Multi-State Insurance: Net Premiums and Policy Reserves

The Equivalence Principle: Calculating Net Premiums

At its core, a net premium is the price that makes an insurance contract "fair" from a purely probabilistic standpoint, ignoring expenses and profit margins. The Equivalence Principle provides the mechanism for calculating this premium. It states that at the inception of a policy (time $t=0$), the Actuarial Present Value of the future premiums to be paid by the policyholder must equal the Actuarial Present Value of the future benefits to be paid by the insurer.

Typically, the premium amount, $P$, is the unknown we need to solve for. The process involves setting up the APV expressions for both sides of the equation using the "triple-product" approach from our last lesson and then algebraically solving for $P$.

Let's see this principle in action with a concrete example involving a discrete-time Markov chain.

Let's break down the logic used to find the premium $P$ in this example.

1. Identify Cash Flows:

   • Benefits (outflow for insurer): A death benefit of $10,000 is paid at the end of the year of death.
   • Premiums (inflow for insurer): A level annual premium of $P$ is payable at the start of each year, for up to 3 years, provided the insured is in the 'Healthy' state (State 0).

2. Set up the APV of Future Premiums:
   The premiums form an annuity-due that is contingent on being in State 0. The APV is:

   $${\text{APV(FP)}}_0=P\left(v^0\cdot {}_0{p}_{00}+v^1\cdot {}_1{p}_{00}+v^2\cdot {}_2{p}_{00}\right)$$

   Where ${}_k{p}_{00}$ is the probability of being in State 0 at time $k$, given a start in State 0. The calculation in the text simplifies this to:

   $${\text{APV(FP)}}_0=P\left(1+\frac{p_{00}}{1.06}+\frac{{}_2{p}_{00}}{(1.06{)}^2}\right)=1.8864P$$

3. Set up the APV of Future Benefits:
   The death benefit is paid upon transition to the 'Dead' state (State 2). We sum the APV of the benefit for death occurring in each of the three years.

   • Death in Year 1: The probability is simply $p_{02}$. The APV is $10000\cdot p_{02}\cdot v^1$.
   • Death in Year 2: The insured must not be dead at time 1, and then die. The probability is $P(Y_1\ne 2,Y_2=2|Y_0=0)$.
   • Death in Year 3: The insured must not be dead at time 1 or 2, and then die. The probability is $P(Y_1\ne 2,Y_2\ne 2,Y_3=2|Y_0=0)$.
     The resource carefully calculates these probabilities by summing over the possible paths (e.g., Healthy -> Healthy -> Dead, or Healthy -> Disabled -> Dead). The total APV of benefits is calculated to be $4056.70.

4. Apply the Equivalence Principle:
   We set the two APVs equal and solve for $P$:

   $$1.8864P=4056.70⟹P=\frac{4056.70}{1.8864}\approx 2150.50$$

This process is the standard method for determining net premiums in multi-state models.

Policy Reserves: A Prospective View

Once a policy is in force, the insurer needs to know how much capital to hold to cover its future obligations for that policy. This amount is the policy reserve (or policy value). A crucial insight is that the required reserve depends on the state of the policyholder at the time of valuation. An insured person who has become disabled, for example, represents a different future liability than one who has remained healthy.

The reserve is calculated using the prospective method, which looks forward from the valuation date $t$.

As the resource states, the reserve at time $t$ for a policyholder in state $j$, denoted ${}_t{V}^{(j)}$, is:

The calculation is identical in principle to what we've already done, with two key differences:

1. The valuation date is now time $t$, not time 0. All cash flows are discounted back to time $t$.
2. The probabilities are conditional on the policyholder being in state $j$ at time $t$.

Let's return to our running example to see how this works.

Let's analyze these two reserve calculations at $t=1$. The premium we calculated earlier was $P=2150.50$.

Case (a): Insured is Healthy (in State 0) at t=1

The reserve is ${}_1{V}^{(0)}$.

• APV of Future Benefits: From $t=1$, death can occur in year 2 or year 3. We calculate the APV of these potential benefits, with probabilities now conditional on being in State 0 at $t=1$. The resource calculates this as $10^4\left(\frac{0.1}{1.06}+\frac{0.18}{(1.06{)}^2}\right)$.
• APV of Future Premiums: From $t=1$, premiums of $P$ are due at the start of year 2 ($t=1$) and year 3 ($t=2$), provided the insured remains healthy. The APV is $P\left(1+\frac{p_{00}}{1.06}\right)$.
• Reserve: ${}_1{V}^{(0)}={\text{APV}}_1(\text{FB})-{\text{APV}}_1(\text{FP})=-822.38$.

A negative reserve might seem counterintuitive. It means that, for a policyholder who is still healthy at time 1, the value of the future premiums they are expected to pay is greater than the value of their expected future benefits. If the policy were to be surrendered at this point, the insurer would realize a profit.

Case (b): Insured is Disabled (in State 1) at t=1

The reserve is ${}_1{V}^{(1)}$.

• APV of Future Benefits: Death can still occur in year 2 or year 3. We now calculate the APV of these benefits using probabilities conditional on being in State 1 at $t=1$. The resource shows this is $10^4\left(\frac{0.4}{1.06}+\frac{0.24}{(1.06{)}^2}\right)$.
• APV of Future Premiums: The policy states premiums are waived if the insured is disabled. Therefore, the APV of future premiums is 0.
• Reserve: ${}_1{V}^{(1)}={\text{APV}}_1(\text{FB})-0=5909.58$.

The reserve is now positive and substantial. The insurer expects no more income from this policy but still has the liability of the potential death benefit. This demonstrates clearly why the reserve must be state-dependent. The insurer needs to hold significantly more funds for the disabled life than for the healthy one.

Continuous-Time Models and Thiele's Equation

The principles of equivalence and prospective reserves apply equally to continuous-time models, where summations are replaced by integrals. The logic remains identical. The provided resource Prof.Mhamed Eddahbi meddahbi@ksu.edu.sa ... contains several excellent examples (e.g., 10.6.3, 10.7.3) of these calculations in a continuous-time framework, which follow the exact same structure.

For continuous models, there is also a powerful alternative for calculating reserves: Thiele's differential equation. Instead of calculating the full APV at each time step, this equation describes the rate of change of the reserve over an infinitesimal time interval, $dt$. For a reserve ${}_t{V}^{(i)}$ in state $i$, the general form is:

Where:

• ${\delta }_t$ is the force of interest.
• $P_t^{(i)}$ is the rate of premium payment in state $i$.
• $B_t^{(i)}$ is the rate of benefit payment while in state $i$.
• ${\mu }_{x+t}^{ij}$ is the force of transition from state $i$ to $j$.
• $S_t^{(ij)}$ is the lump-sum benefit on transition from $i$ to $j$.
• ${}_t{V}^{(j)}$ is the reserve in the destination state $j$.

Given your background, you can see this as a dynamic equation accounting for interest accumulation (${\delta }_t{}_t{V}^{(i)}$), net cash flow ($P_t^{(i)}-B_t^{(i)}$), and the expected cost of transitions to other states. We won't delve into solving these equations now, but it's important to be aware of this alternative and more dynamic approach to reserves.

Conclusion

In this lesson, we have connected the abstract concept of Actuarial Present Value to the practical tasks of pricing and reserving. By mastering these applications, you have acquired the core skills for evaluating the financial dynamics of any insurance product that can be described by a multi-state model.

Key Takeaways:

• The Equivalence Principle is the foundation for calculating net premiums: $\text{APV}(\text{Premiums})=\text{APV}(\text{Benefits})$ at policy inception.
• The Policy Reserve is the net liability an insurer must hold. It is calculated prospectively: ${}_t{V}^{(j)}={\text{APV}}_t(\text{Future Benefits})-{\text{APV}}_t(\text{Future Premiums})$.
• Crucially, the value of the policy reserve, ${}_t{V}^{(j)}$, is dependent on the state, $j$, of the insured at the time of valuation, $t$.

In our next lesson, we will begin to explore time-inhomogeneous processes. We will relax the convenient assumption that transition rates are constant over time and investigate methods for solving models where the forces of transition change with the age of the policyholder or the duration of the policy.