arxiv: 2604.18243 · v2 · submitted 2026-04-20 · 💱 q-fin.MF

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On the market-consistent valuation of health insurance liabilities

Simon Hochgerner , Jonas Ingmanns , Nicole Kastanek

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Pith reviewed 2026-05-10 03:20 UTC · model grok-4.3

classification 💱 q-fin.MF

keywords market-consistent valuationhealth insuranceBest EstimateSolvency IIstochastic interest ratesinflation adjustmentsactuarial equivalence principlevaluation portfolio

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The pith

The Best Estimate for lifelong health insurance liabilities depends on the chosen model for interest and inflation rates, not just spot rate term structures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for lifelong health insurance products with adjustments based on medical inflation via the actuarial equivalence principle, the market-consistent Best Estimate value is not fixed by current nominal and real interest rate curves alone. The value instead varies according to the specific stochastic model selected for how those rates evolve. This means deterministic calculations lack theoretical justification for market-consistent valuation as required under Solvency II. The authors further construct a valuation portfolio that separates deterministic coefficients based on policy data from prices of basis financial instruments independent of individual policies, supporting efficient stochastic evaluation across large portfolios.

Core claim

The Best Estimate of a lifelong health insurance policy depends on the choice of model for the interest and inflation rates. That is, the Best Estimate is not uniquely determined by the currently prevailing term structures of nominal and real spot rates, whence a deterministic calculation is theoretically unjustified. Furthermore, we construct a valuation portfolio such that the Best Estimate valuation decouples into calculations of deterministic coefficients derived from policy data and the prices of basis financial instruments that are independent of the individual policy data.

What carries the argument

The adjustment mechanism from the actuarial equivalence principle interacting with stochastic interest and inflation dynamics, together with a decomposition into a valuation portfolio.

If this is right

Deterministic calculations are theoretically unjustified for assigning Best Estimates to these policies.
Stochastic models for interest and inflation rates are required to obtain market-consistent valuations.
The valuation portfolio decomposition enables efficient computation for large stocks of policies by avoiding per-policy tracking along stochastic paths.
The approach applies directly to adjustment-driven lifelong health insurance products common in European markets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Regulators may need to specify or constrain allowable stochastic models for rate dynamics when enforcing market-consistent reporting for health liabilities.
Model risk in Best Estimate calculations becomes material and may warrant separate disclosure or sensitivity analysis.
Analogous model-dependence issues could appear in other long-duration insurance contracts featuring dynamic economic adjustments.
Numerical experiments comparing deterministic and stochastic Best Estimates on sample portfolios would quantify the practical size of the discrepancy.

Load-bearing premise

The adjustment mechanism derived from the actuarial equivalence principle interacts with stochastic interest and inflation dynamics such that the market-consistent value cannot be recovered from spot rate term structures alone.

What would settle it

Computing the Best Estimate for the same lifelong health insurance policy under two different stochastic models for interest and inflation rates, both calibrated to reproduce identical current nominal and real spot rate term structures, and checking whether the resulting values match.

Figures

Figures reproduced from arXiv: 2604.18243 by Jonas Ingmanns, Nicole Kastanek, Simon Hochgerner.

**Figure 6.1.** Figure 6.1: A comparison of the premium development with a nominal versus a real technical interest rate assumption for an inpatient tariff. Based on typical first order health benefits (KFO x )x for an inpatient tariff and assumptions (q FO x )x for the combined probability of death or surrender, the premiums are calculated with a technical interest rate assumption rcalc = 1% treated as a nominal rate (depicted … view at source ↗

read the original abstract

We are concerned with the market-consistent valuation of lifelong health insurance products, which are subject to adjustments derived from the actuarial equivalence principle and driven by (medical) inflation. Such products are well-established in the European national markets, and the dynamics of the adjustment mechanism is well-understood from an actuarial perspective. However, the question of market-consistent valuation (as is necessary for Solvency II reporting) has not previously been addressed. This gap has led to a situation where some practitioners use stochastic models while others rely on deterministic methods to assign market-consistent values (Best Estimates) to the same type of health insurance liabilities. The purpose of this note is to fill this gap by showing that the Best Estimate of a lifelong health insurance policy depends on the choice of model for the interest and inflation rates. That is, the Best Estimate is not uniquely determined by the currently prevailing term structures of nominal and real spot rates, whence a deterministic calculation is theoretically unjustified. Furthermore, we construct a valuation portfolio such that the Best Estimate valuation decouples into calculations of 1.) deterministic coefficients derived from policy data and 2.) the prices of basis financial instruments that are independent of the individual policy data. Using this decomposition, the policies do not have to be tracked individually along each generated stochastic path. This allows for a more efficient evaluation of the Best Estimate for a large stock of policies with a stochastic model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that best estimates for these inflation-adjusted lifelong health policies depend on the chosen stochastic model for rates rather than just the prevailing term structures, and it supplies a decomposition that makes large-portfolio stochastic valuation feasible.

read the letter

The central claim is that deterministic best-estimate calculations using only current nominal and real spot rates are theoretically unjustified for these products. The actuarial equivalence adjustments, driven by medical inflation, turn the liability into a non-linear, path-dependent functional of the joint interest-inflation process. Two arbitrage-free models that match the same initial curves but differ in volatility or correlation will therefore produce different risk-neutral expectations. That point is internally consistent and does not rest on circular reasoning or extra assumptions.

Referee Report

0 major / 2 minor

Summary. The paper addresses the market-consistent valuation of lifelong health insurance products subject to adjustments derived from the actuarial equivalence principle and driven by medical inflation. It claims that the Best Estimate is not uniquely determined by the prevailing term structures of nominal and real spot rates but depends on the specific stochastic model chosen for interest and inflation rates, rendering deterministic calculations theoretically unjustified. The authors further construct a valuation portfolio allowing the Best Estimate to decouple into (1) deterministic coefficients derived from policy data and (2) prices of basis financial instruments independent of individual policy data, enabling efficient stochastic evaluation for large portfolios without tracking each policy along generated paths.

Significance. If the central claim holds, the result is significant for Solvency II reporting and actuarial practice, as it demonstrates that the non-linear, path-dependent interaction between the equivalence-principle adjustment and stochastic rate dynamics prevents recovery of the market-consistent value from spot-rate term structures alone. The proposed decomposition into policy-specific deterministic coefficients and model-independent basis instruments is a clear practical strength, supporting scalable computation for large books of business. The manuscript fills a documented gap where practitioners have used inconsistent deterministic versus stochastic approaches.

minor comments (2)

The abstract states the central claim at a high level; a brief concrete illustration (e.g., two arbitrage-free models with identical initial term structures but different volatility or correlation producing distinct Best Estimates) would make the dependence on model choice more immediate for readers.
The decomposition into deterministic coefficients and basis instruments is presented as a key contribution; explicit identification of the basis instruments (e.g., which zero-coupon or inflation-linked bonds they correspond to) in the main text would clarify how the approach remains model-independent at the instrument level.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives that Best Estimate valuation of lifelong health insurance policies with actuarial-equivalence adjustments cannot be recovered from nominal and real spot-rate term structures alone, because the adjustment mechanism introduces a non-linear, path-dependent functional of the joint stochastic (interest, inflation) process. This is shown by constructing an explicit valuation portfolio that separates policy-specific deterministic coefficients (from policy data) from prices of basis instruments (independent of individual policies). No load-bearing step reduces to a fitted parameter renamed as prediction, a self-citation chain, or a self-definitional equivalence; the argument follows directly from the problem setup and the distinction between deterministic term structures and arbitrage-free stochastic models that match the same initial curves but differ in volatility or correlation. The decomposition is a direct algebraic consequence of the structure and does not rely on hidden assumptions or prior author results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, new entities, or detailed axioms are described in the provided text.

axioms (1)

domain assumption The actuarial equivalence principle governs the policy adjustments driven by medical inflation.
Referenced in the abstract as the basis for the adjustment mechanism.

pith-pipeline@v0.9.0 · 5549 in / 1178 out tokens · 30062 ms · 2026-05-10T03:20:53.097295+00:00 · methodology

discussion (0)

Reference graph

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