Hi

I am not sure if this will in any way help.

**Please refer to the attached excel** for understanding why (n-1) and not 'n'.

**Assumptions : **
Assuming losses follow Normal distribution with Mean '0' and Standard Deviation =1 and we are computing 95% VaR. Hence, in Excel, you can compute

**[email protected]% = Normsinv(0.95) = 1.64485**.

**(***If you are aware of VaR etc, then actually VaR = -Normsinv(1 - 0.95) = -(-1.64485) = 1.64485)*
As you may be aware, by definition of VaR, this means you are

**95% confident** **that tomorrow if you incur a loss, you won't lose more that 1.64485**. However, it also means that there is

**5% chance that your loss may exceed this VaR value of 1.64485**.

And if that happens, what will be the loss? Unfortunately this is one of the limitations of VaR (other is VaR non sub-additive etc unless you are dealing with elliptical distributions).

VaR doesn't provide the answer to the question "

**Loss beyond VaR?**".

Hence, we use Expected Shortfall measure which is coherent risk measure.

Assuming, the Area under the Standard Normal Curve (Mean = 0, stdev = 1) is divided into

**n = 10** equal parts (also called slices) beyond

[email protected]% i.e. between 95% to 100%.

Hence, the first slice is (95% - 95.5%), second slice is (95.5% - 96%) and so on. Thus the 10th slice is (99.5% t0 100%). As you can observe from the attached excel, you have n = 10 slices,

**but the VaR points (beyond 95% VaR) are only 9 (i.e. = (n-1))**.

As ES is simply the average of these VaR values (exceeding

[email protected]%), you get ES = 2.025.

Please Note that Quantile Value @95% is your regular VaR and hence is not to be included in ES computation as ES by definition is loss exceeding VaR value. Also, as can be seen from the excel, NORMSINV(100%) is not defined. Hence, you are left with only 9 VaR data points (exceeding

[email protected]%), hence the value (n-1) and not n.

Hope this helps.

Cheers!!!

Regards

Ashok

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