Anualidades Contingentes en Seguros de Vida

Summary

The presentation explains how annuities in financial mathematics differ from contingent annuities in actuarial mathematics, and how each can be valued using interest rates and, for actuarial annuities, mortality information.

1. What an annuity is

An annuity is a sequence of payments made at regular intervals—for example, monthly, quarterly, or annually. Two features determine its type:

• Duration
  • Temporary: Payments are made for a specified number of periods.
  • Lifetime: Payments continue while a person is alive.
  • Perpetual: Payments continue indefinitely, as in a financial-mathematics perpetuity.
• Payment timing
  • Ordinary (immediate): Payments occur at the end of each period.
  • Due (in advance): Payments occur at the beginning of each period.

2. Certain and contingent annuities

• Certain annuities, used in financial mathematics, have a payment schedule and duration fixed in advance. Their value depends on the payment amount, timing, term, and interest rate—not on whether a person survives.
• Contingent annuities, used in actuarial mathematics, depend on an event such as the recipient’s survival. In a life-insurance annuity, payments generally stop when the recipient dies.

A life annuity is therefore not the same as a financial perpetuity: a perpetuity has no mortality condition, while a life annuity is payable only while the recipient remains alive.

3. Valuation principles and notation

All payments are brought to a common date, usually the present, using discounting. For an actuarial annuity, each payment is also weighted by the probability that the recipient survives long enough to receive it.

The presentation relates actuarial values to commutation functions derived from mortality tables. In standard notation:

• A temporary certain annuity-immediate for (n) periods has value ((1-v^n)/i), where (v=1/(1+i)).
• A temporary annuity-due shifts each payment one period earlier.
• Actuarial life and temporary annuities use mortality-based functions. For example, standard commutation notation expresses a life annuity-immediate as (N_{x+1}/D_x) and a life annuity-due as (N_x/D_x). Exact symbols can vary by convention.

The lecture derives four main actuarial cases:

• Life annuity-immediate
• Life annuity-due
• Temporary life annuity-immediate
• Temporary life annuity-due

The due version is valued by shifting the payment stream one period earlier. The temporary version stops after the specified term, while the lifetime version continues only as long as the person survives.

4. Building mortality tables and actuarial values

The presenter outlines a workflow for creating or completing mortality-table inputs and using them to value annuities:

1. Start with a cohort and available mortality data. The example uses an initial group of 10,000 people and death probabilities available for a range of ages.
2. Project missing death probabilities. The presentation demonstrates using time-series forecasting in Risk Simulator to extend the data beyond the observed ages. Several smoothing approaches are considered; the presenter says double exponential smoothing gives the best fit in the example.
3. Choose a terminal age. The example uses age 100, while noting that tables may use other limits. At the terminal age, the table assigns a death probability of 1.
4. Calculate survival probabilities. If (q_x) is the probability of death at age (x), then the probability of survival is (p_x=1-q_x).
5. Calculate survivors by age. Apply each year’s survival probability to the number alive at the previous age.
6. Apply discounting. Use the interest rate to calculate the present value of future payments.
7. Build commutation columns. Combine discounted values with the number of survivors and accumulate the resulting quantities. These columns allow the annuity formulas to be evaluated efficiently.

The presenter emphasizes that actuarial calculations depend on both financial discounting and survival probabilities.

5. Comparing costs through examples

The presentation compares:

• A financial perpetuity with annual payments, valued using an interest rate.
• A life-contingent annuity for a person of a specified age, where payments stop at death.
• A temporary annuity-due with payments at the beginning of each year, compared with a corresponding actuarial payment stream.

The main lesson is that actuarial values vary with the recipient’s age and the assumed mortality table. The presenter’s examples describe the relative cost of certain and contingent temporary annuities as changing with age: younger recipients are more likely to survive the term, while older recipients are less likely to do so. The transcript’s numerical comparisons are partly unclear, so it does not support a reliable general numerical conclusion.

6. Net premiums

The lecture introduces two insurance-premium ideas:

• Net single premium: The present actuarial value of the expected insurance benefits, accounting for interest and mortality or survival probabilities.
• Level net premium: A product’s cost paid in equal periodic installments rather than as one lump sum. Because the insured may die before all installments are paid, the payment stream is also affected by survival probabilities.

The broader objective is to show why actuarial pricing can differ by age: people of different ages do not have the same probabilities of survival or death.

Speakers and sources featured

• Gustavo Vadillo — The presenter, identified near the end through the contact details he gives.
• Alicia — Mentioned as having introduced the presenter; no substantive presentation speech from her appears in the subtitles.
• Jose and Daniel — Audience participants thanked or addressed near the end; their comments are not captured in the subtitles.
• Risk Simulator — Software shown as a tool for forecasting missing mortality-table values.