Coherent risk measure

In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.

PropertiesEdit

Consider a random outcome   viewed as an element of a linear space   of measurable functions, defined on an appropriate probability space. A functional    is said to be coherent risk measure for   if it satisfies the following properties:[1]

NormalizedEdit

 

That is, the risk when holding no assets is zero.

MonotonicityEdit

 

That is, if portfolio   always has better values than portfolio   under almost all scenarios then the risk of   should be less than the risk of  .[2] E.g. If   is an in the money call option (or otherwise) on a stock, and   is also an in the money call option with a lower strike price. In financial risk management, monotonicity implies a portfolio with greater future returns has less risk.

Sub-additivityEdit

 

Indeed, the risk of two portfolios together cannot get any worse than adding the two risks separately: this is the diversification principle. In financial risk management, sub-additivity implies diversification is beneficial. The sub-additivity principle is sometimes also seen as problematic.[3][4]

Positive homogeneityEdit

 

Loosely speaking, if you double your portfolio then you double your risk. In financial risk management, positive homogeneity implies the risk of a position is proportional to its size.

Translation invarianceEdit

If   is a deterministic portfolio with guaranteed return   and   then

 

The portfolio   is just adding cash   to your portfolio  . In particular, if   then  . In financial risk management, translation invariance implies that the addition of a sure amount of capital reduces the risk by the same amount.

Convex risk measuresEdit

The notion of coherence has been subsequently relaxed. Indeed, the notions of Sub-additivity and Positive Homogeneity can be replaced by the notion of convexity:[5]

Convexity
 

Examples of risk measureEdit

Value at riskEdit

It is well known that value at risk is not a coherent risk measure as it does not respect the sub-additivity property. An immediate consequence is that value at risk might discourage diversification.[1]Value at risk is, however, coherent, under the assumption of elliptically distributed losses (e.g. normally distributed) when the portfolio value is a linear function of the asset prices. However, in this case the value at risk becomes equivalent to a mean-variance approach where the risk of a portfolio is measured by the variance of the portfolio's return.

The Wang transform function (distortion function) for the Value at Risk is  . The non-concavity of   proves the non coherence of this risk measure.

Illustration

As a simple example to demonstrate the non-coherence of value-at-risk consider looking at the VaR of a portfolio at 95% confidence over the next year of two default-able zero coupon bonds that mature in 1 years time denominated in our numeraire currency.

Assume the following:

  • The current yield on the two bonds is 0%
  • The two bonds are from different issuers
  • Each bond has a 4% probability of defaulting over the next year
  • The event of default in either bond is independent of the other
  • Upon default the bonds have a recovery rate of 30%

Under these conditions the 95% VaR for holding either of the bonds is 0 since the probability of default is less than 5%. However if we held a portfolio that consisted of 50% of each bond by value then the 95% VaR is 35% (= 0.5*0.7 + 0.5*0) since the probability of at least one of the bonds defaulting is 7.84% (= 1 - 0.96*0.96) which exceeds 5%. This violates the sub-additivity property showing that VaR is not a coherent risk measure.

Average value at riskEdit

The average value at risk (sometimes called expected shortfall or conditional value-at-risk or  ) is a coherent risk measure, even though it is derived from Value at Risk which is not. The domain can be extended for more general Orlitz Hearts from the more typical Lp spaces.[6]

Entropic value at riskEdit

The entropic value at risk is a coherent risk measure.[7]

Tail value at riskEdit

The tail value at risk (or tail conditional expectation) is a coherent risk measure only when the underlying distribution is continuous.

The Wang transform function (distortion function) for the tail value at risk is  . The concavity of   proves the coherence of this risk measure in the case of continuous distribution.

Proportional Hazard (PH) risk measureEdit

The PH risk measure (or Proportional Hazard Risk measure) transforms the hazard rates   using a coefficient  .

The Wang transform function (distortion function) for the PH risk measure is  . The concavity of   if   proves the coherence of this risk measure.

 
Sample of Wang transform function or distortion function

g-Entropic risk measuresEdit

g-entropic risk measures are a class of information-theoretic coherent risk measures that involve some important cases such as CVaR and EVaR.[7]

The Wang risk measureEdit

The Wang risk measure is defined by the following Wang transform function (distortion function)  . The coherence of this risk measure is a consequence of the concavity of  .

Entropic risk measureEdit

The entropic risk measure is a convex risk measure which is not coherent. It is related to the exponential utility.

Superhedging priceEdit

The superhedging price is a coherent risk measure.

Set-valuedEdit

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.[8]

PropertiesEdit

A set-valued coherent risk measure is a function  , where   and   where   is a constant solvency cone and   is the set of portfolios of the   reference assets.   must have the following properties:[9]

Normalized
 
Translative in M
 
Monotone
 
Sublinear

General framework of Wang transformEdit

Wang transform of the cumulative distribution function

A Wang transform of the cumulative distribution function is an increasing function   where   and  . [10] This function is called distortion function or Wang transform function.

The dual distortion function is  .[11][12] Given a probability space  , then for any random variable   and any distortion function   we can define a new probability measure   such that for any   it follows that   [11]

Actuarial premium principle

For any increasing concave Wang transform function, we could define a corresponding premium principle :[10] 

Coherent risk measure

A coherent risk measure could be defined by a Wang transform of the cumulative distribution function   if and only if   is concave.[10]

Set-valued convex risk measureEdit

If instead of the sublinear property,R is convex, then R is a set-valued convex risk measure.

Dual representationEdit

A lower semi-continuous convex risk measure   can be represented as

 

such that   is a penalty function and   is the set of probability measures absolutely continuous with respect to P (the "real world" probability measure), i.e.  . The dual characterization is tied to   spaces, Orlitz hearts, and their dual spaces.[6]

A lower semi-continuous risk measure is coherent if and only if it can be represented as

 

such that  .[13]

See alsoEdit

ReferencesEdit

  1. ^ a b Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D. (1999). "Coherent Measures of Risk". Mathematical Finance. 9 (3): 203. doi:10.1111/1467-9965.00068. S2CID 6770585.
  2. ^ Wilmott, P. (2006). "Quantitative Finance". 1 (2 ed.). Wiley: 342. {{cite journal}}: Cite journal requires |journal= (help)
  3. ^ Dhaene, J.; Laeven, R.J.; Vanduffel, S.; Darkiewicz, G.; Goovaerts, M.J. (2008). "Can a Coherent Risk Measure be too Subadditive?". Journal of Risk and Insurance. 75 (2): 365–386. doi:10.1111/j.1539-6975.2008.00264.x. S2CID 10055021.
  4. ^ Rau-Bredow, H. (2019). "Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures". Risks. 7 (3): 91. doi:10.3390/risks7030091.
  5. ^ Föllmer, H.; Schied, A. (2002). "Convex measures of risk and trading constraints". Finance and Stochastics. 6 (4): 429–447. doi:10.1007/s007800200072. S2CID 1729029.
  6. ^ a b Patrick Cheridito; Tianhui Li (2008). "Dual characterization of properties of risk measures on Orlicz hearts". Mathematics and Financial Economics. 2: 2–29. doi:10.1007/s11579-008-0013-7. S2CID 121880657.
  7. ^ a b Ahmadi-Javid, Amir (2012). "Entropic value-at-risk: A new coherent risk measure". Journal of Optimization Theory and Applications. 155 (3): 1105–1123. doi:10.1007/s10957-011-9968-2. S2CID 46150553.
  8. ^ Jouini, Elyes; Meddeb, Moncef; Touzi, Nizar (2004). "Vector–valued coherent risk measures". Finance and Stochastics. 8 (4): 531–552. CiteSeerX 10.1.1.721.6338. doi:10.1007/s00780-004-0127-6.
  9. ^ Hamel, A. H.; Heyde, F. (2010). "Duality for Set-Valued Measures of Risk". SIAM Journal on Financial Mathematics. 1 (1): 66–95. CiteSeerX 10.1.1.514.8477. doi:10.1137/080743494.
  10. ^ a b c Wang, Shaun (1996). "Premium Calculation by Transforming the Layer Premium Density". ASTIN Bulletin. 26 (1): 71–92. doi:10.2143/ast.26.1.563234.
  11. ^ a b Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability. 11 (3): 385. doi:10.1007/s11009-008-9089-z. hdl:10016/14071. S2CID 53327887.
  12. ^ Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (PDF). Archived from the original (PDF) on July 5, 2016. Retrieved March 10, 2012.
  13. ^ Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. ISBN 978-3-11-018346-7.