We have learned about three important <a title="A Comparison of Values-at-Risk (VAR) Methods" href="https://financetrain.com/a-comparison-of-values-at-risk-var-methods/">Value-at-Risk models</a> that are most commonly used by banks and financial institutions, namely, analytical VaR, historical simulation VaR, and Monte Carlo simulation VaR.

None of these models are perfect and have certain assumptions which make their results not entirely suitable to how the financial markets behave.

Here are a few observations and limitations about these VaR models.
<ul>
	<li>Both analytical VaR and Monte Carlo VaR calculations assume that the asset returns are normally distributed. In reality, however, financial returns are not normal, and have tendencies of skewness and leptokurtosis. When applied to a linear portfolio, both these methods will produce quite similar VaR estimates.</li>
	<li>Historical simulation is more popular, and can be more accurate if the P&amp;Ls are non-normal. However, it requires a huge amount of historical data. Even if the data is available, it may not be really directly usable because we will apply the current portfolio weights to derive the price series. This is not correct because we would have had a very different looking portfolio at different times in history (The portfolio weights will not remain the same).</li>
	<li>Even in historical simulation we will make the assumption that returns are independent and identically distributed (i.i.d.) because we will user Square Root of Time rule toi compute the 10-day or one month VaR.</li>
	<li>Bothe analytical VaR and Monte Carlo simulation use Covariance matrix. The covariance matrix can be based on a small amount of historical data. However, having one covariance matrix assumes that all the co-variations between risk factors are linear, which is not always the case.</li>
</ul>
As we can see all the three models have some problems, which require us to investigate more advanced solutions. We will start by looking at the assumption of standard distribution and how it is violated.

Problems with the VaR Models

While calculating VaR using one of the statistical models, we make many assumptions, one of them is that the asset returns are i.i.d. normally distributed. This means that just like a coin toss, each return is an independent draw from the normal distribution.

However, in reality, there is empirical evidence that the financial data is not normally distributed and exhibits properties of skewness or kurtosis. You can take an asset’s daily historical returns for a long period of time and run some statistics. The data analysis will produce results similar to following:
<ul>
	<li>Mean close to zero</li>
	<li>Negative skewness</li>
	<li>Positive kurtosis, indicating heavier tails than normal</li>
</ul>
The results may be different but this is one common pattern. The presence of skewness and kurtosis has already violated the condition of normality. Further the combination of negative skewness and positive kurtosis indicates that there is a high probability of a large negative return than estimated under normal distribution. So, if we are using data that has heavy tails, and still using the standard distribution assumption for calculating VaR, then our VaR will underestimate the actual risk of the portfolio. A lower estimate of VaR leads to a lower/insufficient capital maintained by the bank.

Assume that our analysis is based on 1,000 observations. At 99% confidence level, under normal distribution, 10 of these observations (returns) will be in the lower tail (1% of 2000). However, in reality we have heavy tails which suggests there are more than 10 returns in the lower tail.

What we are interested in is identifying the source of this non-normal behaviour. Our analysis assumes that the volatility over the observation period is constant. However, we know intuitively that it is not true and the volatility would have considerably varied over time, with high volatility in certain periods. This change in volatility apparently contributes maximum to the deviation from normal distribution, and if our models and calculations could account for increases in volatility, then the standard distribution assumption will become more reliable.

When we said that the returns are assumed to be i.i.d. normal, it means that today’s returns are totally independent from yesterday’s returns. However, empirical evidence suggests that this assumption of independence is not real. Today’s returns will be influenced by the size of yesterday’s returns. If yesterday we observed large movement, it will be followed by another large movement today also. This does not concern the direction of movement, however. This concept is known as “Volatility Clustering” which is the most important characteristic of financial data. Financial models have already started accounting for volatility clustering. In the following articles we will learn about how volatility and volatility clustering can be modelled.

VaR Calculation: The Assumptions of Standard Distribution

Advanced Value at Risk Models

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COURSE

Advanced Value at Risk Models

LESSONS