Introduction to Net Premiums and Reserving

In previous chapters, we explored how insurance companies calculate gross premiums—the actual amount a policyholder pays, which includes coverage for benefits, expenses, and a bit of profit. In this chapter, we are stripping all that away to look at net premiums and net premium valuation.

Think of the net premium as the "pure" cost of the insurance. It is the amount needed only to cover the benefits promised, with absolutely no allowance for commissions, office rent, or marketing costs. While insurance companies don't actually sell net premium policies, this concept is vital for reserving (setting aside money to pay future claims) and is often required by regulators to ensure a company remains solvent.

1. What is a Net Premium?

A net premium is calculated using the equivalence principle, but with one major rule: Expenses are ignored.

The fundamental equation is:
\(PV(\text{Net Premiums}) = PV(\text{Benefits})\)

Example: Whole Life Assurance
For a level net premium \(P\) payable annually in advance for life to a life aged \(x\), for a benefit of 1 payable at the end of the year of death:
\(P \cdot \ddot{a}_x = A_x\)
\(\implies P = \frac{A_x}{\ddot{a}_x}\)

Quick Tip: If you see the word "Net" in an exam question, your "expense alarm" should turn off! You do not need to worry about renewal costs, initial commissions, or claim settlement expenses.

Did you know? Net premiums are sometimes called "pure premiums" because they represent the mathematical expectation of the benefit cost alone.

2. Net Premium Valuation (Reserving)

A reserve is the amount of money an insurer must hold today to meet its future obligations. A net premium valuation means we calculate this reserve using only the net premium and the benefits.

Prospective Net Premium Reserve

This is the most common method. We look forward into the future and calculate:
\(\text{Reserve} = PV(\text{Future Benefits}) - PV(\text{Future Net Premiums})\)

For a whole life assurance of 1 issued to \((x)\), the net premium reserve at time \(t\) (denoted as \({}_tV\)) is:
\({}_tV = A_{x+t} - P \cdot \ddot{a}_{x+t}\)

Retrospective Net Premium Reserve

This looks backward at what has happened since the policy started:
\(\text{Reserve} = \text{Accumulated Value of Past Net Premiums} - \text{Accumulated Value of Past Benefits}\)

Don't worry if this seems tricky at first... In a "net" world where the basis (interest and mortality) remains the same as when the premium was calculated, the Prospective Reserve and Retrospective Reserve will always give you the exact same answer!

Key Takeaway: Net premium reserves only consider the "pure" side of the contract. We assume the premium being received is exactly enough to cover the benefits over the life of the policy.

3. Comparing Net and Gross Premium Valuation

It is important to understand how net premium reserves relate to the gross premium reserves you studied previously.

Gross Premium Reserve incorporates:
1. Future Benefits
2. Future Expenses
3. Future Gross Premiums (the actual amount paid)

Net Premium Reserve incorporates:
1. Future Benefits
2. Future Net Premiums

Why use Net Premium Valuation?
  • Consistency: It provides a standardized way to compare different insurance companies.
  • Prudence: Regulators often prefer it because it doesn't allow companies to "front-load" potential future profits from the gross premium into their current accounts.
  • Simplicity: You don't have to estimate future inflation-linked expenses or complex commission structures.

4. Important Relationships and Notation

In CM1, you will often need to manipulate formulas for net premium reserves. One of the most famous identities for a whole life assurance reserve is:
\({}_tV = 1 - \frac{\ddot{a}_{x+t}}{\ddot{a}_x}\)

This formula is beautiful because it shows the reserve only depends on the annuity values at the starting age and the current age. It helps you see that as someone gets older (\(x+t\)), the annuity \(\ddot{a}_{x+t}\) gets smaller, which makes the fraction smaller, and thus the reserve \({}_tV\) grows toward 1 (the sum assured).

Common Mistake to Avoid: When calculating a net premium reserve, never use the Gross Premium in the formula. If the question asks for a "Net Premium Reserve" but gives you the actual premium paid by the policyholder, you must first calculate the Net Premium yourself using the equivalence principle!

5. Summary and Quick Review

To master this chapter, ensure you are comfortable with these three steps:

  1. Calculate the Net Premium: Use \(PV(\text{Premiums}) = PV(\text{Benefits})\) ignoring all expenses.
  2. Choose your method: Usually Prospective (Future Benefits minus Future Net Premiums).
  3. Apply the Valuation: Use the age of the policyholder at the date of valuation (e.g., \(x+t\)).

Key Terms Review:
Net Premium: The pure cost of benefits only.
Equivalence Principle: The basis for setting the net premium so that \(NPV = 0\) at outset.
Net Premium Valuation: A reserving method that ignores expenses and uses the net premium instead of the gross premium.

Note: For further details on how these reserves change year-on-year, see the chapter on "Recursive relationships between successive reserves". Untuk calculating the actual profit or loss compared to these reserves, see "Death strain at risk and mortality profit".