Absolute Value-at-Risk VaR - Mean Zero VaR Reports

Value at Risk (VaR) is computed as the value at a given percentile p where q is the p-th quantile

if the projected portfolio returns are effectively Normally distributed, then the mean is 0 and VaR becomes  simply

where Z(p) is the confidence multiplier required to reach the selected percentile.

We run our portfolio with a 1 day horizon and obtain 1 Mio USD.

Absolute VaR is the mean zero volatility or standard deviation of our projected portfolio  scaled by the confidence interval of the normal distribution's probability density. 

To get a 95% confidence we must scale the projected portfolio's volatility by 1.645.
This multiplicative factor of 1.645 is taken from the standard one tailed normal distribution (A one-tailed cumulative normal distribution with mean zero and unit variance (i.e. one standard deviation) has a 84% likely-hood of taking place so we multiply this 1.645 to reach the 95% confidence level. 

So, if our portfolio has a volatility of 600 000 dollars (i.e. there are 84% chances of loosing 600'000 USD), then we have a 95% chance of loosing 1 Mio USD. (600'000x 1.645).

Justification for a Zero Mean

The mathematical justification for the zero mean comes directly from the Normal distribution (see above).
Intuitively, in a high frequency trading environments where assets are held over short periods of time, the main concern is the price of the asset and not it's payout (which btw would need fiscal adjustment) , especially since the price must, at least in theory, reflect this information.
From a Risk Management perspective this is more conservative as the mean would actually reduce the risk as it moves the market losses rightwards.

What does 1 Mio USD VaR really mean ?

There are two ways of looking at VaR results.

And this really depends on your business line and your investment style.

Is your investments style Tactical or Strategic ?

Are you conservative (defensive) or aggressive ? Are you  trying to minimize market losses or Maximize return on investment.

 Indeed, you can look at the Same 1 Mio USD VaR result in two opposite ways:
 

This also means that 19 out of 20 business days (i.e. 1 Calendar Month). you should NOT loose more than 1 Mio USD.

This is the "Going concern" approach.

As a conservative investor you want to make sure market losses will not impact negatively on your bottom line.

In this framework, you are looking at the probability under the bell shaped distribution of returns.

This tactical approach is typically what a risk officer in the corporate world should be doing.

In this respect, the 1 Mio USD market loss is our WORST estimate for 19 out of 20 working days in the month.

The other side of the same Coin

The 1 Mio USD market loss mentionned previously is our worst estimate for 19 out of the 20 working days in the month.

This also implies there is ONE day where the market loss will be MORE than 1 Mio dollars. 

This is the Extreme Event perspective, typical of the speculator or Risk Taker.

This means that one day out of 20 (one working day in the month), you will be loosing more than 1 Mio USD. 

Knowing the shortcoming of normal distribution assumptions (see six sigma events), we are placing ourselves under the left hand side of the probability distribution of returns (see picture above), which is also commonly known as the TAIL.

A Risk taker is really interested in this side of the coin. What he really wants to know is the loss incurred during these "5% percent of the time" events.
If he can anticipate reasonably well this loss, he can avoid forced liquidation of his positions or even worse, bankruptcy.

Yes,.. but is this our best estimate!

Yes ! this is indeed the biggest flaw and conversely strength of the Normal distribution assumptions.

There has been a lot of debate regarding Normality shortcomings. Newcomers to Risk often assume this tends to invalidates results.

Well, first of all, Normal distribution assumptions hold relatively well when:

-looking at risk from the perspective of a going concern.
-the portfolio is well distributed across  asset classes and market data
-the horizon is short (and the data is sampled daily).
-The portfolio has proportionally few derivatives.


Best Practice Risk Management is actually much more important than the technical intricacies nested in the models used to compute VaR.

Proper Risk Management Infrastructure should always be complemented with multiple adverse Stress Tests combined with Marginal VaR Sensitivity Measures to identify and pinpoint "hot-spots".

Normality assumptions are obviously not valid when assessing extreme events, 
But in most cases this is not as important as most beginners assume since other distribution assumptions can also be used in most models.

What we really need is a clear view of our risks and we can get this by implementing a consistent framework based on combined methodologies. coupled with multiple adverse what-if stress-tests that are established according to multiple factors such as Marginal Risk, Marginal defaults or predictive worst-case scenarios in order to highlight hot spots). 

 Relative VaR