Certain-and-Life Annuities by Decomposition

Why Decompose a Certain-and-Life Annuity?

A certain-and-life annuity combines two promises: payments are guaranteed for an initial period, and payments after that period depend on the annuitant being alive. Valuing the contract directly can make it easy to confuse which payments are guaranteed and which require survival. Decomposition makes the timing explicit: value the guaranteed payments as a certain annuity, then add the expected present value of payments that begin after the guarantee period.

The distinction between annuity-due and annuity-immediate timing matters at the boundary. In an annuity-due, a payment is made at time 0, so a guarantee of $m$ years covers times $0,1,\ldots,m-1$. The first payment after that period is at time $m$. In an annuity-immediate, payments occur at times $1,2,\ldots$, so the $m$ guaranteed payments occur at times $1,2,\ldots,m$, and the life-contingent payments begin at time $m+1$.

Definition: An $m$-year certain-and-life annuity-due pays while the annuitant is alive, but guarantees payments at times $0,1,\ldots,m-1$, whether or not the annuitant is alive. Its life-contingent portion starts at time $m$. An $m$-year certain-and-life annuity-immediate guarantees payments at times $1,2,\ldots,m$; its life-contingent portion starts at time $m+1$. Unless stated otherwise, assume payments are level and annual, the first annuity-due payment is immediate, and payments at a scheduled time are made only if the annuitant is alive unless guaranteed.

The Decomposition

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Write ${}_mp_x$ for the probability that a life aged $x$ survives $m$ years. The annuity factors $\ddot{a}_{x+m}$ and $a_{x+m}$ are valued conditional on being alive at age $x+m$. Thus they describe future payments from that age onward, with due payments beginning immediately and immediate payments beginning one year later, respectively.

Theorem: For a unit annual payment, the present value of an $m$-year certain-and-life annuity-due is $$ \ddot{a}_{\overline{m}|}+v^m\,{}_mp_x\,\ddot{a}_{x+m}. $$ For a unit annual payment, the present value of an $m$-year certain-and-life annuity-immediate is $$ a_{\overline{m}|}+v^m\,{}_mp_x\,a_{x+m}. $$ Multiply the applicable factor by the annual payment amount to obtain the present value of level payments.

Here $\ddot{a}_{\overline{m}|}$ is the value of $m$ certain payments at times $0,\ldots,m-1$, while $a_{\overline{m}|}$ is the value of $m$ certain payments at times $1,\ldots,m$. The factor $v^m{}_mp_x$ discounts to time 0 and weights for survival to time $m$. The remaining life-annuity factor is conditional on survival to that time.

Value the guaranteed payments.
Use the temporary certain annuity factor matching the contract timing: annuity-due or annuity-immediate.

Locate the first life-contingent payment.
For a due annuity it is at time $m$; for an immediate annuity it is at time $m+1$.

Value the life-contingent tail.
Multiply the discount factor $v^m$, the probability ${}_mp_x$ of reaching the start of the tail, and the conditional life-annuity factor at age $x+m$.

Add the parts and scale by the payment.
The two parts cover non-overlapping payment times, so neither payment is omitted or counted twice.

Why the Formula Holds

The decomposition follows by separating the expected present value payment by payment. Consider the annuity-due. At time $k<m$, the payment is guaranteed, so its discounted value is $v^k$, regardless of survival. These payments sum to $\ddot{a}_{\overline{m}|}$. At time $m+j$, for $j=0,1,\ldots$, a payment is made only if the annuitant survives $m+j$ years. By conditioning on survival to time $m$, its probability is ${}_mp_x\,{}_jp_{x+m}$. Its expected present value is therefore $v^{m+j}{}_mp_x\,{}_jp_{x+m}$. Summing this tail gives

\sum_{j=0}^{\infty}v^{m+j}{}_mp_x\,{}_jp_{x+m} = v^m{}_mp_x\sum_{j=0}^{\infty}v^j{}_jp_{x+m} = v^m{}_mp_x\,\ddot{a}_{x+m}.

Adding the guaranteed payments proves the annuity-due formula. For an annuity-immediate, the guaranteed payments are at times $1,\ldots,m$, and the tail payments are at times $m+j$ for $j=1,2,\ldots$. The corresponding conditional sum starts with the one-year payment, so it is $a_{x+m}$, yielding the annuity-immediate formula. These expressions assume annual payments at the stated times; a different payment frequency or timing requires matching the cash-flow schedule to the appropriate factors.

Worked Calculations

Worked Example: An Annuity-Due with a Three-Year Guarantee

A hypothetical contract pays $1,000 annually as an annuity-due. It guarantees three years of payments. The effective annual interest rate is $i=0.05$, and the probability of surviving from issue to the end of year 3 is ${}_3p_x=0.80$. Conditional on reaching age $x+3$, assume a constant annual survival probability of $0.90$ thereafter. Find the present value.

Method. Add the three certain payments at times 0, 1, and 2 to the life-contingent annuity-due beginning at time 3. The method avoids treating the time-3 payment as guaranteed.

First, $v=1/1.05=20/21$. The certain-payment factor is

\ddot{a}_{\overline{3}|}=1+\frac{20}{21}+\left(\frac{20}{21}\right)^2 =\frac{1261}{441}=2.859410\ldots

Under the stated constant survival assumption, the conditional life-annuity-due factor at age $x+3$ is a geometric series with ratio $v(0.90)=6/7$:

\ddot{a}_{x+3}=\sum_{j=0}^{\infty}\left(\frac{6}{7}\right)^j =\frac{1}{1-6/7}=7.

The tail factor at issue is

v^3\,{}_3p_x\,\ddot{a}_{x+3} =\left(\frac{20}{21}\right)^3(0.80)(7) =\frac{44800}{9261}=4.837491\ldots.

The total factor is $2.859410\ldots+4.837491\ldots=7.696901\ldots$. Therefore the present value is $1{,}000(7.696901\ldots)=\$7{,}696.90$, rounded to the nearest cent. As a check, the time-3 payment is included in the tail and weighted by the probability of surviving to year 3.

Worked Example: An Annuity-Immediate with a Two-Year Guarantee

A hypothetical annuity-immediate pays $500 at the end of each year. The first two payments are guaranteed, and payments after that are made only while the annuitant is alive. Let $i=0.05$, ${}_2p_x=0.90$, and assume a constant annual survival probability of $0.90$ after age $x+2$. Calculate the present value.

Method. Value the certain payments at times 1 and 2, then add the life-contingent annuity-immediate beginning at time 3. The common timing error is to start the tail at time 2, which would count the second guaranteed payment again.

With $v=20/21$, the certain factor is

a_{\overline{2}|}=v+v^2 =\frac{20}{21}+\frac{400}{441} =\frac{820}{441}=1.859410\ldots.

For the conditional life-annuity-immediate at age $x+2$, the first payment is one year later. The ratio of successive discounted survival terms is $v(0.90)=6/7$, so

a_{x+2}=\sum_{j=1}^{\infty}\left(\frac{6}{7}\right)^j =\frac{6/7}{1-6/7}=6.

The tail value factor at issue is

v^2\,{}_2p_x\,a_{x+2} =\left(\frac{20}{21}\right)^2(0.90)(6) =\frac{2160}{441}=4.897959\ldots.

Thus the total factor is $(820+2160)/441=2980/441=6.757370\ldots$, and the present value is $500(2980/441)=\$3{,}378.68$, rounded to the nearest cent. The first two payments are represented once in the certain part, and the tail begins after them.

Worked Example: Checking the Guarantee’s Added Value

A hypothetical annuity-due pays $900 annually. The effective annual interest rate is $i=0.25$, and the annual survival probability is $0.80$ at every year of age. Compare the value with a two-year guarantee to the value of an annuity-due that pays only while alive from time 0.

Method. First calculate the certain-and-life value by decomposition, then compare it with the life annuity. This also checks that the guarantee adds value rather than changing payments after the guarantee period.

Here $v=0.80$. The certain two-payment factor is $1+v=1.80$. Conditional on survival to time 2, the life-annuity-due factor is

\ddot{a}_{x+2}=\frac{1}{1-v(0.80)} =\frac{1}{1-0.64}=\frac{25}{9}.

Since ${}_2p_x=0.80^2=0.64$, the tail factor at issue is

v^2\,{}_2p_x\,\ddot{a}_{x+2} =(0.64)(0.64)\frac{25}{9} =\frac{256}{225}.

The certain-and-life factor is $1.80+256/225=661/225=2.937777\ldots$, giving a value of $900(661/225)=\$2{,}644.00$. The life-only annuity-due factor is $1/[1-v(0.80)]=25/9$, so its value is $900(25/9)=\$2{,}500.00$. The guarantee adds $\$144.00$.

This difference is also obtained by noting that only payments made after death during the guaranteed period are extra. The time-0 payment is already certain under either contract; the time-1 payment adds $900v(1-0.80)=\$144.00$. The agreement between the two calculations checks the decomposition.

Common Timing Traps

The formulas are simple once every payment has a time index. Errors usually arise from a mismatch between the contract’s payment timing and the annuity factor, not from the survival calculation itself.

Starting the tail one payment too early. For an annuity-immediate with an $m$-year guarantee, the payment at time $m$ is guaranteed. The life-contingent tail starts at time $m+1$, represented by $v^m{}_mp_xa_{x+m}$.
Using the wrong annuity factor. The conditional factor after the guarantee is $\ddot{a}_{x+m}$ for a due contract and $a_{x+m}$ for an immediate contract. They differ in whether there is a payment at the start of the conditional period.
Forgetting survival to the tail. The conditional factor at age $x+m$ assumes the life has reached that age. From issue, include ${}_mp_x$ as well as $v^m$.
Counting payments twice. Write the guaranteed payment times first, then start the life-contingent list at the next applicable time. In particular, do not include the boundary payment in both components.

A useful check for an annuity-due is to compare it with a life-only annuity-due: the guarantee can add value only for scheduled payments during the guaranteed period that would otherwise be lost because of death. For an annuity-immediate, the same reasoning applies to its guaranteed payment dates. Payments after the guarantee period remain life-contingent in both contracts.

The decomposition is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus because it combines payment timing, discounting, survival probabilities, and life-annuity factors in one valuation. The main practical discipline is to mark the payment dates before selecting a formula.

Check Your Understanding

For each problem, identify the certain-payment period and the first life-contingent payment before calculating.

A hypothetical annuity-due pays $100 annually, with two years certain. Let $i=0.25$, ${}_2p_x=0.64$, and assume annual survival probability $0.80$ after age $x+2$. Find its present value to the nearest cent. (Answer: $293.78)
A hypothetical annuity-immediate pays $250 annually, with two payments certain. Let $i=0.25$, ${}_2p_x=0.64$, and assume annual survival probability $0.80$ after age $x+2$. Find its present value to the nearest cent. (Answer: $542.04)
For a unit annuity-due, let $i=0.10$, and suppose the annual survival probability is $0.90$ at every year of age. Find the increase in value from a three-year guarantee compared with a life-only annuity-due. (Answer: $30/121=0.2479$ approximately)
A hypothetical annuity-due pays $1,200 annually, with three years certain. Let $i=0.25$, ${}_3p_x=0.75$, and assume annual survival probability $0.80$ after age $x+3$. Find its present value to the nearest cent. (Answer: $4,208.00)

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