Mixed-Basis Table Conversions at Fractional Ages

Why the Within-Year Basis Matters

A decrement table may give annual probabilities while a problem asks for a value at a fractional age, or it may combine entries developed under different within-year assumptions. The annual values alone do not determine the fractional-age values: uniform distribution of decrements and constant force give different interpolations between the same annual endpoints. A sound conversion therefore identifies the basis for each step before applying a formula.

This tutorial focuses on conversions between multiple-decrement probabilities and associated single-decrement probabilities. The constant-force conversion relates the annual multiple-decrement table to its associated tables. Fractional-age interpolation is a separate operation, and its basis must be stated separately. If the two operations use different bases, report the result as a mixed-basis table conversion, not as if one assumption had applied throughout.

Definition: For causes \(1,\ldots,m\), let \(q_j\) be the probability that a member of the group exits during one year from cause \(j\), with the causes mutually exclusive. The total annual decrement probability is \(q=\sum_{j=1}^{m}q_j\), and the probability of remaining in the group for the year is \(p=1-q\). The associated single-decrement probability \(q'_j\) is the annual probability of exit from cause \(j\) in a hypothetical table where the other causes have been removed, as determined by the specified conversion assumption.

Annual Conversion Under Constant Forces

Suppose each cause has a constant force of decrement during the year, denoted by \(\lambda_j\). The total force is \(\lambda=\sum_j\lambda_j\). The probability of surviving all causes for one year is \(p=e^{-\lambda}\). The probability of exit from cause \(j\) during the year is \(q_j=(\lambda_j/\lambda)(1-e^{-\lambda})\). Thus the share of total exits attributed to cause \(j\) is \(\lambda_j/\lambda=q_j/q\), provided \(q>0\).

With cause \(j\) acting alone, its annual associated probability is \(q'_j=1-e^{-\lambda_j}\). Since \(\lambda_j/\lambda=q_j/q\), the constant-force conversion is:

q'_j=1-(1-q)^{q_j/q}.

This conversion requires a positive total decrement probability and the stated constant-force model over the year. If \(q=0\), every \(q_j=0\), and there is no decrement to convert. The formula is not a general identity independent of assumptions: a different within-year model can imply a different relationship between multiple-decrement and associated values.

Justification: Under constant forces, \(p=e^{-\lambda}\), so \(\lambda=-\ln p\). The cause share is \(\lambda_j/\lambda=q_j/q\), hence \(\lambda_j=-(q_j/q)\ln p\). The cause-alone survival probability is \(e^{-\lambda_j}=p^{q_j/q}\). Taking one minus this survival probability gives \(q'_j=1-p^{q_j/q}=1-(1-q)^{q_j/q}\).

Fractional Ages: Interpolate After Identifying the Basis

Let \(0\le t\le1\) be the fraction of a year elapsed. For an annual associated probability \(q'_j\), the two common interpolation assumptions give:

\begin{aligned} \text{Uniform distribution of decrements (UDD):}\qquad &q'_j(t)=tq'_j,\\ \text{Constant force:}\qquad &q'_j(t)=1-(1-q'_j)^t. \end{aligned}

Both formulas agree at \(t=0\) and \(t=1\), but generally differ between those endpoints. Under UDD, the probability accumulates linearly with elapsed time. Under constant force, the associated table has survival probability \((1-q'_j)^t\) through time \(t\). Do not use the constant-force annual multiple-decrement conversion formula as though it also specified the interpolation basis for every associated table.

A multiple-decrement table can likewise state its own fractional-age basis. For example, if its entries use UDD, then \(q_j(t)=tq_j\), so an observed fractional-age value can be divided by \(t\) to recover the annual \(q_j\), for \(t>0\). If the multiple-decrement table instead uses constant forces, its cause-specific fractional probability is \(q_j(t)=(q_j/q)\{1-(1-q)^t\}\). These are different models and should not be interchanged.

Read the table basis.
Identify whether each given fractional-age multiple-decrement or associated value is based on UDD or constant force.

Recover or calculate annual values.
For example, under UDD recover \(q_j=q_j(t)/t\). Keep each cause and its unit interval clear.

Apply the annual conversion basis.
If constant-force conversion is specified, use \(q'_j=1-(1-q)^{q_j/q}\) with annual multiple-decrement probabilities.

Interpolate the associated value on its own basis.
Use \(tq'_j\) for UDD or \(1-(1-q'_j)^t\) for constant force, as stated for that associated table.

Check endpoints and totals.
At \(t=1\), a fractional value must equal its annual value. Multiple-decrement annual cause probabilities must sum to \(q\), not to an associated probability.

Worked Conversions

Worked Example: Annual Conversion, Then UDD Interpolation

A hypothetical annual table has two causes, with \(q_A=0.09\) and \(q_B=0.06\). Convert cause A to its associated annual probability under constant forces, then find its value at \(t=0.4\) under UDD interpolation in the associated table. Also compare with constant-force interpolation at \(t=0.4\).

Method and assumptions. The annual multiple-decrement conversion uses constant forces for causes A and B. The requested fractional associated value uses UDD; the comparison value uses constant force. No interest assumption is involved.

First calculate the total decrement probability and annual survival probability:

q=0.09+0.06=0.15,\qquad p=1-0.15=0.85.

The cause A share is \(q_A/q=0.09/0.15=0.6\). Therefore:

q'_A=1-0.85^{0.6} =1-\exp\{0.6\ln(0.85)\} =1-\exp(-0.0975114) \approx 1-0.90709 =0.09291.

Rounded to four decimal places, the associated annual probability is \(0.0929\). Under the specified UDD interpolation:

q'_A(0.4)=0.4(0.09291)=0.037164\approx0.0372.

For comparison, constant-force interpolation of the same associated annual value gives:

q'_A(0.4)=1-(1-0.09291)^{0.4} =1-0.85^{0.24} \approx1-0.96175 =0.03825\approx0.0383.

The two fractional values differ, even though they share the same annual endpoint. Check: both interpolation formulas return \(0.09291\) at \(t=1\). The common mistake avoided here is using constant-force interpolation merely because the annual conversion used constant forces.

Worked Example: Recover Annual Multiple-Decrement Values From UDD Entries

A hypothetical table reports fractional-age multiple-decrement probabilities at \(t=0.4\): \(q_A(0.4)=0.032\) and \(q_B(0.4)=0.024\). These entries use UDD. Recover the annual values, convert cause A to an associated annual value under constant forces, and then find its associated value at \(t=0.4\) under UDD.

Method and assumptions. The fractional multiple-decrement entries are interpreted under UDD, the annual association is calculated under constant forces, and the associated table is interpolated under UDD. This deliberately mixed basis is valid as a conversion convention when those are the stated bases; it does not claim one continuous model generated every step.

Under UDD, divide each fractional value by \(0.4\):

q_A=\frac{0.032}{0.4}=0.08,\qquad q_B=\frac{0.024}{0.4}=0.06.

Thus \(q=0.08+0.06=0.14\) and \(p=1-0.14=0.86\). The cause A share is \(0.08/0.14=4/7\). Its constant-force associated annual value is:

q'_A=1-0.86^{4/7} =1-\exp\{(4/7)\ln(0.86)\} =1-\exp(-0.0861845) \approx1-0.91743 =0.08257.

Applying UDD to this associated annual value gives:

q'_A(0.4)=0.4(0.08257)=0.033028\approx0.0330.

If instead the associated interpolation basis were constant force, the value would be \(1-0.86^{(4/7)(0.4)}=1-0.86^{0.2285714}\approx0.0339\). Check: the recovered annual multiple-decrement probabilities sum to \(0.14\), and their UDD fractional values reproduce \(0.032\) and \(0.024\). The trap is to treat the given fractional entries as annual probabilities without first undoing their UDD scaling.

Worked Example: Recover an Annual Associated Value, Then Reconstruct Causes

A hypothetical associated table gives cause A's value at \(t=0.5\) as \(0.0513167\), using constant-force interpolation. The annual associated probability for cause B is \(q'_B=0.05\). Recover \(q'_A\), then reconstruct the annual multiple-decrement probabilities under the constant-force conversion.

Method and assumptions. Both associated cause values are treated as arising from the constant-force association model. The fractional associated entry uses constant-force interpolation. No rounding is applied until the reported probabilities.

From \(q'_A(0.5)=1-(1-q'_A)^{0.5}=0.0513167\), the associated survival probability for half a year is \(1-0.0513167=0.9486833\). Squaring gives the annual associated survival probability:

1-q'_A=(0.9486833)^2\approx0.900000, \qquad q'_A\approx0.100000.

Under constant forces, the associated force for each cause is \(\lambda_j=-\ln(1-q'_j)\). Hence:

\lambda_A=-\ln(0.90)=0.1053605,\qquad \lambda_B=-\ln(0.95)=0.0512933, $$ $$ \lambda=\lambda_A+\lambda_B=0.1566538.

The multiple-decrement annual survival probability is the product of the associated survivals:

p=e^{-\lambda}=0.90(0.95)=0.855,\qquad q=1-p=0.145.

Allocate the total decrement probability according to the force shares:

q_A=0.145\left(\frac{0.1053605}{0.1566538}\right) \approx0.09752, \qquad q_B=0.145\left(\frac{0.0512933}{0.1566538}\right) \approx0.04748.

To four decimal places, \(q_A=0.0975\) and \(q_B=0.0475\). Check: \(q_A+q_B=0.1450\), so \(p+q_A+q_B=0.855+0.145=1\). A common error is to set \(q_A=q'_A\); the associated probability is not the multiple-decrement probability because the latter accounts for competing exits.

Common Traps and Consistency Checks

Fractional-age calculations often fail because a correct formula is applied to the wrong quantity or under the wrong basis. Before accepting a result, check the following:

Do not confuse annual and fractional values. Under UDD, \(q_j(t)=tq_j\). An observed value at \(t<1\) is not itself the annual probability.
Do not confuse associated and multiple-decrement values. The multiple-decrement probabilities \(q_j\) sum to \(q\). Associated probabilities describe separate hypothetical tables and need not sum to \(q\).
Do not silently transfer a basis. Constant-force annual conversion does not require constant-force interpolation of an associated table if the problem specifies UDD for that table.
Do not use an annual conversion directly on fractional entries. First recover annual values using the stated fractional-age basis, unless a formula has specifically been developed for those fractional entries.

For a constant-force associated table, an effective numerical check is to convert an annual value \(q'_j\) to a fractional value and then back: \(1-q'_j(t)=(1-q'_j)^t\), so the annual survival structure remains consistent. Under UDD, check instead that \(q'_j(t)/t=q'_j\) for \(t>0\). These checks confirm interpolation arithmetic; they do not make different bases interchangeable.

Key takeaway: Keep three items visible in every calculation: the source table's fractional-age basis, the annual multiple-to-associated conversion basis, and the associated table's interpolation basis. Apply each formula only to values on the basis it requires, and verify that fractional values return the annual endpoint at \(t=1\).

Check Your Understanding

For each problem, state the basis used at each stage and retain full precision until the final answer.

A two-cause annual table has \(q_A=0.07\) and \(q_B=0.03\). Under constant-force annual conversion, find \(q'_A\), then find \(q'_A(0.25)\) under UDD interpolation. (Answer: \(q'_A=1-0.90^{0.7}\approx0.0711\); \(q'_A(0.25)\approx0.0178\).)
A table gives \(q_A(0.3)=0.021\) and \(q_B(0.3)=0.009\) under UDD. Recover both annual probabilities and find the constant-force associated annual value for cause B. (Answer: \(q_A=0.0700\), \(q_B=0.0300\), \(q'_B=1-0.90^{0.3}\approx0.0311\).)
An associated table gives \(q'_A(0.4)=0.0400\) under constant-force interpolation. Recover the annual associated probability. (Answer: \(q'_A=1-0.96^{1/0.4}=1-0.96^{2.5}\approx0.0970\).)
Under the constant-force association model, annual associated values are \(q'_A=0.08\) and \(q'_B=0.04\). Reconstruct the annual multiple-decrement probabilities. (Answer: \(p=0.92(0.96)=0.8832\), \(q=0.1168\), \(q_A\approx0.0784\), and \(q_B\approx0.0384\).)
A two-cause annual multiple-decrement table has \(q_A=0.12\) and \(q_B=0.08\). Convert cause A under constant forces, then interpolate its associated table to \(t=0.5\) under UDD. (Answer: \(q'_A=1-0.80^{0.6}\approx0.1253\), so \(q'_A(0.5)\approx0.0627\).)

What You'll Learn