Why Fractional-Age Associated Calculations Need Care
A decrement table often gives probabilities at integer ages, while a question asks about a fractional age such as \(x+0.4\). The calculation depends on the assumed pattern of exits within the year. It also depends on what probability the table represents: an exit from a particular cause in a multiple-decrement setting is not automatically the same as an exit in that cause’s associated single-decrement table.
This tutorial uses annual associated probabilities already obtained from the specified multiple-decrement conversion. Its task is to calculate fractional-age survival and exits from those values. Use the within-year assumption specified for the associated table; do not substitute a multiple-decrement probability for an associated probability or switch interpolation assumptions partway through a calculation.
The distinction matters even when both probabilities carry the same cause label. Conversion from a multiple-decrement table produces the annual associated value \(q'_{x,j}\) on the stipulated conversion basis. Fractional-age calculations then use that associated value and the stipulated within-year assumption. In the formulas below, \(s\) and \(t\) are fractions of a year, with \(0\leq s<t\leq1\).
Fractional Ages Under UDD
Under the uniform distribution of decrements (UDD) assumption for the associated table, the associated decrement count falls linearly through the year. Equivalently, the associated survival probability from age \(x\) to age \(x+r\) is linear in \(r\). For cause \(j\), the survival and exit probabilities from age \(x\) to age \(x+r\) are therefore:
If the starting age is already fractional, the desired probability is conditional on having survived to that starting age. Thus the probability of surviving from \(x+s\) to \(x+t\) is the survival from \(x\) to \(x+t\), divided by the survival from \(x\) to \(x+s\):
Here the notation on the left describes a conditional probability over the interval beginning at \(x+s\). The denominator is essential: the numerator of the exit expression is the probability, measured from age \(x\), of exiting during the interval, while the required probability is conditional on surviving to \(x+s\).
Worked Example: UDD Exit Probability From a Fractional Starting Age
A hypothetical associated table for cause \(J\) has annual probability \(q'_{x,J}=0.20\). The table specifies UDD within the year. Calculate the probability of exit from cause \(J\) between ages \(x+0.25\) and \(x+0.75\), conditional on being alive at age \(x+0.25\). Give the final probability to four decimal places.
Method. Use the conditional UDD formula. The interval length is \(0.75-0.25=0.50\), but the calculation must also condition on survival to \(x+0.25\).
The fraction surviving from age \(x\) to age \(x+0.25\) is \(1-(0.25)(0.20)=1-0.05=0.95\). The fraction surviving from age \(x\) to age \(x+0.75\) is \(1-(0.75)(0.20)=1-0.15=0.85\). Therefore the conditional survival probability over the requested interval is \(0.85/0.95=0.8947368\).
The conditional exit probability is \(1-0.8947368=0.1052632\), or, equivalently, \((0.50)(0.20)/0.95=0.10/0.95=0.1052632\). Rounded to four decimal places, the answer is 0.1053.
Check and common mistake. The unconditional probability of exit during this half-year, measured from age \(x\), is \(0.10\). Dividing by \(0.95\), the fraction surviving to the interval’s start, gives the larger conditional probability \(0.1053\). Using \(0.10\) as the answer omits the conditioning.
Fractional Ages Under a Constant Force
Under a constant force of decrement in the associated table during the year, survival over a fraction of a year is an exponential fraction of annual survival. If annual survival is \(p'_{x,j}=1-q'_{x,j}\), then for \(0\leq r\leq1\):
For an interval beginning at a fractional age \(x+s\), the force is constant throughout the year, so survival over the interval depends on its length \(t-s\). The corresponding conditional probabilities are:
Unlike the UDD expression, the constant-force interval formula does not require a ratio of two linear survival fractions. It follows from the exponential survival model: if the annual associated force is \(\mu'_{x,j}=-\ln(1-q'_{x,j})\), survival for \(t-s\) years is \(e^{-\mu'_{x,j}(t-s)}\), which equals \((1-q'_{x,j})^{t-s}\). This formula assumes a constant force over the relevant year; it is not a substitute for UDD when UDD is specified.
Worked Example: Constant-Force Exit Over Half a Year
A hypothetical associated single-decrement table gives \(q'_{x,K}=0.19\), and the within-year assumption is constant force. Calculate the probability of exit from cause \(K\) between ages \(x+0.20\) and \(x+0.70\), conditional on being alive at \(x+0.20\). Give the final probability to four decimal places.
Method. The interval length is \(0.70-0.20=0.50\) years. Annual associated survival is \(1-0.19=0.81\). Under constant force, survival over half a year is \(0.81^{0.50}=\sqrt{0.81}=0.90\).
The conditional exit probability is \(1-0.90=0.10\). To four decimal places, the answer is 0.1000.
Check. Survival over two successive half-years is \(0.90\times0.90=0.81\), which recovers the annual associated survival. This verifies that the fractional calculation agrees with the table’s annual value.
Common mistake. Applying a linear half-year decrement would give \(0.5(0.19)=0.095\). That calculation assumes UDD, not a constant force, and is therefore inconsistent with the stated basis.
Using Associated Table Values Correctly
A converted table may provide associated annual probabilities directly, or it may provide associated lives from which those probabilities can be calculated. If the number alive in the associated table at age \(x\) is \(l'_{x,j}\), then \(q'_{x,j}=(l'_{x,j}-l'_{x+1,j})/l'_{x,j}\). Once \(q'_{x,j}\) is known, apply the fractional-age formula for the specified within-year assumption. Do not insert the original multiple-decrement cause probability in its place.
The same conditioning principle applies whether the calculation is expressed through probabilities or associated lives. Under UDD, the associated lives at \(x+r\) are \(l'_{x,j}(1-rq'_{x,j})\). The conditional exit probability between \(x+s\) and \(x+t\) is the number exiting in that segment divided by the associated lives at \(x+s\). Under constant force, the ratio of associated lives at \(x+t\) to those at \(x+s\) is \((1-q'_{x,j})^{t-s}\).
Worked Example: Checking a Fractional Calculation Against Associated Lives
A hypothetical associated table for cause \(M\) has \(l'_{x,M}=1{,}000\) and \(l'_{x+1,M}=840\). Assume UDD. Find the conditional probability of exit from cause \(M\) between ages \(x+0.40\) and \(x+0.90\). Give the final probability to four decimal places.
Step 1: Find the annual associated probability. \(q'_{x,M}=(1{,}000-840)/1{,}000=160/1{,}000=0.16\).
Step 2: Find the associated lives at each fractional age. At \(x+0.40\), the associated lives are \(1{,}000[1-(0.40)(0.16)]=1{,}000(0.936)=936\). At \(x+0.90\), they are \(1{,}000[1-(0.90)(0.16)]=1{,}000(0.856)=856\).
Step 3: Calculate the conditional exit probability. There are \(936-856=80\) associated exits between the two ages, out of 936 associated lives at the start. Thus the probability is \(80/936=0.0854701\), which rounds to 0.0855.
Check and common mistake. The ratio \(856/936=0.9145299\) is the conditional survival probability; subtracting it from 1 gives \(0.0854701\), as above. Dividing the 80 exits by 1,000 instead would give an unconditional probability from age \(x\), not the required probability conditional on reaching \(x+0.40\).
A Reliable Calculation Sequence
Fractional-age questions are easiest to organize by separating the annual conversion from the within-year calculation. The conversion establishes the associated table value; the interpolation assumption determines how that value is distributed within the year.
Check whether the question asks for an associated single-decrement probability or a multiple-decrement probability. Use the associated annual value only for the associated-table calculation.
Use UDD formulas for a UDD associated table and constant-force formulas for a constant-force associated table. Do not combine their fractional formulas.
For an interval from \(x+s\) to \(x+t\), calculate its length \(t-s\) and identify whether the probability is conditional on being alive at \(x+s\).
Under either assumption, exit probability equals one minus the conditional survival probability. This gives a simple check on the arithmetic.
For an interval covering a full year, the formula must reproduce \(p'_{x,j}=1-q'_{x,j}\) or \(q'_{x,j}\). For a fractional interval, ensure the result is conditional on survival to its starting age when required.
Common Exam Traps
- Ignoring the condition at a fractional starting age. Under UDD, \((t-s)q'_{x,j}\) is an unconditional probability measured from age \(x\). Divide by \(1-sq'_{x,j}\) to obtain the conditional interval exit probability.
- Using the wrong decrement probability. The multiple-decrement probability \(q_x^{(j)}\) and the associated probability \(q'_{x,j}\) refer to different models. Use the converted associated value when the question asks about the associated table.
- Switching assumptions. Linear interpolation gives UDD results; raising annual survival to a fractional power gives constant-force results. A calculation that uses both without an explicit reason is inconsistent.
- Confusing interval length with endpoint age. For an interval from \(x+0.20\) to \(x+0.70\), the duration is \(0.50\), not \(0.70\). Under constant force, use the duration in the exponent.
- Forgetting what a survival ratio means. A ratio of associated lives at two fractional ages is conditional survival from the first age to the second. Subtract it from 1 only when the requested result is the corresponding conditional exit probability.
Check Your Understanding
For each question, use the stated associated-table basis and give the requested probability to four decimal places.
- An associated table has \(q'_{x,A}=0.24\) under UDD. Find the conditional exit probability between \(x+0.20\) and \(x+0.70\). (Answer: 0.1261)
- An associated table has \(q'_{x,B}=0.36\) under constant force. Find the conditional survival probability from \(x+0.25\) to \(x+0.75\). (Answer: 0.8000)
- Under UDD, an associated table for cause \(C\) has \(q'_{x,C}=0.15\). Calculate the conditional exit probability from \(x+0.40\) to \(x+0.90\). (Answer: 0.0798)
- Under constant force, an associated table gives \(q'_{x,D}=0.19\). Find the conditional exit probability from \(x+0.10\) to \(x+0.60\). (Answer: 0.1000)
- A multiple-decrement table has a probability \(q_x^{(E)}=0.12\) for cause \(E\), while its stipulated conversion gives \(q'_{x,E}=0.10\). Under UDD in the associated table, calculate the conditional exit probability from \(x+0.30\) to \(x+0.80\). (Answer: 0.0515)