Risk measure explained

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 to convex and coherent risk measurement.

Mathematically

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

X is \rho(X) . A risk measure \rho: l{L} \to R \cup \{ +inf ty\} should have certain properties:^[1]

Normalized

\rho( 0 ) = 0

Translative

If a \in R and Z \in l{L} , then \rho(Z + a) = \rho(Z) - a

Monotone

If Z _{1,Z} _{2} \in l{L} and Z _{1} \leq Z _{2} , then \rho(Z _{2)} \leq \rho(Z _{1)}

Set-valued

In a situation with

R ^{d} -valued portfolios such that risk can be measured in m \leq d 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]

Mathematically

A set-valued risk measure is a function

R: L ^{p} _{d} → F _{M} , where L ^{p} _{d} is a d -dimensional Lp space, F _{M} = \{D \subseteq M: D = cl (D + K _{M)\}} , and K _{M} = K \cap M where K is a constant solvency cone and M is the set of portfolios of the m reference assets. R must have the following properties:^[3]

Normalized

K _{M} \subseteq R( 0 ) and R( 0 ) \cap - \operatorname{int}K _{M} = \emptyset

Translative in M

\forall X \in L ^{p,} _{d} \forall u \in M: R(X + u 1 ) = R(X) - u

Monotone

\forall X _{2} - X _{1} \in L ^{p(K)} _{d} ⇒ R(X _{2)} \supseteq R(X _{1)}

Examples

• Value at risk
• Expected shortfall
• Superposed risk measures^[4]
• Entropic value at risk
• Drawdown
• Tail conditional expectation
• Entropic risk measure
• Superhedging price
• Expectile

Variance

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,

Var(X + a) = Var(X) ≠ Var(X) - a for all a \in R , 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

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

R _{A _{R}} (X) = R(X) and A _{R _{A}} = A .^[5]

Risk measure to acceptance set

• If

\rho is a (scalar) risk measure then A _{\rho} = \{X \in L ^{p:} \rho(X) \leq 0 \} is an acceptance set.

• If

R is a set-valued risk measure then A _{R} = \{X \in L ^{p} _{d:} 0 \in R(X)\} is an acceptance set.

Acceptance set to risk measure

• If

A is an acceptance set (in 1-d) then \rho _{A(X)} = inf \{u \in R : X + u 1 \in A\} defines a (scalar) risk measure.

• If

A is an acceptance set then R _{A(X)} = \{u \in M: X + u 1 \in A\} is a set-valued risk measure.

Relation with deviation risk measure

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

\rho where for any X \in l{L} ^{2} D(X) = \rho(X - E [X]) \rho(X) = D(X) - E [X] .\rho is called expectation bounded if it satisfies \rho(X) > E [ - X] for any nonconstant X and \rho(X) = E [ - X] for any constant X.^[6]

Further reading

• Book: Crouhy , Michel . D. Galai . R. Mark . Risk Management . . 2001 . 752 pages . 978-0-07-135731-9 .
• Book: Kevin , Dowd . Measuring Market Risk . 2nd . . 2005 . 410 pages . 978-0-470-01303-8 .
• Book: Hans. Foellmer. Alexander. Schied. Stochastic Finance. Walter de Gruyter. 2004. 978-311-0183467. de Gruyter Series in Mathematics. 27. Berlin. xi+459. 2169807.
• Book: Alexander. Shapiro. Darinka. Dentcheva. Ruszczyński. Andrzej. Andrzej Piotr Ruszczyński. Lectures on stochastic programming. Modeling and theory. Society for Industrial and Applied Mathematics. 2009. 978-0898716870. MPS/SIAM Series on Optimization. 9. Philadelphia. xvi+436. 2562798.

Notes and References

1. Artzner. Philippe. Delbaen. Freddy. Eber. Jean-Marc. Heath. David. 1999. Coherent Measures of Risk. Mathematical Finance. 9. 3. 203–228. February 3, 2011. 10.1111/1467-9965.00068. 6770585 .
2. Jouini. Elyes. Meddeb. Moncef. Touzi. Nizar. 2004. Vector–valued coherent risk measures. Finance and Stochastics. 8. 4. 531–552. 10.1007/s00780-004-0127-6. 10.1.1.721.6338. 18237100.
3. Hamel . A. H. . Heyde . F. . 10.1137/080743494 . Duality for Set-Valued Measures of Risk . SIAM Journal on Financial Mathematics . 1 . 1 . 66–95 . 2010 . 10.1.1.514.8477 .
4. Jokhadze . Valeriane . Schmidt . Wolfgang M. . March 2020 . 10.1142/s0219024920500120 . 2 . International Journal of Theoretical and Applied Finance . 2050012 . 3113139 . Measuring model risk in financial risk management and pricing . 23. free .
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. 10.1007/s11579-011-0047-0. 1011.5986. 154784949.
6. Deviation Measures in Risk Analysis and Optimization. Tyrrell. Rockafellar. Stanislav. Uryasev. Michael. Zabarankin. 22 January 2003. 365640.