COMPARATIVE ANALYSIS OF HISTORICAL VAR AND GARCH VAR METHODS IN ASSESSING THE MARKET RISK OF A CURRENCY PORTFOLIO

ABSTRACT

This article provides a comparative analysis of two popular market risk assessment methods: historical simulation (Historical VaR) and a parametric approach based on conditional heteroskedasticity models (GARCH VaR). The study is based on empirical data of exchange rates (USD, EUR, RUB). Using a 252-day rolling window method, one-step-ahead risk forecasts were generated at a 99% confidence level. Backtesting results showed that both models adequately assess the risk of a diversified currency portfolio; however, the GARCH model demonstrates more conservative estimates for individual currencies during volatile periods by capturing volatility clustering and utilizing the Student's t-distribution.

АННОТАЦИЯ

В данной статье проводится сравнительный анализ двух популярных методов оценки рыночного риска — исторического моделирования (Historical VaR) и параметрического подхода на основе моделей условной гетероскедастичности (GARCH VaR). Исследование базируется на эмпирических данных обменных курсов валют (USD, EUR, RUB). С использованием метода скользящего окна (rolling window) длиной 252 дня были построены прогнозы риска на шаг вперед при доверительном уровне 99%. Результаты бэктестинга показали, что обе модели адекватно оценивают риск диверсифицированного валютного портфеля, однако модель GARCH демонстрирует более консервативные оценки для индивидуальных валют в периоды волатильности благодаря учету эффекта кластеризации и использованию t-распределения Стьюдента.

Keywords:  market risk, currency portfolio, Value at Risk, VaR, GARCH, backtesting, historical simulation.

Ключевые слова: рыночный риск, валютный портфель, Value at Risk, VaR, GARCH, бэктестинг, историческое моделирование.

1. INTRODUCTION

In the context of current global economic instability and high financial market volatility, assessing market risk is a critical task for financial institutions and private investors. Currency risk, as a component of market risk, requires reliable mathematical and statistical tools. The most common metric for quantifying risk is Value at Risk (VaR), which determines the maximum possible loss of a portfolio over a given time horizon with a certain confidence level.

Despite the widespread use of VaR, the choice of the optimal calculation method remains an open question. The traditional historical simulation method (Historical VaR) is simple to implement but relies on the assumption that past return distributions will perfectly repeat in the future. In contrast, parametric models of the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) family can account for such stylized facts of financial time series as volatility clustering and heavy-tailed distributions.

The purpose of this study is an empirical comparison of the effectiveness of the Historical VaR and GARCH VaR methods in assessing the market risk of a multicurrency portfolio based on a backtesting procedure.

2. LITERATURE REVIEW

The evolution of market risk measurement has been extensively documented in academic literature, particularly since the widespread adoption of Value at Risk in the 1990s following the publication of J.P. Morgan’s RiskMetrics methodology and subsequent regulatory frameworks established by the Basel Committee on Banking Supervision. VaR has become the industry standard for risk reporting due to its ability to aggregate multiple risk factors into a single, easily interpretable monetary figure representing potential downside risk.

Historically, the most straightforward approach to estimating VaR has been Historical Simulation (HS). As noted by Beder (1995), the primary advantage of HS is its non-parametric nature; it does not require an assumption about the underlying statistical distribution of asset returns. Instead, it relies entirely on the empirical distribution of historical data. This allows the model to naturally incorporate skewness, excess kurtosis, and linear or non-linear dependencies among assets. However, researchers have consistently pointed out the limitations of the historical approach. Pritsker (2006) emphasized that traditional HS applies equal weights to all historical observations within the selected window, which can lead to the so-called "ghost effects." When a period of extreme volatility exits the historical window, the forecasted VaR drops abruptly, even if current market conditions remain unchanged. Furthermore, HS reacts slowly to sudden structural breaks or fresh volatility spikes in the market.

To overcome the static nature of historical modeling, academic research shifted focus towards parametric models that can capture the dynamic properties of financial time series. It has long been established, dating back to Mandelbrot (1963) and Fama (1965), that financial returns exhibit specific "stylized facts." The most prominent of these are volatility clustering—where large price changes tend to be followed by large changes, and small changes by small ones—and leptokurtosis (fat tails), meaning extreme events occur much more frequently than a normal (Gaussian) distribution would predict.

The foundational breakthrough in modeling time-varying volatility was the Autoregressive Conditional Heteroskedasticity (ARCH) model introduced by Engle (1982). This framework allowed the conditional variance of returns to depend on past squared residuals. Bollerslev (1986) generalized this approach (GARCH) by including lagged values of the conditional variance itself, resulting in a more parsimonious model that fits financial data exceptionally well.

The application of GARCH models to VaR estimation significantly improved the accuracy of risk forecasting. However, early GARCH models assuming a normal distribution for the error term still underestimated the frequency of extreme losses in highly volatile markets. Consequently, Baillie and Bollerslev (1989) and other researchers advocated for the use of the Student's t-distribution within the GARCH framework. This combination directly addresses both volatility clustering (via the GARCH equations) and the heavy-tailed nature of returns (via the degrees of freedom parameter in the t-distribution).

In the specific context of foreign exchange (FX) markets, the comparison between Historical and GARCH-based VaR has been a subject of ongoing debate. Currency markets are highly susceptible to macroeconomic shocks, geopolitical events, and central bank interventions, often resulting in abrupt shifts in volatility regimes. Studies by Jorion (2006) indicate that while parametric models like GARCH provide superior forecasting ability during periods of high turbulence, they can sometimes overestimate risk during tranquil periods, leading to inefficient capital allocation. Conversely, while a diversified portfolio may mask the deficiencies of simple models due to cross-asset correlation offsets, individual currency pairs often expose the weaknesses of the Historical method.

Therefore, empirical backtesting remains the definitive method for validating risk models. This research builds upon the existing literature by conducting a strict empirical comparison between the non-parametric Historical VaR and the highly responsive GARCH VaR with a Student's t-distribution, specifically applied to a modern currency portfolio context encompassing the USD, EUR, and RUB exchange rates.

3. RESEARCH METHODOLOGY

The study considers two approaches to calculating the one-day VaR at a significance level of α = 0.01 (99% confidence level).

3.1. Historical VaR (Historical Simulation). This method does not assume an analytical specification of the return distribution function. The forecasted VaR value at time t+1 is calculated as the α-quantile of the empirical return distribution over the preceding period (in our case, a rolling window of 252 observations).

3.2. GARCH VaR. This approach models the variance of returns as a time-varying quantity. To account for the "heavy tails" of empirical financial data distributions, the study used a GARCH(p,q) model with a Student's t-distribution. The conditional variance equation in the basic GARCH(1,1) model is:

where ω, ,  are estimated parameters, and  are past residuals.

During the study, the optimal parameters (p, q) of the GARCH model were automatically selected at each step using the Bayesian Information Criterion (BIC). VaR was calculated using the following formula:

where  is the quantile of the Student's t-distribution with ν degrees of freedom.

To assess the adequacy of the models, a backtesting procedure based on a rolling window was applied. At each step, a window of 252 log returns was taken, a VaR forecast for t+1 was generated, and then it was compared with the actual return of the following day. If the actual loss exceeded the forecasted VaR, a violation (breach) was recorded.

4. EMPIRICAL DATA

The initial data consisted of foreign currency exchange rates against the national currency: US dollar (USD), euro (EUR), and Russian ruble (RUB). The data were pre-cleaned of weekends and holidays to ensure the continuity of the time series. Simple and logarithmic returns were calculated based on the quotes. A hypothetical currency portfolio (PORTFOLIO) was formed. The out-of-sample test size was 252 days. At a 1% significance level, the expected number of violations for this period is 2.52 ().

5. RESEARCH RESULTS

The results of the backtesting procedure for the aggregate portfolio and individual currencies are presented in Table 1.

Table 1.

Summary of VaR backtesting results (99% confidence level)

Analysis of the results shows that for the aggregate currency portfolio (PORTFOLIO), both models demonstrated equal and high efficiency: exactly 2 violations were recorded against an expected value of 2.52. The violation rate was 0.79%, which almost perfectly aligns with the target significance level of 1%. This indicates that the diversification effect within the portfolio smooths out the extreme price movements of individual assets, allowing even the simpler Historical VaR method to provide accurate estimates.

However, significant differences emerge when analyzing individual currencies. The GARCH VaR model proved to be substantially more conservative. For the US dollar and the Russian ruble, the GARCH model did not record a single violation (0 vs. 2 and 1 for Historical VaR, respectively), and for the euro, it showed 1 violation vs. 3 for the historical method. The absence or reduced number of violations in the GARCH approach is explained by the fact that the specification with the Student's t-distribution incorporates "fatter tails," and the model itself reacts more quickly to an increase in market turbulence, thereby sharply raising the required risk capital estimates.

6. CONCLUSION

The comparative analysis allows us to conclude that the choice of a market risk assessment method should depend on the structure of the assets. For highly diversified portfolios, the Historical VaR method may be preferable due to its computational simplicity and the absence of model risk, as it demonstrates adequate results at the portfolio level.

Nevertheless, when assessing the risk of concentrated positions or individual currencies prone to sudden shifts in volatility regimes, GARCH VaR with a Student's t-distribution is a more reliable tool. This model enables the prompt adaptation of forecasts to current market conditions, safeguarding the investor's capital from unforeseen shocks, although this comes at the cost of excessively conservative estimates (leading to higher capital reserve requirements).

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