The broker model for peer-to-peer insurance: an analysis of its value

Abstract

The increasing role of technology is generating new challenges in fostering innovative insurance products and in offering new means of distribution for existing products. Herein, we focus on peer-to-peer insurance, where technology is used to connect policyholders and the insurance structure recalls its roots based on organised mutual solidarity. The aim is to highlight strengths and weaknesses of the broker model of peer-to-peer insurance. We address the main legal and regulatory issues by referring to European Union regulation on insurance and we develop an actuarial model to support our conclusions and evaluate the convenience of such a business model for policyholders. Although peer-to-peer insurance was created as a fairer alternative to the pool of traditional insurance companies, we see major challenges that need to be solved. The evidence sheds doubt on the convenience of the broker model for peer-to-peer insurance while the regulatory risks seem, paradoxically, certain.

Appendix A: Details on the calculation of P2P premiums in different scenarios

In formula (4), we defined the total fair premiums of the P2P portfolio as:

$${P}_{i,g}^{P2P}={P}_{i,g}^{F}+{P}_{i,g}^{TPI}+{P}_{i,g}^{TPI2}$$

where ${P}_{i,g}^{F}$ is the premiums’ portion kept in the mutual pool, ${P}_{i,g}^{TPI}$ is the fair premium paid to third-party insurance companies and ${P}_{i,g}^{TPI2}$ is used to provide an insurance coverage to cover minor claims in case the mutual pool is empty.

We denote with ${S}_{g}^{L}$ and ${S}_{g}^{S}$ the total aggregate claims amount, for the group $g$, over and below the franchise limit $t$, respectively. These two r.v.s are defined as:

$${S}_{g}^{L}=\left(\left(\sum_{i=1}^{{N}_{g}}{X}_{i,g}\right)|{X}_{i,g}>t\right)\, and\, {S}_{g}^{S}=\left(\left(\sum_{i=1}^{{N}_{g}}{X}_{i,g}\right)|{X}_{i,g}\le t\right).$$

Obviously, we have $S_{g} = S_{g}^{L} + S_{g}^{S}$.

The total expected cost, considering both claims and possible cashback, can be defined as:

$$P^{P2P} = E\left( {\mathop \sum \limits_{g = 1}^{G} S_{g}^{L} } \right) + \mathop \sum \limits_{g = 1}^{G} E\left( {{\max}\left( {S_{g}^{S} - \mathop \sum \limits_{i = 1}^{{r_{g} }} P_{i,g}^{F} ,0} \right)} \right) + \mathop \sum \limits_{g = 1}^{G} E\left( {{\min}\left( {S_{g}^{S} ,\mathop \sum \limits_{i = 1}^{{r_{g} }} P_{i,g}^{F} } \right)} \right) + E\left( {CB} \right).$$

(A.1)

In formula (A.1), we have the following terms:

• The expected value of aggregate amount that regards only claims larger than the limit. This amount is equal to the sum of fair premiums charged by the third-party insurance ${P}_{i,g}^{TPI}$ to each policyholder $i$. In other words, it represents the total premium of a portfolio of homogeneous insurance contracts with a franchise equal to $t$;

• The expected value of the total amount considering only small claims (i.e. lower than the limit) and covering only the case of an empty pool. It represents the fair premium of the specific coverage ${P}_{i,g}^{TPI2}$, aggregated at portfolio level (i.e., the sum of premiums for each group);

• The expected values of claims covered by the pool. For each group $g$, this amount is upper bounded by the initial amount placed in the pool $\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}$;

• The expected cashback. This amount is obviously related to the cashback rule defined by the entity managing the P2P insurance and it will be explored subsequently.

In formula (A.1), we can focus on the second and third terms

$$\sum\limits_{g=1}^{G}E(\mathrm{m}\mathrm{a}\mathrm{x}({S}_{g}^{S}-\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F},0))+\sum\limits_{g=1}^{G}E(\mathrm{m}\mathrm{i}\mathrm{n}({S}_{g}^{S},\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}))=\sum\limits_{g=1}^{G}E\left(\mathrm{max}\left({S}_{g}^{S},\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}\right)\right)+\sum\limits_{g=1}^{G}E(\mathrm{m}\mathrm{i}\mathrm{n}({S}_{g}^{S},\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}))-\sum\limits_{g=1}^{G}\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}=\sum\limits_{g=1}^{G}E\left({S}_{g}^{S}+\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}\right)-\sum\limits_{g=1}^{G}\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}=\sum\limits_{g=1}^{G}E\left({S}_{g}^{S}\right).$$

Given the linearity of the mean, we can rewrite formula (1) as:

$${P}^{P2P}=E\left(\sum_{g=1}^{G}{S}_{g}^{L}\right)+\sum\limits_{g=1}^{G}E\left({S}_{g}^{S}\right)+E\left(CB\right)=E\left(S\right)+E\left(CB\right)= P+E\left(CB\right),$$

(A.2)

where in the last step we have assumed that the expected aggregate amount of claims $E\left(S\right)$ is not affected by the model. In other words policyholders file the same claims in either a P2P model or a traditional stock insurer model.

Different criteria can be set up in order to assess the expected value of the cashback. We provide two different alternatives:

• 1. Any positive amount remaining in the mutual pool is paid back to the policyholders of the group (or to a charity). In this case, we have:

  $$E(CB)=\sum\limits_{g=1}^{G}\left(\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}-E\left(min\left({S}_{g}^{S},\sum_{i=1}^{{r}_{g}}{P}_{i,g}^{F}\right)\right)\right);$$

  (A.3)

• 2. To each policyholder in the group (or to a charity) is paid the difference, if positive, between the Potential Maximum Cashback ($PMCB_{i,g}$), defined by the entity managing the P2P insurance, and the total claim payments related to claims made by policyholders in the group

  $$E\left( {CB} \right) = \mathop \sum \limits_{g = 1}^{G} \left( {\mathop \sum \limits_{i = 1}^{{r_{g} }} PMCB_{i,g} - E\left( {min\left( {S_{g}^{{}} ,\mathop \sum \limits_{i = 1}^{{r_{g} }} P_{i,g}^{F} } \right)} \right)} \right).$$

  (A.4)