fractional-age assumptions (UDD, constant force)
/ UDD = uniform distribution of deaths /
A life table usually only tells you survival at whole ages — q-65, q-66, and so on. But premiums fall due monthly, claims happen on any random day, and you constantly need probabilities like 'surviving four and a half months past age 65'. A fractional-age assumption is a simple rule for filling in the gaps between two integer ages, so the table can answer questions about fractions of a year without inventing new data.
Two assumptions dominate. Under uniform distribution of deaths (UDD), the d-x deaths in a year are spread evenly across that year, like births spread evenly through a calendar. This makes the probability of dying in the first fraction s of the year simply s times q-x — a straight-line interpolation of the survivors l-(x+s). Under the constant-force assumption, the force of mortality is held flat at one value throughout the year, which makes survival within the year exponential: s-p-x = (p-x) to the power s. UDD is the more popular default; constant force is the natural choice when you are thinking in terms of hazard.
These assumptions matter because they quietly decide answers in real pricing: the value of a monthly annuity, the reserve held part-way through a policy year, the cost of a benefit payable at the moment of death rather than year-end. UDD and constant force usually give very close numbers for ordinary mortality, but they are not identical, and quoting a precise figure without stating which assumption you used is a recipe for small, persistent discrepancies between two correct-looking calculations.
With q-70 = 0.02, UDD gives the chance of dying in the first half-year as 0.5 x 0.02 = 0.01; constant force gives 1 - (0.98)^0.5 = 1 - 0.98995 = 0.01005 — close, but not identical.
Two reasonable assumptions, two slightly different fractional-year answers — always state which one you used.
Under UDD the force of mortality actually rises within each year (because the same number of deaths fall on a shrinking pool), whereas constant force keeps it flat. So UDD does not mean a uniform hazard — it means uniformly spread deaths.