Introduction to Recursive Reserve Relationships

In the previous chapters, we learned how to calculate reserves using the prospective method (looking forward to future liabilities) and the retrospective method (looking back at past accumulation). While those methods are great for finding a reserve at a specific point in time, actuaries often need to know how a reserve "evolves" from one year to the next.

The recursive relationship is the "engine room" of actuarial modeling. It explains exactly what happens to the money held for a policy between time \(t\) and time \(t+1\). If you understand this, you understand the fundamental life cycle of an insurance contract!

The Equation of Equilibrium

At its heart, the recursive relationship is an Equation of Value. It assumes that the funds held at the start of the year, plus the income earned, must exactly equal the expected outgoings (claims) plus the funds needed for the survivors at the end of the year.

The General Formula (Gross Premium Reserve)

For a standard life assurance policy where premiums and expenses are paid at the start of the year and death benefits are paid at the end of the year, the relationship is:

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

Where:

  • \(_tV\): The gross premium reserve at time \(t\).
  • \(P\): The gross premium received at the start of the year.
  • \(e\): The expenses incurred at the start of the year.
  • \(i\): The valuation rate of interest.
  • \(S\): The sum assured (benefit) payable at the end of the year if the life dies.
  • \(q_{x+t}\): The probability that a life aged \(x+t\) dies before age \(x+t+1\).
  • \(p_{x+t}\): The probability that a life aged \(x+t\) survives to age \(x+t+1\).
  • \(_{t+1}V\): The gross premium reserve required at time \(t+1\).

Don't worry if this seems tricky at first! Just think of it as a balance sheet for a single year. The left-hand side is the "Money We Have," and the right-hand side is the "Money We Expect to Pay."

Breaking Down the Logic

Let's look at this step-by-step to see how the money moves:

  1. Start with the Reserve: We begin with the money already set aside (\(_tV\)).
  2. Add Premium and Deduct Expenses: At the start of the year, we collect the premium \(P\) and immediately pay the expenses \(e\). This leaves us with \((_tV + P - e)\).
  3. Add Interest: This "pot" of money sits in an investment account for the whole year, earning interest at rate \(i\). By the end of the year, it has grown to \((_tV + P - e)(1+i)\).
  4. Pay the Dead: Some policyholders will die during the year. For each one who dies (probability \(q_{x+t}\)), the company pays the sum assured \(S\).
  5. Set Aside for the Living: For those who survive (probability \(p_{x+t}\)), the company must keep the reserve \(_{t+1}V\) ready for the next year.

Quick Review: The formula essentially says: Income with Interest = Expected Death Claims + Expected Future Reserves.

Variations: Timing is Everything

In the IFoA CM1 exam, the examiners love to change the timing of cashflows. You must adjust your formula accordingly!

1. Death Benefit Payable Immediately

If the sum assured \(S\) is paid immediately on death (instead of at the end of the year), we assume on average it is paid halfway through the year. We must "interest up" the death claim from the middle of the year to the end of the year to keep the equation balanced:

\((_tV + P - e)(1+i) = q_{x+t} \cdot S \cdot (1+i)^{0.5} + p_{x+t} \cdot _{t+1}V\)

2. Annuity Contracts (Reserves for Survival)

For an annuity in payment, there are no premiums coming in (\(P=0\)). Instead, there is an outgo if the person survives. For an annuity payable at the start of the year:

\((_tV - \text{Annuity Payment} - e)(1+i) = p_{x+t} \cdot _{t+1}V\)

Note: There is no \(q_{x+t}\) term here because if the person dies, the insurance company pays nothing and keeps no reserve!

Using the Formula to Calculate Profit

In your Paper B (Excel) exam, you might use these recursive steps to project year-by-year profit. If the actual experience (actual interest, actual deaths) differs from your expected assumptions, the equation won't balance. That "imbalance" is your profit or loss for the year.

Did you know? This relationship is the basis for calculating Mortality Profit (also known as Profit from Mortality Surplus). If fewer people die than expected (\(q_{actual} < q_{expected}\)), the company usually "wins" for assurance products because it pays out fewer claims.

Common Pitfalls to Avoid

  • Expense Timing: Always check if expenses occur at the start of the year (subtract before interest) or the end of the year (subtract after interest).
  • The "p" and "q" logic: Remember that \(p_{x+t} = 1 - q_{x+t}\). Often, the question gives you one, and you need to calculate the other.
  • End of Term: At the very end of the policy term (say time \(n\)), the reserve \(_{n}V\) should equal the final survival benefit (like a maturity value in an endowment) or zero if it's a term assurance.

Key Takeaways

1. Logic: The recursive formula is just a year-to-year "money in = money out" calculation.
2. The Formula: \((_tV + P - e)(1+i) = q_{x+t} \cdot S + p_{x+t} \cdot _{t+1}V\).
3. Purpose: It allows actuaries to calculate reserves iteratively or determine the profit earned during a specific year of a policy's life.