Risk measure

The term risk measure is a collective term for statistical measures with which it is possible to quantitatively describe the uncertainty of an event . For example, the (overall) risk position of a company can be recorded.

General

The application of risk measures to (loss) distribution functions is a sub-task in the risk quantification , an assessment of risks by description using a suitable density or distribution function (or historical data) of the effect of the risk and the assignment of risk measures: First, those that are identified are identified Risks are described quantitatively by suitable distribution functions ( probability distribution ). A risk measure is then used to determine the level of risk from the distribution.

There are several alternative variants for the first step (determining a suitable density function):

1. by means of two distribution functions: one to display the frequency of damage in a period (for example using the Poisson distribution ) and another to display the amount of damage per claim (for example using the normal distribution )
2. by means of a connected distribution function that shows the risk impact in a period

A risk measure (such as the standard deviation or the value at risk ) is an assignment that assigns a real value to a density or distribution function. This value should represent the associated risk. This enables a comparison of risks that are described by different distribution functions. The literature does not see uniformly which properties such an assignment must meet in order to represent a risk measure (see properties of risk measures).

The risk measures can relate to individual risks (e.g. damage to property, plant), but also to the overall scope of risk (e.g. in relation to profit) of a company. The quantitative assessment of an overall risk position requires an aggregation of the individual risks . This is possible, for example, by means of a Monte Carlo simulation , in which the effects of all individual risks are considered in their dependence in the context of planning.

characterization

Risk measures can basically be divided into measures for a single risk (i.e. a risk measure in the narrower sense, such as the standard deviation) or measures that relate the risk of two random variables to one another (i.e. a risk measure in the broader sense such as covariance ).

Another differentiation between risk measures results from the extent to which information from the underlying distribution is taken into account. Two-sided risk measures (such as the standard deviation) take these into account completely, while the so-called downside risk measures (such as the VaR and the LPM measures) only consider the distribution up to a certain limit.

Properties of risk measures

Risk measures (RM) can be characterized by the properties they have (see e.g.). With the help of the corresponding definitions, it is possible to discuss which properties a risk measure should have and which risk measures have which advantages and disadvantages in relation to the description of a risk. A certain class of risk measures is also defined by the sum of risk characteristics.

monotony

Definition: A RM is monotonic if for all applies: . ${\displaystyle X,Y\in L^{0}}$ ${\displaystyle X\preceq _{st}Y\Rightarrow \rho (X)\leq \rho (Y)}$

Monotony means that if the distribution functions are valid for two distributions and at each point (so-called stochastic dominance ), then the RM must be valid . Even if this property seems trivial, it is z. B. not met in terms of volatility. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle F_{X}(x)\geq F_{Y}(x)}$ ${\displaystyle \rho (X)\leq \rho (Y)}$

Translational invariance

Definition: A RM is translation-invariant (or cash-invariant) when for each applies: . ${\displaystyle m\in \mathbb {R} }$ ${\displaystyle \rho (X-m)=\rho (X)-m}$

If the loss distribution is shifted by a fixed, risk-free amount, then - if there is translation invariance - the RM is changed by exactly the same amount.

Positive homogeneity

Definition: A RM is homogeneous positive if for each applies: . ${\displaystyle \alpha \geq 0}$ ${\displaystyle \rho (\alpha X)=\alpha \rho (X)}$

Positive homogeneity means that the loss and the risk measure are scaled to the same extent.

Subadditivity

Definition: An RM is subadditive if: ${\displaystyle \rho (X+Y)\leq \rho (X)+\rho (Y)}$

This means: If the risks of two loss distributions are combined, the risk measure should not show a greater risk. Subadditivity formalizes the basic idea of diversification . Diversification is therefore rewarded with risk measures if they are subadditive. The value at risk is a common measure of risk that is not subadditive.

However, there are also arguments that speak against the usefulness of subadditivity. The division into two separate units may in certain cases have a lower risk. An example would be the division into a " Bad Bank " and a "Good Bank". Splitting up a systemically important bank could also reduce risks for the banking system.

Law invariance (distribution invariance)

An RM is law-invariant if for two identically distributed random variables and applies: . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \rho (X)=\rho (Y)}$

This means: if the same risks exist for two random variables, if law invariance exists, the risk measure shows the same risk.

Comonotonic additivity

An RM is komonoton additive when two komonotone random variable and the following applies: . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \rho (X+Y)=\rho (X)+\rho (Y)}$

Comonotonic additivity means that if the random variables are dependent, no risks are reduced by merging the distributions, i.e. the measured risk does not decrease. If a RM has the property of comonotonic additivity, diversification is only rewarded if the risks are actually reduced.

convexity

An RM is convex if the following applies to each : ${\displaystyle \lambda \in [0,1]}$ ${\displaystyle \rho (\lambda X+(1-\lambda )Y)\leq \lambda \rho (X)+(1-\lambda )\rho (Y)}$

For positively homogeneous risk measures, convexity and subadditivity are obviously equivalent. In particular, subadditivity follows from convexity when set. ${\displaystyle \lambda ={\tfrac {1}{2}}}$

Position-dependent (loss-based) or position-independent (dispersion-based)

Loss-based RM such as value-at-risk or expected shortfall are measured in relation to absolute values ​​and are therefore dependent on the mean value of the distribution

Location-independent risk measures such as volatility measure risk independently of a reference point (solely from the form of the distribution). They are therefore independent of the mean of the distribution.

These two properties can be partially transformed into one another. If, for example, a position-dependent risk measure is not applied to a random variable , but rather to a centered random variable , a position-independent risk measure results. Since the level of the expected value is also included in the calculation of position-dependent risk measures , these can also be interpreted as a type of risk-adjusted performance measure. ${\displaystyle X}$ ${\displaystyle X-E(X)}$ ${\displaystyle E(X)}$

Classification of risk measures

There are different classifications of RM based on the properties described above. These can be described as follows:

• Monetary RM: monotonic, translation invariant
• Convex RM: monotonic, translation invariant, convex
• Coherent RM: monotonic, translation invariant, positive homogeneous, convex

Overview

Standard deviation

The standard deviation as a measure of risk for an insecure payment is calculated as ${\displaystyle \sigma (X)}$ ${\displaystyle X}$

${\displaystyle \sigma (X)={\sqrt {Var(X)}}={\sqrt {E\left(\left(X-E(X)\right)^{2}\right)}}}$,

in which

${\displaystyle E(X)\,}$the expected value of is ${\displaystyle X\,}$

and positive and negative deviations from the expected value are equally recorded. The (apparent) symmetry and identical meaning of opportunities and dangers in risk measurement must, however, be relativized. It also seems to contradict the intuition and risk perception of most people, who rate dangers (possible negative deviations from plan) much higher than equally high opportunities. ${\displaystyle E(X)}$

In order to describe the overall scope of risk, due to the special importance of possible losses, so-called “downside risk measures” are used, which specifically record the possible scope of negative deviations. These include, for example, the Value at Risk, the equity requirement, the LPMs (Lower Partial Moments) and the probability of default. They are useful if the risks are not symmetrical and losses are particularly important.

Value at Risk - VaR

Main article: Value at Risk

In banking and insurance in particular, VaR is often used as a downside risk measure. The VaR explicitly takes into account the consequences - relevant for the KonTraG - of a particularly unfavorable development for the company. The VaR is defined as the amount of damage that will not be exceeded within a certain period with a specified probability (“ confidence level ” ). From a formal point of view, the VaR is the (negative) quantile of a distribution. The x% quantile for a distribution indicates the threshold up to which x% of all possible values ​​lie. If the VaR does not refer to a “value” but, for example, to the cash flow, one sometimes speaks of “ cash flow at risk ”, which, however, means the same risk measure. ${\displaystyle p}$ ${\displaystyle \alpha =1-p}$

The VaR is positively homogeneous, monotonous, translation invariant , but generally not subadditive and consequently not coherent. This allows constellations to be constructed in which the VaR of a financial position combined from two risk positions is higher than the sum of the VaR of the individual positions. This contradicts an intuition shaped by the idea of ​​diversification.

Equity requirements - EKB

The equity requirement EKB (as a special case of risk capital , RAC) is a situation-dependent risk measure related to the VaR, which explicitly relates to the company's earnings. It expresses how much equity (or liquidity) is necessary to bear realistic risk-related losses of a period. It is determined as the maximum of zero and the negative quantile of a random variable , representing the measure of success and denoting the confidence level (security level). ${\displaystyle (1-\alpha )}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \alpha }$

${\displaystyle EKB_{\alpha }(X)=\max(0;-Q_{1-\alpha }(X))\,}$

where applies

${\displaystyle P(X<Q_{1-\alpha }(X))=1-\alpha \,}$.

The following applies in particular to normally distributed payments with expected value and standard deviation : ${\displaystyle E(X)}$ ${\displaystyle \sigma (X)}$

${\displaystyle EKB_{\alpha }(X)=\max \lbrace 0;E(X)+q_{1-\alpha }\sigma (X)\rbrace \,}$.

Lower Partial Moments - LPM

The lower partial moments are risk measures which, as downside risk measures, relate to only part of the total probability density. They only record the negative deviations from a limit (target variable), but here they evaluate the entire information of the probability distribution (up to the theoretically possible maximum damage). ${\displaystyle c}$

The limit can be, for example, the expected value or any deterministic target value (for example plan value) or a required minimum interest rate. A stochastic benchmark is also possible. For example, if you consider a probability distribution for a rate of return , the following limits are possible when calculating an LPM: ${\displaystyle c}$ ${\displaystyle E(X)}$ ${\displaystyle X}$ ${\displaystyle c}$

• ${\displaystyle c=0\,}$ (nominal capital maintenance),
• ${\displaystyle c={\text{Inflationsrate}}\,}$ (real capital preservation),
• ${\displaystyle c=r_{0}\,}$ (risk-free interest) and
• ${\displaystyle c=E(X)\,}$ (expected return)

The understanding of risk corresponds to the perspective of an investor who focuses on the risk of a shortfall , of falling short of a target specified by him (planned return, required minimum return). This is precisely why one speaks here of shortfall risk measures. In general, an LPM measure of order is calculated through ${\displaystyle m}$

${\displaystyle LPM_{m}(c;X)=E({\text{max}}(c-X,0)^{m})\,}$.

In the case of discrete random variables, the relationship shown below results

${\displaystyle LPM_{m}(c;X)=\sum _{j=1}^{K}p_{j}(c-{\bar {x}}_{j})^{m}}$.

Here refers to the possible values that are less than the required barrier are, the number of these values and the probability of the occurrence of . In the case of constant random variables, the calculation rule is ${\displaystyle {\bar {x}}_{j}}$ ${\displaystyle c}$ ${\displaystyle K}$ ${\displaystyle p_{j}}$ ${\displaystyle {\bar {x}}_{j}}$

${\displaystyle LPM_{m}(c;X)=\int \limits _{-\infty }^{c}(c-x)^{m}f(x)dx}$.

The order does not necessarily have to be an integer. They determine whether and how the height of the deviation from the limit should be assessed. The higher the risk aversion of an investor, the greater the choice should be. ${\displaystyle m}$ ${\displaystyle m}$

Usually three special cases are considered in practice:

• the shortfall probability (failure probability), d. H. ${\displaystyle m=0}$

${\displaystyle SW(c;X)=LPM_{0}(c;X)=P(X<c)\,}$,

• the shortfall expectation, d. H. ${\displaystyle m=1}$

${\displaystyle SE(c;X)=LPM_{1}(c;X)=E({\text{max}}(c-X,0))\,}$ and

• the shortfall variance, d. H. ${\displaystyle m=2}$

${\displaystyle SV(c;X)=LPM_{2}(c;X)=E({\text{max}}(c-X,0)^{2})\,}$.

The extent of the risk of falling below the target value is taken into account in various ways. With the shortfall probability, only the probability of falling below plays a role. With the shortfall expectation value, however, the mean undershoot amount is taken into account and with the shortfall variance the mean squared undershoot amount.

The relationship between value at risk and LPM can be described as follows: The value at risk results from the fact that a maximum accepted shortfall probability , i.e. one , is specified for a certain planning period and the corresponding minimum return size of the LPM definition is determined. ${\displaystyle p}$ ${\displaystyle LPM_{0}=p}$ ${\displaystyle (c)}$

The shortfall risk measures can be divided into conditional and unconditional risk measures. While unconditional risk measures (such as the shortfall expectation value or the shortfall variance) disregard the probability of falling below the limit, this is included in the calculation of the conditional shortfall risk measures (such as the conditional value at risk).

Conditional Value at Risk - CVaR

The Conditional Value at Risk (CVaR) and its variants Expected Shortfall (ES) or Expected Tail Loss (ETL) are further risk measures.

Let X be a random variable and . Then ${\displaystyle \alpha \in (0,1)}$

${\displaystyle CVaR_{\alpha }(X):=E(X|X>VaR_{\alpha }(X))=VaR_{\alpha }(X)+\underbrace {E[X-VaR_{\alpha }(X)|X>VaR_{\alpha }(X)]} _{\text{mittlere Überschreitung im Überschreitungsfall}}}$

The conditional value at risk can be interpreted as “quantile reserve (VaR) plus an excess reserve”.

It corresponds to the expected value of the realizations of a risky variable that is above the quantile of the level . The CVaR indicates the deviation when the extreme case occurs, i. H. is to be expected if the VaR is exceeded. The CVaR therefore not only takes into account the probability of a “large” deviation (extreme values), but also the amount of the deviation beyond this. ${\displaystyle \alpha }$

If X has a continuous distribution , the CVaR is positively homogeneous, monotonic, subadditive and translation invariant, i.e. coherent.

Expected Shortfall - ES

In most of the cases considered (random variables with continuous densities) ES and CVaR are identical.

If the distribution function of the random variable has jump points, then also has jump points. ${\displaystyle F_{X}}$ ${\displaystyle X}$ ${\displaystyle \alpha \longmapsto CVaR_{\alpha }(X)}$

The ES, considered as , is a jump-free variant of the CVaR. ${\displaystyle \alpha \longmapsto ES_{\alpha }(X)}$

The only difference between ES and CVaR is the amount . ${\displaystyle \{\alpha \mid F_{X}(VaR(\alpha ))>\alpha \}}$

ES is defined as follows:

${\displaystyle \lambda _{\alpha }:={\frac {P(X>VaR_{\alpha }(X))}{1-\alpha }}={\frac {1-F_{X}(VaR_{\alpha }(X))}{1-\alpha }}}$

${\displaystyle ES_{\alpha }(X):=\lambda _{\alpha }CVaR_{\alpha }(X)+(1-\lambda _{\alpha })VaR_{\alpha }(X)}$

It applies . ${\displaystyle VaR\leq ES\leq CVaR}$

Capital allocation according to Kalkbrener

A company with several divisions provides an ES, this should be assigned to the individual divisions. Kalkbrener suggests the following distribution: Let the loss size of the ith division and X be their sum, i.e. the loss size of the company. ${\displaystyle X_{i}}$

${\displaystyle \lambda _{\alpha }:={\frac {P(X>VaR_{\alpha }(X))}{1-\alpha }}={\frac {1-F_{X}(VaR_{\alpha }(X))}{1-\alpha }}}$

${\displaystyle \Lambda _{ES_{\alpha }}(X_{i},X)=\lambda _{\alpha }E(X_{i}\mid X>VaR_{\alpha }(X))+(1-\lambda _{\alpha })E(X_{i}\mid X=VaR_{\alpha }(X))}$

Each division is assigned the expected value of all damage it has caused, which is caused by events for which the total damage exceeds. ${\displaystyle VaR_{\alpha }(X)}$

If there is only one division in the company, the ES shown above results:

${\displaystyle \Lambda _{ES_{\alpha }}(X,X)=ES_{\alpha }(X)}$

Examples

The conditional value at risk can be explicitly represented especially in the normal distribution case , where the expected value and the variance describe. The following then applies to the VaR: ${\displaystyle X\sim N(\mu ,\sigma ^{2})}$ ${\displaystyle \mu }$ ${\displaystyle \sigma ^{2}}$

${\displaystyle VaR_{\alpha }(X)=E(X)+q_{1-\alpha }\sigma (X)\,}$,

where the quantile denotes the standard normal distribution. If one denotes the density of a standard normally distributed random variable, then the CVaR results: ${\displaystyle q_{1-\alpha }}$ ${\displaystyle (1-\alpha )}$ ${\displaystyle \phi }$

${\displaystyle CVaR_{\alpha }(X)=E(X)+{\frac {\phi (q_{1-\alpha })}{\alpha }}\sigma (X)}$.

Compared to , a higher multiplier of the standard deviation is added, so that is . ${\displaystyle VaR_{\alpha }}$ ${\displaystyle E(X)}$ ${\displaystyle CVaR_{\alpha }\geq VaR_{\alpha }}$

In the case of the log-normal distribution , VaR and CVaR are calculated as follows: ${\displaystyle X\sim N(m,\upsilon ^{2})}$

${\displaystyle VaR_{\alpha }(X)=\exp(m+q_{\alpha }\upsilon )\,}$.

${\displaystyle CVaR_{\alpha }(X)=E(X){\frac {1-\Phi (q_{\alpha }-\upsilon )}{1-\alpha }}}$.

As a relationship to the value at risk, the result is that, in contrast to the case of normal distribution , the surcharge is not additive here, but multiplicative. ${\displaystyle E(X)}$

Sometimes the calculation of the CVaR (and thus the consideration of all possible extreme damage) does not make sense in practice. The damage that leads to bankruptcy of a company more than once is no worse (for the owners) than damage that causes bankruptcy.

Drawdown

The drawdown of a financial investment is a key figure that, depending on the design, describes the maximum possible loss in the past over a period of time. Because of this loss consideration, this risk figure can also be used for asymmetrical distributions. The maximum drawdown is the percentage loss between the highest point and the lowest point of a value development of the investment to be considered in a certain period. Furthermore, an average over the N smallest drawdowns can be formed. For this purpose, the individual returns are sorted according to their size in the period under consideration. The smallest are usually negative. To form an average, the N smallest values ​​are added and divided by N. The number N of values ​​can be chosen freely, whereby it should move within a reasonable range. Another way to use the drawdown to measure risk is to create some kind of variance. For this purpose, the N smallest observed return values ​​of the observation period are squared, then added and finally the sum is square rooted.

References and more detailed information

1. ^ W. Gleißner, F. Romeike: Risk Management. Rudolf Haufe Verlag, Munich 2005, ISBN 3-448-06209-X , p. 211 ff.
2. ^ W. Gleißner, F. Romeike: Risk Management. Rudolf Haufe Verlag, Munich 2005, ISBN 3-448-06209-X , p. 31ff.
3. ↑ P. Artzner et al .: Coherent measures of risk. In: Mathematical Finance. 9 (3), 1999, pp. 203-228.
4. ↑ J. Dhaene, MJ Goovaerts, R. Kaas: Economic capital allocation derived from risk measures. In: North American Actuarial Journal. Vol. 7, No. 2, 2003, pp. 44-56.
5. ^ H. Rau-Bredow: Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures . In: Risks . 7, No. 3, 2019, p. 91. doi : 10.3390 / risks7030091 .
6. ↑ The value at risk at the confidence level is defined as ≥ with ${\displaystyle \alpha }$ ${\displaystyle P(X}$ ${\displaystyle VaR_{\alpha }(X))=\alpha }$ ${\displaystyle 0<\alpha <1.}$
7. ↑ The equity necessary to cover possible losses.
8. ↑ In other words, the probability that a minimum asset or minimum return that is considered necessary will be undershot.
9. ↑ Hersh Shefrin , Meir Statman : Behavioral capital asset pricing theory. In: Journal of Financial and Quantitative Analysis. Vol. 29, No. 3, 1994, pp. 323-349.
10. ↑ The VaR is often applied to a damage distribution, i.e. to a random variable . The VaR can also be used as a location-independent measure of deviation. ${\displaystyle S=E(X)-X}$
11. ↑ A risk measure is called coherent if it is translation invariant, positively homogeneous, monotonic and subadditive.
12. ↑ P. Artzner, F. Delbaen, J. Eber, D. Heath: Coherent Measures of Risk. In: Mathematical Finance. Vol. 9, No. 3, 1999, pp. 203-228.
13. ^ W. Gleißner: Identification, measurement and aggregation of risks. In: G. Meier (Hrsg.): Value-oriented risk management for industry and trade. Gabler Verlag, Wiesbaden 2001, ISBN 3-409-11699-0 , pp. 111-137.
14. ↑ P. Albrecht, R. Maurer, M. Möller: Shortfall risk / excess chance decision calculus: Basics and relationships to the Bernoulli principle. In: Journal for Economics and Social Sciences. 118, 1998, pp. 249-274.
15. ^ P. Albrecht, R. Maurer: Investment and Risk Management. 2nd Edition. Schäffer - Poeschel Verlag, Stuttgart 2005, p. 5ff.
16. ↑ For the moments of the lognormal distribution applies: , and skewness . The log-normal distribution cannot assume negative values. It is mostly used to describe pure damage. In this case, the relevant extreme risk is characterized by exceeding a quantile and not, as in the case of profit / loss distributions, by falling below a quantile. ${\displaystyle E(X)=\exp(m+\upsilon ^{2}/2)}$ ${\displaystyle \sigma ^{2}(X)=\exp(2m+\sigma ^{2})\cdot (\exp(\sigma ^{2})-1)}$ ${\displaystyle \gamma (X)=(\exp(\sigma ^{2})+2)\cdot {\sqrt {\exp(\sigma ^{2})-1}}}$
17. ↑ P. Albrecht, S. Koryciorz: Determination of the Conditional Value at Risk (CVaR) with normal or log-normal distribution. In: Mannheim manuscripts on risk theory. No. 142, 2003, p. 7ff.