In reality- returns distributions

For simplicity and convenience VaR is usually calculated by making an assumptions concerning actual form of the probability distribution of expected return to be normally or log-normally distributed. For example: JP Morgan Risk Metrics (similar to EWMA approach) which is used to forecast the volatility of the portfolio returns, assumes that returns on securities follow a conditionally normal distribution. Additionally Variance Co-variance approach, Historical simulation approach and  Monte Carlo Simulation are all based on distributional assumptions.

However compared to normal (bell-shaped) distribution, actual asset tends to be Fat tailed, Skewed and Time varying. A Fat tailed distribution is characterized by having more probability weight (observations) in its tails relative to the normal distribution. A skewed distribution refers- in this context of financial returns- to the observation that declines in asset prices are more sever than increases. This is in contrast to the symmetry that is built into the normal distribution. Time varying unstable distribution means that the parameters (e.g mean, volatility) vary over time due to variability in market conditions.

Normal Returns	Actual Financial Returns
Symmetrical Distribution	Skewed
“Normal” Tails	Fat-tailed (leptokurtosis)
Stable Distribution	Time-varying parameters

For example: Interest rate distributions are not constant over time
The 10 years interest rate data are collected (1982-1993) and we plot the  daily change in the three-month treasury rate. We observe that the average change is approximately zero, but the probability 'mass' is greater at both tails. It is also greater at the mean; i.e, the actual mean occurs more frequently than predicted by the normal distribution.

The reason that we see fat tails in actual return distribution is because conditional mean and volatility are time varying. It means there is conditional distribution of market returns and it depends on some economic or market or other state. However, given the assumption that markets are efficient time varying conditional mean is refuted by some authors.

Returns are unconditional and normally distributed        Returns are conditional on some information
The implication of heavy tail is that Value at Risk is underestimated.

For example: If normal distribution says VaR is -10% at 95% confidence level, and in the case there is a fat tail distribution, then the expected VAR loss is understated.