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In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement.



A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns. The common notation for a risk measure associated with a random variable   is  . A risk measure   should have certain properties:[1]



In a situation with  -valued portfolios such that risk can be measured in   of the assets, then a set of portfolios is the proper way to depict risk. Set-valued risk measures are useful for markets with transaction costs.[2]


A set-valued risk measure is a function  , where   is a  -dimensional Lp space,  , and   where   is a constant solvency cone and   is the set of portfolios of the   reference assets.   must have the following properties:[3]

Translative in M



Variance (or standard deviation) is not a risk measure in the above sense. This can be seen since it has neither the translation property nor monotonicity. That is,   for all  , and a simple counterexample for monotonicity can be found. The standard deviation is a deviation risk measure. To avoid any confusion, note that deviation risk measures, such as variance and standard deviation are sometimes called risk measures in different fields.

Relation to acceptance setEdit

There is a one-to-one correspondence between an acceptance set and a corresponding risk measure. As defined below it can be shown that   and  .[4]

Risk measure to acceptance setEdit

  • If   is a (scalar) risk measure then   is an acceptance set.
  • If   is a set-valued risk measure then   is an acceptance set.

Acceptance set to risk measureEdit

  • If   is an acceptance set (in 1-d) then   defines a (scalar) risk measure.
  • If   is an acceptance set then   is a set-valued risk measure.

Relation with deviation risk measureEdit

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

  •  .

  is called expectation bounded if it satisfies   for any nonconstant X and   for any constant X.[5]

See alsoEdit


  1. ^ Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk" (PDF). Mathematical Finance. 9 (3): 203–228. doi:10.1111/1467-9965.00068. Retrieved February 3, 2011.
  2. ^ Jouini, Elyes; Meddeb, Moncef; Touzi, Nizar (2004). "Vector–valued coherent risk measures". Finance and Stochastics. 8 (4): 531–552. CiteSeerX doi:10.1007/s00780-004-0127-6.
  3. ^ Hamel, A. H.; Heyde, F. (2010). "Duality for Set-Valued Measures of Risk". SIAM Journal on Financial Mathematics. 1 (1): 66–95. CiteSeerX doi:10.1137/080743494.
  4. ^ Andreas H. Hamel; Frank Heyde; Birgit Rudloff (2011). "Set-valued risk measures for conical market models". Mathematics and Financial Economics. 5 (1): 1–28. arXiv:1011.5986. doi:10.1007/s11579-011-0047-0.
  5. ^ Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization" (PDF). Archived from the original (PDF) on September 16, 2011. Retrieved October 13, 2011.

Further readingEdit