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Let [itex]X[/itex] be a set of [itex]N[/itex] lognormal prices (in dollars), meaning

[tex]\log(X) = Y \sim MN(\mu_Y , \Sigma_Y) ,[/tex]

i.e. the log of [itex]X[/itex] follows a multivariate normal distribution.

Imagine now that one wants to compute various quantiles for this set, e.g. 2.5%, 50% and 97.5\%, and does this by simulating 100k draws from the distribution above.

You then get say a [itex]100k \times N[/itex] matrix, and to get the total value you'd find these three quantiles for each of the [itex]N[/itex] prices, resulting in a [itex]N \times 3[/itex] matrix of quantiles, and then simply add up each of the three columns giving three numbers which represent the total quantiles for the whole set.

So, my question is:

How would one go about transforming the information for the quantiles for the whole set from log-dollars to dollars?

I am of course aware of the exponential function, so what I'm asking is how and where in this process do I use it?

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My main idea is this:

The three final quantiles, 2.5%, 50% and 97.5\%, represent the log-quantiles of the set as a whole, say [itex] V_{2.5\%}, V_{50\%}[/itex] and [itex] V_{97.5\%}[/itex]. They also have their log/exp-counterparts, e.g. [itex] V_{50\%} = log(X_{50\%})[/itex].

Now, the difference between, for example, the 50% and 2.5%, [itex]V_{50\%} - V_{2.5\%}[/itex] could then be represented as

[tex] log(X_{50\%}) - log(X_{2.5\%}) = log\left(\frac{X_{50\%}}{X_{2.5\%}}\right),[/tex]

meaning that the log difference can be interpreted as log-proportional difference. Thus one could just exp and inverse this value and get

[tex] exp \left( log \left( \frac{ X_{50\%} }{ X_{2.5\%} } \right) \right)^{-1} = \frac{X_{2.5\%}}{X_{50\%}} [/tex]

which gives you the proportional difference of the quantiles in dollars, instead of log-dollars. With this information the 2.5% quantile can be obtained by multiplying [itex] X_{2.5\%}[/itex] with the mean of the [itex]N[/itex] prices.

Or am I way off here?

Any help is greatly appreaciated.

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# Quantiles of a log-multivariate-normal-distributed set.

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