Why Within-Year Timing Matters
Annual decrement probabilities specify the chance that an active member exits during a year, and by which cause. They do not, by themselves, specify when during the year an exit occurs. That distinction matters when a pension benefit is paid at the exit date, changes with service or time, or is discounted from the exit date. This tutorial compares two ways to fill in the within-year timing: the uniform distribution of decrements (UDD) and the constant-force assumption.
The comparison is conditional on a member being active at the start of the year. Write \(q_j\) for the probability of exit during the year from cause \(j\), \(Q=\sum_j q_j\) for the total probability of exit, and \(p=1-Q\) for the probability of remaining active to year-end. For a benefit paid at the time of exit, let \(B_j(t)\) be its amount if cause \(j\) occurs at time \(t\), where \(0\le t\le1\). If the effective annual interest rate is \(i\), discounting a payment at time \(t\) uses \(v^t=(1+i)^{-t}=e^{-\delta t}\), where \(\delta=\ln(1+i)\).
Cause-Specific Timing and Payment Values
Under UDD, the probability of cause \(j\) occurring in a small interval from \(t\) to \(t+dt\) is \(q_j\,dt\). The total probability of still being active at time \(t\) is \(1-tQ\). These statements are consistent because the cause-specific probabilities by time \(t\) sum to \(tQ\), leaving probability \(1-tQ\) active. Under the constant-force assumption, the probability of remaining active to time \(t\) is \(e^{-\lambda t}\), and the cause-\(j\) exit density is \(\lambda_j e^{-\lambda t}\).
Consequently, the expected present value (EPV) of a benefit paid immediately at exit is
These expressions assume that the stated benefit is payable only if the corresponding exit occurs during this year, and that the member is active at the start of the year. They also assume the specified within-year decrement pattern and a constant force of interest. A payment due at the end of the year regardless of the exact exit time must instead be discounted for one full year, not for the time of exit.
Calculate \(Q=\sum_jq_j\) and \(p=1-Q\). These are annual multiple-decrement probabilities, not continuous forces.
For UDD, use cause-\(j\) density \(q_j\). For constant forces, calculate \(\lambda=-\ln p\) and \(\lambda_j=(q_j/Q)\lambda\).
For payment at exit, use \(B_j(t)e^{-\delta t}\) inside the timing integral. For payment at year-end, use its full-year discount factor \(v\).
Under constant forces, \(\int_0^1\lambda_j e^{-\lambda t}\,dt=(\lambda_j/\lambda)(1-p)=q_j\). Under UDD, \(\int_0^1q_j\,dt=q_j\).
Deriving the Constant-Force Formula
The constant-force cause-\(j\) density follows from the competing-risk model. At time \(t\), a member must first have survived all causes, with probability \(e^{-\lambda t}\). Conditional on being active at that time, cause \(j\) acts over an infinitesimal interval \(dt\) with probability approximately \(\lambda_jdt\). Multiplying gives the unconditional probability of a cause-\(j\) exit in that interval, \(\lambda_j e^{-\lambda t}dt\). Discounting a payment at that time multiplies its amount by \(e^{-\delta t}\), yielding the constant-force EPV integral above.
To verify that this assumption reproduces the specified annual cause probability, integrate the density:
Here \(e^{-\lambda}=p\) by the definition \(\lambda=-\ln p\), and \(1-p=Q\). Thus the force allocation is not an additional estimate of the annual probabilities: it is the allocation that makes constant forces agree with them. The formula requires \(0<p<1\) and \(Q>0\); if \(Q=0\), there are no exits to value in that year.
Worked Example: A Fixed Benefit Paid at Exit
A hypothetical pension plan has annual exit probability \(q=0.20\) for a specified exit cause, with no other causes. It pays $6,400 immediately on that exit. The effective annual interest rate is \(i=0.05\). Compare the EPV under UDD and constant force. Round money to the nearest cent.
Method. Use the exit-time payment formula under each timing assumption. Since there is one cause, \(Q=0.20\), \(p=0.80\), and the constant force is \(\lambda=-\ln(0.80)=0.2231435513\). The force of interest is \(\delta=\ln(1.05)=0.0487901642\).
UDD. The cause-specific exit density is \(0.20\), so
Since \(e^{-\delta}=1/1.05\), the integral is \((1-1/1.05)/\ln(1.05)\approx0.9759969\). Therefore the EPV is \(1{,}280(0.9759969)\approx\$1{,}249.28\).
Constant force. The exit density is \(\lambda e^{-\lambda t}\), so
Using \(\lambda=0.2231435513\) and \(\delta=0.0487901642\), the timing factor is approximately \(0.19537635\), or \(0.195376\) to six decimal places. The EPV is \(6{,}400(0.19537635)\approx\$1{,}250.41\), rounded to the nearest cent.
The constant-force value is about $1.13 higher. The check is that both models give the same undiscounted probability of exit, \(0.20\); the difference arises only because their exit-time distributions produce different amounts of discounting. A common mistake is to treat \(q=0.20\) as the force in the constant-force integral. The correct force is \(-\ln(0.80)\), not \(0.20\).
Benefits That Change With Exit Time
The same timing distinction applies when the benefit itself depends on the exit date. For example, an exit refund might include a service-related amount that increases during the year. Under UDD, the conditional exit time, given a single decrement during the year, is uniform on \([0,1]\). Under constant force, exits are more concentrated earlier in the year because fewer members remain exposed at later times. A benefit that increases with \(t\) can therefore have a different undiscounted expected amount even when interest is zero.
Worked Example: A Service-Linked Exit Refund Under Competing Causes
A hypothetical plan has two competing exits during a year: withdrawal with \(q_1=0.10\) and death with \(q_2=0.10\). Thus \(Q=0.20\), \(p=0.80\), and there is no interest during this example. A refund is payable only on withdrawal, in the amount \(B_1(t)=\$6{,}000+\$2{,}000t\), where \(t\) is the fraction of the year elapsed. Compare the EPV under UDD and constant forces. Round to the nearest cent.
Method. Integrate the withdrawal benefit against the withdrawal exit density. The total force is \(\lambda=-\ln(0.80)=0.2231435513\). Because the two annual cause probabilities are equal, the withdrawal force is \(\lambda_1=(0.10/0.20)\lambda=0.1115717757\).
UDD. The withdrawal density is the constant \(q_1=0.10\). Therefore
Constant force. The withdrawal density is \(\lambda_1e^{-\lambda t}\). The two required integrals are
The second identity follows by integration by parts: an antiderivative of \(e^{-\lambda t}\) is \(-e^{-\lambda t}/\lambda\), giving \(\int_0^1te^{-\lambda t}dt=-p/\lambda+(1-p)/\lambda^2\), which is the stated expression. Substitution gives
The constant-force EPV is lower than the UDD value because the increasing refund is less likely to be paid late in the year under the constant-force model: the withdrawal density declines as exposure to exit diminishes. The calculation also checks against the annual withdrawal probability: \(\int_0^1\lambda_1e^{-\lambda t}dt=0.10\). A common mistake is to use the total force \(\lambda\) in place of the withdrawal force \(\lambda_1\); that would value a withdrawal benefit as though it were payable on both causes.
When the Timing Assumptions Give the Same Value
Not every pension cash flow depends on the exact exit time. If an amount is payable at the end of the year whenever a specified cause occurs during that year, its present value is based on the annual probability of that cause and a full year of discounting. Since UDD and constant forces are calibrated to the same annual cause probabilities, they give the same value for this particular payment pattern.
Worked Example: Benefits Payable at Year-End
A hypothetical pension arrangement has annual withdrawal and death probabilities of \(0.12\) and \(0.08\), respectively. It pays $3,000 at year-end if withdrawal occurs during the year, or $6,000 at year-end if death occurs. The effective annual interest rate is \(5\%\). Compare the EPV under UDD and constant forces. Round money to the nearest cent.
Method. Because each benefit is paid at year-end rather than at exit, its discount factor is \(v=1/1.05\), regardless of when the exit occurred.
Under either assumption, the expected year-end payment is \(0.12(3{,}000)+0.08(6{,}000)=360+480=\$840\). Discounting for one year gives
The agreement is a useful check: only the annual probabilities and year-end payment amounts enter this valuation. It would be incorrect to discount the benefits to their respective exit dates, because the problem specifies that payment occurs at year-end.
Common Traps and Practical Checks
The comparison is most useful when the benefit timing is stated precisely. In pension valuation, “payable on withdrawal” and “payable at the end of the year of withdrawal” describe different cash flows. The first is discounted from the actual exit time; the second is discounted for a full year. Likewise, a service-linked refund needs the benefit amount evaluated at the service attained at exit, not automatically at the beginning or end of the year.
| Cash-flow feature | UDD treatment | Constant-force treatment |
|---|---|---|
| Benefit paid at exit | Integrate using cause density \(q_j\) | Integrate using cause density \(\lambda_j e^{-\lambda t}\) |
| Benefit amount varies with exit time | Include \(B_j(t)\) inside the integral | Include \(B_j(t)\) inside the integral |
| Fixed benefit paid at year-end after a cause occurs | Use annual cause probability and \(v\) | Use the same annual cause probability and \(v\) |
- Do not confuse \(q_j\) and \(\lambda_j\). \(q_j\) is an annual probability; \(\lambda_j\) is a force. In general, they are not equal.
- Do not discount to the wrong date. A payment at exit uses \(e^{-\delta t}\); a payment at year-end uses \(v\).
- Do not use the total force for a cause-specific benefit. The benefit density must use that cause’s force \(\lambda_j\).
- Check the annual calibration. Integrating each assumed cause density over the year must return its given annual probability.
Check Your Understanding
For each problem, identify the payment timing before calculating. Money answers are rounded to the nearest cent where applicable.
- A hypothetical plan has one decrement cause with annual probability \(0.08\). It pays $15,000 immediately at exit, and the effective annual interest rate is zero. Under UDD, calculate the EPV and state the cause-specific exit density used. (Answer: density \(0.08\); EPV $1,200.00)
- A plan has annual withdrawal and death probabilities of \(0.10\) and \(0.06\). It pays $4,000 at year-end after withdrawal or $9,000 at year-end after death. At an effective annual interest rate of \(5\%\), calculate the EPV under UDD and constant forces. (Answer: $895.24 under either assumption)
- Under UDD, a hypothetical cause has annual probability \(0.10\), while the total probability of all competing decrements is \(0.20\). With zero interest, a $5,000 refund for this cause increases by $1,000 times the fraction of the year elapsed at exit. Calculate its EPV. (Answer: $550.00)
- Use the same probabilities and benefit as in Question 3, but assume constant forces. Find the total force, the force for the refund-triggering cause, and the EPV. Use \(\lambda=-\ln(0.80)\). (Answer: \(\lambda=0.2231436\), \(\lambda_1=0.1115718\), EPV $548.14)
- For total annual decrement probability \(Q=0.20\), compare the probability of remaining active to the midpoint of the year under UDD and constant forces. (Answer: UDD \(0.9000\); constant force \(e^{-[-\ln(0.80)]/2}=0.8944\))