Multiple dimensions of private information in life insurance markets

Abstract

A growing amount of literature has shown the unavailability of a positive correlation between insurance coverage and ex post risk, even though the risk-related adverse selection is confirmed by other evidence. Such puzzles are suggested to be attributed to the coexistence of multi-dimensional private information. This paper develops a method to identify the risk-related private information in this case. To illustrate, we apply data from US life insurance market. Categorizing individuals’ heterogeneous insurance preferences into different groups and using joint maximum likelihood estimation of a mixture density model, we detect a positive correlation between individual mortality and insurance coverage within each group. This result is consistent with the conventional theory regarding the private information of adverse selection.

Appendices

Appendix 1

Our objective function for MLE is:

$$\mathop {\max }\limits_{\theta } \sum\limits_{i = 1}^{N} {\ln f(Die_{i} ,LFI_{i} |X_{i,} W_{i} )}$$

where

$$\begin{aligned} f(Die_{i} ,LFI_{i} ) = \Pr (Die_{i} = 1,LFI_{i} = 1)^{{1(Die_{i} = 1,LFI_{i} = 1)}} \hfill \\*\Pr (Die_{i} = 1,LFI_{i} = 0)^{{1(Die_{i} = 1,LFI_{i} = 0)}} * \hfill \\ \Pr ({\text{Die}}_{i} = 0,{\text{LFI}}_{i} = 1)^{{1({\text{Die}}_{i} = 0,\;{\text{LFI}}_{i} = 1)}} *\Pr ({\text{Die}}_{i} = 0,\;{\text{LFI}}_{i} = 0)^{{1({\text{Die}}_{i} = 0,\;{\text{LFI}}_{i} = 0)}} \hfill \\ \quad \quad \quad = \prod\limits_{\begin{subarray}{l} m = 0,1 \\ n = 0,1 \end{subarray} } {[f({\text{Die}}_{i} = m,{\text{LFI}}_{i} = n)]^{{1({\text{Die}}_{i} = m,\;{\text{LFI}}_{i} = n)}} } \hfill \\ \quad \quad \quad = \prod\limits_{\begin{subarray}{l} m = 0,1 \\ n = 0,1 \end{subarray} } \begin{gathered} \{ \Pr ({\text{Die}}_{i} = m,{\text{LFI}}_{i} = n|H = H_{h} )*\Pr (H = H_{h} ) + \hfill \\ \Pr ({\text{Die}}_{i} = m,{\text{LFI}}_{i} = n|H = H_{l} )*\Pr (H = H_{l} )\}^{{1({\text{Die}}_{i} = m,\;{\text{LFI}}_{i} = n)}} \hfill \\ \end{gathered} \hfill \\ \end{aligned}$$

Let $u_{h}^{*} = \rho_{1} v_{h}^{*} + \eta_{h}$, and $u_{l}^{*} = \rho_{2} v_{l}^{*} + \eta_{l}$; then $\sigma_{{\eta_{h} }}^{2} = 1 - \rho_{1}^{2}$ and $\sigma_{{\eta_{l} }}^{2} = 1 - \rho_{2}^{2}$ by the assumption that $u_{h(l)}^{*} \sim N(0,1)$ and $v_{h(l)}^{*} \sim N(0,1)$. We then write down one of the four cases in our objective function as below:

$$\begin{gathered} f({\text{Die}}_{i} = 1,\;{\text{LFI}}_{i} = 1|X_{i} ,\;W_{i} ) = \Pr ({\text{Die}}_{i} = 1,\;{\text{LFI}}_{i} = 1|H = H_{h} ,X_{i} )*\Pr (H = H_{h} |W_{i} ) \hfill \\ \quad \quad \quad + \Pr ({\text{Die}}_{i} = 1,\;{\text{LFI}}_{i} = 1|H = H_{l} ,X_{i} )*\Pr (H = H_{l} |W_{i} ) \hfill \\ \quad \quad \quad = \Pr (c_{\beta }^{h} + X_{i} \beta_{X} + u_{h}^{*} > 0,C_{\delta }^{h} + X_{i} \delta_{X} + v_{h}^{*} > 0|X_{i} )*\Pr (W_{i} \gamma^{*} + \omega > 0) \hfill \\ \quad \quad \quad + \Pr (c_{\beta }^{l} + X_{i} \beta_{X} + u_{l}^{*} > 0,c_{\delta }^{l} + X_{i} \delta_{X} + v_{l}^{*} > 0|X_{i} )*[1 - \Pr (W_{i} \gamma^{*} + \omega > 0)] \hfill \\ \quad \quad \quad = \int\limits_{{ - c_{\delta }^{h} - X_{i} \delta_{X} }}^{\infty } {} \Phi (\frac{{c_{\beta }^{h} + X_{i} \beta_{X} + \rho_{1} v_{h}^{*} }}{{\sqrt {1 - \rho_{1}^{2} } }})\phi (v_{h}^{*} )dv_{h}^{*} \Phi (W_{i} \gamma^{*} ) \hfill \\ \quad \quad \quad + \int\limits_{{ - c_{\delta }^{l} - X_{i} \delta_{X} }}^{\infty } {} \Phi (\frac{{c_{\beta }^{l} + X_{i} \beta_{X} + \rho_{2} v_{l}^{*} }}{{\sqrt {1 - \rho_{2}^{2} } }})\phi (v_{l}^{*} )dv_{l}^{*} [1 - \Phi (W_{i} \gamma^{*} )] \hfill \\ \end{gathered}$$

Appendix 2

2.1 More specification test

Tables 10, 11, 12, and 13 report the result when proxy variables for risk averse, education, employment status, and financial conditions, respectively, are excluded, where the first column only includes public information, X, and the second column includes both public information, X, and private information on subsequent mortality for the next 10 to 15 years, SS. We see the constants in both equations satisfy the predictors of two-type model, and parameters in both the Die and LFI equations are similar to the corresponding parameters estimated in Table 5 , when all W is used. Moreover, the correlations between the error terms in the Die and LFI equations are still significantly positive in all specifications.

Table 10 Drop ‘risk averse’

Table 11 Drop ‘education’

Table 12 Drop ‘employment status’

Table 13 Drop ‘financial conditions’