Actuarial Mathematics for Modelling · Projecting and valuing cashflows contingent on multiple decrement events

Multiple Decrement Tables and Notation Explained

Updated 11 October 2026 · Fact-checked

A multiple decrement model follows a group of lives that can leave their current status for more than one reason, such as death or withdrawal. The table tracks (al)x, the number in the group at age x, and splits the exits by cause using (ad)x^j and (aq)x^j. You build it by applying each cause's rate to the opening number.

Understand Multiple Decrement Tables and Notation

In a basic life table, a life leaves by one route only: death. Many real contracts and schemes have several routes out. An employee in a pension scheme can die, withdraw, retire early because of ill health, or retire at normal age. A policyholder can die, surrender or lapse. Each route is called a decrement. A multiple decrement model handles all of them together.

The key idea is that the group is in one status, for example 'in service' or 'in force'. The group loses members to different causes over each year of age. The table counts the people in the group at each age and shows how many left for each cause. The notation uses an 'a' in brackets to show that you are looking at all decrements acting together, not one cause alone.

The main symbols are (al)x, the expected number still in the group at exact age x, and (ad)x^j, the expected number who leave between ages x and x+1 by cause j. Adding (ad)x^j over all causes gives (ad)x, the total leaving. Dividing by (al)x gives probabilities: (aq)x^j is the chance that a life aged x in the group leaves within a year by cause j. Here (aq)x is the total chance of leaving for any cause.

There is a second set of rates you must separate from these. The dependent rates (aq)x^j are the rates observed in the multiple decrement table, where all causes compete. The independent rates (also called absolute rates, written qx^j) are the rates that would apply if cause j were the only one acting. Dependent rates are smaller than independent rates for the same cause, because a life may leave by another cause first.

The table is a deterministic summary of expected numbers. You can read survival, exit probabilities by cause, and multi-year probabilities from it, and these feed into the projection and valuation of cashflows that depend on which decrement occurs.

Key rules to remember

Total decrements in a year: (ad)x = Σj (ad)x^j; Sum over all causes j between ages x and x+1.
Dependent decrement rate: (aq)x^j = (ad)x^j ÷ (al)x; Probability a life in the group at age x leaves by cause j within a year, with all causes acting.
Total probability of leaving: (aq)x = Σj (aq)x^j = (ad)x ÷ (al)x; Probability of leaving for any cause within the year.
Probability of staying: (ap)x = 1 − (aq)x = (al)x+1 ÷ (al)x; Probability of still being in the group at age x+1.
Roll-forward of the table: (al)x+1 = (al)x − (ad)x; Use this to build each new row of the table.
Multi-year survival in the group: n(ap)x = (al)x+n ÷ (al)x; Probability that a life in the group at age x is still in it at age x+n.
Force of decrement: μx = Σj μx^j, and t(ap)x = exp(−∫0^t μx+s ds); Total force is the sum of the forces for each cause. This is the continuous-time link.
Probability of exit by cause j within a year: (aq)x^j = ∫0^1 t(ap)x · μx+t^j dt; Survive in the group to time t, then leave by cause j at t.
Independent and dependent rates, constant forces: (ap)x = Πj px^j, and (aq)x^j = (μ^j ÷ μ) × (1 − (ap)x); Valid when each cause has a constant force over the year and the causes act independently. Here px^j = 1 − qx^j.

How to solve Multiple Decrement Tables and Notation questions

Use this method for any question that asks you to build, read or convert a multiple decrement table.

1Identify the status (for example in service or in force) and list every decrement cause. Give each a label such as d, w, i, r.
2Write down what you are given: numbers (al) and (ad)x^j, or rates. Decide whether the rates are dependent (aq)x^j or independent qx^j. The wording and notation tell you which.
3If rates are independent, convert them to dependent rates first. State your assumption, such as constant force for each cause over the year.
4Start from the opening number (al)x. Calculate each (ad)x^j = (al)x × (aq)x^j, and add them to get (ad)x.
5Roll forward: (al)x+1 = (al)x − (ad)x. Repeat for each age.
6To answer a probability question, put the relevant number of exits (or survivors) over the starting number (al) at the age the life is known to be in the group.
7Check that the exits by cause add to the total, and that the row sums to the opening number. Give the answer to a sensible number of decimal places, and say what it means.

Quickest way: Row-by-row table build

When to use it: Use when you are given dependent rates by cause and a starting number, and need several ages of the table.

Set up columns: age, (al)x, one (ad)x^j column per cause, total (ad)x.
Fill each cause's exits as (al)x × rate. Do not round until the end.
Total the exits, subtract from (al)x to get the next opening number.
For a multi-year probability, divide the relevant cumulative number by the first (al). Do not multiply rates unless you have not built the table.
Keep unrounded numbers in your calculator memory so later rows stay accurate.

Common mistakes in Multiple Decrement Tables and Notation

Using independent rates qx^j directly as if they were dependent rates (aq)x^j.
Both are called decrement rates and look similar, and single-decrement tables are more familiar.
Fix: Check the notation. The prefix (a) means dependent. If you are given qx^j, convert it first and state the assumption you use.
Subtracting only one cause's exits when rolling forward to (al)x+1.
You focus on the cause in the question and forget that other causes also remove lives.
Fix: Always subtract the total (ad)x, the sum over all causes, to get the next opening number.
Adding the dependent rates of different ages to get a multi-year probability.
Treating the rates like additive quantities across years.
Fix: Use numbers from the table. Divide the survivors or the exits by the starting (al). Rates apply to the survivors of the earlier year, not to the original group.
Dividing by the wrong (al) in a conditional probability.
The denominator must be the number in the group at the age you are conditioning on, but students use the first row.
Fix: Write the condition in words first, such as 'given in service at 41', then use (al)41 as the denominator.
Assuming (ap)x equals the product of the dependent probabilities of staying for each cause.
The product rule is true for independent rates, and students apply it to dependent ones.
Fix: For dependent rates, use (ap)x = 1 − Σ(aq)x^j. The product rule applies only to the independent rates px^j.
Rounding exits early so the row no longer adds up.
Exits are expected numbers, so they are fractional, and students round them like head counts.
Fix: Carry decimals through the table. Round only the final answer.

Worked examples

Example 1

A group of 10,000 employees is in service at exact age 40. The dependent rates are: at age 40, (aq)^d = 0.002 (death) and (aq)^w = 0.10 (withdrawal); at age 41, (aq)^d = 0.003 and (aq)^w = 0.08. Build the table for ages 40 and 41. Find the probability that an employee in service at 40 is still in service at 42, and the probability that they leave by withdrawal before age 42.

Show the solution

Age 40: (al)40 = 10,000. Deaths (ad)40^d = 10,000 × 0.002 = 20. Withdrawals (ad)40^w = 10,000 × 0.10 = 1,000. Total (ad)40 = 1,020.
Roll forward: (al)41 = 10,000 − 1,020 = 8,980.
Age 41: Deaths (ad)41^d = 8,980 × 0.003 = 26.94. Withdrawals (ad)41^w = 8,980 × 0.08 = 718.40. Total (ad)41 = 745.34.
Roll forward: (al)42 = 8,980 − 745.34 = 8,234.66.
Probability in service at 42 = (al)42 ÷ (al)40 = 8,234.66 ÷ 10,000 = 0.823466.
Withdrawals before 42 = 1,000 + 718.40 = 1,718.40. Probability = 1,718.40 ÷ 10,000 = 0.17184.

Answer: (al)42 = 8,234.66. The probability of still being in service at 42 is about 0.8235. The probability of leaving by withdrawal before 42 is about 0.1718.

Example 2

For a life aged x, the independent (single-decrement) rates are qx^d = 0.02 for death and qx^w = 0.10 for withdrawal. Assume each cause has a constant force of decrement over the year and the causes act independently. Find the dependent rates (aq)x^d and (aq)x^w.

Show the solution

Find the constant forces: μ^d = −ln(0.98) = 0.020203 and μ^w = −ln(0.90) = 0.105361.
Total force: μ = 0.020203 + 0.105361 = 0.125564.
Probability of staying in the group: (ap)x = px^d × px^w = 0.98 × 0.90 = 0.882. So (aq)x = 1 − 0.882 = 0.118.
Share of exits due to death: μ^d ÷ μ = 0.020203 ÷ 0.125564 = 0.16090.
(aq)x^d = 0.16090 × 0.118 = 0.018986.
(aq)x^w = 0.118 − 0.018986 = 0.099014. Check: 0.018986 + 0.099014 = 0.118.

Answer: (aq)x^d ≈ 0.01899 and (aq)x^w ≈ 0.09901. Each dependent rate is smaller than the matching independent rate, since a life can leave by the other cause first.

Exam tips

Write the notation key at the top of your answer: what (al), (ad)x^j and (aq)x^j mean in this question. It shows the examiner you know the model.
In MCQs, check whether the rate given is dependent or independent before doing any arithmetic. Options are often built from this trap.
In written answers, state your assumption (for example constant force for each decrement) before converting rates. Marks are given for stating it.
Always check that causes add up to the total and the row balances. This catches most arithmetic slips quickly.
In the computer-based paper, set up the table in a spreadsheet or R with one column per cause, and use formulas that roll forward so you can extend the table for any number of ages.

Practice questions from Projecting and valuing cashflows contingent on multiple decrement events

Multiple Decrement Tables and Notation: frequently asked questions

What is the difference between dependent and independent decrement rates?

A dependent rate (aq)x^j is the probability of leaving by cause j when all causes act together, and it is what the multiple decrement table shows. An independent rate qx^j is the probability of leaving by cause j if it were the only cause. Dependent rates are never larger than the matching independent rates.

What does (al)x mean in a multiple decrement table?

(al)x is the expected number of lives still in the group at exact age x, before any decrement over the next year. It plays the same role as lx in a life table but counts everyone in the status, not just survivors from death.

How do I construct a multiple decrement table?

Start with a chosen number at the first age. Multiply it by each dependent rate to find the exits by cause, add them to get the total, and subtract the total to get the number at the next age. Repeat for each age.

Why is the notation written with an (a) in brackets?

The (a) signals that the quantity refers to the combined, multiple decrement model rather than a single decrement. It separates (aq)x^j, the dependent rate, from qx^j, the independent rate.