Copula and dependence measures

Suppose that a continuous bivariate random variable (X, Y), representing a joint risk, has marginal risk distribution function F_X and F_Y, respectively. Suppose also that U = F_X(X) and V = F_Y(Y) so that they are uniformly distributed over unit interval, i.e. $$ U,V\sim \mathcal{U}(0,1)$$. Copula C_U,V is a joint distribution function of (U, V) such that

Sklar’s theorem has shown us that the use of copula provides many choices of joint distribution function of (X, Y). The corresponding joint probability function f_X,Y of (X, Y) is given by

To measure dependence between X and Y, the dependence measures of Pearson’s ρ and Kendall’s τ are required. According to Schweizer and Wolff [37], such dependence measures are defined as

It is shown from Eq (5) that Pearson’s ρ_X,Y depends not only on copula but also on marginal of X and Y. Thus, Pearson’s ρ is invariant only under linear transformation. Meanwhile, from Eq (6), it is shown that Kendall’s τ_X,Y depends only on copula. In addition, τ_X,Y = τ_U,V. Thus, Kendall’s τ is invariant under both linear and nonlinear transformations. We may say that Kendall’s τ is copula-based dependence measure. Note that for parameter θ of copula C_U,V, Kendall’s τ_U,V is a function of θ i.e. τ_U,V(θ).

There are several copulas commonly used such as Archimedean and elliptical copulas. Based on Joe [38], examples of Archimedean copulas are Clayton, Gumbel and Frank with the following function and the corresponding density:

Clayton copula: $$ C_{U,V}^{\text{Clayton}}(u,v;\theta )={(u^{-\theta }+v^{-\theta }-1)}^{-\frac{1}{\theta }}$$, where θ ∈ (0, ∞). Its corresponding copula density is



Gumbel copula: $$ C_{U,V}^{\text{Gumbel}}(u,v;\theta )=exp\{-{[{(-lnu)}^{\theta }+{(-lnv)}^{\theta }]}^{\frac{1}{\theta }}\}$$, where θ ∈ [1, ∞). Its corresponding copula density is


where $$ g(u,v)={[{(-lnu)}^{\theta }+{(-lnv)}^{\theta }]}^{\frac{2}{\theta }-2}+(\theta -1){[{(-lnu)}^{\theta }+{(-lnv)}^{\theta }]}^{\frac{1}{\theta }-2}$$.

Frank copula: $$ C_{U,V}^{\text{Frank}}(u,v;\theta )=-\frac{1}{\theta }ln[1+\frac{({\mathrm{e}}^{-\theta u}-1)({\mathrm{e}}^{-\theta v}-1)}{{\mathrm{e}}^{-\theta }-1}]$$, where $$ \theta \in \mathbb{R}\in \backslash \left\{0\right\}$$. Its corresponding density is

Meanwhile, Gaussian and Student’s t copulas are examples of elliptical copulas. Their copula functions are

The dependence measures of Kendall’s τ for Archimedean and elliptical copulas are provided in Table 1. It is shown that Kendall’s τ may be expressed in copula parameter θ. In addition, copula parameter θ may also be expressed in Kendall’s τ, except for Frank copula.

Table 1

Dependence measures of Kendall’s τ for Archimedean and elliptical copulas.

Copula	Kendall’s τU,V(θ)	Parameter θ(τU,V)
Clayton(θ)	$$	$$
Gumbel(θ)	$$	$$
Frank(θ)	$$	$$
Gaussian(θ)	$$	$$
Student’s t(θ, ν)	$$	$$

Note: Kendall’s τ is expressed as function of copula parameter θ and such θ is expressed as function of Kendall’s τ. For Frank copula, its Kendall’s τ is depend on Debye function of $$ D_1\left(\theta \right)=\frac{1}{\theta }{\int }_0^{\theta }\frac{t}{{\mathrm{e}}^t-1}\mathrm{d}t$$ whose inverse function has no explicit form.

For equal value of Kendall’s τ, we present in Fig 1 the contour plot for the density of Archimedean and elliptical copulas. It may be observed that Clayton, Gumbel and Student’s t copulas perform tail dependence. In more detail, Clayton copula is appropriate for lower-tail dependent risks whilst upper-tail dependence may be captured by Gumbel copula. Furthermore, Student’s t copula displays symmetrical lower- and upper-tail dependence. Meanwhile, Frank and Gaussian copulas have symmetrical lower- and upper-tail independence.

Fig 1

Contour plot for the density of Archimedean and elliptical copulas.

The contours show their different dependence structure for equal Kendall’s τ = 0.6.

Copula selection

For a given risk data set, the challenging task is selecting the best copula which fits well to the data. We may do this by considering several criteria. One of them is Akaike Information Criterion (AIC) introduced by Akaike [39]. Suppose that a copula C_U,V(⋅, ⋅;θ) with its corresponding density c_U,V(⋅, ⋅;θ) has parameter θ. Based on data $$ {\left\{(u_i,v_i)\right\}}_{i=1}^n$$ of (U, V), the estimate for such parameter may be obtained through maximum likelihood (ML) method with likelihood function $$ \mathbb{L}\left(\theta \right)={\prod }_{i=1}^n{c}_{U,V}(u_i,v_i;\theta )$$. By replacing θ with its estimate, $$ \widehat{\theta }$$, we have $$ {\widehat{C}}_{U,V}(·,·;\widehat{\theta })$$ as the parametric estimate for C_U,V(⋅, ⋅;θ). The AIC value from such copula is defined as

In addition, we employ other criteria by considering empirical version of copula. The empirical copula $$ {\widehat{C}}_n$$, for the data $$ {\left\{(u_i,v_i)\right\}}_{i=1}^n$$, is defined as

Such empirical function is the nonparametric estimate of distribution function, $$ K(q;\theta )=\mathbb{P}(Q\le q)$$, for the so-called Kendall’s transform, Q = C_U,V(U, V;θ), defining Kendall’s τ in Eq 6. According to Genest and Rivest [40], this leads us to visualize the curve for $$ {\widehat{K}}_n\left(q\right)$$ and its corresponding function of λ, defined by

Pair-copula construction method

Suppose that (X, Y, Z) is a dependent risk model consisting of three continuous random variables with joint probability function

Note that

Since

Now, from Eqs (9) and (10), we obtain

This shows that the trivariate copula density c_{X,Y, Z} for (X, Y, Z) model may be constructed from bivariate copula densities: c_X,Y, c_X,Z, c_Y,Z|X. In other words, it provides flexible way to determine joint distribution of (X, Y, Z). The first two components c_X,Y and c_X,Z show dependence of (X, Y) and of (X, Z), respectively, with dependence measure κ_X,Y and κ_X,Z. Meanwhile, the component c_Y,Z|X shows partial dependence of (Y, Z), given X, with partial dependence measure κ_Y,Z|X. Note that the dependence measure of κ may be either Person’s ρ or Kendall’s τ.

Suppose that F_X, F_Y, F_Z denote marginal distribution functions of (X, Y, Z). The copula density c_{X,Y, Z} in Eq 11 is basically expressed as

This shows that conditional copula density c_Y,Z|X is determined through copula C_X,Y and C_Y,Z. From Eqs (12) and (13), copula density c_{X,Y, Z} is completely given by

According to Joe [38], function of C_Y|X for each copula C_X,Y, from Archimedean and elliptical copulas, is given in Table 2. Now, for u = F_X(x), v = F_Y(y) and w = F_Z(z), copula density c_{X,Y, Z} is given by

Table 2

Function of C_Y|X.

Copula	Function of CY|X(v|u)
Clayton	$$
Gumbel	$$
Frank	e−θu[(e−θ − 1)(e−θv − 1)−1 + (e−θu − 1)]−1
Gaussian	$$
Student’s t	$$

Note: The function of C_Y|X is derived for Archimedean and elliptical copulas.

Graph structure and vine (copula)

Suppose that V denotes an empty and finite set. Suppose also that [V]² = {{v, v′}:v, v′ ∈ V} is a set of all pairs of two unordered elements in V. According to Diestel [43], a graph is a pair (V, E) of sets for some E ⊆ [V]². The element of V is called node whilst the element of E is an edge. A simple graph $$ \mathcal{T}=(V,E)$$ that is connected and has no cycle is called a tree graph.

Based on Bedford and Cooke [44, 45] and Kurowicka and Cooke [46], $$ {\mathcal{V}}_m$$ is called vine on m elements, with $$ E\left({\mathcal{V}}_m\right)=E_1\cup E_2\cup \dots \cup E_{m-1}$$, if

$$ {\mathcal{V}}_m=\{{\mathcal{T}}_1,{\mathcal{T}}_2,...,{\mathcal{T}}_{m-1}\}$$;

$$ {\mathcal{T}}_1$$ is a tree graph with a set of nodes V₁ = {1, 2, …, m} and a set of (m − 1) edges E₁ ⊆ [V₁]²;

for all j = 2, 3, …, m − 1, $$ {\mathcal{T}}_j$$ is a tree graph with a set of nodes V_j = E_j−1 and a set of edges E_j ⊆ [V_j]².

In short, vine is a collection of nested trees where edge of the jth tree is a node of the (j + 1)th tree.

A vine $$ {\mathcal{V}}_m$$ is called a regular vine (R-vine) on m elements if “$$ {\mathcal{V}}_m$$ satisfies proximity condition, that is for all j = 2, 3, …, m − 1, if $$ v_j,v_j^{\prime }\in V_j$$ where $$ \{v_j,v_j^{\prime }\}\in E_j$$, then $$ |v_j\cap v_j^{\prime }|=1$$. In other words, two adjacent nodes in the tree graph $$ {\mathcal{T}}_j$$ are two adjacent edges in the tree graph $$ {\mathcal{T}}_{j-1}$$”. There are two special cases of R-vine that are drawable vine (D-vine) and canonical vine (C-vine). R-vine $$ {\mathcal{V}}_m$$ is a D-vine if all nodes in the tree graph $$ {\mathcal{T}}_1$$ have maximum degrees of 2. Meanwhile, $$ {\mathcal{V}}_m$$ is called a C-vine if the tree graph $$ {\mathcal{T}}_j$$ has exactly one node with degrees of m − j, for j = 1, 2, …, m − 1; in tree graph $$ {\mathcal{T}}_1$$, such node is called a root. For an illustration, it is shown in Fig 2 graph structure of D-vine and C-vine, for m = 4 and $$ {\mathcal{V}}_4=\{{\mathcal{T}}_1,{\mathcal{T}}_2,{\mathcal{T}}_3\}$$.

Fig 2

Graph structure of R-vine.

The R-vine is $$ {\mathcal{V}}_4=\{{\mathcal{T}}_1,{\mathcal{T}}_2,{\mathcal{T}}_3\}$$ as a (a) D-vine and as a (b) C-vine.

Now, an R-vine $$ {\mathcal{V}}_3$$ is employed to represent dependence model (X, Y, Z) determined through trivariate copula by pair-copula construction method. Here, node is a random variable whilst edge is a bivariate copula added by absolute dependence measure as weight. Such weight is used to find appropriate R-vine i.e. an R-vine where each tree graph has maximum of the sum of weights, according to Dißmann et al. [47].

First, define a complete graph $$ {\mathcal{K}}_3$$ (or a cricle graph $$ {\mathcal{C}}_3$$) with three nodes representing random variables of X, Y, Z. Let absolute value of each dependence measure κ_X,Y, κ_X,Z, κ_Y,Z be defined as weight of edge that connects two nodes. Then, select a maximum spanning tree graph, that is a tree subgraph of the complete graph $$ {\mathcal{K}}_3$$ maximizing the sum of weights. We denote the resulted R-vine copula by $$ (\mathbf{F},{\mathcal{V}}_3,\mathbf{C},\mathbf{K})$$, where F is a collection of marginal distribution functions of (X, Y, Z). Meanwhile, C and K consist of bivariate copulas and the corresponding dependence measures, respectively.

Individual VaR forecast

The one-step-ahead VaR forecast is actually a forecast(ing limit) of future risk or return, X_n+1, given previous information up to time n, $$ {\mathcal{F}}_{x;n}$$. At a specified confidence level of 1 − α, it may be calculated through the following formula, see e.g. McNeil et al. [4] and Syuhada [34],

Since we use standard Student’s t for innovation distribution and assume GARCH(1,1) for (negative) return model, the conditional distribution of X_n+1, given $$ {\mathcal{F}}_{x;n}$$, is Student’s t with degrees of freedom ν_x and scale parameter $$ {\sigma }_{x;n+1}\sqrt{\frac{{\nu }_x-2}{{\nu }_x}}$$. This implies that its conditional mean and variance are

Parameter of model, (ω_x, δ_x, β_x, ν_x), may be replaced by its estimate so that we have the estimative one-step-ahead VaR forecast given by

In addition, the estimative ℓ(>1)-step-ahead VaR forecast is as follows

Aggregate VaR forecast

For the case of a portfolio or aggregate risk, we aim to forecast VaR for future aggregate risk of

By employing vine copula, forecasting the one-step-ahead individual VaRs is simultaneously carried out based on the joint distribution of $$ (X_{n+1}^{*},Y_{n+1}^{*},Z_{n+1}^{*})$$. We collect all of them into a vector denoted by

Then, we may forecast the one-step-ahead aggregate VaR, $$ {\text{aggVaR}}_{s;n+1}^{1-\alpha }$$, by taking such individual VaR forecasts into account. We do this by first observing that

This means that the aggVaR forecast above incorporates interactions between different returns by introducing their dependence measures. Note that the estimative aggVaR forecast may be obtained by using the individual estimative VaR forecasts and the estimate of dependence measures.

For perfect (positive) dependence, i.e. κ_x,y = κ_x,z = κ_y,z = 1, it is obvious that the aggVaR forecast is equal to the simple sum of individual VaR forecasts written by

Thus, simplesumVaR may simply be decided as the aggVaR forecast when the worse cases of all of our risks always occur simultaneously, see e.g. Li et at. [49] and Embrechts et al. [2]. In other words, the risks have dependence representation of the form M(u, v, w) = min(u, v, w) which may be called perfect-dependence copula. However, since the dependence measures satisfy |κ_x,y|, |κ_x,z|, |κ_y,z|≤1, we have

This means that the aggVaR forecast in Eq (17) is bounded from above by simplesumVaR in Eq (18). The weaker the dependence among marginal risks, the lower the aggVaR forecast. This is inline with the fact that M(u, v, w) is the upper bound for all classes of copulas, see e.g. Joe [38], including pair-copula which determines the vine copula-based dependence among marginal risks for our portfolio risk.

In addition, it is interesting to investigate diversification benefits of using aggVaR and simplesumVaR. As in Li et at. [49], such diversification benefits may be measured by a so-called diversification coefficient (DC) defined as

From Eq (19), it may be understood that such coefficient is getting higher as the dependence among marginal risks is weaker. This makes the portfolio better diversified.

To find the ℓ-step-ahead aggregate VaR forecast, we consider the vector of ℓ-step-ahead individual VaR forecasts derived simultaneously. We denote it by

The calculation of DC for this aggVaR forecast is analogous to Eq (19) by involving the simple sum of individual VaR forecasts of Eq (20).

Probability-based backtesting

As stated before, the one-step-ahead VaR forecast is the forecasting limit of future risk, given information of risks in the past. To simply asses the accuracy of VaR forecast, we, therefore, may calculate the probability that the actual future risks do not violate the VaR forecast. It is called coverage probability (CP) defined by

The closer the CP to the confidence level, 1 − α, the more accurate the VaR forecast.

When the sequence of actual future risks, $$ {\left\{x_{n+1;k}\right\}}_{k=1}^N$$, is obtained, we may define $$ {\left\{I_{n+1;k}\right\}}_{k=1}^N$$, where $$ I_{n+1;k}={\mathbb{I}}_{\{x_{n+1;k}\le \widehat{\text{VaR}}{}_{x;n+1}^{1-\alpha }\}}$$, for k = 1, 2, …, N. We expect that $$ {\left\{I_{n+1;k}\right\}}_{k=1}^N$$ is the sequence of realizations of Bernoulli random variable with parameter 1 − α. Thus, the value of $$ \frac{1}{N}{\sum }_{k=1}^N{I}_{n+1;k}$$ is required to estimate the CP.

From the sequence of $$ {\left\{x_{n+1;k}\right\}}_{k=1}^N$$, we may also require the sequence of $$ {\left\{I_{n+1;k}^{*}\right\}}_{k=1}^N$$, where $$ I_{n+1;k}^{*}={\mathbb{I}}_{\{x_{n+1;k}>\widehat{\text{VaR}}{}_{x;n+1}^{1-\alpha }\}}$$, for k = 1, 2, …, N. Note that the term 1 refers to failure or violation of the VaR forecast; all terms are expected to be the realizations of Bernoulli random variable with failure proportion equal to α. Comparison of the actual number of failures, $$ {\sum }_{k=1}^N{I}_{n+1;k}^{*}$$, and the expected number of failures, Nα, may be considered to assess the accuracy of VaR forecast. Their ratio is called actual over expected (AE) ratio.

Conditional coverage test

The standard test for assessing the accuracy of VaR forecast is the test whether the failure proportion is equal to α. It is basically carried out through binomial test by considering normal approximation of binomial distribution. However, according to Christoffersen [50], the forecast accuracy may be assessed by determining whether the sequence of $$ {\left\{I_{n+1;k}^{*}\right\}}_{k=1}^N$$ consists of independent and identical realizations with failure proportion equal to α. This means that we need to combine the test of proportion of failure (PoF) and the test whether this sequence satisfies the conditional coverage independence (CCI). Their combination is called conditional coverage (CC) test.

The PoF and CCI tests employ likelihood ratio (LR) test statistic. The LR statistic for PoF test is defined by

Backtesting through loss function

It is known that VaR is basically the quantile of (conditional) distribution of the future risk. In addition to the use of inverse of its distribution function, VaR as quantile may be formulated through different approach. Based on Kuan et al. [51], $$ {\text{VaR}}_{x;n+1}^{1-\alpha }$$ is the minimizer of the loss function defined by

In this case, the loss function evaluated at our VaR forecast may be relatively compared to that evaluated at VaR forecast obtained from other method as benchmark. They are called quantile losses. The value of their ratio below one means that our forecasting method outperforms the benchmark.