Life Table Calculations with Fractional Age Assumptions

Problem Preview: A life table gives $q_{70} = 0.03$. You need the probability a 70-year-old survives exactly 4.5 months. Under UDD, ${}_{0.375}p_{70} = 1 - 0.375 \times 0.03 = 0.98875$. Under constant force, $(0.97)^{0.375} = 0.98861$. Under Balducci, $0.97/(1-0.625 \times 0.03) = 0.98890$. Three assumptions, three different answers.

HIGH-FREQUENCY: UDD is the most commonly tested assumption. Know all three formulas and the force of mortality under each.

For $0 \leq t \leq 1$:

$${}_{t}q_x = t \cdot q_x, \quad {}_{t}p_x = 1 - t \cdot q_x$$

Force of mortality: $\mu_{x+t} = q_x/(1 - tq_x)$, increases within the year.

KEY: Under UDD, ${}_{t}q_x = t \cdot q_x$ for $0 \leq t \leq 1$ only. This formula does NOT extend beyond one year, writing ${}_{2.5}q_x = 2.5 \cdot q_x$ can exceed 1.

NOTE: When computing $\mu_{x+t}$, use $q_x$ from the start of the integer-age interval. If you need $\mu_{80.75}$, use $q_{80}$, not $q_{81}$.

Common mistakes

• Applying UDD beyond one year. ${}_{t}q_x = tq_x$ holds only for $0 \leq t \leq 1$. Writing ${}_{2.5}q_{70} = 2.5 \cdot q_{70}$ can exceed 1. For multi-year fractions, chain integer-year probabilities with fractional adjustments.
• Using $q_{81}$ when $t$ is within the year from 80 to 81. The fractional position is measured from the start of the integer-age interval. If you need $\mu_{80.75}$, use $q_{80}$, not $q_{81}$.
• Confusing Balducci with UDD. UDD: ${}_{t}q_x = tq_x$ from the start. Balducci: ${}_{1-t}q_{x+t} = (1-t)q_x$ from a point to the end. Trap: applying UDD when Balducci is specified.

Bottom line

• UDD: ${}_{t}q_x = t \cdot q_x$, ${}_{t}p_x = 1 - tq_x$. Force increases within each year.
• Constant force: ${}_{t}p_x = (p_x)^t$. Force is flat within each year at $-\ln p_x$.
• Balducci: ${}_{t}p_x = p_x/(1-(1-t)q_x)$. Force decreases within each year.
• All three assumptions agree at integer ages ($t=0$ and $t=1$).

Exam shortcut

Use the fractional-age assumption supplied by the question or its stated table conventions. Integer-age rates alone do not select UDD. When "uniform distribution of deaths" is specified, write ${}_{t}q_x = tq_x$. A useful check: all three assumptions give mortality probability zero at $t=0$ and $q_x$ at $t=1$. For UDD use ${}_tq_x=tq_x$. For constant force use ${}_tp_x=(p_x)^t$. For Balducci, work backward from the end of the age interval.

The full lesson (about 2,661 words, 18 min read) adds 5 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 8d

Browse all free Exam FAM lessons or jump into free Exam FAM practice questions.