Combining Competing Outgoing Transitions

Why Outgoing Transitions Compete

A person or policy in a given state may be able to leave it in several different ways. For example, a hypothetical active policy might lapse, enter a claim state, or terminate on death. At any instant, each possible transition has its own intensity. These alternatives compete to be the first transition out of the current state: if one occurs, the policy has left that state, and the other alternatives no longer describe its immediate exit.

The total exit intensity is therefore obtained by adding the intensities of all the distinct outgoing transition types. This is an addition of rates, not an addition of probabilities. The distinction matters because an intensity is measured per unit of time, whereas a probability is dimensionless and refers to a specified interval.

Definition: Suppose a model is in state $i$ at time $t$, and there are $m$ possible outgoing transition types. Let $\lambda_{ir}(t)$ be the intensity of outgoing type $r$, for $r=1,\ldots,m$. The total exit intensity from state $i$ is $$ \lambda_i^{\mathrm{out}}(t)=\sum_{r=1}^{m}\lambda_{ir}(t). $$ Each $\lambda_{ir}(t)$ is a nonnegative rate per unit of time. The sum is a rate per the same unit of time.

This definition applies when the listed types are the possible immediate exits from the state and their intensities are defined under the same model and time scale. The rates are conditional on being in state $i$ at time $t$; they are not unconditional population frequencies. An outgoing transition whose intensity is zero contributes nothing to the sum. If the total is zero, there is no positive instantaneous exit intensity from that state at that time.

Why the Intensities Add

The addition follows from the definition of transition intensity. Over a short interval of length $h$, the probability of an exit by type $r$, given that the process is in state $i$ at time $t$, is approximately $\lambda_{ir}(t)h$. Because the exit types are distinct alternatives for the first transition, their probabilities add. Events involving multiple transitions within the same very short interval have probability of smaller order than $h$, so they do not affect the intensity.

\begin{aligned} P(\text{exit from }i\text{ in }(t,t+h]\mid X(t)=i) &=\sum_{r=1}^{m}\bigl(\lambda_{ir}(t)h+o(h)\bigr)+o(h)\\ &=\left(\sum_{r=1}^{m}\lambda_{ir}(t)\right)h+o(h). \end{aligned}

Dividing by $h$ and taking the limit as $h$ decreases to zero gives the total exit intensity. This argument assumes the transition intensities exist and that the model has the usual small-interval behavior: the probability of two or more jumps is negligible compared with $h$. It does not require the outgoing intensities to be probabilities, and it does not require treating the transition types as independent random events.

Result: For a short interval of length $h$, the probability of any exit from state $i$ is approximately $\lambda_i^{\mathrm{out}}(t)h$. The approximation is local: it is appropriate for sufficiently small $h$, not automatically an exact probability over a longer interval.

Worked Examples

Worked Example: Combining Three Policy Exit Types

A hypothetical policy is currently active. Its outgoing transition intensities are $0.011$ per year to a disability state, $0.004$ per year to death, and $0.025$ per year to lapse. Find the total exit intensity. Then estimate the probability of each type of exit, and of any exit, over the next $0.02$ years using the small-interval approximation.

Method. Add the outgoing rates to get the total exit intensity, then multiply each rate by the interval length to approximate its transition probability. The given interval is in years, matching the units of the intensities.

Total intensity. $$ \lambda^{\mathrm{out}}=0.011+0.004+0.025=0.040\text{ per year}. $$

Disability exit. The short-interval approximation is $0.011(0.02)=0.00022$.

Death exit. The approximation is $0.004(0.02)=0.00008$.

Lapse exit. The approximation is $0.025(0.02)=0.00050$.

Any exit. Using the total intensity gives $0.040(0.02)=0.00080$. The separate estimates also sum to $0.00022+0.00008+0.00050=0.00080$, as required.

The method avoids the mistake of reporting $0.040$ as a probability. It is an intensity per year; the interval length is needed to obtain a short-interval probability approximation.

Worked Example: Relative Contributions of Competing Exits

A hypothetical member in an active coverage state can enter a disability state at intensity $0.015$ per year, die at intensity $0.005$ per year, or lapse at intensity $0.030$ per year. Find the total exit intensity and the approximate share of immediate exits attributable to each type. Also estimate the probability of any exit in the next $0.01$ years.

Method. First find the sum of the rates. For a very short interval, each exit type contributes approximately its own intensity times the interval length. Dividing that contribution by the total gives its approximate share among exits in the interval.

Total intensity. $$ \lambda^{\mathrm{out}}=0.015+0.005+0.030=0.050\text{ per year}. $$

Disability share. The ratio is $0.015/0.050=0.30$, or 30%.

Death share. The ratio is $0.005/0.050=0.10$, or 10%.

Lapse share. The ratio is $0.030/0.050=0.60$, or 60%. The shares add to $0.30+0.10+0.60=1.00$.

Short-interval exit probability. Over $h=0.01$ years, the approximation is $0.050(0.01)=0.00050$. Separately, the three approximate exit probabilities are $0.015(0.01)=0.00015$, $0.005(0.01)=0.00005$, and $0.030(0.01)=0.00030$. Their sum is $0.00050$, which checks the total.

The shares describe the relative contributions to exits over a sufficiently short interval, or equivalently the instantaneous proportions implied by the rates. They should not be read as probabilities of those exit types over an arbitrary long period.

Worked Example: Two Causes Leading to the Same Destination

A hypothetical active policy can transition to disability at intensity $0.012$ per year or lapse at intensity $0.003$ per year. It can also terminate on death, with two distinguishable causes: accident at intensity $0.006$ per year and illness at intensity $0.009$ per year. Find the total exit intensity, the combined intensity to the death state, and the approximate probabilities of each distinct exit type over $0.1$ years.

Method. Keep the causes distinct when describing exit types, but add intensities when the question concerns all exits to the same destination. Add every outgoing type once to obtain the total exit intensity.

Combined intensity to death. Both death causes lead to the same destination, so the total intensity to death is $0.006+0.009=0.015$ per year.

Total exit intensity. Including disability and lapse gives $$ \lambda^{\mathrm{out}}=0.012+0.003+0.006+0.009=0.030\text{ per year}. $$ Equivalently, the destination-level rates sum to $0.012+0.003+0.015=0.030$ per year.

Approximate probabilities over $0.1$ years. The accident, illness, disability, and lapse estimates are, respectively, $$ 0.006(0.1)=0.0006,\qquad 0.009(0.1)=0.0009,\qquad 0.012(0.1)=0.0012,\qquad 0.003(0.1)=0.0003. $$

The combined death estimate is $0.0006+0.0009=0.0015$. The estimate for any exit is $0.030(0.1)=0.0030$; checking by causes gives $0.0006+0.0009+0.0012+0.0003=0.0030$. This example avoids the mistake of counting the death destination only once while omitting one of its two possible causes.

Common Mistakes and Checks

Combining outgoing transitions is straightforward once the rates have been put on a consistent basis. Several errors recur because intensities and probabilities both describe movement out of a state, but they are not interchangeable.

Adding probabilities instead of intensities. The total exit intensity is the sum of rates. To estimate a short-interval exit probability, multiply the total rate by the interval length.
Mixing time units. A rate per month cannot be added directly to a rate per year. Convert both to the same unit before summing.
Leaving out an outgoing type. List every possible immediate exit from the state before calculating the total.
Double-counting a destination. If two causes lead to the same destination, include both cause intensities in the total exit rate. For a destination-level rate, sum those causes once into that destination's combined rate.
Using a short-interval approximation over a long period without justification. The approximation $\lambda h$ is local. A larger interval requires a model-based calculation rather than treating the approximation as exact.

Key takeaway: For one current state, add the intensities of all distinct outgoing transition types to obtain the total exit intensity. Keep the units consistent, and multiply by a short interval only when an approximate probability is required.

Check Your Understanding

For each problem, identify the relevant outgoing types before combining their rates. Use the small-interval approximation only where requested.

A hypothetical active state has outgoing intensities of $0.009$ per year to disability, $0.006$ per year to death, and $0.015$ per year to lapse. Find the total exit intensity. (Answer: $0.030$ per year.)
A state has exit intensities of $0.018$ per year to one destination and $0.012$ per year to another. Estimate the probability of any exit over $0.04$ years using the small-interval approximation. (Answer: $0.0012$.)
A member can recover at intensity $0.050$ per year, die at intensity $0.020$ per year, or transfer at intensity $0.010$ per year. Find the total exit intensity and the approximate share of immediate exits attributable to recovery. (Answer: $0.080$ per year; $0.625$, or $62.5\%$.)
A hypothetical model gives a lapse intensity of $0.003$ per month and a death intensity of $0.024$ per year. Convert to a common time unit and find the combined exit intensity per year. (Answer: $0.060$ per year.)
Two distinct causes of death have intensities $0.007$ and $0.013$ per year; disability has intensity $0.020$ per year. Find the combined intensity to death, the total exit intensity, and the approximate probability of any exit over $0.01$ years. (Answer: $0.020$ per year to death; $0.040$ per year total; $0.0004$.)

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