Risk Modelling and Survival Analysis · Transition intensities dependent on age (exact or census)

Exact Age and Census Methods of Estimation Explained

Updated 11 October 2026 · Fact-checked

The exposed to risk is the total time (or number of lives) observed at risk, used as the denominator of a mortality rate. The exact method adds up each life's actual time at risk. The census method approximates this using head counts on fixed dates, via the principle of correspondence.

Understand Exact Age and Census Methods of Estimation

To estimate a mortality rate, you need two things: the number of deaths and the amount of risk those deaths came from. The second is the exposed to risk. Without it, a count of deaths means nothing.

There are two versions. The central exposed to risk, Ec(x), is the total time lived by all people while aged x last birthday (or in a stated age range). It pairs with the central rate m(x) or a constant force of mortality μ. The initial exposed to risk, E(x), is the number of lives at the start of the year of age, counting those who die as if they were exposed for the full year. It pairs with q(x).

The exact method needs each life's exact dates of birth, entry, exit and death. You add up the actual time each life spent in the age range. This is accurate but needs full individual records.

The census method is used when you only have counts on census dates, for example the number of policies in force on 1 January each year. You approximate the time lived by integrating the head count over time. A common rule is the trapezium rule, which gives the average of the start and end counts when the counts are taken one year apart.

The principle of correspondence is the key idea. A death must be counted in the numerator only if the life would have been counted in the exposed to risk, and vice versa. The definition of the group (age definition, policy type, calendar period) must be identical in both. Breaking this gives a biased rate.

Key rules to remember

Central rate from data: m̂(x) = μ̂ = d(x) ÷ Ec(x); d(x) is the number of deaths. Estimates a constant force of mortality over the age range. Also the MLE for a constant μ.
Initial rate from data: q̂(x) = d(x) ÷ E(x); E(x) counts each life that dies as exposed for the whole year.
Link between initial and central exposure: E(x) ≈ Ec(x) + ½ d(x); Assumes deaths occur on average halfway through the year of age (uniform distribution of deaths). It is an approximation, not an identity.
Exact central exposure: Ec(x) = Σ (time spent by each life in the age range); Use exact dates. Include lives that die, withdraw or are still alive at the end.
Census approximation of central exposure: Ec ≈ ∫ P(t) dt over the investigation period; P(t) is the number of lives in the group at time t. Approximated with the trapezium rule on census counts.
Trapezium rule, two dates one year apart: Ec ≈ ½ [P(0) + P(1)]; For more dates, use the sum of average counts over each interval multiplied by interval length.
Rate and q relationship, constant force: q(x) = 1 − e^(−μ); Use to convert a central rate to a probability when μ is constant over the year.
Variance of estimator: Var(μ̂) ≈ μ ÷ Ec(x); Large Ec means a more reliable estimate. Approximate, for large samples.

How to solve Exact Age and Census Methods of Estimation questions

Use this method for any question on exposed to risk or on estimating rates from census or exact data.

1Read what data are given: exact dates for each life, or only counts at census dates. This decides exact or census method.
2Define the group clearly: age definition (last birthday, nearest birthday, age at next), the calendar period and any policy class. Check numerator and denominator will match (principle of correspondence).
3Decide the exposure type. Use central exposed to risk if the rate is μ or m(x). Use initial exposed to risk if the rate is q(x).
4For exact data, calculate time spent in the age range for each life, in years, up to exit, death or the end of the period. For census data, apply the trapezium rule to the head counts, matching the age definition.
5Count the deaths that satisfy the same definition as the exposure group.
6Divide deaths by exposure to get the rate. If needed, convert Ec to E(x) using E(x) ≈ Ec(x) + ½ d(x), or μ to q using q = 1 − e^(−μ).
7State your assumptions (constant force over the age, uniform deaths, linear change in head count between census dates) and comment on reliability if asked.

Quickest way: Census trapezium shortcut

When to use it: Use when the question gives head counts at two or more census dates and asks for exposure or a rate, and you are short of time.

Write the counts in a line with the time gap between them.
For each interval, compute (count at start + count at end) ÷ 2 × length of interval.
Add the pieces to get Ec. Check that the counts relate to the same age definition as the deaths.
Rate = deaths ÷ Ec. If q is asked, either use Ec + ½ deaths as E, or compute 1 − e^(−rate), whichever the question's assumption indicates.
Sanity check: the rate for adult ages should be a small number, usually below 0.1 except very old ages.

Common mistakes in Exact Age and Census Methods of Estimation

Using initial exposed to risk with a central rate, or the reverse.
The two exposures have similar names and similar values, so students treat them as interchangeable.
Fix: Match: Ec goes with μ or m(x). E goes with q(x). Write the pairing beside the formula before computing.
Counting deaths from a different group than the exposure (for example deaths at age x last birthday against exposure at age x nearest birthday).
Students take the deaths figure and the head counts from different parts of the data without checking definitions.
Fix: Apply the principle of correspondence: write the age definition and calendar period once and check both numerator and denominator against it.
Forgetting to stop the exposure at death or withdrawal in the exact method.
Students take the full year for every life by default.
Fix: For central exposure, count each life only until the earliest of exit, death or the end of the age range or period. Only the initial exposure counts deaths for the full year.
Using the average of census counts when the dates are not one year apart, or when there are more than two dates.
The simple average ½(P0 + P1) is memorised and applied without checking the interval.
Fix: Apply the trapezium rule to each interval and multiply by its length in years. Then add.
Treating E(x) = Ec(x) + ½ d(x) as exact.
The relationship appears as an equation in notes.
Fix: Say it is an approximation based on deaths being spread evenly across the year. Use exact data when available.
Mixing the census date age with the age definition required for the rate.
A head count taken on a calendar date contains lives of mixed ages in the year of age, so the link to the age definition is lost.
Fix: Adjust the exposure for the shift in age definition. Use the stated fraction of the year that each age definition covers, and show the logic in the answer.

Worked examples

Example 1

An investigation follows five lives, all aged exactly 60 at the start, over one year of age (from exact age 60 to 61). Life A survives the whole year. Life B dies after 0.5 years. Life C withdraws after 0.25 years. Life D dies after 0.75 years. Life E survives the whole year. Calculate the central exposed to risk, the initial exposed to risk, and estimates of μ and q using the exact method.

Show the solution

Central exposure is the total time at risk: A 1 + B 0.5 + C 0.25 + D 0.75 + E 1 = 3.5 years.
Deaths: B and D, so d = 2.
Estimate μ̂ = d ÷ Ec = 2 ÷ 3.5 = 0.5714.
Initial exposure: every life counts for the full year if it dies or survives. C withdrew, so it is exposed for only 0.25 years. Lives A and E count 1 each, B and D die and count 1 each, C counts 0.25. E = 1 + 1 + 1 + 1 + 0.25 = 4.25.
Estimate q̂ = d ÷ E = 2 ÷ 4.25 = 0.4706.

Answer: Ec = 3.5 years, E = 4.25, μ̂ ≈ 0.5714, q̂ ≈ 0.4706.

Example 2

A life office has 1,200 policyholders in force on 1 January 2023, 1,260 on 1 January 2024 and 1,320 on 1 January 2025, all in the same age group. During 2023 and 2024 combined there were 30 deaths. Estimate the central exposed to risk using the census method with the trapezium rule, and estimate the force of mortality. Then estimate the initial exposed to risk for the two years combined using E ≈ Ec + ½ d.

Show the solution

Apply the trapezium rule to each year. Year 2023: ½ (1,200 + 1,260) × 1 = 1,230.
Year 2024: ½ (1,260 + 1,320) × 1 = 1,290.
Total Ec = 1,230 + 1,290 = 2,520 policy years.
Estimate μ̂ = 30 ÷ 2,520 = 0.011905.
Initial exposure: E ≈ 2,520 + ½ × 30 = 2,535.

Answer: Ec ≈ 2,520 policy years, μ̂ ≈ 0.0119, E ≈ 2,535.

Exam tips

Write the pairing at the top of your answer: central exposure with μ, initial exposure with q. Examiners reward clear definitions.
In census questions, always state the assumption that the head count changes linearly between census dates.
Show the principle of correspondence in words whenever the data definitions differ. Short written comments carry marks in Paper A.
In Paper B (R or Excel), compute exposure per life as exit time minus entry time, sum with a formula, then divide deaths by that total. Keep the intermediate values visible.
When asked for a comment on the estimate, mention the sample size and that the standard error is about √(μ ÷ Ec).

Practice questions from Transition intensities dependent on age (exact or census)

Exact Age and Census Methods of Estimation: frequently asked questions

What is the difference between initial and central exposed to risk?

Central exposed to risk is the total time lived by the group while in the age range. Initial exposed to risk is the number of lives at the start of the age range, with deaths counted as exposed for the whole period. Central pairs with μ or m(x), initial pairs with q(x).

How do you calculate central exposed to risk from census data?

Find the number of lives at each census date, then integrate the head count over time, usually with the trapezium rule. For counts one year apart, take the average of the two counts. For several dates, add up the area of each interval.

What is the principle of correspondence?

It says that a life should be counted in the deaths only if it would have been counted in the exposed to risk, and the other way round. The age definition, period and class must be the same in both. If they differ, the estimated rate is biased.

When can I use E(x) ≈ Ec(x) + ½ d(x)?

Use it when you have the central exposure and the number of deaths and need the initial exposure, and deaths are assumed to be spread evenly over the year of age. It is an approximation, so state the assumption in your answer.