Common Errors in Decrement Table Calculations

Why Decrement Calculations Go Wrong

A multiple-decrement table describes a population exposed to several competing causes of exit. A member who exits from one cause is no longer exposed to the other causes in that period. Many errors arise from overlooking that competition: treating a rate as a probability, converting a table without specifying the assumption, or using the wrong group as the denominator in a conditional probability.

Fractional-age calculations require the same care. A table may give annual probabilities, but probabilities within the year depend on an assumption about how decrements are distributed over time. The uniform distribution of decrements (UDD) and constant-force assumptions generally produce different answers. Neither should be inserted silently into a calculation.

A cause-specific force of decrement, written here as \(\mu^{(j)}(t)\), is an instantaneous rate among members still in the group at time \(t\). A force has units of inverse time; it is not itself the probability of exiting from that cause during a year. In general, the annual probability depends on all competing forces operating while a member remains exposed.

Rates, Probabilities, and Conversion Assumptions

When the cause-specific forces are constant over a year, write them as \(\mu_1,\ldots,\mu_m\), and let \(\mu=\sum_j\mu_j\). The probability of surviving all causes for one year is \(e^{-\mu}\), so \(q=1-e^{-\mu}\). The probability of exit from cause \(j\) is not \(1-e^{-\mu_j}\), because that expression acts as if cause \(j\) were the only cause operating.

The integral counts cause-\(j\) exits at time \(t\): the member must first survive all causes to \(t\), with probability \(e^{-\mu t}\), and then exit from cause \(j\) in an infinitesimal interval, at rate \(\mu_j\). This also explains why adding the cause-specific forces and then treating each force as an annual probability is generally wrong.

An associated single-decrement probability asks what the probability would be if one cause operated alone, under a specified model. Under constant forces, removing all causes other than \(j\) gives \(q'^{(j)}=1-e^{-\mu_j}\). That is a different quantity from the multiple-decrement probability \(q^{(j)}\). A conversion from observed multiple-decrement probabilities to associated single-decrement probabilities is not unique unless its assumptions are stated.

A useful diagnostic is to check that the cause-specific probabilities sum to the total probability and that total probability plus survival equals one. These checks detect arithmetic and closure errors, but they do not establish that the conversion assumption was appropriate. The assumed model must also match the question.

Conditional Probabilities Need the Right Denominator

For a decrement table, \(q_x^{(j)}\) is conditional on being in the group at the start of the year. Among those who exit during the year, the conditional probability that cause \(j\) is responsible is instead \(q_x^{(j)}/q_x\), provided \(q_x>0\). Among members who survive to a later year, the later-year decrement probability is conditional on that survival; it is not an unconditional probability for the original cohort.

For multiple years, multiply conditional survival probabilities along the path before applying a later conditional decrement probability. For example, if a member must survive a first year with survival probability \(p_1\) before facing a second-year cause probability \(q_2^{(j)}\), the probability of that sequence is \(p_1q_2^{(j)}\). This does not mean that the second-year probability itself has changed; it means the original cohort includes members who did not reach the second year.

Fractional Ages: Do Not Mix Models

Annual probabilities do not determine fractional-year probabilities without an assumption. Under UDD, decrement counts are assumed to be uniformly distributed through the year: for each cause, the number of decrements accumulated by time \(t\) is \(t\) times its annual count. If annual cause probabilities are \(q^{(j)}\) and total \(q=\sum_jq^{(j)}\), then the probability of surviving to time \(t\) is \(1-tq\), and the probability of cause \(j\) during the first \(t\) years is \(tq^{(j)}\).

The conditional probability of cause \(j\) in the interval from \(s\) to \(t\), given survival to \(s\), is the cause-\(j\) probability over that interval divided by the probability of survival to \(s\): \((t-s)q^{(j)}/(1-sq)\), for \(0\le s<t\le1\). The denominator matters: an interval probability measured against the original cohort is not yet conditional on reaching the interval.

Under constant total force, annual survival \(p=1-q\) instead implies survival to time \(t\) of \(p^t\). If cause-specific forces are constant, their shares of the total force allocate the fractional-period exits. Applying UDD to one part of a calculation and constant force to another generally fails to preserve the stated annual table.

A Practical Error-Checking Sequence

When a calculation appears plausible but its assumptions or denominator are unclear, use a short sequence of checks before accepting it. The aim is to identify the model first, then test whether the resulting quantities agree with that model.

These checks are particularly useful when a problem presents a table and asks for a conditional or fractional-year result. A number can be arithmetically correct under one model but wrong for the model specified in the problem. State the assumption beside the calculation, retain the correct conditioning population, and round only after the checks are complete. This error-diagnosis approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.