UDD Interpolation Across Several Ages

Why Multi-Year UDD Interpolation Needs Care

A life-table probability may begin and end at fractional ages, with one or more integer ages between them. Under the uniform distribution of deaths (UDD) assumption, deaths are spread uniformly within each individual year of age. A multi-year calculation therefore combines a partial first year, any complete intervening years, and a partial final year. The important point is that the first partial year is conditional on having reached the starting fractional age; it is not simply the full-year survival probability.

This tutorial develops a single ratio that handles both endpoints and all intervening years. It also works when the starting and ending ages lie in the same year, provided the ending age is not earlier than the starting age. All mortality values and populations in the examples are hypothetical.

Definition: For an integer age \(i\), let \(q_i={}_1q_i\) be the probability that a life aged \(i\) dies before age \(i+1\), and let \(p_i=1-q_i\). Let \(\ell_i\) be the number of survivors at exact integer age \(i\) in the life table. Under UDD, the number of survivors at age \(i+t\), for \(0\le t\le1\), is \(\ell_{i+t}=\ell_i(1-tq_i)\). This interpolation applies within each year of age; survival across a whole year remains \(p_i\).

The General Fractional-Age Survival Formula

Write the starting age as \(x+r\) and the ending age as \(y+s\), where \(x\) and \(y\) are integers, \(0\le r<1\), \(0\le s<1\), \(y\ge x\), and \(x+r\le y+s\). The last condition is essential: a conditional survival probability must refer to a future age, not an earlier one. If \(x=y\), it requires \(r\le s\).

Under UDD in every year from \(x\) through \(y\), the conditional probability of surviving from age \(x+r\) to age \(y+s\) is

\Pr(T_{x+r}>y+s-(x+r)\mid T_{x+r}>0) = \frac{\ell_{y+s}}{\ell_{x+r}} = \frac{\left(\prod_{i=x}^{y-1}p_i\right)(1-sq_y)}{1-rq_x}.

Here the probability notation on the left means survival from the specified starting age to the specified ending age. Equivalently, it is the probability of being alive at age \(y+s\), conditional on being alive at age \(x+r\). The product is empty, and equals 1, when \(x=y\). A more direct way to write the same conditional probability is

\Pr(\text{survive from age }x+r\text{ to age }y+s \mid \text{alive at age }x+r) = \frac{\left(\prod_{i=x}^{y-1}(1-q_i)\right)(1-sq_y)}{1-rq_x}.

Why the Formula Holds

UDD gives the survivor count at each fractional endpoint. The number alive at age \(x+r\) is \(\ell_x(1-rq_x)\), since a fraction \(r\) of the year's expected deaths has occurred by then. Similarly, the number alive at age \(y+s\) is \(\ell_y(1-sq_y)\). The integer-age life-table relation is \(\ell_{i+1}=\ell_i p_i\), so moving from age \(x\) to age \(y\) gives \(\ell_y/\ell_x=\prod_{i=x}^{y-1}p_i\). Dividing the endpoint survivor counts therefore gives the conditional ratio:

\frac{\ell_y(1-sq_y)}{\ell_x(1-rq_x)} = \frac{\ell_y}{\ell_x}\frac{1-sq_y}{1-rq_x} = \frac{\left(\prod_{i=x}^{y-1}p_i\right)(1-sq_y)}{1-rq_x}.

This derivation also explains the denominator. The life-table value \(\ell_x\) counts survivors at exact age \(x\), not at the later starting age \(x+r\). Dividing by \(1-rq_x\) conditions on survival through that part of the first year. When \(r=0\), the denominator is 1. When \(s=0\), the endpoint is exactly integer age \(y\), so the final-year fractional factor is also 1.

If both ages are in the same year, the product over complete years disappears and the formula reduces to

\Pr(\text{survive from }x+r\text{ to }x+s\mid\text{alive at }x+r) = \frac{1-sq_x}{1-rq_x}, \qquad r\le s.

This is a conditional probability, not the unconditional probability of surviving from age \(x\) to \(x+s\). In particular, it should equal 1 when \(r=s\).

Worked Applications

Worked Example: Crossing Several Complete Age Years

A hypothetical life table has \(q_{42}=0.04\), \(q_{43}=0.05\), \(q_{44}=0.06\), and \(q_{45}=0.08\). Under UDD in each year, calculate the probability that a life aged 42.3 survives to age 45.6. Give the answer to four decimal places.

Method. Use the endpoint survivor ratio. There are three complete integer-age intervals, from 42 to 43, 43 to 44, and 44 to 45, followed by a fractional part of the year of age 45. The starting age is already 0.3 years into age 42, so condition on survival to that age.

First calculate the annual survival probabilities:

p_{42}=1-0.04=0.96,\qquad p_{43}=1-0.05=0.95,\qquad p_{44}=1-0.06=0.94.

The starting-age adjustment is \(1-0.3q_{42}=1-(0.3)(0.04)=0.988\). The final fractional-year factor is \(1-0.6q_{45}=1-(0.6)(0.08)=0.952\). Therefore,

\begin{aligned} \Pr(\text{survive from }42.3\text{ to }45.6) &=\frac{(0.96)(0.95)(0.94)(0.952)}{0.988}\\ &=\frac{(0.85728)(0.952)}{0.988}\\ &=\frac{0.81613056}{0.988}\\ &=0.826043\ldots\\ &=\boxed{0.8260}. \end{aligned}

Check. The result is between 0 and 1, and it is less than the survival probability from exact age 42 to exact age 45, \(0.96(0.95)(0.94)=0.85728\), as expected because the calculation includes additional mortality after age 45 and conditions on starting later than age 42. The common mistake avoided is to omit the starting-age denominator, which would incorrectly treat the life as starting at age 42.

Worked Example: Two Fractional Ages in the Same Year

For a hypothetical mortality table, \(q_{50}=0.10\). Under UDD, find the conditional probability that a life aged 50.2 survives to age 50.7, and then find the probability that the life dies in that interval. Give each answer to four decimal places.

Method. Both endpoints are in the age-50 year, so there are no complete years between them. Apply the same-year endpoint ratio; the condition \(0.2\le0.7\) confirms that the interval is forward in time.

The UDD survivor factors at the start and end are

1-(0.2)(0.10)=0.98,\qquad 1-(0.7)(0.10)=0.93.

Thus,

\Pr(\text{survive from }50.2\text{ to }50.7) = \frac{0.93}{0.98} = 0.948979\ldots = 0.9490.

The probability of dying before age 50.7, conditional on being alive at age 50.2, is the complement:

1-0.948979\ldots=0.051020\ldots=0.0510.

Check. The survival and death probabilities add to \(0.9490+0.0510=1.0000\). The common mistake avoided is using \(1-0.5q_{50}=0.95\) as the conditional survival probability. That is the survivor factor measured from exact age 50, not from age 50.2; the correct calculation divides by the starting-age factor \(0.98\).

Worked Example: Survival and Death Across Multiple Years

A hypothetical table gives \(q_{60}=0.12\), \(q_{61}=0.16\), \(q_{62}=0.20\), and \(q_{63}=0.10\). Assume UDD in every year. Find the probability that a life aged 60.4 dies before reaching age 63.2. Give the answer to four decimal places.

Method. First calculate survival across the interval using the fractional-age ratio, then take its complement. The complete years are ages 60 to 61, 61 to 62, and 62 to 63; the final age-63 segment is 0.2 years.

The full-year survival probabilities are \(p_{60}=0.88\), \(p_{61}=0.84\), and \(p_{62}=0.80\). The endpoint factors are \(1-(0.2)(0.10)=0.98\) and \(1-(0.4)(0.12)=0.952\). Hence,

\begin{aligned} \Pr(\text{survive from }60.4\text{ to }63.2) &=\frac{(0.88)(0.84)(0.80)(0.98)}{0.952}\\ &=\frac{(0.59136)(0.98)}{0.952}\\ &=\frac{0.5795328}{0.952}\\ &=0.6087529\ldots \end{aligned}

Therefore the probability of death before age 63.2 is

1-0.6087529\ldots=0.3912470\ldots=\boxed{0.3912}.

Check. The survival probability \(0.6088\) and death probability \(0.3912\) sum to 1.0000 after rounding. The common mistake avoided is reporting the survival probability as the probability of death; the wording asks for death, so the complement is required.

Boundary Checks and Exam Traps

Before accepting a multi-year calculation, check that each factor represents the correct part of the timeline. The denominator adjusts for the fact that the life is known to have reached the fractional starting age. The product covers only complete years from \(x\) through \(y-1\). The final factor covers only the fractional part \(s\) of the ending age's year. A compact checklist is:

Order the ages.
Confirm \(x+r\le y+s\). If \(x=y\), this specifically requires \(r\le s\).

Identify the complete years.
Use \(p_i=1-q_i\) for \(i=x,\ldots,y-1\), and do not include a full \(p_y\) unless the ending age is at least \(y+1\).

Adjust both endpoints.
Divide by \(1-rq_x\) for the fractional starting age and multiply by \(1-sq_y\) for the fractional ending age.

Interpret the requested event.
For death before the ending age, subtract the survival probability from 1. For survival, report the ratio itself.

For a zero-length interval, the endpoint ages are the same, so the formula gives 1. For a forward interval, the result must be in the range from 0 to 1. A value above 1 often signals that the denominator was omitted, the endpoints were reversed, or the age-year indices were misapplied. A result that requires an unavailable \(q_i\) cannot be determined from the given information; do not silently assume a missing mortality rate.

Key takeaway: Under UDD, a fractional-age survival probability across several years is the ratio of the interpolated survivor count at the ending age to the interpolated survivor count at the starting age. This automatically combines the first partial year, intervening full years, and final partial year.

Check Your Understanding

Use UDD throughout each problem. Each scenario is hypothetical; give probability answers to four decimal places unless a different form is requested.

A table has \(q_{64}=0.08\). Find the probability that a life aged 64.25 survives to age 64.75. (Answer: 0.9592)
For a hypothetical table, \(q_{35}=0.06\), \(q_{36}=0.10\), and \(q_{37}=0.08\). Find the probability that a life aged 35.5 survives to age 37.25. (Answer: 0.8547)
A life aged 71.3 is subject to \(q_{71}=0.14\), \(q_{72}=0.11\), \(q_{73}=0.09\), and \(q_{74}=0.20\). Find the probability of survival to age 74.8. (Answer: 0.6107)
For a hypothetical table, \(q_{52}=0.05\), \(q_{53}=0.08\), \(q_{54}=0.10\), and \(q_{55}=0.12\). Find the probability that a life aged 52.4 survives to age 55.5. (Answer: 0.7545)
A hypothetical table has \(q_{40}=0.05\) and \(q_{41}=0.10\). Under UDD, the probability that a life aged 40.2 survives to age 42.5 is 0.8000. Determine \(q_{42}\). (Answer: 0.1474)

What You'll Learn