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I'm Everette (18) from Calgary, Canada. <br>I'm learning Korean literature at a local high school and I'm just about to graduate.<br>I have a part time job in a post office.<br>xunjie 小売分野に従事する職員があります。
In [[financial mathematics]], a '''deviation risk measure''' is a function to quantify [[financial risk]] (and not necessarily [[downside risk]]) in a different method than a general [[risk measure]].  Deviation risk measures generalize the concept of [[standard deviation]].
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==Mathematical definition==
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A function <math>D: \mathcal{L}^2 \to [0,+\infty]</math>, where <math>\mathcal{L}^2</math> is the [[L2 space]] of [[random variable|random]] [[Portfolio (finance)|portfolio returns]], is a deviation risk measure if
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# Shift-invariant: <math>D(X + r) = D(X)</math> for any <math>r \in \mathbb{R}</math>
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# Normalization: <math>D(0) = 0</math>
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# Positively homogeneous: <math>D(\lambda X) = \lambda D(X)</math> for any <math>X \in \mathcal{L}^2</math> and <math>\lambda > 0</math>
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# Sublinearity: <math>D(X + Y) \leq D(X) + D(Y)</math> for any <math>X, Y \in \mathcal{L}^2</math>
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# Positivity: <math>D(X) > 0</math> for all nonconstant ''X'', and <math>D(X) = 0</math> for any constant ''X''.<ref name="Rockafellar">{{cite journal|title=Deviation Measures in Risk Analysis and Optimization|first1=Tyrrell|last1=Rockafellar|first2=Stanislav|last2=Uryasev|first3=Michael|last3=Zabarankin|year=2002|url=http://papers.ssrn.com/sol3/papers.cfm?abstract_id=365640|accessdate=January 15, 2012}}</ref><ref>{{cite journal|title=Progress in Risk Measurement|first1=Siwei|last1=Cheng|first2=Yanhui|last2=Liu|first3=Shouyang|last3=Wang|year=2004|journal=Advanced Modelling and Optimization|volume=6|number=1}}</ref>
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==Relation to risk measure==
There is a [[bijection|one-to-one]] relationship between a deviation risk measure ''D'' and an expectation-bounded [[risk measure]] ''R'' where for any <math>X \in \mathcal{L}^2</math>
* <math>D(X) = R(X - \mathbb{E}[X])</math>
* <math>R(X) = D(X) - \mathbb{E}[X]</math>.
''R'' is expectation bounded if <math>R(X) > \mathbb{E}[-X]</math> for any nonconstant ''X'' and <math>R(X) = \mathbb{E}[-X]</math> for any constant ''X''.
 
If <math>D(X) < \mathbb{E}[X] - \operatorname{ess\inf} X</math> for every ''X'' (where <math>\operatorname{ess\inf}</math> is the [[essential infimum]]), then there is a relationship between ''D'' and a [[coherent risk measure]].<ref name="Rockafellar" />
 
==Examples==
{{stub-section|examples of measures other than stdev|date=May 2013}}
The [[standard deviation]] is clearly a deviation risk measure.
 
==References==
{{Reflist}}
 
[[Category:Mathematical finance]]
[[Category:Financial risk]]

Latest revision as of 05:56, 21 March 2013

In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation.

Mathematical definition

A function , where is the L2 space of random portfolio returns, is a deviation risk measure if

  1. Shift-invariant: for any
  2. Normalization:
  3. Positively homogeneous: for any and
  4. Sublinearity: for any
  5. Positivity: for all nonconstant X, and for any constant X.[1][2]

Relation to risk measure

There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure R where for any

R is expectation bounded if for any nonconstant X and for any constant X.

If for every X (where is the essential infimum), then there is a relationship between D and a coherent risk measure.[1]

Examples

Template:Stub-section The standard deviation is clearly a deviation risk measure.

References

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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