VaR Updates

 

To consider the VaR constructions in more detail, consider how to define the VaR for a portfolio with just a single asset.  If this asset has a normal distribution then a parametric VaR would say there is a 1% chance that a day's trading will end more than -2.326 standard deviations below average (see the example above for details on calculating V from a Z statistic).  We would use information on the past performance of the asset to determine its average and standard deviation.  A historical simulation VaR would rank, say, the past 4 years of data (approximately 1000 observations) to find the worst 10. 

 

We go back-and-forth between returns (percent changes,  ) and price changes (absolute change, un-scaled).

 

In continuous time  so  so estimate .

 

In discrete time,  so estimate .

 

Recall that the l0g-normal assumption on returns was that the standardized distribution (returns minus their mean, divided by their standard deviation) was distributed normal with zero mean and unit standard deviation, that .  We typically assume that the average change is zero (i.e. that the markets are efficient).  The sigma volatility term should be chosen in units that match the dating of the  -- if they are daily changes then use daily volatility; if they are weekly or monthly or annual returns then use weekly or monthly or annual volatility; and so on.  If  is the annual volatility then remember that  is the volatility over some time period, T – assuming that the changes are uncorrelated over time.  (This is often seen as unreasonable so we might directly find the distribution of changes over the desired time period.)

 

So if we get a critical value corresponding to the expected change in VaR, call it , then  gives the critical value of the percent return.  Using the normal distribution we find either 2.326 for 1% or 1.64 for 5%.  Next either translate this into absolute returns or use the formula with percent returns.  So this is the critical value for percent changes in the return of the underlying security, that we then put into the delta or delta-gamma approximation.

 

If the portfolio is constructed of options then we might use, for example, the same -2.326 standard deviations (assuming we want a 1% VaR) and use the return volatility to decide the critical value of the return, .  Just put these into the equation above, using deltas and gammas of the portfolio, .