UDD Conversion to Associated Single-Decrement Tables

Why Convert Multiple-Decrement Probabilities?

In a multiple-decrement model, each cause can prevent an exit from occurring by another cause. For example, a hypothetical policyholder who lapses during a year cannot also die as an insured policyholder later in that same year. The observed probability of exit by a particular cause therefore reflects both that cause and the presence of competing causes.

An associated single-decrement probability asks a different, counterfactual question: what would the probability of exit from one specified cause be if all other causes were removed? The uniform distribution of decrements (UDD) assumption supplies a way to make this conversion from the multiple-decrement probabilities. It is important to use the probabilities for the other causes in the adjustment for each cause.

Definition: Let $q_x^{(j)}$ be the probability that a life aged $x$ exits during the year from cause $j$, in the presence of all causes in the multiple-decrement model. Let $q_x^{(j)'}$ be the associated single-decrement probability for cause $j$: the probability of exit from cause $j$ if all other causes are eliminated. Under the UDD conversion convention, $$ q_x^{(j)'}= \frac{q_x^{(j)}}{1-\frac{1}{2}\displaystyle\sum_{k\ne j}q_x^{(k)}}. $$

The sum in the denominator includes every competing cause other than $j$, and uses the multiple-decrement probabilities. For two causes, the sum consists of just one probability. For three or more causes, add all the other probabilities before dividing. The adjustment increases the probability for the cause being isolated: the observed multiple-decrement probability is reduced by the presence of competing exits.

Why the UDD Adjustment Has This Form

Under UDD, decrements are spread uniformly through the year. Thus, for a competing cause $k$, the fraction of the original group removed by that cause by time $t$ is $tq_x^{(k)}$, for $0\le t\le1$. Across the year, the average fraction removed by cause $k$ is consequently $q_x^{(k)}/2$: the average of a linear increase from zero to $q_x^{(k)}$.

When cause $j$ is considered on its own, those competing exits are absent. Under the UDD conversion convention, its observed probability is adjusted for the average fraction of the original group removed by the other causes. Summing those average competing fractions gives the UDD exposure factor. In symbols, the conversion identity is

q_x^{(j)} = q_x^{(j)'} \left(1-\frac{1}{2}\sum_{k\ne j}q_x^{(k)}\right).

Solving this identity for $q_x^{(j)'}$ gives the formula in the definition. The division is essential: the competing causes reduce the observed probability, so the associated probability must be larger to represent the cause acting without them. This conversion is specific to UDD; it is not a general identity that can be used without a within-year assumption.

Applying the Formula Cause by Cause

A reliable calculation has four steps: identify all causes, select the cause being converted, add the probabilities of the remaining causes, and apply the formula. The total probability of exit is useful as a data check, but it is not the quantity to place in every denominator.

Check the multiple-decrement data.
Confirm that the cause probabilities are nonnegative and that their sum is at most 1.

Choose one cause.
For cause $j$, exclude $q_x^{(j)}$ from the sum in its denominator.

Calculate the adjustment.
Subtract one-half of the sum for all competing causes from 1, then divide $q_x^{(j)}$ by that factor.

Check the conversion.
Multiply the result by its denominator; the product should reproduce the original $q_x^{(j)}$, subject to rounding.

Worked Example: Converting Three Cause Probabilities

In a hypothetical annual model, the multiple-decrement probabilities are $q_x^{(1)}=0.08$, $q_x^{(2)}=0.12$, and $q_x^{(3)}=0.10$. Assume UDD. Calculate the associated single-decrement probability for each cause. Give probabilities to four decimal places.

Method. Apply the UDD formula separately to each cause, leaving that cause out of its competing-cause sum.

For cause 1, the other-cause sum is $0.12+0.10=0.22$. Half of this is $0.11$, so the denominator is $1-0.11=0.89$. Therefore,

q_x^{(1)'}=\frac{0.08}{0.89}=0.0898876\ldots\approx 0.0899.

For cause 2, the other-cause sum is $0.08+0.10=0.18$. Half is $0.09$, giving a denominator of $0.91$:

q_x^{(2)'}=\frac{0.12}{0.91}=0.1318681\ldots\approx 0.1319.

For cause 3, the other-cause sum is $0.08+0.12=0.20$. Half is $0.10$, giving a denominator of $0.90$:

q_x^{(3)'}=\frac{0.10}{0.90}=0.1111111\ldots\approx 0.1111.

Check. Multiplying back gives $0.0898876(0.89)=0.0800$, $0.1318681(0.91)=0.1200$, and $0.1111111(0.90)=0.1000$, up to rounding. Each associated probability exceeds its corresponding multiple-decrement probability, as expected. The associated probabilities need not sum to 1; they describe separate hypothetical models in which different causes are retained.

Common mistake avoided: using the same denominator for every cause. The denominator changes with the cause being isolated because the competing-cause set changes.

Starting From Decrement Counts

If a table gives numbers of exits rather than probabilities, first divide each cause’s exit count by the number exposed to risk at the start of the year. Then apply the UDD conversion. Do not divide by the number who survive the year: that would use a different denominator and would not produce the annual decrement probability.

Worked Example: Converting Counts From a Hypothetical Table

A hypothetical group has 800 lives at risk at the start of a year. During the year, 40 exit from cause 1, 56 from cause 2, and 24 from cause 3. Assume these are the only exits and that UDD applies. Calculate the associated single-decrement probabilities, to four decimal places.

Step 1: Calculate the multiple-decrement probabilities. Divide each exit count by the initial 800 lives:

q_x^{(1)}=\frac{40}{800}=0.05,\qquad q_x^{(2)}=\frac{56}{800}=0.07,\qquad q_x^{(3)}=\frac{24}{800}=0.03.

Their sum is $0.05+0.07+0.03=0.15$. The implied number remaining is $800(1-0.15)=680$, which also equals $800-(40+56+24)=680$. The counts therefore satisfy the total-exit check.

Step 2: Convert cause 1. The other-cause sum is $0.07+0.03=0.10$, so its denominator is $1-\frac12(0.10)=0.95$:

q_x^{(1)'}=\frac{0.05}{0.95}=0.0526316\ldots\approx0.0526.

Step 3: Convert cause 2. The other-cause sum is $0.05+0.03=0.08$, so its denominator is $1-\frac12(0.08)=0.96$:

q_x^{(2)'}=\frac{0.07}{0.96}=0.0729167\ldots\approx0.0729.

Step 4: Convert cause 3. The other-cause sum is $0.05+0.07=0.12$, so its denominator is $1-\frac12(0.12)=0.94$:

q_x^{(3)'}=\frac{0.03}{0.94}=0.0319149\ldots\approx0.0319.

Check. The reconstructed multiple-decrement probabilities are $0.0526316(0.95)=0.0500$, $0.0729167(0.96)=0.0700$, and $0.0319149(0.94)=0.0300$, to the displayed precision. The conversion uses 800 as the initial exposure for each multiple-decrement probability; the 680 survivors are not the denominator for these calculations.

Common mistake avoided: treating the associated probability as an observed exit proportion from the surviving group. It is a counterfactual probability for a model with other causes removed, not a count divided by year-end survivors.

Two-Cause Case and the Cause-Specific Denominator

With two causes, the conversion is especially transparent. When cause 1 is isolated, cause 2 is the only competitor in the adjustment; when cause 2 is isolated, cause 1 is the only competitor. This is a useful setting for checking whether the cause-specific denominator has been selected correctly.

Worked Example: Avoiding the Total-Probability Denominator

For a hypothetical decrement group, $q_x^{(1)}=0.16$ and $q_x^{(2)}=0.24$ during the year. Assume UDD. Find both associated probabilities to four decimal places, and identify the error in using the total exit probability in each denominator.

For cause 1, only cause 2 is competing. Thus the denominator is $1-\frac12(0.24)=0.88$, and

q_x^{(1)'}=\frac{0.16}{0.88}=0.1818182\ldots\approx0.1818.

For cause 2, only cause 1 is competing. Its denominator is $1-\frac12(0.16)=0.92$, and

q_x^{(2)'}=\frac{0.24}{0.92}=0.2608696\ldots\approx0.2609.

The total multiple-decrement probability is $0.16+0.24=0.40$, but it is not the competing probability for either cause: each cause must be excluded from its own denominator. The incorrect calculation for cause 1 would be

\frac{0.16}{1-\frac12(0.40)} =\frac{0.16}{0.80} =0.2000,

which is too large under the specified conversion. It incorrectly counts cause 1 itself as a competing decrement. The same error for cause 2 would give $0.24/0.80=0.3000$, instead of $0.2609$. Checking the correct results gives $0.1818182(0.88)=0.16$ and $0.2608696(0.92)=0.24$.

Common mistake avoided: using the total decrement probability where the formula requires the sum over causes other than the one being converted.

Conditions, Checks, and Interpretation

The UDD conversion requires multiple-decrement probabilities for the same age interval and a UDD assumption for the within-year distribution. It should not be used just because a table reports annual probabilities. If another within-year assumption is specified, the corresponding conversion method must be used instead. In particular, the result is not an assumption-free estimate of a single-cause probability.

For ordinary valid input data, each $q_x^{(j)}$ is nonnegative and the total $q_x=\sum_j q_x^{(j)}$ satisfies $q_x\le1$. The UDD denominator for cause $j$ is

1-\frac12\sum_{k\ne j}q_x^{(k)}.

Since the sum of the other probabilities is at most 1, this denominator is at least $1/2$, and is therefore positive. This makes the conversion well-defined for a valid multiple-decrement table. It also follows directly from the formula that $q_x^{(j)'}\ge q_x^{(j)}$, because the denominator is no greater than 1. Equality holds when the other causes have zero probability.

The associated probabilities are calculated one cause at a time. They should not be interpreted as a new set of simultaneous, mutually exclusive exits: each is from a separate hypothetical model retaining only its specified cause. Consequently, adding the associated probabilities is not a multiple-decrement closure check. The appropriate check is to multiply each converted value by its cause-specific UDD adjustment and recover the original multiple-decrement probability.

Use $q_x^{(j)}$ in the numerator.
Exclude cause $j$ from the competing-cause sum.
Use multiple-decrement probabilities, not associated probabilities, in the denominator.
Keep the model assumption explicit: the conversion here is under UDD.
Check each cause separately by reconstructing its observed probability.

Key takeaway: Under UDD, convert cause $j$ by dividing its multiple-decrement probability by one minus half the sum of the multiple-decrement probabilities for all other causes. The adjustment is cause-specific; a total-probability denominator is generally wrong.

Check Your Understanding

Assume UDD in each problem. Round numerical probabilities to four decimal places where needed.

A group has two multiple-decrement probabilities, $q_x^{(1)}=0.06$ and $q_x^{(2)}=0.14$. Find both associated probabilities. (Answer: $q_x^{(1)'}=0.0645,\ q_x^{(2)'}=0.1443$)
Three causes have multiple-decrement probabilities $0.08,\ 0.10,\ 0.12$. Find the associated probability for cause 2. (Answer: $0.1111$)
Out of 1,000 lives initially exposed, 45 exit by cause 1, 75 by cause 2, and 30 by cause 3. Find the associated probability for cause 2. (Answer: $0.0779$)
For causes 1, 2, and 3, the multiple-decrement probabilities are $0.11,\ 0.09,\ 0.05$. A calculation for cause 2 uses denominator $1-\frac12(0.11+0.09+0.05)=0.875$. State the correct denominator and associated probability for cause 2. (Answer: denominator $0.92$; $q_x^{(2)'}=0.0978$)
A specified cause has multiple-decrement probability $0.20$. The sum of all other causes’ multiple-decrement probabilities is $0.40$. Find its associated probability. (Answer: $0.2500$)

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