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A long-term care multi-state Markov model revisited: a Markov chain Monte Carlo approach

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Abstract

A multi-state Markov model is calibrated to Austrian data on recipients of long-term care payments the amount of which depends on defined frailty state levels. In contrast to a predecessor paper by one of the authors (see Fleischmann in Eur Actuar J 5(2):327–354, 2015), we are able to allow for different mortality intensities for different frailty states. A correction term is introduced in the mortality intensities’ functional representation to deal with observed mortality humps around the retirement age for certain frailty levels. Parameter calibration is done using MCMC methods (adaptive Metropolis–Hastings-within-Gibbs). The results reveal a considerably better fit of refined to raw prevalence units than the original model of Fleischmann (Eur Actuar J 5(2):327–354, 2015). Finally, the results are used to estimate the remaining healthy lifetime for certain ages, indicating slight but significant increases over the last 4 years.

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Notes

  1. We requested in the specification of the data report, that a person deceased during a year was reported with frailty level i at the beginning of the year.

  2. These units are obtained from comparing the values of ages x and \(x+1\) for a given year.

  3. “Acceptance rate” refers to the number of steps in the algorithm in which the candidate value from the proposal distribution is accepted vs. the number of total steps.

  4. Note that 2014 is the earliest year for which such a comparison is possible due to the restrictions on the data for years before 2014, cf. the discussion in the introduction.

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Authors and Affiliations

  1. Beltios GmbH, Canongasse 17, 1180, Vienna, Austria

    Anselm Fleischmann & Daniel Sirianni

  2. The Boston Consulting Group GmbH, Am Hof 8, 1010, Vienna, Austria

    Jonas Hirz

Authors
  1. Anselm Fleischmann
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  2. Jonas Hirz
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  3. Daniel Sirianni
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Correspondence to Daniel Sirianni.

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Appendix

Appendix

Table 8 contains the numbers of recipients of long-term care benefits as of January 2018 by frailty level v and age x. Table 9 contains numbers of fatalities by frailty level v and age x for 2018. The data in these tables is provided by Dachverband der Sozialversicherungsträger, an umbrella association of Austrian social security institutions. Table 10 is the Austrian unisex mortality table for 2018. Table 11 contains average Austrian population numbers for 2018. The data in these tables is provided by Statistik Austria.

Table 8 Recipients of public welfare LTC benefits, Austria 2018
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Table 9 Deaths per frailty level, Austria 2018
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Table 10 Unisex mortality table, Austria 2018
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Table 11 Average population, Austria 2018
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Fleischmann, A., Hirz, J. & Sirianni, D. A long-term care multi-state Markov model revisited: a Markov chain Monte Carlo approach. Eur. Actuar. J. 12, 215–247 (2022). https://doi.org/10.1007/s13385-021-00285-y

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