Why Fractional-Year Interpolation Matters
A decrement table may provide annual probabilities, while a valuation or analysis requires the probability of decrement over only part of a year. The annual value alone does not determine a unique fractional-year probability: an interpolation assumption is needed. Under a constant force assumption, survival within the year follows an exponential curve. Under UDD, survival decreases linearly. These assumptions generally give different answers, even when they use the same annual probability.
This tutorial focuses on fractional-year probabilities in an associated single-decrement table. For cause \(j\), that table treats cause \(j\) as the only decrement and uses an associated annual probability derived from the multiple-decrement model. It is not the same as simply interpolating the observed annual probability of cause \(j\).
From an Annual Associated Probability to a Fraction of a Year
The constant force assumption says that the force for the associated decrement is unchanged throughout the year. Consequently, the probability of surviving \(t\) years is exponential in \(t\). Since annual survival is \(1-q_x^{\prime(j)}\), the force is chosen to satisfy \(e^{-\mu_x^{\prime(j)}}=1-q_x^{\prime(j)}\). Raising annual survival to the power \(t\) gives survival for \(t\) years.
The decrement probability is the complement of survival. This gives the fractional-year formula directly. The assumption is applied to the associated single-decrement table; it does not say that the observed multiple-decrement probability for cause \(j\) is interpolated by the same formula using its observed annual value.
For comparison, under UDD the associated survival probability is linear within the year:
Both assumptions reproduce the same annual probabilities at \(t=1\), and both give zero decrement at \(t=0\). Between those endpoints, the curves differ. The constant-force curve is exponential; the UDD curve is a straight line.
Worked Example: Interpolating an Associated Annual Probability
A hypothetical associated single-decrement table has annual probability \(q_x^{\prime}=0.36\). Calculate survival and decrement over half a year under constant force, then compare with UDD. State the constant force to four decimal places and probabilities to four decimal places.
Method. Use annual associated survival \(1-q_x^{\prime}=0.64\), raise it to the fractional duration for constant-force survival, and take the complement to obtain decrement.
The constant force is $$ \mu_x^{\prime}=-\ln(0.64)=0.4462871\ldots\approx0.4463. $$ For \(t=0.5\), survival is $$ {}_{0.5}p_x^{\prime}=(0.64)^{0.5}=0.8. $$ Therefore the constant-force decrement probability is $$ {}_{0.5}q_x^{\prime}=1-0.8=0.2. $$ Under UDD, the half-year decrement probability is $$ {}_{0.5}q_x^{\prime}=0.5(0.36)=0.18, $$ and survival is \(1-0.18=0.82\). To four decimal places, constant force gives survival \(0.8000\) and decrement \(0.2000\), while UDD gives survival \(0.8200\) and decrement \(0.1800\).
Check. Under each assumption, half-year survival plus half-year decrement equals \(1\). At a full year, both models return annual decrement \(0.36\). The common mistake avoided here is applying the annual probability linearly when the specified interpolation assumption is constant force.
When the Annual Probability Comes from Multiple Decrements
In a multiple-decrement model, let \(q_x^{(j)}\) denote the observed annual probability of exit from cause \(j\), and let \(q_x=\sum_j q_x^{(j)}\) be the total annual exit probability. Under constant forces for the competing causes, the total force is \(M_x=-\ln(1-q_x)\). The force for cause \(j\) is its share of the observed exits multiplied by the total force:
The annual probability \(q_x^{\prime(j)}\) is the associated single-decrement probability for cause \(j\). Once it is obtained, the fractional-year associated probabilities follow from the exponential formula. The observed annual probabilities \(q_x^{(j)}\) and the associated probabilities \(q_x^{\prime(j)}\) have different roles: the first describe exits with competing causes present, while the second describe a table in which cause \(j\) acts alone.
Worked Example: Fractional Associated Probabilities from Rounded Cause Data
A hypothetical multiple-decrement table reports annual observed probabilities \(q_x^{(1)}=0.157388\) and \(q_x^{(2)}=0.236082\). Treat these displayed probabilities as the inputs. For each cause, calculate the associated half-year decrement probability under constant force and under UDD. Give probabilities to six decimal places.
Method. First calculate the total from the displayed cause probabilities, then find the total force and allocate it by cause share. Because the inputs are rounded, retain the resulting values rather than replacing them with a presumed exact total force.
The total observed annual probability from the stated inputs is $$ q_x=0.157388+0.236082=0.393470. $$ Thus $$ M_x=-\ln(1-0.393470)=-\ln(0.606530) =0.500001087\ldots\approx0.500001. $$ The cause shares are \(0.157388/0.393470=0.4\) and \(0.236082/0.393470=0.6\). Hence $$ \mu_x^{(1)}=0.4(0.500001087\ldots)=0.200000435\ldots, \qquad \mu_x^{(2)}=0.6(0.500001087\ldots)=0.300000652\ldots. $$ The associated annual probabilities are $$ q_x^{\prime(1)}=1-e^{-0.200000435\ldots} =0.181269603\ldots, \qquad q_x^{\prime(2)}=1-e^{-0.300000652\ldots} =0.259182262\ldots. $$ For half a year under constant force, the cause-specific associated probabilities are $$ {}_{0.5}q_x^{\prime(1)} =1-e^{-0.200000435\ldots(0.5)} =0.095162779\ldots, $$ $$ {}_{0.5}q_x^{\prime(2)} =1-e^{-0.300000652\ldots(0.5)} =0.139292304\ldots. $$ Under UDD, multiply each associated annual probability by \(0.5\): $$ {}_{0.5}q_{x,\mathrm{UDD}}^{\prime(1)} =0.5(0.181269603\ldots)=0.090634802\ldots, $$ $$ {}_{0.5}q_{x,\mathrm{UDD}}^{\prime(2)} =0.5(0.259182262\ldots)=0.129591131\ldots. $$ To six decimal places, constant force gives \(0.095163\) for cause 1 and \(0.139292\) for cause 2; UDD gives \(0.090635\) and \(0.129591\), respectively.
Check. The sum of the rounded observed inputs is \(0.393470\). Separately, \(1-e^{-0.5}\approx0.39346934\), not \(0.393470\) to six decimal places. The small difference arises because the supplied cause probabilities are rounded. Using the stated inputs gives \(M_x\approx0.500001\), and therefore \(\mu_x^{(2)}\approx0.300001\), not \(0.300000\). This calculation avoids silently treating rounded inputs as exact values from an unrounded model.
The two associated annual probabilities are not intended to sum to the observed total. Each belongs to a separate single-decrement table. Adding them as though they were mutually exclusive observed probabilities would mix two different interpretations.
Probabilities Within an Interior Part of the Year
The formula for \({}_t q_x^{\prime(j)}\) gives the probability of decrement by time \(t\), measured from the start of the year. A question may instead ask for decrement between times \(s\) and \(s+t\). Under constant force, the probability of surviving to \(s\) and then decrementing during the next \(t\) years is
Here the force is assumed constant over the relevant part of the year, so the same force applies at \(s\). Conditional on having survived to \(s\), the probability of decrement in the next \(t\) years is \(1-e^{-\mu_x^{\prime(j)}t}\). The unconditional probability from the start of the year includes the probability \(e^{-\mu_x^{\prime(j)}s}\) of reaching time \(s\).
Worked Example: Decrement Between Two Fractional Times
An associated table has annual probability \(q_x^{\prime}=0.36\). Calculate the probability of decrement between \(t=0.25\) and \(t=0.75\), both conditional on survival to \(0.25\) and unconditional from the start of the year, under constant force. Compare the unconditional answer with UDD. Give probabilities to four decimal places.
Method. Annual survival is \(0.64\). Under constant force, survival to time \(r\) is \(0.64^r\). First calculate conditional interval survival; then multiply the interval decrement probability by survival to the start of the interval for the unconditional probability.
The conditional probability of decrement during the next \(0.5\) years is $$ 1-\frac{{}_{0.75}p_x^{\prime}}{{}_{0.25}p_x^{\prime}} =1-0.64^{0.75-0.25} =1-0.64^{0.5} =1-0.8=0.2. $$ Survival to \(0.25\) is $$ {}_{0.25}p_x^{\prime}=0.64^{0.25} =\sqrt{0.8}=0.894427191\ldots. $$ Therefore the unconditional probability of decrement in the interval is $$ 0.894427191\ldots(0.2)=0.178885438\ldots\approx0.1789. $$ Under UDD, survival at \(0.25\) is \(1-0.25(0.36)=0.91\), and survival at \(0.75\) is \(1-0.75(0.36)=0.73\). The unconditional interval probability is their difference: $$ 0.91-0.73=0.18. $$ Thus the conditional constant-force probability is \(0.2000\), the unconditional constant-force probability is \(0.1789\), and the unconditional UDD probability is \(0.1800\).
Check. The unconditional probability is smaller than the conditional probability because not everyone survives to time \(0.25\): \(0.894427191\ldots(0.2)=0.178885438\ldots\). The common mistake avoided is reporting the conditional interval probability as though it were unconditional.
Exam Checks and Common Traps
Check whether the input is an observed multiple-decrement probability \(q_x^{(j)}\) or an associated single-decrement probability \(q_x^{\prime(j)}\).
For constant force use \((1-q_x^{\prime(j)})^t\) for survival. For UDD use \(1-tq_x^{\prime(j)}\).
For an interval starting after time zero, distinguish the probability conditional on reaching its start from the unconditional probability from the start of the year.
At \(t=0\), survival is \(1\) and decrement is \(0\); at \(t=1\), the annual associated probability is recovered. Survival plus decrement must equal \(1\).
A frequent error is to apply \(1-(1-q)^t\) directly to an observed cause probability \(q_x^{(j)}\) even though the question asks for an associated probability under a multiple-decrement model. First obtain the associated annual probability, or its associated force, using the stated model. Another error is to use \(tq\) under constant force; that is the UDD rule, not the exponential rule. Finally, when inputs are rounded, calculations from those displayed values may differ slightly from results based on the unrounded underlying data.
Check Your Understanding
For each problem, identify the interpolation assumption and distinguish associated from observed probabilities where relevant.
- An associated single-decrement annual probability is \(0.19\). Find half-year survival and decrement under constant force, and the half-year decrement under UDD. Give answers to four decimal places. (Answer: constant-force survival \(0.9000\), constant-force decrement \(0.1000\), UDD decrement \(0.0950\))
- An associated annual probability is \(0.51\). Find the quarter-year decrement under constant force and under UDD. Give answers to four decimal places. (Answer: constant force \(1-0.49^{0.25}=0.1633\); UDD \(0.1275\))
- An associated decrement force is \(0.4\) per year. Find survival and decrement over three months under constant force. Then find the UDD decrement using the annual probability implied by that force. Give answers to four decimal places. (Answer: survival \(e^{-0.1}=0.9048\), constant-force decrement \(0.0952\), annual probability \(1-e^{-0.4}=0.3297\), UDD decrement \(0.0824\))
- Observed annual cause probabilities are \(0.10\) and \(0.20\), and both causes operate under constant forces. Find the associated half-year decrement probability for the first cause. Give the answer to four decimal places. (Answer: total probability \(0.30\), total force \(-\ln(0.70)=0.3567\), first-cause force \(0.1189\), associated half-year decrement \(1-e^{-0.1189/2}\approx0.0577\))
- An associated annual probability is \(0.36\). Under constant force, find the unconditional probability of decrement between times \(0.25\) and \(0.75\). Then find the corresponding UDD unconditional interval probability. Give answers to four decimal places. (Answer: constant force \(0.64^{0.25}(1-0.64^{0.5})=0.1789\); UDD \((1-0.25(0.36))-(1-0.75(0.36))=0.1800\))