Interpreting Associated Probabilities in Benefit Calculations

Why the Meaning of a Probability Matters

A benefit calculation can depend on the probability that a particular decrement occurs, such as death or entry into a specified state. When several decrements can occur, that probability is affected by the other causes: a person who exits for one cause is no longer exposed to the others. An associated probability answers a different question. It describes the probability for one cause on a hypothetical single-decrement basis, with the competing causes removed.

The distinction is consequential. If a contract pays when a particular cause actually occurs in a population subject to competing decrements, the valuation normally needs the actual cause-specific probability. An associated probability is appropriate when the valuation is expressly on the hypothetical basis that the other decrements do not operate, or when the task specifically asks for a single-decrement valuation. It is not a replacement for the actual probability merely because the benefit names one cause.

Definition: In a multiple-decrement model, the actual cause-specific probability for cause $j$ over a period is the probability that an individual starting the period exits from cause $j$ before the period ends, while all competing causes remain in force. The associated single-decrement probability for cause $j$ is the probability of exit from cause $j$ over the period on a specified basis with all competing causes removed. The associated probability therefore depends on the assumed timing or distribution of decrements used to construct that basis.

The associated probability is not an observed cause-specific probability and does not describe what would actually happen in the multiple-decrement population. It is a model-based counterfactual. That makes the basis part of the meaning of the number: an associated probability under UDD need not equal one under constant forces, even when both are constructed from the same multiple-decrement data.

Constant-Force Interpretation

Suppose there are $r$ competing causes during a year, and each cause $k$ has a constant force $\mu_k$ over that year. The total force is $\mu=\sum_{k=1}^{r}\mu_k$. Starting with one individual, the probability of remaining in the initial state for $t$ years is $e^{-\mu t}$. The cause-specific probability over one year is therefore

q_j=\int_0^1 e^{-\mu t}\mu_j\,dt =\frac{\mu_j}{\mu}\left(1-e^{-\mu}\right).

If all other causes are removed while cause $j$'s force remains unchanged, the single-decrement probability becomes

q'_j=1-e^{-\mu_j}.

These formulas explain why the two probabilities differ. In the multiple-decrement model, cause $j$ can occur only while the individual has survived all causes up to that time. On the associated basis, there is no competing force removing the individual first.

Important condition: The constant-force formulas apply when each cause-specific force is constant over the period. Do not use them merely because annual probabilities are given. Annual probabilities alone do not specify the within-year timing assumption needed to derive associated probabilities.

Worked Example: An End-of-Year Benefit for One Cause

A hypothetical one-year contract pays \$60,000 at the end of the year if cause A occurs during the year. Cause A has constant force $0.12$ per year and a competing cause B has constant force $0.18$ per year. The effective annual interest rate is $i=0.04$. Calculate the expected present value (EPV) using (a) the actual multiple-decrement probability and (b) the associated single-decrement probability for A. Interpret the difference.

Method. First calculate the actual probability of cause A while both forces operate. Then calculate the associated probability after removing cause B. Since payment is at the end of the year, discount either expected payment by $v=1/1.04$.

The total force is $0.12+0.18=0.30$. Thus the actual probability of cause A is

q_A=\frac{0.12}{0.30}(1-e^{-0.30}) =0.4(0.2591817793) =0.1036727117.

The associated probability removes cause B but retains the force for A:

q'_A=1-e^{-0.12}=0.1130795633.

The actual-basis EPV is

60{,}000\left(\frac{1}{1.04}\right)(0.1036727117) =\$5{,}981.12.

The associated-basis EPV is

60{,}000\left(\frac{1}{1.04}\right)(0.1130795633) =\$6{,}523.82.

Both amounts are rounded to the nearest cent. The associated-basis amount is larger because it assumes no exits from cause B. For the stated contract in the population where both causes operate, the actual-basis EPV, \$5,981.12, is the relevant valuation. The other figure answers a hypothetical single-decrement question. The common mistake is to use $q'_A$ simply because the benefit is payable for cause A; naming one benefit cause does not remove the competing cause from the model.

Deriving the Constant-Force Formula

The cause-specific formula follows from the probability of being exposed to cause $j$ at each instant. With total force $\mu$, survival in the initial state to time $t$ is $e^{-\mu t}$. During a short interval of length $dt$, the probability of a cause-$j$ exit, conditional on still being in the state, is approximately $\mu_jdt$. Multiplying these gives the unconditional probability contribution $e^{-\mu t}\mu_jdt$. Integrating over the year gives

\int_0^1 e^{-\mu t}\mu_j\,dt =\mu_j\left[-\frac{1}{\mu}e^{-\mu t}\right]_0^1 =\frac{\mu_j}{\mu}(1-e^{-\mu}).

On the single-decrement basis, the total force is just $\mu_j$. The probability of no cause-$j$ exit during the year is $e^{-\mu_j}$, so its complement, $1-e^{-\mu_j}$, is the associated probability. This derivation also shows why timing matters: the exposure term in the integral changes when the competing forces or their timing change.

Matching the Basis to the Benefit Question

Before calculating, identify what event triggers payment and what population or hypothetical basis the question asks you to value. In particular, distinguish these two questions:

Valuation question	Probability basis	Interpretation
What is the expected payment under the actual multiple-decrement model?	Actual cause-specific probability	Competing decrements remain in force.
What is the single-decrement probability or value for cause $j$ with other causes removed?	Associated probability on the stated basis	Only cause $j$ operates in the hypothetical model.
What is the value of a benefit defined using a specified associated table?	The specified associated basis	Use the table’s assumptions, including its timing convention.

The probability basis must also agree with the payment timing. An end-of-year payment is discounted by $v$ for one year, whereas a payment made at the moment of decrement requires discounting to the time of exit. Even if the benefit amount and cause are unchanged, these payment definitions produce different present values. The associated-versus-actual choice concerns which probability model is being valued; it does not change the contract’s payment timing.

Worked Example: A Benefit Paid at the Time of Exit

A hypothetical contract pays \$60,000 immediately when cause A occurs during the year. Cause A has constant force $0.12$, cause B has constant force $0.18$, and the force of interest is $\delta=0.04$ per year. Calculate the EPV under the actual multiple-decrement model, then compare it with the associated single-decrement EPV for A.

Method. Integrate the discounted cause-A exit density. At time $t$, survival from both decrements is $e^{-0.30t}$, the cause-A exit density is $0.12e^{-0.30t}$, and the discount factor is $e^{-0.04t}$.

\begin{aligned} \operatorname{EPV}_{\mathrm{actual}} &=60{,}000\int_0^1 e^{-0.04t}(0.12e^{-0.30t})\,dt\\ &=60{,}000\frac{0.12}{0.34}(1-e^{-0.34})\\ &=60{,}000\frac{0.12}{0.34}(0.2882296656)\\ &=\$6{,}103.69. \end{aligned}

For the associated basis, cause B is removed. The discounted cause-A exit density is then $0.12e^{-(0.12+0.04)t}$. Hence

\begin{aligned} \operatorname{EPV}_{\mathrm{associated}} &=60{,}000\int_0^1 0.12e^{-0.16t}\,dt\\ &=60{,}000\frac{0.12}{0.16}(1-e^{-0.16})\\ &=60{,}000(0.75)(0.1478562110)\\ &=\$6{,}653.53. \end{aligned}

Both values are rounded to the nearest cent. The actual value includes the possibility that cause B occurs first; the associated value does not. This example also avoids a timing error: using $v$ as though payment were always at year-end would not value an immediate-at-exit benefit.

UDD Requires a UDD-Associated Basis

Under a UDD basis, decrements are treated as uniformly distributed within the year for the relevant conversion. For two causes A and B, if the actual annual multiple-decrement probabilities are $q_A$ and $q_B$, the associated probability for A under UDD is

q'_A=\frac{q_A}{1-\tfrac12 q_B}.

The denominator reflects the assumed average exposure to cause A while cause B is competing uniformly through the year. With more than two causes, replace $q_B$ in the denominator by the sum of the other causes’ actual annual probabilities. This is a UDD conversion, not a universal identity: a constant-force associated probability is calculated using the force-based model instead. The timing assumption used to construct the associated table must match the timing assumption used in the benefit calculation.

Worked Example: Applying the UDD Associated Probability

A hypothetical one-year benefit pays \$100,000 at year-end if cause A occurs. The actual multiple-decrement probabilities are $q_A=0.08$ and $q_B=0.12$. Use a UDD associated basis to calculate the associated probability for A and the EPV on both the actual and associated bases. The effective annual interest rate is $i=0.05$.

Method. Apply the UDD conversion to the actual annual probabilities, then value the end-of-year benefit using the probability appropriate to each basis.

q'_A=\frac{0.08}{1-\tfrac12(0.12)} =\frac{0.08}{0.94} =0.0851063830.

The actual-basis EPV uses $q_A=0.08$:

100{,}000\left(\frac{1}{1.05}\right)(0.08) =\$7{,}619.05.

The associated-basis EPV uses $q'_A=0.0851063830$:

100{,}000\left(\frac{1}{1.05}\right)(0.0851063830) =\$8{,}105.37.

Amounts are rounded to the nearest cent. The actual EPV answers the contract question with both decrements operating. The associated EPV answers the single-decrement question under UDD. A common error is to convert $q_A$ under UDD and then treat the result as the actual probability in the multiple-decrement population. That substitutes the counterfactual probability for the observed cause-specific one.

Exam Traps and a Reliable Decision Process

Associated probabilities are useful because they put one cause on a single-decrement basis, allowing a valuation or comparison that deliberately excludes competing causes. Their usefulness depends on keeping their interpretation intact. A value calculated under UDD should not be silently combined with a constant-force timing model, and an associated probability should not be substituted into a multiple-decrement claim calculation without a reason in the problem statement.

Identify the event.
State which cause triggers the benefit and whether payment is at the end of the period or at the instant of decrement.

Identify the model.
Determine whether competing causes remain in force, and note the assumed within-period timing basis, such as constant forces or UDD.

Select the probability.
Use the actual cause-specific probability for an actual multiple-decrement benefit; use the associated probability only for the specified single-decrement basis.

Apply payment timing and interest.
Discount an end-of-year payment for a full year; for a payment at exit, integrate the discounted exit density or use an equivalent calculation.

Check the interpretation.
Confirm that the result answers the stated valuation question, not merely that the arithmetic is consistent.

A useful check is that, with positive competing forces, the associated probability for a cause will generally exceed its actual multiple-decrement probability over the same period: removing competing exits leaves more exposure to the cause of interest. This is a diagnostic, not a substitute for checking the model assumptions. If competing causes have zero probability, the two probabilities for the remaining cause coincide on a consistent basis.

The central skill is interpretation before calculation. A cause-specific benefit does not by itself imply an associated basis. Decide whether the task concerns actual experience with competition or a hypothetical single-decrement valuation, preserve the stated timing assumption, and then apply the matching probability and discounting method. This distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, decide which basis is requested and keep the timing assumption consistent.

Under constant forces $\mu_A=0.08$ and $\mu_B=0.12$, a hypothetical policy pays \$40,000 at year-end if A occurs. The effective annual interest rate is $i=0.04$. What is the actual multiple-decrement EPV? (Answer: \$2,788.76)
Under UDD, the actual annual probabilities are $q_A=0.06$ and $q_B=0.14$. What is the associated single-decrement probability for A? Give the answer to four decimals. (Answer: 0.0645)
Under constant forces $\mu_A=0.15$ and $\mu_B=0.05$, what are the actual and associated probabilities for A over one year? Give each to four decimals. (Answer: Actual 0.1360; associated 0.1393)
A hypothetical contract pays \$25,000 immediately at a cause-A exit. The constant forces are $\mu_A=0.10$ and $\mu_B=0.10$, and the force of interest is $\delta=0.03$. What are the actual and associated EPVs, rounded to cents? (Answer: Actual \$2,233.33; associated \$2,344.32)
Under UDD, actual annual probabilities are $q_A=0.09$ and $q_B=0.11$. A hypothetical benefit of \$50,000 is paid at year-end on cause A, with $i=0.04$. Calculate the actual and associated EPVs, rounded to cents. (Answer: Actual \$4,326.92; associated \$4,578.75)

What You'll Learn