Academic journal article Journal of Risk and Insurance

An Equilibrium Model of Insurance Pricing and Capitalization

Academic journal article Journal of Risk and Insurance

An Equilibrium Model of Insurance Pricing and Capitalization

Article excerpt

Introduction

This article is concerned with insurer capitalization and its implications for insurance pricing. The theoretical approaches to insurance pricing now most widely accepted are the internal rate of return approach (Cummins, 1990) and that of Myers and Cohn (1981). In each of these cases (and others), the required capitalization of the line of business being priced is taken as the minimum regulatory requirement or determined by reference to market practice.

The degree of capitalization of an insurer will, however, reflect consumer preferences. Increased capital provides policyholders with increased security, but at a cost in the form of an increase of the premium loadings that enable the capital to be serviced. It is therefore of interest to investigate the equilibrium capitalization of various insurers in the total economy.

Turner (1981) investigated the simultaneous equilibrium of stock prices, real asset prices, and insurance prices in a one-period model but without reference to insurer capitalization. Turner's framework is used here but is supplemented by analysis of the manner in which an insurer's surplus affects the security of policyholders and the value of shares in the insurer.

The equilibrium in this augmented one-period model is then investigated in terms of the same variables as in Turner's model and in terms of the equilibrium amount of capital to be held by each insurer in the economy. Of course, equilibrium capitalization affects equilibrium pricing.

Notation and Assumptions

Initially, the model developed here will follow closely Turner's (1981), and the same notation is used to facilitate comparison. Many of the following notational definitions are extracted verbatim from Turner. They relate to an economy observed over a single period.

Subscripts

f = 1, ..., F = insurers.

h = 1, ..., H = individuals or households.

i = 1, ..., I = firms engaged in production (productive firms).

k = 1, ..., K = types of real assets.

Variables

[V.sub.j] = initial (beginning of period) price of one share in productive firm i.

[R.sub.j] = cash flow per share of stock i.

[[V.sub.h][sub.i]] = number of shares in productive firm i owned by household h.

[v.sub.h] = number of shares in productive firm i owned by insurer f.

[A.sub.k] = initial (beginning of period) price of one real asset of type k.

[R.sub.k] = cash flow per unit real asset of type k.

[[X.sub.h][.sub.k]] =individual losses, that is, negative cash flow of real assets, per unit of asset of type k due to individual hazards of household h.

[[a.sub.h][.sub.k]] = number of real assets of type k owned by household h.

[S.sub.f] = initial (beginning of period) price of one share in insurer f.

[Y.sub.f] = cash flow per share of insurance firm f when [K.sub.f], as defined below, is zero (negative values of [Y.sub.f] are permitted, that is, limited liability is not reflected in this variable).

[[S.sub.h][.sub.f]] = number of shares in insurer f owned by household h.

[[P.sub.f][.sub.k]] = premium per unit of insurance in asset type k charged by insurer f.

[[n.sub.h][.sub.f][.sub.k]] = number of units of insurance in asset type k purchased by household h from insurer f.

[W.sub.h] = initial (beginning of period) wealth of household h.

[T.sub.h] = terminal (end of period) wealth of household h.

[C.sub.h] = current consumption of household h.

[[Micro].sub.h] = expected value of terminal wealth of household h.

[v.sub.h] = variance of terminal wealth of household h.

[K.sub.f] = initial (beginning of period) equity (net assets) per share of insurer f.

A number of quantities, derived from those defined above, play a fundamental role in the following analysis. …

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