Life annuities are a cornerstone of actuarial science, particularly in the context of insurance and retirement planning. They represent a contract where an individual (the annuitant) pays a sum of money to an insurance company, and in return, the company promises to make a series of payments to the annuitant for the rest of their life. This learning module will delve into the fundamental concepts, calculations, and applications of life annuities, crucial for understanding actuarial exams like those from the Society of Actuaries (SOA).

At its core, a life annuity is a financial product designed to provide a guaranteed income stream for an individual's lifetime. This income stream can be used for various purposes, most commonly to supplement retirement income, ensuring that individuals do not outlive their savings. The key feature is the 'life contingency' – the payments are contingent upon the annuitant being alive.

In actuarial mathematics, we use specific notation to represent life annuity concepts. This allows for precise calculations and modeling. Understanding these symbols is crucial for solving problems on actuarial exams.

The primary task in actuarial work involving annuities is calculating their present values. This involves considering the timing of payments, the interest rate, and the probabilities of survival. For a life annuity-immediate of $1 per year, the present value is denoted by$a_x$. For a life annuity-due of$1 per year, where payments are made at the beginning of each period, the present value is denoted by $\ddot{a}_x$. The relationship between these two is $\ddot{a}_x = a_x + 1$.

For deferred annuities, we introduce the notation $_n a_x$ for the present value of a life annuity-immediate of $1 per year, where payments begin after$n$years, provided the annuitant is alive at that time. The relationship is$n a_x = v^n \cdot {}{n}p_x \cdot a_{x+n}$. This means the present value is the discount factor for$n$years, multiplied by the probability of surviving$n$years, and then multiplied by the present value of an annuity starting at age$x+n$.

Life annuities are fundamental to several types of insurance products. They are often used in conjunction with life insurance policies, particularly in payout options. For example, a beneficiary of a life insurance policy might choose to receive the death benefit as a lifetime annuity rather than a lump sum. This provides them with a stable income stream. Pension plans also heavily rely on annuity principles to provide retirement income to former employees.

To deepen your understanding, explore the actuarial tables (like the SOA's mortality tables) which provide the probabilities of survival and death at different ages. These tables are the empirical foundation for all life contingency calculations. Understanding the assumptions behind these tables and how they are constructed is also vital for advanced study.