1  Introduction

posts/010-introduction.qmd

Compiled: 2026-02-28 09:46:25.729486800

1.1 Purpose, Scope, and Goals

posts/010-files/introduction.qmd

This monograph presents a practical and tested framework to determine marginal costs by unit based on a business plan, a starting balance sheet, and an overall profit target, and describes how to incorporate those costs into rates reflecting time value of money and loss emergence (reserve) uncertainty. The resulting pricing framework produces explainable costs that respond appropriately to each unit’s ultimate loss volatility, payout pattern, and loss emergence profile under a variety of risk appetite assumptions. It extends well-understood single-period pricing frameworks to the multi-period context inherent in most non-life insurance products. It is homogeneous in volume, ensuring that unit costs “add-up” by Euler’s theorem [ Mildenhall2004]. It is called a pricing framework, despite producing marginal costs, to align with common usage: “pricing actuaries” and “pricing departments” [REF to SECTION].

Marginal costs are the only appropriate decision metric for portfolio management, and are the basis of firm decision making in a competitive market Hirshleifer et al. (2005). They are firm specific and may differ from a cost allocation negotiated between insureds because of their relative market power (Jouini et al. 2008). As such there can be good business reasons why individual accounts are written at premiums that deviate from marginal cost. However, business must be managed by marginal cost and it remains an underwriting input at all levels of aggregation. Allowing cross-business unit subsidies (i.e., non-marginal cost pricing) does not work because manager incentives are not aligned: profits above cost are my skill; claim shortfalls are recouped by supposed linkages to other units’ business. Thus an underwriting manager may deviate from marginal cost pricing within their own book, but they should be charged the marginal cost in aggregate. Marginal cost including the cost of capital is also consistent with the CAS Statement of Principles on Ratemaking (Casualty Actuarial Society 1988).

A pricing framework translates a risk appetite into front-line decisions and therefore steers the supply of insurance. An implicit or explicit framework underpins every major decision an insurer makes, from underwriting a single policy to setting corporate strategy or purchasing reinsurance. It encapsulates the firm’s view of risk and return and shapes both the supply and the price of insurance in the market as it interacts with competing frameworks. Given this central role, it is disappointing that no single, privileged framework exists, in part because the problem is too complex. Nevertheless, it is generally possible to determine a reasonable range of costs across appetites, and ranges are often more informative, persuasive, and useful than point estimates.

The framework is suitable for actuaries working in many different roles. It enables the corporate actuary to estimate an overall return and to determine each business unit’s contribution in an easily communicated way. A business-unit actuary can then translate this guidance into pricing tools and filed rates. And a line-level actuary can use the framework to determine indicated premiums for individual accounts, assess whether reinsurance is economic, and advise on deploying capacity given market rates.

1.2 Motivation

The monograph is an update to the classic pricing papers of the last few decades, notably Robbin (1992), though with a different focus. Its approach is grounded in economic principles and is intended for global application, in contrast to work focused on specific regulatory regimes, such as US statutory rate filings.

Several significant developments since the early 1990s warrant a renewed look at multi-period pricing. The coherent risk measure revolution initiated in the late 1990s provided a new theoretical language for risk. (We regard a pricing measure as an example of a risk measure.) Global regulatory and accounting standards, notably Solvency II and IFRS 17, have fundamentally shifted the industry’s focus to a one-year, economic view of risk and capital. Today, the widespread adoption of sophisticated internal capital models, catastrophe models, and formal risk appetite frameworks ensures the necessary inputs for more advanced pricing models are generally available.

It is also time to move past certain deep-seated but erroneous beliefs. Measures like underwriting profit are not meaningful economic indicators. The idea that discounting losses is a slippery slope to under-pricing is a fallacy; acknowledging the time value of money is a basic economic reality. While some jurisdictions, like the US, remain tied to solvency-driven statutory accounting systems that are ill-suited for pricing—for instance, by carrying nominal reserves, recognizing revenue ahead of risk emergence and consequently lacking a provision risk in reserves—the global trend is clear. Actuaries must recognize the shortcomings of such systems for pricing rather than attempting to rationalize their internal contradictions.

1.3 Why PMIR is Hard

time value of money • information • accounting

Pricing multi-period insurance risk is the quantum gravity of actuarial science, trying to unify static, intra-temporal point-in-time account-level predictive modeling with evolving, inter-temporal reserving and portfolio risk management. A theoretically perfect solution is likely impractical, as it would require computationally intractable path dependent stochastic-on-stochastic nested simulations and my initial ambition to present such a solution proved infeasible. Instead, this monograph presents a framework that is practical, implementable, consistent with modern accounting principles like IFRS 17, and a demonstrable improvement over many current practices. Any model is a simplification, and we will be transparent about those required.

PMIR’s difficulty stems from three intertwined sources: the time value of money (TVM), the flow of information and its effect on probability assessments, and the constraints of accounting.

Superficially, the problem appears to be a TVM exercise, but classical net present value techniques are difficult to apply to the contingent cash flows and risk recognition patterns in insurance in part because the capital flows are undefined. More fundamentally, the difficulty arises from the existence of interim accounting periods. These periods are not merely arbitrary divisions of time; they are moments of mandatory financial disclosure with real economic consequences that can trigger capital calls or violate debt covenants. An old axiom states:

An informed decision maker can always do at least as well as an uninformed decision maker by deciding to ignore the information. (Eeckhoudt et al. 2011)

This principle breaks down for an insurer. Accounting and regulation prohibit ignoring information. Mandatory recognition is the crux of the multi-period problem and explains its inextricable link to accounting. In short: accounting mandates recognition; recognition alters capital; capital has a cost

Consider the challenge of guaranteeing a portfolio will be worth at least $100,000 on each of the next ten annual anniversaries, versus merely guaranteeing that value at the end of ten years. The series of annual tests introduces path dependency and is far more complex and expensive to hedge. It requires a view on future volatility to price the future guarantees—a stochastic-on-stochastic problem analogous to the insurer’s dilemma.

The impact of these multi-period effects is not even directionally obvious. Is risk time cheap and less costly to bear when it emerges over many periods, or time expensive? On one hand, a gauntlet of interim solvency tests suggests a higher price is needed. On the other, the partial information revealed in each period reduces uncertainty, which should lower the risk and therefore the price. This monograph provides a framework to navigate these competing effects without attempting to solve the problem in its full, intractable generality. Thankfully, since most insurance lines are written with thin margins, a pragmatic approach that correctly balances the primary drivers of risk is sufficient for determining useful marginal costs.

1.4 Three Central Ideas

period • profit signature • decoupling

We use three central ideas to help navigate the problem’s complexities.

Period. A single-period model operates within a black hole of time during which no interim evaluation occurs. In reality, insurance risk is multi-period because it is subject to interim evaluations—typically quarterly or annual—driven by financial reporting, capital regulation, or collateral requirements. It is the presence of these mandatory evaluations that drives a wedge between multi- and single-period prices. A multi-period policy is one whose settlement spans multiple accounting periods even when it has a one-year term. The problem is therefore fundamentally an accounting construct, which is why we devote significant attention to accounting frameworks. The tension arises because capital providers, who can exit their investment annually, price risk on a one-year basis, while the insurer must bear that risk for multiple periods.

Profit signature. A concept borrowed from life insurance, the profit signature is the pattern by which accounting recognizes unit-level profits through time under a chosen framework. It separates amount from timing risk and aligns pricing with recognition. Once again, it is inextricably linked to an accounting standard.

Calendar-year loss decoupling. Losses some a single accident-year (AY) are coupled across calendar years through development. Pricing one-year risk requires decoupling AY models into calendar-year (CY) incurred changes: this year’s initial booking on new business plus changes on prior years. In steady state, means and variances align, but higher moments and dependence change. Current practice often splits “pricing risk” and “reserve risk” without checking that the implied ultimate AY variability is coherent; the coupled-vs-decoupled relationship is the hinge in multi-period pricing. The exact nature of the coupling is very subtle, has implications for reserving, and is currently under-appreciated.

1.5 The DMC Framework in a Nutshell

We call the framework DMC, the Decoupled Marginal Cost. It can be regarded as a multi-period extension of the single-period, top-down Bühlmann (1985) approach using spectral risk measures (SRMs) described in Mildenhall and Major (2022). The core idea is to price the risk that is transferred to capital providers on a one-year basis. The framework proceeds as follows:

1. Set a Target: Establish an overall target return for the firm for the upcoming calendar year under the relevant accounting convention.
2. Model CY Losses: Using the calendar year decoupling, transform simulated ultimate loss distributions by business unit (both new business and reserves) into a simulated distribution of total net calendar year incurred loss.
3. Calibrate the SRM: We calibrate a family of SRMs—flexible pricing tools that encode the market’s risk appetite—so that they produce the required profit target when applied to the total CY loss distribution and are consistent with the firm’s capital structure. These SRMs are then held fixed and provide a range of indications.
4. Marginal Costs: Compute marginal costs by business unit using the SRMs’ natural (Euler) allocations. These add up to the total CY profit requirement by Euler’s theorem.
5. Ratemaking: Set rates to produce the required targets when combined with time value of money.

DMC is sensitive to individual unit volatility, payout, and emergence characteristics within the whole portfolio, the market risk appetite encapsulated in each SRM, and prevailing macroeconomic variables. It relies on standard capital model simulations, explicitly accounts for time value of money through discounting, and is consistent with the one-year risk view of modern regulatory (Solvency II) and accounting (IFRS 17) frameworks. Discounting to be “turned off” to replicate US GAAP if desired. The model is typically applied to an adjusted ) portfolio so results are independent of current reserve positions.

Although conceptually straightforward, DMC contains many moving parts—much like a rate level indication—and we employ several simplifications and abstractions to build to it. The most important is a bullet, a unit settles with a single payment at a fixed time. A general unit with a multi-period payout pattern is a weighted combination of bullets; its accounting and cash flows follow directly from those of each bullet. This decomposition preserves additivity and simplifies both valuation and pricing. Our modeling approach therefore begins by understanding pricing for a single bullet. The general case is then built up in four stages, each of which starts with a bullet and then extends to multi-period payout patterns.

1. Use a one-period, zero interest rate model to compute marginal costs from a SRM using Delbaen’s theorem, and interpret the results as an implied capital structure using Jouini’s risk sharing model sec-distortions.
2. Introduce positive interest rates with one-period emergence to study the impact of discounting sec-bullets.
3. Use a zero interest rate, multi-period model to examine the effect of loss emergence sec-emergence.
4. Combine 2 and 3 into a general model and add top-down parameterization sec-general-pricing.

1.6 A Simple Example TODO

As a preview, this section presents a simple example illustrating our most surprising finding: that loss emergence is a rating variable. We believe this is the first explicit interpretation of emergence in this way. More conventional examples focused on discounting and loss payout are given in REF, and combined effects are explored in REF.

Add a minimal two-period, two-outcome example.

1.7 Limitations

standing assumptions for rest of paper

All models are wrong because they choose to ignore certain features to enable better predictions about others. We make the following simplifying assumptions. They make the problem tractable and avoid nested stochastic simulations while preserving the essential economics of multi-period risk recognition.

• Macro Environment: We assume flat, fixed term structures for interest rates and a constant economic backdrop.
• Insurer Operations: We ignore expenses (which can be added separately) and non-insurance businesses. We assume no default risk by restating payouts to available capital. Unit losses are modeled net of unit-level reinsurance but gross of group-level protections, which the model provides an easy way to evaluate.
• Measurement Basis: We use a book value recognition basis consistent with IFRS 17. US GAAP drops out as a special case of the IFRS framework.
• Portfolio State: We use a steady state view rather than the actual current reserve position to ensure prospective prices are state independent.

The following topics, each a significant research area in its own right, are out of scope.

• Market cycle dynamics and franchise value.
• Investment risk modeling beyond the separation of insurance finance costs from investment earnings.
• Long-duration life contracts where policy reserves move premium across periods.
• Business strategy overlays such as cycle management or market share targets.
• Cases where risk and the time value of money are inseparable by construction.

1.8 Context and Literature Review TODO

The monograph stands alone for application but complements PIR and the CMM notes. PIR provides the single-period risk-measure foundations and the case for natural allocation; CMM provides capital-modeling context. A focused literature review covers spectral risk measures and allocation, Solvency II one-year views, IFRS 17 measurement and risk adjustment, and stochastic reserving models related to emergence and development. (REF)

Literature review to follow.

For a review of the literature taken in other approaches, see sec-other-approaches-literature.

1.9 Outline

The remainder of the monograph is organized as follows.

sec-background provides background material, defined as standard results. Readers should scan through it, digging in on items where they are less familiar.

sec-bullets

sec-distortions

sec-emergence

sec-general-pricing

sec-expenses

sec-other-approaches

sec-conclusions offers concluding remarks and suggests areas for future research.

Within these sections there are a series of examples that readers more concerned with implementation may want to read first. This track covers

bullet accounting

discount gaap and ifrs

single period distortions; calibrating to a capital structure

emergence as a rating variable

general pricing

p2p and other examples

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Bühlmann, Hans. 1985. “Premium Calculation from Top Down.” ASTIN Bulletin 15 (2): 89–101. https://doi.org/10.2143/ast.15.2.2015021.

Casualty Actuarial Society. 1988. Statement of Principles Regarding Property and Casualty Insurance Ratemaking. May.

Eeckhoudt, Louis R., Christian Gollier, and Harris Schlesinger. 2011. Economic and financial decisions under risk. Princeton University Press.

Hirshleifer, Jack, Amihai Glazer, and David Hirshleifer. 2005. Price Theory and Applications. https://doi.org/10.1017/cbo9780511813382.

Jouini, E., W. Schachermayer, and N. Touzi. 2008. “Optimal risk sharing for law invariant monetary utility functions.” Mathematical Finance 18 (2): 269–92. https://doi.org/10.1111/j.1467-9965.2007.00332.x.

Mildenhall, Stephen J., and John A. Major. 2022. Pricing Insurance Risk: Theory and Practice. John Wiley & Sons, Inc. https://doi.org/10.1002/9781119756538.

Robbin, Ira. 1992. The Underwriting Profit Provision. Casualty Actuarial Society Study Note.

Tasche, Dirk. 1999. “Risk contributions and performance measurement.” Report of the Lehrstuhl Fur Mathematische Statistik, TU Munchen, 1–26. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.68.9393\&rep=rep1\&type=pdf.