Reserves within a profit test and their effect on profit

Introduction: The Role of Reserves in Profit Testing

In previous chapters, we looked at projecting cashflows to see if a product is profitable. However, an insurance company cannot simply treat all "surplus" cash at the end of a year as immediate profit. Because insurance contracts are long-term, the company must set aside a specific amount of money—a reserve—to ensure it can meet future claims and expenses.

In these notes, we will explore how to incorporate gross premium reserves into our profit test models and, more importantly, how the choice of these reserves changes the "shape" of the profit over time. Don't worry if this seems a bit abstract at first; think of a reserve as a "Rainy Day Fund" that the company is legally required to keep.

1. Why Include Reserves in a Profit Test?

When we perform a basic profit test, we project premiums, expenses, and claims. However, in reality, a company must hold reserves to stay solvent. Including reserves in our projection allows us to see the Emergence of Profit. This tells the company when it can actually "book" the profit and potentially pay it out as dividends to shareholders.

Key Concept: The profit we calculate in a profit test (the Profit Vector) is the profit after setting aside the necessary reserves for the survivors at the end of each year.

2. The Mechanics: How Reserves Fit into the Cashflow

To calculate the profit for a specific year \( t \), we need to consider both the reserve we started with and the reserve we must end with. Let's look at the timeline of a typical policy year:

1. Start of Year: We have the reserve from the previous year, \( _{t-1}V \).
2. Cash In: We receive the premium \( P \).
3. Cash Out: We pay initial or renewal expenses \( E \).
4. Growth: The remaining money earns interest at rate \( i \).
5. Claims: We pay out benefits \( S \) for those who die during the year (probability \( q_{x+t-1} \)).
6. End of Year: We must set aside the required reserve \( _tV \) for everyone who survived (probability \( p_{x+t-1} \)).

The Profit Formula with Reserves

The profit for year \( t \) per policy in force at the start of the year is:

\( PRO_t = (_{t-1}V + P - E)(1+i) - q_{x+t-1}S - p_{x+t-1} \cdot {}_tV \)

Quick Review:
- \( _{t-1}V \): Reserve held at the start of the year.
- \( P \): Gross premium.
- \( E \): Expenses.
- \( i \): Interest rate earned on the company's funds.
- \( q_{x+t-1} \): Probability of death.
- \( S \): Sum assured (death benefit).
- \( p_{x+t-1} \): Probability of survival.
- \( _tV \): Reserve required at the end of the year.

Did you know? In year 1, the starting reserve \( _0V \) is usually zero, because the policy hasn't started yet! However, the first year often shows a loss because the initial expenses and the first year-end reserve \( _1V \) are higher than the first premium.

3. The Effect of Reserves on Profit

It is crucial to understand that reserves do not change the total value of the contract over its lifetime (provided the interest rate earned is the same as the discount rate), but they do change the timing of the profit.

A. Higher Reserves = Delayed Profit

If a regulator requires the company to hold very high ("conservative") reserves, the profit in the early years of the policy will be lower (or more negative). This is because more cash is being "locked away" in the reserve fund rather than being released as profit.

B. Release of Reserves

As the policy approaches maturity, the required reserve eventually falls to zero (or the maturity benefit). As these reserves are "released," the profit in the later years of the contract will be higher.

C. The "Cost of Reserving"

Even though the total nominal profit might stay the same, holding reserves has a real cost. This is because the company could have invested that "locked up" money elsewhere to earn a higher return. In actuarial terms, we say that high reserves reduce the Net Present Value (NPV) of the profit if the Risk Discount Rate (RDR) is higher than the interest rate earned on the reserves.

4. Profit Vector vs. Profit Signature

When we include reserves, we need to distinguish between these two terms:

• Profit Vector: The expected profit at the end of each year, given that the policy was in force at the start of that year. It uses the formula shown in Section 2.
• Profit Signature: The expected profit at the end of each year as seen from the start of the policy. To get this, we multiply the Profit Vector by the probability of the policy actually being in force.

\( \text{Signature}_t = PRO_t \cdot {}_{t-1}p_x \)

Note: For more on these calculations, see the chapter "Profit vector, profit signature, NPV and profit margin".

5. Common Pitfalls and Tips

1. Interest Timing: Always check when premiums are paid and when claims are paid. Usually, premiums are at the start of the year (so they earn a full year of interest) and claims are at the end (so they don't). If claims are paid "immediately on death," you may need to adjust the interest calculation!

2. The Survival Probability: Remember that the end-of-year reserve \( _tV \) is only held for those who survive the year. This is why it is multiplied by \( p_{x+t-1} \).

3. Consistency: Ensure the interest rate used to accumulate the cashflows in the profit test is the rate the company expects to earn on its actual assets.

Memory Aid: Think of the reserve like a security deposit on an apartment. You pay it at the start (reducing your available cash), it sits there during the lease, and you get it back at the end. It doesn't change your total wealth over the whole period, but it definitely changes how much "spending money" you have in month one!

Summary Table: Impact of Increasing Reserves

Key Takeaways

• Reserves are liabilities set aside to meet future obligations.
• Including reserves in a profit test allows us to determine the emergence of profit.
• The formula for profit in year \( t \) accounts for the interest earned on the starting reserve and premiums, less claims and the reserves for survivors.
• Higher reserves defer profit to later years, which generally reduces the Net Present Value of the contract.