Risk Modelling and Survival Analysis · Concepts of survival models

Mortality Laws and Fractional Age Assumptions Explained

Updated 11 October 2026 · Fact-checked

Mortality laws such as Gompertz and Makeham give the force of mortality as a formula in age. Fractional age assumptions (uniform distribution of deaths, or constant force) fill the gaps between integer ages in a life table. You solve questions by choosing the assumption, then using its survival formula.

Understand Mortality Laws and Fractional Age Assumptions

A life table gives you qx only at whole ages. Exam questions often ask about a lapse after 0.4 of a year, or a death at age 60.3. To answer, you need a rule for how mortality behaves inside each year of age. That rule is a fractional age assumption.

There are two you must know. Under the uniform distribution of deaths (UDD), deaths are spread evenly through the year, so the probability of dying within time s is simply s × qx. Under constant force of mortality, the force μ stays fixed through the year, so survival falls exponentially and s years of survival has probability (px)^s. UDD means the force of mortality rises through the year. Constant force means it does not. Both agree at integer ages by design.

A mortality law is different. It is a parametric formula for μx that holds across all ages, with a few parameters fitted to data. Gompertz says μx = B c^x, so mortality grows exponentially with age. Makeham adds a constant A for age-independent risks such as accidents: μx = A + B c^x. Once you have μx, you get survival by integrating: tpx = exp(−∫ μ(x+u) du from 0 to t).

So the two ideas solve different problems. Mortality laws give you a smooth formula for any age, including fractional ones. Fractional age assumptions let you work from a discrete table. In the exam, read which one the question gives you and do not mix them.

Key rules to remember

Gompertz law: μx = B c^x and tpx = exp[ −B c^x (c^t − 1) ÷ ln c ]; Parameters B > 0 and c > 1. Obtained by integrating μ(x+u) = B c^(x+u) from u = 0 to t.
Makeham law: μx = A + B c^x and tpx = exp[ −A t − B c^x (c^t − 1) ÷ ln c ]; Gompertz plus a constant A. Equivalent form: tpx = s^t g^(c^x (c^t − 1)), where s = exp(−A) and g = exp(−B ÷ ln c).
Makeham form of lx: lx = k s^x g^(c^x); Useful for ratios like l(x+t) ÷ lx. The constant k cancels in any survival probability.
UDD: probability of death: s qx = s × qx for 0 ≤ s ≤ 1; So s px = 1 − s qx. Also l(x+s) = (1 − s) lx + s l(x+1), a straight line between integer ages.
UDD: force of mortality: μ(x+s) = qx ÷ (1 − s qx) for 0 ≤ s < 1; Force rises through the year. The density s px × μ(x+s) = qx is constant.
UDD: survival from a fractional age: t q(x+s) = t qx ÷ (1 − s qx) for s + t ≤ 1; If s + t > 1, split at the next integer age and multiply the pieces.
Constant force of mortality: s px = (px)^s = e^(−μ s), with μ = −ln(px) for 0 ≤ s ≤ 1; Force is the same throughout the year. Also t p(x+s) = (px)^t for s + t ≤ 1, which does not depend on s.
Expectation of life under UDD: e°x = ex + ½; Complete expectation equals curtate expectation plus one half, assuming UDD in every year of age.

How to solve Mortality Laws and Fractional Age Assumptions questions

Use this order for any question on mortality laws or fractional ages. It keeps you from mixing formulas.

1Write down what you are asked: a probability t p(x+s), t q(x+s), a force μ(x+s), or an expected value. Note the exact ages involved.
2Identify the information given: a parametric law (A, B, c), or integer-age table values (qx, lx).
3If a law is given, use the closed form of tpx for Gompertz or Makeham. Plug in x, t and the parameters. Do not integrate again unless asked to derive it.
4If a table is given, read the assumption stated: UDD or constant force. If none is stated, say which you assume and why.
5Check whether the interval stays inside one year of age. If s + t ≤ 1, apply the fractional formula directly. If it crosses an integer age, split it into pieces and multiply the survival probabilities.
6For a force of mortality at a fractional age, use qx ÷ (1 − s qx) for UDD, or −ln(px) for constant force.
7Compute carefully, keeping at least five decimal places in intermediate steps. State the assumption and the final answer clearly.

Quickest way: Pick the assumption, then the one-line formula

When to use it: Use this for MCQs and for the opening part of written questions, where you need a fractional survival or death probability fast.

Under UDD, think in terms of deaths: s qx = s × qx. For a start at age x+s, divide by (1 − s qx).
Under constant force, think in terms of survival: s px = (px)^s. For a start at x+s inside the same year, the answer is the same as from age x with the same t.
For a Gompertz or Makeham question, compute B c^x first. Then work out (c^t − 1) ÷ ln c. Then put everything inside exp(−...).
Sanity check: the answer must lie between 0 and 1, and a longer period must give a lower survival probability.
Under both assumptions, the two answers for a short interval are very close. If your UDD and constant force answers differ greatly, recheck.

Common mistakes in Mortality Laws and Fractional Age Assumptions

Using t q(x+s) = t qx under UDD when the start age is not an integer.
The simple rule s qx = s × qx only holds from the start of the year of age. Students apply it from any point.
Fix: Use t q(x+s) = t qx ÷ (1 − s qx). The denominator is the probability of surviving to x+s.
Believing the force of mortality is constant under UDD.
UDD makes the density of death constant, and students confuse density with force.
Fix: Under UDD the density s px × μ(x+s) = qx is constant, but μ(x+s) = qx ÷ (1 − s qx) increases with s. Only the constant force assumption has constant μ.
Applying a fractional formula across an integer age boundary.
Formulas need s + t ≤ 1. Students ignore this when the interval is, say, from age 60.7 for 0.6 years.
Fix: Split at age 61. Find 0.3 p(60.7) and 0.3 p61, then multiply. Apply the assumption to each year separately.
Forgetting the −A t term in Makeham survival, or using Gompertz formula for a Makeham question.
Students memorise the Gompertz expression and treat Makeham as the same.
Fix: Makeham adds A t inside the exponent: tpx = exp[−A t − B c^x (c^t − 1) ÷ ln c]. Check that setting A = 0 gives Gompertz.
Writing ln c as log base 10, or using c^t instead of (c^t − 1).
Calculator habits and loose memory of the integral.
Fix: Use the natural log. Remember the integral of B c^(x+u) is B c^x (c^t − 1) ÷ ln c, which is zero when t = 0.
Rounding px or the intermediate values too early.
Survival probabilities close to 1 make rounding errors visible in the final answer.
Fix: Keep at least five or six decimal places until the last step, especially when taking logs or powers.

Worked examples

Example 1

Mortality follows Makeham's law with A = 0.0004, B = 0.00003 and c = 1.1. Calculate (i) the force of mortality at age 50 and (ii) the probability that a life aged 50 survives to age 52.

Show the solution

(i) μ50 = A + B c^50. First, 1.1^50 = 117.3909 (approximately).
B c^50 = 0.00003 × 117.3909 = 0.0035217.
μ50 = 0.0004 + 0.0035217 = 0.0039217.
(ii) Use 2p50 = exp[−A t − B c^x (c^t − 1) ÷ ln c] with t = 2.
c^t − 1 = 1.1² − 1 = 0.21. ln 1.1 = 0.0953102.
B c^50 × 0.21 = 0.0035217 × 0.21 = 0.00073956. Dividing by 0.0953102 gives 0.0077596.
A t = 0.0004 × 2 = 0.0008.
Exponent = −(0.0008 + 0.0077596) = −0.0085596.
2p50 = exp(−0.0085596) = 0.99148.

Answer: (i) μ50 ≈ 0.003922. (ii) 2p50 ≈ 0.9915.

Example 2

A life table gives q60 = 0.02. For a life aged exactly 60.3, calculate the probability of death within the next 0.4 years, and the force of mortality at age 60.3, (a) assuming UDD and (b) assuming constant force of mortality between integer ages.

Show the solution

Here s = 0.3 and t = 0.4, so s + t = 0.7 ≤ 1. The interval stays within the year of age 60 to 61.
(a) UDD: 0.4 q(60.3) = t qx ÷ (1 − s qx) = (0.4 × 0.02) ÷ (1 − 0.3 × 0.02).
Numerator = 0.008. Denominator = 1 − 0.006 = 0.994.
0.4 q(60.3) = 0.008 ÷ 0.994 = 0.008048.
Force under UDD: μ(60.3) = qx ÷ (1 − s qx) = 0.02 ÷ 0.994 = 0.020121.
(b) Constant force: p60 = 0.98, so μ = −ln(0.98) = 0.020203.
0.4 p(60.3) = (0.98)^0.4 = exp(−0.4 × 0.020203) = exp(−0.0080811) = 0.991952.
So 0.4 q(60.3) = 1 − 0.991952 = 0.008048 (about 0.0080485).

Answer: (a) UDD: probability ≈ 0.008048 and μ(60.3) ≈ 0.02012. (b) Constant force: probability ≈ 0.008048 (0.0080485 before rounding) and μ = 0.02020. The probabilities are almost identical, but the forces differ because UDD has a rising force.

Exam tips

Always state the assumption you use in written answers. If the question does not give one, say so and choose UDD or constant force. This earns method marks.
Check the interval first. Questions often hide a boundary crossing, such as starting at 60.7 for 0.6 years. Split it and multiply.
For Gompertz and Makeham, set up the exponent in pieces: B c^x, then (c^t − 1) ÷ ln c, then A t. Writing each piece lets examiners award marks even if you slip on arithmetic.
MCQs often test conceptual points, such as which assumption gives a constant force, or whether UDD gives an increasing force. Learn these in words.
In Paper B (R or Excel), build the survival function as a formula of x and t, then reuse it. Check that it returns 1 at t = 0 and decreases as t grows.

Practice questions from Concepts of survival models

Mortality Laws and Fractional Age Assumptions: frequently asked questions

What is the difference between Gompertz and Makeham law?

Gompertz law says μx = B c^x, so mortality increases exponentially with age. Makeham law is μx = A + B c^x. The extra constant A represents mortality that does not depend on age, such as accidents.

What is the difference between UDD and constant force of mortality?

Under UDD, deaths are evenly spread through the year, so s qx = s × qx and the force rises as the year goes on. Under constant force, μ is fixed through the year, so s px = (px)^s. Both give the same values at integer ages.

How do I calculate survival probability for fractional ages?

First choose the assumption. Under UDD, t q(x+s) = t qx ÷ (1 − s qx) when s + t ≤ 1. Under constant force, t p(x+s) = (px)^t. If the interval crosses a whole age, split it and multiply the parts.

Does the UDD assumption give a constant force of mortality?

No. UDD gives a constant density of deaths within the year, not a constant force. The force is qx ÷ (1 − s qx), which increases with s.

Which assumption should I use if the question does not say?

Use the one that makes the working simplest and state it clearly. UDD is the usual default for fractional ages in life insurance questions, while constant force is natural when you work with transition intensities. Examiners reward a clearly stated assumption.