Why Check Life-Table Consistency?
A life table presents related quantities in several forms: the number alive at each age, the number dying between successive ages, and probabilities of death and survival. Because these quantities describe the same transitions, they must agree. A single copied or calculated entry can violate an identity even when every number looks plausible by itself.
A consistency check is useful before using a table to calculate survival probabilities or other actuarial quantities. The audit is systematic: confirm the radix, reconcile each pair of adjacent survivor counts with deaths, and compare the resulting probabilities with the reported rates. The checks below apply to an ordinary life table whose entries refer to integer ages.
In this setting, \(q_x\) and \(p_x\) are probabilities conditional on being alive at age \(x\). Thus the relevant denominator for both is \(\ell_x\), the number alive at the start of the interval. The table may use a base age other than zero; the radix check applies at the stated base age.
The Core Consistency Identities
From age \(x\) to age \(x+1\), every life in the table is either among the \(d_x\) deaths or among the \(\ell_{x+1}\) survivors. Therefore the survivor count at age \(x\) must be divided between those two groups.
The one-year death probability is the fraction of those alive at age \(x\) who die during the interval. The survival probability is the fraction who remain alive to age \(x+1\). These definitions give the remaining identities.
These formulas require \(\ell_x>0\) for the interval being checked. For a valid table, survivor counts and death counts cannot be negative, and survivors cannot increase from one age to the next. The one-year probabilities must lie between zero and one, inclusive. A rate reported to a limited number of decimal places may not reproduce the count ratio exactly; in that case, compare it using the table’s stated precision rather than treating ordinary rounding as a contradiction.
The justification is direct: the two groups, deaths and survivors, are disjoint and together contain all \(\ell_x\) lives at age \(x\). Dividing their counts by that same starting total gives their conditional probabilities. Consequently, an audit should not check \(q_x+p_x=1\) alone. Two reported probabilities can sum to one but still fail to match the table’s counts.
At the designated base age, compare the listed survivor count with the stated starting number of lives.
For each interval, calculate \(\ell_x-\ell_{x+1}\) and compare it with the listed \(d_x\).
Use \(d_x/\ell_x\) for \(q_x\) and \(\ell_{x+1}/\ell_x\) for \(p_x\), then check \(q_x+p_x=1\).
Check that counts and probabilities are within their valid ranges, allowing for stated rounding where appropriate.
Worked Audits
Worked Example: Finding a Wrong Survivor Count
A hypothetical life table has radix 1,000 at age 40. It reports \(\ell_{40}=1{,}000\), \(d_{40}=80\), \(q_{40}=0.08\), \(p_{40}=0.92\), and \(\ell_{41}=920\). At age 41 it reports \(d_{41}=46\), \(q_{41}=0.05\), \(p_{41}=0.95\), and \(\ell_{42}=875\). Identify any inconsistent entry.
Method. Check the radix, then test the count identity and probabilities interval by interval.
At the base age, the survivor count is \(1{,}000\), which matches the stated radix. For age 40, the death count implied by the adjacent survivor counts is
This matches the reported \(d_{40}=80\). The probability checks also agree:
For age 41, the reported death probability is consistent with the count at the start of that interval:
The number surviving to age 42 must then be
Equivalently, \(p_{41}=0.95\) implies \(\ell_{42}=0.95(920)=874\). The listed value \(875\) is inconsistent; the corrected entry is \(\ell_{42}=874\). As a check, \(874+46=920\), and \(874/920=0.95\).
Common mistake avoided: dividing deaths by the ending survivor count gives \(46/874\), approximately \(0.0526\), but that is not \(q_{41}\). The conditional probability is based on lives at risk at age 41, so its denominator is \(\ell_{41}=920\), giving \(0.05\).
Worked Example: Checking Counts and Rates Together
A hypothetical table has radix 5,000 at age 50. For the interval from age 50 to 51, it lists \(\ell_{50}=5{,}000\), \(d_{50}=200\), \(q_{50}=0.04\), \(p_{50}=0.96\), and \(\ell_{51}=4{,}800\). For the next interval, it lists \(d_{51}=110\), \(q_{51}=0.025\), \(p_{51}=0.975\), and \(\ell_{52}=4{,}680\). Which entry is inconsistent?
Method. Test each interval independently, using the survivor count at its starting age as the denominator for the probabilities.
The age-50 interval passes the count test:
Its rates also agree with the counts:
At age 51, the adjacent survivor counts imply
The reported death probability gives the same implied number of deaths:
The reported survival probability checks as well:
Thus \(d_{51}=110\) is the inconsistent entry; it should be \(120\). The corrected count satisfies \(4{,}800-120=4{,}680\), and its ratio to the starting count is \(120/4{,}800=0.025\).
Common mistake avoided: checking only that \(q_{51}+p_{51}=1\) would miss the error. The two probabilities do sum to one, but the reported death count does not agree with either the adjacent survivor counts or \(q_{51}\ell_{51}\).
Worked Example: Detecting a Rate That Fails the Count Check
A hypothetical table states a radix of 1,000 at age 60. It lists \(\ell_{60}=1{,}000\), \(d_{60}=50\), \(q_{60}=0.05\), \(p_{60}=0.95\), and \(\ell_{61}=950\). For the interval from age 61 to 62, it lists \(\ell_{61}=950\), \(d_{61}=20\), \(q_{61}=0.03\), \(p_{61}=0.97\), and \(\ell_{62}=930\). Determine whether the table is consistent.
Method. Verify the radix and counts first, then compare each rate with the corresponding ratio.
The radix matches because \(\ell_{60}=1{,}000\). For age 60, the count identity and probabilities all hold:
For age 61, the adjacent counts agree with the listed deaths:
But the reported \(q_{61}=0.03\) does not match those counts:
The survival rate implied by the counts is
Rounded to four decimal places, the rates consistent with the counts are \(q_{61}=0.0211\) and \(p_{61}=0.9789\). The listed rates \(0.03\) and \(0.97\) sum to one, but do not match the count ratios. The count entries and radix are consistent; the reported age-61 rates are not. The check is confirmed by \(20/950+930/950=1\).
Common mistake avoided: treating \(q_{61}+p_{61}=1\) as a complete audit. A pair of complementary probabilities can still be wrong for the survivor and death counts in the table.
Valid Ranges, Radix, and Rounding
The identities supply both equality checks and basic range checks. When \(\ell_x>0\), a valid interval has \(0\leq d_x\leq\ell_x\) and \(0\leq\ell_{x+1}\leq\ell_x\). It follows that \(0\leq q_x\leq1\) and \(0\leq p_x\leq1\). A negative death count, a larger survivor count at the next age, or a probability outside this range signals an error, not an unusual but valid mortality rate.
The radix is a separate check from the transitions. If the table states that its base-age count is 10,000, a listed base-age count of 9,800 fails that statement even if every later row follows the listed counts perfectly. Conversely, matching the radix does not guarantee the later rows are consistent. Check both.
Rounding matters when rates are printed with limited precision. For example, a ratio such as \(20/950\) is approximately \(0.0210526\); a table displaying four decimal places would report \(0.0211\). A small difference between the unrounded ratio and printed rate can be ordinary rounding. A materially different rate cannot be explained by rounding. When entries are explicitly required to be exact, apply the identities exactly; otherwise use the displayed precision to judge agreement.
Check Your Understanding
For each problem, use the stated radix and the one-year identities to decide what is consistent. Rates in the answers are shown to four decimal places when needed.
- A hypothetical table has radix 8,000 at age 25, with \(d_{25}=240\). Calculate \(q_{25}\), \(p_{25}\), and \(\ell_{26}\). (Answer: \(q_{25}=0.0300\), \(p_{25}=0.9700\), and \(\ell_{26}=7{,}760\).)
- At age 40, a table lists \(\ell_{40}=2{,}400\), \(\ell_{41}=2{,}280\), and \(d_{40}=120\). Calculate \(q_{40}\) and \(p_{40}\), and state whether the counts agree. (Answer: The counts agree; \(q_{40}=0.0500\) and \(p_{40}=0.9500\).)
- A table has radix 1,500 at age 50 and lists \(q_{50}=0.06\), \(d_{50}=90\), and \(\ell_{51}=1{,}410\). Find \(\ell_{50}\) and \(p_{50}\), and check consistency. (Answer: \(\ell_{50}=1{,}500\), \(p_{50}=0.9400\), and the entries are consistent.)
- A table lists \(\ell_{70}=1{,}000\), \(d_{70}=40\), \(q_{70}=0.04\), \(p_{70}=0.96\), and \(\ell_{71}=960\). For the next interval it lists \(d_{71}=48\), \(q_{71}=0.05\), \(p_{71}=0.95\), and \(\ell_{72}=913\). Identify the inconsistent entry and its correction. (Answer: \(\ell_{72}=913\) is inconsistent; it should be \(960-48=912\).)
- A table states a radix of 10,000 at age 30 but lists \(\ell_{30}=9{,}800\), \(d_{30}=196\), \(q_{30}=0.02\), \(p_{30}=0.98\), and \(\ell_{31}=9{,}604\). Are the entries consistent, and what does the radix check show? (Answer: The age-30 transition entries are locally consistent, but the table fails the radix check because \(\ell_{30}=9{,}800\), not 10,000.)
These checks are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. On an exam problem, write down the relevant identity before deciding which entry is wrong; this makes the denominator and the age interval explicit and helps prevent a plausible-looking inconsistency from passing unnoticed.