Risk measure

A risk measure is a function that allows you to get an assessment of financial risk for a certain portfolio of assets in quantitative terms (most often in monetary terms). A risk measure is used to determine the amount of reserve capital necessary to meet the requirements of the regulator .

Content

Properties

From the point of view of financial mathematics , a measure of risk is a function that maps a random variable (which, for example, may correspond to the future value of the assets in the portfolio) to a set of real numbers . A common designation for a risk measure associated with a random variable $X$ ${\ displaystyle X}$ , is an $\rho (X)$ ${\ displaystyle \ rho (X)}$ .

Risk measure $\rho :{\mathcal {L}}\to \mathbb {R} \cup \{+\infty \}$ ${\ displaystyle \ rho: {\ mathcal {L}} \ to \ mathbb {R} \ cup \ {+ \ infty \}}$ must satisfy the following properties:

Normalization

\rho (0)=0

{\ displaystyle \ rho (0) = 0}

If the portfolio does not have assets, then it carries no risk.

Continuous broadcast

\mathrm {If} \;a\in \mathbb {R} \;\mathrm {and} \;Z\in {\mathcal {L}},\;\mathrm {Then} \;\rho (Z+a)=\rho (Z)-a

{\ displaystyle \ mathrm {If} \; a \ in \ mathbb {R} \; \ mathrm {and} \; Z \ in {\ mathcal {L}}, \; \ mathrm {Then} \; \ rho ( Z + a) = \ rho (Z) -a}

Adding a risk-free asset to the portfolio (for example, a certain amount of cash) reduces the risk for this portfolio by the value of this asset.

Monotone: $\mathrm {If} \;Z_{1},Z_{2}\in {\mathcal {L}}\;\mathrm {and} \;Z_{1}\leq Z_{2},\;\mathrm {Then} \;\rho (Z_{2})\leq \rho (Z_{1})$ ${\ displaystyle \ mathrm {If} \; Z_ {1}, Z_ {2} \ in {\ mathcal {L}} \; \ mathrm {and} \; Z_ {1} \ leq Z_ {2}, \; \ mathrm {Then} \; \ rho (Z_ {2}) \ leq \ rho (Z_ {1})}$

If the portfolio $Z_{2}$ ${\ displaystyle Z_ {2}}$ always contains more reliable assets than a portfolio $Z_{1}$ ${\ displaystyle Z_ {1}}$ for almost all scenarios, then portfolio risk $Z_{2}$ ${\ displaystyle Z_ {2}}$ should be less than portfolio risk $Z_{1}$ ${\ displaystyle Z_ {1}}$ . For example, $Z_{1}$ ${\ displaystyle Z_ {1}}$ - this is an option to buy shares and $Z_{2}$ ${\ displaystyle Z_ {2}}$ - this is the same option, but with a smaller strike .

Examples

The following risk measures are widely used in practice.

Value at risk
Expected shortfall
Exposure at default

Dispersion as a risk measure

Dispersion (or standard deviation ) is not a measure of risk, since it does not correspond to the above properties of monotony and constant translation. Indeed, $Var(X+a)=Var(X)\neq Var(X)-a$ ${\ displaystyle Var (X + a) = Var (X) \ neq Var (X) -a}$ for all $a\in \mathbb {R}$ ${\ displaystyle a \ in \ mathbb {R}}$ .

Coherent Risk Measure

The concept of a coherent risk measure was introduced by Artzner, Delbin, Eber, and Heath in 1998. A risk measure is considered coherent if, in addition to the above properties, it also satisfies the following requirements:

Sub additivity: $\mathrm {If} \;Z_{1},Z_{2}\in {\mathcal {L}},\;\mathrm {Then} \;\rho (Z_{1}+Z_{2})\leq \rho (Z_{1})+\rho (Z_{2})$ ${\ displaystyle \ mathrm {If} \; Z_ {1}, Z_ {2} \ in {\ mathcal {L}}, \; \ mathrm {Then} \; \ rho (Z_ {1} + Z_ {2} ) \ leq \ rho (Z_ {1}) + \ rho (Z_ {2})}$

Diversification principle: the risk for two assets combined into one portfolio cannot be greater than the total risk for each of their assets separately. The use of netting also reduces the overall risk of the portfolio ^[1] .

Uniformity: $\mathrm {If} \;\alpha \geq 0\;\mathrm {and} \;Z\in {\mathcal {L}},\;\mathrm {Then} \;\rho (\alpha Z)=\alpha \rho (Z)$ ${\ displaystyle \ mathrm {If} \; \ alpha \ geq 0 \; \ mathrm {and} \; Z \ in {\ mathcal {L}}, \; \ mathrm {Then} \; \ rho (\ alpha Z ) = \ alpha \ rho (Z)}$

Roughly speaking, doubling a portfolio, we double the risk.

It can also be said that a risk measure is coherent if it can be represented as a supremum of mathematical expectations of possible losses for a certain family of probability measures P:

\rho (X)=\sup {E_{P}[\rho (X)|P\in Q]}

{\ displaystyle \ rho (X) = \ sup {E_ {P} [\ rho (X) | P \ in Q]}}

Measures P can be considered as scenarios of market developments, and Q - as a set of all possible scenarios. With this interpretation, coherent measures estimate the average loss in the worst case scenario.

Value At Risk is not a coherent measure of risk, since it does not satisfy the sub-additivity property. To illustrate, we use the following example. Suppose we are trying to calculate VaR for a confidence level of 95% and a time horizon of 1 year. The portfolio consists of two zero-coupon bonds that will be redeemed in 1 year. Also suppose that:

The current yield of these bonds is 0%
They have various issuers
The probability of default for each of the bonds over the next year is 4% and these probabilities are independent

Then 95% VaR for the portfolio, which consists of one such bond, is 0, since the probability of default of the bond (4%) is less than the confidence level (5%). However, if we include 50% of each bond in the portfolio, then the probability that at least one of the bonds will be defaulted will be 7.84% and will exceed 5% confidence level, which means VaR will be more than 0. This is a violation of the sub- additivity, as a diversified portfolio should have less risk.

Notes

↑ Alexander J. McNeil, Rüdiger Frey, Paul Embrechts. Basic Concepts in Risk Management // Quantitative Risk Management: Concepts, Techniques and Tools - Revised Edition. - Princeton University Press, 2015 .-- P. 74. - 720 p. - ISBN 1400866286 .

[1] Alexander J. McNeil, Rüdiger Frey, Paul Embrechts. Basic Concepts in Risk Management // Quantitative Risk Management: Concepts, Techniques and Tools - Revised Edition. - Princeton University Press, 2015 .-- P. 74. - 720 p. - ISBN 1400866286 .