CAS Ratemaking Seminar March 2007
Insurance Pricing Models
Richard A. Derrig PhD
President, Opal Consulting
Providence, Rhode Island, U.S.A. richard@derrig.com
.
Insurance Pricing Model Simple Summary
Insurance pricing begins with the fact that both the cost and the expected return (profit) are not known with certainty. Rather, the ultimate cost derives from a stochastic process that commences when the insurance is purchased and is resolved when final claim, expense and tax payments are made. The ultimate profit too results from an interrelated stochastic process which depends on the ultimate costs, the premium charged at the time of sale, and the results of invested premiums and capital. Thus, the determination of an appropriate premium by the actuary or available in the market, prior to sale is key to the opportunity to earn a profit consistent with the risk of adverse development. Of course, in complete or workably competitive markets, where equilibrium supply and demand prices are equal, those prices may or may not agree with the actuary’s calculation of expected profits. We turn to two comprehensive pricing models developed for insurance in the past thirty or so years and illustrate their use in the property-casualty context.
Policyholder Demand Side Model
A policyholder purchases insurance to trade risky and uncertain future adverse financial events for the almost (because of insolvency potential) certainty of a premium payment. In decision theory, the premium would be the certainty equivalent of the uncertain future liabilities.

Insureds should be willing to pay a present value of premiums equal to the present value of expected future claims, expenses and taxes, adjusted in the discount rate for the risk of adverse deviations from the expectations. The key to the demand model is then the identification of all expected payments to (claims), or on behalf of (expenses and taxes), the policyholder. Proper policyholder demand pricing models should not directly include consideration of the insurer’s invested assets and expected investment income above the riskless rate. Policyholders are generally unwilling, except in explicit investment-insurance linked products such as variable annuities, to trade uncertain adverse accidental events for certain premiums plus uncertain insurer investment returns. Policyholder demand models assume the presence of a supplier at the policyholder demand price.
The paradigm demand model, developed at MIT by Stewart C. Myers and Richard A.
Cohn, posits that the appropriate premium P at the beginning of the policy equals the net present value (NPV) of losses, L, expenses E, and taxes T (because of double taxation in the U.S.A.)
  E  T

(MC) where, PV incorporates a negative adjustment to the riskless rate to provide the necessary profit incentive for the insurer assuming the risk. In practice, those risk-adjustments have been difficult to model and calculate from empirical data.
Shareholder Supply Side Model
Insurers offer policies precisely to capitalize on the diversification benefits of pooling uncertain adverse future financial event consequences from personal and commercial risks.
Insurers count on low levels of adverse event correlations among risks (iid is often assumed) to provide a substantial risk spreading benefit across the insured population. A certain premium is
2

traded by the insurer for the risk of collective adverse events in excess of the premiums and expected riskless investment income from premiums (time value of money) over the life of the policy. If the certain premium, combined with the after-tax investment income from the insurer’s asset portfolio, is expected to provide returns to shareholders (investors of capital in the insurer) commensurate with the combined underwriting and investment risk, then the insurer offers the insurance contract at the price that produces expected returns equal to the cost of capital for those risks.
The paradigm supply model is the internal rate of return (IRR) model used for capital budgeting, and project decision making in corporate finance. The fundamental concept is that the flow of invested capital and the return of that capital with realized profit, if any, should be expected to have a net present value of zero when discounted at the cost of capital of the insurance enterprise (the internal rate of return). Nominal policyholder premium, loss, and expense flows and company invested asset flows with after-tax returns are used to estimate (1) the size and timing of the shareholder investment to back the outstanding liabilities, (2) the size and timing of the return of that invested capital to shareholders as liabilities are resolved and paid, and (3) the size and timing of any assets in excess of (1) to be returned as income to shareholders. Figure 1 shows the relative value of the outstanding loss and expense flow for auto bodily injury liability by quarter measured from the issuance of the policy, the commitment of surplus backing those liabilities (proportional with a fixed value 0.67 for all quarters), and the expected profit to be realized by quarter.
3

2.00
1.67
1.00
Bodily Injury
Modeling the Asset / Premium / Surplus / Pre-Tax Profit Flow
Policy Life / Surplus Flow
(Surplus backs outstanding liabilities)
Total Profit at end of period (8.37%)
Surplus (67% * (L+E) )
Underwriting Operation (L+E)
.
1
0.00
0.
25
0.
50
0.
75
1.
00
2.
00
3.
00
4.
00
5.
00
6.
00
Years From Effective Date
Note: Underwriting Operations flows come from 100H-1A Exhibit 4, Column 3
7.
00
8.
00
9.
00
10
.0
0
Figure 1
In regulatory settings such as personal automobile insurance in Massachusetts, the estimate of the required invested capital used is the book value of surplus as a proportion of reserves. This creates a scale mismatch between the proxy for equity capital (book value of surplus) and the data source for estimating the cost of capital the actual returns on the market value of the firm.
There are two principal methods for estimating cost of capital for the entire firm, CAPM and the
Gordon Growth model. At this point in time, the use of empirical data and the simple CAPM discussed above suffers from an omitted or confounding variable problem. The extended three factor CAPM model of Fama and French developed in the 1990s included an important variable omitted in the simple formulation, the size of the insurer. This is well known in finance as the size effect on stock market returns: smaller capitalized stock (need to) earn higher percentage rates of return than large cap stocks. Recent research shows that once the omitted CAPM
4

variables are introduced, the market beta for property-liability companies is about one; i.e., P&C companies are about average risk. The second additional Fama-French factor is the distress factor, as measured by the book to market ratio, also important for insurer cost of capital evaluations because of the central role played in pricing by potential bankruptcy and its costs.
The Gordon Growth Model (GGM) is built on the common assumption that the current per share price P is equal to the present value of all future dividend payments D, discounted at the same cost of capital rate. In a simple GGM formulation, the growth rate of dividends and the cost of capital are assumed constant in perpetuity leading to a simple estimation equation: k 
D
P
 g (GGM) where,
D
= current dividend rate
P k = cost of capital g = dividend growth rate
The dividend growth rate g is, of course, key and many ways of estimation have been used.
Most often, the growth rate is estimated as an average of some historical rate and some forecasted (by analysts) rate. At the heart of all these cost of capital models are probabilistic estimation error models that are rarely calculated in the “real” world.
5

REFERENCES
Automobile Insurers Bureau of Massachusetts, 2005 Underwriting Profit Filing for 2006 Rates,
State of Massachusetts, DOI Docket R2005-09.
Cummins, J. D., “Multi-Period Discounted Cash Flow Ratemaking Models in Property Liability
Insurance,” Journal of Risk and Insurance 57(1), 1990, pp.79-109.
Cummins, J. D. and Phillips, R.D., “Estimating the Cost of Equity Capital for Property-Liability
Insurers”, Journal of Risk and Insurance 72(3), 2005, pp. 441-478
Derrig, R.A.,
" Price Regulation in US Automobile Insurance - A Case Study of Massachusetts
Private Passenger Automobile Insurance 1978-1990", The Geneva Papers on Risk and Insurance
67, 1993, pp.158-173 .
Derrig, R.A., "Theoretical Considerations of the Effect of Federal Income Taxes on Investment
Income in Property-Liability Ratemaking", Journal of Risk and Insurance 61(4), 1994, pp. 691-
709.
Myers, Stewart C. and Richard Cohn, 1987, “Insurance Rate Regulation and the Capital Asset
Pricing Model,” in J.D. Cummins and S.E. Harrington, eds.,
Fair Rate of Return in Property-
Liability Insurance, Kluwer Academic Press, Boston
Taylor, Gregory C., 1994, “Fair Premium Rating Methods and the Relations between Them”,
Journal of Risk and Insurance , 61(4), pp. 592-616.
6