Valuing Benefits Contingent on Decrement Events

Understand Valuing Benefits Contingent on Decrement Events

Key rules to remember

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.

How to solve Valuing Benefits Contingent on Decrement Events questions

1. 1List the decrements and the benefit for each one. Note whether each is paid at the end of the year or at the moment of decrement, and whether the amount changes with time.
2. 2Identify the rates you are given. Check whether they are dependent (absolute) rates q^(j), independent rates q'^(j), or forces μ^(j). Convert to dependent rates if needed before using the formulas.
3. 3Build a year-by-year schedule. For each year, find the probability in force at the start, then the probability of each decrement in that year, by multiplying in-force probability by q^(j).
4. 4Multiply each decrement probability by its benefit, then by the discount factor for that payment date. This gives the EPV for that year and cause.
5. 5Sum across years and causes. If premiums are involved, value them separately as a life-contingent annuity that is paid only while the life is in force.
6. 6For a premium, set EPV of premiums equal to EPV of benefits (plus expenses if stated), and solve for P.
7. 7For a reserve, use the recursion or Thiele's equation from the end of the term backwards, or value future benefits less future premiums prospectively.
8. 8Check: probabilities in each year must add to one (survive plus all decrements), and the EPV must be sensible against the largest benefit.

Quickest way: Table of lives, then one discounting pass

Common mistakes in Valuing Benefits Contingent on Decrement Events

• Using the independent (single-decrement) rate q'^(j) where the dependent rate q^(j) is needed.

  The notation looks similar, and students forget that other decrements can remove the life first.

  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.

  Students think of the benefit cause only and forget that withdrawal also removes lives.

  Fix: Always use the total survival probability p^(τ) = 1 − Σ q^(j) for the in-force population.

• Discounting a benefit paid at the end of the year of decrement by v^t instead of v^(t+1).

  The in-force probability is t p_x, so the index t is carried over to the discount factor.

  Fix: Write the payment date beside each term. A decrement in year t+1 paid at year end is discounted for t+1 years.

• Applying the premium to lives who have already left, for example discounting premiums with the decrement-year probability.

  Students mix up the benefit and premium timing patterns.

  Fix: Premiums at the start of a year need the probability of being in force at that time, t p_x^(τ), with discount v^t. No decrement probability enters.

• Getting the signs wrong in Thiele's equation, especially the term in tV^(j) − tV^(i).

  The sign convention for premiums and benefits is not stated clearly in working.

  Fix: Write the equation in words first: growth in reserve = interest on reserve + premium − benefit payments − expected cost of transitions, where the cost is the lump sum plus the value in the new state minus the current value.

• Treating a refund or surrender benefit as additional to the reserve release, or forgetting that a zero benefit still removes the life.

  Withdrawal with no payment feels like no event.

  Fix: Every decrement removes the life from the in-force group. Give it a benefit of zero in the schedule rather than leaving it out.

Worked examples