Common mistake 1 with Multi-state Markov model

Rows that do not sum to 1. The model rescales them, but it usually means a typing error.

Common mistake 2 with Multi-state Markov model

Ignoring duration dependence. Someone sick for a year is less likely to recover than someone sick for a week, which a simple Markov model cannot see.

Common mistake 3 with Multi-state Markov model

Using a one-year ratings matrix from a calm period to project defaults through a recession.

Common mistake 4 with Multi-state Markov model

Confusing transition probabilities with transition intensities, which are rates in continuous time.

EconStatsCraft › Multi-state Markov model

Multi-state Markov model

Runs a discrete-time Markov chain from a one-year transition matrix: where a life (or a bond) is likely to be each year, the expected time spent in each state, the EPV of a benefit paid while in a state, and multi-year default probabilities for a ratings matrix.

The intuition

Some insurance pays out depending on which state a person is in, not just whether they are alive. Health moves both ways: people fall sick and recover. A multi-state model writes down the chance of each move in a year and lets the arithmetic work out where someone is likely to be later on, how long they will spend in each state, and what a benefit tied to a state is worth today. Credit ratings move the same way, with default as the state nobody leaves.

When to use it

Pricing disability income and long-term care insurance, valuing benefits that depend on health status, projecting credit rating migration and multi-year default probabilities, and the multiple-state model chapters of IFoA CS2 and SOA ALTAM.

Where the estimator comes from

Let P_{ij} be the probability of moving from state i to j in a year. By the Markov property, the two-year probabilities are \sum_k P_{ik}P_{kj}, the matrix product, so t-year probabilities are P^t. Starting from distribution \pi_0, \pi_t=\pi_0P^t. The expected number of years in state j over n years is \sum_{t=1}^n(\pi_t)_j, and a benefit B paid at the end of each year spent in j has EPV \sum_t v^t(\pi_t)_jB.

The model

\pi_{t} = \pi_0\,P^{t},\qquad \text{EPV} = \sum_{t=1}^{n} v^{t}\;{}_tp^{\,0j}\;B,\qquad v=\frac{1}{1+i}

What it assumes

The Markov property: the future depends only on the present state, not on how long you have been in it or how you got there.
Transition probabilities are constant over time. Real sickness and mortality rates rise with age.
Transitions happen at most once a year, at year end, in this discrete version.
Absorbing states (death, default) cannot be left.

Statistical properties

Probabilities in each row of P^t still sum to one.
An absorbing state's probability never falls.
Without duration dependence, the time spent in a state on one visit is geometric.
Cumulative default probabilities from a ratings matrix rise with horizon and with lower starting ratings.

How to read the output

Each row of the matrix is a from-state; the entries are where someone in that state will be a year later, and must sum to 1.
Expected years in a state add up the probability of being there each year. For a sickness benefit, multiply by the benefit and discount.
For a ratings matrix, the last column of P^t gives the cumulative default probability by starting rating.
Absorbing states only grow in the chart; the others eventually drain into them.

A worked reading

A healthy life with a 5% chance of falling sick and 1% of dying each year; the sick recover with 40%, stay sick with 50% and die with 10%. After five years the life is healthy with probability 84.0%, sick 8.5% and dead 7.5%. Over 20 years it expects about 15.3 years healthy and 1.5 years sick. A benefit of 10,000 for each year spent sick is worth 10,249 today at 4%. That is the net single premium for the sickness cover.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Rows rescaled warning	Probabilities do not sum to 1	Check for a typing error in the matrix
Nobody ever leaves a state	A diagonal entry of 1	Correct only for absorbing states
EPV seems small	Benefit state numbered wrongly	States are numbered from 1 in matrix order
Default probabilities too high	One-year matrix from a recession	Use a through-the-cycle matrix

Common mistakes

Rows that do not sum to 1. The model rescales them, but it usually means a typing error.
Ignoring duration dependence. Someone sick for a year is less likely to recover than someone sick for a week, which a simple Markov model cannot see.
Using a one-year ratings matrix from a calm period to project defaults through a recession.
Confusing transition probabilities with transition intensities, which are rates in continuous time.

Where to read this properly

Dickson, Hardy and Waters, Actuarial Mathematics for Life Contingent Risks (2020), ch. 8.
IFoA, CS2 Risk Modelling and Survival Analysis, Markov chains and multiple-state models.
Jarrow, Lando and Turnbull (1997), A Markov Model for the Term Structure of Credit Risk Spreads.

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Multi-state Markov model

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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