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There is indeed some confusion around about the minus sign. This relates to two differnt schools (a) the insureance people: they consider a sochastic variable representing the losses (so higher is worse) and (b) asset managers, they consider the distribution of returns (so lower values are worse).
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for all wells. In fact, the "return" of drinking from all 3 wells is worse than the "return" of drinking from just 1 well with probability 1 (death certain unless all wells clean vs. death certain only in subset of those cases). Therefore by monotonicity we have that
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than another. Unless we have some ordering of investments which is not explained and is totally mysterious. Nor can X be the value of a particular investment, since the risk of an investment (if \rho is indeed risk) is not a function of the value of that investment.
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I think a small text about them would be ok, though it should more be just to comment that other types of risk measures exist. I would not suggest using standard deviation (or variance) since they are not actually
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This article might make sense to someone who is familiar with the broader subject matter, serving to remind them of something they already know, or providing details about something they understand generally.
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In the post, it argues of course you drink from just 1; I agree. However, the author (Glyn Holton) goes from this to stating that subadditivity was violated, which is incorrect. Subadditivity would say
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given. I just read this blog post and the initial example is misinterpreted (which gives rise, I feel, to an incorrect conclusion). I would propose removing this link for this reason.
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The example (from the post) to which I am referring states: There are three wells, one or more of which may be poisoned. Should you drink from just one well or from all three?
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516:{\displaystyle \mathrm {If} \;Z_{1},Z_{2}\in {\mathcal {L}}\;\mathrm {and} \;Z_{1}\leq Z_{2},\;\mathrm {then} \;\rho (Z_{1})\leq \rho (Z_{2})}
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There are many other risk measures that are not cohrent (eg. VAR, standard deviation, etc.). Ok to write a small text about them too?
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Could someone add an example illustrating this as it's not immediately obvious. An example would be of great benefit in my opinion
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1377:{\displaystyle \mathrm {If} \;a\in \mathbb {R} \;\mathrm {and} \;Z\in {\mathcal {L}},\;\mathrm {then} \;\rho (Z+a)=\rho (Z)+a}
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But it offers nothing but confusion and mystery to the non-expert reader who wants to know what a coherent risk measure is.
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The reader wants to assume these are securities or investments, so that \rho(X) is the risk of X. But then we have:
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That is wrong. If all scenarios are worse the risk measure should be higher, not lower, as stated. See for that:
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articles on
Knowledge. If you would like to participate, please visit the project page, where you can join
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1605:{\displaystyle \rho ({\text{drink from all 3}})\leq \sum _{i=1}^{3}\rho ({\text{drink from well }}i)}
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That is, a highly priced item is considered riskier (note the reversal of the implication as well).
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representing (future) values of a risky portfolio, the monotonicity axiom is exactly the opposite:
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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This page is written in the paradigm (b) ... maybe worth to clarify this? Is it ok to do so?
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1205:. Just check Artzner et al. I will correct this if nobody has any objections. --
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The negative sign is correct because the risk is reduced when you add cash.
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Am I missing something? Does anyone object to removing the link? Thanks.
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I don't understand the third rule, translational invariance. What is d?
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I have now changed the sign, since noone commented on this discussion.
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Shouldn't the fourth rule be called 'linearity' not 'homogeneity'?
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Shouldn't it be +a instead of -a after the equality? Like this:
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is a risk-quantification measure, then monotonicity should be:
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http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf
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For instance top page 7 in this article suggests the same:
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The monotonicity axiom is confusing if you don't know what
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http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.3340v1.pdf
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which can trivially be satisfied by situations where
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1147:{\displaystyle \rho (X)\leq \rho (Y)}
1020:{\displaystyle \rho (X)\leq \rho (Y)}
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307:and see a list of open tasks.
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690:Here is another reference:
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1199:for every possible outcome
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634:: more profit, more risk.
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1103:{\displaystyle X\geq Y}
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782:Clarification of
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