Diagnosing Competing-Risk Ancillary Valuations

Why Ancillary Valuations Need an Audit

A valuation can use correct discounting and still be wrong if it leaves out an exit cause, pays a benefit to someone who was not eligible, or counts two mutually exclusive options together. In a competing-risk model, a member can leave through several causes, but can experience only the first exit while covered. The probabilities, eligibility rules, and payment dates must therefore describe the same model.

This tutorial develops a practical audit for hypothetical pension and ancillary-benefit valuations. The aim is not to replace a full valuation: it is to check that its event structure and cash flows are internally consistent before relying on the result.

Definition: In a competing-risk model, a member remains in force until the first of several mutually exclusive exit causes occurs. A cause-specific intensity is the instantaneous rate of exit from that cause, conditional on being in force immediately before the time considered. If the causes are indexed by $j$, with constant intensities $\lambda_j$, the total intensity is $\Lambda=\sum_j\lambda_j$. The probability of remaining in force to time $t$ is $e^{-\Lambda t}$.

For constant intensities, the probability density of exit from cause $j$ at time $t$ is $\lambda_j e^{-\Lambda t}$. If a benefit $B_j$ is paid immediately on an exit from cause $j$, and the force of interest is constant at $\delta$, its expected present value over a term $T$ is the integral of the discounted benefit times that cause-specific exit density. This applies when the causes are mutually exclusive, the stated intensities apply throughout the term, and payment really is made at exit.

\operatorname{EPV}_j = B_j\int_0^T e^{-\delta t}\lambda_j e^{-\Lambda t}\,dt = B_j\frac{\lambda_j}{\delta+\Lambda} \left(1-e^{-(\delta+\Lambda)T}\right).

The associated exit probability is obtained by setting the discount factor to 1. The cause-specific probabilities plus the probability of still being in force must add to 1. That identity is a basic audit check, not an optional approximation.

P(\text{exit from cause }j\text{ by }T) = \frac{\lambda_j}{\Lambda}\left(1-e^{-\Lambda T}\right), \qquad P(\text{in force at }T)=e^{-\Lambda T}.

A Short Derivation and a Practical Audit

To see why the present-value formula works, consider a short interval from $t$ to $t+dt$. The probability of remaining in force to $t$ is $e^{-\Lambda t}$. Conditional on being in force then, the probability of a cause-$j$ exit in the short interval is approximately $\lambda_jdt$. The benefit is discounted to time 0 by $e^{-\delta t}$. Multiplying these terms gives the contribution to expected present value in that interval; integrating over the term gives the formula.

e^{-\delta t}\cdot e^{-\Lambda t}\cdot \lambda_jdt\cdot B_j = B_j\lambda_j e^{-(\delta+\Lambda)t}dt.

For a valuation with eligibility conditions or options, the integral must also reflect whether the member is eligible at the exit time and which benefit is payable for that event. A useful audit proceeds in a fixed order.

List the possible first exits.
Include every cause that removes the member from the covered population during the term, even if some causes do not generate the benefit being valued.

Reconcile the event probabilities.
Check that the cause probabilities and the probability of remaining in force sum to 1. For annual conditional probabilities, multiply later-year probabilities by the probability of reaching that year.

Apply eligibility to the right event.
Identify the service, age, or waiting-period condition and whether it applies to each benefit separately. An exit before eligibility may still qualify for a different benefit.

Specify payment timing and options.
State whether payment is immediate, at the end of the year, or later. For a choice between alternatives, include the selected alternative rather than summing both.

Recalculate and check.
Discount each eligible cash flow from its actual payment date, then compare the result with probability, boundary, and alternative-benefit checks.

Worked Valuation with All Exit Causes

Worked Example: Finding an Omitted Competing Exit

A hypothetical plan member is covered for 2 years. The constant annual intensities are $\lambda_w=0.10$ for withdrawal, $\lambda_d=0.05$ for death, and $\lambda_r=0.05$ for retirement. A benefit of $8,000 is payable on withdrawal, $20,000 on death, and $12,000 on retirement, each paid immediately at exit. Use a constant force of interest $\delta=0.05$. Find the total expected present value and check the probabilities.

Method. Treat the three events as competing exits. Their total intensity is $\Lambda=0.10+0.05+0.05=0.20$. The discounting and survival factor in each benefit integral is therefore $e^{-0.25t}$, since $\delta+\Lambda=0.25$.

Probability check. The probability of any exit by year 2 is $1-e^{-0.20(2)}=1-e^{-0.4}=0.329680$. The cause probabilities are this total multiplied by each cause's share of total intensity: withdrawal $0.329680(0.10/0.20)=0.164840$; death $0.329680(0.05/0.20)=0.082420$; retirement $0.329680(0.05/0.20)=0.082420$. The probability of remaining in force is $e^{-0.4}=0.670320$. The check is $0.164840+0.082420+0.082420+0.670320=1.000000$.

Present value for withdrawal. The common integral is $(1-e^{-0.25(2)})/0.25=(1-e^{-0.5})/0.25=1.573877$. Thus the withdrawal benefit has expected present value $$8,000(0.10)(1.573877)=$1,259.10$.

Present value for death. The expected present value is $$20,000(0.05)(1.573877)=$1,573.88$.

Present value for retirement. The expected present value is $$12,000(0.05)(1.573877)=$944.33$.

Adding the three cause-specific values gives $$1,259.10+$1,573.88+$944.33=$3,777.31$, rounded to the nearest cent. The total is consistent with the reconciled exit probabilities and includes all three exits, although only one can occur first.

Common mistake avoided. Valuing each cause as if it were the only exit would give its own survival probability and then add the separate results. For example, the single-cause withdrawal probability over 2 years would be $1-e^{-0.10(2)}=0.181269$, not the correct competing-risk probability $0.164840$. The other causes reduce the time available for withdrawal to occur; all cause calculations must use the same total intensity.

Eligibility and Payment Timing

Eligibility must be attached to the benefit rule, not applied indiscriminately to the entire member record. A withdrawal benefit with a one-year service threshold can be payable only for withdrawals after that threshold, while a separate death benefit may apply from the start of coverage. In continuous time, a single boundary time has probability zero under continuous intensities, but the threshold still changes the range of integration.

Worked Example: Applying a Service Threshold to One Benefit

A hypothetical member is covered for 2 years. Withdrawal and death intensities are both $0.10$ per year, and the force of interest is $\delta=0.05$. A $10,000 withdrawal benefit is payable only if withdrawal occurs at or after 1 year of service. A $6,000 death benefit is payable on death at any time during coverage. Both benefits are paid immediately at the qualifying exit. Find the expected present value of the package.

Method. The total exit intensity is $\Lambda=0.10+0.10=0.20$, so $\delta+\Lambda=0.25$. The withdrawal benefit is eligible only for exit times from 1 to 2. Its value is $$10,000(0.10)(e^{-0.25}-e^{-0.50})/0.25$. Using $e^{-0.25}=0.778801$ and $e^{-0.50}=0.606531$, this is $$4,000(0.172270)=$689.08$.

The death benefit applies throughout the term. Its expected present value is $$6,000(0.10)(1-e^{-0.50})/0.25 =$2,400(0.393469) =$944.33$. The package value is $$689.08+$944.33=$1,633.41$, rounded to the nearest cent.

Eligibility check. The probability of a withdrawal after the service threshold and by year 2 is $0.10(e^{-0.20}-e^{-0.40})/0.20=0.074205$. This is a probability, not the discounted benefit value; the latter also uses the discount force and the benefit amount.

Common mistake avoided. If the threshold is ignored, the withdrawal benefit would be valued over the full 2 years as $$10,000(0.10)(1-e^{-0.50})/0.25=$1,573.88$. That overstates its value by $$1,573.88-$689.08=$884.80$. Conversely, applying the threshold to the death benefit would wrongly remove valid death claims in the first year.

Payment timing is another independent part of the audit. An immediate payment at exit is discounted using the actual exit time. If a contract instead settles claims at the end of the year of exit, the payment is later; it cannot be valued using the immediate-payment formula without changing the discounting. Likewise, an annual probability given for an exit during a year does not by itself say whether the benefit is paid at the start, midpoint, or end of that year. The stated payment convention must determine the discount factor.

Reconciling Annual Probabilities and Exclusive Options

In an annual model, suppose $q_j$ is the probability of cause $j$ during a year, conditional on being in force at the start of that year. If $p$ is the conditional probability of remaining in force through the year, then $\sum_j q_j+p=1$. These are conditional one-year probabilities, not probabilities from the original valuation date for every year.

Worked Example: Auditing Two-Year Probabilities and Avoiding Double Counting

A hypothetical plan has annual exits at year-end. In year 1, conditional on being in force at the start of the year, withdrawal, death, and retirement probabilities are $0.10$, $0.05$, and $0.05$. In year 2, the corresponding conditional probabilities are $0.08$, $0.04$, and $0.08$. Each year's remaining-in-force probability is the balance. Benefits paid at the end of the year of exit are $5,000 on withdrawal and $20,000 on death. At retirement, the member chooses either a pension with value $12,000 or a cash alternative with value $9,000. Assume the member chooses the pension. Use $i=0.05$.

Method. First reconcile each year's probabilities. Year 1 exit probabilities sum to $0.10+0.05+0.05=0.20$, leaving $0.80$ in force. Year 2 probabilities sum to $0.08+0.04+0.08=0.20$, also leaving $0.80$ in force conditional on reaching year 2.

The unconditional year-2 probabilities must be multiplied by the probability $0.80$ of reaching that year: withdrawal $0.80(0.08)=0.064$, death $0.80(0.04)=0.032$, and retirement $0.80(0.08)=0.064$. The probability of surviving both years in force is $0.80(0.80)=0.64$. Across the full term, withdrawal has probability $0.10+0.064=0.164$, death $0.05+0.032=0.082$, and retirement $0.05+0.064=0.114$. These total $0.360$, and $0.360+0.640=1.000$.

The expected year-1 payment, before discounting, is $0.10($5,000)+0.05($20,000)+0.05($12,000)=$2,100$. Its present value is $$2,100/1.05=$2,000.00$.

Conditional on reaching year 2, the expected payment is $0.08($5,000)+0.04($20,000)+0.08($12,000)=$2,160$. The unconditional expected year-2 payment is $0.80($2,160)=$1,728$. Its present value is $$1,728/(1.05)^2=$1,567.35$, rounded to the nearest cent. The total expected present value is $$2,000.00+$1,567.35=$3,567.35$.

Common mistake avoided. Adding the $9,000 cash option to the $12,000 pension would count two alternatives for the same retirement event. The extra, incorrect present value would be $0.05($9,000)/1.05+0.064($9,000)/(1.05)^2=$951.02$. The result would then be $4,518.37, rather than the correct $3,567.35. If the member's election were uncertain, the valuation would need an explicit election rule or probabilities for the alternatives; simply adding both is not justified.

Common Diagnostic Traps

Audit issue	Incorrect approach	Correction
Omitted exit cause	Use only the intensity or probability for the benefit-producing cause to model survival.	Include every cause that can remove the member from coverage in the total exit probability or intensity.
Inconsistent eligibility	Apply a service threshold to all benefits, or ignore it for one benefit.	State eligibility separately for each benefit and restrict its valuation to qualifying events.
Incorrect payment date	Discount an annual exit benefit to a date that does not match the contract's payment convention.	Use the actual payment date, distinguishing it from the exit date when needed.
Overlapping options	Add pension and cash values even though the member chooses one.	Model the election, or value only the specified selected alternative.
Unreconciled probabilities	Add conditional year-2 probabilities directly to year-1 probabilities.	Multiply each later-year conditional probability by the probability of reaching that year, then check that all outcomes sum to 1.

Key takeaway: A sound ancillary-benefit valuation must agree on four things: which first exit occurs, whether that exit qualifies for the benefit, what cash flow is paid, and when it is paid. Reconcile the probabilities before discounting, and distinguish mutually exclusive alternatives from benefits that genuinely accumulate.

Check Your Understanding

For each hypothetical case, identify the relevant probability or present value and check the event or payment convention.

A one-year continuous competing-risk model has constant intensities $0.06$ for withdrawal and $0.04$ for death. Find the probability of each exit cause by year 1 and the probability of remaining in force. (Answer: withdrawal $0.0571$, death $0.0381$, remaining in force $0.9048$; they sum to $1.0000$.)
For a one-year annual model, conditional probabilities are $0.07$ for withdrawal, $0.03$ for death, and $0.10$ for retirement. What is the probability of remaining in force, and do the probabilities reconcile? (Answer: $0.80$; $0.07+0.03+0.10+0.80=1.00$.)
A hypothetical plan has withdrawal intensity $0.08$, death intensity $0.12$, and force of interest $\delta=0.04$, for a 1.5-year term. A $6,000 withdrawal benefit is available only for exits from time 0.5 onward; a $9,000 death benefit is available throughout. Both are paid immediately at exit. Find the package expected present value. (Answer: withdrawal $378.49, death $1,360.46, total $1,738.94.)
In a two-year annual model, year-1 conditional probabilities are $0.10$ for cause A and $0.10$ for cause B. In year 2, conditional on reaching it, they are $0.05$ and $0.15$. Find the unconditional probabilities of each cause over the two years and the probability of remaining in force throughout. (Answer: cause A $0.14$, cause B $0.22$, remaining in force $0.64$; the outcomes sum to $1.00$.)
At a one-year retirement event, probability $0.06$, a member chooses either an $18,000 benefit or a $14,000 alternative, but not both. If the first alternative is selected and $i=0.05$, find its expected present value. By how much would adding both alternatives overstate that value? (Answer: expected present value $1,028.57; adding the alternative would overstate it by $800.00.)

What You'll Learn