Central Rates and Probabilities in Decrement Tables

Central Rates: Counts Relative to Exposure

A decrement probability measures the proportion of a starting group that exits during an interval. A central rate instead compares the number of exits with the time spent at risk during that interval. These quantities are related, but they are not interchangeable: converting one to the other requires an assumption about when exits occur within the interval.

Consider a group observed over one year, with $l_x$ lives at the start. Let $d_x^{(j)}$ be the number exiting because of cause $j$, and let $E_x$ be the group’s total exposure to risk, measured in person-years. The exposure counts each life’s time in the group until exit or the end of the year. The same total exposure is used for each cause when calculating cause-specific central rates.

Definition: The cause-specific central rate for cause $j$ is $m_x^{(j)}=d_x^{(j)}/E_x$. The total number of exits is $d_x=\sum_j d_x^{(j)}$, and the total central rate is $m_x=\sum_j m_x^{(j)}=d_x/E_x$. The cause-specific probability $q_x^{(j)}$ is the probability that a life present at age $x$ exits from cause $j$ during the year. The total exit probability is $q_x=\sum_j q_x^{(j)}=d_x/l_x$ for a decrement table.

A central rate has units of exits per person-year; a probability is dimensionless and refers to a specified interval. In particular, $m_x^{(j)}$ need not be less than 1, while $q_x^{(j)}$ must lie between 0 and 1. Both the numerator and denominator in a central rate must refer to the same group, causes, time interval, and exposure units.

Converting Rates Under a Uniform Distribution of Decrements

Under the uniform distribution of decrements (UDD) assumption, exits are spread uniformly through the year. For the table lives in a one-year interval, this means that the number remaining in the group declines linearly from $l_x$ at the start to $l_x-d_x$ at the end. Its area over the year, which is the exposure, is the average of those two endpoint counts.

E_x=\frac{l_x+(l_x-d_x)}{2}=l_x-\frac{d_x}{2}

Dividing by $l_x$, the exposure per starting life is $1-q_x/2$, since $q_x=d_x/l_x$. The cause-specific central rate is therefore $m_x^{(j)}=q_x^{(j)}/(1-q_x/2)$. Solving this relationship for each cause gives the UDD conversion.

Theorem: For a one-year multiple-decrement table under UDD, with $m_x=\sum_j m_x^{(j)}$, $$ q_x=\frac{m_x}{1+m_x/2}, \qquad q_x^{(j)}=\frac{m_x^{(j)}}{1+m_x/2}. $$ These conversions require a common interval and common exposure denominator for all causes.

To see why the denominator involves the total rate, note that lives exiting from any cause reduce the exposure available to every cause. Algebraically, $m_x=q_x/(1-q_x/2)$, so $m_x(1-q_x/2)=q_x$, or $m_x=q_x(1+m_x/2)$. Thus $q_x=m_x/(1+m_x/2)$. Since all cause-specific rates share the same exposure factor, $m_x^{(j)}=q_x^{(j)}(1+m_x/2)$, which yields the stated formula for each cause.

Worked Example: Recover Cause Probabilities Under UDD

A hypothetical decrement table starts the year with 1,000 lives. During the year, 80 exit for cause A, 40 for cause B, and 30 for cause C. Assume exits are uniformly distributed through the year. Calculate the exposure, each central rate, and the cause-specific probabilities. Check the result against the table counts.

Method. First calculate total exits and UDD exposure. Then divide each cause count by the common exposure, and use the total central rate in the UDD conversion.

Total exits are $d_x=80+40+30=150$. Hence the exposure is $E_x=1{,}000-150/2=925$ person-years.

The central rates are $m_x^{(A)}=80/925=0.086486$, $m_x^{(B)}=40/925=0.043243$, and $m_x^{(C)}=30/925=0.032432$. Their sum is $m_x=150/925=0.162162$, so the conversion denominator is $1+m_x/2=1+0.162162/2=1.081081$.

The cause probabilities are therefore $q_x^{(A)}=0.086486/1.081081=0.0800$, $q_x^{(B)}=0.043243/1.081081=0.0400$, and $q_x^{(C)}=0.032432/1.081081=0.0300$. The total is $q_x=0.0800+0.0400+0.0300=0.1500$.

Check. The table counts imply probabilities $80/1{,}000=0.0800$, $40/1{,}000=0.0400$, and $30/1{,}000=0.0300$, exactly as recovered. The calculation uses unrounded rates until the displayed results. A common mistake is to use a separate denominator involving each cause’s rate; the UDD exposure is reduced by exits from all causes.

Converting Rates Under Constant Forces

A different assumption is that each cause has a constant force of decrement throughout the year. Write that force for cause $j$ as $\mu_x^{(j)}$, and let $M_x=\sum_j\mu_x^{(j)}$ be the total force. With constant cause forces, the probability of remaining in the group for $t$ years is $e^{-M_xt}$. The exposure per starting life over one year is the integral of this probability from 0 to 1.

\frac{E_x}{l_x}=\int_0^1 e^{-M_xt}\,dt =\frac{1-e^{-M_x}}{M_x}

The cause-$j$ exit probability is the integral of the probability of still being in the group at time $t$, multiplied by the cause-$j$ force. Thus $q_x^{(j)}=\mu_x^{(j)}(1-e^{-M_x})/M_x$. Dividing cause-$j$ exits by exposure shows that its central rate equals its constant force. Consequently, when converting central rates under this assumption, set $M_x=\sum_jm_x^{(j)}$.

Theorem: Under constant cause-specific forces over a one-year interval, $m_x^{(j)}=\mu_x^{(j)}$, and $$ q_x^{(j)}=\frac{m_x^{(j)}}{m_x}\left(1-e^{-m_x}\right), \qquad q_x=1-e^{-m_x}, $$ where $m_x=\sum_jm_x^{(j)}>0$. If $m_x=0$, all cause probabilities are zero.

Worked Example: Convert Constant Central Rates to Probabilities

A hypothetical population has constant cause-specific central rates of $0.08$ per year for cause A and $0.12$ per year for cause B. Assume constant forces over the year. Find each cause probability and the total probability. Also verify the rates using exposure for 10,000 starting lives.

Method. Under constant forces, the central rates equal the forces. The total force is $M_x=0.08+0.12=0.20$ per year. The one-year total exit probability is $1-e^{-0.20}=0.181269$.

The cause shares of the total force are $0.08/0.20=0.4$ and $0.12/0.20=0.6$. Therefore, $q_x^{(A)}=0.4(0.181269)=0.0725077$, and $q_x^{(B)}=0.6(0.181269)=0.1087615$. To four decimal places, these are $0.0725$ and $0.1088$; their unrounded sum is $0.181269$.

For 10,000 starting lives, the exposure predicted by the model is $10{,}000(1-e^{-0.20})/0.20=9{,}063.46$ person-years. The expected cause exits are $10{,}000(0.0725077)=725.08$ and $10{,}000(0.1087615)=1{,}087.62$. Dividing by the exposure gives $725.08/9{,}063.46=0.0800$ and $1{,}087.62/9{,}063.46=0.1200$, to the displayed precision. This checks that the central rates are recovered. A common error is to set each probability equal to its central rate; that would give $0.08$ and $0.12$, rather than the constant-force probabilities.

Check Whether the Assumption Fits the Exposure

A conversion is only as appropriate as its within-year assumption. If starting lives, cause counts, and exposure are all supplied, use them as a consistency check before applying a formula. Under UDD, the exposure must equal $l_x-d_x/2$. Under constant forces with total central rate $m_x$, the model-implied exposure is $l_x(1-e^{-m_x})/m_x$. A noticeable discrepancy means the exact table data do not satisfy that model’s exposure relationship.

For observed experience, central rates are often empirical summaries rather than exact parameters of a fully specified model. A conversion assumption then provides a model-based estimate of probabilities; it does not make the observed data obey that assumption. State the chosen assumption, use the matching formula, and avoid presenting the result as an exact identity unless the data and model are consistent.

Worked Example: Diagnose an Inconsistent Exposure

A hypothetical table has 1,200 lives at the start of a year. It records 72 exits from cause A, 48 from cause B, and total exposure of 1,110 person-years. Assess whether UDD is consistent with these data. Calculate the observed central rates and show what the UDD and constant-force conversions would produce if applied mechanically.

Method. Compare reported exposure with the model-implied exposure before treating a conversion as exact. Then calculate the central rates from the reported exposure.

Total exits are $d_x=72+48=120$. Under UDD, the exposure should be $1{,}200-120/2=1{,}140$ person-years, not 1,110. Therefore the exact data are inconsistent with UDD.

The observed central rates are $m_x^{(A)}=72/1{,}110=0.064865$ and $m_x^{(B)}=48/1{,}110=0.043243$. Their total is $m_x=0.108108$. If the UDD formula were applied mechanically, its denominator would be $1+0.108108/2=1.054054$, giving $q_x^{(A)}=0.0615$ and $q_x^{(B)}=0.0410$, to four decimal places. These sum to $0.1026$, whereas the table counts give $120/1{,}200=0.1000$.

For comparison, mechanically applying constant-force conversion gives $q_x=1-e^{-0.108108}=0.1025$ to four decimal places. The force shares are $72/120=0.6$ and $48/120=0.4$, so the cause probabilities are approximately $0.0615$ and $0.0410$. Constant forces also fail the exposure check: the implied exposure is $1{,}200(1-e^{-0.108108})/0.108108$, approximately 1,137.41 person-years, not 1,110.

Check. The observed cause probabilities are $72/1{,}200=0.0600$ and $48/1{,}200=0.0400$, totaling $0.1000$. Neither mechanical conversion reproduces these probabilities, and neither exposure relationship matches the reported exposure. The exam trap is to select a familiar conversion formula without checking whether its exposure assumption is compatible with the data.

Choosing and Reporting a Conversion

For a one-year interval, the practical distinction is the implied pattern of exposure during the year. UDD makes the number in the group decline linearly; constant forces make survival decline exponentially. Both can be reasonable modeling assumptions in appropriate settings, but they generally produce different probabilities from the same central rates.

Fix the interval and units.
Confirm that counts, exposure, rates, and probabilities all refer to the same interval. A rate per person-year cannot be applied directly to a multi-year interval without a model for the rate over that interval.

Calculate rates from exposure.
For each cause, divide its decrement count by the common total exposure. Sum the cause-specific rates to obtain the total central rate.

State the within-year assumption.
Use the UDD relationship only when uniform exits are assumed; use the exponential relationship only when constant cause-specific forces are assumed.

Check and interpret.
Compare supplied exposure with the model-implied exposure where possible, and verify that cause probabilities sum to the total probability and do not exceed 1.

This distinction matters in decrement-table calculations because probabilities determine how many lives are expected to exit, while central rates summarize exits relative to time at risk. Confusing them can alter cause probabilities and, in turn, benefits or other calculations based on cause of exit. Keeping the exposure definition explicit is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: A central rate is a count divided by exposure, not a probability. For one-year cause-specific rates, UDD gives $q_x^{(j)}=m_x^{(j)}/(1+m_x/2)$; constant forces give $q_x^{(j)}=[m_x^{(j)}/m_x](1-e^{-m_x})$. Choose the relationship that matches the stated assumption, and check exposure consistency.

Check Your Understanding

For each problem, identify the assumption before converting rates or checking exposure.

A hypothetical table starts with 600 lives and records 45 exits from cause A and 15 from cause B. Assume UDD. Find the exposure, both central rates, and both cause probabilities. (Answer: Exposure 570 person-years; rates 0.078947 and 0.026316; probabilities 0.0750 and 0.0250.)
Cause-specific central rates are 0.05 and 0.15 per year under constant forces. Find the total and cause-specific probabilities. (Answer: Total 0.1813; cause probabilities 0.0453 and 0.1360.)
A table starts with 800 lives, has 32 exits from cause A and 48 from cause B, and reports exposure of 760 person-years. Is the exposure consistent with UDD? If so, calculate both central rates and probabilities. (Answer: Yes; rates 0.042105 and 0.063158; probabilities 0.0400 and 0.0600.)
A table has 500 starting lives, 20 exits from cause A, 30 from cause B, and exposure of 470 person-years. Find the observed central rates and decide whether the exact data are consistent with UDD. (Answer: Rates 0.042553 and 0.063830; no, UDD exposure would be $500-50/2=475$ person-years.)
Under constant forces, central rates are 0.02 and 0.03 per year. Find the total probability and the probability for each cause. (Answer: Total 0.0488; cause probabilities 0.0195 and 0.0293.)

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