Introduction to Reserving
Welcome to one of the most fundamental chapters in Actuarial Mathematics! So far, you have learned how to calculate the price (premium) of an insurance contract. But what happens after the policy starts? As time passes, the insurance company needs to know how much money it should hold now to ensure it can pay out all those future claims. This "pot of money" is called a reserve.
Think of a reserve like a savings account for a specific goal. If you are saving for a house, you can check your progress in two ways: by looking at how much you still need to save (Prospective) or by looking at how much you have already put away (Retrospective). In this chapter, we will apply this same logic to insurance contracts.
1. Why do we need reserves?
In most life insurance contracts, the risk of death increases as the policyholder gets older. However, the premium usually stays level throughout the term. This means:
- In the early years, the premium is higher than the actual cost of the insurance risk.
- In the later years, the premium is lower than the actual cost of the risk.
The company must "reserve" the excess money from the early years to cover the shortfall in the later years. This is why we calculate reserves!
2. The Prospective Reserve
The Prospective Reserve is the most common method used in the actuarial profession. It is "forward-looking."
At any time \(t\), the prospective reserve is the Expected Present Value (EPV) of everything the insurer will pay out in the future, minus the EPV of all the premiums they expect to receive in the future.
The Formula:
\({}_{t}V = EPV(\text{Future Benefits}) + EPV(\text{Future Expenses}) - EPV(\text{Future Gross Premiums})\)
Note: All these values are calculated at time \(t\), for a policyholder who is still alive at time \(t\).
Example: Whole Life Assurance
For a policy issued to \((x)\) with a level premium \(G\) payable annually in advance, the prospective reserve at time \(t\) is:
\({}_{t}V = S \cdot A_{x+t} + e \cdot \ddot{a}_{x+t} - G \cdot \ddot{a}_{x+t}\)
Where \(S\) is the sum assured and \(e\) represents future renewal expenses.
3. The Retrospective Reserve
The Retrospective Reserve is "backward-looking." It looks at the history of the policy from its start until the current time \(t\).
It represents the Accumulated Value (AV) of all premiums received, minus the AV of all benefits paid and expenses incurred.
The Formula:
\({}_{t}V = AV(\text{Past Gross Premiums}) - AV(\text{Past Benefits}) - AV(\text{Past Expenses})\)
Wait! Because these are contingent cashflows, we don't just use standard compound interest. We use Expected Accumulated Values. We divide the past cashflows by the probability of survival to time \(t\), which is \({}_{t}p_{x}\), and accumulate them using the interest rate.
A Simple Way to Think About It:
Imagine the retrospective reserve as a "bank balance" for a group of identical policyholders. We add all premiums paid by the group (plus interest) and subtract all the claims paid to those who died (plus interest). What is left over is divided among the survivors.
4. The Equivalence of Reserves
You might be wondering: "Which one should I use?" The good news is that under specific conditions, both methods give the exact same answer.
The Condition for Equivalence:
The prospective reserve and the retrospective reserve will be equal at time \(t\) if, and only if:
- The same actuarial basis (interest rate, mortality table, and expense assumptions) is used for both calculations.
- That same basis was used to calculate the Gross Premium (\(G\)) originally using the Equivalence Principle.
Why does this happen?
At the start of the policy (\(t=0\)), the Equivalence Principle says:
\(EPV(\text{Benefits}) + EPV(\text{Expenses}) = EPV(\text{Premiums})\)
Therefore, the reserve at \(t=0\) is zero. As we move through time, the two methods are simply different ways of splitting that original equation of value.
Quick Tip: If an exam question asks you to "show that the prospective and retrospective reserves are equal," start by writing down the equation for the Gross Premium \(G\) using the equivalence principle!
5. Including Expenses and Varying Benefits
Reserves aren't just for level benefits. We must account for the specific structure of the contract:
Allowance for Expenses
When calculating Gross Reserves, we must include expenses. Usually, these are split into:
- Initial Expenses: Occur at \(t=0\). These only appear in the retrospective reserve calculation (as they are in the past).
- Renewal/Maintenance Expenses: Occur throughout the term. These appear in both, but the prospective method looks at future ones, while the retrospective looks at past ones.
Varying Benefits
If the sum assured increases (e.g., an increasing assurance \((IA)_{x}\)), you simply replace the standard assurance factor with the increasing one in your EPV or AV calculations.
- Prospective: Use the EPV of the remaining future increases.
- Retrospective: Use the AV of the increases that have already occurred.
6. Summary and Key Takeaways
Common Pitfall to Avoid:
When calculating retrospective reserves, students often forget to divide by the survival probability \({}_{t}p_{x}\). Remember: The reserve is held for those who are still alive. If you don't divide by \({}_{t}p_{x}\), you aren't accounting for the "mortality profit" shared among survivors from those who died!
Quick Review Box:
- Prospective Reserve: \(EPV(\text{Future Outgo}) - EPV(\text{Future Income})\). Use this when the future is easier to model.
- Retrospective Reserve: \(AV(\text{Past Income}) - AV(\text{Past Outgo})\). Use this if you are given historical data.
- Equivalence: They match if the premium was calculated on the same basis as the reserve.
- At \(t=0\): Both reserves are typically 0 (if no initial expenses are ignored).
- At \(t=n\) (End of Term): For an endowment, the reserve equals the Sum Assured. For a term assurance, the reserve is 0.
Don't worry if the retrospective accumulation feels a bit messy at first. Practice a few examples with pure endowments (\({}_{n}E_{x}\)) and you'll see the logic click into place!