VaR computation through square root method and assumptions behind it

If historical data (daily returns) have the stationary property of time series, below conditions, then the total variance will rise linearly with time. It means that longer the time period you select more you expect the drift from where you started.

• Constant µ (mean) for all t.
• Constant σ (variance) for all t.
• The autocovariance function between Xt1 and Xt2 only depends on the interval t1 and t2. zero auto correlation.

For stationary process, square root of time rule can be used to calculate the variance of longer or shorter period with given variance. VaR  inherits the property of standard deviation or variance so square root of time rule can be used to convert the VaR from one holding period to another holding period. 

Lot of banks use this property to calculate the var for different holding period. They calculate the var for 1 day and then convert this to different holding periods as required.

But basic assumption that historical returns are stationary are rarely meet in practice. Hence there can be material difference in VaR when directly calculated for a particular holding period and calculated using square root of time rule. 

This was the one of the main reason for the difference in mRWA of different banks for same hypothetical portfolio. The difference in VaR depends upon how co-related series is.
I observed the difference of ~ $12653 in 99 percentile ,10 day VaR for 1 million USD-INR Fx exposure when I calculated it with different methods.