Computing EPVs of Life Contingent Benefits

This module focuses on calculating the present value of benefits that are contingent on human life. These calculations are fundamental to actuarial science, particularly in the pricing and reserving of life insurance and annuity products. We will explore the core concepts and methodologies used to determine these Expected Present Values (EPVs).

Core Concepts: EPVs and Life Contingencies

An Expected Present Value (EPV) represents the average present value of a future payment, taking into account the probability that the payment will be made. In the context of life contingencies, these probabilities are determined by mortality rates. The key elements are: the benefit amount, the timing of the benefit, and the probability of survival to receive that benefit.

Notation and Symbols

Understanding actuarial notation is crucial for working with life contingencies. Key symbols include:

Symbol	Meaning
$A_x$	EPV of a whole life insurance of 1 payable at the end of the year of death.
$A_x^{(m)}$	EPV of a whole life insurance of 1 payable at the end of the m-th period of the year of death.
$A_{x:n\|}$	EPV of an n-year term life insurance of 1 payable at the end of the year of death.
$A_{x:n\|}^1$	EPV of an n-year term life insurance of 1 payable at the end of the year of death, if death occurs within n years.
$I_x$	EPV of an increasing whole life annuity of 1, 2, 3, ... payable at the end of each year of survival.
$a_x$	EPV of a whole life annuity-due of 1 payable at the beginning of each year of survival.
$a_{x:n\|}$	EPV of an n-year term annuity-due of 1 payable at the beginning of each year of survival.

Calculating EPVs for Common Benefits

We can derive formulas for various life contingent benefits using the fundamental principles of probability and discounting. These often involve relationships between insurance and annuity EPVs.

What is the relationship between the EPV of a whole life insurance (

A_x

) and a whole life annuity-due (

a_x

) if the benefit is 1?

$A_x = 1 - a_x$ (assuming benefits are paid at the end of the year of death for insurance and beginning of the year of survival for annuity).

The EPV of a benefit can be calculated by summing the present value of each possible future payment, weighted by the probability of that payment occurring. For example, a whole life insurance pays 1 at the end of the year of death. The probability of dying between age $x+k$ and $x+k+1$ is $q_{x+k}$ . The present value of this payment is $v^{k+1}$ .

The calculation of EPVs involves integrating probabilities of survival and death over time with discount factors. Imagine a timeline where at each point, there's a probability of a life event (death or survival) and a corresponding discount factor. The EPV is the sum of the present values of all potential future benefit payments, each weighted by its probability of occurrence. This process is visualized as a weighted sum of discounted future cash flows, where the weights are derived from mortality tables.

📚

Text-based content

Library pages focus on text content

Approximations and Simplifications

For practical purposes, especially in actuarial exams, we often use approximations or standard formulas. For instance, the EPV of a whole life insurance can be approximated using the EPV of an annuity. The relationship $A_x = 1 - a_x$ is a fundamental identity that simplifies calculations.

Remember that $A_x$ is the EPV of a death benefit paid at the end of the year of death, while $a_x$ is the EPV of an annuity paid at the beginning of each year of survival. The difference in payment timing is crucial for their relationship.

Key Formulas and Relationships

Several key relationships are essential for solving problems involving life contingencies:

Loading diagram...

The EPV of a benefit is the sum of the present value of each potential future payment, weighted by the probability that the payment will occur. This involves understanding mortality tables, discount factors, and the specific structure of the benefit (e.g., whole life, term life, annuity).

What does

v^{k+1}

represent in the EPV calculation for a benefit payable at the end of the year of death?

It represents the present value of a payment made at time $k+1$ (end of the $(k+1)$ -th year).

Learning Resources

Actuarial Mathematics: Life Contingencies(documentation)

This official SOA study note provides foundational material on life contingencies and the calculation of EPVs, directly relevant to actuarial exams.

Introduction to Life Contingencies(documentation)

A comprehensive introduction to life contingencies from the Institute and Faculty of Actuaries, covering EPVs and related concepts.

Actuarial Notation and Life Contingencies(tutorial)

This tutorial offers a clear explanation of actuarial notation and its application in calculating EPVs for life contingent benefits.

Life Contingencies - Actuarial Exam FM/IFM/STAM/LTAM(video)

A video lecture that breaks down key concepts in life contingencies, including EPV calculations, with examples.

Actuarial Mathematics: EPV of Life Insurance(documentation)

This resource from The Actuarial Foundation delves into the EPV of life insurance products, providing detailed explanations and formulas.

Life Contingencies - Expected Present Values(documentation)

A study guide focusing specifically on Expected Present Values in life contingencies, with practice problems and solutions.

Actuarial Mathematics: Annuities(documentation)

This document from The Actuarial Foundation covers the EPV of annuities, which are closely related to life insurance calculations.

Life Contingencies - SOA Exam LTAM(video)

A playlist of video tutorials covering topics relevant to SOA Exam LTAM, including detailed explanations of life contingencies and EPVs.

Actuarial Mathematics: Life Tables and Mortality(documentation)

Understanding life tables and mortality is fundamental to calculating EPVs. This resource provides a solid foundation.

Introduction to Actuarial Science(documentation)

This page offers a broad overview of actuarial science, with links to resources that touch upon life contingencies and EPVs.