2. Vine Copulas

Sklar (Reference Sklar1959) proved that for any multivariate distribution function, there exists a unique copula function $C$ that links the marginal distribution functions, given that those marginal functions are continuous. This theory leads to the convenience that the dependency structure and the marginal distributions can be selected separately. There are three specific aspects broadly defining the dependency structure, namely the overall shape, level of association, and tail dependence. For the bivariate case, there are a large number of choices, including the common ones like the Gaussian copula, t copula, and Archimedean copulas (e.g. Joe, Reference Joe1997; Nelsen, Reference Nelsen1999). For the multivariate case with three or more dimensions, however, there are severe limitations in their initial extensions. Notably, the multivariate Gaussian and t copulas and the multi-dimensional Archimedean copulas give the same shape of dependence and tail dependence for each pair of random variables. Some recently proposed solutions to overcome this inflexibility include the hierarchical (fully or partially nested) Archimedean copulas (e.g. Okhrin et al., Reference Okhrin, Okhrin and Schmid2013) and the vine copulas (e.g. Joe, Reference Joe2014; Czado, Reference Czado2019), the latter of which is adopted here.

Let ${X_1},{\rm{ }}{X_2},\ldots,{\rm{ }}{X_d}$ be a set of d-dimensional multivariate random variables. For instance, when $d$ = 4, the joint density function ${f_{1234}}$ can be expressed in terms of conditional and unconditional densities:

\begin{equation*}{f_{1234}}({x_1},{x_2},{x_3},{x_4}) = {f_{4|123}}({x_4}|{x_1},{x_2},{x_3}){f_{3|12}}({x_3}|{x_1},{x_2}){f_{2|1}}({x_2}|{x_1}){f_1}({x_1}),\end{equation*}

in which ${f_1}$ is the marginal density of ${X_1}$ , ${f_{2|1}}$ is the conditional density of ${X_2}|{X_1}$ , ${f_{3|12}}$ is the conditional density of ${X_3}|{X_1},{X_2}$ , and ${f_{4|123}}$ is the conditional density of ${X_4}|{X_1},{X_2},{X_3}$ . Using the concept of conditional probability $\Pr (A|B) = \Pr (A \cap B)/\Pr (B)$ for two dependent events $A$ and $B$ , the conditional densities above can be expressed for example as the following:

\begin{equation*}{f_{2|1}}({x_2}|{x_1}) = {c_{12}}({F_1}({x_1}),{F_2}({x_2})){f_2}({x_2}),\end{equation*}

\begin{equation*}{f_{3|12}}({x_3}|{x_1},{x_2}) = {c_{13|2}}({F_{1|2}}({x_1}|{x_2}),{F_{3|2}}({x_3}|{x_2})|{x_2}){f_{3|2}}({x_3}|{x_2}),\end{equation*}

\begin{equation*}{f_{3|2}}({x_3}|{x_2}) = {c_{23}}({F_2}({x_2}),{F_3}({x_3})){f_3}({x_3}),\end{equation*}

\begin{equation*}{f_{4|123}}({x_4}|{x_1},{x_2},{x_3}) = {c_{14|23}}({F_{1|23}}({x_1}|{x_2},{x_3}),{F_{4|23}}({x_4}|{x_2},{x_3})|{x_2},{x_3}){f_{4|23}}({x_4}|{x_2},{x_3}),\end{equation*}

\begin{equation*}{f_{4|23}}({x_4}|{x_2},{x_3}) = {c_{24|3}}({F_{2|3}}({x_2}|{x_3}),{F_{4|3}}({x_4}|{x_3})|{x_3}){f_{4|3}}({x_4}|{x_3}),\end{equation*}

\begin{equation*}{f_{4|3}}({x_4}|{x_3}) = {c_{34}}({F_3}({x_3}),{F_4}({x_4})){f_4}({x_4}),\end{equation*}

where $c$ refers to a copula density, $F$ is a distribution function, and the subscripts represent the dimensions and conditions involved. This process is called pair-copula construction (PCC). Note that the joint density function can also be expressed as ${f_{1234}}({x_1},{x_2},{x_3},{x_4}) = {c_{1234}}({F_1}({x_1}),{F_2}({x_2}),{F_3}({x_3}),{F_4}({x_4})){f_1}({x_1}){f_2}({x_2}){f_3}({x_3}){f_4}({x_4})$ . It can then be deduced that a multivariate copula ( ${c_{1234}}$ ) can be decomposed into conditional and unconditional bivariate copulas ( ${c_{12}}$ , ${c_{13|2}}$ , ${c_{23}}$ , ${c_{14|23}}$ , ${c_{24|3}}$ , ${c_{34}}$ ). As each bivariate copula can be selected separately from the others, there is a huge amount of flexibility for modelling the entire dependency structure. However, it is also clear that the derivation above is just one specific example, and there are actually numerous ways to decompose the multivariate copula. For higher dimensions, it is extremely tedious to deal with the different possible combinations.

Vines are a flexible graphical method that can be used for building multivariate copulas based on a cascade of bivariate copulas. A d-dimensional R-vine is a sequence of d – 1 linked layers of trees. Tree 1 has $d$ nodes ${N_1}$ and edges ${E_1}$ . Tree 2 has $(d - 1)$ nodes ${N_2} = {E_1}$ and edges ${E_2}$ , Tree 3 has $(d - 2)$ nodes ${N_3} = {E_2}$ and edges ${E_3}$ , and so on, i.e. an edge in a tree becomes a node in the next tree. There is a proximity condition such that for any two nodes connected by an edge in a tree, they must represent two edges connected to the same node in the previous tree. Figure 1 provides a graphical vine representation of the 4-dimensional example above. The edges 12, 23, and 34 represent the unconditional bivariate copula densities ${c_{12}}$ , ${c_{23}}$ , and ${c_{34}}$ , respectively. The edges 13|2, 24|3, and 14|23 represent the conditional bivariate copula densities ${c_{13;2}}$ , ${c_{24;3}}$ , and ${c_{14;23}}$ . In fact, it is a special case of R-vine called D-vine, where no node is connected to more than two edges (i.e. a path structure). Another special case, called C-vine, is shown in Figure 2, in which tree $i$ has a unique node that is connected to $(d - i)$ edges (i.e. a star structure). Including these two cases, there are indeed a total of $d!{2^{(d - 2)!/(d - 4)!/2 - 1}} = 24$ possible combinations. This way of drawing trees of nodes and edges make the decomposition more manageable, when compared to using the first principles to break down the joint density function asabove.

Figure 1

An example of a 4-dimensional D-vine.

Figure 2

An example of a 4-dimensional C-vine.

Theconstruction of an appropriate R-vine requires making suitable choices on the overall vine structure, the bivariate copulas, and the copula parameters. The state-of-the-art approach in the literature is: (1) compute the copula data using the IFM method; (2) select the spanning tree that maximises the sum of the absolute values of sample Kendall’s tau values across all connected pairs in that tree for each layer; (3) select the bivariate copula with the lowest AIC value for each pair; and (4) use maximum likelihood to estimate the parameters of the selected copulas sequentially. The so-called simplifying assumption is usually taken such that all the bivariate copulas do not depend on the conditioning values directly, though the arguments inside the copulas still depend on the conditioning valuesFootnote ¹. The conditional arguments can be calculated from the corresponding h-functionsFootnote ². More details on the estimation process can be found in Aas et al. (Reference Aas, Czado, Frigessi and Bakken2009) and Dißmann et al. (Reference Dißmann, Brechmann, Czado and Kurowicka2013).

Despite its practical convenience, this current standard approach has a number of potential drawbacks. Firstly, the pre-estimation of the margins may induce bias in the estimation of the copulas. The underlying covariance between the copula parameter estimators and marginal parameter estimators cannot be calculated. Also, the copula parameters are estimated sequentially, which may cause more bias in the higher trees where the parameters are computed later in the sequence. Secondly, the maximum spanning tree serves as a heuristic method to identify the optimal vine structure amongst numerous possible combinations. The identified structure, however, may not represent the truly optimal structure in probabilistic terms. Thirdly, it would be difficult to incorporate parameter and model uncertainties, as well as other relevant information such as experts’ opinions. Omitting these uncertainties and failing to use prior information could lead to a serious underestimation of the aggregate risk. In the next section, we will introduce our Bayesian framework to implement vine copula modelling. This Bayesian approach provides a sound and practical way to address these potential issues.

3. Bayesian Framework

Suppose ${\rm X}$ represents the vector of observations of $d$ multivariate random variables (monthly data breach events of the four breach types, i.e. d = 4), ${{\rm X}_f}$ is the future values, $\Theta $ denotes all the unknown parameters, and ${\rm M}$ refers to the model choice. Under the Bayesian setting, the joint distribution $p({{\rm X}_f},{\rm X},\Theta ,{\rm M})$ is equal to $p({{\rm X}_f},{\rm X}|\Theta ,{\rm M})p(\Theta |{\rm M})p({\rm M})$ . The primary goal is to derive the joint posterior distribution $p({{\rm X}_f},\Theta ,{\rm M}|{\rm X})$ , given the observations ${\rm X}$ . This joint posterior distribution can be used to deduce a variety of very useful Bayesian inferences via conditioning on and marginalising variables, such as the highest posterior probability of $p({\rm M}|{\rm X})$ (i.e. the “best” model), the posterior distribution of the unknown parameters $p(\Theta |{\rm X},{\rm M})$ given a model (or $p(\Theta |{\rm X})$ averaged across all the model candidates), and the predictive distribution of the future values $p({{\rm X}_f}|{\rm X},{\rm M})$ given a model (or $p({{\rm X}_f}|{\rm X})$ averaged across all the model candidates). A comparison between two models ${M_1}$ and ${M_2}$ can be conducted by examining the posterior odds $p({M_1}|{\rm X})/p({M_2}|{\rm X})$ , which is equal to the Bayes factor $p({\rm X}|{M_1})/p({\rm X}|{M_2})$ times the prior odds $p({M_1})/p({M_2})$ .

For complex models, it is generally difficult to derive a closed-form expression for the joint posterior distribution. With the rapidly advancing computing power nowadays, one can use Markov chain Monte Carlo (MCMC) simulation to approximate the joint posterior distribution, in which random samples are simulated from a Markov chain with its stationary distribution being equal to the posterior distribution. In this paper, we use the software JAGS (Plummer, Reference Plummer2017) to perform MCMC simulations. They use the Gibbs sampling (GS) method, which involves successive simulations from the full conditional posterior distribution of each variable in turn. It reduces the multi-dimensional simulation from the joint posterior distribution into a sequence of one-dimensional simulations. The programming language is user-friendly and is highly suitable for building various Bayesian modelsFootnote ³ .

In the literature, there is a long list of bivariate copulas to choose from. We consider the Clayton copula and its rotations, Frank, and Gaussian copulas as the major model choices. The Clayton copula function is $C({u_1},{u_2}) = {(u_1^{ - \theta } + u_2^{ - \theta } - 1)^{ - 1/\theta }}$ for $\theta\,>\,0$ , ${u_1} = {F_1}({x_1})$ , and ${u_2} = {F_2}({x_2})$ , and the 90°, 180°, and 270° rotations can be obtained from ${u_2} - C(1 - {u_1},{u_2})$ , ${u_1} + {u_2} - 1 + C(1 - {u_1},1 - {u_2})$ , and ${u_1} - C({u_1},1 - {u_2})$ , respectively. It has lower tail dependenceFootnote ⁴ (i.e. in the lower-left quadrant), and its rotations have tail dependence in the lower-right, upper-right, and upper-left quadrants. The Frank copula function is given as $C({u_1},{u_2}) = - \ln (1 + {\textstyle{{(\!\exp (\!-\theta {u_1}) - 1)(\!\exp (\!-\theta {u_2}) - 1)} \over {\exp (\!-\theta ) - 1}}})/\theta $ for $ - \infty\,<\,\theta\,<\,\infty $ . It has no tail dependence and is symmetric over all the four quadrants. The Gaussian copula function is expressed as $C({u_1},{u_2}) = {\Phi _2}({\Phi ^{ - 1}}({u_1}),{\Phi ^{ - 1}}({u_2}))$ , in which ${\Phi _2}$ is the distribution function of the standard bivariate normal distribution with association parameter $ - 1\,<\,\theta \,<\,1$ and ${\Phi ^{ - 1}}$ is the inverse distribution function of the standard normal. Like the Frank copula, it has no tail dependence and is symmetric. This list of copulas covers a variety of features which would be suitable for different dependency structures in the data.

To make a Bayesian selection between the model candidates for each pair of random variables, we express the effective copula density as a weighted average of those of the individual models, i.e. $c({u_1},{u_2}) = \sum {{w_j}{c_j}({u_1},{u_2})} $ , in which the weight ${w_j}$ is equal to one when the corresponding copula density ${c_j}$ is chosen and is zero otherwise. The highest posterior probability of $p({\rm M}|{\rm X})$ (reflected by the highest ${\rm E}({w_j}|{\rm X})$ ) would then indicate the most optimal model from a Bayesian perspective. The predictive distribution of the future values can be obtained from either $p({{\rm X}_f}|{\rm X},{\rm M})$ or $p({{\rm X}_f}|{\rm X})$ , depending on the purpose of the analysis. The former (Bayesian model selection, BMS) can be used when the purpose is to identify a single model which provides key insights into the nature of the data or to heuristically collect a few promising models for further examination, while the latter (Bayesian model averaging, BMA) can be adopted if one wants to incorporate more model uncertainty into the prediction. Note that when the data patterns are complex, the model with the highest posterior probability from a finite list of models may not sufficiently represent the true underlying mechanism, and accordingly, one may prefer to use the model averaging approach (over all the models covered or just a few selected models).

Regarding the prior $p({\rm M})$ , a popular choice for model selection is the discrete uniform prior. It is an uninformative prior in the sense that all the model candidates are being favoured equally before the analysis. Since we do not have other information relevant for the data breach events, we employ this uninformative prior in our study. For the prior $p(\Theta |{\rm M})$ , we assume uninformative uniform priors for the copula parameters and uninformative gamma priors for the marginal parametersFootnote ⁵ . In the absence of subjective inputs or experts’ opinions, setting uninformative priors would minimise the influence of the prior assumptions. Moreover, we assume that the priors are independent with one another, as it is often difficult to understand how they are associated a priori.

4. Data

We collect 9,015 data breach events from 2005 to 2019 from the Privacy Rights Clearinghouse (PRC, https://privacyrights.org/) and focus on the non-zero monthly events of four breach types including hacking/malware (H), loss of physical records of data (P), loss of electronic devices containing sensitive data (E), and other unintended disclosure (D)Footnote ⁶ . A frequently used database for empirical cyber risk studies (e.g. Edwards et al., Reference Edwards, Hofmeyr and Forrest2016; Eling & Loperfido, Reference Eling and Loperfido2017; Eling & Jung, Reference Eling and Jung2018), the PRC database stores information of publicly verifiable breach events since 2005 that encroach on the privacy of citizens of the United States. The loss resulting from such incidents is measured by the number of personal records being breached. The breach may occur in any type of organisation or business entity ranging from government bodies to listed companiesFootnote ⁷ . Currently, it is the most comprehensive and biggest data breach database that is free of charge and accessible to the general public. For the purpose of this analysis, we sum up the losses arising from different incidents that are under the same breach type and have happened in the same month and fit our Bayesian model to these monthly aggregated losses. As shown in Table 1, the standard deviations of the losses far exceed the means of the losses. It highlights the critical importance of assessing adequately the level of uncertainty of data breach losses for risk management purposes.

Table 1.

Summary statistics of PRC monthly data breach events from 2005 to 2019.

Figure 3 displays the relationships between the D and H losses and between the P and E losses in logarithmic scale. There is a positive relationship ( $\tau \,=\,0.18$ ) between the H and D losses, which seem to have some form of lower tail dependence (as circled in the graph; non-parametric estimate of 0.34; e.g. Frahm et al., Reference Frahm, Junker and Schmidt2005). By contrast, there is a negative relationship ( $\tau = - 0.04$ ) between the P and E losses, with some tail dependence in the upper-left quadrant (non-parametric estimate of 0.18). These patterns clearly call for different copula functions to be used for the two pairs of random variables and highlight the importance of having the flexibility offered by the vine copula structure under which a specific bivariate copula can be selected for each pair of random variables. For instance, the association between the H and D losses looks like what the Clayton copula can describe, while the association between the P and E losses may be modelled by a suitable rotation of the Clayton copula. Note that these are only very preliminary observations, as the final optimal copula choices depend on formal Bayesian inferencing (Section 5) and also the tree level involved. The variables in the higher trees are conditioned on a larger number of other variables, which would then dilute or distort some of the original patterns in the raw data.

Figure 3

The left graph plots the hacking/malware (H) losses against other unintended disclosure (D) losses; the right graph plots the losses of electronic devices containing sensitive data (E) against the losses of physical records of data (P). The losses are in logarithmic scale.

4.1. GB2 as marginal distribution

The losses of each breach type appear to have a right-skewed and long-tailed distribution. For the margins, we use the generalised beta type II (GB2) distribution (e.g. Chan et al., Reference Chan, Choy, Makov and Landsman2018). The GB2 density function is

\begin{equation*}f(x) = \frac{{\Gamma (p + q)}}{{\Gamma (p)\Gamma (q)}}\frac{{{\textstyle{{|a|} \over b}}{{({\textstyle{x \over b}})}^{ap - 1}}}}{{{{(1 + {{({\textstyle{x \over b}})}^a})}^{p + q}}}},\end{equation*}

in which $x\,>\,0$ , $ - \infty\,<\,a\,<\,\infty $ , $b\,>\,0$ , $p\,>\,0$ , and $q\,>\,0$ . The distribution function can be expressed as

\begin{equation*}F\left( x \right) = {F_{{\rm{Beta}}}}\left( {\frac{{{{\left( {{\textstyle{x \over b}}} \right)}^a}}}{{1 + {{\left( {{\textstyle{x \over b}}} \right)}^a}}}} \right),\end{equation*}

where ${F_{{\rm{Beta}}}}$ is the distribution function of the ${\rm{Beta( }}p,{\rm{ }}q)$ distribution. The GB2 distribution nests a range of commonly used loss distributions as special cases, such as gamma ( $a\,=\,1$ , $b = \beta q$ , $q \to \infty $ ), Weibull ( $b = \beta {q^{1/a}}$ , $p\,=\,1$ , $q \to \infty $ ), lognormal ( $a \to 0$ , $b = {(a\sigma )^{2/a}}{q^{1/a}}$ , $p = {\textstyle{{a\mu \,+\,1} \over {{{(a\sigma )}^2}}}}$ , $q \to \infty $ ), and Pareto ( $a\,=\,1$ , $p\,=\,1$ ). In contrast to the usual frequentist approach of testing a finite set of traditional loss distributions on the data, incorporating the GB2 distribution into the Bayesian framework can take into account the uncertainty in selecting the model distribution, as the GB2 distribution itself generalises a big family of important distributions. In this way, model uncertainty can be integrated coherently within the Bayesian framework and there is no need to choose one (amongst gamma, Weibull, lognormal, Pareto etc.) explicitly based on a certain goodness-of-fit test.

As a preliminary analysis, we fit the GB2 distribution to each breach type separately (using maximum likelihood as a starting point). All the resulting chi-square test values for testing the goodness-of-fit are found to be not statistically significant (p-values of 0.13 (H), 0.10 (P), 0.57 (E), 0.29 (D)) and so the null hypothesis is not rejected. Moreover, Figure 4 shows that the histograms of the copula data (between 0 and 1) from the fitted GB2 distributions under the IFM method look fairly like the uniform distribution, which again suggests that the GB2 is a reasonable distribution for the data.

Figure 4

Histograms of copula data from fitted GB2 distributions under IFM method (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure). The parameter estimates are a = 0.65, b = 6.84, p = 1.61, q = 0.64 for H, a = 1.00, b = 26.79, p = 0.44, q = 1.20 for P, a = 0.28, b = 22.30, p = 4.78, q = 5.69 for E, and a = 1.60, b = 9.40, p = 0.31, q = 0.37 for D.

Figure 5

Autocorrelations of selected copula’s parameter between D and P (left) and cross-correlations between selected copulas’ parameters of DP and DH (right) (H: hacking/malware; P: loss of physical records of data; D: other unintended disclosure).

5. MCMC Simulation Results

For each chain of our MCMC simulations, we omit the first 60,000 iterations and apply a thinning of 40 (i.e. sampling every 40th iteration) when collecting the posterior samples. The resulting 10,000 samples are used for obtaining the required inferences. We notice that the autocorrelations and cross-correlations over successive iterations are negligible and there is no issue in convergence to the stationary distribution. For instance, Figure 5 illustrates that the autocorrelations of the selected copula’s parameter between the D and P losses and also the cross-correlations between the selected copulas’ parameters of the edges DP and DH are insignificant. All the estimated Monte Carlo errors are also within 2% of the sample standard deviations, which further indicate that the level of convergence is satisfactory. Moreover, the results given by different chains with reasonably spaced initial values are very close. It means that the simulations are generally robust to the selection of initial values.

Figure 6 demonstrates the vine copula structure with the lowest Deviance Information Criterion (DIC) value. The DIC is calculated as the posterior mean of $ - 2\ln p({\rm X}|\Theta )$ plus the effective number of parameters. The most optimal structure, amongst all the 24 possible R-vines, turns out to be a C-vine (DIC = 6,941). In Tree 1, D is the unique node connected to the other three nodes. In Tree 2, the edge DH (bivariate copula between D and H) from Tree 1 becomes the unique node. Table 2 gives the posterior mean of the weight ${w_j}$ of bivariate copula density ${c_j}$ for each edge in each tree. In Tree 1, the highest weights (italic figures in Table 2) of the edges DP and DH come from the Clayton copula, and the highest weight of DE comes from the 90° rotation. In Tree 2, the highest weights of HP|D and HE|D belong to the Clayton copula and the 90° rotation, respectively. In Tree 3, the highest weight of PE|DH arises from the 270° rotation. These copulas with the most weights can be seen as the “best” models under Bayesian inferencing. This approach of Bayesian copula selection is more unified with the entire estimation process, when compared to the ad hoc approach of using a standard statistical measure like the AIC to choose the pairwise copulas consecutively.

Figure 6

The most optimal R-vine (C-vine) in terms of DIC (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure).

There are a number of interesting observations on the posterior estimates. First, the final choices for DH in Tree 1 (Clayton) and PE|DH in Tree 3 (270° rotation) are broadly in agreement with the graphs in Figure 3. The former has lower tail dependence while the latter has tail dependence in the upper-left quadrant. Moreover, in Tree 1, the choices are quite certain as the highest weights are 0.78 to 0.86, but in Tree 3, the highest weight is only 0.51 and there is a second highest weight of 0.26. There are a few potential reasons behind the less certain choice in Tree 3. Generally speaking, at the higher trees, the variables are conditioned on a larger number of other variables, which may mix up or dilute the patterns. They tend to have lower dependence and so the effects between fitting different copulas would become less distinct. Furthermore, as shown in Figure 3 (right graph), there seems to be some form of upper tail dependence (opposite to lower tail dependence) as well, which would also make the 180° rotation a competitive candidate. The tail dependence properties of the copula choices for DP, DH, and DE are broadly in line with the coefficients of tail dependence (lower tail of 0.22, lower tail of 0.34, bottom-right of 0.28, respectively) estimated non-parametrically (e.g. Schmidt & Stadtmüller, Reference Schmidt and Stadtmüller2006). Table 3 provides the posterior means of the parameters of the selected copulas from $p(\Theta |{\rm X},{\rm M})$ (i.e. BMS). It is worth noting that while the lower trees appear to have stronger relationships (larger implied $\tau $ ’s from calculated $\theta $ ’s), in line with the rationale of the maximum spanning tree in pulling together those with stronger associations first, there is an exception of HE|D in Tree 2, which has the largest implied $\tau $ (= −0.25) in magnitude. This result pinpoints that though the maximum spanning tree is a convenient and practical method to find a reasonable vine structure, the resulting choice may not be the truly optimal structure overall, at least in terms of the DIC being used here. Dißmann et al. (Reference Dißmann, Brechmann, Czado and Kurowicka2013) also commented that the maximum spanning tree method does not necessarily lead to the global optimum (e.g. likelihood, AIC). Further evidence on this issue using Bayesian inferencing is provided in Section 6.

Table 2.

Posterior means of weights of bivariate copula candidates for each edge in each tree in the most optimal vine copula structure (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure).

Table 3.

Posterior estimates (mean, median, standard deviation, tail quantile, correlation, Kendall’s tau) of parameters of selected copulas in the most optimal vine copula structure (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure).

Table 3 also contains the posterior medians, standard deviations, and tail quantiles of the copula parameters, and the corresponding correlations and Kendall’s tau estimates between the parameters. The posterior distributions of the copula parameters are largely symmetric. Allthe parameters are significantly different from zero (based on the t test at 5% significance level). Since the maximum likelihood estimates correspond roughly to the posterior modes under uniform priors, they are also included in the table for comparison. While their relative sizes are similar to those of the posterior means and medians, they are all smaller by about 0.1 or more. As these maximum likelihood estimates are obtained from the IFM method and then the usual sequential estimation method, the extent that they are smaller would be a reflection of the biases being introduced in the multi-step estimation process. From a Bayesian perspective, it is also desirable to estimate all the parameters jointly such that the associations amongst the parameters can also be assessed. In Table 3, a few posterior correlations are significantly different from zero. This kind of dependence should then be taken into account when using the calibrated model to simulate the future values.

Table 4 shows some additional test results on the goodness-of-fit of the selected copulas in Table 2 above, based on the posterior means of the parameters (BMS), but outside the Bayesian system. The tests include a bivariate version of the Kolmogorov–Smirnov test and the AIC, as follows:

\begin{equation*}\textrm{KS} = \max_j | \widetilde{\rm P}\textrm{r} (X_1 \le x_{1,j}, X_2 \le x_{2,j}) - \widetilde{C}(u_{1,j}, u_{2,j})| \end{equation*}

\begin{equation*}{\rm{AIC}} = - 2\hat l\,+\,2, \end{equation*}

where $\widetilde{P}r(\!\cdot\!)$ denotes the empirical probability, $\widetilde{C}$ represents the selected copula, u _1,j = F ₁(x _1,j ), u _2,j = F ₂(x _2,j ), and $\hat l$ is the computed log-likelihood. The ranking of the test statistic value of each selected copula amongst the initial list of copula choices is given in brackets. The AIC values agree that the selected copulas under the Bayesian framework are the best choices. On the other hand, while the Kolmogorov–Smirnov test values indicate that the selected copulas under the Bayesian framework are one of the two best choices for four of the pairs, the values for DE and PE|DH suggest otherwise. These differences may be caused by the fact that these two pairs have weaker relationships than the other pairs and so their patterns may not be as clear-cut as the others, leading to less consistent conclusions between different statistical criteria. Moreover, this test requires the copula data obtained from the IFM method, which means that the consideration of whether a copula is suitable or not is separate from the consideration of the margins. This view is different to the rationale of coherent selection and estimation within our Bayesian framework. Nevertheless, all the test values are statistically insignificant and none of the selected copulas are rejected under the Kolmogorov–Smirnov test.

Table 4.

Goodness-of-fit test statistics (Kolmogorov–Smirnov and AIC; rankings amongst six copula candidates in brackets) of selected copulas in the most optimal vine copula structure (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure).

We now examine an application to risk management to see the impact of using an appropriate vine copula. Taking reference of the Solvency II regulations in Europe, we calculate the value-at-risk (VaR) of the monthly total data breach losses (aggregating events from all the four breach types into one portfolio)Footnote ⁸ . While the selected copulas in Table 2 are fairly dominating and so using $p({{\rm X}_f}|{\rm X},{\rm M})$ (i.e. BMS) appears to be reasonable, the results from using $p({{\rm X}_f}|{\rm X})$ (i.e. BMA) are also obtained for comparison. Moreover, a few other different assumptions are also included in the analysis. The first is simply assuming the four breach types are independent of one another. The second is adopting a multivariate Gaussian copula as well as a multivariate t copulaFootnote ⁹ , which are restrictive in the sense that they impose the same shape of dependence and tail dependence across all pairs of breach types. Table 5 sets forth the posterior estimates of the risk measures under these five assumptions. First, assuming independent data breach types ignores the underlying relationships and clearly underestimates the VaR. While the multivariate Gaussian copula allows for the relationships between the losses, it does not take tail dependence into account and so the resulting risk measures are still smaller than the others. Compared to the multivariate Gaussian copula, the multivariate t copula, with tail dependence, gives larger estimates only by some extent, probably because it still lacks the flexibility of dealing with each pair differently and the degree of freedom is 14 on average which is quite large. The 99.5% VaR estimates under the BMS produced by our Bayesian model are significantly larger than the three sets of estimates above, highlighting the critical importance of catering for the dependency structure properly when assessing the extreme tail. Further, it is interesting to see that the Bayesian model under the BMA gives the largest 99% and 99.5% VaR estimates. This difference reflects a substantial impact of incorporating copula model uncertainty into the frameworkFootnote ¹⁰ . It suggests that the practice of choosing a copula based on the AIC or similar measures and then relying solely on its results could lead to a significant underestimation of the aggregate risk. Figure 7 plots the extreme right tails of the simulated posterior (kernel) densities of the monthly total data breaches (in logarithmic scale) under the five assumptions. All the five posterior distributions appear to be skewed to the right. The BMA and BMS ones have the thickest tails, followed by those from the multivariate t, multivariate Gaussian, and then independence copulas.

Table 5.

Posterior estimates of monthly value-at-risk for a portfolio aggregating events of all four breach types under five different dependence assumptions, including BMS, BMA, multivariate t, multivariate Gaussian, and independence copulas.

Figure 7

Simulated posterior densities of monthly total data breaches under five different dependence assumptions, including BMS, BMA, multivariate t, multivariate Gaussian, and independence copulas.

Under Solvency II in Europe, all of process error, parameter error, and model error must be taken into account sufficiently. For instance, QIS5 Technical Specifications (CEIOPS, 2010) stated that “it is important to assess the model error that results from the use of a given valuation technique.” It described model error and parameter error as the “uncertainty which is related to the measurement of the risk and is not an intrinsic property of the risk.” It would be inadequate to simply use the empirical or fitted distribution to assess the aggregate tail risk, which then includes only the inherent risk (process error). This full allowance for all uncertainties is particularly important when data are scarce – in our analysis, not just that there are merely about 123 monthly data points, but also that data breaches and cyber losses represent a new kind of insurance opportunities, the development of which is still at its infant stage. Ignoring the potential impact of model uncertainty and parameter uncertainty for a new market is a significant omission in risk management and capital assessment under the current regulations in Europe. Over time, as more data can be collected and more other information can be obtained about data breaches and cyber losses, more confidence can be given to the modelling process and the extent of model error and parameter error would then naturally be reduced.

6. Sensitivity Testing

In this section, we perform a range of sensitivity tests on the Bayesian modelling results discussed in the previous section. Firstly, we experiment with a number of different priors for the copula parameters and the copula selection. While the uniform distribution is often a reasonable default choice, it would be informative to explore the consequences of taking other possible alternatives. Table 6 compares the posterior means and standard deviations of the copula parameters (BMS) obtained from using uninformative logistic, normal, or Cauchy priorsFootnote ¹¹ instead against the previous estimates. Another alternative of assuming uninformative uniform, logistic, normal, or Cauchy priors for Kendall’s tau and then converting it into the copula parameter is also tested.

Table 6.

Posterior estimates (mean, standard deviation) of parameters of selected copulas from eight different priors, using the most optimal vine copula structure (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure).

It can be seen from Table 6 that the posterior means and standard deviations are fairly consistent across different prior assumptions. It means that our Bayesian framework is reasonably robust to the prior settings and that the priors used are adequately vague priors. This result is also reasonable in the sense that as the selected copulas are fairly dominating, the MCMC simulation would give similar results for any reasonable prior choices.

Moreover, we examine the effects of using unequal prior probabilities of $p({\rm M})$ rather than the discrete uniform distribution. Table 7 compares the posterior means of the weights under different settings of prior probabilities. It is observed that when much larger prior probabilities are given to the other copula candidates, the final posterior weights of those pairs with stronger relationships (all of Tree 1 and most of Tree 2 cases) still indicate the same optimal choices as previously, though the weights become a little lower. Despite the reasonable robustness of our results, it is important to note that certain prior information can be incorporated into a more informative prior if it is available and relevant. For instance, a high prior probability can be allocated to a particular copula if some other related data of similar nature but with a bigger size demonstrate the kind of dependence possessed by that copula.

Table 7.

Posterior means of weights of selected copulas (in brackets) for each edge in each tree in the most optimal vine copula structure, using different sets of prior probabilities (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure).

We then compare the situations when model uncertainty and parameter uncertainty do not exist. The former can be removed by using the selected copulas directly without allowing for Bayesian copula selection. The latter condition can be created by removing the prior distributions and treating the parameter estimates as fixed values. Table 8 shows the simulated estimates of the above-mentioned risk measures under the three ways of allowing for uncertainties. Parameter uncertainty clearly makes the most significant contribution to the aggregate risk, as revealed by the much smaller calculated risk measures when it is excluded. Model uncertainty also plays an important part, though to a lesser extent. Insurance loss data are usually limited in size, and it is highly inappropriate to overlook the substantial impact of parameter and model uncertainties.

Table 8.

Simulated estimates of monthly value-at-risk for a portfolio aggregating events from all four breach types under three different ways of allowing for uncertainties, including BMS and BMA, having model error removed, and having both parameter error and model error removed.

We also allow the prior weights of the copulas to be a continuous variable between zero and one instead (with the sum of all the prior weights being one). In this way, the effective copula density becomes a “mixture” of different copula densities, which can offer more flexibility for modelling the dependency structure. Table 9 gives the posterior means of the weights under the new prior assumption. It is interesting to see that while the copulas with the largest weights (italic) are the same as before (except Tree 3, marginally), the weights are more spread out across different candidates. It is a clear sign of this alternative setting being able to exploit and mingle the various features of the different copulas to cope with the data complexity. However, the resulting DIC value increases to 6,956, which implies that the enhanced model sophistication cannot be justified by the better fitting.

Table 9.

Posterior means of weights of bivariate copula candidates for each edge in each tree in the most optimal vine copula structure, using mixtures of copulas (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure).

We further incorporate the selection of vine structure directly into the Bayesian framework. First, we choose the vine structures with the two lowest DIC values and also the optimal structure from the maximum spanning tree. Then, we express the effective joint density as a weighted average of those of the three structures, i.e. $f({x_1},{x_2},{x_3},{x_4}) = \sum {{w_j}{f_j}({x_1},{x_2},{x_3},{x_4})} $ , where the weight ${w_j}$ is equal to one if the corresponding joint density ${f_j}$ is selected and is zero otherwise. From a Bayesian perspective, the best vine structure would have the highest posterior probability. The posterior means of the weights of the three vine structures (from two lowest DICs and maximum spanning tree) included are 0.54, 0.46, and 0, respectively, and the results of which are consistent with those from the use of DIC values (6,941, 6,942, 6,946). (Based on maximum likelihood, given the margins, their corresponding log-likelihood values are 17.69, 16.77, 15.33; the AIC values are −23.37, −21.54, −18.65; and the BIC values are −6.50, −4.67, −1.78, respectively.) This test reemphasises the fact that the maximum spanning tree may not identify the truly optimal structure, despite being a convenient heuristic method which is widely applied.

We also replace the GB2 distribution with a mixture of the gamma, Weibull, lognormal, and Pareto distributions, using the same rationale in the copula selection and vine structure selection above. Both dummy prior weights (0 or 1) and continuous prior weights (0 to 1) are tested. Correspondingly, Table 10 provides the resulting posterior means of the weights of different marginal distributions for each breach type. Amongst the four two-parameter distribution candidates, the Pareto or Lognormal distribution is clearly the optimal choice when the prior selection can only be one distinct distribution. However, when a mixture distribution is allowed in the prior, the posterior weights of the individual candidates are more dispersed (except for Weibull). Interestingly, the optimal choices under the two prior assumptions match with each other for the P and D losses but not for the H and E losses. Furthermore, we realise that the DIC value of the latter is a lot smaller than that of the former. These results imply that the two-parameter distributions may be insufficient to capture all the salient features of the data breach losses. Practitioners trying to model cyber losses with two-parameter distributions should thus exercise extra caution, especially for pricing and reserving purposes, as the losses may then be substantially underestimated. The consequent effects on the vine copula model are also examined. Although the best copula choices remain the same as before, the copula parameter estimates (BMS) are significantly, systematically smaller under the dummy prior weights. This deviation further confirms the insufficiency of using the two-parameter distributions for analysing the data breach events in this paper.

Table 10.

Posterior means of weights of marginal distribution candidates based on dummy prior weights and continuous prior weights, using the most optimal vine copula structure (H: hacking/malware; P: loss of physical records of data; E: loss of electronic devices containing sensitive data; D: other unintended disclosure).

Furthermore, we incorporate a time trend into the GB2 distribution in order to allow for possible changes in data breach sizes over time. Since the mean of the GB2 distribution is equal to $b{\textstyle{{\Gamma \left( {p + {\textstyle{1 \over a}}} \right)\Gamma \left( {q - {\textstyle{1 \over a}}} \right)} \over {\Gamma (p)\Gamma (q)}}}$ , we set $b(t) = b(t - 1) + \mu + \varepsilon (t)$ , in which $b(t)$ becomes a time-varying parameter, $\mu $ is the drift term, and $\varepsilon (t)$ is the normal error term (with zero mean) at month t. Uninformative uniform, normal, and inverse gamma priors are used for $b(1)$ , $\mu $ , and the error term’s variance, respectively. Approximate t test statistics are then calculated on $\mu $ for the four breach types via dividing the posterior means by the posterior standard deviations. It turns out that the time drifts of the P and E losses are statistically significant while those of the D and H losses are not. At the same time, the posterior means of some copula parameters are somewhat smaller than previously. It appears that some of the potential relationships between the data breach losses across time are now captured by the newly added temporal trends and accordingly the levels of association reflected in some of the copulas are reduced.

We also experiment with adding a time trend into the dependency structure to allow for possible changes in dependency over time. As an illustration, we assume a time-varying $\tau = {\textstyle{{\exp (2s) - 1} \over {\exp (2s)\,+\,1}}}$ for HE|D in Tree 2 (with the largest copula parameter estimate in magnitude in Table 3), where $s(t) = s(t - 1) + \mu + \varepsilon (t)$ , $\mu $ is the drift term, and $\varepsilon (t)$ is the normal error term at month t. The priors are set in the same way as above. Based on the t test statistic calculated (−1.37), the time drift is not statistically significant, which means that there is no need to use a dynamic copula in this case.

Finally, we test two simplified vine structures (i.e. truncated vines), one of which has the variables set simply as independent in Tree 3, and the other having all the variables in Trees 2 and 3 being independent with one another (e.g. Brechmann et al., Reference Brechmann, Czado and Aas2012). Their DIC values are 6,942 and 7,051, which are higher than that of the optimal full model above (6,941), suggesting that these two vine structures are rather over-simplified and comparatively not as suitable for modelling the breach data.

7. Further Analysis on Department of Health and Human Services (HHS) Data set

In this section, we study a new data set containing healthcare data breaches from 2009 to 2020, obtained from the ongoing public database maintained by the U.S. Department of Health and Human Services (HHS). Recently, this database has been analysed by McLeod and Dolezel (Reference McLeod and Dolezel2018) in the field of cyber-analytics. There are 3,241 data breach incidents reported during the investigation period with the incident description, number of individuals affected, and type of breach made known to the public. We examine the severity of the monthly data breach losses in terms of the number of affected individuals from all the incidents reported in the month by four major breach types, as well as the dependency structure between the different breach types. The four major breach categories include the losses that are mainly caused by theft (T), hacking (H), unauthorised access (U), and mishandling of documents (M). The category for mishandling of documents includes incidences that can be mainly described as either loss of documents or improper disposal of documents. Essentially, the first three data breach categories are concerned with malicious breach attempts, whereas the last category covers negligence in handling confidential or sensitive information. Categorisation of the incidents depends on both the type of breach as specified in the database and incident description provided in the database. Altogether, the number of incidents covered by the four categories accounts for 95% of all the data breach events reported during the study period. Table 11 provides a summary of the descriptive statistics of the monthly losses by breach type. Again, the standard deviations of the losses are much larger than the means, which suggest a high level of uncertainty in the future outcomes. Compared to the summary statistics in Table 1, however, the coefficients of variation of the losses in the HHS data tend to be smaller than those of the PRC data. For both data sets, the hacking breach type contributes to a larger number of losses than the other types of losses, indicating that hacking is a very serious issue in data security.

Table 11.

Summary statistics of HHS monthly data breach events from 2009 to 2020 (T: theft; H: hacking; U: unauthorised access; M: mishandling of sensitive/confidential documents).

The most optimal structure for the HHS data set is a C-vine (DIC = 1,593). In Tree 1, T is the unique node linked to the others. In Tree 2, the edge TU from Tree 1 becomes the unique node. Table 12 provides the posterior weight of bivariate copula density for each edge in each tree. In Tree 1, the highest weights (italic figures in Table 12) of TH, TU, and TM come from the Frank copula, Gaussian copula, and 90° Clayton rotation. In Tree 2, the highest weights of HU|T and UM|T belong to the Clayton copula and the 270° rotation. In Tree 3, the highest weight of HM|TU arises from the Gaussian copula. It can be seen that different pairs of random variables call for different copula choices under the Bayesian setting.

Table 12.

Posterior means of weights of bivariate copula candidates for each edge in each tree in the most optimal vine copula structure (T: theft; H: hacking; U: unauthorised access; M: mishandling of sensitive/ confidential documents).

There are some similarities between the HHS results here and the PRC results in Section 5. First, the optimal structure is a C-vine in both cases, while the other structures like a D-vine are suboptimal. Second, the hacking breach type, despite having the largest number of losses, is not chosen as the unique node in Tree 1 under the Bayesian framework for both data sets. It reflects that whether a breach type would be selected as the unique node depends mainly on how it is associated with the other breach types but less on its size. Nevertheless, there are significant differences in the components of the two vines. Firstly, all the selected copulas for the PRC data have tail dependence, but only half of the selected copulas for the HHS data have tail dependence. Moreover, the posterior weights of the selected copulas are lower for the HHS data than for the PRC data. These results are in line with the fact that the HHS data have comparatively fewer extreme losses than the PRC data do. For instance, as noted earlier, the coefficients of variation of the losses in the HHS data are smaller. So, using the HHS data, those copulas with tail dependence would be less needed while the other copulas without tail dependence would become more competitive in the Bayesian selection process.

Table 13 gives the posterior means, medians, standard deviations, and tail quantiles of the parameters of the selected copulas from the BMS. The copula parameters are significantly different from zero, except for TM and HM|TU. As a comparison, the corresponding maximum likelihood estimates are quite close to the posterior means and medians. The lower trees tend to have stronger relationships (larger implied $\tau $ ’s from calculated $\theta $ ’s) than the higher ones.

Table 13.

Posterior estimates (mean, median, standard deviation, tail quantile) of parameters of selected copulas in the most optimal vine copula structure (T: theft; H: hacking; U: unauthorised access; M: mishandling of sensitive/ confidential documents).

Table 14 (top panel) reports the posterior estimates of the VaR of the monthly total data breach losses under the five assumptions of BMS, BMA, multivariate t, multivariate Gaussian, and independence copulas for the HHS data. We adopt the method in Jacobs (Reference Jacobs2014) and convert the number of events into dollar terms. It is a regression model given as dollar loss = exp[7.68 + 0.76 × ln(events)]. This empirically deduced equation can serve as a rough approximation of the real loss, as there are no available data in monetary terms and there is no general method to estimate the loss. As in the previous section, treating different data breach types as independent omits the underlying relationships and produces the smallest VaR. For the 90% and 95% VaR, assuming any of the copula models considered leads to about 20% increase in the tail estimate, when compared to the independence copula. For the 99% and 99.5% VaR, the multivariate Gaussian and t copulas leads to about 60% increase in the tail estimate. Under the BMS and BMA, the 99% and 99.5% VaR estimates are larger by about 90% or more, in which the BMA still produces the largest tail estimates. Table 14 (bottom panel) also presents the posterior VaR estimates for the PRC data used in Section 5. Similarly, for the 90% and 95% VaR, using any of the copula models brings about an 20% increase in the tail estimate, and for the 99% and 99.5% VaR, the multivariate Gaussian and t copulas generates a 60% increase in the tail estimate. Under the BMS and BMA, the 99% and 99.5% VaR estimates increase by around 80% or more from those under the independence copula. These results illustrate that both allowing for the dependency structure adequately and incorporating model uncertainty can play a vital role in the assessment of capital. Without addressing these uncertainties properly, there could be significant underestimation of the aggregate risk level and insufficient capital for covering adverse events.

Table 14.

Posterior estimates of monthly value-at-risk for a portfolio aggregating events of all four breach types under five different dependence assumptions, including BMS, BMA, multivariate t, multivariate Gaussian, and independence copulas.

8. Concluding Remarks

In this paper, we have adopted a Bayesian framework to implement the vine copula structure for modelling data breach events jointly. We have observed varying levels of tail dependence between different breach types, based on both the optimal copulas selected and non-parametric estimation of tail dependence coefficients. An insurer providing cover on these types of losses must make a sufficient allowance for the dependency structure when calculating premiums and reserves. As demonstrated earlier, while the estimated severities of data breaches at 90% confidence level are not too different (within 20%) from one another regardless of whether they are simulated by the vine copula, or the multivariate elliptical copulas, or the independence model, the differences are drastic for the 99% and 99.5% VaR. They range from about 50% when comparing between the vine and multivariate t copulas to being more than double between the vine copula and the independence model. Therefore, the assumptions made on the underlying dependencies between risks could have huge impacts on the required capital and other risk management considerations. Regulators and insurers should be keenly aware of model risk and take proper measures to prevent underestimation of risks, especially given the short history of development in the line of cyber insurance business.

With the continual advancement in technology in our fast-paced society, the cyber insurance market is destined for rapid expansion in the foreseeable future, and more knowledge and data in this arena would become accessible over time. Considering our evolving understanding and the dynamic nature of cyber claims, the proposed Bayesian modelling framework is naturally tailored for businesses interested in cyber risk estimation. Besides being capable of incorporating exclusive information provided by subject matter experts, where available, this Bayesian approach allows for a simultaneous determination of the vine structure, copulas, and margins and estimation of all the unknown parameters in a coherent way. It not only avoids introducing biases as under the IFM and sequential estimation methods but also integrates process, parameter, and model uncertainties in one unified setting. A proper allowance for all uncertainties is of utmost importance in the area of risk management under current regulations. Moreover, we also note that the numerical results are quite robust to the prior specifications and that a mixture of copulas can provide more modelling flexibility.

It is often quite difficult to use the maximum likelihood to estimate the copula and marginal parameters jointly. The Bayesian approach does not have this problem and can generate all the estimates coherently from MCMC simulation. Furthermore, by taking all kinds of uncertainties into account, the proposed Bayesian model provides an opportunity to improve the forecasting performance and the computation of risk measures. Particularly, failing to assess model uncertainty appropriately may lead to a serious miscalculation of such regulatory reporting items as reserves and capital.

We have shown how to implement the selection between a few “best” vine structures in the Bayesian framework. However, as the number of possible combinations grows rapidly with the number of dimensions, the scalability of our approach is limited. Our suggestions are that the proposed approach is more suitable for lower dimensions and that for higher dimensions it can be applied in conjunction with other ways of identifying the vine structure such as the standard information criteria, maximum spanning tree, and subjective inputs of grouping the variables in the order of importance. Recently, Gruber and Czado (Reference Gruber and Czado2018) developed a reversible jump MCMC algorithm for moving within a large model candidate space and showed that it can achieve quick convergence to desirable models. They claimed that their algorithm is suitable for up to around 10 dimensions, and so future research for more scalable Bayesian methods is called for. Moreover, further work is needed to make this kind of advanced algorithms more accessible to industry practitioners and other academics, like the user-friendly platforms offered by WinBUGS (Spiegelhalter et al., Reference Spiegelhalter, Thomas, Best and Lunn2003) and JAGS, for wider use in actuarial practice and research.