Identify and apply Thiele's differential equation in a single life or multiple state setting.

Thiele's equation says reserves grow with interest, premiums in, and benefits out, with one extra term for the risk released or absorbed at every potential transition. Master that one sentence and the formula writes itself in both single-life and multi-state form.

Reserves change for three reasons. Interest credits the existing reserve at force $\delta_t$. Net cash flow (premium in minus continuous benefit out) adjusts the balance. Decrements release the reserve to pay a benefit or jump to a new state. Thiele packages all three into one differential equation.

KEY: The "net amount at risk" is what the insurer truly loses on the decrement. It equals the benefit paid plus the reserve assumed for the new state minus the reserve already held. Multiplying by the transition intensity gives the expected loss rate.

Let ${}_tV$ denote the reserve at duration $t$ for a fully continuous policy on $(x)$. Premium rate is $P_t$, continuous annuity benefit rate is $b_t$, lump-sum death benefit is $S_t$, and force of mortality is $\mu_{x+t}$.

Common mistakes

• Wrong sign on the decrement term. The mortality term $-\mu(S - V)$ is subtracted because death drains the reserve. Writing $+\mu(S - V)$ flips every slope.
• Confusing the $B^{(i)}$ sign. If $B^{(i)}$ is defined as net OUT, premiums enter as a negative $B$. Some sources use net IN with a sign flip on every term. Pick a convention and freeze it.
• Skipping the multi-state net-amount-at-risk structure. The correct bracket is $S^{ij} + {}_tV^{(j)} - {}_tV^{(i)}$. Using just $S^{ij}$ ignores the reserve switch and badly misvalues recoveries.

Bottom line

• Single-life form: $\frac{d}{dt} {}_tV = \delta_t \cdot {}_tV + P_t - b_t - \mu_{x+t}(S_t - {}_tV)$.
• Multi-state form: one ODE per non-absorbing state; for state $i$, $\frac{d}{dt} {}_tV^{(i)} = \delta_t \cdot {}_tV^{(i)} - B_t^{(i)} - \sum_{j\ne i} \mu_{x+t}^{ij}\bigl(S_t^{ij} + {}_tV^{(j)} - {}_tV^{(i)}\bigr)$.
• Single-life net amount at risk is $S_t - {}_tV$ (death benefit minus the current reserve).
• Multi-state net amount at risk for the $i \to j$ transition is $S_t^{ij} + {}_tV^{(j)} - {}_tV^{(i)}$.

Exam shortcut

Write the rate-of-change sentence in English first: interest, plus premium, minus benefit, minus risk cost. Then translate term by term. Sign errors evaporate when you narrate before formulating. DECISION: Single decrement, lump-sum-only benefit goes to single-life Thiele. Continuous state-dependent annuities or returns from disabled to healthy go to multi-state Thiele. For multi-state problems, build a small table: states down the left, columns for $B^{(i)}$, each $\mu^{ij}$, each $S^{ij}$, each ${}_tV^{(i)}$.

The full lesson (about 1,968 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2e

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