Management of commissions to meet the regulatory requirements: the case of property–casualty insurance in China

Abstract

We investigate how the 2009 regulatory change to the method of calculating combined ratios in the Chinese property–casualty insurance industry affected the relationship between commissions and combined ratios. We find that since the 2009 reform, the industry has shown a non-linear relationship between commissions and combined ratios. The relationship is negative (positive) when the combined ratio is higher (lower) than the regulatory threshold. Before 2009, this relationship was linear. Since 2009, when commissions increased, the combined ratio converges to the threshold. As the volatility of the combined ratio is positively related to the statutory capital required, this change provides incentives for insurers to decrease the combined ratio and/or its volatility as they seek to manage their commissions to approximate the threshold without jeopardising compliance with other regulatory requirements.

Appendix

Proof of Proposition 1

• (1) According to Table 1, in the pre-NEAC era,

$$UPR_{t} = PW_{t} \cdot \theta_{t} \,,\,\,PE_{t} = PW_{t} - (UPR_{t} - UPR_{t - 1} ) = PW_{t} (1 - \theta_{t} ) + PW_{t - 1} \theta_{t - 1} ,$$

$$\begin{aligned} CR_{t} & = \frac{{PC_{t} + DLR_{t} }}{{PE_{t} }} + \frac{{Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PW_{t} }}, \\ & \Rightarrow \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} = \frac{1}{{PW_{t} }} > 0, \\ \end{aligned}$$

i.e., the combined ratios and the commissions are positively correlated.

• (2) In the post-NEAC era,

$$UPR_{t} = (PW_{t} - Fc_{t} - E_{t} ) \cdot \theta_{t} ,$$

$$\begin{aligned} PE_{t} & = PW_{t} - (UPR_{t} - UPR_{t - 1} ) \\ & = PW_{t} (1 - \theta_{t} ) + (Fc_{t} + E_{t} )\theta_{t} + (PW_{t - 1} - Fc_{t - 1} - E_{t - 1} )\theta_{t - 1} \\ & = PW_{t} (1 - \theta_{t} ) + (Fc_{t} + E_{t} )\theta_{t} + (PW_{t - 1} - f^{ - 1} (Fc_{t} ) - E_{t - 1} )\theta_{t - 1} \\ \end{aligned}$$

$Fc_{t} = f(Fc_{t - 1} )$ is a functional relationship between the commissions of year $t$ and year $t - 1$. $Fc_{t - 1} = f^{ - 1} (Fc_{t} )$ is the inverse function.¹⁸

$$\begin{aligned} CR_{t} & = \frac{{PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PE_{t} }} = \frac{{C_{t} }}{{PE_{t} }}, \\ & \Rightarrow \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} = \frac{{PE_{t} - C_{t} [\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} ]}}{{PE_{t}^{2} }} \\ & = \frac{1}{{PE_{t} }}\left[ {1 - \frac{{C_{t} }}{{PE_{t} }} \cdot (\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} )} \right] \\ & = \frac{1}{{PE_{t} }}\left[ {1 - CR_{t} \cdot (\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} )} \right] \\ & = \frac{1}{{PE_{t} }}\left[ {1 - CR_{t} \cdot \frac{1}{{\mu_{t} }}} \right] \\ \end{aligned}$$

where $\mu_{t} = \displaystyle \frac{1}{{\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} }}$.

Obviously, $PE_{t} > 0$, $\theta_{t} \in (0,1)$.¹⁹

$$\frac{{dFc_{t} }}{{dFc_{t - 1} }} = f^{\prime}(Fc_{t - 1} ) < 0 \Rightarrow (f^{ - 1} (Fc_{t} ))^{\prime} = \frac{1}{{f^{\prime}(Fc_{t - 1} )}} < 0$$

So $0 < \mu_{t} = \displaystyle \frac{1}{{\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} }} < \frac{1}{{\theta_{t} }}$.

Therefore,

when $CR_{t} < \mu_{t}$, $\displaystyle \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} > 0$, $CR_{t}$ is monotonically increasing with $Fc_{t}$, i.e., the higher the fees and commissions, the higher the combined ratio;

when $CR_{t} > \mu_{t}$,$\displaystyle \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} < 0$, $CR_{t}$ is monotonically decreasing with $Fc_{t}$, i.e., the higher the commissions, the lower the combined ratio;

and when $Fc_{t} \to + \infty$, $CR_{t} \to \mu_{t} = \displaystyle \frac{1}{{\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} }}$.

Proof of Proposition 2

• (1) In the pre-NEAC era,

$$\begin{aligned} ER_{t} & = \frac{{Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PW_{t} }}, \\ & \Rightarrow \frac{{\partial ER_{t} }}{{\partial Fc_{t} }} = \frac{1}{{PW_{t} }} > 0 \\ \end{aligned}$$

i.e., the expense ratios and commissions are positively correlated.

• (2) In the post-NEAC era,

$$\begin{aligned} ER_{t} & = \frac{{Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PE_{t} }}, \\ & \Rightarrow \frac{{\partial ER_{t} }}{{\partial Fc_{t} }} = \frac{{PE_{t} - (Fc_{t} + E_{t} + E^{\prime}_{t} )[\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} ]}}{{PE_{t}^{2} }} \\ & = \frac{1}{{PE_{t} }}\left[ {1 - \frac{{Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PE_{t} }} \cdot (\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} )} \right] \\ & = \frac{1}{{PE_{t} }}\left[ {1 - ER_{\text{t}} \cdot \frac{1}{{\mu_{t} }}} \right] \\ \end{aligned}$$

where $\mu_{t} = \displaystyle \frac{1}{{\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} }}$.

Therefore,

when $ER_{t} < \mu_{t}$, $\displaystyle \frac{{\partial ER_{t} }}{{\partial Fc_{t} }} > 0$, $ER_{t}$ is monotonically increasing with $Fc_{t}$, i.e., the higher the fees and commissions, the higher the expense ratio;

when $ER_{t} > \mu_{t}$, $\displaystyle \frac{{\partial ER_{t} }}{{\partial Fc_{t} }} < 0$, $ER_{t}$ is monotonically decreasing with $Fc_{t}$, i.e., the higher the commissions, the lower the expense ratio;

and when $Fc_{t} \to + \infty$, $ER_{t} \to \mu_{t}$.

Proof of Proposition 3

According to Table 1, in the pre-NEAC era,

$$\begin{aligned} Profit_{t} & = PE_{t} - C_{t} \\ & = PW_{t} (1 - \theta_{t} ) + PW_{t - 1} \cdot \theta_{t - 1} \\ & \quad - \left( {PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} } \right) \\ \end{aligned}$$

$$\frac{{\partial Profit_{t} }}{{\partial Fc_{t} }} = - 1 < 0.$$

In the post-NEAC era,

$$\begin{aligned} Profit_{t} & = PE_{t} - C_{t} \\ & = PW_{t} (1 - \theta_{t} ) + (Fc_{t} + E_{t} )\theta_{t} + (PW_{t - 1} - f^{ - 1} (Fc_{t} ) - E_{t - 1} )\theta_{t - 1} \\ & \quad - \left( {PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} } \right) \\ \end{aligned}$$

$$\displaystyle \frac{{\partial Profit_{t} }}{{\partial Fc_{t} }} = - \left( {1 - \frac{1}{{\mu_{t} }}} \right)$$

.

As in Proposition 1, we have proven that $0 < \mu_{t} < \frac{1}{{\theta_{t} }}$, so $1 - \frac{1}{{\mu_{t} }} < 1$;

i.e., in the post-NEAC era, profits and commissions are still negatively and linearly correlated, but the sensitivity between them is decreased.

Proof of Proposition 4

According to Table 1, in the pre-NEAC era,

$$\begin{aligned} CR_{t} & = \frac{{PC_{t} + DLR_{t} }}{{PE_{t} }} + \frac{{Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PW_{t} }} \\ & \Rightarrow \frac{{\partial CR_{t} }}{{\partial DLR_{t} }} = \frac{1}{{PE_{t} }}. \\ \end{aligned}$$

In the post-NEAC era,

$$\begin{aligned} CR_{t} & = \frac{{PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PE_{t} }} \\ & \Rightarrow \frac{{\partial CR_{t} }}{{\partial DLR_{t} }} = \frac{1}{{PE_{t} }} \\ \end{aligned}$$

Therefore, in both the pre- and the post-NEAC eras, loss reserves and combined ratios are positively and linearly correlated. However, in the post-NEAC era, the premiums earned, $PE_{t}$, increases, i.e., the sensitivity of the combined ratios to loss reserves decreases due to the initial expenses such as commissions.

It is worth noting that premiums written and premiums earned are not affected by the DLR. Premiums earned is only affected by the UPR, and premiums written is not affected by any other items.

Proof of Proposition 5

• (1) According to Table 1, in the pre-NEAC era,

$$\begin{aligned} APE_{t} & = PW_{t} - UPR_{t} = PW_{t} (1 - \theta_{t} ),\,\,\,\,ACR_{t} = \frac{{PC_{t} + DLR_{t} }}{{APE_{t} }} + \frac{{Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PW_{t} }}, \\ & \Rightarrow \frac{{\partial ACR_{t} }}{{\partial Fc_{t} }} = \frac{1}{{PW_{t} }} > 0, \\ \end{aligned}$$

i.e., the adjusted combined ratios and commissions are positively and linearly correlated.

• (2) In the post-NEAC era,

$$\begin{aligned} APE_{t} & = PW_{t} - UPR_{t} = PW_{t} (1 - \theta_{t} ) + (Fc_{t} + E_{t} )\theta_{t} , \\ ACR_{t} & = \frac{{PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} }}{{APE_{t} }} = \frac{{C_{t} }}{{APE_{t} }}, \\ & \Rightarrow \frac{{\partial ACR_{t} }}{{\partial Fc_{t} }} = \frac{{APE_{t} - C_{t} \cdot \theta_{t} }}{{APE_{t}^{2} }} = \frac{1}{{APE_{t} }}\left( {1 - \frac{{C_{t} }}{{APE_{t} }} \cdot \theta_{t} } \right) = \frac{1}{{APE_{t} }}\left( {1 - ACR_{t} \cdot \theta_{t} } \right) \\ \end{aligned}$$

Obviously, $APE_{t} > 0$, $\theta_{t} \in (0,1)$, so

when $\displaystyle ACR_{t} < \frac{1}{{\theta_{t} }}$, $\displaystyle \frac{{\partial ACR_{t} }}{{\partial Fc_{t} }} > 0$, $ACR_{t}$ is monotonically increasing with $Fc_{t}$, i.e., the higher the commissions, the higher the combined ratio;

when $\displaystyle ACR_{t} > \frac{1}{{\theta_{t} }}$, $\displaystyle \frac{{\partial ACR_{t} }}{{\partial Fc_{t} }} < 0$, $ACR_{t}$ is monotonically decreasing with $Fc_{t}$, i.e., the higher the commissions, the lower the combined ratio;

and when $Fc_{t} \to + \infty$, $ACR_{t} \to \frac{1}{{\theta_{t} }}$.

• (3) According to Table 1, in the post-NEAC era,

$$\begin{aligned} \frac{1}{{CR_{t} }} & & = \frac{1}{{ACR_{t} }} + \frac{{UPR_{t - 1} }}{{C_{t} }} & = \frac{1}{{ACR_{t} }} + \frac{{(PW_{t - 1} - Fc_{t - 1} - E_{t} ) \cdot \theta_{t - 1} }}{{PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} }} \\ & & & \Rightarrow \displaystyle \mathop {\lim }\limits_{{Fc_{t} \to + \infty }} \frac{1}{{CR_{t} }} = \displaystyle \mathop {\lim }\limits_{{Fc_{t} \to + \infty }} \left[ {\frac{1}{{ACR_{t} }} + \frac{{(PW_{t - 1} - Fc_{t - 1} - E_{t} ) \cdot \theta_{t - 1} }}{{PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} }}} \right]. \\ \end{aligned}$$

As has been proven, when $Fc_{t} \to + \infty$, $ACR_{t} \to \frac{1}{{\theta_{t} }}$,

$${\text{i}}.{\text{e}}.,\,\,\,\mathop {\lim }\limits_{{Fc_{t} \to + \infty }} \frac{1}{{ACR_{t} }} = \theta_{t}$$

(i)

and $\frac{{dFc_{t} }}{{dFc_{t - 1} }} < 0$, when $Fc_{t} \to + \infty$,$Fc_{t - 1} = f^{ - 1} (Fc_{t} ) \to - \infty$.

According to L’Hospital’s rule,

$$\begin{aligned} & \Rightarrow \mathop {\lim }\limits_{{Fc_{t} \to + \infty }} \frac{{(PW_{t - 1} - Fc_{t - 1} - E_{t} ) \cdot \theta_{t - 1} }}{{PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} }} & = \mathop {\lim }\limits_{{Fc_{t} \to + \infty }} \frac{{(PW_{t - 1} - f^{ - 1} (Fc_{t} ) - E_{t} ) \cdot \theta_{t - 1} }}{{PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}_{t} }} \\ & & = \mathop {\lim }\limits_{{Fc_{t} \to + \infty }} \left[ { - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} } \right] \\ \end{aligned}$$

(ii)

According to (i) and (ii), we can see that when $Fc_{t} \to + \infty$,

$$\frac{1}{{CR_{t} }} \to \theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1}$$

$${\text{i}}.{\text{e}}.,\,\,\,\,CR_{t} \to \frac{1}{{\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} }} = \mu_{t}$$

(iii)

Therefore, according to (i) and (iii), when $ACR_{t} \to \frac{1}{{\theta_{t} }}$, $CR_{t} \to \mu_{t}$,

i.e., the threshold of the adjusted combined ratio $\frac{1}{{\theta_{t} }}$ and that of the combined ratio $\mu_{t}$ have a one-to-one correspondence.

Proof of Proposition 6

Assume that the premium growth rate is $\lambda_{t}$, i.e., $PW_{t} = (1 + \lambda_{t} )PW_{t - 1}$

$$\begin{aligned} APE_{t} & = \frac{1}{2}PW_{t} ,\,UPR_{t} = \frac{1}{2}PW_{t} \\ CR_{t} & = \frac{{C_{t} }}{{PE_{t} }} = \frac{{C_{t} }}{{PW_{t} - UPR_{t} + UPR_{t - 1} }} = \frac{{C_{t} }}{{\frac{1}{2}PW_{t} + \frac{1}{2}PW_{t - 1} }} \\ & = \frac{{C_{t} }}{{\frac{1}{2}PW_{t} \cdot \left( {1 + \frac{1}{{1 + \lambda_{t} }}} \right)}} = \left( {\frac{{1 + \lambda_{t} }}{{2 + \lambda_{t} }}} \right)\frac{{C_{t} }}{{\frac{1}{2}PW_{t} }} \\ & = \left( {\frac{{1 + \lambda_{t} }}{{2 + \lambda_{t} }}} \right)\frac{{C_{t} }}{{APE_{t} }} = \left( {\frac{{1 + \lambda_{t} }}{{2 + \lambda_{t} }}} \right)ACR_{t} \\ \end{aligned}$$

When $\lambda_{t} = 6.91\%$ and $ACR_{t} = \frac{1}{{\theta_{t} }} = 2$, $CR_{t} = \mu_{t} = 1.03$.

Proof of Proposition 7

We denote the adjusted combined ratios in the pre- and post-NEAC eras as $ACR_{t}^{old}$ and $ACR_{t}^{new}$, respectively, and $\alpha_{t}$ is the proportion of initial expenses to premiums written.

$$\begin{aligned} ACR_{t}^{old} & = \frac{{C_{t} }}{{APE_{t}^{old} }} = \frac{{C_{t} }}{{(1 - \theta_{t} )PW_{t} }}, \\ ACR_{t}^{new} & = \frac{{C_{t} }}{{APE_{t}^{new} }} = \frac{{C_{t} }}{{PW_{t} - (PW_{t} - \alpha_{t} PW_{t} )\theta_{t} }}. \\ \end{aligned}$$

When $ACR_{t}^{new} = \displaystyle \frac{1}{{\theta_{t} }}$, we have $\displaystyle \frac{{C_{t} }}{{PW_{t} }} = \frac{{1 - (1 - \alpha_{t} )\theta_{t} }}{{\theta_{t} }}$.

Therefore, $ACR_{t}^{old} = \frac{1}{{\theta_{t} }} + \frac{{\alpha_{t} }}{{1 - \theta_{t} }}$.

Proof of Proposition 8

If NISR had not been implemented, then

$$CR_{t} = \frac{{PC_{t} + DLR_{t} }}{{PE_{t} }} + \frac{{Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PW_{t} }}$$

In the pre-NEAC era, $PE_{t} = PW_{t} - \left( {UPR_{t} - UPR_{t - 1} } \right)$, $UPR_{t} = PW_{t} \cdot \theta_{t}$

then, $\displaystyle \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} = \frac{1}{{PW_{t} }}$

In the post-NEAC era, $PE_{t} = PW_{t} - \left( {UPR_{t} - UPR_{t - 1} } \right)$, $UPR_{t} = \left[ {PW_{t} - \left( {Fc_{t} + E_{t} } \right)} \right] \cdot \theta_{t} + RM_{t}$

then $\begin{aligned} \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} & = - \frac{{PC_{t} + DLR_{t} }}{{PE_{t}^{2} }}\left( {\theta_{t} - (f^{ - 1} (Fc_{t} ))^{\prime} \cdot \theta_{t - 1} } \right) + \frac{1}{{PW_{t} }} \\ & = - \frac{{LR_{t} }}{{PE_{t} \cdot \mu_{t} }} + \frac{1}{{PW_{t} }}, \\ \end{aligned}$

where $LR_{t} = \displaystyle \frac{{PC_{t} + DLR_{t} }}{{PE_{t} }}$, i.e., Loss Ratio

Obviously, even if we do not change NISR, in the post-NEAC era,

when $LR_{t} < \displaystyle \frac{{PE_{t} }}{{PW_{t} }} \cdot \mu_{t}$, $\displaystyle \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} > 0$, $CR_{t}$ is monotonically increasing with $Fc_{t}$, i.e., the higher the fees and commissions, the higher the combined ratio;

when $LR_{t} > \displaystyle \frac{{PE_{t} }}{{PW_{t} }} \cdot \mu_{t}$, $\displaystyle \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} < 0$, $CR_{t}$ is monotonically decreasing with $Fc_{t}$, i.e., the higher the commissions, the lower the combined ratio.

Proof of Proposition 9

If NEAC had not been implemented, then

$$PE_{t} = PW_{t} - \left( {UPR_{t} - UPR_{t - 1} } \right),\,\,\,UPR_{t} = PW_{t} \cdot \theta_{t}$$

In the pre-NISR era, $CR_{t} = \displaystyle \frac{{PC_{t} + DLR_{t} }}{{PE_{t} }} + \frac{{Fc_{t} + E_{t} + E^{\prime}_{t} }}{{PW_{t} }}$

then $\displaystyle \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} = \frac{1}{{PW_{t} }}$

In the post-NISR era, $CR_{t} = \displaystyle \frac{{PC_{t} + DLR_{t} + Fc_{t} + E_{t} + E^{\prime}}}{{PE_{t} }}$

then, $\displaystyle \frac{{\partial CR_{t} }}{{\partial Fc_{t} }} = \frac{1}{{PE_{t} }}$

Obviously, $\displaystyle \frac{1}{{PE_{t} }} > \frac{1}{{PW_{t} }}$

Therefore, if NEAC had not been implemented (in either the pre- or the post-NISR era), then the commissions and the combined ratios would be linearly and positively correlated. However, in the post-NISR era, the combined ratios would be more sensitive to commissions.