Long-term care transition matrix and three-year projection
A long-term care insurer uses a three-state annual Markov chain with states Healthy (H), Care (C) and Dead (D). The estimated one-year transition matrix is: from H, 0.85 to H, 0.10 to C and 0.05 to D; from C, 0.20 to H, 0.55 to C and 0.25 to D; from D, 1 to D. A 65-year-old policyholder is currently healthy. The insurer wants the probability of being in care at age 68 and the expected number of years spent in care over the next three years. The matrix was estimated from five years of data, and transitions are assumed to depend only on current state and age. A reviewer questions whether the Markov assumption is realistic because recovery from care depends on duration in care and on age.
1. Compute the probability that the policyholder is in care at age 68 and the expected number of years in care over the three years. Show your working.
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Let v_t be the state distribution at age 65+t. v_0=(1,0,0). v_1=(0.85,0.10,0.05). For v_2: H=0.85×0.85+0.10×0.20=0.7425; C=0.85×0.10+0.10×0.55=0.1400; D=0.1175. For v_3: H=0.7425×0.85+0.14×0.20=0.659125; C=0.7425×0.10+0.14×0.55=0.15125; D=0.189625. So the probability of being in care at age 68 is 0.15125, or 15.13%. Under the annual-chain convention, time spent in care during the three years is the sum of the probability of being in care at the start of each year: 0+0.10+0.14=0.24 years. If transitions can occur within a year, a continuous-time Markov jump process would be needed and the expected occupancy would differ; the valuation should state which convention is used.
2. Critique the Markov assumption and propose how the model could be extended. Include the data you would require and the effect on reserves.
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The Markov assumption says the next transition depends only on current state and age, not on duration in state or history. That is questionable for care: recovery probability typically falls sharply with duration, and mortality depends on duration and age. A semi-Markov or duration-dependent multi-state model would be more appropriate. Required data include exact transition dates, duration in each state, age, sex and exit reasons, with sufficient counts of H→C, C→H and C→D by duration and age. If recovery is overstated, expected care time and premiums or reserves are understated; if duration dependence makes care more persistent, reserves should increase. Also check that the matrix is valid, with non-negative entries and rows summing to one, and that the five-year estimate is not dominated by one calendar period or by a change in assessment criteria.
3. Explain how you would estimate the transition matrix from five years of data and what biases might arise.
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With annual snapshots, count transitions between states at successive anniversaries and divide by the number at risk in each origin state. For example, the H→C probability is the number of healthy members who are in care one year later divided by the number healthy at the start, allowing for deaths and exits. If exact dates are available, estimate transition intensities from waiting times and transition counts, allowing for censoring at death or withdrawal and for new entrants. Bias can arise from interval censoring, because multiple transitions within a year are missed; from heterogeneity by age, sex and duration; and from a five-year window that is not representative. A multinomial or Poisson GLM with age and duration can be used, and the fitted matrix should be validated against independent data or a holdout period.