR X BRL, BRL X JPY, and BRL X CNY currency pairs for the period from 3 January 2000 to 4 May 2023. To analyze the effectiveness of the log return indicators, data from 6 May 2023 to 1 August 2025 was analyzed.

With this data in hand, it was possible, using the values in Table 3a, to identify, within these histories, the number of times in which the exchange rate change exceeded the limits set by the Expected Shortfall.

Table 3. (a) The Expected Shortfalls assessed in the empirical analyses from 3 January 2000 to 5 May 2023. (b) Number of times the exchange rate in (a) exceeded ES limits from 6 May 2023 to 1 August 2025. (c) The Expected Shortfalls assessed in the empirical analyses from 11 March 2020 to 5 May 2023. (d) Number of times the exchange rate in (c) exceeded ES Limits from 6 May 2023 to 1 August 2025. (e) Variations of the ES in the results of analyses from different time windows. (f) Variations in the number of times the exchange rate exceeded ES limits in the results of analyses from different time windows.

(a)
ES	USD BRL	EUR USD	USD JPY	USD CNY
ES ND	0.035846707	0.159036488	0.171660549	0.03559427
ES POT	0.122616969	0.935038618	0.80321275	0.912192718
ES H	0.028628472	0.016088149	0.00581079	0.028344639
ES H1	0.032015293	0.040353727	0.031720879	0.055325994
ES H2	0.032869974	0.061561664	0.045680888	0.072121177
ES H3	0.042726746	0.572794416	0.479247393	0.205114255
ES J1	0.548030742	0.033013892	0.036567388	0.514159589
ES J2	0.50966859	0.03070292	0.03583604	0.508110653
${ES}_{\gamma }^{MV}$	0.169050436	0.231073734	0.201217085	0.291370412
(b)
ES	USD	EUR	JPY	CNY
ES ND	0	0	0	0
ES POT	0	0	0	0
ES H	1	17	126	1
ES H1	0	0	0	0
ES H2	0	0	0	0
ES H3	0	0	0	0
ES J1	0	0	0	0
ES J2	0	1	0	0
${ES}_{\gamma }^{MV}$	0	0	0	0
(c)
ES	USD BRL	EUR USD	USD JPY	USD CNY
ES ND	0.058605654	0.07721881	0.016033748	0.057616854
ES POT	0.07256674	0.181712228	0.235314753	0.04095985
ES H	0.051083129	0.050921489	0.007651133	0.050855172
ES H1	0.078650785	0.12700765	0.063293722	0.065260405
ES H2	0.0515264	0.056935107	0.018611104	0.050855172
ES H3	0.063973296	0.129395932	0.144249305	0.049865639
ES J1	0.00769453	0.00770241	0.012241941	0.173299995
ES J2	0.007540639	0.007548362	0.011997102	0.169833995
${ES}_{\gamma }^{MV}$	0.048955146	0.079805248	0.063674101	0.082318385
(d)
ES	USD	EUR	JPY	CNY
ES ND	0	0	15	0
ES POT	0	0	0	0
ES H	0	0	88	0
ES H1	0	0	0	0
ES H2	0	0	9	0
ES H3	0	0	0	0
ES J1	134	127	33	0
ES J2	136	130	35	0
${ES}_{\gamma }^{MV}$	0	0	15	0
(e)
ES	USD	EUR	JPY	CNY
ES ND	63.49%	−51.45%	−90.66%	61.87%
ES POT	−40.82%	−80.57%	−70.70%	−95.51%
ES H	78.43%	216.52%	31.67%	79.42%
ES H1	145.67%	214.74%	99.53%	17.96%
ES H2	56.76%	−7.52%	−59.26%	−29.49%
ES H3	49.73%	−77.41%	−69.90%	−75.69%
ES J1	−98.60%	−76.67%	−66.52%	−66.29%
ES J2	−98.52%	−75.41%	−66.52%	−66.58%
${ES}_{\gamma }^{MV}$	−71.04%	−65.46%	−68.36%	−71.75%
(f)
ES	USD	EUR	JPY	CNY
ES ND	0.00%	0.00%	100.00%	0.00%
ES POT	0.00%	0.00%	0.00%	0.00%
ES H	0.00%	0.00%	−43.18%	0.00%
ES H1	0.00%	0.00%	0.00%	0.00%
ES H2	0.00%	0.00%	100.00%	0.00%
ES H3	0.00%	0.00%	0.00%	0.00%
ES J1	100.00%	100.00%	100.00%	0.00%
ES J2	100.00%	99.23%	100.00%	0.00%
${ES}_{\gamma }^{MV}$	0.00%	0.00%	0.00%	0.00%

Source: prepared by the authors.

Table 3b shows, based on the ES estimators calculated, how many times the daily exchange rate change exceeded the limits imposed by the ES. Table 3b shows that the limits were breached across all parities, but at a lower volume for USD. Regarding USD parity, the breaches occurred in just one of the non-parametric Expected Shortfall–ES H. Regarding EUR, the breaches occurred in one non-parametric ES, ES H, and in one semi-parametric ES–ES J2, i.e., in the ES H with 17 breaches, and in the ES J2 with 1 breach. In JPY parity, the breaches occurred in one non-parametric Expected Shortfall, i.e., in the ES H, with 126 breaches. Regarding CNY parity, the breaches occurred in one non-parametric Expected Shortfall; i.e., in the ES H, with 1 breach. The large volume of breaches in the JPY non-parametric estimators can be explained by its historic low volatility in a COVID scenario.

Table 3c shows the calculations made in the empirical analyses using the exchange rates between the USD X BRL, EUR X BRL, BRL X JPY, and BRL X CNY currency pairs for the period from 11 March 2020 to 4 May 2023. To analyze the effectiveness of the log return indicators, data from 6 May 2023 to 1 August 2025 was collected.

With this data in hand, it was possible, using the values in Table 3c, to identify, within these histories, the number of times in which the exchange rate change exceeded the limits set by the Expected Shortfall.

Table 3d shows, based on the ES estimators calculated, how many times the daily exchange rate change exceeded the limits imposed by the ES. Table 3d shows that the limits were breached for all parities, but at a lower volume for USD. Regarding USD parity, the breaches occurred in two semi-parametric Expected Shortfall models, i.e., in ES J1, with 134 breaches, and in ES J2, with 136 breaches. For EUR parity, breaches occurred in two semi-parametric ES, i.e., in ES J1, with 127 breaches, and in ES J2, with 130 breaches. Regarding JPY parity, the breaches occurred in one parametric Expected Shortfall, in two non-parametric Expected Shortfalls, and two semi-parametric Expected Shortfalls, i.e., in ES ND, with 15 breaches; in ES H, with 88 breaches; in ES H2, with 9 breaches; in ES J1, with 33 breaches; and ES J2, with 35 breaches. This parity is the only one of the groups that breached ${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{M}\mathit{V}}$, with 15 breaches. Regarding CNY parity, the breaches did not occur in any Expected Shortfall.

With this data in hand, it was possible, using the values in Table 3c, to identify the number of times in which the exchange rate variation exceeded the limits established by the Expected Shortfall, as shown in Table 3d.

Table 3e shows the variance between the Expected Shortfall stipulated in Table 3a,c. The variations in the empirical analyses from the exchange rates between the analyses done in the time windows of the USD X BRL, EUR X BRL, BRL X JPY and BRL X CNY currency pairs for the period from 3 January 2000 to 5 May 2023, and 11 March 2020 to 5 May 2023 are shown in Table 3e. These variations show the difference between the estimators calculated in the short-term and most recent analysis and the long-term and most complete analysis. The semiparametric estimators tend to be lower in smaller and most recent time windows than in the larger and more complete time window, and the non-parametric estimators tend to be larger in the smaller and most recent time window than in the larger and more complete time window. And the average of the eight estimators tends to be larger in the smaller and most recent time window than in the large and most complete time window.

The same process was done to analyze the effectiveness of the estimators from 11 March 2020 to 1 August 2025, and it is shown in Table 3f. This table shows the inconsistency in the frequency with which the exchange rate exceeds the ES limits, comparing longer, more complete periods with shorter, more recent ones.

These critical points occurring across the variation of time windows for the period of exchange rate volatility tend to be higher in the time windows of the EJ1 estimator of USD, EUR, and JPY. The ES was not reached in the large, most complete time window, but it was reached in the smaller, more recent time window. The same happened in ES J2 of USD and JPY. There were no occurrences in ES ND of USD, EUR and CNY in both period windows. The same happened in ES POT, ES H1 and ES H3 of the four parities. And there was no variance in ES H and ES H2 of USD, EUR and CNY. But there was 99.23% variance in ES J2 of EUR in the number of times the exchange rate exceeded ES limits in the results of analyses from different time windows. Furthermore, the variation in semi-parametric estimators tends to be more sensitive to different time windows when the exchange rate volatility exceeds the ES limits for EUR, and less sensitive in ES H for JPY.

These variance timetables show that, as indicated in Table 3e, the variance of the ES estimators is, most of the time, positive. As a positive variation, the estimators of the smaller and most recent period tend to be higher than the larger and more complete period. This was to be expected, given that, during this period, there was the COVID crisis, and the financial risk was higher than usual. Table 3f reflects the variance in the number of times the exchange rate exceeded the ES limits, and it shows that, for most of them, there were no variations. Variations occurred, often because there were no variations in the larger period, but there were in the smaller one. There were two occurrences in the larger period that indicate variations, compared to the smaller period—the ES H of YEN—43.18% smaller, and ES J2—99.23 larger than the larger period. Most of the indicators did not have any occurrence, and, when they did, most of them happened in the smaller and most recent analyses.

These analyses show that the ES is a good indicator for smaller and more recent analyses, and this estimator is better for more recent analyses than for larger and more complete analyses.

5. Conclusions

The empirical analyses show that the variation in the BRL exchange rates tends to be more volatile in parametric and non-parametric estimators when comparing the larger time window, from 3 January 2000 to 5 May 2023, to the smaller time window, from 11 March 2020 to 5 May 2023, when analyzing the four parities. These events show that the ES calculated by non-parametric and semi-parametric methodologies is more sensitive to reaching the ES zone than that calculated using parametric methodologies.

The limits imposed by the semi-parametric ES are violated more in smaller and most recent time windows, and the limits imposed by the non-parametric ES are not violated in smaller and most recent time windows of EUR and CNY. Unlike the behavior of other exchange rates, the USD did not show changes in the number of ES occurrences when compared to the smaller and more recent time window.

Due to the tail behavior of ES on different exchange rates, hedging strategies should be different, but they can use derivatives to build these strategies. ${\mathit{E}\mathit{S}}_{\mathit{\gamma }}^{\mathit{M}\mathit{V}}$ is the average of the eight ES models demonstrated, and it shows that the smaller and most recent time window from volatility analysis is better than the larger and more complete volatility analysis. Its results show that the most recent time windows reflect better than the most complete volatility analyses. These risk estimators show a higher correlation with more recent data than with a longer historical base, so the operational cost should be lower.

The limitations of this study indicate that, since Expected Shortfall occurs during periods of financial crisis, in calmer periods, Expected Shortfall may not reflect the true risk of the operation due to its reliance on daily data. Therefore, it is important that investors and policymakers use these estimators to prevent financial disasters.