Construct and deconstruct multiple decrement tables using the associated single decrement models and appropriate fractional age assumptions. | Free Exam ALTAM Lesson

A pension plan loses members to death, disability, withdrawal, and retirement. Each cause competes; removing one changes the rates of the others. The associated single decrement (ASD) model is the bookkeeping device that lets you toggle between the joint table and a what-if world where only one cause operates.

A life age x faces m mutually exclusive causes of exit. Notation:

$μ_{x}^{(j)}$ is the force of decrement j at age x.
$μ_{x}^{(τ)} = \sum_{j} μ_{x}^{(j)}$ is the total force.
$q_{x}^{(j)}$ is the probability of exit by cause j within one year in the actual joint table.
$q_{x}^{(τ)} = \sum_{j} q_{x}^{(j)}$ is the probability of any exit; $p_{x}^{(τ)} = 1 - q_{x}^{(τ)}$ .

The ASD for decrement j is the hypothetical world where only j operates. Its rates carry a prime: $q_{x}^{' (j)}, p_{x}^{' (j)}$ . Because competitors are removed, fewer lives are stolen before j strikes, so the prime is always at least as large.

$q_{x}^{' (j)} \geq q_{x}^{(j)}, p_{x}^{(τ)} = j \prod p_{x}^{' (j)} (constant force)$

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Common mistakes

Using $p^{(τ)} = \prod p^{' (j)}$ under UDD in ASD. Exact only for constant force. The polynomial expansion is required; the bare product yields a death rate near 0.00460 instead of 0.00475.
Swapping the exponent ratio. Correct form is $p_{x}^{' (j)} = (p_{x}^{(τ)})^{q^{(j)} / q^{(τ)}}$ . The ratio is decrement-to-total, not the reverse.
Mixing the two UDDs. UDD in MDT splits a given joint rate. UDD in ASD combines given singles. Crossing them produces inconsistent rates.

Bottom line

Dependent probability $q_{x}^{(j)}$ lives in the joint table. Independent probability $q_{x}^{' (j)}$ lives in the ASD where only decrement j operates.
Constant force: $p_{x}^{(τ)} = \prod_{j} p_{x}^{' (j)}$ is exact, $q_{x}^{' (j)} = 1 - (p_{x}^{(τ)})^{μ^{(j)} / μ^{(τ)}}$ , and $q_{x}^{(j)} = (μ^{(j)} / μ^{(τ)}) q_{x}^{(τ)}$ .
UDD in MDT: $q_{x}^{' (j)} = 1 - (p_{x}^{(τ)})^{q_{x}^{(j)} / q_{x}^{(τ)}}$ .
UDD in ASD (three decrements): $q_{x}^{(1)} = q_{x}^{' (1)} [1 - \frac{1}{2} (q_{x}^{' (2)} + q_{x}^{' (3)}) + \frac{1}{3} q_{x}^{' (2)} q_{x}^{' (3)}]$ , cross term added.

Exam shortcut

When all rates are below 5 percent, constant-force and UDD-in-MDT deconstructions agree to four decimals. Use whichever computes faster. DECISION: Given MDT, need ASD, use the UDD-in-MDT exponent formula. Given ASD, need MDT, use the UDD-in-ASD polynomial. Given forces, use constant-force ratios. Given Makeham parameters, integrate $A + B c^{x}$ directly. Given a discrete-age decrement, partition the year at the jump.

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