Understanding Probabilities of Survival and Death

In the realm of actuarial science, particularly for competitive exams like those administered by the Society of Actuaries (SOA), understanding the probabilities of survival and death is fundamental. These probabilities form the bedrock for calculating insurance premiums, annuities, and other financial products that depend on human lifespan. This module will introduce you to the core concepts and notation used to quantify these life contingencies.

Key Concepts and Notation

The study of life contingencies relies on a standardized set of notations to represent probabilities related to a person's life. We typically focus on a single individual, often denoted as 'x', who is currently alive at age x. The primary probabilities we deal with are the probability that an individual aged x will survive for a certain number of years, and the probability that they will die within a certain period.

What does the notation

_5 p_{30}

represent?

The probability that a person aged 30 will survive for 5 years (i.e., reach age 35).

What is the relationship between

_n p_x

and

_n q_x

$_n q_x = 1 - _n p_x$ (The probability of dying within n years is 1 minus the probability of surviving n years).

Mortality Tables: The Foundation

Actuarial calculations for survival and death probabilities are heavily reliant on mortality tables. These tables, often derived from historical data and actuarial assumptions, provide a statistical representation of death rates across different ages. They are essential tools for actuaries to estimate future mortality trends and their financial implications.

Mortality tables, also known as life tables, are statistical tables that show the probability of death (mortality rate) and the probability of survival for a given population at different ages. A typical life table includes columns for: age (x), number of lives alive at the beginning of the year ( $l_x$ ), number of deaths during the year ( $d_x$ ), and probabilities of survival ( $_n p_x$ ) and death ( $_n q_x$ ). The values are usually derived from observed mortality data and smoothed using actuarial techniques. For example, if $l_{30} = 100,000$ and $l_{35} = 95,000$ , then $_5 p_{30} = 95,000 / 100,000 = 0.95$ . The number of deaths in that period would be $d_{30..35} = l_{30} - l_{35} = 100,000 - 95,000 = 5,000$ . The probability of death within these 5 years would be $_5 q_{30} = 5,000 / 100,000 = 0.05$ , which aligns with $1 - 0.95 = 0.05$ .

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Applications in Insurance

The probabilities of survival and death are directly applied in the pricing of life insurance products. For instance, a term life insurance policy's premium is calculated based on the probability that the insured will die within the policy term. Similarly, annuities, which provide a stream of payments for life, depend on the probability of survival to ensure the payments are made for the expected duration.

Understanding these probabilities is not just theoretical; it's the core of actuarial risk assessment and financial product design in the insurance industry.

Further Exploration

To excel in actuarial exams, a deep understanding of these concepts and their application is crucial. The following resources will provide further insights, examples, and practice opportunities.

Learning Resources

SOA Exam FM/IFM Study Notes - Life Contingencies(documentation)

Official study notes from the Society of Actuaries covering life contingencies, essential for understanding survival and death probabilities.

Introduction to Life Contingencies - Actuarial Outpost(blog)

A forum discussion providing introductory explanations and common questions about life contingencies, including survival and death probabilities.

Life Contingencies - Actuarial Society of South Africa(documentation)

Comprehensive notes on life contingencies from the Actuarial Society of South Africa, offering a structured approach to the topic.

Mortality Tables Explained(wikipedia)

An explanation of mortality tables, their purpose, and how they are used in finance and insurance to estimate life expectancies and probabilities.

Actuarial Mathematics: Life Table Functions(paper)

A mathematical paper detailing life table functions and their applications in actuarial science, including survival and death probabilities.

Probability and Statistics for Actuarial Science - Life Contingencies(documentation)

A PDF document covering life contingencies within the context of probability and statistics for actuarial science, with relevant formulas.

Introduction to Actuarial Science - Life Contingencies(documentation)

An introductory guide to actuarial science focusing on life contingencies, explaining core concepts like survival and death probabilities.

Actuarial Exam P/1 - Probability Concepts(tutorial)

Sample chapter from a study manual for Actuarial Exam P, covering fundamental probability concepts that underpin life contingencies.

Understanding Life Tables - Society of Actuaries(documentation)

A document from the SOA explaining the construction and interpretation of life tables, crucial for understanding survival and death probabilities.

Actuarial Mathematics: Survival Models(documentation)

A resource detailing survival models used in actuarial science, which directly relate to the probabilities of survival and death over time.