Chapter 1: Foundations of Multiple State Models#

Imagine a life insurance policy that pays out not only when you die, but also if you become permanently disabled while the policy is active. To design such a contract, we need a way to describe all the possible situations you can be in—alive and healthy, disabled, dead—and the rules for how money moves between the insurance company and you as these situations change. That’s exactly what a multiple state model does.

The Big Picture#

Multiple state models give us a clear way to describe life events—like getting sick, getting better, retiring, or dying—and the money promises that go with them. Instead of treating a policy as a simple bet on whether you die or survive, we think of your journey as a set of states (like “healthy”, “disabled”, “dead”) and moves between them called transitions. This chapter builds the basics: we learn to draw the model as a picture, sort its states, and describe how benefits and premiums depend on where you are and how you move. By the end, you’ll see how a few simple ideas can describe disability insurance, critical illness cover, long‑term care, and many other real‑world contracts.

States and Transitions: A Graphical Model#

We start with two basic ideas: a state is a label for your situation at a given moment, and a transition is a sudden change from one state to another. We can picture the whole process as a diagram with boxes and arrows.

State: A clear label for a situation you can be in at any time (like “Alive and healthy”, “Temporarily disabled”, “Dead”).

Transition: A direct move from one state to another. It happens in a flash; you are never partly in two states at the same time.

Think of a board game where your piece sits on a square that shows your current health. Every so often, a dice roll tells you to move to a different square, maybe “Broken leg” or “Back to health”. The squares are the states; the allowed moves are the transitions. Our job as actuaries is to put a price on the money consequences of these moves.

A simple example is the sickness‑death model with three states:

• State 1: Healthy (active)
• State 2: Sick (disabled)
• State 3: Dead

The graph might look like this:

(Healthy) ──→ (Sick) ──→ (Dead)
  │                        ↑
  └────────────────────────┘

Arrows show possible transitions: you can fall ill (Healthy → Sick), recover (Sick → Healthy), die while healthy (Healthy → Dead), or die while sick (Sick → Dead). A more realistic picture might add permanent disability, allowing no recovery, or a state for retirement. Whatever the detail, the graph is our map.

A transition can happen at any moment, so we think of the process as running in continuous time. We write for the state you are in at time . If you are in state at time and later in state at time , we say a transition from to happened somewhere in between, maybe with stops in other states. Most actuarial work then comes down to answering questions like:

• Given you are in state now, what is the probability of being in state at a future time?
• Given you just entered state , how much money should the insurer now pay, and for how long?

By drawing a clear diagram, we can immediately see what paths are allowed and what benefits or premiums might be triggered. This picture is so helpful that we will always start a new product design by sketching its state‑transition diagram.

📝 Section Recap: A multiple state model is a diagram of boxes (states) and arrows (transitions) that shows all the possible situations and moves of a person over time. The diagram is the starting point for both probability calculations and cash‑flow design.

Transient and Absorbing States#

Not all states are the same. Some states, once you enter them, you can never leave. Others you pass through and eventually leave. This difference is key to how we value the benefits.

Absorbing state: A state you can never leave once you enter it. The chance of leaving is zero.

Transient state: A state you can leave; it is not absorbing. In a model with a limited number of states, you will eventually leave all transient states and end up in an absorbing state.

The classic absorbing state is Dead. Unless we model resurrection (which actuaries wisely avoid), a person who enters the Dead state stays there. In some contracts, “Permanent disability” or “Matured endowment” (a savings plan that has reached its end) can also be absorbing; once you are permanently disabled, you never recover. A transient state is one like “Healthy” or “Temporarily sick”—you may stay there for a while, but you will eventually move on (perhaps to death).

Why does this matter? When we price a benefit, we need to know how long someone will stay in a state. If a state is absorbing, any regular benefit paid while you are in that state could go on forever (or until the contract ends). If it is transient, the stay is temporary. Also, some payments are triggered by entering a state (a transition benefit), while others are paid just for being there (a state‑contingent benefit). Knowing if the destination is absorbing helps us decide whether the benefit should be a one‑off lump sum or a lifelong income, for example.

A helpful picture is a funnel. The model has several transient sections at the top, each standing for a possible condition, and a single absorbing section “Death” at the bottom. Everyone eventually trickles through the funnel and ends in the absorbing state. In maths, we often work with the chance of moving from a transient state to an absorbing state by time , or the chance of still being in the transient set.

In a full model, we can compute all these chances from the transition intensities (the speed of movement at each instant)—a topic we’ll explore later. For now, just remember: when you see a diagram, immediately spot the absorbing states, because they shape the long‑term cash flows.

📝 Section Recap: An absorbing state is one you can never leave (like “Dead”); a transient state is one you can leave. This sorting is essential for designing benefits and knowing whether a payment stream will stop or go on forever.

Cash Flows in a Multiple State Model#

Now we get to the money part. An insurance contract is a set of promises: you pay premiums, and in return the insurer pays benefits. Both depend on the states and transitions. We need a clear way to describe these cash flows, so we can later work out their present values.

We think of cash flows as being triggered by two kinds of event:

1. Being in a particular state at a given time.
2. Crossing a particular transition at the moment it happens.

Each cash flow has an amount and a pattern: it can be a steady stream (continuous), a payment at set times, a one‑off lump sum, or a regular repeating amount. We’ll look at benefits first, then premiums.

Benefits#

A benefit is money the insurer pays to you (or your family) when certain conditions are met. In a multiple state model, these conditions are tied to the current state or to a recent transition. Three common types are:

Continuous annuity while in a state: A payment that flows at a steady rate over time (say, £1 per year) for as long as you stay in a given state.

Lump sum on transition: A one‑off payment of a fixed amount made the moment a specific transition happens (like becoming disabled, or dying).

Pure endowment: A lump sum paid only if you are in a particular state on a set future date; if you are not in that state at that time, nothing is paid.

A real disability income policy often mixes these: a lump sum when you first become disabled, then a continuous annuity (the monthly income) while you stay disabled, and a lump sum on death. Each can be described neatly as “£5000 on transition from Healthy to Sick”, “£1000 per month while in Sick”, and “£20 000 on transition from any state to Dead”. See how the language of states and transitions makes the contract’s promises crystal clear.

A pure endowment is a bit different because it is not tied to a transition but to being in a particular state on a fixed date, with no payment for changes before then. For example, a policy might pay £10 000 if you are still alive and healthy on your 65th birthday; if you get sick or die before that, you get nothing. This is different from term insurance, which pays on death (a transition benefit), and from a whole‑life annuity, which pays continuously while you are alive.

Premiums#

Premiums are the money you pay, and they can also be structured in many ways. The same cash‑flow logic applies, but now the money flows from you to the insurer.

Single premium: A one‑off lump‑sum payment made at the start of the contract.

Continuous premium: A premium paid at a steady rate over time (like an income stream to the insurer) for as long as you are in a given state or set of states.

Discrete premium: A regular payment made at set intervals (yearly, monthly) as long as you are in a specified state.

Recurrent premium: Another name for a stream of premium payments, stressing that they are paid repeatedly over time; often used interchangeably with “periodic premium”.

The premium depends on your state. You normally only pay premiums while you are alive and (in some policies) not yet disabled. A typical disability insurance contract might have monthly premiums payable while you are in the “Healthy” state, but they stop if you become disabled or die. In model terms, we write: “premium at rate per year continuously while in state Healthy”. If premiums stop on disability, that’s a waiver of premium feature—a benefit in itself, which we can also show as a state‑contingent cash flow.

Sometimes premiums are not level. A single premium means one payment at the start, no matter what happens later. Recurrent premiums might go up according to some index or schedule. The beauty of the multiple state framework is that all these variations can be handled by saying, for each state, whether a premium is due and at what rate.

Bringing It All Together#

Let’s put this into a tiny example. Take a simple 3‑state sickness model: Healthy (H), Sick (S), Dead (D). A compact contract might have:

• A continuous annuity of £500 per month while Sick.
• A single premium of £12 000 paid at the start.
• No premiums after that, and no benefit on death (to keep it simple).

The cash flows would be described as:

• Premium: £12 000 at time 0 if in H (the only possible starting state).
• Benefit: a continuous payment at rate £6 000 per year while in S, stopping on recovery or death.

The actuarial problem is then to find the premium that makes the expected present value of premiums equal to the expected present value of benefits, given a set of transition intensities and an interest rate. That’s the core of pricing and reserving, which we’ll tackle later. The key here is that the multiple state model gives us the framework to hang all the cash flows on.

A quick note on notation: for a continuous benefit while in state , we might write for an annuity payable for the next years while in state , or an infinite one if . For a lump sum on transition from to , we often use a capital sum notation. The exact symbols matter less than the idea that they all refer back to state occupation or transitions.

When we later compute premiums, we’ll treat future cash flows as random amounts that depend on the path takes. The present value of a continuous annuity while in state is worked out by adding up the discounted payments over the random time spent in . Transition‑triggered lump sums are like one‑off jumps. Don’t worry if that sounds technical now—it will become clear when we work through examples.

📝 Section Recap: Cash flows in a multiple state model are defined by states and transitions: benefits can be continuous annuities, lump sums on transition, or pure endowments; premiums can be single, continuous, discrete, or recurrent. All are written as conditions on the random process , which lets us value them precisely later.

Summary#

We’ve seen that a multiple state model is simply a picture of possible life situations, with arrows showing allowed moves. We sort each state as transient (you can leave it) or absorbing (you stay there forever), because that shapes how long payments last. Finally, we learned to describe the money promises—benefits and premiums—in terms of the states and transitions: an annuity pays while you are in a state, a lump sum pays on a transition, and a pure endowment pays if you are in a specific state on a fixed date. Premiums mirror these forms but in reverse. This vocabulary is the foundation for every insurance product we’ll study later.