Retrospective Reserves (Contingencies: Actuarial Mathematics)

Summary

This video explains retrospective reserves and how they relate to prospective reserves. Retrospective reserves are the accumulated value of premiums received (allowing for interest and survivorship) minus accumulated benefits and expenses paid — essentially the contract’s self-funding history. The instructor shows how to move between expected present values and accumulated values using discounting and npx, and gives an intuitive explanation for dividing by npx. He proves prospective and retrospective reserves are equal when the same basis is used for both reserves and premium calculation, using a whole-life example. The video closes by noting recursive reserves will be covered next.

Takeaways

• 😀 **Prospective reserves** focus on setting aside reserves for future liabilities, calculated based on the expected present value (EPV) of outgo minus income.
• 😀 **Retrospective reserves** involve looking back at the actual premiums received and benefits paid, accumulating these values over time, with interest and survivorship adjustments.
• 😀 The main difference between **prospective** and **retrospective reserves** is that prospective reserves predict future requirements, while retrospective reserves rely on past data and accumulated premiums.
• 😀 **Retrospective reserves** are calculated by subtracting the accumulated value of benefits and expenses from the accumulated value of premiums received, adjusted for interest and survivorship.
• 😀 To accumulate the present value to the future value, we apply compound interest and adjust for survival probability using actuarial notation (npx).
• 😀 **Interest and survivorship** factors (like npx) are essential in calculating accumulated reserves, as they account for both time value of money and the probability of survival for policyholders.
• 😀 The **equivalence principle** is used to calculate premiums, which must be consistent with the reserve calculations, ensuring that the **prospective** and **retrospective** reserves are equal under the same assumptions.
• 😀 **Prospective reserves** are easier to calculate because they only require predicting future liabilities, while **retrospective reserves** require accumulated historical data on premiums, benefits, and expenses.
• 😀 The **equality of prospective and retrospective reserves** holds if both are calculated on the same basis (same assumptions and actuarial model) and the premiums are calculated based on the same principles as the reserves.
• 😀 The key to understanding **retrospective reserves** in life insurance contracts lies in breaking down the contract into parts like **temporary annuities** or **term assurances**, especially when applying retrospective calculations to a whole life insurance contract.

Q & A

What is the main difference between prospective and retrospective reserves?

Prospective reserves focus on the future, estimating the future liabilities based on expected income and outgo. Retrospective reserves, on the other hand, look at the past, calculating the accumulated value of premiums received minus benefits paid to date, considering interest and survivorship.

How are reserves calculated in the context of prospective reserves?

Prospective reserves are calculated by taking the expected present value (EPV) of the future outgo (claims and expenses) and subtracting the EPV of future income (premiums). The result represents the amount the insurer must reserve to meet future liabilities.

What does 'self-funding' mean in the context of insurance contracts?

'Self-funding' means that the premiums collected from an insurance contract should be sufficient to cover all future liabilities related to that contract. It's not about every individual contract being self-sufficient, but rather the entire book of contracts collectively being able to cover claims.

What role does survivorship play in calculating retrospective reserves?

Survivorship accounts for the probability that policyholders will survive to a future point. When calculating retrospective reserves, the accumulated premiums and outgo need to be adjusted based on the probability of survival, which is reflected by the term 'npx'.

Why is there no actuarial notation for the accumulation of assurance benefits?

Assurance benefits cannot be accumulated because the contract either terminates upon death or reaches full value at the end. Unlike annuities, which accumulate over time, assurance benefits are more binary, either paid out or not, which makes accumulation conceptually unnecessary.

How does the calculation of retrospective reserves incorporate interest and probability?

Retrospective reserves are calculated by accumulating the premiums received, factoring in compound interest over time, and adjusting for the probability of survival using the term 'npx'. This ensures that the reserve accurately reflects the value of premiums collected and the benefits paid out over time.

When will prospective and retrospective reserves be equal?

Prospective and retrospective reserves will be equal when they are calculated using the same actuarial assumptions and premium basis. These reserves are conceptually different, but under these conditions, they will yield the same value.

What is the equivalence principle, and how does it relate to reserve calculations?

The equivalence principle ensures that the present value of premiums equals the present value of benefits and expenses. This principle is used to calculate reserves, ensuring that the premiums collected will be sufficient to cover future liabilities, which is the basis for both prospective and retrospective reserve calculations.

What is the significance of multiplying by factors like '1 + i' and 'npx' in reserve calculations?

Multiplying by '1 + i' accounts for the accumulation of value due to interest over time, while multiplying by 'npx' adjusts for the probability of survival. These factors ensure that the reserve calculations accurately reflect both the time value of money and the likelihood of policyholders surviving to claim benefits.

Why are temporary annuities and term assurances used in retrospective reserve calculations for a whole life insurance contract?

Temporary annuities and term assurances are used in retrospective reserve calculations for whole life insurance contracts because the reserve is calculated retrospectively from a specific point in time (T years). Even though the policy is whole life, the calculation only considers the contract's history up to that point, which requires applying terms from temporary annuities and term assurances.