Introduction to Health Insurance Structures

Welcome! In this chapter, we take the theoretical "Multiple State Models" we've learned and apply them to a real-world scenario: Health Insurance. If you've ever wondered how insurance companies decide what to charge for Income Protection or Critical Illness cover, this is where the magic happens.

Don't worry if the transition symbols look like a different language at first. We are simply moving from "Living or Dead" (the single decrement model) to a world where we can be "Healthy," "Sick," or "Dead." By the end of these notes, you'll be able to build the equations needed to value these complex-looking cashflows with confidence.

The Basic Three-State Model

The foundation of most health insurance mathematics is the Healthy-Sick-Dead Model (often called the Sickness-Death model). It usually consists of three states:

  • State H (Healthy): The starting point. The policyholder is working and paying premiums.
  • State S (Sick): The policyholder is ill and receiving a sickness benefit.
  • State D (Dead): The absorbing state. No more premiums or sickness benefits are paid.

Transition Forces: We use \(\mu_{x+t}^{jk}\) to represent the "force" or instantaneous rate of moving from state \(j\) to state \(k\) at age \(x+t\). In health insurance, the most common transitions are:

  • \(\mu_{x+t}^{HS}\): The force of becoming sick (becoming a claimant).
  • \(\mu_{x+t}^{SH}\): The force of recovery (returning to health).
  • \(\mu_{x+t}^{HD}\): The force of mortality for a healthy person.
  • \(\mu_{x+t}^{SD}\): The force of mortality for a sick person (which might be higher than for a healthy person!).

Quick Review: In CM1, we often assume these forces are constant over certain intervals (like single years of age) to make the integration easier to handle!

Types of Health Insurance Benefits

In this chapter, we focus on two primary ways a policy pays out:

1. Sickness Benefits (Income Protection)

This is a continuous payment made while the person is in the "Sick" state. Think of it like a "salary replacement" that stays switched on as long as the person is ill.

The Expected Present Value (EPV) of a sickness benefit of \(S\) per annum payable for \(n\) years is:

\(EPV = \int_{0}^{n} S \cdot e^{-\delta t} \cdot {}_t p_x^{HS} dt\)

Where \({}_t p_x^{HS}\) is the probability that someone currently healthy (age \(x\)) will be in the sick state at time \(t\).

2. Lump Sum Benefits (Critical Illness/Death)

These are lump sums paid the exact moment a transition occurs. For example, a Critical Illness policy might pay a sum \(B\) the moment someone moves from "Healthy" to "Sick."

The EPV for a benefit \(B\) payable immediately on transition from Healthy to Sick is:

\(EPV = \int_{0}^{n} B \cdot e^{-\delta t} \cdot {}_t p_x^{HH} \cdot \mu_{x+t}^{HS} dt\)

Key Takeaway: If the benefit is for being in a state, we use the probability of being there. If the benefit is for moving between states, we use the probability of being in the starting state multiplied by the force of transition.

Premium Structures

Insurance isn't free! The policyholder must pay premiums to keep the cover active. In health insurance models, we usually assume:

  • Premiums are paid continuously at a rate of \(P\) per annum.
  • Premiums are only paid while the person is Healthy (State H).
  • If the person becomes sick, premiums are usually waived (they stop paying while they are receiving benefits).

The EPV of these premiums for a term of \(n\) years is:

\(EPV(Premiums) = \int_{0}^{n} P \cdot e^{-\delta t} \cdot {}_t p_x^{HH} dt\)

Did you know? "Waiver of Premium" is a common feature in real-life insurance. It means the insurance company effectively pays the premium for you while you are too ill to work.

Applying the Equation of Value

To find the Net Premium (the premium that perfectly balances the cost of benefits without including expenses or profit), we use the Equivalence Principle:

EPV of Premiums = EPV of Benefits

Step-by-Step Calculation Guide:
  1. Identify the States: Usually Healthy, Sick, and Dead.
  2. Identify the Cashflows:
    • Outflow: Premiums while Healthy (\(P \times {}_t p_x^{HH}\)).
    • Inflow: Sickness benefits while Sick (\(S \times {}_t p_x^{HS}\)).
    • Inflow: Lump sum on sickness transition (\(B \times {}_t p_x^{HH} \mu_{x+t}^{HS}\)).
  3. Set up the Integrals: Discount each cashflow using \(e^{-\delta t}\).
  4. Solve for the Unknown: Usually, you will be asked to find the annual premium rate \(P\).

Common Mistakes to Avoid

1. Mixing up probabilities: Using \({}_t p_x^{HS}\) for a lump sum benefit on transition. Remember: \({}_t p_x^{HS}\) means the person is already sick at time \(t\). For a lump sum on the moment of falling ill, you need the force \(\mu\).

2. Forgetting the "Alive" requirement: For a premium to be paid, the person must be Healthy (\(HH\)). If they are Sick or Dead, the premium income stops.

3. Integration Limits: Always check if the benefit is for the whole life or a fixed term (\(n\)).

Summary Table: EPV Formulas

Cashflow Type Trigger Mathematical Term inside Integral
Continuous Premium Being in State H \(P \cdot {}_t p_x^{HH}\)
Continuous Benefit Being in State S \(S \cdot {}_t p_x^{HS}\)
Lump Sum on Sickness Transition H \(\to\) S \(B \cdot {}_t p_x^{HH} \cdot \mu_{x+t}^{HS}\)
Lump Sum on Death Transition H \(\to\) D \(D \cdot {}_t p_x^{HH} \cdot \mu_{x+t}^{HD}\)

Note on recovery: If the model allows recovery (\(S \to H\)), the probability \({}_t p_x^{HH}\) becomes more complex because it must account for people who were sick and then got better! These probabilities are usually calculated using the methods found in the "Multiple-state Markov models" chapter.

Key Takeaway for Exams: Most CM1 exam questions on this topic will provide you with the transition probabilities or ask you to define the integral. Focus on understanding why a specific probability (\(p\)) or force (\(\mu\)) is being used for a particular benefit.