We have learned about three important Value-at-Risk models 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.

• 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.
• Historical simulation is more popular, and can be more accurate if the P&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).
• 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.
• 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.

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.