Intensities Versus Transition Probabilities in a Disability Model
A disability income insurer models a portfolio of 10,000 active lives using a three-state Markov model: Active (A), Disabled (D), Dead (X). The annual transition intensities are estimated as: A→D 0.018, D→A 0.22, D→X 0.035, A→X 0.004. The pricing team built a spreadsheet that applies these as annual transition probabilities in a discrete-time chain and computes the expected number of years spent disabled over a 10-year term by summing the probability of being in D at each year-end. The reserving team instead used a semi-Markov model because recovery rates fall sharply after the first year of disability: the D→A intensity is 0.30 in the first year and 0.12 thereafter. A reinsurer argues the semi-Markov structure is unnecessary because the portfolio is small.
1. The pricing spreadsheet uses the A→D intensity 0.018 as a one-year transition probability. Explain the error and compute a better approximation for the one-year probability of moving from Active to Disabled, stating assumptions.
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In a continuous-time Markov jump process the intensities are instantaneous rates, not one-year probabilities. For Active lives the total transition intensity out of A is 0.018 + 0.004 = 0.022. Under a constant-intensity competing-risk calculation, the probability of the first transition being to D and occurring within one year is: p_AD(1) = λ_AD/(λ_AD + λ_AX) × [1 − exp(−(λ_AD + λ_AX))] = 0.018/0.022 × (1 − e^(−0.022)) = 0.8182 × 0.02176 ≈ 0.0178. Using 0.018 as the probability overstates the one-year A→D probability slightly. The calculation also ignores recovery from D back to A within the year, so it is only an approximation to the probability of being in D at the end of the year; the exact answer is the (A,D) entry of the matrix exponential exp(Q), where Q is the generator. The spreadsheet should either use the matrix exponential or, for small rates, the approximation p ≈ I + Q rather than copying intensities into probability cells.
2. Explain why the semi-Markov model may materially change expected disabled years and reserves. Advise the reinsurer on whether portfolio size justifies ignoring duration.
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The Markov model assumes the chance of recovery depends only on current state, so every disabled life has the same annual D→A intensity of 0.22. The data show recovery is 0.30 in the first year and 0.12 thereafter. A semi-Markov model with duration-dependent intensities therefore predicts longer disability spells for lives who remain disabled beyond year one, and more expected disabled years over a 10-year term. The Markov model will understate the number of long-duration disabled lives and hence understate income benefit reserves and the cost of the disabled state. Portfolio size does not justify ignoring a structural duration effect: with 10,000 lives the insurer has enough exposure to estimate at least two duration bands, and even with a small portfolio the bias is one-directional. I would advise the reinsurer to use a staged or semi-Markov model, e.g. D1 for duration 0–1 and D2 for duration 1+, with transition intensities estimated separately and smoothed. If data are thin, a parametric duration model can be used. As a check, compare projected recovery rates by duration with observed rates.