Actuarial
Fair Premiums, Risk Margins, and Solvency Considerations in Commercial Insurance Pricing
Gordon Thompson | June 6, 2025
Footnotes
1 Hans Bühlmann, Mathematical Methods in Risk Theory, Section 4.1.3, 1970.
2 Stephen J. Mildenhall and John A. Major, Pricing Insurance Risk: Theory and Practice, 2022.
3 Mildenhall and Major, Pricing Insurance Risk, pg. 1.
4 Mildenhall and Major, Pricing Insurance Risk, pg. 238.
In a previous Expert Commentary, "The Evolution of Commercial Insurance Pricing," we outline changes in actuarial pricing methodology and theory over time, and we end with the necessary consideration of capital adequacy in pricing starting in the 1980s–1990s. In 2022, Stephen J. Mildenhall and John A. Major synthesized the best of statistical theory and presented a practical approach to pricing that can be widely applied. This approach was then adopted (blessed?) by the Casualty Actuarial Society and put on the syllabus for the advanced pricing exam. This means that their approach is required reading for any actuarial student who wants to become a fellow.
In this article, we discuss fair premiums, risk margins, and solvency considerations, starting with the approach developed by Professors Stewart Myers and Richard Cohn (the "Myers–Cohn approach") from the 1980s–1990s and ending with the even more modern Mildenhall–Major approach published in 2022.
Is It Fair to Charge a Risk Margin?
Commercial insurance premiums pay for insurance company expenses and a profit margin, and then, the lion's share of the premium goes to losses and a risk margin. Expenses? Sure, the company needs a margin to cover the expense of operating in this highly regulated space. A profit margin? Sure, each company needs to be able to gather profits to have some return on capital invested in the enterprise. Losses? That's the tricky part.
When the insurance entity is large and writes thousands (or tens of thousands) of policies, the "expected losses" may be a very stable number that is easy to estimate, and "risk margins" can be very small. If, however, the insurance entity is small, or if the policy limits are incredibly large, then the "expected losses" are unstable, and risk margins must be large enough to cover a large proportion of the possible loss scenarios. In the small insurer or large policy limits cases, risk margins increase. Actuaries and statisticians have worked for decades to determine a fair way to apply these risk margins.
Risk Margins Are Central to Actuarial Premiums
First, we note that the introduction of risk margins into the pricing formula is not a new actuarial concept. In Mathematical Methods in Risk Theory, Hans Bühlmann discusses the principles of premium calculation. 1 On page 86, he replaces "expected costs" with "expected loss + risk margin." When using a general industry average, we must be careful to always add a significant risk margin when we are looking at only one insured, one policy, or one risk.
The purpose of the inclusion of a risk margin in the price is to cover the "specter of large loss" that could result in the economic ruin of the insurance program. Premiums require risk transfer. Risk transfer occurs when premiums are less than the policy limits. So, while losses may be less than the total premiums written in a risk collective, there must be a reasonable probability that losses may also exceed premiums. The risk margin should reflect the uncertainty associated with the risk collective and a reasonable return on capital for the equity supporting the insurance coverage limits.
Modern Insurance Pricing Theory
The view of fair market premiums introduces capital considerations into the "modern insurance pricing theory." In developing the modern insurance pricing model, I agree with the approach advanced in Pricing Insurance Risk: Theory and Practice. 2 The approach demonstrates the theoretical concept associated with a "fair rate of return" for insurers.
This pricing theory includes the cost of equity, which supports the policy coverage limits. It is similar to the well-known premium conditions advocated by Myers–Cohn in 1987. The Myers–Cohn fair-premium condition indicates as follows.
Myers–Cohn inclusion of an insurer's "equity" support for a policy is central to the determination of the "fair-premium" for an insurance risk collective.
"Classical insurance pricing models" do not directly consider the insurer's equity position in supporting the coverage. Historically, the large writers in the commercial lines insurance industry write at significantly low premium-to-surplus ratios (.65 to 1), while small insurance operations may write at a 10 to 1.
Solvency considerations are not typically considered as the large insurers are monitored annually for capital adequacy. However, for small insurers (insurers writing less than $50 million in total annual premium, of which there are hundreds in the National Association of Insurance Commissioners and thousands in the captive insurance market), this is not a dominant monitoring consideration. The small insurance program covers a volatile and incredibly variable book of risks, and hence, if losses exceed premiums to the extent that ruin occurs, regulators realize there is a significant risk of nonpayment. To offset this insolvency risk, premium levels must represent higher confidence levels for losses while maintaining risk transfer.
Regardless, insurance regulators are required to manage the "insolvency risk" of their domicile's insurance companies. The Actuarial Standards Board, in its Actuarial Standard of Practice (ASOP), also maintains a direct tie to solvency in ASOP No 55 (2019), "Capital Adequacy Assessment." Financial viability (i.e., solvency) for the insurance program is also a consideration under basic ratemaking principles highlighted in the "Statement of Principles Regarding Property and Casualty Insurance Ratemaking." It states the following.
Increase the Risk Margin and Decrease the Probability of Ruin
In the regulation of personal lines insurance, the introduction of risk margins (also known as profit loads) is assumed to meet the needs of annual claim count fluctuations (for stability based on volume). But even for personal lines coverages, reinsurance costs associated with claims costs often have risk margins of 50 percent or more (see Pricing Insurance Risk: Theory and Practice 3). By extension, commercial lines of risk pricing on small insurers with limited capital (or large policy limits) and risk margins need to be sufficient to have a lower probability of ruin.
The standard for this "probability of ruin" is typically negotiated with the insurer's regulator. Often, with the insurers we have worked with, regulators require probabilities of ruin of 5–10 percent in year one of operations, working towards 1 percent or less over time as capital within the insurance company builds. This is a generality, as regulators vary, and the risk being reviewed varies (for example, size of policy limits, credibility/representativeness of data, and quality of loss control would each play a vital role.).
Small insurance entities (like captives, which make up more than half of the commercial insurance market) must find the "Goldilocks" premium amount that is large enough for solvency concerns but also small enough to be "not excessive" and to "transfer risk." Threading that needle is difficult for insurance enterprises with scarce data and historically volatile results. A common actuarial solution is to model the situation (in a collective risk model) so it can be stress tested.
The Pricing Formulas
Modern portfolio pricing theory introduces the concept of "fair market returns" into the pricing theory for insurance risks. This theory recognizes the aggregation of individual policy coverages into aggregate risk collectives. It recognizes the purpose of an insurer's capital is necessary to support claims when actual claim costs exceed the selected loss costs. Under modern pricing theory, supporting equity is also due a fair return.
Under traditional pricing models, the expected value principle is the basis for a common variation of the premium formula as follows.
In developing the actuarial premium under modern pricing models, the traditional pricing model is modified to include an amount for an economic return on the supporting capital.
This modern pricing model is slightly modified from the model displayed on page 238 of Pricing Insurance Risk: Theory and Practice. 4 We believe these two models to be comparable. Mildenhall and Major's model is as follows.
This formula meets the Myers–Cohn fair premium condition without tax and has various other advantages (to be discussed in a future article).
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