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 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 through Risksvr with a 1 day horizon and obtain 1 Mio USD.

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

By using a 95% confidence interval, we have scaled the portfolio's volatility by 1.645. (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). 

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).

Why do we use a mean return of Zero?


 

What DOES  1Mio USD VaR really mean?

There are two ways of looking at VaR results. And this really depends on your approach towards dealing with risk, i.e. your management "style". Defensive or aggressive ? Tactical or Strategic ?

 You can look at the Same 1 Mio USD VaR result in two 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: The Investor or Risk Avoider looks at his daily business and how risk might impact negatively his bottom line.
He places himself under the probability distribution of returns (the right hand bell shape of the picture above). This tactical approach is typically what a risk officer in the corporate world should be doing.

The 1 Mio USD example is our WORST estimate of our 19 days in the month. This also implies there is ONE day where we will loose MORE than 1 Mio dollars. 

You can also look at the same result the other way round:

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 ! the biggest flaw and conversely strength of VaR lies in 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 quite 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).


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

Proper Risk Management should ALWAYS be Complemented with Stress Tests and Marginal Sensitivity Measures to identify and pinpoint so called "hot-spots".

Normality assumptions are obviously not valid when assessing extreme events, 
But in most cases this is not important at all. What we need is a clear view of our risks and we can get that by following a disciplined approach that defines a  combined methodology policy (i.e. Parametric for linear, well balanced short term view, Monte-Carlo for complex Derivative Portfolios ? Historical for specific spread risks and a series of scenarios to highlight hot spots). 

By implementing a combined approach you can build upon the strength of each model and forget their shortcomings! 

 Relative VaR