Limitations of the Markov Assumption

The standard Markov assumption requires that transition intensities depend only on the current state and the individual's age. However, in many insurance contexts, the duration spent in a state also matters. A person who has been disabled for 6 months has different recovery and mortality prospects than someone who just became disabled yesterday. The Markov model cannot capture this duration dependence because it has no memory of when the current state was entered. Semi-Markov models address this limitation by incorporating duration as an additional variable.

Semi-Markov Models

In a semi-Markov model, the transition intensity from state i to state j depends on three variables: the current age x, the current state i, and the duration z in the current state. The notation mu_{x,z}^{ij} makes this explicit. When z = 0, the individual has just entered state i; as z increases, the transition rates change to reflect the evolving characteristics of someone who has been in that state for a while.

For disability insurance, the most important duration dependence is in the recovery rate. Empirical data consistently shows that the probability of recovering from disability decreases with the duration of disability. A person disabled for 3 months might have a recovery rate of 0.50 per year, while someone disabled for 3 years might have a recovery rate of only 0.05 per year. This steep decline in recovery rates with duration is one of the strongest arguments for using semi-Markov rather than Markov models for disability products.

Mathematical Framework

The transition probabilities in a semi-Markov model are more complex than in the Markov case because the duration variable must be tracked. For an individual currently in state i with duration z, the probability of being in state j at time t is written as t_p_{x,z}^{ij}. If the individual transitions to a new state, the duration clock resets to zero in that new state.

Kolmogorov-type equations still govern the transition probabilities, but they now involve partial derivatives with respect to both time and duration. The forward equation for the occupancy probability in the same state (no transition) satisfies: (d/dt) t_p_{x,z}^{ii} = minus t_p_{x,z}^{ii} * sum over j (j not equal to i) of mu_{x+t, z+t}^{ij}. Note that both age and duration advance together at the same rate when the individual remains in state i, so age becomes x+t and duration becomes z+t.

Reserve Calculations

Reserves in a semi-Markov model depend on the current state and the duration in that state: t_V^{(i,z)} denotes the reserve at policy duration t given the individual is in state i with duration z in that state. Thiele's equation extends naturally: (d/dt) t_V^{(i,z+t)} includes the same terms as the Markov version (interest, premiums, benefits, and expected transition costs), but all transition intensities are evaluated at the current age and duration.

For example, in a disability income product, the disabled life reserve t_V^{(1,z)} depends on the duration z of the current disability spell. A recently disabled individual (small z) has a higher expected recovery rate, meaning they are more likely to stop claiming benefits soon. This leads to a lower disabled reserve compared to someone who has been disabled for a long time (large z) and is unlikely to recover.

Practical Implications for Disability Insurance

The choice between Markov and semi-Markov models has significant financial implications. A Markov model that uses an average recovery rate (averaging over all disability durations) will overestimate recovery for long-duration disabled lives and underestimate it for short-duration disabled lives. This leads to underestimation of reserves for the existing disabled population (where many individuals have been disabled for years) and overestimation of the cost of new disability claims.

In practice, disability income products often use a semi-Markov model where the transition intensities out of the disabled state (recovery and disabled mortality) depend on the duration of disability, while the transition intensities out of the healthy state (disability incidence and healthy mortality) depend only on age. This hybrid approach captures the most important duration effect while keeping the healthy-state calculations straightforward.

Non-Homogeneous Models

Non-homogeneous models introduce calendar time as an additional variable affecting transition intensities. The motivation is that transition rates change over time due to medical advances, changes in occupational risk, evolving diagnostic criteria, and other temporal factors. In a non-homogeneous model, the transition intensity mu_{x,t_cal}^{ij} depends on age x and calendar time t_cal.

The most common application is mortality improvement. If the base mortality rate at age x in a reference year is mu_x^{base}, and the annual improvement rate at age x is r_x, then the mortality rate in calendar year t is mu_x(t) = mu_x^{base} * (1 minus r_x)^t. For multi-state models, improvement factors can be applied to each transition intensity separately, recognizing that disability incidence, recovery rates, and mortality may improve at different rates.

Common Pitfalls

Students sometimes treat semi-Markov and non-homogeneous models as interchangeable, but they address different issues. Semi-Markov adds duration dependence within the current state; non-homogeneous adds calendar-time dependence. A model can be both semi-Markov and non-homogeneous simultaneously, though such models are complex and rarely tested in full on Exam LTAM.

When working with semi-Markov models, a common error is forgetting to reset the duration clock upon a state transition. If an individual recovers from disability and later becomes disabled again, the second disability spell starts at duration z = 0, not at the duration accumulated during the first spell. Each entry into a state begins a fresh duration count.

Study Tips

Exam LTAM tests semi-Markov models primarily in the context of disability insurance with duration-dependent recovery rates. Focus on understanding how to set up the reserve equations when recovery intensity depends on disability duration. Practice computing disabled life reserves for individuals at different disability durations and compare the results to see the effect of duration dependence. For non-homogeneous models, focus on applying improvement factors to base transition intensities and understanding how mortality improvement affects insurance and annuity values. Be prepared to explain qualitatively why semi-Markov models are more appropriate than Markov models for disability products.

Semi-Markov and Non-Homogeneous Models for LTAM