# Risk measures: a generalization from the univariate to the matrix-variate

## María A. Arias-Serna, Francisco J. Caro-Lopera and Jean-Michel Loubes

#### Need to know

• This paper proposes a method to calculate matrix-variate value-at-risk.
• This paper develops a method for estimating the value-at-risk and the conditional value-at-risk when the underlying risk factors follow a beta distribution in a univariate and matrix-variate setting.
• Analytical expressions of the risk measures are developed.
• A numerical solution for the risk measures for any parameterization of beta distributed loss variables is presented.
• Of fundamental importance is the application of computer-based algorithms for solving classically analytic problems in financial risk management. The data we acquired from Colombian financial institutions are considered using both algorithmic and analytic methods. Our results demonstrate a correspondence between the two. Although our results are motivated by problems in finance, we believe that our methods may well more general applications as well.

#### Abstract

This paper develops a method for estimating value-at-risk and conditional value-at-risk when the underlying risk factors follow a beta distribution in a univariate and a matrix-variate setting. For this purpose, we connect the theory of the Gaussian hypergeometric function of matrix argument and integration over positive definite matrixes. For certain choices of the shape parameters, a and b, analytical expressions of the risk measures are developed. More generally, a numerical solution for the risk measures for any parameterization of beta-distributed loss variables is presented. The proposed risk measures are finally used for quantifying the potential risk of economic loss in credit risk.

## 1 Introduction

Measuring the risk of a portfolio basically involves determining its distribution function or the functionals describing this distribution function, such as its mean, variance or $\alpha$th percentile (Rockafellar and Uryasev 2002). Perhaps the most commonly used risk measure in finance is value-at-risk (VaR), which has received the honor of being included in industry regulations (see, for instance, Wagalath and Zubelli 2018; McNeil et al 2015; Gamboa et al 2016; Jorion 2007; Pflug 2000; JPMorgan 1996).

Although risk measures based on loss distributions is a subject that has been widely investigated (Stavroyiannis et al 2012), and the inadequacy of Gaussian laws to model the distribution of risk factors, especially in view of applications to risk modeling, is well documented in the literature (Marinelli et al 2012), few studies address the estimation of such risk measures for specific probability distributions. These include the standard normal distribution (Jorion 2007; Alexander 2008), which does not account for fat tails and is symmetric; the Student $t$ distribution (Lin and Shen 2006), which is fat-tailed but symmetric; and the generalized error distribution (GED), which is more flexible than the Student $t$ distribution because it includes both fat and thick tails. Since it is generally very difficult, if not impossible, to obtain analytically tractable expressions for many distribution functions, risk measures are usually estimated by generating random samples from a distribution and computing the corresponding empirical quantiles.

The aim of this paper is to calculate some risk measures when the underlying risk factors follow a beta distribution in a univariate and a matrix-variate setting. As stated (Johnson 1997), this is seen as a suitable model in risk analysis since it models a wide range of data with different shapes in closed domains. The beta distribution is of special interest both in the field of finance and in the field of insurance because it allows us to model the loss associated with a loss or the fraction of loss associated with a risk event (see, for example, Wang 2005). The distribution can be strongly right-skewed or less skewed as the parameters approach each other. (The distribution would be left-skewed if the parameters’ values were switched.)

When univariate VaR is considered for matrix-variate distributions, a number of problems arise. First, we cannot easily define an interpretable event associated with the given probability $\alpha$ or $1-\alpha$ for the risk. Second, if we have a well-defined event for the risk, then the problem resides in the computation of the probability. Both of these problems are also interesting in matrix-variate distribution theory in a different context. First, a restriction over the space of any matrix into the full linear group (invertible matrixes) and then into the study of positive definite symmetric matrixes enables us to interpret the events. When positive definite matrixes are considered, space is reduced to cones, which are modeled by the corresponding positive eigenvalues, and then a more manageable space appears. The events can be ordered in such a way that they have a geometrical meaning, ie, if we have two $m\times m$ random positive definite matrixes $\bm{V}$ and $\bm{W}$, we can ask for the probability that $\bm{V}\leq\bm{W}$ or $\bm{W}\leq\bm{V}$. Second, we can compute the probability of an event involving multiple integrals with respect to the Lebesgue measure. The computation of VaR in this context opens an interesting line of research. In this paper, we introduce two alternative extensions of classical univariate VaR for matrix-variate beta distributions, connecting the theory of zonal polynomials and integration over positive definite matrixes.

The rest of this paper is organized as follows. Section 2 presents the description of the univariate approach, provides analytical solutions for the values of VaR and conditional VaR (CVaR), and provides numerical solutions for the values of these risk measures. Section 3 presents a description of the matrix-variate approach. Section 4 presents a case study on the credit risk framework. In Section 5, some conclusions are outlined as well as some possible directions for future work. Finally, the proofs of our theorems and propositions are presented in the online appendix.

## 2 Risk measures univariate under a family of beta distributions

Given a real random variable $X$ on a probability space $(\varOmega,\mathfrak{F},P)$, let $f_{X}(x)$ denote its density function and let $F_{X}(x)=P(X\leq x)$ denote its associated cumulative distribution function. VaR $(V_{\alpha}(X))$ for the random variable $X$ at the confidence level $\alpha\in(0,1)$ is defined by Rockafellar and Uryasev (2000) as

 $\mathrm{VaR}_{\alpha}(X)=\min\{x\in\mathbb{R}\mid P(X\leq x)\geq\alpha\}$

or, equivalently,

 $\mathrm{VaR}_{\alpha}(X)=\sup\{x\in\mathbb{R}\mid P(X\geq x)\geq 1-\alpha\}.$

In the univariate case, the above problems are equivalent and have received considerable interest in the literature (see, for instance, JPMorgan 1996; Rockafellar and Uryasev 2000; Pflug 2000; Embrechts and Puccetti 2006; Alexander 2008; McNeil et al 2015; Wagalath and Zubelli 2018). If $F$ is strictly increasing, then VaR is the unique threshold $\mathrm{VaR}_{\alpha}(X)$ at which $F_{X}(\mathrm{VaR}_{\alpha}(X))=\alpha$: in other words, VaR is a real number such that

 $P(X\leq\mathrm{VaR}_{\alpha}(X))=\alpha.$ (2.1)

The classical two-parameter probability density function of the beta distribution with shape parameters $a$ and $b$ is given by

 $\frac{\varGamma(a+b)}{\varGamma(a)\varGamma(b)}x^{a-1}(1-x)^{b-1},\quad 0\leq x% \leq 1,\,a>0,\,b>0.$ (2.2)

The beta distribution is of special interest both in the field of finance and in the field of insurance because it allows us to model the fraction of loss associated with a risk event. In credit risk, for example, consider that $Y_{1}$, $Y_{2}$ represent the losses associated with the first and second borrowers, respectively, and that their amounts follow a gamma distribution $\varGamma(c,a_{i})$, $i=1,2$. In this case, the random variable $X_{1}=Y_{1}/(Y_{1}+Y_{2})$ represents the fraction of loss associated with the first borrower, while $X_{2}=Y_{2}/(Y_{1}+Y_{2})$ represents the fraction associated with the second borrower. Then, the distribution of $X_{1}$ is a beta distribution of parameters $a_{1}$, $a_{2}$, while that of $X_{2}$ is a beta distribution of parameters $a_{2}$, $a_{1}$. Another application is in insurance. For example, consider a policy that covers two types of independent claims $Y_{1}$, $Y_{2}$ and suppose that their amounts follow a gamma distribution $\varGamma(c,a_{i})$, $i=1,2$. In this case, $Y_{1}/(Y_{1}+Y_{2})$ follows a beta distribution of parameters $a_{1}$, $a_{2}$.

Now, since our objective is to find VaR for a random variable that is distributed as a beta, we are interested in calculating $P(X\leq\mathrm{VaR}_{\alpha}(X))$. By definition, if $X\sim\operatorname{Beta}(a,b)$, then $\mathrm{VaR}_{\alpha}(X)$ is a real number such that

 $P(X\leq\mathrm{VaR}_{\alpha}(X))=\frac{\varGamma(a+b)}{\varGamma(a)\varGamma(b% )}\int_{0}^{\mathrm{VaR}_{\alpha}(X)}x^{a-1}(1-x)^{b-1}\mathrm{d}x=\alpha.$

That is, what is relevant to calculate VaR is the calculation of the integral, which, in turn, is closely related to the incomplete beta function, which is given by

 $\int_{0}^{x}t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{x^{a}}{a}{}_{2}F_{1}(a,1-b;a+1% ;x),$

where

 $\displaystyle{}_{2}F_{1}(a,b;c;x)$ $\displaystyle=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^{2}}{2!}+\cdots$ $\displaystyle=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{(c)_{k}k!}x^{k},\quad|x% |<1,$ (2.3)

is the Gaussian hypergeometric function and $(a)_{k}=a(a+1)\cdots(a+k-1)=\varGamma(a+k)/\varGamma(a)$ is Pochhammer’s symbol. The hypergeometric function is important in both pure and applied mathematics, since many elementary functions are special cases of the hypergeometric function, eg, ${}_{2}F_{1}(a,b;b;x)=(1-x)^{-a}$, $-x_{2}F_{1}(1,1;2;x)=\ln(1-x)$. Some other functions, such as the incomplete gamma function, are also defined in terms of hypergeometric functions (see Erdelyi (1995), Andrews (1998) and the references therein).

Now we are ready to propose the main theorem of this section. We show that finding the VaR of a beta distribution is equivalent to finding the zeros of the Gaussian hypergeometric function. This result will also allow us to find the VaR in the matrix setting.

###### Theorem 2.1.

Let $X\sim\operatorname{Beta}(a,b)$ with $a>0$, $b>0$. The univariate VaR $(\mathrm{VaR}_{\alpha}(X))$ of the beta distribution at probability level $\alpha\in(0,1)$ is a unique solution, in the interval $[0,1]$, of the following hypergeometric equation:

 $\frac{\varGamma(a+b)}{\varGamma(a+1)\varGamma(b)}\mathrm{VaR}_{\alpha}(X)^{a}{% }_{2}F_{1}(a,1-b;a+1;\mathrm{VaR}_{\alpha}(X))=\alpha.$ (2.4)

Under the previous theorem, some classical properties of $\mathrm{VaR}_{\alpha}(X)$ are satisfied.

###### Proposition 2.2.

Let $X\sim\operatorname{Beta}(a,b)$ and $Y\sim\operatorname{Beta}(a,b)$ for $\alpha\in(0,1)$. The univariate VaR satisfies the following properties.

1. (1)

Monotonicity: if $X\leq Y$, $\mathrm{VaR}_{\alpha}(X)\leq\mathrm{VaR}_{\alpha}(Y)$.

2. (2)

Positive homogeneity: for all $\lambda\geq 0$, $\mathrm{VaR}_{\alpha}(\lambda X)=\lambda\mathrm{VaR}_{\alpha}(X)$.

3. (3)

Translation invariance: for $c\in\mathbb{R}$, $\mathrm{VaR}_{\alpha}(X+c)=\mathrm{VaR}_{\alpha}(X)+c$.

Although VaR, by definition, is able to calculate risk, it lacks some desirable properties such as subadditivity, which is a mathematical statement of the response of risk concentration, a common reality in risk management. Among other objections raised against $\mathrm{VaR}_{\alpha}(X)$, we should also mention that it is unable to account for the consequences of the established threshold being surpassed, and it is generally not continuous on the parameter $\alpha$ (Arias et al 2016). A measure of risk closely related to VaR is CVaR, defined as the conditional expected value of the $(1-\alpha)$ tail. It is defined by Rockafellar and Uryasev (2000) as follows:

 $\mathrm{CVaR}_{\alpha}(X)=\frac{1}{1-\alpha}\int_{\mathrm{VaR}_{\alpha}}^{% \infty}xf_{X}(x)\mathrm{d}x.$ (2.5)

As an immediate consequence of the previous theorem, we can find the CVaR of the beta distribution.

###### Corollary 2.3.

Let $X\sim\operatorname{Beta}(a,b)$ with $a>0$, $b>0$. The CVaR $(\mathrm{CVaR}_{\alpha}(X))$ of the beta distribution at probability level $\alpha\in(0,1)$ is given by

 $\displaystyle\mathrm{CVaR}_{\alpha}(X)$ $\displaystyle=\frac{1}{(1-\alpha)}\frac{\varGamma(a+b)}{\varGamma(a)\varGamma(% b)}$ $\displaystyle\quad\times\bigg{[}\frac{\varGamma(a+2)\varGamma(b)}{\varGamma(a+% b+1)}-V_{\alpha}(X)^{a+1}{}_{2}F_{1}(a+1,1-b,a+2,V_{\alpha}(X))\bigg{]}.$ (2.6)

### 2.1 Analytical expressions of risk measures

In this subsection, we consider some specific values of the parameters $a$ and $b$ that enable us to more precisely analyze the problem. That is, we are seeking values for which it is possible to obtain families of variable changes that provide analytical results on the zeros of the Gaussian hypergeometric function.

An important property of the hypergeometric function is that if $a=-m$ and/or $b=-m$, where $m=0,1,2,\dots$, the series (2.3) terminates and reduces to a polynomial of degree $m\in Z$: the so-called hypergeometric polynomial of grade $m$ (see Erdelyi 1995). For instance, if $b=-m$, the hypergeometric function is reduced to the next polynomial

 $\displaystyle{}_{2}F_{1}(a,b;c;x)$ $\displaystyle=\sum_{k=0}^{m}\frac{(a)_{m}(-m)_{m}}{(c)_{m}m!}x^{m}$ $\displaystyle=1-\frac{am}{c}x-\frac{a(a+1)m(1-m)}{c(c+1)}\frac{x^{2}}{2!}+% \cdots+\frac{(a)_{k}(-m)_{m}}{(c)_{k}}\frac{x^{m}}{m!}.$

Then, applying this property to Theorem 2.1, if $1-b$ is a negative integer, the hypergeometric series (2.3) is reduced to a polynomial. Next, we use the fact that $(-m)_{k}=(-1)^{k}(m!/((m-k)!))$, $0\leq k\leq m$ (Driver and Möller 2001), such that, by (2.4), $\mathrm{VaR}_{\alpha}(X)$ is reduced and we are able to solve the following polynomial equation:

 $\frac{\varGamma(a+b)}{\varGamma(a+1)\varGamma(b)}\mathrm{VaR}_{\alpha}(X)^{a}% \sum_{k=0}^{b-1}\frac{(-1)^{k}}{(a+k)\varGamma(b-k)}\frac{\mathrm{VaR}_{\alpha% }(X)^{k}}{k!}-\alpha=0.$ (2.7)

For example, if $b=2$ and $a=1$, (2.7) is equivalent to

 $-\mathrm{VaR}_{\alpha}(X)^{2}+2\mathrm{VaR}_{\alpha}(X)-\alpha=0,$

whose zeros are given by

 $\mathrm{VaR}_{\alpha}(X)=1\pm\sqrt{1-\alpha}.$

As $0<\alpha<1$, $1-\alpha>0$, and thus $1-\sqrt{1-\alpha}\in(0,1)$.

In particular, note that when $a=b=1$, the function $\operatorname{Beta}(a,b)=U(0,1)$, and then

 $\mathrm{VaR}_{\alpha}(X)=\alpha.$

In the following proposition, we will deal with the analytical calculation of $\mathrm{VaR}_{\alpha}(X)$ seen as real zeros of the hypergeometric polynomial expression of (2.7).

###### Proposition 2.4.

$\mathrm{VaR}_{\alpha}(X)$ for $a,b\in Z$ such that $a+b\leq 5$ are given by the values shown in Table 1.

###### Remark 2.5.

To find the

 $\mathrm{CVaR}_{\alpha}(X)=\frac{1}{1-\alpha}\int_{\alpha}^{1}\mathrm{VaR}_{\nu% }(X)\mathrm{d}\nu,$

simply calculate the integral of the previous expressions. Continuing with the previous example, if $b=2$ and $a=1$, then $\mathrm{VaR}_{\alpha}(X)=1-\sqrt{1-\alpha}$; thus,

 $\mathrm{CVaR}_{\alpha}(X)=\frac{1}{1-\alpha}\int_{\alpha}^{1}(1-\sqrt{1-\nu})% \mathrm{d}\nu=\frac{3+2\sqrt{1-\alpha}}{3}.$

### 2.2 Numerical solutions

According to Theorem 2.1 and Corollary 2.3, $\mathrm{VaR}_{\alpha}(X)$ and $\mathrm{CVaR}_{\alpha}(X)$ for a beta distribution do not always have a closed expression; this is because, by Abel’s theorem, there is no formula describing the roots of any general polynomial of degree greater than or equal to $5$. In this subsection, we will deal with the numerical calculation of the risk measures, which will be calculated from the calculation of the zeros of the hypergeometric function. A number of methods for the calculation of zeros of the hypergeometric function have been proposed in the literature. Some of these proposed methods are the Newton method and the method based on the division algorithm (Dominici et al 2013), asymptotic estimates (Srivastava et al 2011; Duren and Guillou 2001) and matrix methods (Ball 2000). We base our work on the Newton method. This method has the advantage of being easy to understand and has also been completely implemented in the routine packages of R software. In addition, it has internal routines for evaluating the hypergeometric function (see Welbers et al 2017).

Our proposed algorithm to compute $\mathrm{VaR}_{\alpha}$ and $\mathrm{CVaR}_{\alpha}$ is the following.

### The algorithm

Input: $a,b,\alpha,l$

$x\leftarrow\mathrm{seq}(0,1,l)$

$\mathrm{poly}\leftarrow\mathrm{function}(a,b,x,\alpha)$ $\bigg{\{}\dfrac{\varGamma(a+b)}{\varGamma(a)\varGamma(b)}\dfrac{x^{a}}{a}% \mathrm{hypergeo}(a,1-b,a+1,x)-\alpha\bigg{\}}$

$\mathrm{fun}\leftarrow\mathrm{function}(x)\{\mathrm{Re}(\mathrm{poly}(a,b,x,s))\}$

$V_{\alpha}\leftarrow\mathrm{uniroot.all}(\mathrm{fun},c(0,1))$

$\mathrm{CV}_{\alpha}\leftarrow\dfrac{1}{(1-\alpha)}\dfrac{\varGamma(a+b)}{% \varGamma(a)\varGamma(b)}\bigg{[}\dfrac{\varGamma(a+2)\varGamma(b)}{\varGamma(% a+b+1)}-V_{\alpha}^{a+1}{}_{2}F_{1}(a+1,1-b,a+2,V_{\alpha})\bigg{]}.$

In Table 2, we provide the risk measures. We have considered different values for the parameters $a$ and $b$. The first rows (marked in bold) are contrasted with the results obtained using the analytical expressions obtained in the previous section.

## 3 Generalization to a matrix-variate approach

In the previous section, we developed a methodology to find univariate VaR as the zeros of the Gaussian hypergeometric function. In this section, we will show that the philosophy of our proposed method can be extended to a matrix setting. As a generalization of the univariate beta distribution, Olkin and Rubin (1964) derived a beta distribution generalizing the ratio $X/(Y+X)$, where $X$ and $Y$ follow the matrix-variate gamma distributions $\varGamma_{m}(a,\bm{I}_{m})$ and $\varGamma_{m}(b,\bm{I}_{m})$, respectively, resulting in a matrix-variate beta distribution with parameters $a$ and $b$. A random symmetric positive definite matrix $\bm{X}_{m\times m}$ is said to have a matrix-variate beta distribution with parameters $a$ and $b$, and we will write that $\bm{X}$ is $\operatorname{Beta}_{m}(a,b),$ if its probability density function is given by

 $\displaystyle\frac{\varGamma_{m}(a+b)}{\varGamma_{m}(a)\varGamma_{m}(b)}|\bm{X% }|^{a-((m+1)/2)}|\bm{I}_{\bm{m}}-\bm{X}|^{b-((m+1)/2)},$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad{}\bm{0}<\bm{X}<\bm{I}_{\bm{m% }},\,a>\frac{m-1}{2},\,b>\frac{m-1}{2},$

where $\bm{0}<\bm{X}<\bm{I}_{\bm{m}}$ means that $\bm{X}>0$ and $\bm{I}_{\bm{m}}-\bm{X}>0$ (ie, $\bm{X}$ and $\bm{I}_{\bm{m}}-\bm{X}$ are positive definite matrixes).

As in the univariate case, we are interested in calculating $P(\bm{X}\leq\mathrm{VaR}_{\alpha}(\bm{X}))$ and $P(\bm{X}\geq\mathrm{VaR}_{\alpha}(\bm{X}))$ when $\bm{X}\sim\mathrm{Beta}_{m}(a,b)$. For this, it is necessary to keep in mind that a unique definition of multivariate VaR does not exist because there are different possible definitions of multivariate quantiles. In the last decade, many extensions to multidimensional settings have been investigated, and recent papers suggest alternative ways of measuring risk for multivariate portfolios. For instance, Embrechts and Puccetti (2006) used the notion of a quantile curve to define both the multivariate lower orthant VaR and the multivariate upper orthant VaR at probability level $\alpha$ for an increasing function, which is represented by an infinite number of points. Cousin and Di Bernardino (2013) proposed two alternative extensions of the multivariate VaR for continuous vectors based on the level surfaces provided in Embrechts and Puccetti (2006). Torres et al (2015) introduced a directional multivariate VaR based on the concept of the directional multivariate quantile. They considered the multivariate VaR as a vector-valued point that defines the vertex of an oriented orthant in the direction of analysis. They also presented comparisons in terms of robustness with the alternative multivariate VaR introduced by Cousin and Di Bernardino (2013).

If we search for a matrix-variate extension for VaR, the finance literature does not provide us with any approaches. However, from a mathematical point of view, VaR just requires meaningful percentiles in the context of matrix cumulative density functions. The theory behind the random matrix setting has been thoroughly studied by Muirhead (2005). In particular, that paper provided a formulation for calculating $P(\bm{X}\leq\bm{V})$ and $P(\bm{X}\geq\bm{V})$ when $\bm{X}$ follows a Wishart distribution and $\bm{V}$ is a positive definite matrix. They also demonstrated that its cumulative distribution function can be expressed in terms of a Gaussian hypergeometric function of matrix argument. Supported by Theorem 7.2.10 of Muirhead (2005), we provide a couple of theorems that will allow us to find $P(\bm{X}\leq\bm{V})$ and $P(\bm{X}\geq\bm{V})$, where $\bm{X}\sim\mathrm{Beta}_{m}(a,b)$ and $\bm{V}$ is a positive definite matrix.

###### Theorem 3.1.

If $\bm{X}\sim\mathrm{Beta}_{m}(a,b)$, with $a>(m-1)/2$, $b>(m-1)/2$, and $\bm{V}$ is an $m\times m$ positive definite matrix $(\bm{V}>\bm{0})$, then

 $\displaystyle P(\bm{X}<\bm{V})$ $\displaystyle=\frac{\varGamma_{m}(a+b)\varGamma_{m}((m+1)/2)}{\varGamma_{m}(b)% \varGamma_{m}(a+((m+1)/2))}$ $\displaystyle\qquad\times|\bm{V}|^{a}{}_{2}F_{1}\bigg{(}a,\frac{m+1}{2}-b;a+% \frac{m+1}{2};\bm{V}\bigg{)},$ (3.1)

where

 ${}_{2}F_{1}(a,b;c;\bm{X})=\sum_{k=0}^{\infty}\dfrac{1}{k!}\sum_{\kappa}\frac{(% a)_{k}(b)_{k}}{(c)_{\kappa}}C_{\kappa}(\bm{X})$

is the Gaussian hypergeometric function of matrix argument. The series converges for $\|\bm{X}\|<1$, where $\|\bm{X}\|$ denotes the maximum of the absolute values of the eigenvalues of $\bm{X}$;

 $(n)_{\kappa}=\pi^{m(m-1)/4}\frac{\prod_{i=1}^{m}\varGamma[n+\kappa_{i}-\tfrac{% 1}{2}(i-1)]}{\varGamma_{m}(n)}$

is the generalized hypergeometric coefficient, $\sum_{\kappa}$ denotes summation over all partitions $\kappa=(\kappa_{1},\dots,\kappa_{m})$, $\kappa_{1}\geq\dots\geq\kappa_{m}$, and $C_{\kappa}(\bm{X})$ denotes the zonal polynomial (see Muirhead (2005) for more details).

We now introduce $P(\bm{X}>\bm{V})$. We will use the same argument as in Muirhead (2005) to prove that in the case where $r=a-((m+1)/2)$ is a positive integer, an expression can be obtained in terms of a finite series involving zonal polynomials.

###### Theorem 3.2.

Let $\bm{X}\sim\mathrm{Beta}_{m}(a,b)$, with $a>(m-1)/2$, $b>(m-1)/2$, and $V$ is an $m\times m$ positive definite matrix $(\bm{V}>\bm{0})$. If $r=a-((m+1)/2)$ is a positive integer, then

 $P(\bm{X}>\bm{V})=|\bm{I}-\bm{V}|^{b}\sum_{k=0}^{mr}\frac{1}{k!}\sum_{\kappa^{*% }}(-r)_{\kappa}\frac{((m+1)/2)_{\kappa}}{(((m+1)/2)+b)_{\kappa}}C_{\kappa}(-(% \bm{V}^{-1}-\bm{I})),$ (3.2)

where the summation runs over partitions $\kappa^{*}=(k_{1},\dots,k_{m})$ of $k$ such that $k_{1}\leq r$.

Starting from the previous theorems, we now introduce two alternative extensions of the VaR measure. We denote by $\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})$ our matrix upper VaR associated with $P(\bm{X}<\mathrm{VaR}_{\alpha}(\bm{X}))$, and by $\underline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})$ our matrix lower VaR associated with $P(\bm{X}>\mathrm{VaR}_{\alpha}(\bm{X}))$.

###### Definition 3.3.

Let $\bm{X}$ be a random real matrix, if $\bm{X}\sim\mathrm{Beta}_{m}(a,b)$ with $a>(m-1)/2$, $b>(m-1)/2$. The matrix upper VaR for the matrix variate beta distribution at probability level $\alpha\in(0,1)$ is the solution of the hypergeometric equation of matrix argument:

 $\displaystyle\frac{\varGamma_{m}(a+b)\varGamma_{m}((m+1)/2)}{\varGamma_{m}(b)% \varGamma_{m}(a+((m+1)/2))}$ $\displaystyle\qquad\times|\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})|^{a}{}% _{2}F_{1}\bigg{(}a,\frac{m+1}{2}-b;a+\frac{m+1}{2};\overline{\textbf{VaR}_{\bm% {\alpha}}}(\bm{X})\bigg{)}=\alpha,$ (3.3)

$\bm{0}<\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})<\bm{I}_{\bm{m}}$.

Now we introduce another possible generalization of the VaR based on $P(\bm{X}>\mathrm{VaR}_{\alpha}(\bm{X}))$.

###### Definition 3.4.

Let $\bm{X}$ be a random real matrix, if $\bm{X}\sim\mathrm{Beta}_{m}(a,b)$ with $a>(m-1)/2$, $b>(m-1)/2$, and let $r=a-((m+1)/2)$ be a positive integer. The matrix lower VaR for the matrix variate beta distribution at probability level $\alpha\in(0,1)$ is the solution of the hypergeometric equation of matrix argument:

 $|\bm{I}-\underline{\bm{V}_{\bm{\alpha}}}(\bm{X})|^{b}\sum_{k=0}^{mr}\frac{1}{k% !}\sum_{\kappa^{*}}(-r)_{\kappa}\frac{((m+1)/2)_{\kappa}}{(((m+1)/2)+b)_{% \kappa}}C_{\kappa}(-(\underline{\bm{V}_{\bm{\alpha}}}(\bm{X})^{-1}-I))=1-\alpha.$ (3.4)

$\bm{0}<\underline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})<\bm{I}_{\bm{m}}$.

Under the previous definitions the classical properties of VaR are satisfied.

###### Proposition 3.5.

Let $\bm{X}\sim\operatorname{Beta}(a,b)$ and $\bm{Y}\sim\operatorname{Beta}(a,b)$. For $\alpha\in(0,1)$, the matrix upper VaR and matrix lower VaR satisfy the following properties.

1. (1)

Monotonicity: if $\bm{X}\leq\bm{Y}$, then

 $\overline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{X})\leq\overline{\mathbf{VaR}_{\bm{% \alpha}}}(\bm{Y}),\qquad\underline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{X})\leq% \underline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{Y}).$
2. (2)

Positive homogeneity: for all symmetric matrixes $\bm{\varOmega}\geq 0$,

 $\overline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{\varOmega X})=\bm{\varOmega}% \overline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{X}),\qquad\underline{\mathbf{VaR}_{% \bm{\alpha}}}(\bm{\varOmega X})=\bm{\varOmega}\underline{\mathbf{VaR}_{\bm{% \alpha}}}(\bm{X}).$
3. (3)

Translation invariance: for all symmetric matrixes $\bm{\varOmega}\geq 0$,

 $\overline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{X}+\bm{\varOmega})=\overline{\mathbf% {VaR}_{\bm{\alpha}}}(\bm{X})+\bm{\varOmega},\qquad\underline{\mathbf{VaR}_{\bm% {\alpha}}}(\bm{X}+\bm{\varOmega})=\underline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{X% })+\bm{\varOmega}.$

As in the univariate case, the properties of monotonicity, positive homogeneity and translation invariance are satisfied since these properties are conserved for matrix-variate beta distributions (see Gupta and Nagar 2000).

To find the matrix upper VaR, Algorithm 4.2 of Koev and Edelman (2006) can be used. They have made an algorithm that efficiently approximates the hypergeometric function of matrix argument through its expansion as a series of zonal polynomials. The implementation of the algorithms in MATLAB (current version: 1.5, February 12, 2018) is available at http://www.math.sjsu.edu/~koev/. Basically, the algorithm computes the truncated hypergeometric function ${}_{p}F_{q}$ as a series of zonal polynomials, truncated for partitions of size not exceeding $M$. However, as in the univariate case, an important property of the hypergeometric function is that if $b=(m+1)/2$, then the previous series is reduced to 1, and we can obtain a closed expression for the calculation of $\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})$.

###### Corollary 3.6.

Let $\bm{X}$ be a random real matrix if $\bm{X}\sim\mathrm{Beta}_{m}(a,((m+1)/2))$ with $a>(m-1)/2$. The matrix upper VaR at probability level $\alpha\in(0,1)$ is the solution of

 $|\overline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{X})|^{a}=\alpha,\qquad\bm{0}<% \overline{\mathbf{VaR}_{\bm{\alpha}}}(\bm{X})<\bm{I}_{\bm{m}}.$ (3.5)

If $m=2$, then $\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})$ is a $2\times 2$ positive definite matrix such that $\bm{0}<\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})<\bm{I}_{2}$ and $v_{11}v_{22}-v_{12}^{2}=\alpha^{1/a}$, whose solution is in the region of $R^{3}$ described by the inequalities $0 and $1-v_{11}-v_{22}+v_{11}v_{22}-v_{12}^{2}>0$. Let $v_{11}=\sqrt{\alpha}$ as $v_{12}^{2}=\sqrt{\alpha}v_{22}-\alpha^{1/a}$; then, $\sqrt{\alpha}v_{22}-\alpha^{1/a}\geq 0$, and thus $v_{22}\geq\alpha^{1/a-1/2}$. Let $v_{22}=\alpha^{1/a-1/2}+t$, $t\in\mathbb{R}^{+}$; then, $v_{12}=\sqrt{t\sqrt{\alpha}}$, and then

 $\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})=\begin{bmatrix}\sqrt{\alpha}&% \sqrt{t\sqrt{\alpha}}\\ \sqrt{t\sqrt{\alpha}}&\alpha^{1/a-1/2}+t\end{bmatrix}.$

As $1-v_{11}-v_{22}+v_{11}v_{22}-v_{12}^{2}>0$, $t$ is such that $t<1-\alpha^{1/2}-\alpha^{1/a-1/2}+\alpha^{1/a}$.

Inspired by Theorem 3.3.3 of Muirhead (2005), and focusing on $m=2$, for an intuitive interpretation, the following theorem states that the distribution of a matrix beta can be decomposed into the product of three independent univariate beta distributions. We consider the exact case for $b=3$ in order to apply the previous corollary as a solution for the power determinantal equation. If we start with $a=1$ in Corollary 3.6, then $|\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})|=\alpha$. Then, the Cholesky decomposition and the previous corollary provide the following relation with the equations underlying the univariate VaR.

###### Theorem 3.7.

Let $\bm{X}\sim B_{2}(a,\tfrac{3}{2})$ and $\bm{X}=\bm{T}^{\prime}\bm{T}$, where $\bm{T}=(t_{ij})$ is an upper triangular matrix with $t_{ij}>0$, $i=1,2$. Then, the distribution of a matrix beta can be decomposed into the product of three independent univariate beta distributions.

A plausible general result using the above argument should hold for any order $m$. However, this involves cumbersome algebra that will be part of a future work, because we require new representations for nested determinants according to the definition of positive matrixes. The generalization should claim that the distribution of any beta matrix $\bm{X}$ can be decomposed into a product of independent univariate beta distributions indexed by the nested determinants $x_{11}>0$, $|\bm{X}_{2}|>0$ and $|\bm{X}_{3}|>0,\dots,|\bm{X}|>0$, where $|\bm{X}_{j}|$ is the determinant of the $(m-j)\times(m-j)$ left upper corner submatrix of $\bm{X}$ with $j=m-1,m-2,\dots,1,0$.

Once the general representation is derived, $\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})$ can be interpreted in the same way as for the case $m=2$, but with a more complex Cholesky indexed by the same subdeterminants and univariate VaRs. This opens an interesting line of future work because the interpretation of the VaR matrix is possible not only in the beta case but also in the $F$ case in terms of univariate betas or $F$ distributions. Finally, note that by replacing $b=3$ and $\bm{X}=\bm{I}-\bm{X}$, the probability beta in $|\bm{I}-\bm{X}|=\beta$ can be easily justified and interpreted in the proof of the Cholesky decomposition of $\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})$.

## 4 An application to credit risk

In this section, the proposed risk measures are used for quantifying the potential risk of economic loss in credit risk. We define credit risk as the risk that the value of a portfolio changes due to unexpected changes in the credit quality of issuers or trading partners. Only default risk is modeled, not downgrade risk. The following is assumed.

• For a loan, the probability of default in a given period is the same as that in any other comparable period.

• For a large number of obligors, the probability of default by any particular obligor is small, and the number of defaults that occur in any given period is independent of the number of defaults that occur in any other period (Crouhy et al 2000).

In the case of a portfolio with $N$ borrowers, the portfolio loss variable $X_{N}$ is defined as

 $X_{N}=\sum_{n=1}^{N}\mathrm{EAD}_{n}\times\mathrm{LGD}_{n}\times\mathrm{PD}_{n},$

where $X_{N}$ is loss, the amount that an institution is contractually owed but does not receive because of the default of the borrower or borrowers; $\mathrm{EAD}_{n}$ is exposure at default, the total amount of the institution’s liability to a borrower; $\mathrm{LGD}_{n}$ is loss given default, the fraction of the exposure that is actually lost given the default of the borrower; and $\mathrm{PD}_{n}$ is the probability of default, the binomially random variable that measures whether a borrower has defaulted or not. This takes the value one in the case of default, and zero otherwise.

As in the model by Ward and Lee (2002), we work with the credit loss rate, which is the total credit loss of the institution divided by the total exposure. Because the default is modeled as a Bernoulli and does not allow firms to default repeatedly without curing, the sum of a correlated portfolio of loans follows a beta distribution, with the result that $X=(X_{n}/X_{N})\in[0,1]$ (Ward and Lee 2002). We will consider the consumer portfolio model designed by the Superintendencia Financiera de Colombia (SFC), which is used in Colombia for the evaluation and supervision of internal models presented by financial institutions. In our study, the balance of the debt at the time of calculation was considered as the exposure at default. The probability of default was chosen from a rating system (Table 3) proposed by the SFC. The loss given default was chosen from the values by the SFC (Table 4) (Superintendencia Financiera de Colombia 2008).

### 4.1 The data

The data under consideration is a portfolio of more than 14 000 different loans with an average portfolio balance of USD41 000 000 in a Colombian financial institution over a period of 12 months (January 2019 to December 2019 inclusive). In Table 5, we show the exposure at default, the probability of default, the loss given default, credit loss and the credit loss rate obligor.

### 4.2 The implementation

After calculating the loss rate derived from the credit risk (last column of Table 5), the parameters of the beta distribution of each month analyzed were calculated in R. Having estimated the unknown parameters of the model, $\mathrm{VaR}_{\alpha}(X)$ and $\mathrm{CVaR}_{\alpha}(X)$ of the beta distribution can be easily calculated using our algorithm (see Section 2). We will also calculate the economic capital $(\mathrm{EC}_{\alpha}(X)=E(X)-\mathrm{VaR}_{\alpha}(X))$, which is a value that is commonly used in the context of credit risk and can be easily calculated with the help of $\mathrm{VaR}_{\alpha}(X)$. The results obtained were converted into monetary units to give us a better idea of the potential losses that the company could incur in each month of the year under consideration. From Table 6, it is possible to infer the following: on average, for the year 2019, the total expected credit loss was USD720 495 087; the maximum loss was USD2 185 686 058 with up to 95% confidence; the economic capital needed as a buffer against unexpected losses was USD1 465 190 972; and the expected loss amount beyond VaR at probability level 95% was USD4 787 439 041.

The motivation for using matrix-variate VaR measures can be exemplified below.

A financial institution generally offers different lines of credit according to the needs of its clients. Each line of credit is exposed to a random total loss, and these losses must be below the given level while simultaneously having a high probability.

If we were to consider a single line of credit and observe its total losses in subsequent periods of time, we would have a random vector for each line of credit and could therefore think of a matrix whose rows are the total losses per period of time and whose columns are the lines of credit. It is reasonable to require that subsequent portfolio losses must be below a given level with a high probability $\alpha$. In this case, the matrix-variate VaR provides us with complete information on the maximum loss that can occur in each period of time for each line of credit.

In order to ensure this, we look at the total losses per period of time as components of a random matrix. We find a matrix-variate VaR to know which points are in the $m\times r$-dimensional space (where $m$ is the number of periods and $r$ is the number lines of credit); these must exceed the matrix of total losses to guarantee the given level.

Now, suppose that we have the information on expected losses for two lines of credit in two periods of time and that we have this information for two different banks. Let $\bm{Y}_{1}$ and $\bm{Y}_{2}$ represent the losses associated with the first and second banks, respectively, and assume their amounts follow a multivariate gamma distribution. Let us now assume, just for the purposes of illustration, that $\bm{Y}_{1}\sim\varGamma_{2}(1,\bm{I}_{2})$ and $\bm{Y}_{2}\sim\varGamma_{2}(1,\bm{I}_{2})$. In this case, the random matrix $\bm{X}_{1}=\bm{Y}_{1}(\bm{Y}_{1}+\bm{Y}_{2})^{-1}$ represents the fraction of loss associated with the first bank, and $\bm{X}_{1}\sim\mathrm{Beta}_{2}(1,1)$. By Corollary 3.6, $\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})$ is a $2\times 2$ matrix such that $|\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})|=\alpha$.

If, for instance, $\alpha=0.95$, then

 $\overline{\textbf{VaR}_{\bm{\alpha}}}(\bm{X})=\begin{bmatrix}0.9747&0.0242\\ 0.0242&0.9753\end{bmatrix}.$

## 5 Concluding remarks

In the current literature, multivariate risk measures are related to a specific partial order, or to the property of the univariate risk measures that it is desirable to extend. From this perspective, in this work, we have considered the Loewner order and have developed a methodology to calculate univariate VaR seen as the zeros of a hypergeometric function. This methodology could be generalized to take advantage of the definition of a hypergeometric function of matrix argument.

We have developed computational procedures and analytical solutions to estimate the univariate risk measures, and, under parametric restrictions, some analytical expressions can be found. In other cases, we introduce a numerical algorithm that allows us to computate these risk measures. For the matrix case, we used the algorithms proposed by Koev and Edelman (2006), which calculate hypergeometric functions with matrix arguments.

The methodology developed here may be applicable to other distribution functions whose risk measures are defined in terms of hypergeometric functions, such as the Wishard, gamma and $F$ distributions. This is a topic of ongoing research.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

## Acknowledgements

The authors would like to thank the editor-in-chief and anonymous referees for careful reading and helpful suggestions. This work was supported by the Doctoral School of Mathematics, IT and Telecommunications, University of Toulouse, France, and the Doctorate in Modelling and Scientific Computing of University of Medellín, Colombia.

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