IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Valuing cashflows contingent on multiple transition events: formula sheet

Full chapter guide

Key formulas

Markov property: P[X(t) = j | X(s) = i, X(r) = x(r) for all r < s] = P[X(t) = j | X(s) = i]; State this in words too: the future depends only on the present state, not on the history.
Transition probability: t p_x^ij = P[X(x + t) = j | X(x) = i]; j may equal i. It includes paths that leave i and return.
Probabilities sum to one: Σ over all states j of t p_x^ij = 1; Use this to find a missing probability or to check your answers.
Chapman-Kolmogorov equations: s+t p_x^ij = Σ over k of ( s p_x^ik × t p_{x+s}^kj ); Sum over every state k the life could be in at the intermediate time. The second factor uses age x + s.
Transition intensity: μ_x^ij = lim as h→0 of ( h p_x^ij ÷ h ), for i ≠ j; Defined for different states. It is a rate, not a probability, and can exceed 1.
Holding probability: t p_x^īi = exp( − ∫ from 0 to t of Σ over j≠i of μ_{x+s}^ij ds ); Probability of remaining continuously in state i. With constant intensities it is exp(−t × total intensity out of i).
Probability of a first move to j: P[in j at time t, having first left i directly to j] = ∫ from 0 to t of ( s p_x^īi × μ_{x+s}^ij × (t−s) p_{x+s}^jj ) ds; Condition on the time s of leaving i. If j is absorbing, the last factor is 1. In general t p_x^ij also includes routes that go from i to j via other states, so add those paths. The integral equals t p_x^ij only when j can be reached from i by this direct route alone.
Definition of transition intensity: μ_ij(x) = lim (h→0+) [ h p_x^ij ÷ h ], for i ≠ j; Assumes a Markov model with probabilities that are differentiable in time.
Small-interval approximation: h p_x^ij = h × μ_ij(x) + o(h), for i ≠ j; o(h) is a term that goes to zero faster than h. Use it to build the equations.
Staying probability (waiting time): t p_x^īi = exp( −∫₀ᵗ Σ_{j≠i} μ_ij(x+s) ds ); Probability of remaining continuously in state i from x to x+t. It equals t p_x^ii only if you cannot return to i after leaving.
Kolmogorov forward equation: d/dt [ t p_x^ij ] = Σ_{k≠j} ( t p_x^ik × μ_kj(x+t) − t p_x^ij × μ_jk(x+t) ); Start state i is fixed. Inflow to j minus outflow from j.
Euler step: (t+h) p_x^ij ≈ t p_x^ij + h × d/dt [ t p_x^ij ]; Evaluate the derivative at time t using the current probabilities and intensities at age x+t.
Probability check: Σ_j t p_x^ij = 1; The derivatives across all j must sum to zero. Use this to check your working.
EPV of continuous annuity while in state j, starting in state i: ∫₀ⁿ v^t × tp_x^ij × B dt; B is the annual rate of payment. Uses occupancy probability tp_x^ij, not the probability of staying continuously in j.
EPV of discrete annuity-due while in state j: Σ (t = 0 to n−1) B × v^t × tp_x^ij; Payments at the start of each year, made only if in state j at that time.
EPV of lump sum S on transition from state j to k, starting in i: ∫₀ⁿ S × v^t × tp_x^ij × μ_x+t^jk dt; Here tp_x^ij is the probability of being in j at time t (not necessarily continuously), because the life may have moved in and out of j earlier.
Probability of remaining in state i throughout: tp_x^ii-bar = exp(−∫₀ᵗ Σ(k≠i) μ_x+s^ik ds); Total force of leaving state i. Use this for waiting periods or other conditions that need an unbroken stay in state i. It equals the occupancy probability tp_x^ii when re-entry to state i is impossible.
Policy value at start: EPV benefits − EPV premiums; Net premium: choose the premium so this equals zero on the premium basis. Gross premium: include expenses.
Thiele's equation (general state i): dV_i(t)/dt = δ V_i(t) + P_i(t) − B_i(t) − Σ_{j≠i} μ_ij(t) [ S_ij(t) + V_j(t) − V_i(t) ]; P_i is the premium rate paid while in state i. B_i is the benefit rate paid while in state i. S_ij is the lump sum paid on a jump from i to j. δ is the force of interest.
Two-state alive-dead term assurance: dV(t)/dt = δ V(t) + P − μ_{x+t} (S − V(t)); Lump sum S on death, and V in the dead state is zero. Boundary condition V(n) = 0 for a term assurance.
Backward Euler step: V_i(t − h) ≈ V_i(t) − h × dV_i(t)/dt; Evaluate the derivative at t, using the V_j(t) values you already have. Smaller h gives better accuracy.
Improved step using the derivative at t − h: V_i(t − h) ≈ V_i(t) − h × [ derivative at t + derivative at t − h ] ÷ 2; Optional improvement when a question asks for it. The derivative at t − h needs V_i(t − h) itself, so it is implicit and needs rearranging.
Boundary conditions: Term contract ending at n: V_i(n) = 0. Death state: V = 0 unless a benefit is payable there.; Check the contract. For an endowment, V_i(n) equals the maturity benefit while the life is in the benefit state.
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.

Quick revision

Markov property: the future depends only on the current state, not on the past path.
μ_ij(x) is the transition intensity from state i to state j at age x, where i ≠ j.
Over a small time h, P(move i to j) ≈ h × μ_ij(x), with an error that is o(h).
Forward equation: the rate of change of p_ij is inflow to j minus outflow from j.
Boundary condition for the forward equation: p_ii(0) = 1 and p_ij(0) = 0 for i ≠ j.
Expected present value = Σ or ∫ of payment × discount factor × probability of the right state or transition.
Thiele: dV_i/dt = δ V_i + P_i − B_i − Σ over j≠i of μ_ij (S_ij + V_j − V_i).
Solve Thiele backwards from the known terminal reserve, not forwards from time zero.
Euler's method with a small step gives an approximate reserve; state the step size you use.
In a multiple decrement model, the total force of decrement is the sum of the individual forces.
For joint life problems, state independence assumptions clearly before using products of probabilities.
Always draw the diagram and state your assumptions before writing any equation.

Common mistakes

Confusing t p_x^ii with the holding probability t p_x^īi. Fix: Ask whether leaving and returning is allowed. If yes, use t p_x^ii. If the life must stay in i throughout, use the holding probability with the bar.
Treating intensities as probabilities, or adding them to 1. Fix: Intensities are rates per year. Use them only inside exp(−…) or integrals, or as h × μ for a very small h.
Treating μ_ij(x) as a probability and capping it at 1. Fix: Remember that it is a rate per year. Only h × μ_ij(x) is approximately a probability, and only for small h.
Using the outflow intensity of the wrong state in the negative term. Fix: Outflow is always from state j, so the negative term is t p^ij × μ_jk. Check the arrow direction on your diagram.
Using tp_x^ii-bar when the question needs tp_x^ii. Fix: Use the bar version only when the payment requires an unbroken stay. For an annuity paid while in a state at time t, use plain occupancy probability.
Forgetting the intensity μ in transition payments. Fix: Multiply the occupancy probability of the starting state, tp_x^ij, by the intensity μ_x+t^jk, then integrate with v^t over time.
Stepping forwards instead of backwards, so V(t + h) is computed as V(t) + h × derivative. Fix: Thiele's equation starts at the end of the term. Always use V(t − h) = V(t) − h × dV/dt.
Writing μ_ij × S_ij only and leaving out V_j − V_i. Fix: A jump pays S_ij and also changes the reserve you must hold from V_i to V_j. Always write S_ij + V_j − V_i.
Using l(x+k) as the denominator when finding the chance of exit in a later year from age x. 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). 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.

Exam tips

Start every written answer with a state diagram and a one-line statement of the Markov assumption. These are quick marks and they organise your thinking.
In MCQs, check first whether the question asks for t p^ii or the holding probability. The two options often differ only by the return paths.
Always run the sum-to-1 check on your probabilities. It catches arithmetic and missed-path errors in seconds.
Keep notation exact: superscripts for states, subscript for age, and a bar for holding probabilities. Say which intensities are constant.
For Paper B work in R or Excel, set up the transition matrix cleanly, label states, and show the matrix multiplication or formula you used.
Always draw the state diagram first. Most lost marks come from a missing or wrongly directed arrow.
Show the general forward equation in standard notation before you substitute numbers. Method marks are given for it.
In Euler questions, set out a small table of t, probabilities and derivatives. Check that the probabilities sum to 1 at each step.