Life Annuities: Whole Life, Temporary, and Deferred

Life Annuity Fundamentals

A life annuity is a series of payments contingent on the survival of the annuitant. Unlike a certain annuity (which makes a fixed number of payments regardless of survival), a life annuity ceases upon death. Life annuities are the financial mechanism behind pension benefits, structured settlements, and the payout phase of retirement products. Computing their actuarial present values is a core skill for Exam LTAM.

Life annuities come in several varieties distinguished by payment timing and duration. The annuity-due makes payments at the beginning of each period, while the annuity-immediate makes payments at the end. The annuity may be whole life (continuing for the annuitant's entire remaining lifetime), temporary (limited to a specified number of years), or deferred (with payments beginning after a waiting period). Each variety has its own APV formula, but all are interconnected through fundamental relationships.

Whole Life Annuity-Due

A whole life annuity-due pays 1 at the beginning of each year while (x) survives. The first payment is immediate (at time 0), and subsequent payments occur at times 1, 2, 3, ... as long as the annuitant is alive. The APV is a-ddot_x = sum from k=0 to infinity of v^k * k_p_x. The first term (k=0) equals 1 (certain first payment), and each subsequent term is smaller due to both discounting and mortality.

The present value random variable for the annuity-due is Y = a-ddot_{K_x+1} = (1 - v^{K_x+1})/d, where K_x is the curtate future lifetime. This connects directly to the whole life insurance present value Z = v^{K_x+1} through the relationship Y = (1-Z)/d. Taking expectations: a-ddot_x = (1 - A_x)/d, or equivalently A_x = 1 - d*a-ddot_x. This fundamental identity connects insurance and annuity values and is one of the most important formulas on Exam LTAM.

For the variance: Var(Y) = Var((1-Z)/d) = Var(Z)/d^2 = (^2A_x - A_x^2)/d^2. This shows that the annuity variance is directly proportional to the insurance variance, scaled by 1/d^2.

Whole Life Annuity-Immediate and Continuous

A whole life annuity-immediate pays 1 at the end of each year while (x) survives. The first payment is at time 1, and the relationship to the annuity-due is a_x = a-ddot_x - 1. This simply reflects that the annuity-due has one extra payment at time 0 (the immediate first payment) that the annuity-immediate lacks.

A continuous whole life annuity pays at a continuous rate of 1 per year while (x) survives. The APV is a-bar_x = integral from 0 to infinity of v^t * t_p_x dt. The fundamental identity for the continuous case is A-bar_x = 1 - delta*a-bar_x, where delta = ln(1+i). Under UDD, a-bar_x is approximately a-ddot_x - 1/2, reflecting that continuous payments average about half a year earlier than beginning-of-year payments.

Under constant force of mortality mu, the continuous annuity has a particularly simple form: a-bar_x = 1/(mu + delta). Combined with A-bar_x = mu/(mu + delta), the identity A-bar_x + delta*a-bar_x = mu/(mu+delta) + delta/(mu+delta) = 1 is confirmed.

Temporary Annuities

An n-year temporary annuity-due pays 1 at the start of each year for at most n years while (x) survives. Payments occur at times 0, 1, 2, ..., n-1, but stop earlier if the annuitant dies. The APV is a-ddot_{x:n} = sum from k=0 to n-1 of v^k * k_p_x. The corresponding insurance-annuity identity is A_{x:n} = 1 - d*a-ddot_{x:n}.

Temporary annuities are essential for computing net premiums: when premiums are paid as an n-year temporary annuity-due, the APV of premiums is P * a-ddot_{x:n}. Setting this equal to the APV of benefits gives the net premium P = A / a-ddot_{x:n}, where A is the insurance APV.

The continuous temporary annuity APV is a-bar_{x:n} = integral from 0 to n of v^t * t_p_x dt. It satisfies A-bar_{x:n} = 1 - delta*a-bar_{x:n}, paralleling the whole life identity.

Deferred Annuities

An n-year deferred whole life annuity-due begins payments at time n (instead of time 0) and continues for the remainder of the annuitant's life. The APV is n|a-ddot_x = sum from k=n to infinity of v^k * k_p_x = n_E_x * a-ddot_{x+n}. The factor n_E_x = v^n * n_p_x discounts the annuity value at age x+n back to the present, accounting for both interest and the probability of surviving to the deferred start date.

The whole life annuity decomposes into temporary plus deferred components: a-ddot_x = a-ddot_{x:n} + n|a-ddot_x. This identity is invaluable for computation. If you know any two of the three quantities, you can find the third. For example, n|a-ddot_x = a-ddot_x - a-ddot_{x:n}.

Deferred annuities are important in pension mathematics, where an employee contributes during working years and receives a pension annuity beginning at retirement. The deferred annuity APV combines the deferral (representing the working period) with the payment period (representing retirement).

Key Relationships and Conversions

Several relationships tie the different annuity types together. The fundamental insurance-annuity identities are: A_x = 1 - d*a-ddot_x (discrete whole life), A_{x:n} = 1 - d*a-ddot_{x:n} (discrete temporary/endowment), A-bar_x = 1 - delta*a-bar_x (continuous whole life). These allow instant conversion between insurance and annuity values.

Between annuity types: a_x = a-ddot_x - 1 (immediate from due), a-bar_x is approximately a-ddot_x - 1/2 (continuous from due, under UDD). For 1/m-thly annuities under UDD: a-ddot_x^{(m)} is approximately a-ddot_x - (m-1)/(2m). As m increases to infinity, this approaches a-ddot_x - 1/2 = a-bar_x.

Woolhouse's formula provides a better approximation: a-ddot_x^{(m)} is approximately a-ddot_x - (m-1)/(2m) - (m^2-1)/(12m^2)*(delta + mu_x). The three-term Woolhouse approximation accounts for the slope of the survival function and is more accurate at older ages where mortality changes rapidly.

Common Pitfalls

A frequent error is confusing annuity-due (payments at the beginning of the period, denoted with double dots) and annuity-immediate (payments at the end, without double dots). The two differ by exactly one payment at time 0. Another common mistake is applying the whole life identity A_x = 1 - d*a-ddot_x when the problem involves a temporary annuity, or vice versa. Each identity matches a specific insurance-annuity pair.

When computing deferred annuities, students sometimes forget the survival probability in the deferral factor. The deferred annuity n|a-ddot_x = v^n * n_p_x * a-ddot_{x+n}, not just v^n * a-ddot_{x+n}. The mortality factor n_p_x is essential because the annuitant must survive the deferral period to receive any payments.

Study Tips

Exam LTAM problems frequently require computing one annuity or insurance value from others using the fundamental identities. Memorize the key relationships and practice applying them in both directions. Work through problems that combine multiple annuity types (computing a deferred annuity from whole life and temporary values, or finding a premium using a temporary annuity). The insurance-annuity identities save enormous computation time and should become second nature.