Cohort Probability Updates Across Subintervals

Updating a Cohort Across Consecutive Intervals

In a competing-decrement model, a cohort can leave through several causes, while the intensities of those causes change over time. A calculation over consecutive subintervals must carry forward the proportion still exposed at the start of each interval. Applying each interval’s exit probabilities to the original cohort, rather than to the remaining cohort, overstates exits.

The central task is to update two things at every interval boundary: the proportion still in the group and the cumulative proportion that has exited through each cause. This recursion is useful whenever the intensity assumptions are piecewise constant, and it also provides the framework for numerical calculations when intensities vary within an interval. The treatment here is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Consider a cohort with initial proportion 1, exposed to $m$ mutually exclusive exit causes. Let $\mu_k(t)$ be the cause-specific intensity for cause $k$ at time $t$, and let $\mu(t)=\sum_{k=1}^{m}\mu_k(t)$ be the total intensity. Write $R(t)$ for the proportion that has experienced no exit by time $t$, and $C_k(t)$ for the cumulative proportion that has exited by cause $k$ by time $t$. Then $R(t)+\sum_{k=1}^{m}C_k(t)=1$.

This is a continuous-time model: an intensity is a rate per unit of time, not an interval probability. The causes are assumed to be mutually exclusive first exits, so an individual who exits through one cause is no longer at risk of a later exit through another cause in this model. The time unit for the intensities and interval lengths must agree.

The Recursion for One Subinterval

Suppose an interval runs from $t_j$ to $t_{j+1}$, has length $\Delta_j=t_{j+1}-t_j$, and begins with remaining proportion $R_j=R(t_j)$. Let $R(t)$ and $\mu_k(t)$ describe the cohort throughout this interval. The remaining proportion at the end and the increment in cumulative exits through cause $k$ are

R_{j+1}=R_j\exp\left(-\int_{t_j}^{t_{j+1}}\mu(s)\,ds\right), \qquad \Delta C_{k,j} = R_j\int_{t_j}^{t_{j+1}} \exp\left(-\int_{t_j}^{t}\mu(s)\,ds\right)\mu_k(t)\,dt.

The exponential inside the cause-exit integral is the proportion of the interval’s starting cohort still present at time $t$, relative to $R_j$. The cause intensity then determines the rate of exits through cause $k$ among those still present. Multiplying by $R_j$ puts the increment on the scale of the original cohort.

Theorem: If all cause-specific intensities are constant within subinterval $j$, with values $\mu_{k,j}$, then $\mu_j=\sum_{k=1}^{m}\mu_{k,j}>0$, and $$ R_{j+1}=R_j e^{-\mu_j\Delta_j}, \qquad \Delta C_{k,j} = R_j\frac{\mu_{k,j}}{\mu_j}\left(1-e^{-\mu_j\Delta_j}\right). $$ The cumulative update is $C_{k,j+1}=C_{k,j}+\Delta C_{k,j}$. If $\mu_j=0$, all cause intensities are zero and the interval produces no exits: $R_{j+1}=R_j$ and $\Delta C_{k,j}=0$.

The cause share $\mu_{k,j}/\mu_j$ in this formula is valid for the interval’s constant intensities. If intensities vary within an interval, the cause share can vary too; in general, the cause increment must be found from the integral rather than by applying one constant share to the total exits.

Why the Recursion Holds

The survival differential equation for competing decrements is $dR(t)/dt=-\mu(t)R(t)$. Separating variables gives $dR(t)/R(t)=-\mu(t)\,dt$. Integrating from $t_j$ to $t$, and using $R(t_j)=R_j$, yields the remaining proportion at any point in the interval:

R(t)=R_j\exp\left(-\int_{t_j}^{t}\mu(s)\,ds\right).

The rate of exits through cause $k$ at time $t$ is $R(t)\mu_k(t)$. Integrating that rate over the interval gives the cause-exit increment. For constant intensities, substitute the expression for $R(t)$ and integrate:

\begin{aligned} \Delta C_{k,j} &=R_j\mu_{k,j}\int_{t_j}^{t_{j+1}}e^{-\mu_j(t-t_j)}\,dt\\ &=R_j\frac{\mu_{k,j}}{\mu_j}\left(1-e^{-\mu_j\Delta_j}\right). \end{aligned}

The interval exit increments conserve probability. Summing over all causes gives

\sum_{k=1}^{m}\Delta C_{k,j} = R_j\left(1-e^{-\mu_j\Delta_j}\right) = R_j-R_{j+1}.

Thus each interval removes exactly the amount from the cohort that it allocates to cause-specific exits. Repeating the update preserves the total probability across the entire sequence of intervals.

Applying the Updates in Sequence

For piecewise-constant intensities, a reliable calculation follows the interval order. At each step, calculate the interval’s total intensity, use the current remaining proportion as the starting amount, update each cause’s cumulative exit proportion, and carry the new remaining proportion into the next interval.

Record the interval assumptions.
Write the interval length and all cause-specific intensities, using consistent time units.

Find the total intensity.
Add the cause-specific intensities to obtain $\mu_j$.

Update remaining and exits.
Use the current $R_j$, not the initial cohort proportion, in every cause-exit increment.

Carry forward and check.
Set the next interval’s starting remaining proportion to $R_{j+1}$, then check that remaining plus cumulative exits equals 1.

Worked Example: Two Causes With a Change in Rates

A hypothetical cohort starts with proportion 1. During the first half-year, the constant intensities are $0.2$ per year for cause A and $0.3$ per year for cause B. During the following year, they are $0.1$ and $0.4$ per year, respectively. Find the final remaining proportion and cumulative exits by cause. Use years as the time unit.

Method. Apply the constant-intensity recursion in each interval, carrying the remaining proportion from the first interval into the second.

First interval. Its length is $0.5$, and the total intensity is $0.2+0.3=0.5$. The interval’s total exit proportion, measured from the original cohort, is $1-e^{-0.5(0.5)}=1-e^{-0.25}=0.2211992$. Cause A’s share is $0.2/0.5=0.4$, and cause B’s share is $0.3/0.5=0.6$. Therefore,

\Delta C_{A,1}=0.4(0.2211992)=0.0884797,\quad \Delta C_{B,1}=0.6(0.2211992)=0.1327195,\quad R_1=e^{-0.25}=0.7788008.

The first-interval check is $0.0884797+0.1327195+0.7788008=1.0000000$, apart from rounding.

Second interval. The starting remaining proportion is $R_1=0.7788008$, not 1. The total intensity is again $0.5$, and the interval length is 1. Thus the total increment in exits is $0.7788008(1-e^{-0.5})=0.3064342$. Cause A’s share is $0.1/0.5=0.2$, and cause B’s share is $0.4/0.5=0.8$:

\Delta C_{A,2}=0.2(0.3064342)=0.0612868,\quad \Delta C_{B,2}=0.8(0.3064342)=0.2451474,\quad R_2=0.7788008e^{-0.5}=0.4723666.

The cumulative exits are $C_A=0.0884797+0.0612868=0.1497665$ and $C_B=0.1327195+0.2451474=0.3778669$. Checking gives $0.1497665+0.3778669+0.4723666=1.0000000$. To four decimal places, the final remaining proportion is 0.4724, and the cause exits are 0.1498 and 0.3779.

Common mistake avoided. Using the second interval’s exit proportion $1-e^{-0.5}$ as if the full cohort were still present would ignore the first interval. The second interval’s exits must be multiplied by $0.7788008$.

Worked Example: Three Subintervals and Cumulative Cause Exits

A hypothetical cohort begins with proportion 1 and is exposed to two causes. For a quarter-year, the intensities are $0.4$ and $0.4$ per year. For the next half-year, they are $0.2$ and $0.3$ per year. For the final quarter-year, they are $0.1$ and $0.1$ per year. Calculate the final remaining proportion and cumulative exit proportions.

Method. Compute the total intensity and cause allocation in each interval, using the updated remaining proportion each time.

In the first interval, the total intensity is $0.8$, so the integrated total intensity is $0.8(0.25)=0.2$. The total exits are $1-e^{-0.2}=0.1812692$. The causes have equal intensities, so each receives half, or $0.0906346$. The remaining proportion is $e^{-0.2}=0.8187308$.

In the second interval, the total intensity is $0.5$, the length is $0.5$, and the starting remaining proportion is $0.8187308$. The total increment in exits is $0.8187308(1-e^{-0.25})=e^{-0.2}-e^{-0.45}=0.1811026$. The cause shares are $0.2/0.5=0.4$ and $0.3/0.5=0.6$. Hence the increments are $0.0724410$ for the first cause and $0.1086616$ for the second. The remaining proportion becomes $e^{-0.45}=0.6376282$.

In the final interval, the total intensity is $0.2$ over $0.25$ years. Total exits are $0.6376282(1-e^{-0.05})=e^{-0.45}-e^{-0.5}=0.0310975$. The causes again have equal intensities, so each receives $0.0155487$. The final remaining proportion is $e^{-0.5}=0.6065307$.

Adding increments across all three intervals gives cumulative exits of $0.0906346+0.0724410+0.0155487=0.1786243$ for the first cause, and $0.0906346+0.1086616+0.0155487=0.2148449$ for the second. The check is $0.1786243+0.2148449+0.6065307=1.0000000$, apart from rounding. To four decimal places, the final remaining proportion is 0.6065 and the cause exits are 0.1786 and 0.2148.

Common mistake avoided. The cause exits are cumulative sums of interval increments. The final interval’s increments alone are not the cohort’s cumulative cause probabilities.

Worked Example: Updating a Cohort That Has Already Experienced Exits

At a particular time, a hypothetical cohort has cumulative exit proportions 0.18 for cause A and 0.10 for cause B, leaving 0.72 still in the group. Over the next half-year, the intensities are $0.2$ for A and $0.2$ for B per year. Over the following half-year, they are $0.1$ for A and $0.3$ for B per year. Find the updated cumulative exits and final remaining proportion.

Method. Start the new intervals with $R_0=0.72$, while retaining the exits that have already accumulated.

For the first half-year, the total intensity is $0.4$, so the integrated intensity is $0.2$. The interval’s total exits are $0.72(1-e^{-0.2})=0.1305139$. Equal cause intensities allocate $0.0652569$ to each cause. The remaining proportion becomes $0.72e^{-0.2}=0.5894861$.

For the second half-year, the total intensity is also $0.4$, and the starting remaining proportion is $0.5894861$. The total increment in exits is $0.5894861(1-e^{-0.2})=0.1068557$. The cause shares are $0.1/0.4=0.25$ and $0.3/0.4=0.75$, so the increments are $0.0267139$ for A and $0.0801418$ for B. The final remaining proportion is $0.5894861e^{-0.2}=0.4826304$.

The updated cumulative exits are $0.18+0.0652569+0.0267139=0.2719708$ for A and $0.10+0.0652569+0.0801418=0.2453987$ for B. The check is $0.2719708+0.2453987+0.4826304=1.0000000$, apart from rounding. To four decimal places, the cumulative exits are 0.2720 and 0.2454, and the remaining proportion is 0.4826.

Common mistake avoided. The starting remaining proportion is 0.72, not 1. The earlier exits remain in the cumulative totals, but they are no longer exposed to the new intensities.

Common Errors and Probability Checks

A useful final check is to add the ending remaining proportion to every cumulative cause exit. For an initial cohort proportion of 1, the total must be 1, subject only to rounding. If the starting cohort proportion is less than 1 because earlier exits have already been recorded, include those earlier cumulative exits in the check as well.

A frequent error is to treat an intensity as an interval probability. For example, an intensity of $0.3$ per year over one year does not mean that the exit probability is $0.3$. With a single constant cause, the probability of exit over the year is $1-e^{-0.3}$, approximately $0.2592$. With competing causes, the total exit probability is $1-e^{-\mu\Delta}$, and it is allocated between causes according to their relative intensities only when those intensities are constant throughout the interval.

Check time units. Multiply an annual intensity by a length expressed in years. A half-year interval has length $0.5$, not 6.
Use the current remaining proportion. Every interval’s cause increments are based on the cohort still present at that interval’s start.
Keep increments and cumulative exits distinct. Add each interval’s increment to the cause’s existing cumulative total.
Do not assume fixed cause shares when rates vary within an interval. Use the cause-exit integral, or subdivide the interval appropriately.

Key takeaway: For each consecutive interval, update the remaining proportion by the total intensity, calculate cause-specific exits from the proportion exposed during that interval, and add those increments to the cumulative cause totals. Probability conservation is the final check.

Check Your Understanding

For each problem, assume mutually exclusive first exits and constant cause-specific intensities within each stated interval. Give final proportions to four decimal places.

A cohort starts at 1. Over 0.5 years, cause A has intensity 0.2 per year and cause B has intensity 0.6 per year. Find the remaining proportion and both cause exits. (Answer: Remaining 0.6703; A 0.0824; B 0.2473.)
Over the first year, intensities are 0.1 for A and 0.2 for B per year. Over the next half-year, they are 0.4 for A and 0.1 for B per year. Find the final remaining proportion and cumulative exits. (Answer: Remaining 0.5770; A 0.2175; B 0.2056.)
For a two-year interval, three cause intensities are constant at 0.1, 0.2, and 0.3 per year. Find the remaining proportion and the exit proportion for each cause. (Answer: Remaining 0.3010; cause exits 0.1165, 0.2330, and 0.3495.)
A cohort has already accumulated exits of 0.25 and 0.15 through causes A and B, leaving 0.60. Over the next year, the intensities are 0.2 for A and 0.3 for B per year. Find the updated cumulative exits and remaining proportion. (Answer: A 0.3444; B 0.2916; remaining 0.3639.)
Cause intensities are 0.3 and 0.1 per year for the first half-year, then 0.1 and 0.3 per year for the next half-year. Starting from 1, find the final remaining proportion and cumulative exits by cause. (Answer: Remaining 0.6703; first cause 0.1731; second cause 0.1566.)

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