Multiple Decrement Tables and Joint Life Applications

Understand Multiple Decrement Tables and Joint Life Applications

Key rules to remember

Total survival and decrement identity

l(x+1) = l(x) − Σ d(x, j); p(x, τ) = l(x+1) ÷ l(x); q(x, j) = d(x, j) ÷ l(x)

Dependent rates use l(x), the whole group at age x, as the denominator, not the number left by one cause.

Probability of surviving t years in the table

t p(x, τ) = l(x+t) ÷ l(x)

Use it to find the chance of still being in the group at a later age.

Probability of exit by cause j in year k+1

(k p(x, τ)) × q(x+k, j) = d(x+k, j) ÷ l(x)

Survive k years in the group, then leave by cause j in the next year.

Expected present value of a benefit paid at year end of exit

EPV = Σ over k and j of B(j, k+1) × v^(k+1) × d(x+k, j) ÷ l(x)

B is the benefit for cause j. Use v = 1 ÷ (1 + i), and adjust the power if payment is at the moment of exit.

Force of decrement link

q(x, j) = ∫ from 0 to 1 of s p(x, τ) × μ(x+s, j) ds; s p(x, τ) = exp(−∫ from 0 to s of μ(x+r, τ) dr); μ(τ) = Σ μ(j)

The total force is the sum of the forces for each cause. If the forces are constant over the year, q(x, j) = (μ(j) ÷ μ(τ)) × (1 − e^(−μ(τ))).

Independent and dependent rates

p(x, τ) = Π p'(x, j) if the causes are independent; p'(x, j) = exp(−∫ from 0 to 1 of μ(x+s, j) ds)

This holds only when the causes are independent. The table alone does not tell you whether they are.

Joint life survival (independent lives)

t p(xy) = t p(x) × t p(y)

Assumes the future lifetimes are independent. State this assumption.

Last survivor survival

t p(x̄ȳ) = t p(x) + t p(y) − t p(xy)

At least one life alive. For independent lives, also equals 1 − t q(x) × t q(y).

Assurances on two lives

A(x̄ȳ) = A(x) + A(y) − A(xy)

Holds for assurances paid at the same time point after the last or first death, and is valid without independence.

Annuities on two lives

ä(x̄ȳ) = ä(x) + ä(y) − ä(xy); reversionary annuity to y after x = ä(y) − ä(xy)

Annuity while exactly one life is alive = ä(x) + ä(y) − 2 × ä(xy).

Assurance and annuity link

A(xy) = 1 − d × ä(xy), where d = i ÷ (1 + i)

Works for whole life joint and last survivor statuses with the same timing basis.

Equivalence principle premium

P × ä(premium status) = EPV of benefits

Use the premium status for the premium term and the benefit status for the benefit.

How to solve Multiple Decrement Tables and Joint Life Applications questions

1. 1Identify the lives and the statuses. Write down the decrements or the states, for example death, withdrawal, joint life, last survivor, both alive.
2. 2List each cashflow and say exactly when it is paid and on which event. Mark premiums separately from benefits.
3. 3Choose the probability you need. For tables, use d(x+k, j) ÷ l(x) for exit by cause j in year k+1. For two lives, use joint or last survivor survival probabilities.
4. 4State the assumptions you rely on, such as independence of lives, independence of causes, uniform distribution of decrements, or constant forces.
5. 5Apply the discount factor for each payment time. Match the factor to the timing, either year end or moment of exit.
6. 6Sum the terms to get the expected present value. Use the equivalence principle if you need a premium.
7. 7Check the result. Probabilities should not exceed 1, and the last survivor value should be no smaller than the joint life value.
8. 8Write the final answer with units, and give a one-line comment if the question asks for interpretation.

Quickest way: Table columns and status identities

Common mistakes in Multiple Decrement Tables and Joint Life Applications

• Using l(x+k) as the denominator when finding the chance of exit in a later year from age x.

  The table gives d ÷ l at each age, and students read it as the probability from the start.

  Fix: Divide by l at the starting age. The chance of exit by cause j in year k+1 is d(x+k, j) ÷ l(x). Or multiply k p(x, τ) by q(x+k, j).

• Treating the dependent rate q(x, j) as if it were the independent rate q'(x, j).

  Both are labelled as the rate for cause j, and a single decrement table uses q' only.

  Fix: Check which one the question gives. Dependent rates come from the multiple decrement table. Independent rates apply when the cause acts alone. Convert only with a stated assumption.

• Multiplying survival probabilities of two lives without stating independence.

  The product formula t p(xy) = t p(x) × t p(y) is so common that it becomes automatic.

  Fix: Write the assumption. If the question gives a joint model with dependence, such as a four-state model with different intensities after one death, do not use the product.

• Writing the last survivor value as the sum of the single life values, forgetting to subtract the joint value.

  Students remember a plus sign but not the correction for double counting the period when both are alive.

  Fix: Use ä(x̄ȳ) = ä(x) + ä(y) − ä(xy). Then check that the result is above both single life values.

• Valuing benefits with the wrong timing of payment, such as year end instead of moment of exit.

  Tables give yearly decrements, and the discount power is not adjusted.

  Fix: Read the payment basis. For year-end payment, use v^(k+1). For payment at exit with uniform distribution, apply the adjustment the question's basis requires, such as a half-year factor.

• Adding probabilities of the same event under different causes to get a rate of exit by one cause.

  Students mix up total decrement with one cause.

  Fix: Sum d(x, j) over all causes to get total exits. Use a single cause only when the benefit depends on that cause.

Worked examples