IAI Actuarial Core Principles · Risk Modelling and Survival Analysis

Transition Intensities Dependent on Age: Exact Age or Census Methods

A transition intensity μ_x^ij is the instantaneous rate of moving from state i to state j at age x. For a constant intensity, the MLE is the number of transitions ÷ time spent in the state. Exact age uses exact dates for that time; census estimates it from census counts, often with the trapezium rule (linear change between dates).

What this chapter covers

This chapter extends the two-state alive-dead model into models with several states, such as healthy, sick and dead. Movement between states is described by transition intensities, written μ_x^ij. You learn to write the probability of staying in a state or moving between states, and how to set up the Kolmogorov forward equations for these probabilities. Solving them is usually done numerically (for example with Euler's method) or only for simple cases.

The second half is about estimation. You treat the time each life spends in a state as random and write down a likelihood. If the intensity is assumed constant over an age range, the maximum likelihood estimator is the observed number of transitions divided by the waiting time (central exposed to risk). You then learn how to measure that waiting time from real data: the exact age method uses exact dates of entry and exit, and the census method approximates the exposure from counts of lives on census dates, often using the trapezium rule, which assumes the counts change linearly between dates.

This chapter links to the rest of CS2 survival models: Markov jump processes in stochastic processes, the Cox model and graduation. It also feeds CM1, where multiple state models are used in pricing and reserving for health and disability benefits. Expect both MCQs on definitions and written questions that need derivations and numerical estimates.

This chapter sits in survival models, which carry a large share of the 2026 CS2 syllabus (25%). The material is very testable: a written question can ask for a derivation of the MLE, a numerical estimate from data, a variance or confidence interval, and a comment on assumptions, all in one. The Paper B computer-based exam also tends to use mortality data and R calculations of exposures and rates, so the same ideas earn marks twice. Once you can do the method cleanly, the marks are predictable.

Transition intensities dependent on age (exact or census): topics in the order to study them

1Two-State Model and Force of MortalityStart with the simplest model so the ideas of intensity, survival probability and the link μ_x = −(d/dt) ln(ₜpₓ) evaluated at t = 0 are secure before adding more states.
2Multiple State Models and Transition IntensitiesNow generalise to several states, define μ_x^ij, and learn the Markov assumption and the forward equations that all later estimation relies on.
3Likelihood Estimation of Constant Transition IntensitiesWith the model defined, derive the likelihood in terms of transitions and waiting time and get the MLE, its asymptotic variance and standard error.
4Exact Age and Census Methods of EstimationLast, apply the MLE to real data: you need to know how to calculate the waiting time under each method and which approximations each one makes.

How to prepare Transition intensities dependent on age (exact or census)

Treat this chapter as theory first, then calculation, then data handling. Short daily sessions work well if you study alongside work, since each topic builds on the previous one.

Write the definition of μ_x^ij and the two-state link between μ_x and survival probabilities from memory until you can do it without notes.
Draw the state diagram for every model you meet (healthy-sick-dead, marriage, disability income) and write the forward equations directly from the diagram.
Derive the likelihood for a constant intensity yourself: include the term for each transition and the exp(−μ × waiting time) term. Then differentiate to get the MLE.
Practise the estimator results: the MLE is the count of transitions ÷ total waiting time, and its variance is approximately μ² ÷ the expected number of transitions. Use them for confidence intervals.
Work through data questions on both methods. For each one, write out the exact exposure or census-based exposure, state the assumption, and then compute the rate.
Do past-paper questions under time, and write one-line comments on assumptions such as constant intensity and the Markov property.
Repeat the same calculations in R for Paper B so you can compute exposures and rates from a dataset.

Common mistakes in Transition intensities dependent on age (exact or census)

Dividing the number of transitions by the number of lives instead of the waiting time.
Fix: Ask what the estimate is. A transition intensity is a rate, so the divisor is time spent in the state.
Forgetting a term in the likelihood when a life is censored or has no transition.
Fix: Every life contributes exp(−μ × time in state). Only lives that move also contribute a factor μ.
Writing forward equations with a wrong sign or missing a state.
Fix: Draw the diagram first. For each state, add inflows from all other states and subtract outflows to all other states.
Using the variance of μ̂ without stating it is approximate.
Fix: Say it is asymptotic, give μ̂² ÷ d as the estimate, and note it needs a reasonably large number of transitions.
Not stating the assumptions of the census method.
Fix: Write the assumption (for example how movements are spread between census dates) before computing, and comment on its effect in the final line.
Treating a constant intensity as true for all ages.
Fix: State the age range the estimate covers and say intensities are assumed constant only within it.

Last-day revision: Transition intensities dependent on age (exact or census)

μ_x^ij is the instantaneous rate of moving from state i to state j at age x.
Two-state model: μ_x = −d/dx ln(ₜpₓ) evaluated at t = 0, and ₜpₓ = exp(−∫ μ_{x+s} ds) from 0 to t.
The Markov assumption: future moves depend only on the current state and age, not on the past path.
Forward equations: d/dt ₜp_x^ij = Σ_{k≠j} (ₜp_x^ik μ_{x+t}^kj − ₜp_x^ij μ_{x+t}^jk). The first term is the flow into state j from state k. The second is the flow out of state j to state k.
For small h, the probability of a move from i to j in time h is approximately h × μ_x^ij.
Likelihood for constant μ: μ^d × exp(−μ × v), where d is the number of transitions and v is total waiting time.
MLE: μ̂ = d ÷ v. The asymptotic variance of μ̂ is μ ÷ E[V] = μ² ÷ E[D], where V is the random total waiting time and D is the random number of transitions. It is estimated by μ̂² ÷ d.
Approximate 95% interval: μ̂ ± 1.96 × μ̂ ÷ √d.
Exact age method: use exact dates of entry and exit to get each life's time in the state.
Census method: approximate the exposure by integrating the census counts of lives over time, for example with the trapezium rule between census dates. This assumes the number of lives changes linearly between dates.
Estimates from constant intensities apply to the age range in question, not to a single exact age.
Check units: waiting time in years gives a rate per year.

Transition intensities dependent on age (exact or census) practice questions

Transition intensities dependent on age (exact or census): frequently asked questions

What is the difference between the exact age and census methods?

The exact age method uses precise entry and exit dates to work out each life's time in the state. The census method approximates the exposure from counts of lives on census dates, often using the trapezium rule, which assumes the count changes linearly between census dates. Exact age needs more detailed data but needs fewer approximations.

Why is the MLE of a constant transition intensity the number of transitions divided by waiting time?

The likelihood is proportional to μ^d × exp(−μv). Taking logs gives d ln μ − μv. Setting the derivative d/μ − v to zero gives μ̂ = d ÷ v.

Do I need to memorise the forward equations?

You should be able to write them from a state diagram rather than memorise each one. Inflows to a state are added and outflows subtracted, with the right intensity in each term.

Is this chapter tested in Paper B as well?

Yes, it can be. Paper B is computer-based, and you may need to compute exposures, rates or intervals from a dataset using R. Practise the same calculations by hand and in code.