**1. Introduction**

We start the introduction with a quick outline of the main result of this paper. The conditional value-at-risk (CVaR) is a popular risk measure. It is called expected shortfall (ES) in financial applications and it is included in financial regulations. This paper provides algorithms for the estimation of CVaR with linear regression as a function of factors. This task is of critical importance in practical applications involving low probability events.

By definition, CVaR is an integral of the value-at-risk (VaR) in the tail of a distribution. VaR can be estimated with the quantile regression by minimizing the Koenker–Bassett error function. This paper shows that CVaR can be estimated by minimizing a mixture of the Koenker–Bassett errors with an additional constraint. This mixture is called the Rockafellar error and it has been earlier used for CVaR estimation without a rigorous mathematical justification. One more equivalent variant of CVaR regression can be done by minimizing a mixture of CVaR deviations for finding all coefficients, except the intercept. In this case, the intercept is calculated using an analytical expression, which is the CVaR of the optimal residual without an intercept. The new mathematical result links quantile and CVaR regressions and shows that convex and linear programming methods can be straightforwardly used for CVaR estimation. Mathematical justification of the results involves a risk quadrangle concept combining regret, error, risk, deviation, and statistic notions.

Quantiles evaluating di fferent parts of a distribution of a random value are quite popular in various applications. In particular, quantiles are used to estimate tail of a distribution (e.g., 90%, 95%, and 99% quantiles). This paper is motivated by finance applications, where a quantile is called VaR. Risk measure VaR is included in finance regulations for the estimation of market risk. VaR has several attractive properties, such as the simplicity of calculation, stability of estimation, and availability of quantile regression, for the estimation of VaR as a function of explanatory factors. The quantile regression (see Koenker and Bassett (1978), Koenker (2005)) is an important factor supporting the popularity of VaR. For instance, a quantile regression was used by Adrian and Brunnermeier (2016) to estimate institution's contribution to systemic risk.

However, VaR also has some undesirable properties:


Shortcomings of VaR led financial regulators to use an alternative measure of risk, which is called conditional value-at-risk (CVaR) in this paper. This risk measure was introduced in Rockafellar and Uryasev (2000) and further studied in Rockafellar and Uryasev (2002) and many other papers. CVaR for continuous distributions equals the conditional expectation of losses exceeding VaR. An important mathematical fact is that CVaR is a coherent risk measure (see Acerbi and Tasche (2002), Rockafellar and Uryasev (2002)). Ziegel (2014) shows that CVaR is elicitable in a week sense. Fissler and Ziegel (2015) proved that (VaR, CVaR) is jointly elicitable, meaning elic(CVaR) ≤ 2, and more generally, that spectral risk measures have a low elicitation complexity. These results clarify the regression procedure of Rockafellar et al. (2014); their algorithm implicitly tracks the quantiles suggested by elicitation complexity.

Rockafellar and Uryasev (2000, 2002) have shown that CVaR of a convex function of variables is also a convex function. Due to this property, CVaR optimization problems can be reduced to convex and linear optimization problems.

This paper is based on risk quadrangle theory, which defines quadrangles (i.e., groups) of stochastic functionals Rockafellar and Uryasev (2013). Every quadrangle contains risk, deviation, error, and regre<sup>t</sup> (negative utility). These elements of the quadrangle are linked by the statistic function.

The relation of quantile regression and CVaR optimization was explained using a quantile quadrangle (see Rockafellar and Uryasev (2013)). It was shown that the Koenker–Bassett error function and CVaR belong to the same quantile quadrangle. By minimizing the Koenker–Bassett error function with respect to one parameter, we obtain the CVaR deviation (which is the CVaR for the centered random value). The optimal value of the parameter, which is called statistic, equals VaR. Therefore, the linear regression with the Koenker–Bassett error estimates VaR as a function of factors. The fact that statistic equals VaR and is also used for building the optimization approach for CVaR (see Rockafellar and Uryasev (2000, 2002)).

Another important contribution that takes advantage of quadrangle theory is the regression decomposition theorem proved in Rockafellar et al. (2008). With this decomposition theorem, the regression problem is decomposed in two steps: (1) minimization of deviation from the corresponding quadrangle, and (2) calculation of the intercept by using statistic from this quadrangle. For instance, by applying the decomposition theorem to the quantile quadrangle, we can do quantile

regression by minimizing CVaR deviation for finding all regression coefficients, except the intercept. Then, the intercept is calculated by using VaR statistic.

CVaR can be approximated using the weighted average of VaRs with different confidence levels, which is called the mixed VaR method. Rockafellar and Uryasev (2013) demonstrated that mixed VaR is a statistic in the mixed-quantile quadrangle. The error function, corresponding to this quadrangle (called the Rockafellar error) can be minimized for the estimation of the mixed VaR with linear regression. The Rockafellar error is a solution of a minimization problem with one linear constraint. Linear regression for estimating mixed VaR can be done by minimizing the Rockafellar error with convex and linear programming (Appendix A contains these formulations). Alternatively, this regression can be done in two steps with the decomposition theorem. The deviation in the mixed-quantile quadrangle is the mixed CVaR deviation, therefore all regression coefficients, except the intercept, can be found by minimizing this deviation. Further, the intercept can be found by using statistic, which is the mixed VaR.

Rockafellar et al. (2014) developed the CVaR quadrangle with the statistic equal to CVaR. Risk envelopes and identifiers for this quadrangle were calculated in Rockafellar and Royset (2018). This CVaR quadrangle is a theoretical basis for constructing the regression for estimating CVaR. Rockafellar et al. (2014) called the linear regression for estimation of CVaR using superquantile (CVaR) regression. The superquantile is an equivalent term for CVaR. Here we use the term "CVaR regression". The CVaR regression plays a major role in various engineering areas, especially in financial applications. For instance, Huang and Uryasev (2018) used CVaR regression for the estimation of risk contributions of financial institutions and Beraldi et al. (2019) used CVaR for solving portfolio optimization problems with transaction costs.

This paper considers only discrete random values with a finite number of equally probable atoms. This special case is considered because it is needed for the implementation of the linear regression for the CVaR estimation. We have explained with an example how parameters of the optimization problems are calculated.

The equal probabilities property was used for calculating parameters of optimization problems. It is possible to calculate parameters with non-equal probabilities of atoms, but this is beyond the scope of the paper, which is focused on the linear regression.

We suggested two sets (Sets 1 and 2) of parameters for the mixed-quantile quadrangle. Set 1 corresponds to the two-step implementation of the CVaR regression in Rockafellar et al. (2014), and Set 2 is a new set of parameters. We proved that with Set 1, the statistic, risk, and deviation of the mixed-quantile and CVaR quadrangles coincide. Therefore, CVaR regression can be done by minimizing the Rockafellar error with convex and linear programming. For Set 2, the mixed-quantile and CVaR quadrangle share risk and deviation parameters. Also, the statistic of this mixed-quantile quadrangle (which may not be unique) includes statistic of the CVaR quadrangle. Therefore, minimizing the Rockafellar error correctly calculates all regression coefficients, but may provide an incorrect intercept. This is actually not a big concern because we know that the intercept is equal to the CVaR of an optimal residual without intercept.

Also, we demonstrated that the CVaR regression can be done in two steps with the decomposition theorem by using parameters from Sets 1 and 2 in the mixed-quantile deviation. A similar two-step procedure was used for CVaR regression in Rockafellar et al. (2014). Here we justify this two-step procedure through the equivalence of deviations in CVaR and mixed-quantile quadrangles with parameters from Sets 1 and 2.

This paper is organized as follows. Section 2 provides general results about quadrangles. In particular, we considered quantile, mixed-quantile, and CVaR quadrangles. Sections 3 and 4 introduced and investigated the parameters from Sets 1 and 2, respectively. Section 5 provided optimization problem statements based on CVaR and mixed-quantile quadrangles and described the linear regression for CVaR estimation. Section 6 presented a case study and applied CVaR regression to the financial style classification problem. The case study is posted on the web with codes, data, and solutions. Appendix A provides convex and linear programming problems for minimization of the Rockafellar error; Appendix B provides Portfolio Safeguard (PSG) codes implementing regression optimization problems.

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

Rockafellar and Uryasev (2013) developed a new paradigm called the risk quadrangle, which linked risk management, reliability, statistics, and stochastic optimization theories. The risk quadrangle methodology united risk functions for a random value *X* in groups (quadrangles) consisting of five elements:


These elements of a risk quadrangle are related as follows:

$$\mathcal{V}(X) = \mathcal{E}(X) + E(X)$$

$$\mathcal{R}(X) = \mathcal{D}(X) + E(X)$$

$$\mathcal{R}(X) = \min\_{\mathbb{C}} \{ \mathbb{C} + \mathcal{V}(X - \mathbb{C}) \}$$

$$\mathcal{D}(X) = \min\_{\mathbb{C}} \{ \mathcal{E}(X - \mathbb{C}) \}$$

$$\operatorname\*{argmin}\_{\mathbb{C}} \{ \mathbb{C} + \mathcal{V}(X - \mathbb{C}) \} = \mathcal{S}(X) = \operatorname\*{argmin}\_{\mathbb{C}} \{ \mathcal{E}(X - \mathbb{C}) \}$$

where *E*(*X*) denotes the mean of *X* and the statistic, S(*X*), can be a set, if the minimum is achieved for multiple points.

Further, we use the following notations. The cumulative distribution function is denoted by *FX*(*x*) = *prob*{*<sup>X</sup>* ≤ *<sup>x</sup>*}. The positive and negative part of a number are denoted using:

$$[t]^{+} = \begin{cases} \ t, \text{ for } t > 0\\ 0, \text{ for } t \le 0 \end{cases} \text{ and } [t]^{-} = \begin{cases} -t, \text{ for } t < 0\\ 0, \text{ for } t \ge 0 \end{cases}$$

The lower and upper VaR (quantile) are defined as follows: lower VaR:

$$VaR^-\_{\alpha}(X) = \begin{cases} \sup\{\mathfrak{x}, F\chi(\mathfrak{x}) < \alpha\} for \, 0 < \alpha \le 1\\ \inf\{\mathfrak{x}, F\_X(\mathfrak{x}) \ge \alpha\} \, for \, \alpha = 0 \end{cases}$$

upper VaR:

$$V a R\_{\alpha}^{+}(X) = \begin{cases} \inf \{ \mathfrak{x}, F\_X(\mathfrak{x}) > \alpha \} \text{ for } 0 \le \alpha < 1\\ \sup \{ \mathfrak{x}, F\_X(\mathfrak{x}) \le \alpha \} \text{ for } \alpha = 1 \end{cases}$$

VaR (quantile) is a set if the lower and upper quantiles do not coincide:

$$VaR\_{\alpha}(X) = \left[ VaR\_{\alpha}^{-}(X), VaR\_{\alpha}^{+}(X) \right]$$

otherwise VaR is a singleton *VaR*α(*X*) = *VaR*<sup>−</sup>α (*X*) = *VaR*+α (*X*).

Conditional value-at-risk (CVaR) with the confidence level α ∈ (0, 1) can be defined in many ways. We prefer the following constructive definition:

$$C VaR\_{\alpha}(X) = \min\_{\mathbb{C}} \left\{ \mathbb{C} + \frac{1}{1 - \alpha} E[X - \mathbb{C}]^{+} \right\}$$

In financial applications, however, the most popular definition of CVaR is

$$CVaR\_{\alpha}(X) = \frac{1}{1-\alpha} \int\_{\alpha}^{1} VaR\_{\beta}^{-}(X) d\beta.$$

For α = 0, *CVaR*0(*X*) is defined as *CVaR*0(*X*) = lim ε→0*CVaR*ε(*X*) = *<sup>E</sup>*(*X*).

For α = 1, *CVaR*1(*X*) is defined as *CVaR*1(*X*) = *VaR*− 1(*X*) if a finite value of *VaR*− 1(*X*) exists.

Quadrangles are named after statistic functions. The most famous quadrangle is the quantile quadrangle (see Rockafellar and Uryasev (2013)), named after the VaR (quantile) statistic. This quadrangle establishes relations between the CVaR optimization technique described in Rockafellar and Uryasev (2000, 2002) and quantile regression (see Koenker and Bassett (1978), Koenker (2005)). In particular, it was shown that CVaR minimization and the quantile regression are similar procedures based on the VaR statistic in the regre<sup>t</sup> and error representation of risk and deviation.

Here is the definition of the quantile quadrangle for α ∈ (0, <sup>1</sup>):

### **Quantile Quadrangle** (Rockafellar and Uryasev (2013))

Statistic: Sα(*X*) = *VaR*α(*X*) = VaR (quantile) statistic. Risk: Rα(*X*) = *CVaR*α(*X*) = min *C C* + Vα(*<sup>X</sup>* − *C*) = CVaR risk. Deviation: Dα(*X*) = *CVaR*α(*X*) − *E*[*X*] = min *C* Eα(*<sup>X</sup>* − *C*) = CVaR deviation.

Regret: Vα(*X*) = 1 <sup>1</sup>−<sup>α</sup>*<sup>E</sup>*[*X*] + = average absolute loss, scaled. Error: Eα(*X*) = *E* α 1−<sup>α</sup> [*X*] + + [*X*] − = Vα(*X*) − *E*[*X*] = normalized Koenker–Bassett error.

The quantile quadrangle sets an example for development of more advances quadrangles. The following mixed-quantile quadrangle includes statistic, which is equal to the weighted average of VaRs (quantiles) with specified positive weights. Therefore, the error in this quadrangle can be used to build a regression for the weighted average of VaRs (quantiles). Since CVaR can be approximated by a weighted average of VaRs, the error function in this quadrangle can be used to build linear regression for the estimation of CVaR.

### **Mixed-Quantile Quadrangle** (Rockafellar and Uryasev (2013)).

Confidence levels α*k* ∈ (0, <sup>1</sup>), *k* = 1, ... ,*r*, and weights λ*k* > 0, *r k*=1 λ*k* = 1. The error in this quadrangle is called the Rockafellar Error.

Statistic: S(*X*) = *r k*=1 <sup>λ</sup>*kVaR*<sup>α</sup>*k* (*X*) = mixed VaR (quantile). Risk: R(*X*) = *r k*=1 <sup>λ</sup>*kCVaR*<sup>α</sup>*k* (*X*) = mixed CVaR. Deviation: D(*X*) = *r k*=1 <sup>λ</sup>*kCVaR*<sup>α</sup>*k* (*X* − *E*[*X*]) = mixed CVaR deviation. Regret: V(*X*) = min *B*1,...,*Br r k*=1 <sup>λ</sup>*k*V<sup>α</sup>*k* (*X* − *Bk*) *r k*=1 λ*kBk* = 0 = the minimal weighted average of regrets V<sup>α</sup>*k* (*X* − *Bk*) = 1 1−α*<sup>k</sup> E*[*X* − *Bk*] + satisfying the linear constraint on *B*1, ... , *Br*. Error: E(*X*) = min *B*1,...,*Br r k*=1 <sup>λ</sup>*k*E<sup>α</sup>*k* (*X* − *Bk*) *r k*=1 λ*kBk* = 0 = Rockafellar error = the minimal weighted average of errors E<sup>α</sup>*k* (*X* − *Bk*) = *E* <sup>α</sup>*k* 1−α*<sup>k</sup>* [*X* − *Bk*] + + [*X* − *Bk*] − satisfying the linear constraint on *B*1, ... , *Br*.

The following CVaR quadrangle can be considered as the limiting case of the mixed-quantile quadrangle when the number of terms in this quadrangle tends to infinity. The statistic in this quadrangle is CVaR; therefore, the error in this quadrangle can be used for the estimation of CVaR with linear regression.

**CVaR Quadrangle** (Rockafellar et al. (2014)) for α ∈ (0, <sup>1</sup>).

Statistic: Sα(*X*) = *CVaR*α(*X*) = CVaR. Risk: Rα(*X*) = 1 1−<sup>α</sup> 1 α *CVaR*β(*X*)*d*β = CVaR2 risk. Deviation: Dα(*X*) = Rα(*X*) − *E*[*X*] = 1 1−<sup>α</sup> 1 α *CVaR*β(*X*)*d*β − *E*[*X*] = CVaR2 deviation. Regret: Vα(*X*) = 1 1−<sup>α</sup> 1 0 *CVaR*β(*X*) + *d*β = CVaR2 regret. Error: Eα(*X*) = Vα(*X*) − *E*[*X*] = CVaR2 error.

The following section proves that for a discretely distributed random value with equally probable atoms, the CVaR quadrangle is "equivalent" to a mixed-quantile quadrangle with some parameters in the sense that statistic, risk, and deviation in these quadrangles coincide. This fact was proved for a set of random values with equal probabilities and variable locations of atoms.

The set of parameters considered in the following section is used in two-step CVaR regression in Rockafellar et al. (2014).

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

Set 1 of parameters for the mixed-quantile quadrangle for a discrete uniformly distributed random value *X* consists of confidence levels α*k* ∈ (0, <sup>1</sup>), *k* = 1, ... ,*r* and weights λ*k* > 0 such that *r k*=1 λ*k* = 1. Parameter *r* depends only on the number of atoms in *X* and the confidence level α of the CVaR quadrangle. We proved that statistic, risk, and deviation of the mixed-quantile quadrangle with the Set 1 of parameters coincide with the statistic, risk, and deviation of the CVaR quadrangle.

Let *X* be a discrete random value with support *xi* and Prob*X* = *xi* = 1/ν for*i* = 1, 2, ... , ν, where ν is the number of atoms. Denote *xmax* = max *i*=1,...,<sup>ν</sup>*<sup>x</sup>i*. For this random value, *CVaR*1(*X*) = *VaR*− 1 (*X*) = *<sup>x</sup>max*.

### **Set 1 of parameters:**


**Lemma 1.** *Let X be a discrete random value with* ν *equally probable atoms. Then, statistic, risk, and deviation of the CVaR quadrangle for X are given by the following expressions with parameters specified by Set 1:*

*1. CVaR statistic:*

$$\overline{\mathcal{S}}\_a(X) = \text{CVaR}\_a(X) = \sum\_{i=\nu\_a}^{\nu} p\_i VaR\_{\mathcal{V}\_i}(X) \tag{1}$$

*2. CVaR2 risk:*

$$\overline{\mathcal{R}}\_{a}(X) = \frac{1}{1 - a} \int\_{a}^{1} \text{CVaR}\_{\beta}(X) d\beta = \sum\_{i = \nu\_{a}}^{\nu} p\_{i} \text{CVaR}\_{\mathcal{V}\_{i}}(X) \tag{2}$$

*3. CVaR2 deviation:*

$$\overline{\mathcal{D}}\_a(X) = \frac{1}{1 - a} \int\_a^1 \mathbb{C}VaR\_{\beta}(X) d\beta - E[X] = \sum\_{i = \nu\_a}^{\nu} p\_i \mathbb{C}VaR\_{\gamma\_i}(X) - E[X] \tag{3}$$

**Proof.** Appendix C contains proof of the lemma. -

**Note.** Expression (1) is valid for arbitrary γ*i* ∈ (β*<sup>i</sup>*−1, β*i*), *i* = να, ... , ν. Equations (2) and (3) are valid for arbitrary γν ∈ [βν−1, 1].

We want to emphasize that the statement of Lemma 1 is valid for any discrete random value with equally probable atoms. The statement does not depend upon atom locations.

**Corollary 1.** *For the random value X defined in Lemma 1, statistic, risk, and deviation of the CVaR quadrangle coincide with statistic, risk, and deviation of the mixed-quantile quadrangle with r* = ν − να + 1*,* λ*k* = *p*να−1+*k,* α*k* = γνα−1+*k, k* = 1, ... ,*r*.

**Proof.** Right hand sides in Equations (1)–(3) define statistic, risk, and deviation of the mixed-quantile quadrangle because *pi* > 0, *i* = να, ... , ν, <sup>ν</sup>*i*=να*pi* = 1 and *VaR*γ*i*(*X*), *i* = να, ... , ν, are singletons. -

**Example 1.** *Let X be a discrete random value with five atoms (*−*40;* −*10; 20; 60 100) and equal probabilities* = *0.2.*

Figure 1 explains how to calculate statistic in the CVaR quadrangle and mixed-quantile quadrangle with Set 1 of parameters for α = 0.5. Bold lines show *VaR*<sup>−</sup>α (*X*) as a function of α. *CVaR*0.5(*X*) equals the dark area under the *VaR*α(*X*) divided by 1 − α. CVaR can be calculated as integral of *VaR*α(*X*) or as the sum of areas of rectangles. Figure 2 explains how to calculate risk in the CVaR quadrangle and mixed-quantile quadrangle with Set 1 of parameters for α = 0.5. The bold continuous curve shows *CVaR*α(*X*) as a function of α. Risk R0.5(*X*) is equal to the area under the CVaR curve divided by 1 − α. This area can be calculated as the integral of CVaR or as the sum of areas of rectangles. The area of every rectangle is equal to the area under CVaR in the appropriate range of α. The equality of areas defines values of γ*i*. Parameters *pi*, γ*i* do not depend on the values of atoms.

**Figure 1.** Five equally probable atoms. VaR and CVaR with Set 1 of parameters for α = 0.5.

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

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

This section gives an alternative expression for the risk Rα(*X*) and deviation Dα(*X*) in the CVaR quadrangle for a discrete uniformly distributed random variable. This expression is based on the following Set 2 of parameters. This set of parameters has the same number of parameters as Set 1 but different values of weights and confidence levels. Similar to Section 3, let *X* be a discrete random value with support, *xi*, *i* = 1, 2, ... , ν, and Prob*X* = *xi* = 1/ν for *i* = 1, 2, ... , ν. Denote *xmax* = max *<sup>i</sup>*=1,...,ν*xi*. For this random value *CVaR*1(*X*) = *VaR*<sup>−</sup>1(*X*) = *<sup>x</sup>max*.

### **Set 2 of parameters:**


$$\begin{array}{l} q\_{\boldsymbol{\nu}} = \boldsymbol{0};\\ q\_{\boldsymbol{\nu}-1} = \frac{\boldsymbol{\delta}}{1-\boldsymbol{a}} \times \Big[\Big[}{2\ln(2)}\big] \times \frac{\boldsymbol{\delta}}{1-\boldsymbol{a}} \times 1.386294361, (\text{if } \boldsymbol{\nu} - 1 > \boldsymbol{\nu}\_{a})\\ q\_{\boldsymbol{\nu}-2} = \frac{\boldsymbol{\delta}}{1-\boldsymbol{a}} \times 2 \Big[\Big[}{3\ln\Big(\frac{3}{2}\big)} + \Big[\Big{1}\Big{(}\frac{1}{2}\big)\Big] \times \frac{\boldsymbol{\delta}}{1-\boldsymbol{a}} \times 1.0446496288, (\text{if } \boldsymbol{\nu} - 2 > \boldsymbol{\nu}\_{a})\\ q\_{\boldsymbol{\nu}-j} = \frac{\boldsymbol{\delta}}{1-\boldsymbol{a}} \times \Big[\Big[}{(j+1)\ln\Big(\frac{j+1}{j}\big)} + (j-1)\ln\Big(\frac{j-1}{j}\big)\Big], (\text{if } j > 2, \boldsymbol{\nu} - j > \boldsymbol{\nu}\_{a})\\ q\_{\boldsymbol{\nu}\_{a}} = \frac{\boldsymbol{\delta}}{1-\boldsymbol{a}} \times \Big[\Big[}{}\boldsymbol{\delta} - \boldsymbol{\delta}\_{a} + (j+1)\ln\Big(\frac{1-\boldsymbol{a}}{\delta\big$$

**Lemma 2.** *Let X be a discrete random value with* ν *equally probable atoms. Then, risk and deviation of the CVaR quadrangle for X are given by the following expressions with parameters from Set 2.*

*1. CVaR2 Risk:*

$$\overline{\mathcal{R}}\_{\mathfrak{a}}(X) = \frac{1}{1 - \alpha} \int\_{a}^{1} \mathrm{CVaR}\_{\beta}(X) d\beta = \sum\_{i = \nu\_{a} - 1}^{\nu - 1} q\_{i} \mathrm{CVaR}\_{\beta\_{i}}(X) \tag{4}$$

*2. CVaR2 Deviation:*

$$\overline{\mathcal{D}}\_{a}(X) = \frac{1}{1 - a} \int\_{a}^{1} \text{CVaR}\_{\boldsymbol{\theta}}(X) d\boldsymbol{\beta} - E[X] = \sum\_{i = \nu\_{a} - 1}^{\nu - 1} q\_{i} \text{CVaR}\_{\boldsymbol{\beta}\_{i}}(X) - E[X] \tag{5}$$

**Proof.** Appendix D contains proof of the lemma. -

**Note.** Equations (4) and (5) are valid if *CVaR*βν−<sup>1</sup>is replaced by *CVaR*γ with an arbitrary γ ∈ [βν−1, 1].

**Corollary 2.** *For the random value X defined in Lemma 2, risk and deviation of the CVaR quadrangle coincide with risk and deviation of the mixed-quantile quadrangle with r* = ν − να + 1*,* λ*k* = *q*να−2+*k,* α*k* = βνα−2+*k, k* = 1, ... ,*r*.

**Proof.** Right hand sides in Equations (4) and (5) define risk and deviation of the mixed-quantile quadrangle because *qi* > 0, *i* = να − 1, ... , ν − 1, and <sup>ν</sup>−<sup>1</sup> *i*=να−1 *qi* = 1. -

**Lemma 3.** *Let X be a discrete random value with equally probable atoms xi, i* = 1, 2, ... , ν*, ProbX* = *xi*= 1/<sup>ν</sup>. *Then, statistic of the mixed-quantile quadrangle defined by the Set 2 of parameters is a range containing statistic of the CVaR quadrangle.*

**Proof.** Appendix E contains proof of the lemma. -

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

This section formulates regression problems using the CVaR quadrangle and mixed-quantile quadrangle. For discrete final distributions with equally probable atoms, we prove some equivalence statements for the CVaR and mixed-quantile quadrangles. Further, we demonstrate how to estimate CVaR by using the linear regression with error and deviation from the mixed-quantile quadrangle.

We want to estimate variable *V* using a linear function *f*(*Y*) = *C*0 + *CTY* of the explanatory factors *Y* = (*<sup>Y</sup>*1, ... ,*Yn*). Let E be an error from some quadrangle (further we consider, mixed-quantile and CVaR quadrangles), and D and S be a deviation and a statistic, respectively, corresponding to this quadrangle. Below we consider optimization statements for solving regression problems.

### **General Optimization Problem 1**

Minimize error E and find optimal *<sup>C</sup>*<sup>∗</sup>0, *C*<sup>∗</sup>

$$\min\_{\mathcal{C}\_0 \in \mathbb{R}, \ C \in \mathbb{R}^m} \overline{\mathcal{E}}\left(Z(\mathcal{C}\_{0\prime}\mathcal{C})\right)$$

where <sup>Z</sup>(*<sup>C</sup>*0,*<sup>C</sup>*) = *V* − *C*0 − *CTY*.

### **General Optimization Problem 2**

• *Step 1. Find an optimal vector C*<sup>∗</sup> *by minimizing deviation:*

$$\min\_{\mathbf{C}\in\mathbb{R}^{m}} \tilde{\mathcal{D}}(Z\_{0}(\mathbf{C})) $$

*where <sup>Z</sup>*0(*C*) = *V* − *CTY*.

• *Step 2. Assign <sup>C</sup>*<sup>∗</sup>0:

$$\mathcal{C}\_0^\* \in \overline{\mathcal{S}}(Z\_0(\mathcal{C}^\*))$$

Error E (*X*) is called nondegenerate if:

$$\inf\_{X:EX = D} \overline{\mathcal{E}}\left(X\right) > 0 \text{ for constants } D \neq 0.$$

Rockafellar et al. (2008), p. 722, proved the following decomposition theorem.

**Theorem 1.** *(Error-Shaping Decomposition of Regression). Let* E *be a nondegenerate error and* D = min *C* E(*X* − *C*) *be the corresponding deviation, and let* S *be the associated statistic. Point (C*<sup>∗</sup>0*, C*<sup>∗</sup>*) is a solution of the General Optimization Problem 1 if and only if C*<sup>∗</sup> *is a solution of the General Optimization Problem 2, Step 1 and C*∗0∈ S (*<sup>Z</sup>*0(*C*<sup>∗</sup>)) *with Step 2.*

According to the decomposition theorem, when E, D , and S are elements of the CVaR quadrangle, the following Optimization Problems 1 and 2 are equivalent.
