Projecting and valuing cashflows contingent on multiple decrement events: formula sheet

Key formulas

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.

Total force of decrement

μ_{x+t} = Σ_j μ^j_{x+t}

The total force is the sum of the forces for each cause.

Survival probability

t p_x = exp(−∫₀ᵗ μ_{x+s} ds)

This is the probability of remaining in the group, not leaving by any cause.

Dependent (absolute) rate

t q^j_x = ∫₀ᵗ s p_x μ^j_{x+s} ds

s p_x is the survival probability in the multiple decrement table, so all causes are included.

Independent (net) rate

q'^j_x = 1 − exp(−∫₀¹ μ^j_{x+t} dt)

The rate in the associated single decrement table for cause j.

Product of single survival probabilities

p_x = Π_j (1 − q'^j_x)

Holds because the total force is the sum of the individual forces.

Total of dependent rates

Σ_j q^j_x = 1 − p_x

Use this as a check.

UDD in the multiple decrement table

s p_x μ^j_{x+s} = q^j_x for 0 ≤ s ≤ 1, so the t-year probability t q^j_x = t × q^j_x

On the left, t q^j_x is the probability of leaving by cause j within time t (the t is the prefix). On the right, t multiplies the one-year rate q^j_x. Under UDD in the multiple decrement table the t-year probability is linear in t, for 0 ≤ t ≤ 1.

Independent rate from dependent rates (UDD in the multiple decrement table)

q'^j_x = 1 − (1 − Σ_i q^i_x)^(q^j_x ÷ Σ_i q^i_x)

Equivalently 1 − p_x^(q^j_x ÷ (1 − p_x)). Use when all dependent rates are given. Why it works: under UDD in the multiple decrement table, s p_x μ^j_{x+s} = q^j_x and s p_x μ_{x+s} = 1 − p_x. So μ^j_{x+s} ÷ μ_{x+s} = q^j_x ÷ (1 − p_x), a constant. Then ∫₀¹ μ^j_{x+s} ds = (q^j_x ÷ (1 − p_x)) × (−ln p_x), which gives the formula.

Dependent rate from independent rates, two causes (UDD in each single decrement table)

q^1_x = q'^1_x × (1 − ½ q'^2_x)

Exact when each cause is uniform in its own single decrement table. For more than two causes, q^j_x = ∫₀¹ Π_{i≠j} (1 − t q'^i_x) × q'^j_x dt, with UDD in each single table. With two causes this integral gives the formula above.

Number in force after one year

(al)_{x+1} = (al)_x − Σ_j (ad)^j_x

Sum over all causes j. Use the same year's decrements for all causes.

Decrements by cause

(ad)^j_x = (al)_x × (aq)^j_x

Use the dependent (all causes acting) rate (aq)^j_x. This gives the expected number leaving by cause j between age x and x+1.

Rate from table

(aq)^j_x = (ad)^j_x ÷ (al)_x

Use this when the question gives a multiple decrement table instead of rates.

Probability of staying in force one year

(ap)^τ_x = 1 − Σ_j (aq)^j_x = (al)_{x+1} ÷ (al)_x

τ means all causes combined.

Probability in force t years ahead

t(ap)^τ_x = (al)_{x+t} ÷ (al)_x = (ap)^τ_x × (ap)^τ_{x+1} × … × (ap)^τ_{x+t−1}

Multiply the one-year survival probabilities of consecutive ages.

Expected cashflow for cause j in year t+1 (single policy)

t(ap)^τ_x × (aq)^j_{x+t} × B_j

B_j is the benefit payable on exit by cause j. Payment is at the time stated, for example the end of the year of exit.

Expected present value

EPV = Σ_t E(CF_t) × v^t

Use v = 1 ÷ (1 + i). If exits happen mid-year, discount from the actual payment time, using the stated timing.

Total survival probability (one year)

p_x^(τ) = 1 − Σ_j q_x^(j)

Here q_x^(j) is the dependent (absolute) probability of leaving by cause j in the year, so the causes add up. Do not use independent rates here unless you have converted them first.

Probability of surviving t years

t p_x^(τ) = p_x^(τ) × p_{x+1}^(τ) × … × p_{x+t−1}^(τ)

In terms of lives in force, t p_x^(τ) = l_{x+t}^(τ) ÷ l_x^(τ).

EPV of a benefit on cause j, paid at end of year of decrement

EPV = Σ_{t≥0} v^(t+1) × t p_x^(τ) × q_{x+t}^(j) × B_{t+1}^(j)

B_{t+1}^(j) is the benefit if the life leaves by cause j in year t+1. For several causes, sum the EPVs.

Same EPV using a table of lives

EPV = Σ_{t≥0} v^(t+1) × d_{x+t}^(j) × B_{t+1}^(j) ÷ l_x^(τ)

Here d^(j) = l^(τ) × q^(j) is the expected number leaving by cause j.

Continuous-time EPV

EPV = ∫ from 0 to ∞ of v^t × t p_x^(τ) × μ_{x+t}^(j) × B_t dt

Here t p_x^(τ) = exp(−∫ from 0 to t of Σ_j μ_{x+s}^(j) ds). The force μ^(j) is the transition intensity for cause j.

One-year recursion for an EPV

A_x = v × [ Σ_j q_x^(j) × B^(j) + p_x^(τ) × A_{x+1} ]

Works for benefits paid at end of the year of decrement. Start at the final age, where A = 0 if cover ends.

Policy value recursion (premium at start, benefits at end of year)

(tV + P_t) × (1 + i) = Σ_j q_{x+t}^(j) × B^(j) + p_{x+t}^(τ) × t+1V

Here tV is the policy value for a life in force. Benefits on decrement are the amounts paid, which may include a policy value release if the benefit is a refund.

Thiele's equation for state i

d/dt tV^(i) = δ × tV^(i) + P^(i) − b^(i) − Σ_{j≠i} μ_{x+t}^(ij) × ( S^(ij) + tV^(j) − tV^(i) )

Here P^(i) is the premium rate in state i, b^(i) the benefit payment rate in state i, S^(ij) the lump sum on moving from i to j, and tV^(j) the value in the state moved to. If j is a terminal state such as death, tV^(j) = 0.

Quick revision

• 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.

Common mistakes

• Using independent rates qx^j directly as if they were dependent rates (aq)x^j. 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. Fix: Always subtract the total (ad)x, the sum over all causes, to get the next opening number.
• Treating dependent rates as if they were independent and multiplying (1 − q^j) together. Fix: Only independent rates multiply: p_x = Π(1 − q'^j_x). Dependent rates add: Σ q^j_x = 1 − p_x.
• Using the (1 − ½ q') formula when the question states UDD in the multiple decrement table. Fix: Underline the assumption in the question. Multiple decrement table UDD gives the exponent formula. Single table UDD gives the (1 − ½ q') formula.
• Applying the rate to the original group every year instead of the survivors. Fix: Always use the number in force at the start of that year. Update it every row.
• Paying a benefit for every life who leaves, ignoring which cause it was. Fix: Compute exits separately by cause and multiply each by its own benefit.
• Using the independent (single-decrement) rate q'^(j) where the dependent rate q^(j) is needed. Fix: Check the notation in the question. In EPV and survival formulas use the dependent rates, which add up to 1 − p^(τ). Convert independent rates first if that is what you are given.
• Using only the survival probability for one cause, for example (1 − q^d), when working out who is still in force. Fix: Always use the total survival probability p^(τ) = 1 − Σ q^(j) for the in-force population.

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.
• Read the assumption wording first. The phrase 'in the multiple decrement table' or 'in each single decrement table' decides the formula.
• In a written answer, define your notation. The IAI marks method, so state which rate is dependent and which is independent.
• Use the check Σ q^j = 1 − p_x. It catches most arithmetic and formula errors in under a minute.