Pricing for Multiline Insurer: Frictional Costs, Insolvency, and Asset Allocation

• DOI: http://doi.org/10.1111/j.1540-6296.2012.01214.x
• Author: Li Zhang,Norma Nielson
• Published date: 01 September 2012
• Date: 01 September 2012

Risk Management and Insurance Review

C

Risk Management and Insurance Review, 2012, Vol.15, No. 2, 129-152

DOI: 10.1111/j.1540-6296.2012.01214.x

FEATURE ARTICLES

PRICING FOR MULTILINE INSURER:FRICTIONAL COSTS,

INSOLVENCY,AND ASSET ALLOCATION

Li Zhang

Norma Nielson

ABSTRACT

This article examines multiline insurance pricing based on the contingent claim

approach in a limited liability and frictional costs environment. Capital allo-

cation is based on the value of the default option, which satisfies the realistic

assumption that each distinct line undertakes a pro rata share of deficit caused

by insurer insolvency.Premium levels, available assets, and default risk interact

with each other and reach equilibrium at the fair premium. The assets avail-

able to pay for liabilities are not predetermined or given; instead, the premium

income and investment income jointly influence the available assets. The re-

sults show that equity allocation does not influence the overall fair premium.

For a given expected loss, the premium-to-expected-loss ratio for firms offering

multiple lines is higher than that for firms only offering a single line, due to

the reduced risk achieved through diversification. Premium-to-expected-loss

ratio and equity-to-expected-loss ratio vary across lines. Lines having a higher

possibility or claim amount not being paid in full exhibit lower premium-to-

expected-loss ratio and higher equity-to-expected-loss ratio. Positive correlation

among lines of business results in lower premium-to-expected-loss ratio than

when independent losses are assumed. Positive correlation between investment

return and losses reduces the insolvency risk and leads to a higher premium-

to-expected-loss ratio.

INTRODUCTION

Setting a fair or competitive premium plays an important role in the insurance industry.

Capital is invested or retained in the insurance industry only if the return provided by

the insurance industry is comparable to that offered by other industries. Determining

an appropriate insurance premium has been the subject of extensive scrutiny over the

last several decades among both academia and industry practitioners. Starting from

the earliest attempt to determine the fair premium—the Target Underwriting Profit

Li Zhang is an Assistant Professor at G.R. Herberger College of Business, St. Cloud State Uni-

versity, 720 Fourth Avenue, South St. Cloud, MN 56301; phone: 320-308-3876; fax: 320-308-4973;

e-mail: lzhang@stcloudstate.edu. Norma Nielson holds the Chair in Insurance and Risk Man-

agement at Haskayne School of Business, University of Calgary, 2500 University Drive N.W.,

Calgary,AB T2N 1N4, Canada. T hisarticle was subject to double-blind peer review.

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130 RISK MANAGEMENT AND INSURANCE REVIEW

Margin promulgated by the National Convention of Insurance Commissioners in 1921—

a variety of insurance pricing models have been proposed and applied, including the

capital asset pricing model (e.g., Fairley, 1979; Hill, 1979; Cummins and Harrington,

1985; Hill and Modigliani, 1987), the internal rate of return approach (e.g., Cummins,

1990), the discounted cash flow approach (e.g., Myers and Cohn, 1987; Cummins, 1990;

D’Arcy and Garven, 1990), the arbitrage pricing model (e.g., Kraus and Ross, 1982;

Urrutia, 1987), and the option pricing model (e.g., Doherty and Garven, 1986; D’Arcy

and Garven, 1990; Phillips et al., 1998; Sherris, 2006; Ibragimov et al., 2010). Such financial

insurance pricing models have the strength that they incorporate the capital market into

insurance pricing and could provide nonarbitrage insurance pricing.

D’Arcy and Garven (1990) compared the major property–liability insurance pricing

models, including target underwriting profit margin method, total rate of return model,

capital asset pricing model (CAPM), and option pricing model (OPM), over the 60-year

period from 1926 through 1985. Their results showed that the total rate of return model

and option pricing model usually produced a better fit, but the relative goodness of fit

of the these models was not stable over time. Their results also found that the option

pricing model was particularly sensitive to changes in tax-related parameters, making it

a good tool to carefully examine the effects of taxation on underwriting profit marginand

insurance premium. Garven (1992) concluded several important practical advantages of

the option pricing model. OPM can explicitly quantify the value of insolvency risk and

the effects of underutilized tax shields.

Since the 1970s, the financial field has witnessed tremendous growth in the application

of the OPM (Campbell et al., 1997; McNeil et al., 2005). Unexceptionally OPM has re-

ceived increasing attention among both insurance academia and industry practitioners

(e.g., Doherty and Garven, 1986; Cummins, 1988; Derrig, 1989; D’Arcy and Garven,

1990; Garven, 1992; Wang, 2000; Sherris 2006; Ibragimov et al., 2010). The rationale for

applying OPM in insurance pricing is that insurance policies can be viewed as a pack-

age of contingent payments depending on the insurer’s underwriting and investment

performance, and the value of the contingent payments can be estimated within the

framework of OPM.

In early insurance applications of the Black–Scholes model, many studies assumed that

insurers provide only one line of business (or viewed the total business as one single

line). For example, Merton (1977) applied the OPM to estimate the pricing of loan guar-

antees and deposit insurance. Doherty and Garven (1986) modeled the contingent claims

to shareholders, policyholders, and tax authorities by using European options to esti-

mate the insurance premium and underwriting profit margin. Sommer (1996) applied

the OPM framework to measure insolvency risk and derived that insurance price was the

present value of loss claims minus the value of an insolvency put option that captured

the insolvency risk of insurer.The empirical results from his regression model supported

the hypotheses derived from the theoretical framework that insolvency risk was nega-

tively related to insurance price.1

1Motivated by the problems caused by the flat rate guarantee fund premium scheme, Cummins

(1988) developed a risk-based premium estimation technique for insurance guaranty funds. The

value of the insurance guaranty fund was modeled using a put option with the value of the