Deriving the Continuous Cause-Specific Valuation Integral

Why a Cause-Specific Integral Is Needed

When a member can leave active service through several competing causes, a benefit contingent on one cause is paid only if two events occur in sequence: the member remains active until the relevant time, and then exits by the specified cause. The cause-specific expected present value (EPV) must account for both events as well as the time value of money. A cause intensity by itself is not the probability that the cause occurs at time $t$; it applies only among members who are still active at that time.

This tutorial derives the valuation integral from the probability of being active immediately before an exit and the probability of a specified exit in a short interval. The derivation applies to hypothetical pension or other long-term benefit arrangements in which benefits are paid at the instant of exit.

Definition: Let $T$ denote time since the valuation date until exit from active status. For each exit cause $j$, the cause-specific intensity $\lambda_j(t)$ is defined so that, conditional on being active at time $t$, $$ P(\text{exit by cause }j\text{ in }[t,t+h)\mid\text{active at }t) =\lambda_j(t)h+o(h) $$ as $h$ tends to zero. The total exit intensity is $\lambda(t)=\sum_j\lambda_j(t)$. Write $p_A(t)=P(T>t)$ for the probability of remaining active through time $t$.

The intensities describe competing exits from active status. They are conditional rates, not unconditional probabilities, and the sum of the cause-specific intensities is the total rate of leaving active status. The derivation does not require independent latent times to each cause: it uses the cause-specific intensities conditional on current active status.

From a Short Interval to the Valuation Integral

Suppose a benefit of amount $B_j(t)$ is paid immediately when exit occurs by cause $j$ at time $t$. Let the force of interest be deterministic and possibly time-varying, $\delta(s)$. The discount factor from time 0 to time $t$ is $\exp(-\int_0^t\delta(s)\,ds)$. First consider a short interval $[t,t+h)$.

Be active at the start of the interval.
The probability of being active at time $t$ is $p_A(t)$.

Exit by the specified cause.
Conditional on being active at $t$, the probability of cause-$j$ exit in the interval is $\lambda_j(t)h+o(h)$. Thus the joint probability of being active at $t$ and exiting by cause $j$ in the interval is $p_A(t)\lambda_j(t)h+o(h)$.

Discount the payment.
For an interval of vanishing width, a payment during that interval has discount factor approaching $\exp(-\int_0^t\delta(s)\,ds)$.

Sum across time.
Adding the expected discounted payments over a partition of the valuation period and taking the limit gives the continuous-time integral.

The short-interval argument also identifies the density of an exit by a specified cause. For $t\geq 0$, that unconditional density is $p_A(t)\lambda_j(t)$: the member must first still be active, then leave by cause $j$. Therefore the cause-specific EPV over a term $T_0$ is

\operatorname{EPV}_j =\int_0^{T_0} \exp\!\left(-\int_0^t\delta(s)\,ds\right) B_j(t)\,p_A(t)\lambda_j(t)\,dt.

If the benefit amount is random conditional on the exit time and cause, replace $B_j(t)$ in the integral by its conditional expected amount given exit by cause $j$ at time $t$. The formula values payments made at exit. It does not, without modification, value a payment delayed until a later date or an additional benefit paid to members who remain active at the end of the term.

Active-Member Survival and a Useful Special Case

The probability of remaining active follows from the total intensity. Over a short interval, the conditional probability of remaining active is $1-\lambda(t)h+o(h)$. Taking the continuous-time limit gives the survival equation $p_A'(t)=-\lambda(t)p_A(t)$, with $p_A(0)=1$. Its solution is

p_A(t)=\exp\!\left(-\int_0^t\lambda(s)\,ds\right) =\exp\!\left(-\int_0^t\sum_j\lambda_j(s)\,ds\right).

For constant cause intensities and a constant force of interest, this reduces to a particularly convenient expression. If $B_j$ is also constant and the valuation term is $T_0$, define $\lambda=\sum_j\lambda_j$. Then

\operatorname{EPV}_j =B_j\lambda_j\int_0^{T_0}e^{-(\delta+\lambda)t}\,dt =B_j\lambda_j\frac{1-e^{-(\delta+\lambda)T_0}}{\delta+\lambda}.

The sum $\delta+\lambda$ appears because discounting contributes $e^{-\delta t}$ while remaining active contributes $e^{-\lambda t}$. If $\delta+\lambda=0$, the integral is $T_0$, obtained as the limit of the displayed quotient. For a finite term, the formula is valid for constant rates and a benefit paid immediately at the relevant exit. For an unlimited term, the same expression has a finite limit when $\delta+\lambda>0$.

Worked Example: Constant Intensities over a Finite Term

A hypothetical plan has two exit causes: withdrawal and death. Assume constant withdrawal intensity $0.03$ per year, constant death intensity $0.02$ per year, and a constant force of interest $\delta=0.05$ per year. The plan pays $5,000 at withdrawal and $12,000 at death, immediately upon exit. Find the combined EPV over 10 years.

Method. Use the constant-intensity formula separately for each cause, then add the unrounded values. The total exit intensity is $\lambda=0.03+0.02=0.05$ per year, so $\delta+\lambda=0.10$ per year. The common integral factor is \[ \frac{1-e^{-0.10(10)}}{0.10} =\frac{1-e^{-1}}{0.10} =10(0.6321205588) =6.321205588. \]

For withdrawal, the benefit-intensity product is $5{,}000(0.03)=150$. Thus \[ \operatorname{EPV}_{w}=150(6.321205588)=948.1808382. \] For death, the benefit-intensity product is $12{,}000(0.02)=240$. Thus \[ \operatorname{EPV}_{d}=240(6.321205588)=1{,}517.089341. \] Adding the unrounded cause values gives $948.1808382+1{,}517.089341=2{,}465.2701792$. The combined EPV is $2,465.27, rounded to the nearest cent.

Check. The total benefit-intensity product is $150+240=390$, so the combined calculation is $390(6.321205588)=2{,}465.270179$, agreeing with the sum of the two cause-specific values. This check also confirms that both causes use the same active-member survival and discount factor.

Common mistake avoided: applying discounting but omitting active-member survival would use $e^{-0.05t}$ in place of $e^{-0.10t}$, producing $390(1-e^{-0.5})/0.05=$3,069.06$. That overstates the value because it treats members who exited earlier as if they could still generate later exits.

Piecewise-Constant Intensities

When cause intensities change at specified times, the same integral applies, but active-member survival must be carried forward from each interval. On a new interval, do not reset the probability of being active to 1. Instead, use the survival accumulated up to the interval’s start, together with discount accumulated to that time. The following example makes that continuation explicit.

Worked Example: Two Intervals with Different Cause Intensities

A hypothetical plan pays $1,000 on withdrawal and $2,000 on death at the instant of exit. The force of interest is $0.10$ per year. From time 0 to year 1, the withdrawal and death intensities are $0.10$ and $0.10$ per year. From year 1 to year 2, they are $0.20$ and $0.10$ per year. Find the combined EPV through year 2.

Method. On the first interval, the total exit intensity is $0.20$, so discounting and active survival together give exponent rate $0.10+0.20=0.30$. The sum of benefit-intensity products is $1{,}000(0.10)+2{,}000(0.10)=300$. The first-interval EPV is therefore \[ 300\int_0^1e^{-0.30t}\,dt =300\frac{1-e^{-0.30}}{0.30} =1{,}000(1-e^{-0.30}) =259.1817793. \]

At year 1, active-member survival is $e^{-0.20}$, and the discount factor is $e^{-0.10}$. Their product is $e^{-0.30}$. In the second interval, the total exit intensity is $0.30$, so the combined exponent rate is $0.10+0.30=0.40$. The benefit-intensity products sum to $1{,}000(0.20)+2{,}000(0.10)=400$. Hence the second-interval EPV is \[ 400e^{-0.30}\int_0^1e^{-0.40u}\,du =400e^{-0.30}\frac{1-e^{-0.40}}{0.40} =1{,}000(e^{-0.30}-e^{-0.70}) =244.2329169. \] Here $u$ measures time elapsed since the start of the second interval.

Adding the unrounded interval values gives $259.1817793+244.2329169=503.4146962$. The combined EPV is $503.41, rounded to the nearest cent. As a check, the expression simplifies to $1{,}000(1-e^{-0.70})=503.4146962$, the same result. The key point is that the second interval retains the survival and discount accumulated in the first year.

Unlimited-Term Valuation

For constant intensities and a constant force of interest, the unlimited-term integral is the limit of the finite-term integral as $T_0$ tends to infinity. If $\delta+\lambda>0$, then $e^{-(\delta+\lambda)T_0}$ tends to zero. For constant benefits $B_j$, this gives

\operatorname{EPV}_j =\frac{B_j\lambda_j}{\delta+\lambda}.

The positive denominator matters: it ensures the discounted probability density has a finite integral over an unlimited term. If $\delta+\lambda=0$, the integral does not converge for a positive constant benefit and positive cause intensity. The following example illustrates the finite case and the importance of including all competing causes in $\lambda$.

Worked Example: Two Exit Causes over an Unlimited Term

A hypothetical plan has constant withdrawal intensity $0.02$ per year and death intensity $0.03$ per year. Its force of interest is $0.01$ per year. It pays $6,000 on withdrawal and $9,000 on death immediately at exit. Find the combined EPV over an unlimited term.

Method. The total exit intensity is $0.02+0.03=0.05$, and the combined denominator is $\delta+\lambda=0.01+0.05=0.06$. The withdrawal value is \[ \frac{6{,}000(0.02)}{0.06} =\frac{120}{0.06} =2{,}000. \] The death value is \[ \frac{9{,}000(0.03)}{0.06} =\frac{270}{0.06} =4{,}500. \] Therefore the combined EPV is $2{,}000+4{,}500=$ $6,500.00.

Check. Combining the benefit-intensity products first gives $(120+270)/0.06=390/0.06=6{,}500$. The denominator includes discounting and the total intensity of both competing exits. Using only the cause-specific intensity in the denominator would incorrectly ignore the possibility of leaving through the other cause.

Common Exam Traps and Interpretation

The cause-specific valuation integral is useful because it separates three distinct ingredients: the probability of reaching time $t$ in active status, the rate of the specified exit at that time, and the present-value adjustment. Keeping those ingredients visible is especially important when rates change over time or when several benefits depend on different causes.

Omitting active-member survival: the factor $p_A(t)$ is required because the cause intensity is conditional on active status. Discounting alone does not account for exits that occurred earlier.
Using the wrong total intensity: survival uses $\sum_j\lambda_j(t)$, not just the intensity of the cause whose benefit is being valued.
Resetting survival at a rate-change date: each later interval starts with the probability of having survived active to that date, not with probability 1.
Changing payment timing without changing the formula: the integral above values a payment at the instant of exit. If payment is made later, the discount factor must reflect that delay.
Rounding components too early: retain unrounded interval and cause values through addition, then round the final EPV to cents.

Key result: For exit by cause $j$, the unconditional probability density at time $t$ is $p_A(t)\lambda_j(t)$. Multiplying this density by the benefit paid at exit and its discount factor, then integrating over the valuation period, gives the cause-specific EPV.

Check Your Understanding

For each problem, assume benefits are paid immediately at exit and use the stated constant or piecewise-constant intensities.

A hypothetical plan has withdrawal intensity $0.04$, death intensity $0.01$, and force of interest $0.05$, all per year. It pays $3,000 on withdrawal and $10,000 on death. Find the combined EPV over five years. (Answer: $865.63)
A hypothetical arrangement has constant intensities $0.03$ for cause A and $0.02$ for cause B, with force of interest $0.05$. It pays $4,000 on cause A and $8,000 on cause B over an unlimited term. Find the combined EPV. (Answer: $2,800.00)
A plan pays $2,000 for cause A and $4,000 for cause B. The force of interest is $0.05$. During year 1, the cause intensities are $0.05$ and $0.05$; during year 2, they are $0.10$ and $0.05$. Find the EPV through year 2. (Answer: $590.62)
For an unlimited term, cause A has constant intensity $0.02$, cause B has constant intensity $0.01$, and the force of interest is $0.02$. Benefits are $5,000 for cause A and $15,000 for cause B. Calculate the cause-A EPV and the combined EPV. (Answer: cause A $2,000.00; combined $5,000.00)

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