Risk measure

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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 set[edit]

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 .[5]

Risk measure to acceptance set[edit]

  • 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 measure[edit]

  • 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 measure[edit]

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.[6]

See also[edit]


  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. S2CID 6770585. 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. S2CID 18237100.
  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. ^ Jokhadze, Valeriane; Schmidt, Wolfgang M. (2018). "Measuring model risk in financial risk management and pricing". SSRN. doi:10.2139/ssrn.3113139. S2CID 169594252. {{cite journal}}: Cite journal requires |journal= (help)
  5. ^ 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. S2CID 154784949.
  6. ^ 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. {{cite journal}}: Cite journal requires |journal= (help)

Further reading[edit]