IAI Actuarial Core Principles · Actuarial Mathematics for Modelling
Projecting and Valuing Cashflows with Multiple Decrements
A multiple decrement model lets a life leave its current state by more than one cause, such as death, withdrawal or retirement. You build a table of survivors and decrements, project expected cashflows for each cause, then discount and sum them. Each benefit is paid only on its own cause of exit.
What this chapter covers
This chapter extends the single-life survival model in CM1 to cases where a person can exit a state in several ways. A pension scheme member may die, leave service, become disabled or retire. Each cause has its own rate, and each may trigger a different benefit. The key question is how to combine these causes in one consistent model.
You start with the notation and the multiple decrement table: (al)x⁽τ⁾ for the number still in the state at age x, (ad)x⁽ʲ⁾ for the number leaving by cause j, and μx⁽ʲ⁾ for the force of decrement. You then separate the dependent rates (the real, observed rates in the presence of all causes) from the independent (single-decrement) rates, and learn how to move between them under stated assumptions.
The last two topics are applications. You project expected cashflows, such as contributions, benefits and expenses, for a group of lives. Then you value benefits that depend on the cause of exit. This links to the theory of interest, equation of value, and the pricing and reserving work elsewhere in CM1. It also feeds into pensions and health-insurance questions in written papers and in the computer-based Paper B.
CM1 gives a large share of its syllabus weighting to pricing and reserving, and to decrement and multiple life models. Multiple decrement questions are a natural way to test both. They appear in written answers where you must show notation, working and a final value, and they suit spreadsheet or R work in Paper B. The ideas are mechanical once understood, so careful practice turns them into reliable marks. Slips in notation or in which rate to use cost marks quickly, so the chapter rewards disciplined method.
Projecting and valuing cashflows contingent on multiple decrement events: topics in the order to study them
- 1Multiple Decrement Tables and NotationEverything else uses this notation and the survivor and decrement layout, so learn it first.
- 2Dependent and Independent Decrement RatesYou need to know which rate to use and how to convert between them before you build any projection.
- 3Projecting Cashflows with Multiple DecrementsProjection puts the table and rates to work, giving expected cashflows year by year.
- 4Valuing Benefits Contingent on Decrement EventsValuation discounts the projected cashflows, so it comes last and combines all earlier skills.
How to prepare Projecting and valuing cashflows contingent on multiple decrement events
Treat this chapter as one workflow: set up the table, choose the right rates, project, then discount. Practise the workflow end to end, not just each piece.
- Write out the notation on one page and say what each symbol means in words: survivors, decrements by cause, total force of decrement.
- Build a small multiple decrement table by hand from given rates. Check that the decrements by cause add up to the total decrements.
- Practise converting between dependent and independent rates. Always note the assumption given, such as uniform distribution of each decrement in the associated single-decrement table.
- Project cashflows for a cohort of lives. Use the number in force at the start of each year, and apply each benefit only to its own cause.
- Value the benefits by discounting each expected cashflow at the stated interest rate. Pay attention to timing, whether at the year end or at the moment of exit.
- Redo a past-paper question under timed conditions. Then repeat the calculation in a spreadsheet so you are ready for Paper B.
Common mistakes in Projecting and valuing cashflows contingent on multiple decrement events
Using independent rates as if they were dependent rates in a projection
Fix: Check the definition. Projections of actual numbers need dependent rates. Convert first if you are given independent ones.
Not stating the assumption when converting between rates
Fix: Write the assumption in one line, then show the formula that follows from it. The marker looks for both.
Paying a benefit on the wrong cause or on all exits
Fix: Project decrements by cause separately and attach each benefit only to its own cause.
Applying decrements to the wrong starting population
Fix: Set out the table with a clear starting number each year and subtract that year's total decrements to get the next.
Getting the timing of discounting wrong
Fix: Read the benefit terms first and mark the payment time on a timeline. If exit is mid-year, use the stated assumption for the average time.
Skipping a reasonableness check on the final value
Fix: Check that cause-wise decrements sum to the total and that the value is plausible against a rough estimate.
Last-day revision: Projecting and valuing cashflows contingent on multiple decrement events
- A multiple decrement model allows more than one cause of exit from a state.
- (al)x⁽τ⁾ is the number in the state at age x; (ad)x⁽ʲ⁾ is the number leaving by cause j.
- Total decrements in a year equal the sum of decrements over all causes.
- μx⁽τ⁾ = Σ μx⁽ʲ⁾, the total force of decrement is the sum of the forces by cause.
- Dependent rates apply when all causes act together; independent rates apply as if only one cause acted.
- Conversion between the two needs an explicit assumption, so state it.
- Project from the number in force at the start of each period.
- Each benefit is paid only on its own cause of exit.
- Value = expected payment × discount factor, summed over all times and causes.
- Check timing: year-end payments versus payments at the moment of exit.
- Sanity-check that survivors never go negative and probabilities sum to at most 1.
Projecting and valuing cashflows contingent on multiple decrement events practice questions
- In a double-decrement service table, l_x^(τ) = 10,000 lives are in service at age x. During the year, d_x^(1) = 300 leave by withdrawal and …
- In a multiple decrement table for active employees with two causes of exit, death (d) and withdrawal (w), the force of decrement from cause …
- A policy is in its final year, with no further premiums. At the end of the year it pays 50,000 on death, 10,000 on withdrawal and 20,000 on …
- A member earns an annual salary of Rs 5,00,000. Over the next year the probabilities of decrement while in service are: death 0.02 and withd…
- In a double decrement model, the force of decrement from cause 1 is a constant 0.02 per annum and from cause 2 is a constant 0.03 per annum.…
- Rs 1,000 is paid at the start of each year for up to three years while an employee remains in service. The probability of being in service i…
- In a double decrement table for a pension scheme, l_60 = 1000 members are in service at age 60. During the following year 12 members leave b…
- A service table for Indian employees shows 2,000 active members at age 60. During the following year, 30 members die in service and 90 withd…
Projecting and valuing cashflows contingent on multiple decrement events: frequently asked questions
Is this chapter tested in both Paper A and Paper B of CM1?
Yes, the ideas can appear in either. Paper A tests notation, derivations and written working. Paper B is computer-based, so you may need to build the table and valuation in a spreadsheet or in R.
What is the difference between dependent and independent decrement rates?
A dependent rate is the probability of leaving by a given cause when all causes act together. An independent rate is the probability of leaving by that cause if it were the only one acting. Converting between them needs a stated assumption.
Do I need to memorise formulas for this chapter?
Learn the core notation and the relationships between total and cause-specific decrements. Most other results can be derived from these and from the assumption you are given, which is safer than rote memory.
How should I practise valuation questions?
Draw a timeline of cashflows, list the probability of each cause at each time, then discount and sum. Doing the same question by hand and in a spreadsheet helps you catch errors and prepares you for Paper B.