Mortality: survival curves, q(x) and term vs whole-of-life pricing
Interactive Gompertz-Makeham mortality explorer. Change age, term and discount rate, switch between an illustrative basis and the Standard Ultimate Survival Model, and see the survival curve, annual death probabilities q(x), life expectancy and net premiums for term and whole-of-life assurance.
Life assurance, pensions, annuities and family takaful all rest on one curve: the probability of surviving from one age to the next. The Gompertz-Makeham law models the force of mortality as a constant background hazard plus a term that grows exponentially with age. It describes adult mortality well across most of the adult age range, though at the very oldest ages observed death rates rise more slowly than the formula predicts.
From the survival function S(x) everything else follows: the one-year death probability q(x), the curtate life expectancy at any age, and the expected present value of a benefit paid on death. For a $100,000 benefit the module compares the level annual net premium for term assurance, which pays only if death falls inside a fixed term, with whole-of-life assurance, which is certain to pay eventually. At the defaults (age 35, a 20-year term, a 4% discount rate) the term premium is about $424 a year against about $1,289 for whole of life.
Two mortality bases are offered. The site default is an illustrative calibration chosen to make the shape easy to see; its age-related hazard doubles about every 7.3 years. The second is the Standard Ultimate Survival Model of Dickson, Hardy and Waters (Makeham with A = 0.00022, B = 2.7×10⁻⁶, c = 1.124), the basis of the SOA's Standard Ultimate Life Table used in actuarial exams. It is much lighter: its hazard doubles about every 5.9 years, and a 35-year-old can expect about 51 more years against 38 under the site default. Neither is a pricing basis for a real portfolio.
Gompertz-Makeham survival function
μ(x) = A + B·e^{Cx}, S(x) = exp(−A·x − (B/C)(e^{Cx} − 1))
μ(x) is the force of mortality (the instantaneous death rate) at age x and S(x) the probability of surviving from birth to x. A is the age-independent hazard; B sets the level and C the slope of the ageing term, which doubles every ln 2 / C years. In textbook notation B·cˣ, c = e^C.
Frequently asked questions
What is q(x) in actuarial notation?
q(x) is the probability that a life aged exactly x dies before age x + 1. It equals 1 − S(x + 1) / S(x), and a life table is essentially a list of q(x) values by age.
Why is whole-of-life assurance more expensive than term assurance?
A term contract pays only if death occurs inside the term, and most insured lives survive it (about 91% of 35-year-olds survive a 20-year term under the site default). A whole-of-life contract will certainly pay at some point, so its premium must fund a benefit with a far higher expected present value, offset only by discounting over a longer horizon.
How does the Gompertz-Makeham law relate to takaful and conventional pricing?
The mortality mathematics is identical; what differs is the contract structure: risk pooling and surplus sharing in family takaful versus risk transfer in conventional assurance. Both need the same survival curve to set contributions or premiums.
Interactive explorer
The live model with sliders, charts and real-time statistics loads on this page. If it does not appear, enable JavaScript or open https://www.beyondaverages.org/mortality in a modern browser.
Related modules
- Dice & convolution: Interactive 3D dice simulator. Roll up to five dice and watch the empirical sum histogram converge to the exact convolution probabilities: uniform to triangular to bell-shaped, the central limit theorem made visible.
- Cards & hypergeometric: Interactive card-dealing simulator. Deal hands from a 3D deck and compare empirical frequencies of aces, pairs and colors against exact hypergeometric probabilities, and see why a deck's memory matters.
- Frequency: Interactive Poisson distribution explorer for insurance claim counts and hospital admissions. Move the rate λ and watch P(X=k), the mode, and the probability of zero or extreme counts update live.
- Severity: Interactive loss severity explorer comparing Log-Normal, Pareto (heavy-tail) and Gamma distributions. Adjust parameters and see the mean, median, 95th and 99th percentile claim sizes, and the share of total cost from the largest 5% of claims, update in real time.