Why Compare the Interpolation Assumptions?
A life table gives survival information at integer ages, but many calculations begin or end between birthdays. The annual probability of death fixes the survival probability over a full year; it does not, by itself, specify when deaths occur within that year. Uniform distribution of deaths (UDD) and constant force of mortality are two different assumptions for filling in that missing detail.
Both assumptions reproduce the same annual life-table probability. They can nevertheless give different answers for survival over part of a year, because they describe different within-year mortality patterns. The important exam skill is to identify the assumed pattern, use its corresponding formula, and condition correctly when the starting age is already fractional.
The Two Fractional-Age Formulas
Write \(u\) for the fraction of the year already elapsed since age \(x\). The probability of surviving from age \(x\) to age \(x+u\) is the starting point for both methods. Under UDD, the proportion of the original cohort still alive is linear in \(u\). Under constant force, survival is exponential in \(u\).
These formulas agree at the start and end of the year: at \(u=0\), survival is 1; at \(u=1\), survival is \(1-q_x=p_x\). Between those endpoints, the formulas need not agree. If the starting age is \(x+s\), rather than \(x\), use conditional survival:
where \(0\leq s\leq s+t\leq1\). The UDD expression is a ratio because it compares the surviving cohort at \(x+s+t\) with the cohort already alive at \(x+s\). Under constant force, the exponential survival law has the same conditional form over any subinterval.
What Changes Within the Year?
The assumptions agree on the total number dying during the year but distribute those deaths differently in time. Under UDD, the number of deaths per unit of age, measured relative to the original cohort, is constant. The conditional force of mortality is not constant: it increases as the year progresses, because the same number of deaths per unit time is drawn from a shrinking group of survivors. Under constant force, the force is fixed throughout the year, while the number dying per unit of age from the original cohort decreases as fewer remain alive.
For \(q_x>0\), the UDD force starts at \(q_x\) and rises with \(u\). The constant force equals \(-\ln(1-q_x)\) throughout the year. The conditional probability of death in a particular part of the year depends on the whole survival pattern, so it should not be inferred just by comparing the two forces at one age.
| Feature | UDD | Constant force |
|---|---|---|
| Survival from age \(x\) to \(x+u\) | \(1-uq_x\) | \((1-q_x)^u\) |
| Within-year force | \(q_x/(1-uq_x)\), increasing in \(u\) | \(-\ln(1-q_x)\), constant |
| Deaths per unit age from the original cohort | Constant | Decreasing through the year |
| Survival over the full year | \(1-q_x\) | \(1-q_x\) |
Worked Example: Comparing the First and Second Half of a Year
A hypothetical life table has \(q_x=0.40\). Find the probability of death during the first half-year for a life aged \(x\), under each interpolation assumption. Then find the probability of death in the second half, conditional on surviving to age \(x+0.5\). State the method and check that the two half-year survival factors reproduce the annual survival probability.
Method. Calculate survival to age \(x+0.5\) and, for the second half, divide survival to age \(x+1\) by survival to age \(x+0.5\). This avoids treating a conditional probability as though it were measured from age \(x\).
UDD, first half. Survival to \(x+0.5\) is \(1-(0.5)(0.40)=0.80\). Therefore the probability of death in the first half is \(1-0.80=0.20\).
UDD, second half conditional on survival. Survival from \(x+0.5\) to \(x+1\) is \(0.60/0.80=0.75\). The conditional probability of death in that half is \(1-0.75=0.25\).
Constant force, first half. Survival to \(x+0.5\) is \(0.60^{0.5}=\sqrt{0.60}=0.7745967\). The probability of death in the first half is \(1-0.7745967=0.2254033\), or \(0.2254\) to four decimals.
Constant force, second half conditional on survival. The conditional survival probability over another half-year is also \(\sqrt{0.60}=0.7745967\). Thus conditional death probability is \(1-0.7745967=0.2254033\), or \(0.2254\).
Check and interpretation. Under UDD, survival through both halves is \(0.80(0.75)=0.60\). Under constant force it is \(\sqrt{0.60}\sqrt{0.60}=0.60\). Each method matches the annual survival, but UDD assigns fewer deaths to the first half and more to the second; constant force gives equal conditional death probabilities in the two half-years.
Calculations Across an Integer Age
When the interval crosses an integer age, split it at that age. Use the annual \(q_x\) for the part inside \([x,x+1]\), and use \(q_{x+1}\) for the part inside \([x+1,x+2]\). Multiply the segment survival probabilities: the second segment is entered only if the life survives the first. This multiplication applies under either interpolation assumption.
Split the requested interval wherever it crosses a birthday represented by a new annual life-table probability.
For a segment within \([y,y+1]\), use \(q_y\) in the UDD or constant-force formula for that year.
The result is the probability of surviving all segments in succession; do not apply one annual probability to an interval that spans different ages.
Worked Example: Survival Across Two Annual Age Intervals
A hypothetical table has \(q_{80}=0.20\) and \(q_{81}=0.40\). Find survival from age 80.25 to age 81.5 under UDD and constant force. Give probabilities to four decimals. The common mistake is to use the age-80 probability for the entire interval; instead, split at age 81.
UDD, age 80.25 to 81. This is the final \(0.75\) of the age-80 year. Survival from age 80 to 81 is \(1-0.20=0.80\); survival from age 80 to 80.25 is \(1-(0.25)(0.20)=0.95\). The conditional survival for the segment is \(0.80/0.95=0.8421053\).
UDD, age 81 to 81.5. The first half of the age-81 year has survival \(1-(0.50)(0.40)=0.80\). Total survival is \(0.8421053(0.80)=0.6736842\), or 0.6737 to four decimals.
Constant force, age 80.25 to 81. The segment length is \(0.75\), so survival is \(0.80^{0.75}=0.8458970\).
Constant force, age 81 to 81.5. The segment length is \(0.50\), so survival is \(0.60^{0.50}=0.7745967\). Total survival is \(0.80^{0.75}(0.60)^{0.50}=0.6552\) to four decimals.
Check. Each calculation uses the relevant annual probability in each age interval. In particular, the constant-force product is approximately \(0.8458970(0.7745967)=0.6552\), not a single fractional power based on just one of the annual probabilities.
Worked Example: Converting Life-Table Counts and Comparing Conditional Survival
A hypothetical table records \(\ell_x=1{,}000\) and \(\ell_{x+1}=700\). Find survival from age \(x+0.4\) to \(x+0.9\) under UDD and constant force. Also compare the UDD force at the start and end of that interval with the constant force. Give survival probabilities to four decimals and forces to four significant figures.
Method. First obtain the annual death probability from the counts. Then calculate conditional survival within the year and evaluate each force at the stated fractional ages.
Annual probability. The number dying is \(d_x=1{,}000-700=300\). Thus \(q_x=d_x/\ell_x=300/1{,}000=0.30\), and \(p_x=0.70\).
UDD conditional survival. Survival from age \(x+0.4\) to \(x+0.9\) is \(\{1-(0.9)(0.30)\}/\{1-(0.4)(0.30)\}=0.73/0.88=0.8295455\), or 0.8295. The corresponding conditional death probability is \(1-0.8295455=0.1704545\), or \(0.1705\).
Constant-force conditional survival. The interval is \(0.5\) years, so survival is \(0.70^{0.5}=\sqrt{0.70}=0.8366600\), or 0.8367. Conditional death probability is \(1-0.8366600=0.1633400\), or \(0.1633\).
Force comparison. At \(x+0.4\), the UDD force is \(0.30/(1-0.4(0.30))=0.30/0.88=0.3409\). At \(x+0.9\), it is \(0.30/(1-0.9(0.30))=0.30/0.73=0.4110\). The constant force is \(-\ln(0.70)=0.3567\) at both ages. The UDD force rises over the year, whereas the constant-force value does not.
Check. Both formulas give annual survival \(0.70\) over the full year. The UDD and constant-force fractional-age survival estimates differ because the assumptions assign deaths differently within that year.
Common Mistakes and When the Difference Matters
The interpolation assumption matters whenever a calculation uses a fractional age, such as survival to a fractional-age valuation date or a probability over part of a year. A full-year calculation alone cannot distinguish UDD from constant force: both reproduce \(q_x\). In applications, use the assumption specified in the problem rather than choosing the method that gives the more convenient result.
- Using the unconditional UDD expression from a fractional starting age. The expression \(1-tq_x\) measures survival from age \(x\), not from age \(x+s\). From age \(x+s\), divide by \(1-sq_x\).
- Using the same annual probability on both sides of a birthday. Split the interval and use the annual probability for each year.
- Assuming the two methods must agree between birthdays. Matching at integer ages does not imply matching at fractional ages.
- Confusing force with annual death probability. Under constant force the within-year force is \(-\ln(1-q_x)\), not \(q_x\), except as an approximation when \(q_x\) is small.
Check Your Understanding
For each problem, use the stated interpolation assumption and round probabilities to four decimals.
- A hypothetical life table has \(q_x=0.25\). Find survival from age \(x+0.2\) to \(x+0.6\) under UDD and constant force. (Answer: UDD \(0.8947\); constant force \(0.8913\).)
- For \(q_x=0.36\), find the probability of death during the first quarter-year under UDD and constant force, conditional on being alive at age \(x\). (Answer: UDD \(0.0900\); constant force \(0.1056\).)
- A hypothetical table has \(q_x=0.10\) and \(q_{x+1}=0.20\). Find survival from age \(x+0.5\) to age \(x+1.25\) under both assumptions. (Answer: UDD \(0.9000\); constant force \(0.8972\).)
- A hypothetical table has \(q_x=0.50\). Compare the probability of death in the first half-year under UDD and constant force, then give the conditional probability of death in the second half-year under each assumption. (Answer: First half—UDD \(0.2500\), constant force \(0.2929\); second half conditional—UDD \(0.3333\), constant force \(0.2929\).)
- A hypothetical table has \(\ell_x=1{,}000\) and \(d_x=240\). Find survival from age \(x+0.3\) to age \(x+0.8\) under UDD and constant force. (Answer: UDD \(0.8707\); constant force \(0.8718\).)