Multiple State Model

Study guide: Exam ALTAM

A Multiple State Model describes a life that moves in continuous time among a finite set of states, such as healthy, sick and dead, with Y(x) the state occupied at age x. Moves are governed by transition intensities \mu^{ij}_{x}, and under the Markov assumption the future depends only on the current state and age. Benefits and premiums attach to being in a state or to moving between states, which makes it the model for state-dependent insurance, long-term health insurance and joint lives.

{}_tp^{ij}_x = \Pr\left[Y(x+t) = j \mid Y(x) = i\right]

{}_tp^{\overline{ii}}_x = \exp\left(-\int_0^t \sum_{j \neq i} \mu^{ij}_{x+s}\,ds\right)

{}_tp^{01}_x = \int_0^t {}_sp^{\overline{00}}_x\,\mu^{01}_{x+s}\;{}_{t-s}p^{\overline{11}}_{x+s}\,ds

• Symbols: Following Dickson, Hardy & Waters (chapter 8), {}_tp^{ij}_x is the probability that a life aged x in state i is in state j at age x+t, and {}_tp^{\overline{ii}}_x is the probability that it stays in state i throughout. The third block applies when state 1 cannot be left except to an absorbing state: the life stays in 0 until s, moves at s, and then stays in 1.
• Solving for probabilities: In general {}_tp^{ij}_x includes every path that ends in j, such as healthy, sick, healthy, so it seldom has a simple closed form. It is found from Kolmogorov's forward equations, usually numerically. The discrete-time version of the same model is the multi-state model built from a transition probability matrix and the Chapman-Kolmogorov equations.
• Standard models: The alive–dead model has \mu^{01}_x = \mu_x, the force of mortality. The accidental death model has two dead states. The permanent disability model has healthy, disabled and dead states with no recovery, and the disability income model adds recovery. The joint life model has four states for two lives, and a CCRC model has states for independent living, assisted living, nursing care and death. A multiple decrement model is the special case with one active state and several exits.
• Critiquing the assumptions: The Markov assumption says that the time already spent in a state does not matter. In practice, recovery is most likely early in a sickness and mortality is higher soon after disablement, so a model can split a state by duration, at the cost of more states and less data per state. The model also treats every life in a state as having the same intensities, which ignores heterogeneity such as health differences that underwriting did not capture.

State Probabilities in the Permanent Disability Model A healthy life is subject to constant intensities of disablement \mu^{01} = 0.02 and death while healthy \mu^{02} = 0.01. A disabled life does not recover and dies with intensity \mu^{12} = 0.07. Find the probabilities that the life is healthy, disabled or dead 10 years from now.

Answer A healthy life leaves state 0 at total rate 0.03, so {}_{10}p^{00} = {}_{10}p^{\overline{00}} = e^{-0.3}. For state 1, use the third block with {}_{10-s}p^{\overline{11}} = e^{-0.07(10-s)}.

\begin{align*}
{}_{10}p^{00} &= e^{-0.3} \\
&= 0.740818 \\
{}_{10}p^{01} &= \int_0^{10} e^{-0.03s}(0.02)\,e^{-0.07(10-s)}\,ds \\
&= \frac{0.02}{0.04}\left(e^{-0.3} - e^{-0.7}\right) \\
&= 0.5(0.740818 - 0.496585) \\
&= 0.122116 \\
{}_{10}p^{02} &= 1 - 0.740818 - 0.122116 \\
&= 0.137066
\end{align*}

About 12% of lives are disabled at time 10. More than that became disabled during the 10 years, because some disabled lives have since died.

Two Lives Exposed to a Common Accident Lives (x) and (y) are modelled with four states: 0 (both alive), 1 (only x alive), 2 (only y alive) and 3 (both dead). While both are alive, y dies with intensity 0.03, x dies with intensity 0.02, and a common accident kills both at once with intensity 0.005. After one death, the survivor dies with intensity 0.02 for x or 0.03 for y. Find {}_{10}p^{00}, {}_{10}p^{01} and the probability that both die in a common accident within 10 years.

Answer State 0 is left at total rate 0.055, and state 1 is left at rate 0.02.

\begin{align*}
{}_{10}p^{00} &= e^{-0.55} \\
&= 0.576950 \\
{}_{10}p^{01} &= \int_0^{10} e^{-0.055s}(0.03)\,e^{-0.02(10-s)}\,ds \\
&= 0.03\,e^{-0.2}\left(\frac{1 - e^{-0.35}}{0.035}\right) \\
&= 0.207241 \\
\Pr(\text{common accident}) &= \int_0^{10} e^{-0.055s}(0.005)\,ds \\
&= \frac{0.005}{0.055}\left(1 - e^{-0.55}\right) \\
&= 0.038459
\end{align*}

If each life were independent with its own force including the accident, 0.025 and 0.035, both would be alive at 10 with probability e^{-0.6} = 0.548812. The model gives more, because one accident ends both lives at once and is counted only once.

How it appears on the exam

• ALTAM asks candidates to apply Markov multiple state models to state-contingent life insurance, long-term health insurance and CCRCs, and to calculate state-dependent probabilities such as {}_tp^{ij}_x and {}_tp^{\overline{ii}}_x from given intensities.
• Critique: ALTAM's objectives also ask candidates to understand and critique the assumptions behind the models, such as the Markov property and the absence of duration dependence, and to say how a model could be changed to remove them.
• Joint lives: ALTAM asks how joint-life mortality can be modelled as a multiple state model, as well as through the time to failure of a status as in Joint Life. A common shock enters the model as a direct transition from both alive to both dead.

Exam traps

• Being in a state versus staying in it: {}_tp^{00}_x includes lives that were sick and recovered, while {}_tp^{\overline{00}}_x does not. The exponential formula gives only the second, so it understates {}_tp^{00}_x in a model with recovery.
• Every exit from the state: The exponent in {}_tp^{\overline{ii}}_x sums the intensities of every transition out of state i. Leaving out death from the sick state, for example, overstates the probability of staying sick.
• The intensity at the right age: In the path integral, the intensity out of state 1 applies from age x+s onward, after the move. Using the intensities at age x for the whole period is only correct when they are constant.