Actuarial Mathematics for Modelling · Valuing cashflows contingent on multiple transition events

Valuing Cashflows Using Multiple State Models

Updated 11 October 2026 · Fact-checked

The expected present value (EPV) of a multi-state cashflow is the sum of each payment times its discount factor times the probability that the payment is made. Annuities while in a state use occupancy probabilities. Lump sums on transition use transition intensities integrated over time. Premiums are valued the same way.

Understand Valuing Cashflows Using Multiple State Models

A multiple state model describes a policyholder moving between states, such as Healthy, Sick and Dead. Payments depend on where the life is, or on a move between states. A disability income policy might pay a benefit while Sick, and a lump sum on death.

There are two kinds of cashflow. A state-based payment is made while the life is in a given state, for example an annuity while sick. A transition-based payment is made at the moment of a move, for example a lump sum on moving from Healthy to Dead.

The EPV is the expected value of the discounted payments. For each payment you need the probability it happens. For payments at a point in time t, this is the probability of being in the required state at time t, written tp_x^ij, starting from state i. Here tp_x^ij is the probability of being in state j at age x+t given state i at age x. Note that this is not the same as staying in state i throughout, which is tp_x^ii-bar.

For a transition payment on a move from state j to state k, the probability density of making the move at time t is tp_x^ij × μ_x+t^jk. Here tp_x^ij is the occupancy probability: the probability of being in state j at time t, not necessarily continuously. You then integrate over time with the discount factor v^t.

The continuous-stay probability tp_x^ii-bar is defined only when the life starts in state i and must stay in state i throughout. It never replaces tp_x^ij when i and j are different states. Use it when the contract needs an unbroken stay, such as a waiting period. If re-entry to state i is impossible, tp_x^ii-bar equals tp_x^ii, so the two give the same value.

In practice, the policy is usually valued from a known starting state. Premiums are paid while in certain states, often only while healthy, and stop on becoming sick. The value of the policy is EPV of benefits minus EPV of premiums, using the stated basis.

Key rules to remember

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.

How to solve Valuing Cashflows Using Multiple State Models questions

Use the same method for every multi-state valuation question. Work out what is paid, when, and with what probability.

1Draw the state diagram and mark the starting state, the states in which payments are made, and the transitions that carry lump sums.
2List each cashflow separately: benefit annuities, transition lump sums, premiums, and expenses if given.
3For each cashflow, write the payment condition. Ask: do I need to be in the state at time t, or have stayed continuously in it, or make a move at time t?
4Write the probability term: tp_x^ij for occupancy, tp_x^ii-bar for a continuous stay only when the contract needs an unbroken stay, and tp_x^ij × μ_x+t^jk for a transition from j to k.
5Add discounting v^t and the payment amount. Write the EPV as a sum for discrete payments or an integral for continuous payments or transitions.
6Evaluate using the given probabilities or intensities. If needed, approximate integrals by a sum over short steps or solve with Kolmogorov equations.
7Combine: EPV benefits minus EPV premiums, or solve for the premium that makes the EPV of premiums equal the EPV of benefits.
8Check the answer for sense: it should be positive, below the sum insured for lump sums, and consistent with the survival probabilities.

Quickest way: Table-and-sum method for discrete payments

When to use it: Use when the question gives yearly occupancy probabilities or asks for annual payments, or when integration is not practical.

Set up a table with columns: t, v^t, probability of being in the payment state, payment, and product.
Fill in each row using the given probabilities. Do not recompute them if the question gives them.
Sum the product column to get the EPV.
For a transition lump sum with only annual data, approximate each year's contribution as the payment × the occupancy probability of state j at the relevant time × μ^jk × the step length (1 year). Discount to mid-year or year-end, as the stated assumption says. Alternatively, use payment × (probability of being in state j at the start of the year) × (one-year probability of moving from j to k during that year) × the discount factor. Use the one-year probability only when the model allows just a direct path from j to k in the year, with no moves through other states or re-entry. Do not drop the start-of-year occupancy factor.
State your assumptions, such as payments at year start or mid-year.

Common mistakes in Valuing Cashflows Using Multiple State Models

Using tp_x^ii-bar when the question needs tp_x^ii.
The two look alike, and students forget that a life can leave and return to a state.
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.
Students treat the lump sum like a payment at a fixed time and multiply by a probability only.
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.
Including premiums in the wrong states.
The policy wording says premiums stop on sickness but students keep them going.
Fix: Write the premium condition explicitly: premiums are paid at time t only if in the premium-paying state at time t.
Mixing starting states.
Probabilities from a table are quoted for different starting states.
Fix: Label every probability with i and j, and check that i matches the policyholder's state at the valuation date.
Discounting at the wrong time.
Payments at mid-year or at end-of-year of transition are treated as at start.
Fix: State the timing assumption and use v^t for the actual payment time.

Worked examples

Example 1

A policyholder is Healthy at age x. A benefit of ₹60,000 per year is paid annually in advance while Sick, for at most 2 years. Interest is 5% per year. Given 0p_x^HS = 0, 1p_x^HS = 0.10 and the policy lasts exactly two payment dates (t = 0 and t = 1). Calculate the EPV.

Show the solution

Payments are at t = 0 and t = 1, only if Sick at that time.
At t = 0 the life is Healthy, so the probability of being Sick is 0p_x^HS = 0. The EPV of that payment is 0.
At t = 1 the probability of being Sick is 1p_x^HS = 0.10 and v = 1 ÷ 1.05 = 0.952381.
EPV = 60,000 × 0.952381 × 0.10 = 5,714.29.

Answer: EPV ≈ ₹5,714

Example 2

A Healthy life is exposed to a constant force of death μ = 0.02 per year from the Healthy state and a constant force of sickness σ = 0.03 per year. There is no recovery: once a life leaves Healthy it never returns. A lump sum of ₹1,00,000 is paid on death from Healthy, for a 2-year term. Nothing is paid on death from Sick. Force of interest δ = 0.04. Calculate the EPV.

Show the solution

Total force of leaving Healthy = 0.02 + 0.03 = 0.05.
With no recovery, occupancy of Healthy equals continuous stay, so the probability of being Healthy at time t = e^(−0.05t).
EPV = 1,00,000 × ∫₀² e^(−0.04t) × e^(−0.05t) × 0.02 dt.
Combine exponents: e^(−0.09t). The integral of e^(−0.09t) from 0 to 2 = (1 − e^(−0.18)) ÷ 0.09.
e^(−0.18) = 0.835270, so 1 − 0.835270 = 0.164730. Divide by 0.09 to get 1.83033.
EPV = 1,00,000 × 0.02 × 1.83033 = 3,660.66.

Answer: EPV ≈ ₹3,661

Exam tips

Write the state diagram first. It earns marks and prevents wrong probabilities.
Show the formula in notation before substituting numbers, so method marks are secured even if arithmetic slips.
State assumptions on timing of payments and on the use of occupancy versus continuous-stay probabilities.
In computer-based papers, set up the table of time steps clearly and check one row by hand.
Check premium conditions: premiums usually stop when the life becomes sick or dies.

Practice questions from Valuing cashflows contingent on multiple transition events

Valuing Cashflows Using Multiple State Models: frequently asked questions

How do I value an annuity paid while in a state?

Multiply each payment by its discount factor and by the probability of being in that state at the payment time. Sum for discrete payments or integrate for continuous ones. Use occupancy probability, not continuous stay, unless the wording demands it.

When do I use the force of transition μ?

Use it for lump sums paid on a move between states. The payment density at time t is the probability of being in the starting state at t times the intensity of the move.

What is the difference between tp_x^ij and tp_x^ii-bar?

tp_x^ij is the probability of being in state j at time t, allowing any path. tp_x^ii-bar is the probability of staying in state i throughout the period without leaving.

How is the premium found in a multi-state model?

Set the EPV of premiums equal to the EPV of benefits on the stated basis, with expenses added for a gross premium. Then divide by the EPV of a unit premium annuity in the premium-paying states.