2.1 Preliminaries

Consider an insurance market with two types of policies, A and B. These policies are from two different business lines, and it is assumed that there is a single risk profile of policyholders in each business line. The per-policy present values of future payments to policyholders in each business line are denoted by $V_{A}$ and $V_{B}$ , respectively. The positive and finite expected values of $V_A$ and $V_B$ are denoted by $\pi_A$ and $\pi_B$ , respectively. The corresponding positive and finite standard deviations are denoted by $\sigma_A$ and $\sigma_B$ . Without loss of generality, the ratio $b=\frac{\sigma_B\pi_A}{\sigma_A\pi_B}$ is assumed to satisfy $b\geq1$ , i.e. $\frac{\sigma_B}{\pi_B}\geq\frac{\sigma_A}{\pi_A}$ , which means that business line B has a higher standard deviation per unit of average benefit. Business line B is sometimes said to be riskier than business line A. The correlation coefficient between $V_A$ and $V_B$ is denoted by $\rho$ , with $-1\leq \rho<1$ ; the liabilities $V_A$ and $V_B$ are not perfectly positively correlated.

The policies are underwritten in exchange for single premiums paid at policy issue. The premiums are composed of a pure premium derived from the actuarial equivalence principle, and a safety loading, or risk premium. This loading allows the insurer to compensate for the un-diversifiable part of the risk and takes into account the fact that the actual realisations of $V_A$ and $V_B$ are likely to depart from their expected values; more on premium principles can be found in Kaas et al. (Reference Kaas, Goovaerts, Dhaene and Denuit2008) and Zweifel and Eisen (Reference Zweifel and Eisen2012), among others. Here, the loading is assumed to be proportional to the expected value.

Two actuarial pricing strategies are considered. The first one is when the loadings are determined separately based on the features of each contract, and this approach is referred to as stand-alone pricing. The second one is when an insurer active on both business lines exploits their offsetting relationship by charging the same loading for both contracts, and this approach is referred to as joint pricing. The two approaches are described in the remainder of this section.

2.2 Stand-alone pricing

The pure premiums correspond to the expected present values of the contracts, i.e. $\pi_A$ and $\pi_B$ . Charging a loading allows the insurance company to reduce its exposure to a certain level. In case of stand-alone pricing, the loadings are set according to the specific risk of each contract. The loss random variables without loadings are given by $V_A-\pi_A$ and $V_B-\pi_B$ . It is assumed that the insurer determines the loaded premiums such that the loss, measured by some appropriate risk measure, for each contract separately is reduced by a certain factor chosen by the insurer. This implies that the insurance company charges policyholders for (part of) the risk in the form of a safety loading.

The stand-alone loaded premiums for contracts A and B are denoted by $P_{A}^{\textit{sa}}$ and $P_{B}^{\textit{sa}}$ , respectively, with:

\begin{equation*} P_A^{\textit{sa}} = \left(1 + \psi_A\right)\pi_A, \qquad \text{ and } \qquad P_B^{\textit{sa}} = \left(1 + \psi_B\right)\pi_B,\end{equation*}

where $\psi_A$ and $\psi_B$ are the loadings for each contract. Denoting by $\zeta\in(0,1)$ the risk reduction factor set by the insurer, and by $\varphi$ the risk measure, the risk reduction equations of $P_{A}^{\textit{sa}}$ and $P_{B}^{\textit{sa}}$ are written as follows:

\begin{equation*} \varphi\left[V_{A} - P_{A}^{\textit{sa}}\right]=\left(1-\zeta\right)\varphi\left[V_{A} - \pi_{A}\right],\end{equation*}

and

\begin{equation*} \varphi\left[V_B - P_B^{\textit{sa}}\right]=\left(1-\zeta\right)\varphi\left[V_B - \pi_B\right].\end{equation*}

These equations mean that the insurer’s risk with the loading (i.e. $\varphi\left[V-P^{sa}\right]$ ) is reduced by a factor $\zeta$ compared to the case where there is no loading (i.e. $\varphi\left[V-\pi\right]$ ).

Different risk measures can be used in practice. For instance, the value-at-risk would allow to reduce the insurer’s risk for some specific quantile level. This measure is used in practice to determine regulatory capital for financial institutions in many countries. It is however also subject to some criticisms, for instance because it is not sub-additive, i.e. the quantile of the sum is not necessarily less than the sum of the quantiles. The conditional value-at-risk, prescribed in, e.g., the Swiss Solvency Test, is sub-additive. However, the conditional value-at-risk would not allow to derive explicit closed-form expressions, especially in the first part of the present analysis.

This paper uses the mean-standard deviation (MSD) risk measure, defined for some random variable V as follows:

\begin{equation*}\varphi[V]=\pi+\gamma\sigma,\qquad \text{for }\gamma>0,\end{equation*}

where $\pi$ and $\sigma$ are the expectation and standard deviation of V, respectively. The choice of this risk measure is justified by four arguments. First, it satisfies positive homogeneity (i.e. $\varphi[aV]=a\varphi[V]$ for $a>0$ ), translation invariance (i.e. $\varphi[V+a]=\varphi[V]+a$ ), as well as sub-additivity. Second, it is convenient to manipulate and provides deeper insights into the dynamics of the offsetting relationship between the contracts. Third, taking into account that it is sometimes used as a premium principle, the standard deviation is meaningful in the context of pricing, as in Chen et al. (Reference Chen, Cheung, Choi and Yam2020); see also Asimit and Boonen (Reference Asimit and Boonen2018) for a recent applications. Fourth, as illustrated in section 5 and depending on the marginal distributions of $V_A$ and $V_B$ , the coefficient $\gamma$ can be tuned to match with great precision the results from other risk measures, such as the (conditional) value-at-risk. Nevertheless, this later property does not hold for all marginal distributions of $V_A$ and $V_B$ .

Solving the risk reduction equations using the MSD risk measure leads to:

\begin{equation*}P_{A}^{\textit{sa}} = \left(1+\zeta\gamma \frac{\sigma_{A}}{\pi_A}\right)\pi_{A}, \qquad \text{ and } \qquad P_B^{\textit{sa}} = \left(1+\zeta\gamma \frac{\sigma_{B}}{\pi_B}\right)\pi_B,\end{equation*}

and the notations $\psi_A=\zeta\gamma\frac{\sigma_A}{\pi_A}$ and $\psi_B=\zeta\gamma\frac{\sigma_B}{\pi_B}$ are introduced. Note that since $b\geq 1$ , it immediately follows that $\psi_B\geq \psi_A$ .

2.3 Joint pricing

Consider now the case where the insurer exploits the offsetting relationship between the two business lines. The notations $N_{A}$ and $N_B$ , with $N_A,N_B>0$ , are used for the number of underwritten policies in business lines A and B, respectively. The proportion of policies sold in business line B is denoted by $n=\frac{N_B}{N_{A}+N_B}$ . Note that there are two possible assumptions regarding the set to which $N_A$ and $N_B$ belong. Namely, $N_A,N_B \in \mathbb{N}^{*}$ , or $N_A,N_B \in \mathbb{R}^{*}_+$ . The analysis in section 3 focuses essentially on the proportion n, and hence, specifying the choice of the assumption is not necessary. On the other hand, in section 4, the model used for the numbers $N_A$ and $N_B$ assumes that they belong to $\mathbb{R}^{*}_+$ .

Determining the loaded premiums when the contracts are priced jointly requires some knowledge about the numbers $N_{A}$ and $N_B$ . However, these quantities are unknown when the loaded premiums are set, and are likely to be impacted by the prices. Therefore, the loaded premiums in the present and following sections shall be interpreted as the conditional premiums associated with some values of $N_{A}$ and $N_B$ . These conditional premiums may as well be interpreted as the required premiums associated with the value of n as it unfolds. Section 4 introduces further assumptions on these numbers.

Let $P_{A}^{\textit{jp}}(n)$ and $P_{B}^{\textit{jp}}(n)$ be the required loaded premiums when the insurer prices the two business lines jointly, such that:

\begin{equation*}P_{A}^{\textit{jp}}(n)=\left(1+\psi(n)\right)\pi_{A}, \qquad \text{ and } \qquad P_B^{\textit{jp}}(n)=\left(1+\psi(n)\right)\pi_B,\end{equation*}

where $\psi(n)$ is the required loading proportion associated with n and is the same for both contracts.

The required loaded premiums $P_{A}^{\textit{jp}}(n)$ and $P_B^{\textit{jp}}(n)$ can be determined analogously to the stand-alone case, implying that the insurer has the same risk reduction regardless of the pricing method. In particular, the loading for the combined portfolio is such that the overall loss is reduced by the same factor $\zeta$ . Thus, $\psi(n)$ satisfies:

\begin{equation*} \varphi\left[N_{A}\left(V_{A}-P_{A}^{\textit{jp}}(n)\right) + N_B\left(V_B-P_B^{\textit{jp}}(n)\right)\right]=(1-\zeta)\varphi\left[N_{A}\left(V_{A}-\pi_{A}\right) + N_B\left(V_B-\pi_B\right)\right].\end{equation*}

Using the MSD risk measure, the required conditional risk premium is given by:

(2.1) \begin{equation} \psi(n) = \psi_A\left(\lambda_1\tilde{n}^2 - 2\lambda_2\tilde{n} + 1\right)^{\frac{1}{2}},\end{equation}

where $\tilde{n} = \frac{n\pi_B}{(1-n)\pi_A+n\pi_B}$ , and

\begin{equation*}\lambda_1 = 1 + b^2 - 2b\rho, \qquad \text{ and } \qquad \lambda_2 = 1 - b\rho.\end{equation*}

The overall risk of the joint pricer is also reduced under the present setting. In particular, the difference between the overall portfolio risk when the contracts are priced separately and the overall portfolio risk when the contracts are priced jointly is:

\begin{equation*}(1-\zeta)\gamma\left(N_A\sigma_A+N_B\sigma_B - \sigma_{\textit{ptf}}\right),\end{equation*}

where $\sigma_{\textit{ptf}}$ is the standard deviation of $N_AV_A+N_BV_B$ . Since the standard deviation is sub-additive, this difference is always positive. Thus, on top of the potential competitive advantage that the insurer could achieve with the joint loading $\psi(n)$ , there is also a reduction in the level of risk. In an alternative setting, it is possible to include this difference in the joint pricing, which would accentuate the potential competitive advantage of the joint pricer. Note that depending on the marginal distributions of $V_A$ and $V_B$ , this observation may not hold if the MSD risk measure is replaced by the value-at-risk.

3.1 Competitiveness region

The joint pricer is said to have a competitive advantage over stand-alone pricers if the policies can be more favorably priced using the offsetting relationship between their liabilities. This would be the case if there exist values of $n\in(0,1)$ such that $\psi(n)<\psi_A$ and $\psi(n)<\psi_B$ . Since $\psi(0)=\psi_A$ , $\psi(1)=\psi_B$ and $b\geq1$ , a competitiveness region exists if the minimum value of $\psi(n)$ is less than $\psi_A$ .

The first question investigated in this section is whether such a competitiveness region exists. Namely, whether there exists a range of values of the proportion n such that the joint loading $\psi(n)$ is smaller than the smallest stand-alone loading $\psi_A$ . The following lemma provides a condition on the contracts’ features such that a competitiveness region always exists, i.e. such that there exists n satisfying $\psi(n)<\psi_A$ . A proof can be found in Appendix A.1.

Lemma 3.1. A competitiveness region exists for the joint pricer if and only if:

(3.1) \begin{equation}b\rho<1.\end{equation}

Under Condition (3.1), the minimum loading proportion $\psi^{\min}$ is given by:

(3.2) \begin{equation}\psi^{\min} = \psi(n^{\min}) =\psi_A\sqrt{\frac{\lambda_1-\lambda_2^2}{\lambda_1}}<\psi_A,\end{equation}

where $n^{\min}=\frac{\lambda_{2}\pi_A}{\lambda_{2}\pi_A+\left(\lambda_1-\lambda_2\right)\pi_B }\in(0,1)$ , with $n^{\min}=0$ for $b\rho=1$ and $n^{\min}=1$ for $b=\rho=1$ , whereas the maximum loading proportion is given by $\psi^{\max}=\psi(1)=\psi_B$ .

Along the lines of previous research on optimal product mix, Lemma 3.1 states that there exists a unique proportion of underwritten businesses which minimises the value of the required loaded premium. Through the lens of pricing, this result implies the existence of a competitiveness region for the joint pricer. More specifically, in order for the premium to be sufficient and for the joint pricer to gain a competitive advantage over stand-alone pricers, the actual premium for each business line should be at least equal to the minimum required loaded premium, and at most equal to the stand-alone premium. Note that for $\pi_A=\pi_B$ and $\sigma_A = \sigma_B$ , it follows that $n^{\min}=\frac{1}{2}$ . This means that when the contracts have identical expected values and standard deviations, the minimum required loading is reached when the portfolio is equally balanced. If in addition, the two business lines exhibit an extreme negative correlation $\rho=-1$ , then it follows from the expressions of $\lambda_1$ and $\lambda_2$ that $\psi^{\min}=0$ , i.e. the minimum loaded premiums correspond to the pure premiums.

A sufficient condition for the existence of the competitiveness region is a negative correlation between the liabilities of the business lines. Indeed, Condition (3.1) is always satisfied for $\rho\leq0$ . A prime example is when one business line pays a survival benefit and the other pays a death benefit. This is the typical case on which most of the literature on natural hedging has focused. Lemma 3.1, however, states that the existence of the competitiveness region is not limited to contracts with negatively correlated liabilities. Joint pricing can be implemented when there is a positive correlation as well, as long as Condition (3.1) is satisfied.

Recall that by definition, $b=\frac{\sigma_B \pi_A}{\sigma_A \pi_B}$ with $b\geq 1$ , i.e. $\frac{\sigma_B}{\pi_B}\geq \frac{\sigma_A}{\pi_A}$ . Condition (3.1) gives $\frac{\sigma_B}{ \pi_B}\rho<\frac{\sigma_A}{\pi_A}$ . Thus, Condition (3.1) can be interpreted as a requirement that the diversification effect between the business lines, which is captured in the correlation $\rho$ , should decrease the risk of the riskiest business line B relatively to business line A in order to limit the effect of subsidisation across the two lines. An example where this cannot occur is when the liabilities $V_A$ and $V_B$ are highly positively correlated. In which case, the diversification effect would be too small to reduce the gap of riskiness between the two business lines. For instance, the competitiveness region does not exist for $\rho=1$ .

Let $P^{\kern1pt\star}_A$ and $P^{\star}_B$ be the actual loaded premiums set by the joint pricer for the respective contracts, such that $P^{\star}_A=(1+\psi^{\star})\pi_A$ and $P^{\kern1pt\star}_B=(1+\psi^{\star})\pi_B$ , with $\psi^{\star}\in\left[\psi^{\min},\psi^{\max}\right]$ . In particular, $\psi^{\star}$ is a loading under joint pricing, which may be set within the competitiveness region, i.e. $[\psi^{min},\psi_A)$ , or outside of that region, i.e. $[\psi_A,\psi_B]$ . Suppose that the condition $b\rho<1$ is satisfied. Despite the potential competitive advantage of the joint pricer, setting a portfolio loading $\psi^{\star}$ within the competitiveness region raises two issues, to which the remainder of this section is devoted. The first one is portfolio monitoring, which originates from the convexity of the function $\psi(n)$ . The second one is the existence of a critical threshold beyond which the joint pricer loses its competitive advantage on business line A, and this arises when $\psi_A\neq\psi_B$ as well.

3.2 Portfolio monitoring

One implication of the convexity of $\psi(n)$ for joint pricers is that the choice of competitiveness comes with the burden of portfolio monitoring. More specifically, in order for the required premium to be lower than or equal to the actual premium (i.e. $P^{\kern1pt\star}_A$ or $P^{\star}_B$ ), the proportion n has to be maintained within the interval $[n^{\star}_{l},n^{\star}_{u}]$ , where $n^{\star}_{l}$ and $n^{\star}_{u}$ are solutions of the equation:

\begin{equation*} \psi\left(n\right) = \psi^{\star}.\end{equation*}

Solving the second-order equation $\psi(\tilde{n})^2=\left(\psi^{\star}\right)^2$ for $\tilde{n}$ , and then determining the corresponding solution for n, leads to:

\begin{equation*}n^{\star}_l = \frac{\tilde{n}^{\star}_l\pi_A}{\tilde{n}^{\star}_l\pi_A+(1-\tilde{n}^{\star}_l)\pi_B} \qquad \text{ and } \qquad n^{\star}_u = \frac{\tilde{n}^{\star}_u\pi_A}{\tilde{n}^{\star}_u\pi_A+(1-\tilde{n}^{\star}_u)\pi_B},\end{equation*}

with

(3.3) \begin{equation}\tilde{n}^{\star}_{l} = \frac{\lambda_2}{\lambda_1} - \sqrt{ \frac{\lambda_2^2}{\lambda_1^2} - \frac{1}{\lambda_1}\left(1-\left(\frac{\psi^{\star}}{\psi_{A}}\right)^2\right)},\end{equation}

and

(3.4) \begin{equation}\tilde{n}^{\star}_{u} = \frac{\lambda_2}{\lambda_1} + \sqrt{ \frac{\lambda_2^2}{\lambda_1^2} - \frac{1}{\lambda_1}\left(1-\left(\frac{\psi^{\star}}{\psi_{A}}\right)^2\right)}.\end{equation}

In case the proportion n remains within the interval $[n^{\star}_{l},n^{\star}_{u}]$ , the actual loaded premiums $P^{\kern1pt\star}_A$ and $P^{\star}_B$ would be higher than their required counterparts $P^{\textit{jp}}_A(n)$ and $P^{\textit{jp}}_B(n)$ . Otherwise, the joint pricer would be underwriting at a loss for a given risk reduction factor $\zeta$ .

The flexibility of the insurer in terms of portfolio monitoring can be measured by the length of the interval $[n^{\star}_{l},n^{\star}_{u}]$ , i.e. $n^{\star}_u-n^{\star}_l$ . Suppose that the joint pricer sets the actual premiums equal to the lowest possible ones given by the competitiveness region, i.e. $\psi^{\star}=\psi^{\min}$ . In this case, $\tilde{n}^{\star}_u=\tilde{n}^{\star}_l=\frac{\lambda_2}{\lambda_1}$ , and in particular, $n^{\star}_u=n^{\star}_l=n^{\min}$ . Hence, the length of this interval is 0. This means that setting the premium equal to the lower bound of the competitiveness region, as this may arise from an analogy with previous studies on optimal product mix, is in fact the most risky choice for the insurer. More precisely, setting $\psi^{\star}=\psi^{\min}$ implies that there is only a single proportion n for which the conditional loaded premium matches the actual one.

3.3 Critical threshold

For $\psi^{\star}\in[\psi^{\min},\psi_A)$ , the joint pricer has a competitive advantage over stand-alone pricers on both business lines. For $\psi^{\star}\in(\psi_A,\psi_B)$ , the joint pricer has a competitive advantage on business line B, and a competitive disadvantage on business line A. Therefore, the required loaded premium for the contract with the highest standard deviation per unit of average benefit will always be lower under joint pricing compared to its stand-alone counterpart. In contrast, for the contract with the lowest risk per unit of average benefit, there always exists a critical threshold in the proportion n beyond which the required loaded premium under joint pricing is higher than its stand-alone counterpart. In other words, beyond the critical threshold, the safer business line subsidises the riskier one. This is formally stated in the following lemma; see Appendix A.2 for a proof.

Lemma 3.2. For $b> 1$ , there always exists $n_{\textit{ct}}\in(0,1)$ given by:

(3.5) \begin{equation}n_{\textit{ct}} = \frac{2 \lambda_2\pi_A}{2 \lambda_2\pi_A + (\lambda_1-2 \lambda_2)\pi_B},\end{equation}

such that:

\begin{equation*}\left\{\begin{array}{l} \psi(n)< \psi_A, \qquad \text{ for } n <n_{\textit{ct}},\\ \psi(n)>\psi_A, \qquad \text{ for } n >n_{\textit{ct}}, \end{array}\right.\end{equation*}

with $\psi(n_{\textit{ct}})=\psi_A$ .

The existence of the critical threshold is all the more pertinent given that the competitive advantage of the joint pricer is higher on business B than on business line A. Indeed, denoting again by $P^{\star}_A$ and $P^{\star}_B$ the actual loaded premiums set by the joint pricer for the respective contracts, an immediate consequence of $b>1$ is that:

\begin{equation*}\frac{P^{\star}_A - P^{\textit{sa}}_A}{P^{\textit{sa}}_A}>\frac{P^{\star}_B - P^{\textit{sa}}_B}{P^{\textit{sa}}_B}.\end{equation*}

This implies that the joint pricer could attract more policyholders on business B than on business line A. Thus, the actual proportion n may turn out to be close to 1, and hence, beyond the critical threshold.

A practical solution for the joint pricer to cope with the issues raised by the critical threshold consists in constraining some contract features (e.g. the benefit amounts) such that $b=1$ , or equivalently, $\psi_A=\psi_B$ . In this case, the joint pricer has the same competitive advantage on both business lines, leading to $n_{\textit{ct}}=1$ , which means that there is no critical threshold. In practice, it may not always be possible to control all contract features. Nevertheless, insurers could still identify pairs of business lines satisfying $\psi_A\approx \psi_B$ and $b\rho<1$ without constraining contract features.

4.1 Market model

Consider a market with $N^{T}_{A}$ and $N^{T}_{B}$ policyholders willing to buy contracts A and B, respectively. The proportion $w^{d}=\frac{N^{T}_{B}}{N^{T}_{A}+N^{T}_{B}}$ represents the market proportion of demand for business line B. The market is composed of $k_{A}$ providers of contract A and $k_{B}$ providers of contract B, with $2\leq k_{A}<N^{T}_{A}$ and $2 \leq k_{B}<N^{T}_{B}$ . Among these insurers, it is assumed that there is a single insurer considering the joint pricing strategy. Thus, there are $k_{A}-1$ insurers providing contract A at its stand-alone price $P^{\textit{sa}}_A$ , and $k_{B}-1$ insurers providing contract B at its stand-alone price $P^{\textit{sa}}_B$ .

The joint pricer sets the premiums $P^{\star}_{A}$ and $P^{\kern1pt\star}_B$ for the respective business line, such that:

\begin{equation*}P^{\star}_{A}=\left(1+\psi^{\star}\right)\pi_{A}, \qquad \text{ and } \qquad P^{\star}_{B}=\left(1+\psi^{\star}\right)\pi_{B},\end{equation*}

where $\psi^{\star}\in\left[\psi^{\min},\psi_B\right]$ is derived according to the risk reduction constraint introduced in section 2. The quantities $c^{\star}_{A}$ and $c^{\star}_B$ are defined as follows:

\begin{equation*}c^{\star}_{A}=\frac{P^{\kern1pt\star}_{A}}{P^{\textit{sa}}_{A}}-1, \qquad \text{ and }\qquad c^{\star}_B=\frac{P^{\star}_B}{P^{\textit{sa}}_B}-1.\end{equation*}

The joint pricer is said to have a competitive advantage over stand-alone pricers on one of the business lines if the corresponding $c^{\star}$ is negative. The lower the values of $c^{\star}_{A}$ or $c^{\star}_B$ , the higher the competitive advantage on the corresponding business line.

Let $N^{\star}_A$ and $N^{\star}_B$ be the numbers of policyholders attracted by the joint pricer for the loading $\psi^{\star}$ . Using the MSD risk measure, and following the reasoning that leads to (2.1), the portfolio loading proportion $\psi^{\star}$ solves the equation:

(4.1) \begin{equation}\psi^{\star}=\psi_A\left(\lambda_1\tilde{n}^{\star2} - 2\lambda_2\tilde{n}^{\star} + 1\right)^{\frac{1}{2}},\end{equation}

with $\tilde{n}^{\star}=\frac{n^{\star}\pi_B}{(1-n^{\star})\pi_A+n^{\star}\pi_B}$ , and $n^{\star}=\frac{N^{\star}_B}{N^{\star}_A+N^{\star}_B}$ is the proportion of policies B underwritten by the insurance company active on both business lines.

The loading proportion $\psi^{\star}$ depends on the numbers of policyholders who buy each contract, which in turn depend on $\psi^{\star}$ . In particular, the loading $\psi^{\star}$ appears in both sides of Equation (4.1). Different functions could be applied to model this relationship. It is assumed that for $c^{\star}_{A}=c^{\star}_B=0$ , the total number of policyholders is shared equally among all insurance companies. This means that in case the insurance company active on both business lines prices the contracts separately, then all companies active on A underwrite $\frac{N^{T}_{A}}{k_{A}}$ contracts, and all those active on B underwrite $\frac{N^{T}_{B}}{k_{B}}$ contracts.

A candidate function satisfying this property is the logistic function, such that the numbers $N^{\star}_{A}$ and $N^{\star}_{B}$ of contracts A and B sold by the joint pricer are given by:

\begin{equation*}\begin{array}{ll}N^{\star}_A &= N^T_A \left(1 - \frac{k_A-1}{k_A-1 +\exp\left(-q_Ac^{\star}_A\right)}\right),\\[5pt] N^{\star}_B &= N^T_B \left(1 - \frac{k_B-1}{k_B-1 +\exp\left(-q_Bc^{\star}_B\right)}\right).\end{array}\end{equation*}

where $q_A>0$ and $q_B>0$ are the reaction factors of policyholders in each business line. These reaction factors are related to the joint pricer’s price elasticity of demand, such that a 1% change in the price of a given contract is perceived by policyholders willing to buy the corresponding contract from the joint pricer as a $q\%$ change.

The logistic function is bounded from below by 0, and from above by the total market demand. In other words, the number of policyholders attracted by the joint pricer is at least 0 for $\psi^{\star}\rightarrow +\infty$ , and at most equal to $N^T_A$ and $N^{T}_B$ for $\psi^{\star}\rightarrow -\infty$ . However, these bounds are unlikely to be reached. Indeed, under the present setting, and provided the insurer does not underwrite at a loss, realistic values of $\psi^{\star}$ lie within the interval $\left[\psi^{\min},\psi_B\right]$ , which is expected to be relatively narrow. Thus, for sufficiently small values of $c^{\star}_A$ and $c^{\star}_B$ , the cumbersomeness of the logistic function can be circumvented by applying a Taylor expansion, which leads to the following conveniently linear demand functions:

\begin{equation*}N^{\star}_{A} = \frac{N^{T}_{A}}{k_A}\left(1 - \frac{k_{A}-1}{k_{A}}q_{A}c^{\star}_{A}\right), \qquad \text{ and } \qquad N^{\star}_{B} = \frac{N^{T}_{B}}{k_B}\left(1 - \frac{k_{B}-1}{k_{B}}q_{B}c^{\star}_{B}\right).\end{equation*}

4.2 Pricing decision

The decision criterion used to determine the optimal pricing strategy is the total collected premiums. In case the insurer prices the contracts jointly, the total premiums collected in each business line are $N_{A}^{\star}P^{\star}_{A}$ and $N_{B}^{\star}P^{\star}_{B}$ . In case the insurer prices the contracts separately, the total market demand is shared equally among all insurers, and hence, the total premiums collected in each business line by each of the $k_A$ and $k_B$ insurers are $\frac{N^{T}_{A}}{k_A}P^{\textit{sa}}_{A}$ and $\frac{N^{T}_{B}}{k_B}P^{\textit{sa}}_B$ , respectively. Therefore, the difference between the total collected premiums with and without joint pricing is given by:

\begin{equation*}\mathcal{D}_{\textit{ptf}} = \left(N_A^{\star}P^{\kern1pt\star}_A - \frac{N^{T}_A}{k_A}P^{\textit{sa}}_A\right)+\left(N_B^{\star}P^{\star}_B - \frac{N^{T}_B}{k_B}P^{\textit{sa}}_B\right).\end{equation*}

Consider the proportion $\eta$ defined as follows:

(4.2) \begin{equation}\eta = \frac{N^T_B\frac{1}{k_B}\pi_B(\psi_B-\psi^{\min})}{N^T_A\frac{1}{k_A}\pi_A(\psi_A-\psi^{\min})+N^T_B\frac{1}{k_B}\pi_B(\psi_B-\psi^{\min})}.\end{equation}

Consider as well the critical threshold $w_{\textit{ct}}$ in the proportion of market demand for contracts B:

(4.3) \begin{equation} w_{\textit{ct}}=\frac{n_{\textit{ct}}}{n_{\textit{ct}} + (1-n_{\textit{ct}})\left(1+\frac{k_B-1}{k_B}\frac{\psi_B-\psi_A}{1+\psi_B}q_B\right)\frac{k_A}{k_B}},\end{equation}

where $n_{\textit{ct}}$ is the critical threshold derived from the conditional analysis in (3.5). In particular, $w_{\textit{ct}}$ corresponds to the proportion of market demand for contracts B (i.e. $w^{\textit{d}}$ ) such that $\psi^{\star}=\psi_{A}$ . It follows that $\psi^{\star}<\psi_A$ for $w^{\textit{d}}<w_{\textit{ct}}$ , and $\psi^{\star}>\psi_A$ for $w^{\textit{d}}>w_{\textit{ct}}$ .

The following theorem provides sufficient conditions on policyholders’ reaction factors to support the decision on the pricing strategy. These conditions involve information on the market demand and supply, as well as on the contracts’ features. A proof is given in Appendix A.3, which also contains the derivations for $\eta$ and $w_{\textit{ct}}$ .

Theorem 4.1. An insurer active on both business lines A and B collects more premiums at portfolio level by pricing them jointly rather than separately, if the following conditions are satisfied:

(4.4) \begin{equation}q_B >\frac{k_B}{k_B-1}\frac{1+\psi_B}{1+\psi_A},\end{equation}

and

(4.5) \begin{equation}\left\{\begin{array}{ll} (1-\eta)q_A\frac{k_A-1}{k_A}\frac{1+\psi^{\min}}{1+\psi_A}+\eta q_B\frac{k_B-1}{k_B}\frac{1+\psi^{\min}}{1+\psi_{B}}>1, & \qquad \text{for } w^d<w_{\textit{ct}},\\ q_A <\frac{k_A}{k_A-1}\frac{1+\psi_A}{1+\psi_B},& \qquad \text{for } w^d>w_{\textit{ct}}, \end{array}\right.\end{equation}

and inequality (4.4) is sufficient for $w^d=w_{\textit{ct}}$ .

An insurer active on both business lines A and B collects more premiums at portfolio level by pricing them separately rather than jointly, if the following conditions are satisfied:

(4.6) \begin{equation}q_B<\frac{k_B}{k_B-1},\end{equation}

and

(4.7) \begin{equation}\left\{\begin{array}{ll}q_A<\frac{k_A}{k_A-1},& \qquad \text{for } w^d<w_{\textit{ct}},\\ q_A>\frac{k_A}{k_A-1},& \qquad \text{for } w^d>w_{\textit{ct}},\end{array}\right.\end{equation}

and inequality (4.6) is sufficient for $w^d=w_{\textit{ct}}$ .

Theorem 4.1 states that the choice of the pricing strategy can be inferred from the values of the reaction factors $q_A$ and $q_B$ . In general, it is possible to determine the values of the contract features, the number of competitors $k_A$ and $k_B$ , and the proportion of market demand $w^d$ . Thus, the remaining task of the insurer is to estimate $q_A$ and $q_B$ based on its own experience.

These conditions imply that the insurer should reduce (resp. increase) its premiums only if the reaction of policyholders is high (resp. low) enough in order for the premium reduction (resp. increase) to result in an increase of the total premiums. In case the joint pricer does not expect to attract a sufficient number of policyholders in business line B, then joint pricing is not necessarily desirable. Another important implication of Theorem 4.1 is that even if business line A subsidises B (i.e. beyond the critical threshold), joint pricing may still be rewarding for the insurer in terms of total collected premiums.

The following corollary, whose proof can be found in Appendix A.4, provides simpler rules to support the decision on the pricing strategy.

Corollary 1. An insurer active on both business lines collects more premiums at portfolio level by pricing both contracts jointly rather than separately, if the following conditions are satisfied:

\begin{equation*}q_B >2\frac{1+\psi_B}{1+\psi^{\min}},\end{equation*}

and

\begin{equation*}\left\{\begin{array}{ll} q_A>2\frac{1+\psi_A}{1+\psi^{\min}}, & \qquad \text{for } w^d<w_{\textit{ct}},\\ q_A<\frac{1+\psi_A}{1+\psi_B},& \qquad \text{for } w^d>w_{\textit{ct}}. \end{array}\right.\end{equation*}

An insurer active on both business lines collects more premiums at portfolio level by pricing both contracts separately rather than jointly, if the following conditions are satisfied:

\begin{equation*}q_B<1,\end{equation*}

and

\begin{equation*}\left\{\begin{array}{ll}q_A<1,& \qquad \text{for } w^d<w_{\textit{ct}},\\ q_A>2,& \qquad \text{for } w^d>w_{\textit{ct}}.\end{array}\right.\end{equation*}

From Corollary 1, the results of this section can be summarised as follows. Consider an insurance company active on business lines A and B, such that $\psi_A\leq\psi_B$ . Suppose that this insurance company aims at increasing its total collected premiums by pricing business lines A and B jointly. The company could do so if:

1. • business line B is relatively elastic, and

2. • business line A is relatively elastic if it has a lower demand than B, or is relatively inelastic if it has a higher demand than B.

5.1 Business lines, simulation models, and data

The random variables $V_A$ and $V_B$ are the per-policy random present values at policy issue of the annuity and assurance business lines, respectively, such that:

\begin{equation*} V_{A}=C_{A} \sum_{k=1}^{T_{A}}\text{ }_kp_x^{(A)}v(0,k),\end{equation*}

and

\begin{equation*} V_{B}=C_{B} \sum_{k=1}^{T_{B}}\text{ }_{k-1}p_y^{(B)}\left(1-p_{y+k-1}^{(B)}\right)v(0,k),\end{equation*}

where $T_A$ and $T_B$ are the terms of the annuity and the assurance, respectively, $C_A$ is the yearly annuity benefit payable at the end of the year, and $C_B$ is the death benefit payable at the end of the year of death. The probability $\text{ }_kp_x^{(A)}$ is the k-year survival probability in the term annuity business line, where all policyholders are aged x. The probabilities $\text{ }_{k-1}p_y^{(B)}$ and $p_{y+k-1}^{(B)}$ are the $(k-1)$ -year and 1-year survival probabilities, respectively, in the term assurance business line. In order to take into account longevity and mortality risks, all probabilities are assumed to be random variables. Market risk is not included in this analysis, and the discounting factor v(0,k) is assumed to be constant, with $v(0,k)=v^k$ .

The group of policyholders in the annuity business line (i.e. business line A) are all aged $x=60$ , whereas the group of policyholders in the assurance business line (i.e. business line B) are all aged $y=30$ . All contracts are assumed to expire after 30 years, i.e. $T_A=T_B=30$ .

The distributions of $V_A$ and $V_B$ are obtained by simulating the survival probabilities $p^{(A)}$ and $p^{(B)}$ from the Li-Lee two-population model (Li and Lee Reference Li and Lee2005), which allows to take into account the dependence between the mortality in the two business lines. Specifically, under the Li-Lee model, the central death date $\mu^{(i)}_{x,t}$ at age x and time t in population i is given by:

(5.1) \begin{equation} \mu^{(i)}_{x,t} = \exp\left(\alpha_x^{(i)} + \beta_x^{(i)}\kappa_t^{(i)} + \beta_x\kappa_t\right),\end{equation}

where $\beta_x\kappa_t$ represents the common mortality improvement for a given age x, and $\beta_x^{(i)}\kappa_t^{(i)}$ represents the population-specific mortality improvement for that age. The processes $\kappa_t^{(A)}$ , $\kappa_t^{(B)}$ , and $\kappa_t$ are simulated from correlated random walks.

The English and Welsh population data are used for the annuity business line, and the US population data are used for the term assurance business line. Using data from two different populations allows to incorporate in the analysis the fact that mortality experience in annuity and assurance businesses tend to be different. The data sets cover the period 1950–2018 and ages 30–90 and were obtained from the Human Mortality Database (www.mortality.org). The models are estimated using singular value decomposition.

5.2 Competitiveness region

Table 1 reports the values of the pure premiums per unit of benefits (i.e. $\pi_A$ and $\pi_B$ for $C_A=C_B=1$ ), the corresponding standard deviations, the ratio of the pure premiums and standard deviations (i.e. $\psi$ for $\zeta \gamma=1$ ), and the correlation coefficient $\rho$ . The table shows that the business lines satisfy the condition $\psi_A<\psi_B$ , i.e. $b>1$ . Further, since the correlation coefficient is negative, the condition $b\rho<1$ from (3.1) is satisfied, meaning that the competitiveness region exists.

Table 1. Values of the pure premiums per unit of benefits (i.e. $\pi_A$ and $\pi_B$ for $C_A=C_B=1$ ), the corresponding standard deviations, the ratio of the pure premiums and standard deviations (i.e. $\psi$ for $\zeta\gamma=1$ ), and the correlation coefficient $\rho$ .

Figure 1 displays the premium loadings under joint pricing $\psi(n)$ in function of the proportion of term assurances n for four combinations of the benefits $C_A$ and $C_B$ and the reduction factor $\zeta$ . The coefficient $\gamma$ is equal to $1.686$ , such that the loading $\psi(n)$ from the MSD risk measure (black curve) is approximately equal to the loading obtained numerically from the value-at-risk at the confidence level $0.95$ (blue circles). Note that the black curve and the blue circles are very close, which shows the flexibility of the MSD risk measure under the present assumptions. The horizontal dashed lines are the stand-alone loadings $\psi_A$ and $\psi_B$ , with $\psi_A<\psi_B$ .

Figure 1 Premium loadings under joint pricing $\psi(n)$ in function of the proportion of term assurances n for four combinations of the benefits $C_A$ and $C_B$ and the reduction factor $\zeta$ . The coefficient $\gamma$ is equal to $1.686$ , such that the loading $\psi(n)$ from the MSD risk measure (black curve) is approximately equal to the loading from the value-at-risk at the confidence level $0.95$ (blue circles). The horizontal dashed lines are the stand-alone loadings $\psi_A$ and $\psi_B$ , with $\psi_A<\psi_B$ .

The results from Figure 1 show how the competitiveness region is influenced by the relative levels of the benefits $C_A$ and $C_B$ . In particular, the minimum loading $\psi^{\min}$ as well as the critical threshold changes depending on the benefits. For instance, in case $C_B=100C_A$ (i.e. bottom-left panel), the critical threshold is lower than $0.5$ , which limits the flexibility of the joint pricer. On the other hand, for $C_B=10C_A$ (i.e. top- and bottom-right panels), the critical threshold is closer to 1, which means that setting the actual loading within the competitiveness region in this case requires less monitoring. Regarding the effect of the risk reduction factor $\zeta$ , the figure shows that it is essentially a scaling factor. For instance, comparing the top- and bottom-right panels where $C_B=10C_A$ for both, the critical threshold remains unchanged. Section 6 contains another illustration of the competitiveness region with further discussion.

5.3 Total collected premiums

Figure 2 displays the relative difference of the total collected premiums with and without joint pricing in function of the proportion $w^d$ of market demand for term assurances (business line B) for different combinations of the reaction factors $q_A$ and $q_B$ , where $C_B=10C_A$ , $\zeta=0.5$ , $\gamma=1.686$ , and $k_A=k_B=10$ . Positive (resp. negative) values indicate that joint pricing leads to higher (resp. lower) total collected premiums compared to stand-alone pricing.

Figure 2 Relative difference of the total collected premiums with and without joint pricing in function of the proportion $w^d$ of market demand for term assurances (business line B) for different combinations of the reaction factors $q_A$ and $q_B$ , where $C_B=10C_A$ , $\zeta=0.5$ , $\gamma=1.686$ , and $k_A=k_B=10$ . Positive (resp. negative) values indicate that joint pricing leads to higher (resp. lower) total collected premiums compared to stand-alone pricing.

For $q_A=q_B=0.5$ (black curve with circles), the total collected premiums are higher under stand-alone pricing when $q_B<1$ and $q_A<1$ for $w^d<w_{ct}$ , which is formally stated as a sufficient condition in Corollary 1. The figure also shows that stand-alone pricing remains the preferred strategy when $q_A<1$ for $w^d\geq w_{ct}$ , although it is not formally stated as a sufficient condition in the corollary. For $q_A=q_B=3$ (blue curve with squares), the opposite holds. Note that $2\frac{1+\psi_B}{1+\psi^{\min}}\approx 2.12$ and $2\frac{1+\psi_A}{1+\psi^{\min}}=2.01$ , which means that the conditions of Corollary 1 under which joint pricing is more favourable are satisfied for $w^d<w_{ct}$ . Again, even though $\frac{1+\psi_A}{1+\psi_B} \approx 0.95$ , and hence $q_A>\frac{1+\psi_A}{1+\psi_B}$ for $w^d>w_{ct}$ , joint pricing still leads to higher total collected premiums. A similar conclusion holds for $q_A=0.5$ and $q_B=3$ (red curve with triangles), where joint pricing leads to higher collected premiums for all $w^d$ . The case where $q_A=3$ and $q_B=0.5$ (magenta curve with stars) is the only one in Figure 2 where the choice of the pricing strategy depends on the critical threshold. In particular, for low proportions of market demand for term assurances, $q_A=3$ and $q_B=0.5$ leads to higher total collected premiums under joint pricing, whereas for high proportions of market demand for term assurances, stand-alone pricing leads to higher total collected premiums. Note that the latter case is stated as a sufficient condition in Corollary 1.

6.1 Data description

The Superintendence of Private Insurers (SUSEP), which is the executive arm of the national Brazilian insurance regulator, provides the Sistema de Estatísticas da SUSEP, or SUSEP Statistics System (SES) data set freely accessible online (see www.susep.gov.br). It consists in aggregate losses for different business lines obtained from SES at different frequencies. The frequency used in this analysis is bi-annual. The data cover the period from July/December 2006 to July/December 2019, with 27 observations per business line.

The raw data contain information for up to 154 business lines depending on the time period. Only non-life lines are studied. Further, business lines that were introduced or removed during the observation period were discarded. Among the remaining lines, only the top 10 business lines in terms of total collected premiums were selected. Table 2 lists the selected lines, together with their rank in terms of total collected premiums. The advantage of working with this relatively small subset of business lines is that it allows for an easier reporting. Indeed, the total number of possible pairs of business lines is 45.

Table 2. Business lines ordered based on the ratio $\psi=\sigma/\pi$ , where the business line on the first row has the highest $\psi$ . The first column corresponds to the identification code attributed by SUSEP. The second column contains the name of the business line translated to English from SUSEP website. The third column contains the rank of the business line in terms of total collected premiums at market level, where “1” is the highest.

All time series were de-trended, after which all passed the KPSS test of stationarity (Kwiatkowski et al. Reference Kwiatkowski, Phillips, Schmidt and Shin1992)Footnote ¹ . For each pair of business lines, Condition (3.1) is assessed to determine whether a competitiveness region exists. The random variable V stands for the aggregate losses, from which the expected values (i.e. $\pi$ ) and the standard deviations (i.e. $\sigma$ ) of the individual series are calculated to obtain the ratios $\psi=\frac{\sigma}{\pi}$ . The pairwise correlations $\rho$ are also calculated. For A and B such that $\psi_A\leq \psi_B$ , if $b\rho<1$ , where $b=\frac{\psi_B}{\psi_A}$ , the competitiveness region of these two contracts is said to exist, indicating that insurers active on these two business lines can potentially gain a competitive advantage by pricing them jointly.

6.2 Analysis of business lines

The evolution of the aggregate losses over time is displayed in Figure 3, and the corresponding ratios $\psi$ are displayed in Figure 4. The minimum value of pairwise correlations is $-0.34$ and is found between comprehensive residential insurance (0114) and carrier insurance (0654). The maximum correlation is $0.86$ between optional auto liability (0553) and auto property damage (0531). The high positive correlation for the latter pair complies with the intuition.

Figure 3 Evolution of the aggregate losses over time for different business lines using the SES data set, expressed in million Brazilian Reals, where the business lines are identified by their SUSEP code (see Table 2).

Figure 4 Ratio $\psi=\frac{\sigma}{\pi}$ , i.e. the standard deviation per unit of expected loss, for all business lines, where the business lines are identified by their SUSEP code (see Table 2).

Table 3 contains the output of the analysis. Among the 45 possible pairs, 34 pairs satisfy the condition $b\rho<1$ . This means that a large majority of $75.55\%$ of the possible pairs have a positive competitiveness region, and hence, can be priced jointly. Two business lines can be priced jointly with any of the other business lines included in the analysis, namely insurance for comprehensive residential insurance (0114) and lender insurance (0977). Commercial multiple peril insurance (0118) can be priced jointly with any other business line, except motor third party liability (0588). The latter has the lowest number of matches, where it can be priced jointly with only three among the nine business lines. Specifically, motor third party liability (0588) can be priced jointly only with comprehensive residential insurance (0114), passenger private auto (0982), or lender insurance (0977).

Table 3. Assessment of the condition $b\rho<1$ for all pairs of business lines, identified by their SUSEP code (see Table 2). The symbol (+) indicates that there exists a competitiveness region for the corresponding pair (i.e. the condition $b\rho<1$ is satisfied). The symbol (–) indicates that there is no competitiveness region for the corresponding pair (i.e. the condition $b\rho<1$ is not satisfied).

The four non-commercial auto insurance business lines are all in the top five lines in terms of total collected premiums. The top two business lines from this category are auto property damage (0531) and optional auto liability (0553), and they are also the least risky ones. These two lines have a positive competitiveness region, which is rather surprising given that they also exhibit the highest pairwise correlation of 0.86. The other two lines are motor third party liability (0588) and private passengers auto (0982), and are the most risky ones. These two lines have a positive competitiveness region with a relatively low pairwise correlation of 0.18. Other combinations involving these four lines do not have a competitiveness region. This means that besides the two pairs reported above, one line would necessarily subsidise the other if they were to be priced jointly.

The conclusion from this analysis is that there is room for insurance companies to price many business lines jointly, even when the pairwise correlation is high. Some pairs of business lines in related market segments, especially in auto insurance, do not have a competitiveness region. Nevertheless, candidate business lines for joint pricing do not necessarily need to be from the same market segment. In fact, it may be argued that unrelated lines are more likely to have a low correlation. But even for related lines with strong positive correlation, the competitiveness region may still exist, which is the case of auto property damage (0531) and optional auto liability (0553).

Another relevant point to recall is that the decision on the pricing strategy should be supplemented by information about the price elasticity of demand in each line. To illustrate, consider the case of compulsory motor third party liability (0588) and optional auto liability (0553). The latter pays a benefit in excess of the cover provided by the former. Insurers are likely to be active on both of these business lines, but the analysis suggests that there exists no competitiveness region between them. As a consequence, pricing them jointly implies that the least risky business line (i.e. optional auto liability) would subsidise the riskier one (i.e. compulsory auto liability). In this case, and based on the analysis from Section 4, the insurer could still price them jointly if two conditions are satisfied. The first condition is that the compulsory line should be relatively elastic. The second condition is on the reaction of policyholders on the optional line, and this condition is revealed from the relative demand of this line compared to the compulsory cover. Note that it is not unrealistic to infer that the compulsory cover is likely to have the highest demand among the two, and hence, that the demand for the compulsory cover may be such that $w^d<w_{ct}$ . If this is the case, then on top of the high elasticity on the compulsory business line, joint pricing would also require a high elasticity on the optional one.

6.3 Illustration of the competitiveness region

Two pairs of business lines with $\psi_A\leq \psi_B$ from the SES data set are used to illustrate the competitiveness region. For the first pair, business line A corresponds to optional auto liability (0553) and business line B corresponds to motor third party liability (1198). This pair is chosen because it has the highest value of $b\rho$ , with $b\rho\approx0.49$ . For the second pair, business line A corresponds to mortgage insurance (1068) and business line B corresponds to commercial multiple peril insurance (0118). This pair is selected because it has the lowest difference $\psi_B-\psi_A \approx 0.0069$ , and in addition, satisfies $b\rho<1$ , with $b\rho \approx 0.31$ .

For each pair, the function $\psi(n)$ from (2.1) is determined. Recall that this function corresponds to the required joint loading conditionally on the proportion of contracts in business line B. Figure 5 displays the loaded premiums under joint pricing per unit of expected benefit, i.e. $1+\psi(n)$ . The dashed horizontal lines are the stand-alone prices $1+\psi_A$ and $1+\psi_B$ .

Figure 5 Loaded premiums under joint pricing per unit of expected benefit in function of the proportion of contracts in business line B, i.e. $1+\psi(n)$ , for two selected pairs of business lines. The dashed horizontal lines are the stand-alone prices $1+\psi_A$ and $1+\psi_B$ . The dashed vertical line on the right panel is the critical threshold.

The left panel of Figure 5 corresponds to the case where the competitiveness region does not exist. Moreover, due to the fact that condition $b\rho<1$ is not satisfied and that $\psi_A<\psi_B$ , business line A subsidises business line B. Indeed, the lowest possible price for business line A is its stand-alone price, whereas any proportion n leads to a lower premium for business line B under joint pricing.

The right panel of Figure 5 illustrates a favourable situation for two business lines to be priced jointly. The existence of the competitiveness region is ensured by $b\rho<1$ , whereas the critical threshold $n_{\textit{ct}}\approx 1$ is ensured by the fact that $\psi_A\approx\psi_B$ . The remaining task for the insurer is to control the proportion n in order for the actual loading to be above the required one. Alternatively, the insurer can gain more flexibility in terms of portfolio monitoring if the commercial premiums under joint pricing are set by taking into account information on market structure and policyholders’ behaviour in an appropriate way.