VaR Calculation: The Assumptions of Standard Distribution

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:

  • Mean close to zero
  • Negative skewness
  • Positive kurtosis, indicating heavier tails than normal

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.

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