Life Table Calculations with Fractional Age Assumptions | Free Exam FAM Lesson

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 $0 \leq t \leq 1$ :

$_{t} q_{x} = t \cdot q_{x},_{t} p_{x} = 1 - t \cdot q_{x}$

Force of mortality: $μ_{x + t} = q_{x} / (1 - t q_{x})$ , 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 $μ_{x + t}$ , use $q_{x}$ from the start of the integer-age interval. If you need $μ_{80.75}$ , use $q_{80}$ , not $q_{81}$ .

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Common mistakes

Applying UDD beyond one year. $_{t} q_{x} = t q_{x}$ 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 $μ_{80.75}$ , use $q_{80}$ , not $q_{81}$ .
Confusing Balducci with UDD. UDD: $_{t} q_{x} = t q_{x}$ 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: $_{t} q_{x} = t \cdot q_{x}$ , $_{t} p_{x} = 1 - t q_{x}$ . 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

Most FAM questions use UDD, it is the default unless stated otherwise. When you see "uniform distribution of deaths," immediately write $_{t} q_{x} = t q_{x}$ . A useful check: all three assumptions agree at $t = 0$ (giving 0) and $t = 1$ (giving $q_{x}$ ). If your answer violates this, you used the wrong formula. "UDD = times t." "Constant force = power rule: $(p_{x})^{t}$ ." "Balducci = backward UDD." Force ordering at $t > 0.5$ : "BaCoU".

The full lesson (about 2,295 words, 15 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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