Talk:Coherent risk measure

Monotonicity

 * Monotonicity
 * $$\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} \;\mathrm{and}\; Z_1 \leq Z_2 ,\; \mathrm{then} \; \rho(Z_1) \leq \rho(Z_2)$$

That is, if portfolio $$Z_2$$ has better values than portfolio $$Z_1$$ under all scenarios then the risk of $$Z_2$$ should be bigger than the risk of $$Z_1$$: more profit, more risk.

That is wrong. If all scenarios are worse the risk measure should be higher, not lower, as stated. See for that: http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf page 7 —Preceding unsigned comment added by 91.34.52.136 (talk) 13:58, 29 March 2010 (UTC)

91.34.52.136 (talk) 13:58, 29 March 2010 (UTC)

Agreed! Smc2911 (talk) 07:09, 28 April 2010 (UTC)

Here is another reference: Convex measures of risk and trading constraints

Smc2911 (talk) 07:14, 28 April 2010 (UTC)

Incohorent risk measure
The page gives no clue what X and Y are.

The reader wants to assume these are securities or investments, so that \rho(X) is the risk of X. But then we have:

"Monotonicity   \rho(X) \leq \rho(Y) whenever Y \leq X "

So X can't be an investment: one investment is not less 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.

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.

But it offers nothing but confusion and mystery to the non-expert reader who wants to know what a coherent risk measure is.

Joaquin

Third rule
I don't understand the third rule, translational invariance. What is d?

Shouldn't the fourth rule be called 'linearity' not 'homogeneity'?

Clarification of $$rho$$, $$X$$, and $$Y$$
The monotonicity axiom is confusing if you don't know what $$\rho$$ represents. Is $$\rho$$ an antitone function? Seems to me that if $$X$$ and $$Y$$ represent the value of a risky item, and $$\rho$$ is a risk-quantification measure, then monotonicity should be:

if $$X \leq Y$$ then $$\rho(X) \leq \rho(Y)$$

That is, a highly priced item is considered riskier (note the reversal of the implication as well).

--203.185.215.144 23:44, 28 February 2007 (UTC) Greg


 * With $$X$$ and$$Y$$ representing (future) values of a risky portfolio, the monotonicity axiom is exactly the opposite:
 * if $$X \geq Y$$ then $$\rho(X) \leq \rho(Y)$$,
 * meaning if $$X$$ portfolio value is higher than $$Y$$ portfolio value for every possible outcome (state of the world), then its risk (identified with the cash amount that has to be added to the portfolio to become acceptable) should be smaller. Just check Artzner et al. I will correct this if nobody has any objections. -- Zsolt Tulassay 15:09, 15 June 2007 (UTC)

Non-coherance of VaR
Could someone add an example illustrating this as it's not immediately obvious. An example would be of great benefit in my opinion —Preceding unsigned comment added by Red.devil.ade (talk • contribs) 09:30, 18 February 2009 (UTC)

Translation invariance, which sign?
Shouldn't it be +a instead of -a after the equality? Like this:


 * Translation invariance
 * $$\mathrm{If}\; a \in \mathbb{R} \; \mathrm{and} \; Z \in \mathcal{L} ,\;\mathrm{then}\; \rho(Z + a) = \rho(Z) + a $$

For instance top page 7 in this article suggests the same:

http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.3340v1.pdf

Or am I wrong? —Preceding unsigned comment added by MVjensen (talk • contribs) 11:02, 12 November 2009 (UTC)

I have now changed the sign, since noone commented on this discussion.

MVjensen (talk) 20:40, 7 December 2009 (UTC)

The negative sign is correct because the risk is reduced when you add cash.

91.34.52.136 (talk) 13:55, 29 March 2010 (UTC)

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). This page is written in the paradigm (b) ... maybe worth to clarify this? Is it ok to do so? Phdb (talk) 14:22, 12 August 2011 (UTC)

Other non-coherent measures than VaR
There are many other risk measures that are not cohrent (eg. VAR, standard deviation, etc.). Ok to write a small text about them too? Phdb (talk) 14:22, 12 August 2011 (UTC)


 * 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 risk measures. Zfeinst (talk) 17:34, 12 August 2011 (UTC)

External link: The Case for Incoherence
In external links there is the page
 * Glyn Holton: The Case for Incoherence

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.

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?

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
 * $$\rho(\text{drink from all 3}) \leq \sum_{i = 1}^3 \rho(\text{drink from well }i)$$

which can trivially be satisfied by situations where $$\rho(\text{drink from well }i) < \rho(\text{drink from all 3})$$ 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 $$\rho(\text{drink from well }i) \leq \rho(\text{drink from all 3})$$ for all wells.

Am I missing something? Does anyone object to removing the link? Thanks. Zfeinst (talk) 04:06, 12 March 2012 (UTC)