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### An analytical/semi-analytical analysis of the characteristics of the Expected Worst Loss in T realisations for Normal random variables

Suppose we have  which are independent identically distributed Normal random variables, i.e. . The worst loss in  realisations is .

What, in such circumstances is the expected worst loss (in  realisations)?

Without much loss of generality, we can focus on , i.e. unit Normal random variables, as .

If  then all of the  must also satisfy . As the  are independent and identically distribution, this means that:

where  is the cumulative distribution function of the unit Normal distribution.

So the probability density function of  is  where:

The expected value of , i.e. the expected worst loss in  realisations for a unit Normal distribution, is then , i.e.:

Using a symbolic algebra engine we find that there are analytical formulae  for  = 1, 2 or 3 but not thereafter:

 1 2 3

As  increases   becomes more negative, but only relatively slowly:

 1 0 2 -0.5641895835 -0.5641895835 4 -1.029375373 -0.4651857895 8 -1.423600306 -0.394224933 16 -1.765991393 -0.342391087 32 -2.069668828 -0.303677435 64 -2.343733465 -0.274064637 128 -2.594597369 -0.250863904 256 -2.826863279 -0.232265910

We may check that the values shown above are reasonable using a simulation approach, e.g. using VBA code in Microsoft Excel as per VBA code that can be used to check this analysis.

The worst loss can also be thought of as a specific quantile of a sample of Normal random variables. It might be viewed as the ’th quantile (i.e. half way between 0 and , given that the sample has  observations) . We might therefore expect its expected value to be similar to the corresponding quantile point of the Normal distribution. However, the Normal probability density function is not normally approximately flat at the relevant quantile point (and is instead upward sloping), so the Expected Worst Loss in T realisations is normally somewhat above the ’th Normal quantile point:

 c.f. 1 0 0 2 -0.5641895835 -0.67449 4 -1.029375373 -1.15035 8 -1.423600306 -1.53412 16 -1.765991393 -1.86273 32 -2.069668828 -2.15387 64 -2.343733465 -2.41756 128 -2.594597369 -2.66007 256 -2.826863279 -2.88563