Deriving UDD Survival Interpolation

Why Interpolate Between Integer Ages?

A life table commonly gives survivor counts at integer ages, such as \(\ell_x\) and \(\ell_{x+1}\). A calculation may instead require the probability that a life aged \(x\) survives to age \(x+r\), where \(r\) is a fraction of a year. The uniform distribution of deaths assumption (UDD) supplies a simple interpolation between those two integer-age entries.

The key idea is that, under UDD, deaths are spread uniformly over the year of age from \(x\) to \(x+1\). Consequently, the proportion of the year’s deaths that has occurred by fraction \(r\) of the year is \(r\). This makes the survivor count decrease linearly between the two endpoint counts. The resulting formula applies within one integer-age year, for \(0\le r\le1\).

Definition: Under the uniform distribution of deaths assumption for the year of age from \(x\) to \(x+1\), the fraction of that year’s deaths occurring by age \(x+r\) is \(r\), for \(0\le r\le1\). Write \(d_x=\ell_x-\ell_{x+1}\) for the number of deaths in the year, and \(q_x=d_x/\ell_x\) for the annual probability of death among lives aged \(x\). The fractional-age survival probability is \({}_r p_x=\ell_{x+r}/\ell_x\).

Deriving the UDD Survival Interpolation

At age \(x\), the life table has \(\ell_x\) survivors. By age \(x+1\), it has \(\ell_{x+1}\) survivors, so \(d_x=\ell_x-\ell_{x+1}\) deaths occur during the year. Under UDD, the fraction of these deaths occurring during the first \(r\) years of the age interval is \(r\). Thus \(r d_x\) deaths have occurred by age \(x+r\), and the number remaining is \(\ell_x-rd_x\).

\ell_{x+r}=\ell_x-rd_x =\ell_x-r(\ell_x-\ell_{x+1}) =(1-r)\ell_x+r\ell_{x+1}.

To obtain a survival probability, divide the number alive at age \(x+r\) by the number alive at age \(x\). Substituting \(d_x/\ell_x=q_x\) gives two equivalent expressions:

{}_r p_x =\frac{\ell_{x+r}}{\ell_x} =\frac{\ell_x-rd_x}{\ell_x} =1-rq_x =(1-r)+r\frac{\ell_{x+1}}{\ell_x}, \qquad 0\le r\le1.

The division by \(\ell_x\) matters: \(\ell_{x+r}\) is a survivor count on the life table’s scale, whereas \({}_r p_x\) is a probability conditional on being alive at age \(x\). Life table counts may use any convenient radix; the ratio, and therefore the probability, does not depend on that scale.

Result: Under UDD, interpolate the survivor count linearly between the two integer-age counts: \(\ell_{x+r}=(1-r)\ell_x+r\ell_{x+1}\). The probability of survival from age \(x\) to age \(x+r\) is \({}_r p_x=1-rq_x\). Both formulas are for a fraction \(r\) within the single year from age \(x\) to \(x+1\).

Properties and Checks

The endpoints provide a quick check on both formulas. When \(r=0\), the interpolated count must be \(\ell_x\), and survival over zero years must be 1. When \(r=1\), the count must be \(\ell_{x+1}\), and the survival probability must be \(\ell_{x+1}/\ell_x=1-q_x\). The expressions above satisfy both checks:

\ell_{x+0}=\ell_x,\qquad {}_0p_x=1; \qquad \ell_{x+1}=\ell_{x+1},\qquad {}_1p_x=1-q_x.

For \(0<r<1\), the survivor count is a weighted average of the endpoint counts, with nonnegative weights \(1-r\) and \(r\) that sum to 1. If \(\ell_{x+1}\le\ell_x\), the interpolated count must therefore lie between them. This is another useful arithmetic check.

The result relies on the UDD assumption. It is not a general interpolation rule that can be applied to any mortality model. In particular, it specifies a linear path for the survivor count within this year; it does not assert that every possible measure of mortality is constant during the year. Also, a single application uses the entries at the two ends of the same age interval. If the target age is outside that interval, do not extend this straight line beyond its endpoints.

Worked Example: Survival Through Part of a Year

A hypothetical life table has \(\ell_{60}=8{,}000\) and \(\ell_{61}=7{,}600\). Assume UDD from age 60 to age 61. Find the number of survivors at age 60.35 on the life table scale and the probability that a life aged 60 survives to age 60.35.

Method. First find the annual deaths and annual death probability. Then use the UDD interpolation at \(r=0.35\), dividing the interpolated count by \(\ell_{60}\) to obtain the probability.

The number of deaths during the year is

d_{60}=\ell_{60}-\ell_{61}=8{,}000-7{,}600=400.

Therefore, the annual probability of death is

q_{60}=\frac{d_{60}}{\ell_{60}} =\frac{400}{8{,}000}=0.05.

By age 60.35, UDD assigns \(0.35\) of the year’s deaths to the elapsed portion of the year. That is \(0.35(400)=140\) deaths, giving

\ell_{60.35}=\ell_{60}-0.35d_{60} =8{,}000-140=7{,}860.

The survival probability is the fraction of the age-60 lives still represented at age 60.35:

{}_{0.35}p_{60} =\frac{7{,}860}{8{,}000} =0.9825.

Check and trap avoided. The count 7,860 is between 8,000 and 7,600, as it must be. The probability also equals \(1-0.35(0.05)=0.9825\). Using the full-year survival probability \(1-q_{60}=0.95\) for this partial year would incorrectly give 0.95 instead of 0.9825.

Worked Example: Interpolating From an Annual Death Probability

A hypothetical life table has \(\ell_{45}=15{,}000\), and the annual probability of death at age 45 is \(q_{45}=0.06\). Assume UDD during this year. Find \(\ell_{45.4}\) and the probability that a life aged 45 survives to age 45.4.

Method. Recover the age-46 survivor count from \(q_{45}\), then apply the fractional-age formula. The given \(q_{45}\) is an annual probability, so first use it to find the full-year number of deaths.

By definition, \(q_{45}=d_{45}/\ell_{45}\). Hence

d_{45}=q_{45}\ell_{45} =0.06(15{,}000)=900.

The age-46 count is consequently

\ell_{46}=\ell_{45}-d_{45} =15{,}000-900=14{,}100.

At \(r=0.4\), the number of deaths assigned to the elapsed portion is \(0.4(900)=360\). Therefore,

\ell_{45.4}=15{,}000-360=14{,}640.

Dividing by the starting count gives

{}_{0.4}p_{45} =\frac{14{,}640}{15{,}000} =0.976.

Check and trap avoided. The same result follows directly from \(1-rq_{45}=1-0.4(0.06)=0.976\). A common error is to use \(q_{45}=0.06\) as the number of deaths; it is a probability, not a count. The count is \(q_{45}\ell_{45}=900\).

Worked Example: Finding the Fraction of a Year

A hypothetical table has \(\ell_{72}=10{,}000\) and \(\ell_{73}=9{,}400\), with UDD between these ages. At what fractional age \(72+r\) is the interpolated survivor count 9,760? Also find the survival probability from age 72 to that age.

Method. Use the linear survivor-count formula and solve for \(r\). Then divide the target count by the starting count to obtain the survival probability.

The annual decrement is

d_{72}=\ell_{72}-\ell_{73} =10{,}000-9{,}400=600.

Under UDD, the fractional-age count is \(\ell_{72+r}=10{,}000-600r\). Set this equal to the target and solve:

9{,}760=10{,}000-600r, \qquad 600r=240, \qquad r=\frac{240}{600}=0.4.

The target occurs at age \(72.4\). The corresponding probability is

{}_{0.4}p_{72} =\frac{9{,}760}{10{,}000} =0.976.

Check and trap avoided. Since \(r=0.4\), the interpolated count is \(60\%\) of the way from 10,000 down to 9,400: \(10{,}000-0.4(600)=9{,}760\). The value \(r=0.4\) is the elapsed fraction of the year, not the remaining fraction \(0.6\).

Common Mistakes and Practical Use

Using a full-year probability for a partial year. Under UDD, survival over fraction \(r\) is \(1-rq_x\), not \(1-q_x\) unless \(r=1\).
Confusing a death count with a death probability. The annual count is \(d_x=\ell_x-\ell_{x+1}\); the annual probability is \(q_x=d_x/\ell_x\). Keep the scale clear before calculating.
Forgetting the conditional starting population. The survival probability is \(\ell_{x+r}/\ell_x\), not just the interpolated count \(\ell_{x+r}\).
Reversing elapsed and remaining time. In \(x+r\), \(r\) is the time since age \(x\). Thus \(r=0.25\) means one quarter of the year has elapsed.
Extrapolating outside the age interval. The formula is justified for \(0\le r\le1\) using the endpoints \(\ell_x\) and \(\ell_{x+1}\), not for fractional ages beyond age \(x+1\).

The interpolation is useful whenever a calculation needs a life table count or survival probability at a fractional age while the supplied table reports integer-age entries. Before applying it, confirm that the mortality assumption is UDD for the relevant age year, identify the correct two endpoint counts, and translate the target age into an elapsed fraction \(r\). The resulting calculation is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Key takeaway: Under UDD, a fraction \(r\) of the year accounts for a fraction \(r\) of that year’s deaths. Therefore \(\ell_{x+r}=(1-r)\ell_x+r\ell_{x+1}\), and the conditional survival probability is \({}_r p_x=\ell_{x+r}/\ell_x=1-rq_x\).

Check Your Understanding

For each problem, assume UDD over the stated age year. Keep survivor counts and probabilities distinct, and check that each interpolated count lies between its integer-age endpoints.

A hypothetical table has \(\ell_{50}=9{,}000\) and \(\ell_{51}=8{,}820\). Find \(\ell_{50.25}\) and \({}_{0.25}p_{50}\). (Answer: \(\ell_{50.25}=8{,}955\); \({}_{0.25}p_{50}=0.9950\))
At age 38, a hypothetical table has \(\ell_{38}=15{,}000\) and \(q_{38}=0.06\). Find \(\ell_{38.4}\) and \({}_{0.4}p_{38}\). (Answer: \(\ell_{38.4}=14{,}640\); \({}_{0.4}p_{38}=0.9760\))
A hypothetical table has \(\ell_{64}=7{,}200\) and \(\ell_{65}=6{,}840\). Find the survival probability from age 64 to age 64.75 and the interpolated survivor count. (Answer: \({}_{0.75}p_{64}=0.9625\); \(\ell_{64.75}=6{,}930\))
A hypothetical table has \(\ell_{80}=12{,}000\) and \(\ell_{81}=11{,}100\). Find the fraction \(r\) for which \(\ell_{80+r}=11{,}340\), and give the corresponding survival probability. (Answer: \(r=0.7333\) to four decimal places; \({}_r p_{80}=0.9450\))
A hypothetical table has \(\ell_{55}=10{,}000\), \(\ell_{56}=9{,}600\), and a target survival probability of 0.985 from age 55. Find the fractional age at which that probability is attained and the corresponding survivor count. (Answer: age \(55.375\); \(\ell_{55.375}=9{,}850\))

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