Conditional Value at Risk

Differentiation of the CVaR from the VaR

The Conditional Value at Risk (CVaR) represents a conditional shortfall risk measure and has been developed from the Value at Risk (VaR). Further variants of this risk measure are the expected shortfall (ES) and the tail conditional expectation (TCE). In some cases, this risk measure is also identical to the Average Value at Risk (e.g. for all continuous loss distributions).

Basics

The CVaR represents the expected value of the realization of a risky variable that is below the quantile for the level (confidence level:) . The CVaR thus corresponds to the average loss in the event of a loss that was triggered by exceeding the VaR. While the VaR represents the maximum loss that will not be exceeded with a certainty probability of, the CVaR implies the average loss outside the safety level (i.e. in all other bad cases). If you put z. For example, based on a confidence level of 95%, the CVaR is the average maximum loss of the 5% worst cases. If, for example, a 1% VaR is interpreted as a 100-year loss (i.e., on average, the 1% VaR is only exceeded once every 100 years), then the 1% CVaR can be the mean level of the 100-year loss. Damage to be considered. ${\displaystyle \alpha }$ ${\displaystyle \alpha =1-p}$ ${\displaystyle \alpha }$ ${\displaystyle (1-\alpha )\cdot 100\,\%}$

In order to determine the CVaR of a financial position, the classic VaR (= critical amount of loss) is first calculated using a fixed time interval and a specified confidence level. In all cases in which the period loss is greater than the VaR, the CVaR represents the mean amount of loss. A conditional expected value is formally formed for this. To calculate the CVaR, add the VaR and the mean excess of the VaR (mean conditional excess). The CVaR is therefore always higher than the classic VaR. Portfolios with a low CVaR therefore always have a low VaR. This risk measure is often interpreted as a quantile reserve plus an excess reserve.

A VaR-inclined investor would ask: “How often could my portfolio lose at least € 100,000? "A CVaR-inclined investors whereas asks: " If my portfolio loses more than € 100,000, how much could I lose? " . In addition to the probability of large deviations, the CVaR also takes into account their size. This is a great advantage over VaR, as it only looks at the probability of loss, but not the amount of default. The CVaR is monotonic, positively homogeneous, subadditive and translation invariant and therefore also coherent. In the case of VaR, on the other hand, the property of subadditivity is not guaranteed. Subadditivity means that by pooling risk collectives, the total risk capital based on the CVaR is reduced. Therefore, the CVaR is seen as a more consistent measure compared to the VaR. However, the CVaR is only a coherent risk measure if it has a distribution with a density function (“continuous distribution”). However, if there is a discrete distribution, modification is required to get a coherent risk measure. ${\displaystyle X}$

When the VaR became more and more unattractive due to the financial crisis in 2007–2008, the CVaR moved more and more into focus, as it also takes into account very rare and very large losses. The CVaR is increasingly being used in both risk management and portfolio management. For example, CVaR is used in portfolio optimization. Depending on the investment class and the type of risk, risk managers use various mathematical methods to calculate the CVaR:

• Monte Carlo simulation

• Pricing and valuation of financial derivatives

• Econometric models (e.g. market interest method )

The CVaR is a measure of significant and undesirable changes in the value of a portfolio. In practice, however, the calculation of the CVaR does not always make sense, as, for example, damage that leads to bankruptcy more than once is no worse for the owner than damage that only leads to bankruptcy.

The term Conditional Value at Risk could lead to misunderstandings, however, since it is an expected value and not the VaR that results from a conditional distribution. In terms of content, the abbreviation CVaR must not be confused with the credit value at risk or the so-called component value at risk, for which this abbreviation is also common.

Formal definition

Let be a random variable with the distribution function , the inverse of the distribution function and the confidence level with . Then the VaR is defined as follows: ${\displaystyle X}$ ${\displaystyle F(X)}$ ${\displaystyle F_{X}^{-1}}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha ~\in ~(0,1)}$

${\displaystyle VaR_{\alpha }(X)=F_{X}^{-1}(\alpha )}$

If it is normally distributed with , where corresponds to the expected value and the standard deviation, then the VaR can be defined as follows: ${\displaystyle X}$ ${\displaystyle N(\mu ,\sigma )}$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$

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

With the help of the VaR, the CVaR can be formally defined:

${\displaystyle CVaR_{\alpha }(X)=E[X|X>VaR_{\alpha }]=E[X|X>F_{X}^{-1}(\alpha )]}$

Under the condition that it is a continuous random variable: ${\displaystyle X}$

${\displaystyle CVaR_{\alpha }(X)=VaR_{\alpha }(X)+E[X-VaR_{\alpha }|X>VaR_{\alpha }]}$

If is normally distributed, then we get: ${\displaystyle X}$

${\displaystyle CVaR_{\alpha }(X)=E(X)+{\dfrac {\varphi (F_{X}^{-1}(\alpha ))}{1-\alpha }}\sigma (X)}$

Where the density function of the normal distribution corresponds. ${\displaystyle \varphi }$

Calculation example

In a portfolio, for example, there are 3 positions A, B and C, which have achieved the returns shown in the table below over the last 10 days. The probability of loss specified by the board of directors for plants A to C should be 20%, which corresponds to a confidence level of 80%.

Day	Share A	Share B	Share C
1	2.00%	3.50%	1.00%
2	-0.89%	1.62%	5.62%
3	3.17%	2.36%	4.63%
4th	1.24%	-4.51%	3.80%
5	8.67%	-4.23%	3.62%
6th	-11.21%	26.80%	-1.25%
7th	-8.40%	12.52%	-2.31%
8th	-16.26%	-1.62%	1.25%
9	12.02%	-1.80%	-2.25%
10	9.62%	1.02%	-1.89%

Table 1: Returns on investments A, B and C.

In order to clearly show the difference between CVaR and VaR, the VaR should first be determined in the following with a confidence level of 80%. The VaR corresponds to the best return value of the 20% worst cases. Due to the simply chosen example with only 10 data points (in practice there are usually significantly more data available), the VaR is therefore the better of the 2 worst return cases (20% of 10 data points):

${\displaystyle VaR_{80\,\%}^{\text{Aktie A}}=-11{,}21\,\%}$

${\displaystyle VaR_{80\,\%}^{\text{Aktie B}}=-4{,}23\,\%}$

${\displaystyle VaR_{80\,\%}^{\text{Aktie C}}=-2{,}25\,\%}$.

Assuming, for example, that capital of € 1,000,000 was invested, the VaR key figure says that the potential loss of the risk positions under consideration is € 112,100 in 80% of all cases (= € 1,000,000 x 11, 21%, share A), € 42,300 (share B) or € 22,500 (share C).

The CVaR, on the other hand, now corresponds to the average loss amount in the event of a loss event triggered by exceeding the VaR. With a confidence level of 80%, the CVaR therefore corresponds to the mean value of the 20% worst returns, which in this example corresponds to the mean value of the 2 worst data points. The following applies:

${\displaystyle CVaR_{80\,\%}^{\text{Aktie A}}=-0{,}5\cdot (-16{,}26\,\%-11{,}21\,\%)=13{,}74\,\%}$

${\displaystyle CVaR_{80\,\%}^{\text{Aktie B}}=-0{,}5\cdot (-4{,}51\,\%-4{,}23\,\%)=4{,}37\,\%}$

${\displaystyle CVaR_{80\,\%}^{\text{Aktie C}}=-0{,}5\cdot (-2{,}31\,\%-2{,}25\,\%)=2{,}28\,\%}$.

If one again assumes an invested capital in the amount of 1,000,000 €, then in the 20% worst cases with systems A, B and C with an average loss of € 137,400 (= € 1,000,000 x 13.74%) ), € 43,700 and € 22,800 respectively.

The CVaR is always positive (note the sign when calculating the mean value) and greater than the VaR. The latter can be justified by the fact that the CVaR can also be calculated equivalently as the sum of the VaR and the mean excess in the event of excess (the sign must also be observed here):

${\displaystyle CVaR_{80\,\%}^{\text{Aktie A}}=11{,}21\,\%+0{,}5\cdot (16{,}26\,\%-11{,}21\,\%)=13{,}74\,\%}$

${\displaystyle CVaR_{80\,\%}^{\text{Aktie B}}=4{,}23+0{,}5\cdot (4{,}51\,\%-4{,}23\,\%)=4{,}37\,\%}$

${\displaystyle CVaR_{80\,\%}^{\text{Aktie C}}=2{,}25+0{,}5\cdot (2{,}31\,\%-2{,}25\,\%)=2{,}28\,\%}$.

Differentiation of the CVaR from other risk measures

CVaR and Value at Risk

According to the formula given in the formal definition, the VaR is the maximum damage in the cases (e.g. in 99% of the cases). In comparison with the already defined CVaR, it can be seen that the CVaR in principle leads to a higher risk. As already mentioned and shown in the calculation example, in addition to the probability of loss of the VaR, the mean amount of the loss when it occurs is also taken into account. ${\displaystyle \alpha \cdot 100\,\%}$

It should be noted that the VaR and the CVaR do not represent a general, coherent risk measure. This is due to the fact that subadditivity does not generally exist . This means that diversification does not necessarily reduce risk. The risk of a portfolio is therefore not always smaller than the individual risks of the alternatives.

Nevertheless, there is coherence in individual cases for both risk measures. In the case of VaR, this is the case when applied to the normal distribution, provided that it is less than 0.5. The subadditivity is thus served. In CVaR, the distribution function must have a density. ${\displaystyle \alpha }$

The property of subadditivity is valid under more general conditions, which suggests that the CVaR is advantageous over the VaR.

The choice of risk level is often u. a. depends on the consistency of statistical estimates, mathematical properties or the complexity of optimization procedures.

CVaR and expected shortfall

${\displaystyle ES_{\alpha }(X)={\dfrac {1}{\alpha }}\int _{0}^{\alpha }VaR_{u}(X)du={\dfrac {1}{\alpha }}\int _{1-\alpha }^{1}Q_{u}(X)du}$

According to the formula, the ES can be understood as the average of the VaR values. It should be noted that the interpretation, contrary to the interpretation of the VaR, is the maximum damage in the cases (for example in 1% of the cases). ${\displaystyle (1-\alpha )\cdot 100\,\%}$

CVaR and ES coincide when the distribution function has a density and is therefore continuous. In this case, the CVaR also represents a coherent risk measure and the interpretation of the ES can be adopted. In the case of discrete random variables, there is no coherence in CVaR, which is why it makes more sense from the point of view of coherence to use the ES or equivalent risk measures. The ES is more complicated, however, so it is beneficial to use VaR or CVaR when possible.

Nevertheless, the ES has two decisive advantages:

1. Compared to VaR and CVaR: It fulfills the subadditivity condition , which is why it can be referred to as a coherent risk measure .
2. Compared to VaR: Extreme losses are explicitly taken into account.

In the literature, both terms are sometimes used as synonyms. However, since the ES is more versatile due to its generally applicable subadditivity and is based on a different calculation, both risk measures can also be viewed separately from one another.

CVaR and Tail Conditional Expectation

${\displaystyle TCE_{\alpha }(X)=-E[X|X\leq -VaR_{\alpha }(X)]}$

The risk measure of the TCE, which can also be found under the names Tail Value at Risk and Conditional Tail Expectation (CTE), is very similar to the CVaR and is also based on the VaR. The TCE and the Worst Conditional Expectation (WCE) can be described as preliminary considerations of the CVaR.

With the TCE, all values ​​(and thus losses) that are below the VaR are taken into account in a probability-weighted manner in accordance with the formula when calculating the expected value. This can remedy a weakness in the VaR. Extreme losses, which have a lower cumulative probability of occurrence than the VaR itself and are neglected in the VaR, are taken into account. This takes place as a result of the expected value resulting from losses exceeding or reaching the VaR. However, it is not an advantage over the CVaR.

However, through the following part of the equation , the TCE takes into account realizations of random variables and not of environmental conditions. As a result, it is not possible for the TCE to consider a constant proportion of realizations across the totality of the financial positions. This, in turn, is due to the sometimes ambiguous definition of the VaR with constant courses of discrete distributions. ${\displaystyle X\leq -VaR_{\alpha }(X)}$

As with VaR, the property of general subadditivity is not given, from which it can be concluded that the TCE is not a coherent risk measure. In contrast, the CVaR can be coherent under relatively general conditions. Due to the weaknesses of the TCE, the use of the WCE is suggested.

CVaR and Worst Conditional Expectation

${\displaystyle WCE_{\alpha }(X)=-\min\{E(X|M)|M\in F,P(M)>\alpha \}}$

The WCE also takes up the concept of the conditional expected value on which the TCE is based. However, the conditions under which the formation of the expected values ​​takes place are designed according to the representation theorem of coherent risk measures. Firstly, this includes the consideration of environmental conditions instead of random variables, which results in subadditivity , and secondly, it is expressed in the maximum of the negative expectation values ​​of different probability measures / scenarios. The WCE thus combines the empirical probability function with the condition that the probability of an event from a set of events is greater than the confidence level . In this way, scenarios can be generated for which the expected value should then be calculated. The minimum expected value of all scenarios determines the level of risk. The connection to environmental conditions instead of random variables, as is the case with TCE, thus represents an advantage or a further development compared to the TCE. ${\displaystyle M}$ ${\displaystyle F}$ ${\displaystyle P(M)>\alpha }$

The WCE can be described as a coherent risk measure, which is why it is more advantageous than the CVaR, as well as the TCE and VaR. However, it also has two major disadvantages:

1. It can hardly be used in practice because it requires knowledge of the entire underlying probability space.
2. As with the TCE, often not only the smallest realizations are taken into account when determining the risk. An event can consist of several elements or realizations. Since the probability of the event occurring is greater than the confidence level and this is the case, for example, by 2%, the smallest realizations are taken into account. This weakness is addressed by the CVaR. ${\displaystyle \alpha \cdot 100\,\%}$ ${\displaystyle \alpha \cdot 100\,\%+2\,\%}$

literature

• Acerbi, C. & Taschen, D. (2002): On the coherence of expected shortfall. Journal of Banking and Finance , Vol. 26, Iss. 7, pp. 1487-1503.
• Albrecht, P. (2003): To measure financial risks. Mannheim Manuscripts on Risk Theory, Portfolio Management and Insurance Industry , No. 143.
• Albrecht, P. & Koryciorz, S. (2003): Determination of the Conditional Value at Risk (CVaR) with normal or log-normal distribution. Mannheim manuscripts on risk theory, portfolio management and the insurance industry, No. 142, pp. 1–15.
• Albrecht, P. (2018): Conditional Value at Risk (CVaR) , accessed on July 12, 2018.
• Albrecht, P., R. Maurer (2002): Investment and Risk Management: Models, Methods, Applications, Schäffer-Poeschel: Stuttgart.
• Andersson, F., Mausser, H., Rosen, D. & Uryasev, S. (2001): Credit risk optimization with Conditional Value-at-Risk criterion. Mathematical Programming . Vol. 89, Iss. 2, pp. 273-291.
• Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999): Coherent measures of risk. Mathematical Finance , Vol. 9, Iss. 3, pp. 203-228.
• Bhattacharya, J. (2016): Conditional Value at Risk Calculator , accessed July 12, 2018.
• Brandtner, M. (2012): Modern methods of risk and preference measurement - conception, decision-theoretical implications and financial applications , Wiesbaden: Springer Gabler.
• Gleißner, W. (2006): Series: Risk Measures and Evaluation - Part 2: Downside Risk Measures - Risk Measures, Safety-First Approaches and Portfolio Optimization . Risk manager . Issue 13, pp. 17-23.
• Gleißner, W. (2017): Fundamentals of Risk Management , 3rd edition, Stuttgart: Vahlen.
• Hanisch, J. (2004): Risk Measurement with Conditional Value-at-Risk: Decision-Theoretical Foundations and Implications for Risk Management, Dissertation, 2004.
• Huschens, S. (2017): Risk Measures. Dresden contributions to quantitative methods , No. 68/17, pp. 1–184.
• Hürlimann, W. (2002): Analytical Bounds for Two Value at Risk Functionals. ASTIN Bulletin: The Journal of the IAA , Vol. 32, Iss. 2, pp. 235-265.
• Landsman, ZM & Valdez, EA (2003): Tail Conditional Expectation for Elliptical Distributions. North American Actuarial Journal , Vol. 7, Iss. 4, pp. 55-123.
• Lim, AEB, Shanthikumar, JG & Vahn, G.-Y. (2011): C onditional value-at-risk in portfolio optimization: Coherent but fragile. Operations Research Letters . Vol. 39, Iss. 3, pp. 163-171.
• Mathworks (2018): Conditional Value-at-Risk , accessed July 12, 2018.
• Rockafellar, RT & Uryasev, S. (2000): Optimization of Conditional Value at Risk. Journal of Risk . Vol. 2, No. 3, pp. 21-41.
• Sarykalin, S., Serraino, G. & Uryasev, S. (2008): Value-at-Risk vs. Conditional Value-at-Risk in Risk Management and Optimization. Tutorials in Operations Research , INFORMS. Pp. 270-294.
• Schulz, MT & Mader, W. (2016): Modern Risk Management. Wisu - Das Wirtschaftsstudium , Vol. 45, Iss. 11, pp. 1209-1211.
• Wagner, F. (2018): Value at Risk (VaR) , accessed on July 14, 2018.

Individual evidence

1. ↑ Gleißner, W. (2006): Series: Risk Measures and Evaluation - Part 2: Downside Risk Measures - Risk Measures, Safety-First Approaches and Portfolio Optimization . Risk manager . Issue 13, p. 20.
2. ↑ Albrecht, P. (2018): Conditional Value at Risk (CVaR) , accessed on July 12, 2018.
3. ↑ Albrecht, P. & Koryciorz, S. (2003): Determination of the Conditional Value at Risk (CVaR) with normal or log-normal distribution. Mannheim manuscripts on risk theory, portfolio management and the insurance industry, No. 142, p. 2.
4. ↑ Huschens, S. (2017): Risk Measures. Dresden Contributions to Quantitative Methods , No. 68/17, p. 95.
5. ↑ Gleißner, W. (2006): Series: Risk Measures and Evaluation - Part 2: Downside Risk Measures - Risk Measures, Safety-First Approaches and Portfolio Optimization . Risk manager . Issue 13, p. 20.
6. ↑ Albrecht, P. (2018): Conditional Value at Risk (CVaR) , accessed on July 12, 2018.
7. ↑ Hanisch, J. (2004): Risk measurement with the conditional value-at-risk: Decision-theoretical bases and implications for risk management, dissertation, 2004, p. 30.
8. ↑ Andersson, F., Mausser, H., Rosen, D. & Uryasev, S. (2001): Credit risk optimization with Conditional Value-at-Risk criterion. Mathematical Programming . Vol. 89, Iss. 2, p. 274.
9. ↑ Albrecht, P. (2018): Conditional Value at Risk (CVaR) , accessed on July 12, 2018.
10. ↑ Albrecht, P. (2018): Conditional Value at Risk (CVaR) , accessed on July 12, 2018.
11. ↑ Rockafellar, RT & Uryasev, S. (2000): Optimization of Conditional Value at Risk. Journal of Risk . Vol. 2, No. 3, p. 21.
12. ↑ Gleißner, W. (2006): Series: Risk Measures and Evaluation - Part 2: Downside Risk Measures - Risk Measures, Safety-First Approaches and Portfolio Optimization . Risk manager . Issue 13, p. 20.
13. ↑ Bhattacharya, J. (2016): Conditional Value at Risk Calculator , accessed July 12, 2018.
14. ↑ Gleißner, W. (2006): Series: Risk Measures and Evaluation - Part 2: Downside Risk Measures - Risk Measures, Safety-First Approaches and Portfolio Optimization . Risk manager . Issue 13, p. 20.
15. ↑ Wagner, F. (2018): Value at Risk (VaR) , accessed on July 14, 2018.
16. ↑ Gleißner, W. (2006): Series: Risk Measures and Evaluation - Part 2: Downside Risk Measures - Risk Measures, Safety-First Approaches and Portfolio Optimization . Risk manager . Issue 13, p. 20.
17. ↑ Wagner, F. (2018): Value at Risk (VaR) , accessed on July 14, 2018.
18. ↑ Albrecht, P. (2018): Conditional Value at Risk (CVaR) , accessed on July 12, 2018.
19. ↑ Andersson, F., Mausser, H., Rosen, D. & Uryasev, S. (2001): Credit risk optimization with Conditional Value-at-Risk criterion. Mathematical Programming . Vol. 89, Iss. 2, p. 274.
20. ↑ Albrecht, P. & Koryciorz, S. (2003): Determination of the Conditional Value at Risk (CVaR) with normal or log-normal distribution. Mannheim manuscripts on risk theory, portfolio management and the insurance industry, No. 142, p. 3.
21. ↑ Lim, AEB, Shanthikumar, JG & Vahn, G.-Y. (2011): C onditional value-at-risk in portfolio optimization: Coherent but fragile. Operations Research Letters . Vol. 39, Iss. 3, p. 163.
22. ↑ Mathworks (2018): Conditional Value-at-Risk , accessed on July 12, 2018th
23. ↑ Mathworks (2018): Conditional Value-at-Risk , accessed on July 12, 2018th
24. ↑ Andersson, F., Mausser, H., Rosen, D. & Uryasev, S. (2001): Credit risk optimization with Conditional Value-at-Risk criterion. Mathematical Programming . Vol. 89, Iss. 2, p. 274.
25. ↑ Gleißner, W. (2006): Series: Risk Measures and Evaluation - Part 2: Downside Risk Measures - Risk Measures, Safety-First Approaches and Portfolio Optimization . Risk manager . Issue 13, p. 20.
26. ↑ Huschens, S. (2017): Risk Measures. Dresden Contributions to Quantitative Methods , No. 68/17, p. 98.
27. ↑ Albrecht, P. (2003): To measure financial risks. Mannheim manuscripts on risk theory, portfolio management and the insurance industry , No. 143, p. 27.
28. ↑ Gleißner, W. (2017): Fundamentals of Risk Management , 3rd edition, Stuttgart: Vahlen, p. 207.
29. ↑ Artzner, P., Delbaen, F. Eber, J.-M. & Heath, D. (1999): Coherent measures of risk. Mathematical Finance , Vol. 9, Iss. 3, p. 223.
30. ↑ Albrecht, P., R. Maurer (2002): Investment and Risk Management: Models, Methods, Applications, Schäffer-Poeschel: Stuttgart, p. 675.
31. ↑ Gleißner, W. (2017): Fundamentals of Risk Management , 3rd edition, Stuttgart: Vahlen, p. 209.
32. ^ Schulz, MT & Mader, W. (2016): Modern Risk Management. Wisu - Das Wirtschaftsstudium , Vol. 45, Iss. 11, pp. 1209-1210.
33. ↑ Albrecht, P. (2003): To measure financial risks. Mannheim manuscripts on risk theory, portfolio management and the insurance industry , No. 143, p. 32.
34. ↑ Artzner, P., Delbaen, F. Eber, J.-M. & Heath, D. (1999): Coherent measures of risk. Mathematical Finance , Vol. 9, Iss. 3, p. 216.
35. ↑ Albrecht, P. (2003): To measure financial risks. Mannheim manuscripts on risk theory, portfolio management and the insurance industry , No. 143, p. 32.
36. ↑ Rau-Bredow, H. (2020): Value at Risk and Diversification
37. ↑ Landsman, ZM & Valdez, EA (2003): Tail Conditional Expectation for Elliptical Distributions. North American Actuarial Journal , Vol. 7, Iss. 4, p. 56.
38. ↑ Albrecht, P. (2003): To measure financial risks. Mannheim manuscripts on risk theory, portfolio management and the insurance industry , No. 143, pp. 31–32.
39. ↑ Sarykalin, S., Serraino, G. & Uryasev, S. (2008): Value-at-Risk vs. Conditional Value-at-Risk in Risk Management and Optimization. Tutorials in Operations Research , INFORMS. P. 270.
40. ↑ Albrecht, P. (2003): To measure financial risks. Mannheim manuscripts on risk theory, portfolio management and the insurance industry , No. 143, pp. 31–32.
41. ↑ Hürlimann, W. (2002): Analytical Bounds for Two Value at Risk Functionals. ASTIN Bulletin: The Journal of the IAA , Vol. 32, Iss. 2, p. 239.
42. ↑ Acerbi, C. & Taschen, D. (2002): On the coherence of expected shortfall. Journal of Banking and Finance , Vol. 26, Iss. 7, p. 1488.
43. ↑ Brandtner, M. (2012): Modern methods of risk and preference measurement - conception, decision-theoretical implications and financial applications , Wiesbaden: Springer Gabler, pp. 103-105.
44. ↑ Brandtner, M. (2012): Modern methods of risk and preference measurement - conception, decision-theoretical implications and financial applications , Wiesbaden: Springer Gabler, pp. 105-107.
45. ↑ Acerbi, C. & Taschen, D. (2002): On the coherence of expected shortfall. Journal of Banking and Finance , Vol. 26, Iss. 7, p. 1488.