Diagnosing Mortality Assumption Mismatches

Why the Scope of a Mortality Assumption Matters

A mortality rate is not a complete description of survival at every point between integer ages. An annual probability \(q_x\) gives the probability that a life aged \(x\) dies before age \(x+1\). To calculate survival over only part of that year, an additional interpolation assumption may be needed. Likewise, select rates apply by duration since selection for a stated select period; they do not automatically replace or extend ultimate rates at every age.

The key diagnostic is to mark the boundaries before calculating: identify each age interval, the assumption specified there, and—when select mortality is used—the duration since selection. Then calculate each segment using only the assumption that applies to it. An assumption stated for one interval is not a license to extend it into another.

Definition: Under a constant-force assumption within the year of age \(x\), the force of mortality is constant over \([x,x+1)\). Given annual death probability \(q_x\), survival for \(t\) years from age \(x\), where \(0\leq t\leq1\), is \((1-q_x)^t\). Under a uniform distribution of deaths (UDD) assumption within that year, the fraction of the original year’s deaths occurring by time \(t\) is \(t\). In particular, survival from age \(x+t\) to \(x+s\), for \(0\leq t\leq s\leq1\), conditional on being alive at \(x+t\), is \((1-sq_x)/(1-tq_x)\). Each assumption applies only on the interval where it is specified.

For select-and-ultimate mortality, \(q_{[x]+j}\) denotes the death probability in select duration \(j\) for a life selected at age \(x\). If the select period has length \(m\), the select rates apply for durations \(j=0,\ldots,m-1\). After that period, the ultimate rate is indexed by attained age: at duration \(m\), the relevant rate is \(q_{x+m}\). The select period and any interpolation assumption within an age year are separate matters; one does not determine the other.

A Boundary-First Calculation

Suppose a requested survival probability crosses one or more boundaries. Split the request at every boundary where the applicable rate or assumption changes. For successive segments, conditional survival probabilities multiply. The multiplication is valid because each segment’s survival probability is conditional on having survived to the start of that segment.

{}_tp_x = P(T_x>t) = P(T_x>t_1) P(T_x>t_2\mid T_x>t_1) \cdots P(T_x>t\mid T_x>t_{k-1}), \qquad 0<t_1<\cdots<t_{k-1}<t.

This decomposition explains why an interval-specific formula cannot simply be used across the whole requested duration. Each factor must be calculated from the assumption that applies on its own segment. The following examples show how to diagnose the boundaries before performing the arithmetic.

Worked Example: Switching Interpolation Assumptions at an Age Boundary

A hypothetical mortality basis gives \(q_{60}=0.10\) and specifies constant force only on ages 60 to 61. It gives \(q_{61}=0.20\) and specifies UDD only on ages 61 to 62. Calculate the probability that a life aged 60 survives to age 61.5. Compare it with the result from incorrectly extending the age-60 constant-force assumption. Give probabilities to four decimals.

Method. Split the time at age 61, where the specified interpolation assumption changes. On ages 60 to 61, the full-year survival probability is \(1-q_{60}=0.90\). On ages 61 to 61.5, UDD gives conditional survival \(1-0.5q_{61}\).

P(T_{60}>1.5) = (1-0.10)(1-0.5(0.20)) = 0.90(0.90) = 0.8100.

The correct probability is \(0.8100\). As a check, the first segment has survival \(0.90\), the second has conditional survival \(0.90\), and their product is the stated result.

Incorrect extension. Applying the age-60 constant-force formula to all 1.5 years gives \(0.90^{1.5}=0.8538\) to four decimals. That result uses \(q_{60}\) beyond the interval on which its constant-force assumption was specified and ignores the age-61 rate and UDD condition. This is the common mistake the boundary split avoids.

Worked Example: Switching from Select to Ultimate Mortality

A hypothetical select-and-ultimate table has a select period of two years. For a life selected at age 50, \(q_{[50]}=0.02\) and \(q_{[50]+1}=0.04\). The ultimate rates are \(q_{51}=0.10\) and \(q_{52}=0.08\). Find the probability of surviving three years from selection.

Method. The first two years are select durations 0 and 1. At duration 2 the select period has ended, and the attained age is 52, so the third-year rate is the ultimate rate \(q_{52}\). No fractional-year interpolation is needed because the requested duration ends at an integer age.

P(T_{50}>3) = (1-q_{[50]})(1-q_{[50]+1})(1-q_{52}) = (0.98)(0.96)(0.92).

First, \(0.98(0.96)=0.9408\). Then \(0.9408(0.92)=0.865536\), so the probability is \(0.8655\) to four decimals. The result is checked by using exactly two select-year survival factors and one ultimate-year factor.

A common indexing error is to switch to ultimate mortality after only one year and use \(q_{51}\) for the second year. That calculation gives \(0.98(0.90)(0.92)=0.81144\), or \(0.8114\) to four decimals. It is wrong because duration 1 is still inside the two-year select period; the correct second-year rate is \(q_{[50]+1}=0.04\), not \(q_{51}=0.10\).

Worked Example: Recognizing When the Data Do Not Determine the Answer

A hypothetical table supplies only \(q_{70}=0.20\). A calculation asks for the probability that a life aged 70 survives to age 70.5, but gives no interpolation assumption. Is there a unique answer? Calculate the result under constant force and under UDD to show the effect of the missing condition.

Method. First recognize that an annual death probability alone does not specify how deaths are distributed within the year. Calculate the requested probability separately under each possible stated assumption.

Under constant force, half-year survival is \((1-q_{70})^{0.5}\):

(0.80)^{0.5} = \sqrt{0.80} \approx 0.8944.

Under UDD, half-year survival is \(1-0.5q_{70}\):

$$ 1-0.5(0.20)=0.9000. $$

These are different probabilities, so the given data do not determine a unique answer. The defensible response is to state that an interpolation assumption is required, then provide a conditional answer if an assumption is supplied. A common mistake is to silently choose a familiar interpolation rule and present its result as implied by \(q_{70}\) alone.

A Practical Diagnostic Sequence

A short, disciplined check often catches a mismatch before it affects several later calculations. Use the following sequence whenever rates, age intervals, and durations appear together.

Mark the requested ages and durations.
Write down the starting age, ending age, and any integer-age boundaries between them. For select mortality, also record time since selection.

Attach each rate to its correct index.
Use a select rate for its stated select duration. Once the select period ends, use the ultimate rate indexed by attained age.

Check the interpolation condition on each partial year.
Use constant force or UDD only on the age interval where it is specified. If neither is specified, an annual \(q_x\) alone may not answer a fractional-age question.

Calculate segments and multiply conditional survivals.
For a request crossing a boundary, calculate each segment separately and multiply the resulting conditional probabilities in order.

Check the answer against its assumptions.
Confirm that every factor uses the intended rate, time interval, and assumption, and that each survival probability lies between 0 and 1.

Two formulas that produce reasonable numbers can both be mathematically valid under different assumptions. The issue is not which formula is generally preferable; it is whether the problem specifies that formula for the interval being calculated. Similarly, a select rate and an ultimate rate may both be listed at a related age, but their indices describe different things: select duration versus attained age.

Common Exam Traps

Extending constant force past its stated interval. A constant-force formula based on \(q_x\) describes the year of age \(x\) only when that is the stated assumption. Recheck the rate and assumption at the next integer age.
Treating UDD as a universal fractional-age rule. UDD is an assumption, not a consequence of knowing \(q_x\). Do not apply it unless the problem supplies it for the relevant interval.
Switching to ultimate rates too early. Count select durations from selection. If the select period is \(m\) years, durations \(0\) through \(m-1\) remain select durations.
Using attained age to index a select rate. A rate such as \(q_{[50]+1}\) refers to the second select year for a life selected at age 50; it is not the ultimate rate \(q_{51}\).
Ignoring an underspecified interval. If the requested probability depends on fractional-age survival and no interpolation condition is given, state the limitation instead of silently inventing an assumption.

Key takeaway: A mortality calculation is only as sound as its indexing and interval boundaries. Before applying a formula, verify both what its rate refers to and where its assumption applies. When the stated information does not cover the requested interval, the correct diagnosis may be that more information is needed.

Check Your Understanding

For each problem, identify the applicable assumption and respect its stated age or duration range.

A hypothetical basis gives \(q_{45}=0.12\) and specifies constant force only on ages 45 to 46. Find the probability of survival from age 45 to 45.25. (Answer: \(0.9685\), to four decimals.)
A hypothetical table specifies UDD on ages 62 to 63 and gives \(q_{62}=0.15\). Find the probability of surviving from age 62.2 to age 62.7, conditional on being alive at age 62.2. (Answer: \(0.9227\), to four decimals.)
A hypothetical select table has a three-year select period, with \(q_{[40]}=0.03\), \(q_{[40]+1}=0.05\), \(q_{[40]+2}=0.07\), and ultimate \(q_{43}=0.09\). Find the probability of surviving four years from selection. (Answer: \(0.7799\), to four decimals.)
A hypothetical table gives \(q_{70}=0.16\) but states no fractional-age assumption. Find the half-year survival under constant force and under UDD, and say whether the supplied table entry alone determines one answer. (Answer: \(0.9165\) under constant force and \(0.9200\) under UDD; no unique answer is determined.)
A hypothetical basis gives \(q_{60}=0.10\) with constant force on ages 60 to 61, and \(q_{61}=0.25\) with UDD on ages 61 to 62. Find survival from age 60.5 to 61.5. (Answer: \((0.90)^{0.5}(1-0.5(0.25))=0.8301\), to four decimals.)

What You'll Learn