# «Value-at-Risk: Strengths, Caveats and Considerations for Risk Managers and Regulators Master Thesis by Bogdan Izmaylov Supervisor: Thomas Berngruber ...»

Daníelsson (2002) examines the limitations of the modern risk models both for risk management and regulation. In particular, he studies the precision of the models at different confidence levels and finds that in terms of precision, GARCH and RiskMetrics models are the best at 95% level, but their performance diminishes at 99%. The BCBS capital requirements based on VaR are criticized as arbitrary and ineffective. The author believes that when VaR is targeted by institutions in order to meet capital requirements, it poses risks of manipulation of the measures and escalates the losses in times of crises. He also proposes a

**corollary of Goodhart’s Law applied to VaR:**

“A risk model breaks down when used for regulatory purposes” The scaling of VaR for different time horizons and distributions has been addressed in another paper by Daníelsson and Zigrand (2006), where the authors conclude that the square-root-of-time rule leads to underestimation of risk. This bias is very small for the 10-day period (which coincides with the requirements of BCBS), but increases at an increasing rate for longer horizons.

In his book on financial risk forecasting, Daníelsson (2011) has combined the methodology of VaR with the versatility and processing power of modern software (R and MATLAB). The comprehensive guide on implementation of risk

articles. The author stresses on the importance of the assumptions made for each model and the critical interpretation of results. For the calculation of highconfidence VaR, EVT methodology is used to calculate VaR in the extreme regions of the distribution tails. This approach mitigates the underestimation of risk, since the distribution of returns in the tails is better captured by the EVT.

The endogenous prices in financial markets are pointed out as the cases, when the RMPs need to utilize common sense and intuitive understanding of risk measures to be able to make rational decisions.

Nocera (2009), in the Risk Mismanagement article, presents us with an excellent discussion of opinions from VaR opponents and advocates. The main arguments presented in the article against VaR are that it cannot predict extreme risks and that it is valid for distributions which are not common for financial returns. The responses of the VaR proponents are suggesting that the mistakes in risk management were primarily in the managerial decisions and not in the models.

Risk management experts share in the interviews with the author, that VaR measure has been manipulated by managers in order to create “asymmetric risk positions”, which increased the losses that were not captured by the models. VaR also did not capture the effects of leverage. The author concludes that the latest crisis has shown once again the critical need for better understanding of risk.

Guégan and Tarrant (2012) resent in their paper theoretically the insufficiency of up to four risk measures combined in order to capture the risk from peculiar, although possible, loss distributions. Since the measures can produce the same risk estimates for different distributions, the proposed solution for assessment of risk profile is to use 95% and 99% VaR and TCE in combination with Maximum Loss (ML). The authors suggest that the banks should be required to submit multiple risk measures to the regulatory bodies.

for calculating VaR and the daily capital charges for authorized deposit taking institutions. The study found that there is no single best model, but each model performs best at certain time periods (stages of the economic cycle). Before the GFC, GARCH is the best in terms of days with minimized capital charges.

Riskmetrics model performs best in the beginning of the crisis, until September

2009. Exponential GARCH with shocks following the Student-t distribution performs best until the end of 2009. The authors conclude that the institutions should use multiple risk models in order to minimize the capital charges and thus the penalties from VaR violations.

In the “BlackRock: The monolith and the markets” (2013), a concern on the widespread use of same risk models in the financial markets is presented. The BlackRock’s risk-management system popularity can be compared with the wide-spread use of VaR since the RiskMetrics release and its implementation in the Basel accords. The main concern is connected to the limitations of the models, also as pointed out by Danielsson (2002), which are used by the increasingly large groups of people. Even the systems much more sophisticated than VaR, like the BlackRock’s “Aladdin”, raise concerns when used by large group of market participants.

VaR. The ambition of the study is to provide an overview of caveats and strengths of VaR use in risk management practice, as well as to provide some suggestions for mitigation of possible issues with the use of this risk measure.

4.1 VaR as a measure for non-normally distributed returns.

Assumption of normality is crucial for some methods of calculating VaR. This is quite similar to many other financial models, since normality greatly simplifies

incomprehensively complex. Since the normal distribution can be described by only 2 moments – the mean and the standard deviation, the VaR calculation is the simplest when the assumption of normality holds. But the methodology does not break up when the returns are not normally distributed, even though VaR may lose some desirable properties (Artzner et al., 1999) One of the main critiques of VaR is that it relies on parametric distributions, which can be a poor fit for the market data. Danielsson (2011) emphasizes in his

**book the stylized facts about financial returns:**

** Figure 6. Daily S&P 500 returns 1989-2009 density plot and the normal distribution fit (red line).**

Source: own calculations.

Figures 5 and 6 clearly show the differences between normally distributed returns and the distribution of S&P500 data. The real probability is higher-thannormal near the mean of the distribution, it is lower when the distance from the mean increases, and it is again higher-than-normal in the tails of the distribution, which is a very important fact for risk analysis.

- Volatility estimation models have to be flexible enough in order to capture the clusters – high/low volatility has to be followed by

- Linear dependency can approximate the joint movements in the centre of the distributions, but it becomes progressively less precise for

The obvious choice and common practice for calculating VaR in the case of on non-normally distributed returns is the Historical Simulation, Nonparametric VaR. But such approach raises another question, whether it is realistic to assume that the history repeats itself and as such whether the past contains enough information for prediction of future events. There is also a trade-off between the size of the estimation window (the number of observations used to forecast VaR) and the speed at which the forecasts will adjust to new information. The years of returns data before the first part of 2008 (before the GFC) are clearly a poor fit for the VaR forecast of returns in the second part of 2008. The HS VaR is a relatively simple and precise tool when there are no or very little structural changes in risk, like in the years before the GFC (Daníelsson, 2011).

To account for non-normality of returns while allowing for losses that are higher than the ones from the estimation window, either a parametric distribution of choice (e.g. Student-t) should be used, or the Generalized Pareto Distribution applied to model the distribution in the tails, while accounting for skewness and kurtosis of the data.

In order to incorporate risk changes quickly and precisely into VaR forecasts,

1998; Jorion, 2007; Danielsson, 2011).

Nassim Taleb criticizes VaR methods for using past return data (HS) and relying on normal distribution (parametric methods), concluding that VaR cannot predict extreme events (“Black Swans”) and is harmful because it creates a false sense of security and confidence in RMPs. The obvious counter-argument is that the non-parametric and parametric methods should be used in the context of the market situation. This is where the managerial discretion is important – it is up to the risk manager to choose the best appropriate method. The choice of distribution for calculation of VaR is also important and the normal distribution should not be used to simplify calculations at the cost of precision. Existing methods are sufficiently precise given that the choice of method is justified by the data and market conditions (Daníelsson, 2011; Jorion, 2007). By definition, VaR gives the maximum loss for a given confidence interval, given normal market conditions (when the markets are abnormal, VaR estimates are not valid).

The limitations of models also have to be taken into account and the manager’s understanding of these models and the risks they forecast is crucial for the quality of VaR forecasts.

Given the reality of the modern financial markets, it is unrealistic to assume normal distribution of financial returns, meaning that more sophisticated VaR methods should be used by RMPs, combined with improved risk understanding.

Using of HS in order to capture the non-normality should be done in the context of the fact that such calculations do not provide sufficient information about the future volatility (Fong Chan & Gray, 2006; Pérignon & Smith, 2010).

4.2 The quality of high confidence level VaR forecasts – 95%, 99% VaR and higher.

Due to the nature of VaR as a quantile risk measure, the precision of the estimates

which can be used to estimate VaR, as well as from the fact that the financial data distribution has fat tails. The problem is even worse for estimation of ES, since the measure itself is not a quantile, but the mean of all the quantiles beyond α.

This is confirmed by the empirical findings of Giannopoulos & Tunaru (2005) and Yamai & Yoshiba (Yamai & Yoshiba, 2005). There are several most common ways

**of overcoming such limitations in practice:**

4) Applying EVT to model the tails of the distribution (99%+ VaR) Gathering more data can be problematic, and the trade-off between the estimation window length and speed of adjustment to new information is an important concern. Daily VaR is a commonly calculated measure, and gathering more daily data may not be very challenging in most situations. But if the weekly, 10-day or even monthly VaR needs to be calculated, it can be impossible to obtain enough observations in order to get sufficiently precise estimates.

If the available data is limited, bootstrapping can provide more precise estimates of VaR, ES and their standard errors, without any assumptions about the distribution of returns. The observations are drawn randomly with replacement from the collected sample. This method also adjusts slowly to new information in the sample and relies on the assumption that the history repeats itself, but improves on HS by providing more precise estimates. Pritsker (2006) confirms that bootstrapping is superior to HS by analysing the returns of the exchange rates of 10 currencies versus the USD.

If the distribution of returns can be approximated and defined in the simulation

generator, the number of simulations, and the quality of the transformation method. The transformation method is the way of converting the randomly generated numbers into the random numbers from the distribution of interest.

Since the number of calculations can be very high for a big financial institution, pseudo-random number generators can be used. These produce a pre-defined sequence of “random” numbers, which allows the simulation to converge faster and produce accurate results with fewer simulations. Even with these tweaks, Monte Carlo (MC) is the most computationally demanding method.

Another way to speed up simulations and VaR calculation is to use factor models.

Such models assume that the changes in risk are driven by changes in one or several risk factors. Each position is modelled according to its exposure to these risk factors, thus only a handful of factors need to be simulated by using MC.

For calculation of high-precision VaR with confidence levels above 99%, the EVT can be applied. It accounts for the shape of the tail of the distribution, usually producing more precise estimates. EVT should be used with caution, since the precision of the estimates is even more sensitive to the underlying distribution of returns in comparison to lower confidence level VaR. Danielsson (2011) suggests using EVT with sample sizes above 1000 and for confidence levels of 99,6% and higher. (Lin & Shen, 2006) also provide evidence on improved performance of VaR when calculated based on student-t or EVT modelled tails, in comparison to normal distribution tails for confidence level of 98,5% and higher. An easy approximation would be to use normal or student-t tails up to 95% confidence level, and use student-t for 99% or when the number of observations does not allow to implement EVT efficiently.

There is no method superior to others in all situations (although usually MC can better forecast the tails of returns distributions at the cost of computation time),

the forecast and fundamental understanding of the models and nature of risk.

4.3 VaR as an aggregate measure of risk.