Asymmetric Information, Self‐selection, and Pricing of Insurance Contracts: The Simple No‐Claims Case

• DOI: http://doi.org/10.1111/j.1539-6975.2013.01520.x
• Date: 01 December 2014
• Published date: 01 December 2014

©

DOI: 10.1111/j.1539-6975.2013.01520.x

757

ASYMMETRIC INFORMATION,SELF-SELECTION,AND

PRICING OF INSURANCE CONTRACTS:THE SIMPLE

NO-CLAIMS CASE

Catherine Donnelly

Martin Englund

Jens Perch Nielsen

Carsten Tanggaard

ABSTRACT

This article presents an optional bonus-malus contract based on aprioririsk

classification of the underlying insurance contract. By inducing self-selection,

the purchase of the bonus-malus contract can be used as a screening device.

This gives an even better pricing performance than both an experience rating

scheme and a classical no-claims bonus system. An application to the Danish

automobile insurance market is considered.

INTRODUCTION

If one tells the truth, one is sure, sooner or later, to be found out.

Oscar Wilde

In a world with two levels of risk and asymmetric information, where the insurance

company cannot distinguish between customers from the two risk groups, there

is no pooling equilibrium and there may not be an equilibrium at all (Rothschild

and Stiglitz, 1976). In a pooling equilibrium, the lower risk policyholders subsidize

the higher risk policyholders. If the policyholders are aware of their risks and the

difference is too large between risk and price, the lower risk individuals will not

Catherine Donnelly is at the Department of Actuarial Mathematics and Statistics, and

the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14

4AS, United Kingdom. Donnelly can be contacted via e-mail: C.Donnelly@hw.ac.uk.

Martin Englund is at the Department of Economics and Business, Aarhus University, Den-

mark, and Codan Insurance, part of the RSA Group, Denmark. Englund can be con-

tacted via e-mail: eld@codan.dk. Jens Perch Nielsen is at Cass Business School, City

University, 106 Bunhill Row, London EC1Y 8TZ, United Kingdom. Nielsen can be con-

tacted via e-mail: Jens.Nielsen.1@city.ac.uk. Carsten Tanggaard is at CREATES, Aarhus

University, Denmark. Tanggaard can be contacted via e-mail: ctanggaard@creates.au.dk.

The authors wish to express their gratitude to an anonymous referee whose comments greatly

improved the article. Carsten Tanggaard acknowledges support from CREATES, funded by

the Danish National Research Foundation.

The Journal of Risk and Insurance, 2013, Vol. 81, No. 4, 757–779

758 THE JOURNAL OF RISK AND INSURANCE

buy insurance. Thus, either the market for insurance breaks down or each type of

policyholder buys an insurance contract with a payoff that caters to the specific

riskiness of the policyholder. This idea of sufficiently tailor-made (i.e., differentiated)

contracts may not fit exactly with what we see in current insurance markets. Insurance

contracts offered by insurance companies are only to some extent differentiated,

depending on the sophistication of the models, the available data, and regulations.

However, Allard, Cresta, and Rochet (1997) show that pooling equilibria may exist if

even the slightest distributional cost exists. Differences in risk aversion can also make

policyholders with different risks accept pooling (at least to some extent), according

to Rothschild and Stiglitz (1976).

Covariate-based regression is generally used to differentiate pricing and the most

widely used models for this are generalized linear models (GLMs); see Pinquet (2001)

for applications in actuarial science and McCullagh and Nelder (1989) for a general

discussion of the statistical theory of GLMs. One way to improve the aprioripricing

is to make use of a posteriori corrections based on experience rating, such as credi-

bility theory and bonus-malus systems (BMS). Simple time-independent credibility

models with extensions to both time-dependence and multivariate experience rating

are found in Pinquet, Guill´

en, and Bolanc´

e (2001) and Englund et al. (2009). Brouhns

et al. (2003) provide a quick survey of different types of BMS. Unfortunately, these

experience-based methods have a limited impact when individual claim informa-

tion is rare, as for new policyholders, and it may take several years of observation

before the precision is reasonable. However, Donnelly, Englund, and Nielsen (2013)

find evidence of adverse selection within the Danish automobile insurance market,

meaning that the degree of coverage chosen by the policyholder is based on the ex

ante assessment that a policyholder makes of his riskiness and wealth. Therefore, we

seek new ways to differentiate the policyholders by risk at the very outset, that is, at

the time of purchase of the insurance product.

This article gives examples of an insurance contract that may induce self-selection

within the existing risk classes of the rating scheme. The new type of insurance

contract has some features that are not found in contracts of standard BMS. Similar

to BMS, the payoff that is offered to policyholders depends on the actual number

of experienced claims during each insurance period, but the contract has a more

general form of payoff than a standard contract with bonus. The payoff can be tailor-

made to individual policyholders’ needs or preferences. Implicit in this is the idea that

insurance companies should offer a menu of contracts in such a way that self-selection

is induced among the policyholders. Hence, we allow for an extended individual

choice compared to standard BMS.

With such a contract the insurer will be able to differentiate the insurance pre-

mium for each period of time to an even greater extent, which makes it more com-

petitive due to the separating effect. This is desirable since adverse selection can

cause inefficiency in insurance contracts, reducing the benefit of taking an insur-

ance for the lower classes of risk since the price will be too high in relation to the

risk (Akerlof, 1970; Rothschild and Stiglitz, 1976; Stiglitz, 1977). For recent contri-

butions on the subjects of asymmetric information, adverse selection, deductibles,

and bonus systems, see Snow (2009), Spreeuw and Karlsson (2009), and Kim et al.

(2009).