# CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles

^{1}

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^{*}

## Abstract

**:**

## 1. Introduction

- Lack of convexity: portfolio diversification may increase VaR.
- VaR is not sensitive to outcomes exceeding VaR, which allows for stretching of the distribution without an increasing of the risk measured by VaR.
- VaR has poor mathematical properties, such as discontinuity with respect to (w.r.t.) portfolio positions for discrete distributions based on historical data.

## 2. Quantile, Mixed-Quantile, and CVaR Quadrangles

- Risk $\mathcal{R}\left(X\right)$, which provides a numerical surrogate for the overall hazard in $X$.
- Deviation $\mathcal{D}\left(X\right)$, which measures the “nonconstancy” in $X$ as its uncertainty.
- Error $\mathcal{E}\left(X\right)$, which measures the “nonzeroness” in $X$.
- Regret $\mathcal{V}\left(X\right)$, which measures the “regret” in facing the mix of outcomes of $X$.
- Statistic $\mathcal{S}\left(X\right)$ associated with $X$ through $\mathcal{E}$ and $\mathcal{V}$.

- Statistic: ${\mathcal{S}}_{\alpha}\left(X\right)=Va{R}_{\alpha}\left(X\right)$ = VaR (quantile) statistic.
- Risk: ${\mathcal{R}}_{\alpha}\left(X\right)=CVa{R}_{\alpha}\left(X\right)=\underset{C}{\mathrm{min}}\left\{C+{\mathcal{V}}_{\alpha}\left(X-C\right)\right\}=$ CVaR risk.
- Deviation: ${\mathcal{D}}_{\alpha}\left(X\right)=CVa{R}_{\alpha}\left(X\right)-E\left[X\right]=\underset{C}{\mathrm{min}}\left\{{\mathcal{E}}_{\alpha}\left(X-C\right)\right\}=$ CVaR deviation.
- Regret: ${\mathcal{V}}_{\alpha}\left(X\right)=\frac{1}{1-\alpha}E{\left[X\right]}^{+}=$ average absolute loss, scaled.
- Error: ${\mathcal{E}}_{\alpha}\left(X\right)=E\left[\frac{\alpha}{1-\alpha}{\left[X\right]}^{+}+{\left[X\right]}^{-}\right]={\mathcal{V}}_{\alpha}\left(X\right)-E\left[X\right]=$ normalized Koenker–Bassett error.

- Statistic: $\mathcal{S}\left(X\right)={\sum}_{k=1}^{r}{\lambda}_{k}Va{R}_{{\alpha}_{k}}\left(X\right)=$ mixed VaR (quantile).
- Risk: $\mathcal{R}\left(X\right)={\sum}_{k=1}^{r}{\lambda}_{k}CVa{R}_{{\alpha}_{k}}\left(X\right)=$ mixed CVaR.
- Deviation: $\mathcal{D}\left(X\right)={\sum}_{k=1}^{r}{\lambda}_{k}CVa{R}_{{\alpha}_{k}}\left(X-E\left[X\right]\right)=$ mixed CVaR deviation.
- Regret: $\mathcal{V}\left(X\right)=\underset{{B}_{1},\dots ,{B}_{r}}{\mathrm{min}}\left\{{\sum}_{k=1}^{r}{\lambda}_{k}{\mathcal{V}}_{{\alpha}_{k}}\left(X-{B}_{k}\right)|{\sum}_{k=1}^{r}{\lambda}_{k}{B}_{k}=0\right\}=$ the minimal weighted average of regrets ${\mathcal{V}}_{{\alpha}_{k}}\left(X-{B}_{k}\right)=\frac{1}{1-{\alpha}_{k}}E{\left[X-{B}_{k}\right]}^{+}$ satisfying the linear constraint on ${B}_{1},\dots ,{B}_{r}$.
- Error: $\mathcal{E}\left(X\right)=\underset{{B}_{1},\dots ,{B}_{r}}{\mathrm{min}}\left\{{\sum}_{k=1}^{r}{\lambda}_{k}{\mathcal{E}}_{{\alpha}_{k}}\left(X-{B}_{k}\right)|{\sum}_{k=1}^{r}{\lambda}_{k}{B}_{k}=0\right\}=$ Rockafellar error $=$ the minimal weighted average of errors ${\mathcal{E}}_{{\alpha}_{k}}\left(X-{B}_{k}\right)=E\left[\frac{{\alpha}_{k}}{1-{\alpha}_{k}}{\left[X-{B}_{k}\right]}^{+}+{\left[X-{B}_{k}\right]}^{-}\right]$ satisfying the linear constraint on ${B}_{1},\dots ,{B}_{r}.$

- Statistic: ${\overline{\mathcal{S}}}_{\alpha}\left(X\right)=CVa{R}_{\alpha}\left(X\right)$ = CVaR.
- Risk: ${\overline{\mathcal{R}}}_{\alpha}\left(X\right)=\frac{1}{1-\alpha}{\int}_{\alpha}^{1}CVa{R}_{\beta}\left(X\right)d\beta =$ CVaR2 risk.
- Deviation: ${\overline{\mathcal{D}}}_{\alpha}\left(X\right)={\overline{\mathcal{R}}}_{\alpha}\left(X\right)-E\left[X\right]=\frac{1}{1-\alpha}{\int}_{\alpha}^{1}CVa{R}_{\beta}\left(X\right)d\beta -E\left[X\right]=$ CVaR2 deviation.
- Regret: ${\overline{\mathcal{V}}}_{\alpha}\left(X\right)=\frac{1}{1-\alpha}{\int}_{0}^{1}{\left[CVa{R}_{\beta}\left(X\right)\right]}^{+}d\beta $ = CVaR2 regret.
- Error: ${\overline{\mathcal{E}}}_{\alpha}\left(X\right)={\overline{\mathcal{V}}}_{\alpha}\left(X\right)-E\left[X\right]$ = CVaR2 error.

## 3. Set 1 of Parameters for Mixed-Quantile Quadrangle

**Set 1 of parameters:**

- partition of the interval $\left[\alpha ,1\right]$: ${\beta}_{{\nu}_{\alpha}-1}=\alpha $, and ${\beta}_{i}=i\delta $, for $i={\nu}_{\alpha},{\nu}_{\alpha}+1,\dots ,\nu $, where $\delta =1/\nu $, ${\nu}_{\alpha}=\lfloor \nu \alpha \rfloor +1$, with $\lfloor z\rfloor $ being the largest integer less than or equal to $z$; ${\delta}_{\alpha}={\beta}_{{\nu}_{\alpha}}-\alpha $.
- weights: ${p}_{{\nu}_{\alpha}}=\frac{{\delta}_{\alpha}}{1-\alpha},{p}_{i}=\frac{\delta}{1-\alpha},i={\nu}_{\alpha}+1,\dots ,\nu $.
- confidence levels: ${\gamma}_{i}=1-\frac{{\beta}_{i}-{\beta}_{i-1}}{\mathrm{ln}\left(\frac{1-{\beta}_{i-1}}{1-{\beta}_{i}}\right)},i={\nu}_{\alpha},\dots ,\nu -1$; ${\gamma}_{\nu}=1$.

**Lemma**

**1.**

- 1.
- CVaR statistic:$${\overline{\mathcal{S}}}_{\alpha}\left(X\right)=CVa{R}_{\alpha}\left(X\right)={\sum}_{i={\nu}_{\alpha}}^{\nu}{p}_{i}Va{R}_{{\gamma}_{i}}\left(X\right)$$
- 2.
- CVaR2 risk:$${\overline{\mathcal{R}}}_{\alpha}\left(X\right)=\frac{1}{1-\alpha}{\int}_{\alpha}^{1}CVa{R}_{\beta}\left(X\right)d\beta ={\sum}_{i={\nu}_{\alpha}}^{\nu}{p}_{i}CVa{R}_{{\gamma}_{i}}\left(X\right)$$
- 3.
- CVaR2 deviation:$${\overline{\mathcal{D}}}_{\alpha}\left(X\right)=\frac{1}{1-\alpha}{\int}_{\alpha}^{1}CVa{R}_{\beta}\left(X\right)d\beta -E\left[X\right]={\sum}_{i={\nu}_{\alpha}}^{\nu}{p}_{i}CVa{R}_{{\gamma}_{i}}\left(X\right)-E\left[X\right]$$

**Proof.**

**Note.**

**Corollary**

**1.**

**Proof.**

**Example**

**1.**

## 4. Set 2 of Parameters for the Mixed-Quantile Quadrangle

**Set 2 of parameters:**

- partition of the interval $\left[{\beta}_{{\nu}_{\alpha}-1},1\right]$: $\delta =1/\nu $, ${\beta}_{i}=i\delta $, for $i={\nu}_{\alpha}-1,{\nu}_{\alpha},\dots ,\nu $, where ${\nu}_{\alpha}=\lfloor \nu \alpha \rfloor +1$, with $\lfloor z\rfloor $ being the largest integer less than or equal to $z$; ${\delta}_{\alpha}={\beta}_{{\nu}_{\alpha}}-\alpha $.
- confidence levels: ${\beta}_{i}$, $i={\nu}_{\alpha}-1,{\nu}_{\alpha},\dots ,\nu $.
- weights:${q}_{\nu}=0$;${q}_{\nu -1}=\frac{\delta}{1-\alpha}\times \left[2\mathrm{ln}\left(2\right)\right]\approx \frac{\delta}{1-\alpha}\times 1.386294361$, $(\mathrm{if}\nu -1{\nu}_{\alpha})$${q}_{\nu -2}=\frac{\delta}{1-\alpha}\times 2\left[3\mathrm{ln}\left(\frac{3}{2}\right)+\mathrm{ln}\left(\frac{1}{2}\right)\right]\approx \frac{\delta}{1-\alpha}\times 1.046496288$, $(\mathrm{if}\nu -2{\nu}_{\alpha})$${q}_{\nu -j}=\frac{\delta}{1-\alpha}\times j\left[\left(j+1\right)\mathrm{ln}\left(\frac{j+1}{j}\right)+\left(j-1\right)\mathrm{ln}\left(\frac{j-1}{j}\right)\right]$, $(\mathrm{if}j2,\nu -j{\nu}_{\alpha})$${q}_{{\nu}_{\alpha}}=\frac{\delta}{1-\alpha}\times j\left[\delta -{\delta}_{\alpha}+\left(j+1\right)\mathrm{ln}\left(\frac{1-\alpha}{\delta j}\right)+\left(j-1\right)\mathrm{ln}\left(\frac{j-1}{j}\right)\right]$, ($\mathrm{if}{\nu}_{\alpha}\nu -1,j=\nu -{\nu}_{\alpha})$${q}_{{\nu}_{\alpha}-1}=\frac{\delta}{1-\alpha}\times j\left[{\delta}_{\alpha}+\left(j-1\right)\mathrm{ln}\left(\frac{\delta \left(j-1\right)}{1-\alpha}\right)\right]$, $(\mathrm{if}{\nu}_{\alpha}-1\nu -1,j=\nu -{\nu}_{\alpha}+1$)if ${\nu}_{\alpha}=\nu -1$, then ${q}_{{\nu}_{\alpha}}=\frac{\delta}{1-\alpha}\times 2\left[1+\mathrm{ln}\left(\frac{1-\alpha}{\delta}\right)\right]-1$, ${q}_{{\nu}_{\alpha}-1}=\frac{\delta}{1-\alpha}\times 2\left[\mathrm{ln}\left(\frac{\delta}{1-\alpha}\right)-1\right]+2$if ${\nu}_{\alpha}=\nu $, then ${q}_{{\nu}_{\alpha}-1}=1.$

**Lemma**

**2.**

- 1.
- CVaR2 Risk:$${\overline{\mathcal{R}}}_{\alpha}\left(X\right)=\frac{1}{1-\alpha}{\int}_{\alpha}^{1}CVa{R}_{\beta}\left(X\right)d\beta ={\sum}_{i={\nu}_{\alpha}-1}^{\nu -1}{q}_{i}CVa{R}_{{\beta}_{i}}\left(X\right)$$
- 2.
- CVaR2 Deviation:$${\overline{\mathcal{D}}}_{\alpha}\left(X\right)=\frac{1}{1-\alpha}{\int}_{\alpha}^{1}CVa{R}_{\beta}\left(X\right)d\beta -E\left[X\right]={\sum}_{i={\nu}_{\alpha}-1}^{\nu -1}{q}_{i}CVa{R}_{{\beta}_{i}}\left(X\right)-E\left[X\right]$$

**Proof.**

**Note.**

**Corollary**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

## 5. On the Estimation of CVaR with Mixed-Quantile Linear Regression

**General Optimization Problem 1**

**General Optimization Problem 2**

- Step 1. Find an optimal vector${\mathit{C}}^{*}$by minimizing deviation:$$\underset{\mathit{C}\in {\mathbb{R}}^{m}}{min}\tilde{\mathcal{D}}\left({Z}_{0}\left(\mathit{C}\right)\right)$$
- Step 2. Assign${C}_{0}^{*}$:$${C}_{0}^{*}\in \tilde{\mathcal{S}}\left({Z}_{0}\left({\mathit{C}}^{*}\right)\right)$$

**Theorem**

**1.**

**Optimization Problem 1**

**Optimization Problem 2**

- Step 1. Find an optimal vector${\mathit{C}}^{*}$by minimizing deviation from the CVaR quadrangle:$$\underset{\mathit{C}\in {\mathbb{R}}^{m}}{min}{\overline{\mathcal{D}}}_{\alpha}\left({Z}_{0}\left(\mathit{C}\right)\right)$$
- Step 2. Calculate:$${C}_{0}^{*}=CVa{R}_{\alpha}\left({Z}_{0}\left({\mathit{C}}^{*}\right)\right)$$

**Optimization Problem 3**

**Optimization Problem 4**

- Step 1. Find an optimal vector${\mathit{C}}^{*}$by minimizing deviation from the mixed-quantile quadrangle:$$\underset{\mathit{C}\in {\mathbb{R}}^{m}}{min}\mathcal{D}\left({Z}_{0}\left(\mathit{C}\right)\right)$$
- Step 2. Assign:$${C}_{0}^{*}\in \mathcal{S}\left({Z}_{0}\left({\mathit{C}}^{*}\right)\right)$$

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

- Step 1. Find an optimal vector${\mathit{C}}^{*}$by minimizing deviation from the mixed-quantile quadrangle:$$\underset{\mathit{C}\in {\mathbb{R}}^{m}}{min}\mathcal{D}\left({Z}_{0}\left(\mathit{C}\right)\right)$$
- Step 2. Calculate${C}_{0}^{*}=CVa{R}_{\alpha}\left({Z}_{0}\left({\mathit{C}}^{*}\right)\right)$.

**Proof.**

## 6. Case Study: Estimation of CVaR with Linear Regression and Style Classification of Funds

`cvar2_err`) and CVaR2 deviation (

`cvar2_dev`) from the CVaR quadrangle, Rockafellar error (

`ro_err`) from the mixed-quantile quadrangle, and CVaR deviation (

`cvar_dev`) from the quantile quadrangle.

- Minimization of the CVaR2 error (PSG function
`cvar2_err`). - Two-step procedure with CVaR2 deviation (PSG function
`cvar2_dev`). - Minimization of the Rockafellar error (PSG function
`ro_err`) with the Set 1 of parameters. - The two-step procedure using mixed CVaR deviation with the Set 1 and Set 2 of parameters. This is calculated as a weighted sum of CVaR deviations (PSG function
`cvar_dev`) from the quantile quadrangle.

^{8}observations), the standard PSG solver VAN will dramatically outperform the Gurobi linear programming implementation. In this case, Gurobi may not even start on a small PC because of a shortage of memory. Nevertheless, if the number of observations is small (e.g., 10

^{3}) and the number of factors is very large (e.g., 10

^{7}), it is recommended that the linear programming formulation is used.

- Optimization Problem #: Optimization Problem number, as denoted in Section 5; also it is the problem number in the case study posted online, see, Case Study (2016).
- Set #: Set of parameters for the mixed-quantile quadrangle.
- Objective: Optimal value of the objective function.
- RLG: coefficient for the Russell Value Index.
- RLV: coefficient for the Russell 1000 Value Index.
- RUJ: coefficient for the Russell Value Index.
- RUO: coefficient for the Russell 2000 Growth Index.
- Intercept: regression intercept.
- Solving Time: solver optimization time.

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. CVaR Regression with Rockafellar Error: Convex and Linear Programming

#### Appendix A.1. Convex Programming Formulation for CVaR Regression

#### Appendix A.2. Linear Programming Formulation for CVaR Regression

## Appendix B. Codes Implementing Regression Optimization Problems

**Optimization Problem 1**

`minimize``cvar2_err(0.75,matrix_s)`

`minimize`” indicates that the objective function is minimized. The objective function

`cvar2_err(0.75,matrix_s)`calculates error ${\overline{\mathcal{E}}}_{\alpha}\left(Z\left({C}_{0},\mathit{C}\right)\right)$ in CVaR quadrangle with confidence level $\alpha $ = 0.75. The

`matrix_s`contains scenarios of residual of the regression $Z\left({C}_{0},\mathit{C}\right)=V-{C}_{0}-{\mathit{C}}^{T}\mathit{Y}$.

**Optimization Problem 2**

`minimize``cvar2_dev(0.75,matrix_s)``value:``cvar_risk(0.75,matrix_s)`

`minimize`” indicating that the objective function is minimized. It implements Step 1 of the Optimization Problem 2, which minimizes deviation from the CVaR quadrangle (for determining the optimal vector ${\mathit{C}}^{*}$ of regression coefficients without an intercept). The PSG function

`cvar2_dev(0.75,matrix_s)`calculates deviation ${\overline{\mathcal{D}}}_{\alpha}\left({Z}_{0}\left(\mathit{C}\right)\right)$ with $\alpha $ = 0.75. The

`matrix_s`contains scenarios of the residual of the regression without an intercept ${Z}_{0}\left(\mathit{C}\right)=V-{\mathit{C}}^{T}\mathit{Y}$.

`value`.” This part implements Step 2 of the Optimization Problem 2 for calculating the optimal value of intercept ${C}_{0}^{*}$. The PSG function

`cvar_risk(0.75,matrix_s)`calculates $CVa{R}_{\alpha}\left({Z}_{0}\left({\mathit{C}}^{*}\right)\right)$ with $\alpha $ = 0.75 at the optimal point ${\mathit{C}}^{*}$.

**Optimization Problem 3**

`minimize``ro_err(matrix_s, matrix_coeff)`

`minimize`” indicates that the objective function is minimized. The objective function

`ro_err(matrix_s, matrix_coeff)`calculates the Rockafellar error $\mathcal{E}\left(Z\left({C}_{0},\mathit{C}\right)\right)$ in the mixed-quantile quadrangle. The

`matrix_s`contains scenarios of the residual of the regression $Z\left({C}_{0},\mathit{C}\right)=V-{C}_{0}-{\mathit{C}}^{T}\mathit{Y}$. The

`matrix_coeff`includes vectors of weights and confidence levels for Set 1 with $\alpha $ = 0.75.

**Optimization Problem 4**

`minimize``vector_c*cvar_dev(vector_a, matrix_s)``value:``cvar_risk(0.75,matrix_s)`

`minimize`” indicating that the objective function is minimized. It implements Step 1 of the Optimization Problem 4, which minimizes deviation from the mixed-quantile quadrangle (for determining optimal vector ${\mathit{C}}^{*}$ of regression coefficients without intercept). The inner product

`vector_c*cvar_dev(vector_a, matrix_s)`calculates the mixed CVaR deviation $\mathcal{D}\left({Z}_{0}\left(\mathit{C}\right)\right).$ The function

`cvar_dev`corresponds to the CVaR deviation from the quantile quadrangle. Vector

`vector_c`contains weights for CVaR Deviation mix corresponding to Set 1. Vector

`vector_a`contains confidence levels defined by Set 1. The

`matrix_s`contains scenarios of the residual of the regression $Z\left({C}_{0},\mathit{C}\right)=V-{C}_{0}-{\mathit{C}}^{T}\mathit{Y}$.

`value`.” This part implements Step 2 of the Optimization Problem 4, calculating the optimal value of intercept ${C}_{0}^{*}$. The PSG function

`cvar_risk(0.75,matrix_s)`calculates $CVa{R}_{\alpha}\left({Z}_{0}\left({\mathit{C}}^{*}\right)\right)$ with $\alpha $ = 0.75 at the optimal point ${\mathit{C}}^{*}$.

## Appendix C. Proof of the Lemma 1

**Proof.**

## Appendix D. Proof of Lemma 2

**Proof.**

- ${q}_{\nu}=0$ in both cases, when $\nu >{\nu}_{\alpha}$ and $\nu ={\nu}_{\alpha}$.
- ${q}_{i}=\frac{1}{1-\alpha}\times \frac{{\sigma}_{i}}{\delta}\left[{\sigma}_{i-1}\mathrm{ln}\left(\frac{{\sigma}_{i-1}}{{\sigma}_{i}}\right)\right]$ for $i$ such that ${\nu}_{\alpha}<i=\nu -1$.
- ${q}_{i}=\frac{1}{1-\alpha}\times \frac{{\sigma}_{i}}{\delta}\left[{\sigma}_{i-1}\mathrm{ln}\left(\frac{{\sigma}_{i-1}}{{\sigma}_{i}}\right)+{\sigma}_{i+1}\mathrm{ln}\left(\frac{{\sigma}_{i+1}}{{\sigma}_{i}}\right)\right]$ for $i$ such that ${\nu}_{\alpha}<i<\nu -1$.
- ${q}_{i}=\frac{1}{1-\alpha}\times \frac{{\sigma}_{i}}{\delta}\left[\delta -{\delta}_{\alpha}+{\sigma}_{i-1}\mathrm{ln}\left(\frac{1-\alpha}{{\sigma}_{i}}\right)+{\sigma}_{i+1}\mathrm{ln}\left(\frac{{\sigma}_{i+1}}{{\sigma}_{i}}\right)\right]$ for $i={\nu}_{\alpha}<\nu -1$.
- ${q}_{i}=\frac{1}{1-\alpha}\times \frac{{\sigma}_{i}}{\delta}\left[{\delta}_{\alpha}+{\sigma}_{i+1}\mathrm{ln}\left(\frac{{\sigma}_{i+1}}{1-\alpha}\right)\right]$ for $i={\nu}_{\alpha}-1<\nu -1$.

- ${q}_{i}=\frac{1}{1-\alpha}\times \frac{{\sigma}_{i}}{\delta}\left[\delta -{\delta}_{\alpha}+{\sigma}_{i-1}\mathrm{ln}\left(\frac{1-\alpha}{{\sigma}_{i}}\right)\right]$ for $i={\nu}_{\alpha}=\nu -1$.
- ${q}_{i}=\frac{1}{1-\alpha}\times \frac{{\sigma}_{i}}{\delta}\left[{\delta}_{\alpha}\right]=1$ for $i={\nu}_{\alpha}-1=\nu -1$.

## Appendix E. Proof of the Lemma 3

**Proof.**

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**Figure 2.**Five equally probable atoms. Risk in the CVaR quadrangle and mixed-quantile quadrangle with Set 1 of parameters for $\alpha =0.5.$

Optimization Problem # | Set # | Objective | RLG | RLV | RUJ | RUO | Intercept | Solving Time (s) |
---|---|---|---|---|---|---|---|---|

1 | N/A | 0.01248 | 0.486 | 0.581 | −0.0753 | −6.22 × 10^{−3} | 6.98 × 10^{−3} | 0.02 |

2 | N/A | 0.01248 | 0.486 | 0.582 | −0.0753 | −6.22 × 10^{−3} | 6.98 × 10^{−3} | 0.02 |

3 | Set 1 | 0.01247 | 0.486 | 0.582 | −0.0753 | −6.22 × 10^{−3} | 6.96 × 10^{−3} | 0.03 |

4 | Set 1 | 0.01251 | 0.486 | 0.582 | −0.0752 | −6.22 × 10^{−3} | 6.98 × 10^{−3} | 0.11 |

4 | Set 2 | 0.01248 | 0.486 | 0.582 | −0.0753 | −6.23 × 10^{−3} | 6.98 × 10^{−3} | 0.03 |

Optimization Problem # | Set # | Objective | RLG | RLV | RUJ | RUO | Intercept | Solving Time (s) |
---|---|---|---|---|---|---|---|---|

1 | N/A | 0.016656 | 0.472 | 0.606 | −0.078 | −7.052 × 10^{−3} | 1.05 × 10^{−2} | 0.02 |

2 | N/A | 0.016656 | 0.472 | 0.606 | −0.078 | −7.052 × 10^{−3} | 1.05 × 10^{−2} | 0.02 |

3 | Set 1 | 0.016656 | 0.472 | 0.606 | −0.078 | −7.052 × 10^{−3} | 1.05 × 10^{−2} | 0.02 |

4 | Set 1 | 0.016656 | 0.472 | 0.606 | −0.078 | −7.052 × 10^{−3} | 1.05 × 10^{−2} | 0.11 |

4 | Set 2 | 0.016656 | 0.472 | 0.606 | −0.078 | −7.052 × 10^{−3} | 1.05 × 10^{−2} | 0.04 |

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## Share and Cite

**MDPI and ACS Style**

Golodnikov, A.; Kuzmenko, V.; Uryasev, S.
CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles. *J. Risk Financial Manag.* **2019**, *12*, 107.
https://doi.org/10.3390/jrfm12030107

**AMA Style**

Golodnikov A, Kuzmenko V, Uryasev S.
CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles. *Journal of Risk and Financial Management*. 2019; 12(3):107.
https://doi.org/10.3390/jrfm12030107

**Chicago/Turabian Style**

Golodnikov, Alex, Viktor Kuzmenko, and Stan Uryasev.
2019. "CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles" *Journal of Risk and Financial Management* 12, no. 3: 107.
https://doi.org/10.3390/jrfm12030107