UDD Feasibility and Boundary Checks

Why Feasibility Checks Matter

A UDD conversion can produce algebraic answers without producing a valid set of associated probabilities. A negative root, a probability above one, or a negative observed probability signals that something is inconsistent: the data, the arithmetic, or the use of the UDD assumption. Boundary cases need particular care. For example, a zero associated probability always produces a zero observed probability, even if its adjustment factor is negative.

The aim here is to check a proposed conversion rather than to repeat the full UDD conversion procedure. The checks are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: For a given year with mutually exclusive causes of decrement, let $q_j$ be the observed probability of exit from cause $j$, and let $p_j$ be the associated single-decrement probability for cause $j$. Under UDD, the conversion equations are $$ q_j=p_j\left(1-\frac12\sum_{k\ne j}p_k\right). $$ Each $q_j$ is an observed cause-specific probability, whereas $p_j$ is a probability in the associated table for cause $j$ considered alone. In particular, the sum of the $p_j$'s is not automatically a probability and need not be at most one.

For a valid set of observed probabilities, every $q_j$ must be in $[0,1]$, and the total $\sum_j q_j$ must be at most one because the observed exits are mutually exclusive. Each associated probability must also satisfy $0\le p_j\le1$. These are necessary checks, but they do not by themselves guarantee that the UDD equations can reproduce the observed probabilities.

What the Adjustment Factor Can Tell You

The factor $1-\frac12\sum_{k\ne j}p_k$ adjusts the probability for the presence of competing decrements. If $p_j>0$, the equation implies that the factor equals $q_j/p_j$. Thus, for $q_j$ to be nonnegative and no greater than $p_j$, the factor must be in $[0,1]$. Given nonnegative associated probabilities, it is automatically no greater than one. Its nonnegativity requires $\sum_{k\ne j}p_k\le2$ for each cause with $p_j>0$.

Do not apply that necessity blindly when $p_j=0$: the equation then gives $q_j=0$ regardless of the factor. A modeller may impose nonnegative adjustment factors for every cause as an additional feasibility convention, but that is a stricter condition than requiring the UDD equations to produce nonnegative observed probabilities.

Key check: If $p_j>0$, a negative adjustment factor would make $q_j<0$, so the proposed vector is infeasible. If $p_j=0$, the product is zero even when the factor is negative; check the equation itself before rejecting the vector.

A Two-Cause Boundary Check

For two causes, write $Q=q_1+q_2$, $d=q_1-q_2$, and $P=p_1+p_2$. The conversion equations simplify to $Q=P-p_1p_2$ and $d=p_1-p_2$. Since $p_1p_2=(P^2-d^2)/4$, a candidate total $P$ must satisfy

P^2-4P+4Q-d^2=0, \qquad P=2\pm\sqrt{4-4Q+d^2}.

This calculation is a useful diagnostic, not permission to accept either root automatically. For each root, recover $p_1=(P+d)/2$ and $p_2=(P-d)/2$, then check their bounds and substitute them into the original equations. Reject a root that gives a negative probability or one above one. A root that passes every check is a solution; if both pass, the observed data alone have not selected a unique associated vector.

Worked Example: Accepting One Root and Rejecting the Other

Two causes have observed probabilities $q_1=0.285$ and $q_2=0.085$. Find the associated probabilities under UDD, check the algebraic roots, and verify the accepted answer in the original equations.

Method. Use the two-cause total-and-difference equations, then check each root against the probability bounds and the original conversion equations.

First, $Q=0.285+0.085=0.370$ and $d=0.285-0.085=0.200$. The expression under the square root is $4-4(0.370)+(0.200)^2=4-1.480+0.040=2.560$, whose square root is $1.600$. Therefore $P=2\pm1.600$, giving $P=0.400$ or $P=3.600$.

For $P=0.400$, $p_1=(0.400+0.200)/2=0.300$ and $p_2=(0.400-0.200)/2=0.100$. Both lie in $[0,1]$. For $P=3.600$, the recovered values are $p_1=1.900$ and $p_2=1.700$, so this root is invalid: both exceed one.

Now verify the accepted vector. For cause 1, the adjustment factor is $1-\frac12(0.100)=0.950$, so $q_1=0.300(0.950)=0.285$. For cause 2, it is $1-\frac12(0.300)=0.850$, so $q_2=0.100(0.850)=0.085$. The total observed probability is $0.370\le1$, as required.

Check and common mistake. The result reproduces both inputs and satisfies the probability bounds. Do not accept a root just because it solves the quadratic; the larger root fails the individual bounds.

Necessary Checks Are Not Sufficient

Observed probabilities may be nonnegative and sum to at most one but still be impossible under UDD. One useful diagnostic for comparing causes follows directly from the equations. For $i\ne j$,

q_i-q_j=(p_i-p_j)\left(1-\frac12\sum_{k\ne i,j}p_k\right).

This identity follows by expanding both equations: the terms involving $p_i p_j$ cancel, leaving the difference in associated probabilities multiplied by the adjustment for the other causes. With three causes, the factor is $1-p_k/2$, which is at least $1/2$ because $0\le p_k\le1$. Therefore, with three causes, equal observed probabilities require equal associated probabilities. That implication can expose infeasibility even when the observed total looks reasonable.

Worked Example: Rejecting a Plausible-Looking Three-Cause Set

A hypothetical portfolio has three observed cause probabilities, each equal to $0.300$. Their total is $0.900$, so the basic observed-probability checks pass. Determine whether a UDD associated vector can produce them.

Method. Use the difference identity to establish equality of the associated probabilities, then test the common value in the UDD equation.

Since each observed probability is equal, the difference identity gives, for any pair, $0=(p_i-p_j)(1-p_k/2)$. The second factor is at least $1/2$, so it is positive. Thus $p_1=p_2=p_3=x$.

The UDD equation for any cause now becomes $0.300=x(1-x)$, because the other two associated probabilities sum to $2x$. But $x(1-x)\le0.250$ for $0\le x\le1$, as $x(1-x)=\frac14-(x-\frac12)^2$. It cannot equal $0.300$. No feasible associated vector exists.

Check and common mistake. The observed total being at most one is necessary, not sufficient. Do not conclude that this set is feasible just because its three entries are valid probabilities and sum to $0.900$; the UDD equations impose additional restrictions.

Zero Probabilities and Boundary Interpretation

A zero observed probability does not always imply that the associated probability is zero. If $p_j>0$, the UDD equation gives $q_j=0$ exactly when its adjustment factor is zero. Conversely, if $p_j=0$, the observed probability is zero whatever the factor. This is why a zero entry deserves substitution into the equation rather than a conclusion based on the factor alone.

Worked Example: Checking a Zero-Probability Cause

Consider four causes and the proposed associated vector $(p_1,p_2,p_3,p_4)=(0,1,1,1)$. Check the implied observed probabilities and decide whether a negative factor for the zero cause invalidates the equations.

Method. Evaluate each UDD product separately, paying particular attention to whether its associated probability is zero.

For cause 1, the factor is $1-\frac12(1+1+1)=1-1.5=-0.5$. Its observed probability is nevertheless $q_1=0(-0.5)=0$.

For cause 2, the other associated probabilities sum to $0+1+1=2$, so the factor is $1-\frac12(2)=0$, and $q_2=1(0)=0$. The same calculation gives $q_3=0$ and $q_4=0$. All four associated probabilities lie in $[0,1]$, and all four equations are satisfied.

Check and common mistake. The proposed vector reproduces the observed vector $(0,0,0,0)$. It would be incorrect to reject it solely because cause 1 has a negative factor: multiplying by $p_1=0$ still gives $q_1=0$. A separate rule requiring every factor to be nonnegative would reject this vector, but that rule is an additional condition, not a consequence of the product equation for a zero $p_j$.

A Practical Feasibility Sequence

When a conversion looks suspicious, use the checks in order. This makes it easier to locate whether the issue is in the observed data, the algebra, or the interpretation of a boundary case.

Check the observed probabilities.
Confirm each $q_j\ge0$ and $\sum_j q_j\le1$. If either fails, recheck the counts, exposure denominator, cause labels, and time interval.

Check each algebraic candidate.
Require $0\le p_j\le1$ for every associated probability. Do not discard a valid candidate merely because another root is invalid, and do not assume there is only one valid candidate.

Check the UDD factors where needed.
For $p_j>0$, a negative factor is inconsistent with $q_j\ge0$. For $p_j=0$, evaluate the product directly and distinguish the equation from any stricter all-factors-nonnegative convention.

Substitute into every original equation.
Recompute each $p_j(1-\frac12\sum_{k\ne j}p_k)$ and compare with the stated $q_j$. If a check fails, revisit arithmetic, the input data, and whether UDD is the intended assumption.

These checks also help diagnose what to revisit. An observed total above one points first to the data or denominator. A recovered $p_j$ outside $[0,1]$ points to an invalid algebraic root or an inconsistent input. A negative factor paired with $p_j>0$ contradicts nonnegative observed probabilities under the stated UDD equations. A boundary case with $p_j=0$, by contrast, must be judged from the product equation itself.

Key takeaway: Feasibility is more than solving equations. Check the observed probability bounds, the associated probability bounds, and the original UDD equations. Treat zero associated probabilities separately when assessing adjustment factors, and investigate rather than conceal an inconsistency by clipping a result to zero or one.

Check Your Understanding

For each item, apply the probability bounds and UDD equations rather than relying on a single check.

Two causes have associated probabilities $p_1=0.400$ and $p_2=0.200$. Find the observed probabilities and check their total. (Answer: $q_1=0.360,\ q_2=0.160$, total $0.520$.)
For two causes, $q_1=q_2=0.250$. Find the feasible associated vector with equal entries, to four decimal places. (Answer: $p_1=p_2=1-\sqrt{0.5}\approx0.2929$.)
Three causes have observed probabilities $0.240$ each. Find the equal-associated-probability solutions, if any. (Answer: $(p_1,p_2,p_3)=(0.400,0.400,0.400)$ or $(0.600,0.600,0.600)$.)
For four causes, test $p=(0,1,1,1)$. What observed vector results, and does the negative factor for the first cause alone invalidate the UDD equations? (Answer: $q=(0,0,0,0)$; no, because $p_1=0$, so its product is zero. An extra all-factors-nonnegative rule would reject it.)
Two mutually exclusive observed causes are reported with probabilities $0.620$ and $0.410$. Can these be used as observed probabilities for one year without further correction? (Answer: No; their sum is $1.030>1$, so recheck the inputs or denominator before attempting a UDD conversion.)

What You'll Learn