Introduction to EPVs in Multiple State Models
Welcome! In our journey through CM1, we have already looked at the "building blocks" of multiple state models—understanding how people move between states like Healthy, Sick, and Dead. Now, we are going to do something very practical with that knowledge: we are going to put a price tag on it!
In this chapter, we learn how to calculate the Expected Present Value (EPV) of cashflows that depend on these transitions. Whether it is a sickness benefit paid while someone is ill or a lump sum paid the moment they are diagnosed with a condition, these techniques are the bread and butter of modern health insurance pricing.
Note: This chapter builds directly on the previous chapter "Multiple-state Markov models: forces and probabilities of transition." We assume you are comfortable with the idea of transition probabilities \( {}_t p_x^{ij} \) and forces of transition \( \mu_{x+t}^{ij} \).
The Three Pillars of Cashflows
When dealing with multiple state models (like health insurance), cashflows usually fall into three categories:
- Premiums: Usually paid continuously while the policyholder is in the Healthy state.
- Annuity-type Benefits: Paid continuously while the policyholder is in a specific state (e.g., Sick).
- Lump Sum Benefits: Paid immediately at the moment a specific transition occurs (e.g., moving from Healthy to Dead).
To find the EPV of any of these, we follow a simple golden rule: Value = (Amount) \(\times\) (Discount Factor) \(\times\) (Probability). Because these cashflows happen over time, we use calculus (integrals) to sum them up.
1. EPV of Benefits Payable while in a State
Imagine a policy that pays a sickness benefit at a rate of \( \$1 \) per year while the life is in the Sick (s) state. If the person starts in the Healthy (h) state at age \( x \), how much is that worth today?
We need to sum up the discounted probability of being in the "Sick" state at every possible moment \( t \).
The Formula:
The EPV of a continuous annuity of \( 1 \) per annum payable while in state \( j \), given the life is currently in state \( i \), is denoted as \( \bar{a}_x^{ij} \):
\( \bar{a}_x^{ij} = \int_{0}^{n} e^{-\delta t} \cdot {}_t p_x^{ij} \, dt \)
Breaking it down:
- \( e^{-\delta t} \): This is our discount factor (bringing the future money back to today).
- \( {}_t p_x^{ij} \): This is the probability that a person aged \( x \) in state \( i \) will be in state \( j \) at time \( t \).
- \( \int_{0}^{n} \dots dt \): We are adding up all these tiny slices of value from time \( 0 \) to the end of the contract at time \( n \).
Quick Tip: If the benefit is paid while the person stays in their original state (e.g., premiums paid while healthy), the formula uses \( {}_t p_x^{ii} \).
2. EPV of Benefits Payable on Transition
Some benefits aren't paid for being in a state; they are paid for the act of moving between states. For example, a lump sum of \( \$10,000 \) paid the moment a policyholder moves from Healthy (h) to Dead (d).
To value this, we look at the probability of being in state \( i \) at time \( t \), and then "jumping" to state \( j \) at that exact instant.
The Formula:
The EPV for a benefit of \( 1 \) payable immediately on transition from state \( i \) to state \( j \):
\( EPV = \int_{0}^{n} e^{-\delta t} \cdot {}_t p_x^{ii} \cdot \mu_{x+t}^{ij} \, dt \)
Why this formula?
Think of it as a sequence of events:
1. The person must be in state \( i \) at time \( t \). Probability: \( {}_t p_x^{ii} \).
2. They must transition to state \( j \) at that exact moment. Force: \( \mu_{x+t}^{ij} \).
3. We discount the payment back to today: \( e^{-\delta t} \).
Common Mistake to Avoid: Students often confuse \( {}_t p_x^{ij} \) with \( {}_t p_x^{ii} \mu_{x+t}^{ij} \). Remember:
- \( {}_t p_x^{ij} \) is the probability of being in state \( j \) at time \( t \).
- \( {}_t p_x^{ii} \mu_{x+t}^{ij} \) relates to the instant of moving into state \( j \).
3. Calculating Premiums (The Equivalence Principle)
Just like in simpler life insurance models, we use the Equivalence Principle to find the premium. This principle states:
EPV of Premiums = EPV of Benefits
Suppose we have a "Healthy-Sick-Dead" model. A policyholder pays a continuous premium \( P \) while Healthy (h). In return, they get a continuous benefit \( B \) while Sick (s) and a lump sum \( S \) if they Die (d) directly from the Healthy state.
The equation of value would look like this:
\( P \cdot \bar{a}_x^{hh} = B \cdot \bar{a}_x^{hs} + S \cdot \int_{0}^{n} e^{-\delta t} \cdot {}_t p_x^{hh} \cdot \mu_{x+t}^{hd} \, dt \)
Step-by-Step Approach:
1. Identify all possible states and transitions.
2. List the cashflows (Which state are they in? Which transition triggers them?).
3. Write down the integral for each cashflow.
4. Set the Total EPV (Income) = Total EPV (Outgo) and solve for the unknown.
Simple Cases & Constant Forces
The syllabus mentions you only need to calculate these in simple cases. Usually, this means cases where the forces of transition (\( \mu \)) and the force of interest (\( \delta \)) are constant.
When these are constant, the integrals become much easier to solve. For example, if \( \mu \) is the only exit from a state, the probability of remaining in that state is \( e^{-\mu t} \). The integral then looks like this:
\( \int_{0}^{\infty} e^{-\delta t} \cdot e^{-\mu t} \, dt = \int_{0}^{\infty} e^{-(\delta + \mu) t} \, dt = \frac{1}{\delta + \mu} \)
Did you know? This is mathematically identical to the formula for a continuous perpetuity \( \bar{a}_{\infty} = 1/\delta \), just with an "adjusted" interest rate that accounts for the risk of leaving the state!
Quick Review: Key Takeaways
The "States" Rule: If money is paid for occupying a state, use the probability of being in that state (\( {}_t p_x^{ij} \)).
The "Transitions" Rule: If money is paid for moving between states, use the probability of being in the first state multiplied by the force of transition (\( {}_t p_x^{ii} \cdot \mu_{x+t}^{ij} \)).
The "Discounting" Rule: Always include the discount factor \( e^{-\delta t} \) inside your integral.
The "Equivalence" Rule: EPV of Income = EPV of Outgo.
Summary & Next Steps
Calculating EPVs in multiple state models might look intimidating because of the integrals, but it is just a logical extension of basic life contingency principles. Once you identify whether a benefit is for staying or leaving, the formula writes itself!
In the next chapter, we will look at Multiple Decrement Tables, which are a special type of multiple state model where transitions only go one way (no recovery!). Understanding the EPV logic here will make that much easier to grasp.