Deferred Life Annuity¶

Definition¶

A unit annual life annuity-due deferred $u$ years, for integer $u\ge0$ $0$ , begins payments at time $u$ conditional on survival and then continues while the annuitant lives. Its present value is $\displaystyle Y=\sum_{k=u}^{\infty}v^k\indicator_{\{T_x>k\}}$ $k = u \sum \infty v^{k} 𝟙 1_{{T_{x} > k}}$ , and its actuarial present value (APV) is denoted ${}_{u|}\ddot a_x=\mathbb{E}[Y]$ $E [Y]$ .

Interpretation¶

A pension can be viewed as an annuity whose first payment waits until retirement. Survival to retirement and discounting both affect its issue-date value.

Broken timeline for a life aged 40: no payments before time 20, unit payments from time 20 worth 12 at age 60, carried back by 20E40 = 0.3 to an APV of 3.6.

The figure uses the Examples case: a unit pension for a life aged 40, deferred $u=20$ years (the time axis is broken between 2 and 19). No payments are made before time 20. From then on, unit payments are made while the annuitant is alive; for an annuitant alive at age 60 they are worth $\ddot a_{60}=12$ at time 20. Multiplying by ${}_{20}E_{40}=0.3$ $0.3$ , which discounts over 20 years and allows for survival to age 60, gives ${}_{20|}\ddot a_{40}=3.6$ at issue.

Move $u$ : no payment is made before time $u$ , and ${}_{u|}\ddot a_{40}={}_uE_{40}\ddot a_{40+u}=\ddot a_{40}-\ddot a_{40:\actuarialterm{u}}$ $\overset{a}{¨}_{40 : u}$ . The defaults give ${}_{20}E_{40}\approx0.30$ and $\ddot a_{60}\approx12.0$ $12.0$ , close to the Examples. The conditional later-life annuity is defined only when ${}_up_{40}>0$ $0$ ; with zero survival the unconditional deferred APV is still zero.

The pure endowment factor ${}_uE_x$ discounts the start date and accounts for survival to it. Conditional on reaching age $x+u$ $u$ , the remaining payments form a whole life annuity-due, with value $\ddot a_{x+u}$ at that date. This gives the first identity below. Alternatively, remove the first $u$ payments from a whole life annuity-due at age $x$ : those payments form a temporary life annuity, giving the second identity.

Key Results¶

The first factorisation below requires ${}_up_x>0$ $0$ . If survival to the start has probability zero, the issue-date deferred APV is zero and the conditional later-life annuity is undefined.

\begin{aligned} {}_{u|}\ddot a_x&={}_uE_x\ddot a_{x+u},\\[4pt] {}_{u|}\ddot a_x&=\ddot a_x-\ddot a_{x:\actuarialterm{u}}. \end{aligned}

When ${}_up_x>0$ $0$ , the first identity follows by writing ${}_{u+j}p_x={}_up_x{}_jp_{x+u}$ in the payment sum.
For a deferred annuity-immediate, specify that the first payment is at $u+1$ $1$ ; the factorization uses $a_{x+u}$ .

Examples¶

If ${}_{20}E_{40}=0.3$ and $\ddot a_{60}=12$ $12$ , a unit pension starting at age 60 has issue-date APV $0.3(12)=3.6$ $3.6$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 6

Aliases: deferred annuity due, pension deferment, deferred annuity, deferred whole life annuity, deferred life annuity-due Keywords: pension starting at retirement age, u| ä x = u E x ä x+u, whole life annuity minus temporary annuity, deferment period Symbols: infinity, sum

Last updated: 27 October 2026 11:52