Life Annuity

Study guides: Exam FAM, Exam MAS-I

A Life Annuity pays periodic benefits contingent on the survival of one or more lives. A life-annuity-due pays 1 at the start of each year while $(x)$ is alive; its actuarial present value (EPV) is $\ddot{a}_x$ . A life-annuity-immediate pays 1 at the end of each year, provided $(x)$ is still alive then; its EPV is $a_x$ .

$\ddot{a}_x = \sum_{k=0}^{\infty} v^k\,{_k}p_x$

$\ddot{a}_x = 1 + v\,p_x\,\ddot{a}_{x+1}$

$\ddot{a}_x = \frac{1 - A_x}{d}$

Recursion: an annuity-due on $(x)$ is a payment of 1 now plus, if $(x)$ survives the year, an annuity-due on $(x+1)$ delivered in one year
Relationship to Whole Life Insurance: $d\,\ddot{a}_x + A_x = 1$ , where $d = 1 - v = i/(1+i)$ is the effective rate of discount — 1 today is worth $d$ at the start of each year while $(x)$ lives plus the 1 returned at the end of the year of death
The annuity-immediate is the annuity-due without its first payment: $a_x = \ddot{a}_x - 1$
Temporary annuity (pays until death or $n$ years, whichever comes first): $\ddot{a}_{x:\overline{n}|} = \ddot{a}_x - v^n\,{_n}p_x\,\ddot{a}_{x+n}$ . Guaranteed for $n$ years (whichever comes last): $\ddot{a}_{\overline{n}|} + v^n\,{_n}p_x\,\ddot{a}_{x+n}$
Paid continuously (continuous variant): $\bar{a}_x = \int_0^\infty e^{-\delta t}\,{_t}p_x\,dt = (1 - \bar{A}_x)/\delta$ ; under a constant force of mortality $\mu$ , $\bar{a}_x = 1/(\mu + \delta)$

Annuity-Due from the Illustrative Life Table, Two Ways The Illustrative Life Table ( $i = 0.06$ ) gives $1000q_{40} = 2.78$ , $\ddot{a}_{41} = 14.6864$ and $1000A_{40} = 161.32$ . Calculate $\ddot{a}_{40}$ by the recursion and from $A_{40}$ .

Answer $\ddot{a}_{40} = 1 + \frac{0.99722 \times 14.6864}{1.06} = 1 + 13.8166 = 14.8166$ $\ddot{a}_{40} = \frac{1 - 0.16132}{0.06/1.06} = \frac{0.83868 \times 1.06}{0.06} = 14.8167$ The table lists $\ddot{a}_{40} = 14.8166$ ; the second route is off in the last digit only because $A_{40}$ is rounded to five places.

Temporary Annuity-Due to Age 65 Using the Illustrative Life Table ( $\ell_{40} = 9{,}313{,}166$ , $\ell_{65} = 7{,}533{,}964$ , $\ddot{a}_{40} = 14.8166$ , $\ddot{a}_{65} = 9.8969$ , $i = 0.06$ ), find the EPV of an annuity-due of 1 a year on $(40)$ that stops at death or at age 65, whichever comes first.

Answer Give a life annuity on $(40)$ now and take back, in 25 years, a life annuity on $(65)$ if $(40)$ is still alive: $\ddot{a}_{40:\overline{25}|} = \ddot{a}_{40} - v^{25}\,\frac{\ell_{65}}{\ell_{40}}\,\ddot{a}_{65}$ $= 14.8166 - 0.23300 \times 0.80896 \times 9.8969 = 14.8166 - 1.8654 = 12.9512$

Continuous Annuity Under Constant Force of Mortality A life has constant force of mortality $\mu = 0.03$ and the force of interest is $\delta = 0.05$ . Calculate $\bar{a}_x$ .

Answer $\bar{a}_x = \frac{1}{\mu + \delta} = \frac{1}{0.03 + 0.05} = \frac{1}{0.08} = 12.5$