4.1. Risk Theory

Aggregate loss, S, represents the total amount for a given period and group of risks,

(1) $$S = {X_1} + {X_2} +, \ldots, + {X_N},$$

where N and $ {X}_{i}$ can be defined from two perspectives of risk theory, namely:

• Collective Risk (CR): loss count, $N$ , and (non-negative) severities, ${X_1}, \ldots, {X_N}$ , are random variables with independence assumptions as follows: N does not depend on the severity of loss; given $N$ , ${X_i}{\rm{s}}$ are i.i.d., independently with respect to count

• Individual Risk (IR): here, N denotes a fixed number of risks with respective losses, ${X_i}{\rm{s}}$ that are independently distributed (as opposed to i.i.d.) random variables with mixed CDFs that may have mass at point zero (i.e. for the probability of no loss)

In terms of (1) – IR, CR models – determining the ALD is one of the classical problems in the realm of risk theory. As there is generally no closed-form solution (Shevchenko, Reference Shevchenko2010, Section 1) various techniques have been deployed: Fast Fourier Transform (FFT) can be used to reconstruct the density with the aid of the transforms (e.g. characteristic function, CF; moment or probability generating functions, MGF, PGF respectively – provided these exist) – (Kaas et al., Reference Kaas, Goovaerts, Dhaene and Denuit2008, Section 2.1).

4.2. Increased Limit Factors

An ILF is a multiplicative factor that is applied to the premium at a basic limit to determine the premium at an increased limit. Basic limits typically refer to the lowest levels of coverage provided, (Werner & Modlin, Reference Werner and Modlin2010). However, in principle, any non-negative limit can be contemplated for this purpose (hereafter, the term base limit is used instead of basic limit).

Limit definitions

A policy limit refers to the maximum amount payable under an insurance policy, either overall, or in respect of a particular section of a policy (Lloyd’s, 2019), hereafter, known as “coverage section.” This may be expressed on several bases; for demonstrative purposes, the following are assumed applicable for losses associated with classes A–E:

• Per-loss: applies to individual costs (i.e. classes A–D)

• Per-occurrence: applies to total cost (i.e. class E)

The limited random variable ${X^{(b)}}$ is defined as follows:

(2) $${X^{(b)}} = \min\, \left( {X,b} \right),$$

where $X$ is a random variable and $\{ b:b \gt 0\} $ is some limit. More generally, consider the limited variable ${X^{(b)}}$ (2), and suppose $X$ has a CDF and PDF denoted by $F$ and $f$ respectively; the limited k^th-order moment of $X$ , when limit $b$ applies, can then be expressed in terms of the Riemann-Stieltjes integral:

(3) $${\rm{E}}\left( {{X^{(b)k}}} \right) = {\rm{E}}\left( {\min {{(X,b)}^k}} \right) = \int\limits_0^b {{x^k}dF{{(x)}}} dx + {b^k}(1 - F(b)) = \int\limits_0^b {k{x^{k - 1}}{S_X}{{(x)}}} dx,$$

where ${S_X} = 1 - F$ and ( $k = 1$ yields the LEV). Refer to Lee (Reference Lee1988) for a graphical illustration and Klugman et al. (Reference Klugman, Panjer and Willmot2004) for a mathematical proof. Now let the aggregate loss in respect of limited severities (hereafter, Limited Aggregate Severity, LAS) be $S(b)$ defined by:

(4) $$S(b) = \sum\limits_{i = 1}^N {X_i^{(b)}}, $$

where $b \gt 0$ is a given limit, and ${X_i}$ s are severities, and $N$ is the loss count, as for the aggregate loss in (2). This gives rise to the following definition: let limit factor, $\gamma, $ for a given base limit, $a,$ be defined by:

(5) $$\gamma (b): = \gamma (b;a) = {{{\rm{E}}\left( {S(b)} \right)} \over {{\rm{E}}\left( {S(a)} \right)}},{\rm{E}}\left( {S(a)} \right),{\rm{E}}\left( {S(b)} \right) \gt 0,$$

where $a,b \gt 0$ . The term limit factor, for the purpose of the present research, refers to both discount factors and ILFs, defined as follows:

• Discount factor: $(a \gt b \gt 0) \Rightarrow \gamma (b;a) \in (0,1)$ ; in this case, $a$ could represent the highest limit of coverage, or, in the context of coverage without-limits, $a \to \infty $

• ILF: $(0 \lt a \le b) \Rightarrow \gamma (b;a) \ge 1$ ; the conventional definition of an $ILF$ , where $a$ and $b$ represent “basic” and increased limits respectively

In terms of (5), CR independence assumptions lead to the following expression for limit factors:

(6) $$\gamma (b) = {{{\rm{E}}\left( {{X^{(b)}}} \right)} \over {{\rm{E}}\left( {{X^{(a)}}} \right)}}.$$

Consistency properties

Limit factors satisfy consistency properties if they are asymptotically constant, have a monotonically decreasing and positive gradient, and are concave down.

4.3. Risk Adjustments

Process risk refers to the inherent variability associated with the stochastic nature of frequency and severity of losses. To allow for this, actuaries may incorporate a risk adjustment into ILF calculations to achieve a certain risk margin (percentage increase in LAS).

Let ${\pi _{{\mathop{\rm var}} }}(S;w)$ denote the variance-adjusted (pure-risk) premium with respect to the aggregate loss amount, $S,$ and a risk parameter, $w \gt 0,$ be defined by:

(7) $${\pi _{{\mathop{\rm var}} }}\left( {S;w} \right) = {\rm{E}}\left( S \right) + w{\rm{Var}}\left( S \right),$$

then the variance-adjusted limit factor, ${\gamma _S}$ , can be defined as:

(8) $${\gamma _S}(b;a,w) = {{{\pi _{{\rm{var}}}}(S(b);w)} \over {{\pi _{{\rm{var}}}}(S(a);w)}},$$

where $S(a)$ and $S(b)$ are LASs (4) with limits $a,b \gt 0$ respectively. Independence assumptions (1) concerning loss count and i.i.d. severity (i.e. $N$ , $Y$ respectively), with Poisson $N,$ simplify the risk-adjusted limit factor, ${\gamma _S}$ (8) to the following:

(9) $${\gamma _Y}(b;a,w) = {{\pi _{{\rm{var}}}^*({Y^{(b)}};w)} \over {\pi _{{\rm{var}}}^*({Y^{(a)}};w)}},\pi _{{\rm{var}}}^*({Y^{(b)}};w) = {\pi _{{\rm{var}}}}({Y^{(b)}};w) + w{\left( {{\rm{E}}\left( {{Y^{(b)}}} \right)} \right)^2}.$$

For a compound Poisson “excess” LAS, $S = \sum\nolimits_{i = 1}^N {{Y_i}} $ , where ${Y_i} = \max (0,v{X_i}^{({\textstyle{b \over v}})} - d)$ , $i = 1,2, \ldots, N$ (i.e. $N\sim Poisson)$ , under CR independence assumptions (1), limits $a,b \gt 0;$ deductible $d,$ s.t. $0 \le d \lt \min (a,b)$ , and constant inflation $v \gt 1$ , the variance-adjusted limit factor, ${\gamma _Y}$ (9), with parameter $w$ (as before), becomes:

(10) $${\gamma _Y}(b;a,d,w,v) = {{\pi _{{\rm{var}}}^*\left( {{X^{\left( {{b \over v}} \right)}},vw} \right) - \pi _{{\rm{var}}}^*\left( {{X^{\left( {{d \over v}} \right)}},vw} \right) - 2dw\left[ {{\rm{E}}\left( {{X^{\left( {{b \over v}} \right)}}} \right) - {\rm{E}}\left( {{X^{\left( {{d \over v}} \right)}}} \right)} \right]} \over {\pi _{{\rm{var}}}^*\left( {{X^{\left( {{a \over v}} \right)}},vw} \right) - \pi _{{\rm{var}}}^*\left( {{X^{\left( {{d \over v}} \right)}},vw} \right) - 2dw\left[ {{\rm{E}}\left( {{X^{\left( {{a \over v}} \right)}}} \right) - {\rm{E}}\left( {{X^{\left( {{d \over v}} \right)}}} \right)} \right]}},$$

where $\pi _{{\mathop{\rm var}} }^*$ is defined as previously. This can be shown through substitution $x = y{v^{ - 1}}$ .

Let ${\pi _{PH}}$ be the mean in respect of the PH transform defined by:

(11) $${\pi _{PH}}({Y^{(b)}};b,w) = \int_0^b {{S_Y}{{(x)}^{{\textstyle{1 \over w}}}}dx}, $$

where $b$ is a given non-negative limit, and, and $w \ge 1$ (Wang, Reference Wang1995, Reference Wang, Connell, Crifo, Dove, Edlefson, Gardiner, Golz, Josephson, Kufera, Lewis, Moody, Schwab and Turnacioglu1999a).

Let $\gamma ({2^k}a,a) = {(1 + r)^k}$ , where $\gamma $ is the ILF in respect of an increased limit and base limit, in this case, ${2^k}a$ and $a$ respectively, with $a \gt 0,$ $r \in (0,1),$ and $k \gt 1$ . It follows that:

(12) $$\gamma (b;a,w) = {(1 + r)^{{{\log }_2}(b{a^{ - 1}})}} = {(b{a^{ - 1}})^{{{\log }_2}(1 + r)}} = {(b{a^{ - 1}})^w},$$

where $w = {\log _2}(1 + r).$ Refer to Mack and Fackler (Reference Mack and Fackler2003) for details including origin.

4.4. Correlated ALDs

This section describes CFs for correlated aggregate loss and count, based on pioneering contributions by Wang (Reference Wang1998, Reference Wang, Connell, Crifo, Dove, Edlefson, Gardiner, Golz, Josephson, Kufera, Lewis, Moody, Schwab and Turnacioglu1999a) and conventional techniques for mixture models (Klugman et al., Reference Klugman, Panjer and Willmot2004; Mildenhall, Reference Mildenhall2005).

For random variables ${X_i}$ and ${X_j}$ , with Pearson correlation coefficient ${\rho _{ij}}$ , means $ {\mu }_{i}$ and standard deviations, $ {\sigma }_{i}$ , the covariance coefficient ${\kappa _{ij}}$ is given by:

(13) $${\kappa _{ij}} = {{{\rm{Cov}}({X_i},{X_j})} \over {{\mu _i}{\mu _j}}} = {{{\rho _{ij}}{\sigma _i}{\sigma _j}} \over {{\mu _i}{\mu _j}}}.$$

The range of ${\kappa _{ij}}$ (13) depends on the shape of marginal distributions for ${X_i}$ and ${X_j}$ .

CFs for correlated aggregate loss

Define the joint CF, ${C_{\bf{S}}}: = {C_{{S_1}, \ldots, {S_m}}}$ , for $m \in {{\bf Z}^ + }$ random variables, ${\bf{S}} = \left[ {{S_1}, \ldots, {S_m}} \right],$ by:

(14) $${C_{\bf{S}}}[{\bf{t}}] = \left( {1 + \sum\nolimits_{i \lt j}^{} {{\kappa _{ij}}\left( {1 - {C_i}[{t_i}]} \right)\left( {1 - {C_j}[{t_j}]} \right)} } \right)\prod\nolimits_{i = 1}^m {{C_i}[{t_i}]}, $$

where ${S_i},{S_j} \in {\bf{S}}$ have respective CFs ${C_i},{C_j},$ and covariance coefficient ${\kappa _{ij}},$ $1 \le i \lt j \le m;$ and $ t=[{t}_{1}, \ldots, {t}_{m}]$ , Wang (Reference Wang1998, pt. IV). Following (14), let the univariate CF of

$S = {S_1} +, \ldots, {S_m}$ be ${C_{S}}$ , then:

(15) $${C_S}[t] = \left( {1 + \sum\nolimits_{i \lt j}^{} {{\kappa _{ij}}\left( {1 - {C_i}[t]} \right)\left( {1 - {C_j}[t]} \right)} } \right)\prod\nolimits_{k = 1}^m {{C_k}[t]}, $$

where ${\kappa _{ij}}{\rm{s}}$ and ${C_i}{\rm{s}}$ are defined as previously. The mean and variance of aggregate loss, $S,$ is:

(16) $$\eqalign{ & \mu : = {\rm{E}}\left( S \right) = {\rm{E}}\left( {{S_1} +, \ldots, + {S_m}} \right), \cr & {\rm{Var}}\left( S \right) = {\sigma ^2} + 2\sum\nolimits_{i \lt j}^{} {{\kappa _{ij}}{\rm{E}}\left( {{S_i}} \right){\rm{E}}\left( {{S_j}} \right)} \cr} $$

where ${\sigma ^2} = \sum\nolimits_{j = 1}^m {{\rm{Var}}\left( {{S_j}} \right)} $ , Wang (Reference Wang1998). The univariate CF (15) is apparently less restrictive, in terms of covariance coefficients (for valid PDF), than is the case for the joint CF (16).

CFs for correlated loss count

Often, there is an exogenous cause for uncertainty regarding the extent or number of losses. This is referred to as parameter risk in the context of stochastic models (Freifelder, Reference Freifelder1979, cited by Miccolis (Reference Miccolis1978). To reflect such uncertainty, a secondary mixture CDF can be incorporated within the model. In this section, a joint PGF for correlated aggregate loss count variables is built up using Poisson mixtures. Refer to Klugman et al. (Reference Klugman, Panjer and Willmot2004, Section 4.6.10) for examples of various other mixtures with theoretical underpinnings.

Poisson mixture models

Let ${\bf{N}} = [{N_1}, \ldots, {N_m}]$ be a vector of $m$ discrete random variables with joint PGF given by ${P_{\bf{N}}}: = {P_{{N_1}, \ldots, {N_n}}}$ and assume there exists a random variable $\theta $ with MGF ${M_\theta }$ such that $({N_j}|\theta = \omega )\sim Poisson({\lambda _j}\omega )$ (B.3) where ${\rm{E}}{N_j}\left( {\theta = \omega } \right) = \omega {\lambda _j}$ , $j = 1, \ldots, m.$ The marginal PGF of ${N_j}|(\theta = \omega )$ is then ${P_{{N_j}|\theta = \omega }}[{t_j}] = {e^{w{\lambda _j}({t_j} - 1)}}$ , which leads to the following joint PGF for ${\bf{N}}$ :

(17) $${P_{\bf{N}}}[{\bf{t}}] = {{\rm{E}}_\theta }\left( {{\rm{E}}\left( {{t_1}^{{N_1}} \ldots {t_m}^{{N_m}}|\theta } \right)} \right) = {{\rm{E}}_\theta }\left( {\exp (\theta {\bf{\lambda }} \cdot ({\bf{t'}} - {{{\bf{1'}}}_m}))} \right) = {M_\theta }[{\bf{\lambda }} \cdot ({\bf{t'}} - {{\bf{1'}}_m})],$$

where ${\bf{\lambda }} = [{\lambda _1}, \ldots, {\lambda _m}],{\rm{ }}{\bf{t}} = [{t_1}, \ldots, {t_m}],$ and ${{\bf{1}}_m}$ is a (row) vector with $m$ ones.

Suppose $\theta \sim Gamma(\alpha, 1),$ for some $a \gt 0$ , has MGF ${M_\theta }[t] = {(1 - t)^{ - \alpha }},$ then the joint PGF in (17) becomes ${P_{\bf{N}}}[{\bf{t}}] = {(1 - {\bf{\lambda }} \cdot ({\bf{t'}} - {{\bf{1'}}_m}))^{ - \alpha }}$ . This specifies a form of multivariate negative binomial CDF

where marginals, ${N_j}\sim NB(\alpha, {\lambda _j})$ (B.4), have respective PGFs, ${P_{{N_j}}},\, j = 1, \ldots, m{\rm{ }},$ defined by:

(18) $${P_{{N_j}}}[{t_j}] = {(1 - {\lambda _j}({t_j} - 1))^{ - \alpha }},$$

(Wang, Reference Wang, Connell, Crifo, Dove, Edlefson, Gardiner, Golz, Josephson, Kufera, Lewis, Moody, Schwab and Turnacioglu1999b). Refer to Mildenhall (Reference Mildenhall2005) for MGFs with alternative parameterisations, and Reshetar (Reference Reshetar2008) for practical application in the context of OR.

From Example 4.5, let ${N_j}\sim NB({a_j},{\lambda _j})$ – the joint PGF, ${P_{\bf{N}}}$ , is now:

(19) $${P_{\bf{N}}}[{\bf{t}}] = {({{\bf{1}}_m}{ \cdot }{\bf{k'}} - m + 1)^{ - {\rm{ }}{1 \over w}}},$$

where ${\bf{t}} = \left[ {{t_1}, \ldots, {t_m}} \right],${{\bf{1}}_m}$ is a row vector of $m$ ones; ${\bf{k}} = [{k_1}, \ldots, {k_m}]$ with ${k_j} = {(1 - {\lambda _j}({t_j} - 1))^{{\alpha _j}w}},\,j = 1, \ldots, m$ ; and $w \ne 0.$ This specifies a family of MNB CDFs, with marginals ${N_j}\sim NB({\alpha _j},{\lambda _j}),$ in either of the following cases:

1. 1. $0 \lt w \lt \mathop {\min }\limits_{j \in [1,m]} \{ {\alpha _j}^{ - 1}\} $

2. 2. $w \lt 0$ s.t. ${P_{\bf{N}}}[{{\bf{0}}_m}] \gt 0$ and $ - {\textstyle{1 \over w}} \in {{\bf Z}^ + }$

where ${{\bf{0}}_m}$ is a row vector of $m$ zeros, (Wang, Reference Wang1998).

Here, the random vector ${\bf{N}}$ follows an MNB distribution, denoted by ${\bf{N}}\sim {\rm{MNB(}}{\bf{\alpha }},\,{\bf{\lambda }},\,w)$ with vector parameters $\alpha = \left[ {{\alpha _1}, \ldots, {\alpha _m}} \right]$ and ${\bf{\lambda }} = [{\lambda _1}, \ldots, {\lambda _m}].$ Suppose ${S_1}, \ldots, {S_m}$ represent $m \in {{\bf Z}^ + }$ CR loss models (1) that are specified by their severities and loss count variables, $({X_i},{N_i}),$ $i = 1, \ldots, m,$ and only correlated through ${\bf{N}} = [{N_1}, \ldots, {N_m}]\sim {\rm{MNB \,(}}{\bf{\alpha }},{\bf{\lambda }},w)$ (Example 4.6). Accordingly, the CF for the overall aggregate loss, ${C_S}: = {C_{{S_1} +, \ldots, + {S_m}}}$ , is defined by:

(20) $${C_S}[t] = {({{\bf{1}}_m}{ \cdot}{\bf{y'}} - m + 1)^{ - {\rm{ }}{1 \over w}}},$$

where ${{\bf{1}}_m}$ is a row vector of $m$ ones, ${\bf{y}} = [{y_1}, \ldots, {y_m}]$ with ${y_j} = {(1 - {\lambda _j}({C_j} - 1))^{{\alpha _j}w}}$ , ${C_j}$ is the CF of ${X_j}$ $j = 1, \ldots, m$ (Meyers & Heckman, Reference Meyers and Heckman1984; Wang, Reference Wang1998). As such, FFT reconstructs the CDF of $S = {S_1} +, \ldots, + {S_m}$ , from transforms ${C_S}$ (20). The mean and variance of $S$ can be determined using (16) – substituting ${\kappa _{ij}}$ with $w$ , the correlation parameter in (20).

4.5. Severity Model

Define a two component spliced model in terms of $n$ observed severities, ordered as ${x_1} \lt {x_2} \lt, \ldots, \lt {x_n}.$ Losses in the interval $[0,\tau ]$ , for a given non-negative threshold, $\tau $ (i.e. “splicing point”), are assumed to follow a small loss CDF (in this case, estimated by the empirical CDF, ${F_n}$ ). To cover the interval $(\tau, \infty )$ , a parametric distribution G is estimated using (observed) losses greater than $\tau $ . Now let $H$ be the spliced distribution in question:

(21) $$\eqalign{ 1 - H(x) & = \left\{ {\matrix{ {1 - {F_n}(x)} \hfill & { x \le \tau } \hfill \cr {\left( {1 - {F_n}(\tau )} \right)\left( {1 - {{G(x) - G(\tau )} \over {1 - G(\tau )}}} \right)} \hfill & { x \gt \tau } \hfill \cr } } \right. \cr & = \left\{ {\matrix{ {1 - {F_n}(x)} \hfill & { x \le \tau } \hfill \cr {\left( {1 - {F_n}(\tau )} \right)\left( {{{1 - G(x)} \over {1 - G(\tau )}}} \right)} \hfill & { x \gt \tau } \hfill \cr } } \right.{\rm{ }} \cr} $$

where the first component CDF, ${F_n}\left( x \right)/{F_n}\left( \tau \right)$ (for $x \le \tau $ ) and second component CDF, $\left( {G\left( x \right) - G\left( \tau \right)} \right)/\left( {1 - G\left( \tau \right)} \right)$ , for $x \gt \tau, $ are spliced with weights ${F_n}(\tau )$ and $1 - {F_n}(\tau )$ respectively (Klugman et al., Reference Klugman, Panjer and Willmot2004).

Selection (large-loss model)

The following steps are used to select a large-loss CDF, from a set of $k \in {{\bf Z}^ + }$ candidate models (e.g. Burr, Weibull, Pareto, etc.) and identify a suitable threshold for application of the spliced model in (21) (i.e. given ${{\bf{x}}_n} = [{x_1}, \ldots, {x_n}]$ ):

Step 1 Fit $m \gt 1$ CDFs, $G_{i1}^{}, \ldots, {G_{im}},$ to the largest $n - i + 1$ severities, for some $i = 2,3, \ldots, n - k - 1,$ where $k \le n - 2$ is the minimum number parameter estimates for each CDF (e.g. based on Maximum Likelihood Estimation, MLE).

Step 2 Let $G_i^* = \mathop {\min }\limits_j \{ {c_j}\} $ , where ${c_j}$ is the AIC^c for ${G_{ij}}$ , $j = 1, \ldots, m$ .

Step 3 Calculate $B_i^*$ , the KS-ratio (ratio of the Kolmogorov Smirnov test statistic (Glivenko-Cantelli – van der Vaart (Reference van der Vaart1998)) to the critical value at the specified level) for $G_i^*$ .

Steps 1–3 have the following outputs: the large-loss distribution, $G_i^*,$ empirical threshold, ${x_i}$ , and KS-ratio, $B_i^*$ (valid scores require $i = 2, \ldots, n - k - 1$ , as in step 1). In terms of the spliced model, $H$ (21), $G_i^*\left( {\rm{x}} \right) = {{\left( {G\left( x \right) - G\left( \tau \right)} \right)} \mathord{\left/ {\vphantom {{\left( {G\left( x \right) - G\left( \tau \right)} \right)} {\left( {1 - G\left( \tau \right)} \right)}}} \right. } {\left( {1 - G\left( \tau \right)} \right)}},x \gt \tau $ and ${x_i} \le \tau \lt {x_{i + 1}}$ – if $\tau \lt {x_2}$ or $\tau \gt {x_{n - k - 1}}$ , then the unconditional CDFs, $G$ and ${F_n}$ respectively, might be used. The threshold itself can be expressed in terms of the empirical rank as follows:

(22) $$j = n{F_n}(\tau ),$$

where $j = 1, \ldots, n$ ; ${F_n}$ , $\tau, $ and ${x_1}$ are defined as previously (21).

Threshold determination

A score-based approach (Klugman et al., Reference Klugman, Panjer and Willmot2004, Section 13.5.3) is adopted using a similar set-up adopted for Maximum Likelihood by Ralucavernic (Reference Ralucavernic2009), but with greater emphasis being placed on tail fit and limit-factor consistency. Differentiability and continuity requirements (Cerchiara & Acri, Reference Cerchiara and Acri2016) are not explicitly allowed for, however, model selection incorporates the corrected Akaike Information Criteria (AIC^C); refer to Akaike (Reference Akaike, Parzen, Tanabe and Kitagawa1998) and Burnham and Anderson (Reference Burnham and Anderson2002) for details. This provides a practical and simplified means to identify both parametric CDF and threshold – additional considerations pertain to limit-factor consistency and mean excess (ME) plots.

Criteria 4.1 Splicing point

The following criteria are contemplated for determining threshold, $\tau $ , in terms of output from steps 1–3:

1. 1. $\tau, $ with the greatest rank, $i$ .

2. 2. $\tau, $ with the lowest KS-ratio, $B_i^*$ .

In this way, larger thresholds are favoured through the first criterion, whilst the second attempts to optimise tail fit. Upper bounds are established subjectively by considering ME plots.

Normalising scores

According to the set of Criteria 4.1, preference is given to higher and lower values of ${x_i}{\rm{s}}$ and $B_i^*{\rm{s}}$ respectively (i.e. steps 1–3). Equivalently, higher values of ${\alpha _i}$ and ${\beta _i},$ defined as follows, are favoured over lower values:

(23) $${\alpha _i} = {{{x_i}} \over {{x_{n - k - 1}}}}{\rm{, }}{\beta _i} = {{\mathop {\min }\limits_{i \in [1,n]} \{ B_i^*\} } \over {B_i^*}}{\rm{ }}\forall {\rm{ }}B_i^* \gt 0$$

The weighted average score, ${z_i},$ with respect to measures ${\alpha _i}$ and ${\beta _i}$ (23), is determined by:

(24) $${z_i} = {w_i}{\alpha _i} + (1 - {w_i}){\beta _i},$$

With $i$ defined as previously in step 1; thus ${\alpha _i},{\beta _i} \in (0,1]$ are on the same scale. $w_i^{(2)}$ , which reduces as $\tau $ increases, can be defined as follows:

(25) $$w_i^{(2)} = {{n - i} \over n}$$

Algorithm 1 Optimal threshold and large-loss CDF

For a given group (i.e. class) of $n$ ordered, homogeneous, and independent severities, ${x_1}, \ldots, {x_n}$ , with empirical CDF ${F_n} = {1 \over n}\sum\limits_{i = 1}^n {{1_{\{ {x_i} \le x\} }}} $ ; steps 1-3 (p. 14) are run for each $i \in [2,n]$ to produce the following input vectors for this algorithm:

• ${\bf{G}} = [G_2^*,G_3^*, \ldots, G_{n - k - 1}^*]$ (i.e. selected large-loss distributions from step 3).

• ${\bf{x}} = [x_2^*, \ldots, x_{n - k - 1}^*]$ (i.e. vector of “thresholds”).

• ${\bf{B}} = [B_2^*,B_3^*, \ldots, B_{n - k - 1}^*]$ (i.e. associated vector of KS-ratios).

Next, (23) is applied to ${\bf{x}}$ and ${\bf{B}}$ (element by element) to obtain the vector of scores ${\bf{\alpha }} = [{\alpha _2}, \ldots, {\alpha _n}]$ and ${\bf{\beta }} = [{\beta _2}, \ldots, \beta _n^{}]$ respectively. For a given vector of weights ${\bf{w}} = [{w_2}, \ldots, {w_n}],$ where ${w_i} \in (0,1){\rm{ }}\forall {\rm{ }}i = 2,3, \ldots, n$ , the vector of (calculated) weighted scores, ${\bf{z}} = [{z_2}, \ldots, {z_n}]$ , is determined using (24). The optimal threshold, ${\tau ^*}$ , is ${x_{i*}}$ , where ${i^*} \in \{ 2,3, \ldots, n\} $ is the optimal index value that yields the solution to the following:

(26) $${z_{i*}} = \max \{ {z_i}:i = 2,3, \ldots, n\}.$$

The corresponding (parameterised) optimal distribution is then $G_{n{F_n}(\tau *)}^* = G_{i*}^*$ (which follows from (22) with $j: = {i^*}).$ Thus, the outputs of this algorithm are the optimal threshold, optimal index value, and optimal distribution (i.e. ${\tau ^*}$ , ${i^*}$ , and $G_{i*}^*$ respectively).

Algorithm 2 Model confidence sets – Kullback-Leibler

This algorithm follows the bootstrap approach of Burnham & Anderson Reference Burnham and Anderson2002 (Section 4.5), which is based on essential Kullback & Leibler (Reference Kullback and Leibler1951) theory associated with AIC and other such information criteria. For each candidate CDF (i.e. parametric family), $G_i^{},$ and bootstrap sample indexed $i = 1, \ldots, m$ and $j = 1, \ldots, M$ respectively, $m,{\rm{ }}M \gt 2,$ determine Akaike differences, ${\delta _{ij}}$ , in relation to the minimum AIC^C, $A_j^* = \mathop {\min }\limits_{i = 1,..,m} \{ {A_{ij}}\} $ , and associated Akaike weights, ${w_{ij}}$ (that sum to one for each sample) as follows:

(27) $${\delta _{ij}} = {A_{ij}} - A_j^*\,\,\,\,\,\,{w_{ij}} = {{\exp ( - 0.5{\delta _{ij}})} \over {\sum\nolimits_{u = 1}^m {\exp ( - 0.5{\delta _{ij}})} }}$$

where ${A_{ij}}$ is the AIC^C score for CDF ${G_i}$ , parameterised (e.g. using MLE) in respect of data for sample $j \in \{ 1, \ldots, M\} $ . Differences and weights accompanying the $M$ samples can provide insight into model (in this case, CDF) selection uncertainty. For instance, in terms of the following “model confidence set” and selection probability estimates:

• The 100 $\alpha {\rm{\% }}$ “Kullback-Leibler” (KB) confidence set, for specified CDF with (common) index $s \in \{ 1, \ldots, m\}, $ comprises the set of candidate CDFs with corresponding Akaike differences below the $100\alpha {\rm{\% }}$ empirical quantile, ${q^{(\alpha )}},$ of Akaike differences for the specified CDF; the probability that CDF indexed $i = 1, \ldots, m$ is in such a confidence set, $c_i^{(\alpha )},$ can be estimated from the samples as follows: $\hat c_i^{(\alpha )} = {M^{ - 1}}\sum\nolimits_{j = 1}^M {{1_{\{ {A_{ij}} - A_j^* \le {q^{(\alpha )}}\} }}} $ (where, in general, indicator ${1_{\{ A\} }} = 1$ if a given event $A$ occurs – failing which, ${1_{\{ A\} }} = 0$ ).

• Correspondence between the average weight, ${\hat w_i} = {M^{ - 1}}\sum\nolimits_{j = 1}^M {{w_{ij}}} $ , for a given CDF with index $i = 1, \ldots, m$ , and the proportion of $(M)$ minimum Akaike scores that correspond to the CDF in question, ${\hat \pi _i} = {M^{ - 1}}\sum\nolimits_{j = 1}^M {{1_{\{ {\delta _{ij}} = 0\} }}}, $ attests to the veracity of the (aforementioned) KL confidence set, and associated model inference uncertainty.

4.6. Limit Factor and Aggregate Loss Models

This section describes and formulates various models, which are grouped in Figure 2 according to whether correlation is recognised, and how loss count, $N$ , is modelled:

• IR framework: $N{\rm{ }} = n$ is given.

• CR framework: $N$ is a random variable with a given PDF.

Figure 2. Flow chart for Models 4.1–4.6. Models 4.4–4.5 and Model 4.6 assume correlated aggregate loss amounts and counts (classes A–D) respectively. Adjustments (e.g. inflation, risk) may apply to limit factors based on any of these models.

In this way, IR represents a special type of CR, where $N$ has a degenerate distribution such that $Pr\left( {N = n} \right) = 1$ , as contemplated by Klugman, Panjer & Willmot (Reference Klugman, Panjer and Willmot2004, Section 6.1).

Models 4.1 and 4.2 Limit factors for independent, individual classes (IR model)

The following is an overview of Models 4.1–4.6, as depicted in Figure 2:

• Models 4.1–4.2 model aggregate losses in respect of small and large severities, using empirical CDFs and the spliced-severity model (Section 4.5); relevant limited moments (3) are used to determine the risk-adjusted LAS (7) and limit factors (8) in an IR framework with consideration for possible application in a CR framework.

• Model 4.3 (IR and CR) derives ALDs in respect of classes A–D (subject to per-loss limits), and class E (subject to a per-occurrence limit) from which limit factors are determined in respect of ground-up or excess losses; inflation and risk adjustments (10).

• Models 4.4 and 4.5 rely on given covariance coefficients between aggregate losses in classes A–D (15).

• Model 4.6 applies (20) with relevant parameters for the (correlated) marginal loss count CDFs (NB, Table B.2: B.4).

Models 4.1 and 4.2 are formerly defined in this section; Models 4.3–4.6 are more descriptive in nature and are framed in the context of tailored FFT steps, with compound Poisson and negative binomial applications for Models 4.5–4.6.

Assumptions for ILFs

The “top-slicing” method is used to determine ILFs; in respect of the risk premium. This assumes that severities, by class, are homogenous, independent, and independent of loss count; non-risk elements (e.g. expenses) are negligible (or proportional); and there is no anti-selection (e.g. by size of limit).

Variables and definitions

Define the following for a given class with $n$ observed severities:

• ${F_n}$ and $\tau $ : empirical CDF and splicing point respectively.

• ${x_1} \le, \ldots, \le {x_u} \le \tau $ : the smallest, ordered, $u$ (i.i.d.) severities with LAS, LEV, and “limited” variance denoted by ${Z_S}(b) = \sum\nolimits_{i = 1}^u {x_i^{(b)}}, {\mu _{S;b}} = {\rm{E}}\left( {X_S^{(b)}} \right) = {\textstyle{1 \over u}}\sum\nolimits_{i = 1}^u {x_i^{(b)}}, $ and $\sigma _{S;b}^2 = {\rm{Var}}\left( {X_S^{(b)}} \right) = {\textstyle{1 \over u}}\sum\nolimits_{i = 1}^u {{{(x_i^{(b)} - {\mu _{S;b}})}^2}} $ respectively, where $b \gt 0$ is a single limit that applies to severity (maximum payable in respect of individual claims); $u = n{F_n}(\tau ) \in \{ 0,1, \ldots, n\} $ ; ${X_S} \in \{ {x_1}, \ldots, {x_u}\} $ is the small severity random variable: ${x_i}{\sim ^d}{X_S},i = 1, \ldots, u$ , ${X_S}\sim {F_n}$ .

• ${X_1}, \ldots, {X_{n - u}}$ : $n - u$ random variable “large” severities with LAS, LEV, and limited variance ${Z_L}(b) = \sum\nolimits_{i = 1}^u {X_i^{(b)}}, $ ${\mu _{L;b}} = {\rm{E}}\left( {X_L^{(b)}} \right)$ , and $\sigma _{L;b}^2 = {\rm{Var}}\left( {X_L^{(b)}} \right)$ respectively, where ${X_i}$ are i.i.d. such that ${X_i}{\sim ^d}{X_L},i = 1, \ldots, n - u$ ; ${X_L}\sim G$ , where ${X_L}$ and $G$ are large severity and its CDF (unconditional with respect to $\tau ),$ respectively; ${X_L} \bot {X_S};$ $b$ is the limit as before.

Thus, ${\mu _{L;b}} = \int_0^b {{S_X}(x)dx} $ and $\sigma _{L;b}^2 = 2\int_0^b {x{S_X}(x)dx} - {\mu _{L;b}}^2$ , which follows from Equation (3) with $k = 1,2$ respectively, and ${S_X} = 1 - {F_X}$ where $F_X^{}\left( {\rm{x}} \right) = {{\left( {G\left( x \right) - G\left( \tau \right)} \right)} \mathord{\left/ {\vphantom {{\left( {G\left( x \right) - G\left( \tau \right)} \right)} {\left( {1 - G\left( \tau \right)} \right)}}} \right. } {\left( {1 - G\left( \tau \right)} \right)}},$ $x \gt \tau $ $\left( {{F_X}(x) = 0,{\rm{ }}x \le \tau } \right)$ . The overall aggregate loss, $Z$ , its mean, ${\mu _Z}$ , variance, $\sigma _Z^2$ , and associated (variance principle) risk-adjusted LAS, ${\pi _Z}: = {\pi _{{\mathop{\rm var}} }}$ ((7), $S = Z$ ), and limit factor, ${\gamma _Z}: = {\gamma _S}$ ((8), $S = Z$ ), are defined by Models 4.1–4.2, in an IR framework, as follows:

(28) $$\eqalign{ Z(b) & = {Z_S}(b) + {Z_L}(b) = \sum\limits_{i = 1}^u {x_i^{(b)}} + \sum\limits_{i = 1}^{n - u} {X_i^{(b)}} \cr {\rm{E}}\left( {Z(b)} \right) & = {\mu _{Z;b}} = u{\mu _{S;b}} + (n - u){\mu _{L;b}}{\rm{ }};{\rm{ Var}}\left( {Z(b)} \right) = \sigma _{Z;b}^2 = u\sigma _{S;b}^2 + (n - u)\sigma _{L;b}^2 \cr {\pi _{Z;b}} & = {\mu _{Z;b}} + w\sigma _{Z;b}^2{\rm{ }};{\rm{ }}{\gamma _{Z;a,b}} = {{{\pi _{Z;b}}} \over {{\pi _{Z;a}}}} \cr} $$

where $a,b \gt 0$ ; ( $u,$ $n,$ $b,$ ${Z_S}$ , ${Z_L},$ ${\mu _{S;b}},$ ${\mu _{L;b}},$ $\sigma _{L;b}^2$ ) as before; and ${\rm{Cov}}{\left( {{X_S}X} \right)_L} = 0.$ Models 4.1–4.2 can now be distinguished from one another as follows:

• Model 4.1 – by setting $u = n$ (or equivalently, $\tau \ge {x_n}$ , the maximum observed severity), ${X_L}$ and associated terms in Equation (28) become redundant and $Z$ , ${\mu _Z}$ , $\sigma _Z^2$ ,. ${\pi _Z},$ . and ${\gamma _Z}$ are expressed solely in terms of ${x_i}$ , $i = 1, \ldots, n$ ) and calculated numerically.

• Model 4.2 – this relies on the spliced-severity model (and associated algorithms) developed in Section 4.5, by setting $\tau $ , $u$ , and $G$ to the optimal outputs (i.e. threshold ${\tau ^*}$ , index ${i^*}$ , and large-loss CDF, $G_{i*}^*$ respectively); analytical solutions, are checked using Model Risk by Vose (2019), risk analysis software and simulation (e.g. Appendix C)

ILFs and associated measures for Models 4.1–4.2 can then be determined for a range of different splicing points and associated (small and large) severity CDFs. Model 4.2 can easily be amended to cater for the CR framework.

Attention is now turned to Models 4.3–4.6, which utilise FFT as summarised in Table 3.

Table 3. FFT steps for ALDs (Models 4.3–4.6) (✓) if step is relevant, (x) otherwise

Model 4.3 ALD for independent classes (IR, CR models)

Of Models 4.3–4.6, this model represents the most straightforward application of FFT. In terms of steps 1–4 (Table 3), consider a class with $n$ observed severities. Model 4.3 (IR) proceeds with step 1 by discretising the spliced-severity distribution (of limited severities) using the rounding method. The corresponding vector of CFs (determined in step 2) are raised to the power of $n$ (element by element) to obtain CFs in respect of ALDs (step 3a), which are yielded using the inverse Fourier transform (step 4). Model 4.3 (CR) is very similar except, instead of raising severity CFs to the power of $n$ (step 3a), the PGF of an assumed loss count CDF (in this case, Poisson) is incorporated (steps 1, 2, and 4 remain otherwise unchanged.

Model 4.4 ALD for correlated aggregate losses (IR model)

Step 3a is relevant for Model 4.4 as this is based on the IR framework which assumes each class has a (deterministic) loss count, $n$ . The CF for each of the classes A–D is thus raised to the power of $n$ (element by element) to obtain corresponding (class-level) CFs in respect of their marginal ALDs. Step 3b combines these using (15) (with $m = 4,$ and assumed covariance coefficients, ${\kappa _{ij}} = \kappa ),$ before taking the inverse Fourier transform in step 4 to yield the aggregate loss CDF (i.e. joint CDF for correlated marginal ALDs with respect to classes A–D).

Model 4.5 ALD for correlated aggregate losses (CR model)

Model 4.5 is the CR analogue to Model 4.4. Instead of raising the CFs in each of the classes A–D to the power of a deterministic count parameter, $n$ , as is the case for Model 4.4 in step 3a, the PGF of an assumed loss count variable is incorporated within the CF (element by element). This yields the CFs in respect of the (marginal) ALDs for each of the classes A–D. Step 3b (i.e. application of (15) with given marginals and covariance coefficients) and step 4 (i.e. inverse Fourier transform) used in this model are otherwise identical to those used for Model 4.4. For variance principle adjustments regarding limit factors, (16) is utilised.

Model 4.6 ALD for correlated loss count (CR model)

Model 4.6 utilises a (multivariate) mixture model, as considered for Example 4.6.

In particular, step 3a assumes that the class has random variable loss count, ${N_j}$ , with $NB({a_j},{\lambda _j})$ CDF and specified parameters ${a_j},{\lambda _j}$ , $j = 1,2,3,4$ (Table B.2: B.4). The associated PGF is thus incorporated (element by element) within CFs in step 2 to produce (class-level) vectors of CFs (step 3a) for respective ALDs. These are then combined using (15) (with $m = 4,$ and assumed correlation parameter, $w)$ in step 3b, before using the inverse Fourier transform to yield the aggregate loss CDF in step 4 (i.e. joint CDF in respect of classes with correlated aggregate NB loss count variables).

6.1. Conclusions

Conclusions, some of which are data or model dependent (i.e. not necessarily applicable in every situation) include:

• Severity distributions, based on data breach costs, were heavy tailed in the main, although D, representing business interruption, often affiliated with issues such as interdependence in the realm of insurance, was found to be light-tailed.

• Correlation between D and other classes (i.e. A–C) was found to have the greatest impact on the ALD in its tail (in the case where the aggregate loss model was used, the peak of the second mode of a bimodal distribution was intensified). The Value at Risk, however, was less affected by this compared to other risk measures (e.g. standard deviation).

• Empirical evidence suggests insurers are indeed avoiding volatile severity risk associated with increased cover limits, not only through low upper limits, but through increasing implied risk margins. Reducing Riebesell parameters support this view; in some (isolated) cases, this led to ILF consistency not being observed.

Enriched empirical data, as a basis for actuarial experience rating, may represent a source of value, despite the notion that it “quickly goes stale” due to the dynamic nature of the technological environment. This is demonstrated by reconciling modelled (i.e. “experience-based”) and insurer (exposure-based) ILFs, and introduces the recommendations made in Section 6.2.

6.2 Recommendations

Wider audience

Onus should be placed on all stakeholders concerned to establish a unified approach to deal with common cyber-risk management issues – whilst industry groups and international initiatives are reportedly underway; actions to “better” address basic data issues are still highly anticipated.

Developing an anonymised “community-wide” data base (with key elements for quantifying cyber-risk) may be fraught with wider issues concerning cooperation, funding, administration, and governance. However, there would appear to be some incentive to collaborate more effectively, given the $600bn (and growing) cyber-cost estimate previously mentioned (Section 1).

This would align with academic interests in support of such an initiative – although a unified approach may also be required here – possibly through a multidisciplinary academic interest group. Such cross-pollination would accelerate the development of cyber-risk and associated pricing models.

Specific directions – academia

There were only two “actuarial” contributions (according to title) that featured in the model review, neither of which appeared to have emerged from that domain. Given this, it is worth emphasising that further actuarial contribution to this specialised field of academia is essential.

Specific areas that warrant greater input include the following:

• Correlation and interdependence: risks within a class were assumed to be independent – simulation (e.g. common shock model) would be useful for understanding interdependence with respect to business interruption.

• Information asymmetry: anti-selection (e.g. different limits attracting different types, levels of risk) could be explored using SERFF ILFs (e.g. Hanover, 2015) which differentiate by turnover; or considering class D divided by customer churn); empirical insight into the notion of secondary loss (Bandyopadhyay et al., Reference Bandyopadhyay, Mookerjee and Rao2010) and associated asymmetries (e.g. insureds’ claiming strategy) could be investigated in terms of “retention factors” (for pricing different deductibles).