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Loss of earning capacity in Denmark -- an actuarial perspective

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For pricing at inception, claim-settlement delays can be ignored; for ongoing reserving and risk management they cannot.

desk verdict A clear, well-written review of Danish disability insurance that makes a plausible case for information-aware multistate models; the central quantitative conclusion rests on unvalidated independence assumptions, but the paper is honest about that. read the letter →

arxiv 2501.11578 v2 pith:7TTI2FN3 submitted 2025-01-20 q-fin.RM stat.AP

C. Furrer , O.L. Sandqvist This is my paper · ORCID
classification q-fin.RMstat.AP MSC 91B3062P05
keywords disabilityinsurancelossofearningcapacitymultistatemodelspublicbenefitsclaimsettlementdelaysclaimsreservingpreventioninitiativescausalinference
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that Danish disability insurance, where public benefits, claim settlement, and prevention programs interact, cannot be handled by copying classic life-insurance methods. Its central claim is that reporting and adjudication delays do not matter for pricing at contract inception, but do matter for later reserving and risk management: once coverage is underway, reserves must be computed from the insurer's actual information and realized cash flows, including backpay, rather than from the insured's unseen health state. The paper develops this through a multistate framework—tracking an insured across states such as active, disabled, and dead—in which the operational reserve decomposes into a classic active-life reserve plus corrections for disabilities that have occurred but not yet been reported, with separate formulas for claims that are reported but unpaid, currently paid, and previously paid. It also claims that raw claim data are biased by delays and rejections, so disability and reactivation hazards need a two-step correction, and that prevention initiatives need causal-inference evaluation. A sympathetic reader would care because the wrong reserve size and timing distorts solvency, hedging, and the measured value of prevention.

What carries the argument

The central object is the operational filtration $\mathcal{G}_t$ — the insurer's actual available claim-level information, including reports, adjudications, payments, and backpay — together with the realized cash flow $d\bar{B}$, as opposed to the classic filtration $\mathcal{F}_t=\sigma\{Y(s):s\le t\}$ generated by the hidden biometric state process and the contractual payments $dB$. The reserve that carries the argument is $\bar{V}(t)=E[\bar{P}(t)|\mathcal{G}_t]$, the conditional expectation of the present value of realized future benefits given what the insurer genuinely knows. Structural independence assumptions let the paper write this operational reserve in closed form: for an insured with no reported claim it is the classic active reserve plus an integral over unreported disabilities weighted by the probability that the reporting delay exceeds the elapsed time; for reported claims it uses the probability that the claim is (re)awarded times the corresponding classic reserve. This machinery turns a textbook reserve that is not computable in practice into a computable one, and it feeds a two-step estimator in which award probabilities are estimated first and disability and reactivation hazards second.

What would settle it

On a Danish disability portfolio with average reporting delays of several months and a material share of initially rejected claims, compute reserves under the classic biometric filtration and under the operational filtration with realized cash flows; if the gap between the two is negligible relative to the portfolio's stochastic fluctuation, the paper's claim that delays matter after inception is wrong, while a large gap explained by backpay and unreported disabilities would confirm it.

Watch

Extended reading notes

Core claim

The core discovery is that the question 'Can the claim-settlement process be ignored?' has a time-dependent answer. At time zero the operational reserve equals the classic reserve, because nothing has been reported and no backpay can relate to events before inception; this makes the classic models adequate for pricing and initial reserving. After inception, the classic reserve is built on information the insurer does not actually have: the insured may be disabled before time $t$ but report only later, or have a claim awarded after a rejection, triggering interest-bearing backpay. The paper therefore defines the reserve as the conditional expectation of the realized cash flow given the insurer's operational information and, following results from the recent multistate literature it surveys, expresses it as the classic reserve plus a correction for unreported disabilities, while reported claims fall into three cases with reserves proportional to the probability of (re)award. In estimation, the paper claims that reporting delays and incomplete adjudications bias the observed sample, and that unbiased disability and reactivation hazards require scaling exposures by reporting-delay survival and outcomes by award and reaward probabilities.

Load-bearing premise

The load-bearing premise is that reporting delays and claim adjudication outcomes are conditionally independent of the insured's true health trajectory and of future claims once all the insurer's information is known.

Editorial extensions

If this is right

  • At contract inception, premiums and initial reserves can still be set with the classic multistate models, because no backpay can relate to events before inception.
  • Once coverage is underway, reserves that ignore reporting and adjudication delays will be wrong in both size and timing, so risk-management and hedging decisions based on them are not reliable.
  • Disability and reactivation hazards estimated from raw observed claim data are biased; the two-step correction using reporting-delay survival and award probabilities is needed before the fitted model is used for reserving.
  • Multistate individual reserving remains a computationally moderate route, so insurers already using multistate models can adopt the delay-adjusted reserve more easily than switching to high-dimensional non-life-style models.
  • Prevention programs should be evaluated with causal-inference designs, such as no-unmeasured-confounding plus positivity or regression discontinuity, rather than by comparing high-risk participants with non-participants.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the independence assumptions fail, the reserve formulas may still be approximately right in expectation but will understate reserve uncertainty; a natural extension is to model reporting and adjudication jointly with the biometric process rather than conditionally independently.
  • The same filtration gap should appear in other insurance products with long settlement lags, such as income protection or critical illness, so the framework likely transfers beyond the Danish disability market.
  • Because backpay accrues interest over the delay, interest-rate stress tests should be part of reserve validation: a rising rate path makes the delay correction larger than it looks in a level-rate calculation.
  • Insurers could intentionally design prevention targeting rules that create regression discontinuities or exogenous assignment, turning 'happy accidents' into planned experiments.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. This paper is a review and position statement on loss-of-earning-capacity (disability) insurance in Denmark. It describes the interplay among the insured, the insurer, the employer, and the public benefit system, using two stylized cases, then formalizes contractual versus realized payment streams. The central technical argument is that classic multistate reserving models, which condition on the filtration generated by the biometric state process, are not operational once reporting delays and claim adjudication are present: the biometric state may not be observed at time t, and backpay makes the classic present value differ from the realized one. The paper reviews non-life individual reserving, aggregate reserving, and a recently proposed multistate framework based on an operational filtration G_t and realized cash flows, with formulas for the corrected reserve in four states (no reported claim, reported-but-not-paid, currently eligible, previously eligible). It also discusses two-step estimation of disability and reactivation hazards and argues that prevention initiatives should be evaluated through causal inference. The main conclusion is that for pricing at inception the classic model suffices, but for reserving and risk management during the coverage period the classic model fails unless reporting and adjudication delays are very short.

Significance. If the central claim is accepted, the paper fills a gap in the actuarial literature by systematically connecting the Danish institutional setting to formal reserving methodology. Its strength lies in a clear articulation of why the filtration sigma{Y(s): s <= t} is not observable under reporting delays and why realized cash flows with backpay change the reserve dynamics. The paper also gives credit where due to the authors' own recent work (Buchardt et al. 2023, 2025; Sandqvist 2025) and to the non-life reserving literature. At the same time, the headline recommendation is not empirically calibrated: no quantitative threshold for 'very short' delays is given, and the corrected reserve formulas require conditional-independence assumptions that are stated but not validated. As a review paper, it will be useful for practitioners entering the Danish disability market, but the strength of the 'resounding no' should be tempered or supported with data or simulation.

major comments (3)
  1. [Section 3.2 and Section 4.4.1] The central claim that classic multistate models are inadequate for reserving 'unless the reporting and adjudication delays are very short' is not quantified. The manuscript does not provide a numerical threshold for 'very short,' nor does it show with Danish delay data or a sensitivity analysis that the difference between the classic reserve (conditioned on the biometric filtration) and the G_t reserve (conditioned on the operational filtration with realized cash flows) is material for typical delay distributions. The qualitative operational argument is sound, but the practical conclusion ('a resounding no') needs at least an illustrative calibration to be actionable. Please add a numerical example or explicit bounds on delay durations under which the classic model is adequate.
  2. [Section 4.4.1 and Section 4.4.2] The corrected reserve formulas and the two-step hazard estimation procedure are not derived in this manuscript; they are taken from Sandqvist (2025) and Buchardt et al. (2025), which are authored by the same group. As the manuscript itself concedes, these formulas require 'rather strong independence assumptions' (Section 4.4.1), and if reporting delays or adjudication outcomes are correlated with disability severity or recovery, both the size and the timing of the reserves are affected. The paper's own Charlie case (Section 2.2) illustrates that adjudication can be informative about recovery. No Danish data or simulation is provided to support the conditional independence of reporting delays and adjudication outcomes given G_t, so the reliability of the headline recommendation is not demonstrated. The authors should either provide evidence for the assumptions or explicitly frame the recommendation as conditional on them, with a discussion of the likely direction and magnitude of bias under plausible violations.
  3. [Section 4.4.2] The 'approximately' in the reserve formulas (for example, V(t) = Va(t) + integral term and V(t) = P(Claim is awarded | G_t) Vi(G(t),0)) is not defined. Since the paper's conclusion that the classic model is inadequate depends on the magnitude of the G_t correction, the order of the approximation and the conditions for its accuracy should be stated, at least by reference to the relevant theorems in Sandqvist (2025). Without this, the reader cannot assess when the correction is a first-order effect versus a second-order refinement.
minor comments (7)
  1. [Section 2.1] In the sentence 'Disabilities can very in form, degree, and duration,' 'very' should be 'vary.'
  2. [Section 2.5] The phrase 'The emphasize will be' should read 'The emphasis will be.'
  3. [Section 3.2] In the description of reporting delay, 'until the insured notifies the insured' should be 'until the insured notifies the insurer.'
  4. [Section 4.4.2] The notation for the two reserves (the classic reserve and the G_t-based reserve) is visually indistinguishable in the text; please use distinct symbols, for example V^F and V^G, throughout the discussion.
  5. [Table 1] The caption contains the typo 'T able 1'; it should be 'Table 1.' In addition, the entries such as 'decent/good' and 'poor/good' are subjective and would benefit from a one-sentence definition of the scale in the text.
  6. [References] The entry 'Delong, /suppress L.' appears corrupted; it should read 'Delong, L.'
  7. [Section 4.4.2] In the exposure formula for the two-step estimation, 'P(Reporting delay <= t - t_i)' should be written conditionally on the operational filtration (for example, P(Reporting delay <= t - t_i | G_t)) to be consistent with the conditioning used elsewhere in the same subsection.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the key conceptual claim is argued directly, and the cited reserve formulas are transparent attributions to prior same-group work, not hidden redefinitions or fitted predictions.

full rationale

This is a review/perspective paper rather than a self-contained derivation. Its central operational claim—that under reporting and adjudication delays the classic filtration F_t=sigma{Y(s):s<=t} is not usable for in-force reserving—is argued on its own terms in Section 4.4.1: the information may not be available at time t and the insurer may not have paid B(t), so the classical present value is not the relevant one. That argument does not depend on the authors' prior theorems. The quantitative reserve and estimation formulas in Section 4.4.2 are explicitly attributed to Sandqvist (2025) and Buchardt et al. (2025), which are same-group preprints; this is heavy self-citation, but it is transparent attribution to the original sources rather than an unacknowledged reuse, and this review does not pass the formulas off as newly derived here. The paper also openly states the main caveat (Section 4.4.1: 'rather strong independence assumptions which if violated may impair both the size and timing of the reserves'); that is a validity/robustness limitation, not a circularity, and the absence of a quantitative threshold for 'very short' delays is a calibration gap rather than a circular step. No fitted parameter is renamed as a prediction, no definition is fixed in terms of the target, and no uniqueness theorem from the same authors is invoked to forbid alternatives. Under the stated evidence, there is no circular step that reduces a claimed result to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities are introduced. The paper's claims rest on standard actuarial mathematics, domain assumptions about Danish public benefits, and independence assumptions inherited from the authors' own multistate framework.

assumptions (6)
  • standard math Reserve V(t)=E[P(t)|F_t] in Eq. (1)-(2) is the conditional expectation of discounted future cash flows under a market-consistent measure.
    Used in Section 4.1 as the starting point for all reserving paradigms.
  • domain assumption Contractual payment streams can be modeled in continuous time with rates dB(t), treating monthly payments as a high-frequency approximation.
    Section 3.1 introduces dB(t)=1{Y(t)=2} b dt and the min/max disability annuity; this continuous-time idealization underlies the subsequent reserve formulas.
  • standard math By the law of iterated expectations, reserves can be based on average future payments rather than explicit forecasts of earnings capacity e_t and social security offsets c_t.
    Section 3.1.2 states that it is not necessary to forecast e_t and c_t to compute premiums and reserves.
  • domain assumption The modeling framework of Buchardt et al. (2023) and Sandqvist (2025) relies on structural independence assumptions between the biometric process, reporting delays, and adjudication outcomes.
    Section 4.4.1 notes the framework 'necessitates rather strong independence assumptions' and that violations can be accommodated only with added complexity; the formulas in Section 4.4.2 inherit those assumptions.
  • domain assumption Causal inference for prevention requires either no unmeasured confounding plus positivity, or a valid regression discontinuity design.
    Section 5 states these conditions as prerequisites for identifying CATE and CATE(w0); they are standard causal-inference conditions, not proved here.
  • domain assumption Claim settlement information is not available to the insurer at time t before reporting; the filtration G_t is the relevant operational information.
    Section 4.4.1 argues F_t = sigma{Y(s): s<=t} is not operational because disabilities can occur before reporting; this motivates the entire 'transaction time' approach.

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Cite this review

Pith. "Pith review of Loss of earning capacity in Denmark -- an actuarial perspective." pith.science (2026). https://pith.science/paper/7TTI2FN3

@misc{pith2026250111578,
  author       = {Pith},
  title        = {Pith review of: Loss of earning capacity in Denmark -- an actuarial perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TTI2FN3}},
  note         = {Machine review of arXiv:2501.11578}
}
read the original abstract

We describe challenges and opportunities related to risk assessment and mitigation for loss of earning capacity insurance with a special focus on Denmark. The presence of public benefits, claim settlement processes, and prevention initiatives introduces significant intricacy to the risk landscape. Accommodating this requires the development of innovative approaches from researchers and practitioners alike. Actuaries are uniquely positioned to lead the way, leveraging their domain knowledge and mathematical-statistical expertise to develop equitable, data-driven solutions that mitigate risk and enhance societal well-being.

Figures

Figures reproduced from arXiv: 2501.11578 by the authors.

Figure 1
Figure 1. provides a schematic representation of the interactions between the insured, the insurer, the employer, and the public system for disability insurance, where by an inter￾action we refer to a direct exchange between two or more agents. One central observation is that the graph is almost fully connected, which contributes to the many-faceted com￾plexity of the situation. The only exception is that exchange between the… view at source ↗
Figure 2
Figure 2. Primary systematic and idiosyncratic (unsystematic) risks for disability insurance. 3 Product design In this section, disability insurance coverages are formalized and compared with a fo￾cus on the Danish market. Our approach is initially descriptive, seeking to explore and understand the characteristics of existing products rather than discuss optimal design. Further, we are looking for common instead of distinguis… view at source ↗
Figure 3
Figure 3. State space J for a classic disability model. The arrows represent the possible transi￾tions. Defining the disabled state as being eligible for disability benefits has the effect of placing the difficulty in modeling onto the probabilities of transitioning to and from the disabled state, rather than placing it in the payment rates. This implies, among other things, that the time of disablement and the transition pro… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: State space for an alternative disability model with separate reactivated state. The arrows represent the possible transitions. When disability benefits are constant, underwriters should manually take people’s finan￾cial situation into account and anticipate what benef…

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