Counting Payments in Temporary Life Annuities

Why Payment Counting Matters

In a temporary life annuity, the term tells you how long the payment schedule runs, but the payment timing determines which dates belong to that schedule. For an annuity-due, the first payment is made at the start of the term. For an annuity-immediate, the first payment is made one period later. That shift changes both the survival probabilities and the discount factors used in the value.

The most reliable method is to mark the payment dates before writing a formula. A temporary annuity-due with an $n$-year term has potential payment dates $0,1,\ldots,n-1$. A temporary annuity-immediate with the same term has potential payment dates $1,2,\ldots,n$. A payment is made only if the annuitant is alive on its payment date. Thus, the immediate annuity includes a possible payment at time $n$, while the due annuity does not.

Definition: Assume annual payments, an effective annual interest rate $i$, and an annuitant aged $x$ at time 0. Let $v=1/(1+i)$, and let ${}_kp_x$ be the probability that the annuitant survives $k$ years from age $x$, with ${}_0p_x=1$. A unit temporary life annuity-due for $n$ years pays at times $0,1,\ldots,n-1$, provided the annuitant is alive at each payment date. A unit temporary life annuity-immediate for $n$ years pays at times $1,2,\ldots,n$, provided the annuitant is alive at each payment date.

The standard present values per unit annual payment are

\ddot{a}_{x:\overline{n}|} =\sum_{k=0}^{n-1}v^k\,{}_kp_x, \qquad a_{x:\overline{n}|} =\sum_{k=1}^{n}v^k\,{}_kp_x.

The term $v^k{}_kp_x$ is the expected present value of a unit payment at time $k$: the payment is discounted by $v^k$, and it is received only if the annuitant survives to that date. For payments of amount $R$, multiply the annuity value by $R$. These formulas assume level payments at the stated annual dates, with no payment made after death. They do not describe a contract that pays a death benefit or makes payments at other times.

Read the Timeline Before the Formula

For either annuity, the term $n$ is the length of the payment period, not an instruction to start counting from the first payment. The payment dates make the distinction clear:

Contract	Potential payment dates	Survival conditions	Number of potential payments
Temporary annuity-due, term $n$	$0,1,\ldots,n-1$	${}_0p_x,{}_1p_x,\ldots,{}_{n-1}p_x$	$n$
Temporary annuity-immediate, term $n$	$1,2,\ldots,n$	${}_1p_x,{}_2p_x,\ldots,{}_np_x$	$n$

“Potential” matters: the schedule contains $n$ dates, but the actual number of payments can be smaller if the annuitant dies before one or more dates. In particular, the time-0 payment of a due annuity is made to an annuitant alive at issue. It is not multiplied by a probability of surviving a year. Conversely, the immediate annuity has no time-0 payment, but it can pay at time $n$ if the annuitant survives the full term.

Worked Example: Counting and Valuing Three-Year Payments

A hypothetical contract pays $R=\$1{,}000$ annually for at most three years while an annuitant aged $x$ is alive. Payments are annual and level. The effective annual interest rate is $i=5\%$. Assume the conditional probability of surviving each successive year is $0.90$, so ${}_kp_x=0.90^k$ for $k=0,1,2,3$. Find the present value of the due and immediate versions.

Method: list dates, then weight each payment by survival and discounting. For the due version, the three dates are $0,1,2$. The survival probabilities are $1,0.90,0.81$, respectively. Here $v=1/1.05=0.9523809524$, $v^2=0.9070294785$, and the value per dollar of annual payment is

\begin{aligned} \ddot{a}_{x:\overline{3}|} &=1+(0.90)(0.9523809524)+(0.81)(0.9070294785)\\ &=1+0.8571428571+0.7346938776\\ &=2.5918367347. \end{aligned}

The contract pays $\$1{,}000$, so its due value is $1{,}000(2.5918367347)=\$2{,}591.84$, rounded to the nearest cent.

For the immediate version, the three dates are $1,2,3$. The survival probabilities are $0.90,0.81,0.729$. Since $v^3=0.8638375985$,

\begin{aligned} a_{x:\overline{3}|} &=(0.90)(0.9523809524)+(0.81)(0.9070294785) +(0.729)(0.8638375985)\\ &=0.8571428571+0.7346938776+0.6297376093\\ &=2.2215743440. \end{aligned}

The immediate value is therefore $1{,}000(2.2215743440)=\$2{,}221.57$, rounded to the nearest cent. As a check, the due schedule has a certain time-0 payment, whereas the immediate schedule’s last payment is at time 3 and requires survival for all three years. The method avoids treating the two schedules as if they shared the same endpoints.

Deriving the Relationship Between the Values

The two schedules have the same payments at dates $1$ through $n-1$. The due annuity has an additional time-0 payment, and the immediate annuity has an additional possible payment at time $n$. This observation gives a direct relationship between their values.

Write the two date ranges.
The due value sums dates $0,\ldots,n-1$; the immediate value sums dates $1,\ldots,n$.

Cancel the common dates.
For each date $1,\ldots,n-1$, both sums contain the same term $v^k{}_kp_x$.

Keep the two endpoint terms.
The due annuity contributes $1$ at time 0; the immediate annuity contributes $v^n{}_np_x$ at time $n$.

Subtracting the immediate value from the due value therefore yields

\ddot{a}_{x:\overline{n}|}-a_{x:\overline{n}|} =1-v^n{}_np_x.

The first endpoint is worth 1 because the annuitant is alive at time 0 and no discounting is needed. The last endpoint is worth $v^n{}_np_x$ because the payment is discounted $n$ years and is made only if the annuitant survives to time $n$. The relation is useful when one temporary value is known and the other is required; it does not require the individual intermediate survival probabilities.

Worked Example: Converting an Immediate Value to a Due Value

A hypothetical three-year temporary life annuity-immediate has a value of $2.1000$ per unit annual payment. The effective annual interest rate is $i=4\%$, and the mortality information supplied is ${}_3p_x=0.50$. Find the corresponding three-year annuity-due value, to four decimal places. The given immediate value is to be treated as exact for this calculation.

Method: apply the endpoint relationship. Here $n=3$, $v=1/1.04$, and $v^3=1/(1.04^3)=1/1.124864$. The discounted value of the time-3 survival-contingent payment is

v^3{}_3p_x =\frac{0.50}{1.124864} =0.44449818\text{ (approximately)}.

Rearrange the relationship to obtain the due value:

\begin{aligned} \ddot{a}_{x:\overline{3}|} &=a_{x:\overline{3}|}+1-v^3{}_3p_x\\ &=2.1000+1-0.44449818\\ &=2.65550182\\ &=2.6555\quad\text{to four decimal places}. \end{aligned}

Only the three-year survival probability is needed because all intermediate terms cancel in the difference. A common mistake is to use $2.1000+1-0.50$, omitting discounting on the time-3 payment. The endpoint must be both discounted and survival-weighted.

Endpoint Checks and Common Traps

A one-year term is a quick way to check whether the dates in a formula have been counted correctly. The due annuity has one payment at time 0, so $\ddot{a}_{x:\overline{1}|}=1$. The immediate annuity has one possible payment at time 1, so $a_{x:\overline{1}|}=v\,{}_1p_x$. If a proposed formula gives either version two payments, or gives the due version a time-1 payment instead of a time-0 payment, the dates have been misread.

Worked Example: Checking a Four-Year Annuity Against Its Dates

A hypothetical contract pays $\$800$ annually for at most four years. The annuitant is aged $x$ at issue, the effective annual interest rate is $i=4\%$, and the survival probabilities at the payment dates are ${}_0p_x=1$, ${}_1p_x=0.95$, ${}_2p_x=0.84$, ${}_3p_x=0.70$, and ${}_4p_x=0.55$. Each payment is made only if the annuitant is alive on its scheduled date. Find the due and immediate values.

Method: use a date-by-date timeline. For the due contract, the dates are $0,1,2,3$, so the time-4 survival probability is not used. With $v=1/1.04$, the needed discount factors are $v=0.9615384615$, $v^2=0.9245562130$, and $v^3=0.8889963587$. Thus

\begin{aligned} \ddot{a}_{x:\overline{4}|} &=1+(0.95)(0.9615384615)+(0.84)(0.9245562130) +(0.70)(0.8889963587)\\ &=1+0.9134615385+0.7766272189+0.6222974511\\ &=3.3123862085. \end{aligned}

The due value is $800(3.3123862085)=\$2{,}649.91$ to the nearest cent.

For the immediate contract, the dates are $1,2,3,4$, including the possible payment at the end of year 4. Since $v^4=0.8548041910$,

\begin{aligned} a_{x:\overline{4}|} &=(0.95)(0.9615384615)+(0.84)(0.9245562130) +(0.70)(0.8889963587)+(0.55)(0.8548041910)\\ &=0.9134615385+0.7766272189+0.6222974511+0.4701423051\\ &=2.7825285136. \end{aligned}

The immediate value is $800(2.7825285136)=\$2{,}226.02$ to the nearest cent. As a check, the difference in values per dollar is $1-(0.55)(0.8548041910)=0.5298576949$, which agrees with $3.3123862085-2.7825285136=0.5298576949$ apart from displayed rounding. The main trap avoided here is including year 4 in the due schedule or omitting it from the immediate schedule.

A useful final audit is to check three things: the first date, the last date, and the survival probability attached to each. Due payments begin at time 0 and end at $n-1$; immediate payments begin at time 1 and end at $n$. The survival term must match the date of its payment, not merely the number of years named in the contract.

Key takeaway: For a term of $n$ annual payments, count $n$ dates in either schedule. A temporary annuity-due uses dates $0$ through $n-1$; a temporary annuity-immediate uses dates $1$ through $n$. Marking those dates first prevents most off-by-one errors.

Check Your Understanding

For each problem, identify the relevant dates before calculating. Values are per unit annual payment unless a payment amount is specified.

A six-year temporary annuity-due pays annually while the annuitant is alive. State its potential payment dates and its number of potential payments. (Answer: Dates $0,1,2,3,4,5$; 6 potential payments.)
A four-year temporary annuity-immediate pays only if the annuitant is alive at the payment date. Is a payment at time 4 included, and what survival probability weights it? (Answer: Yes; it is weighted by ${}_4p_x$.)
A two-year temporary life annuity has $i=5\%$, ${}_1p_x=0.80$, and ${}_2p_x=0.60$. Find both values to four decimal places. (Answer: $\ddot{a}_{x:\overline{2}|}=1.7619$ and $a_{x:\overline{2}|}=1.3061$.)
A three-year temporary life annuity has $i=0\%$, ${}_1p_x=0.90$, ${}_2p_x=0.75$, and ${}_3p_x=0.60$. Find the due and immediate values to four decimal places. (Answer: $\ddot{a}_{x:\overline{3}|}=2.6500$ and $a_{x:\overline{3}|}=2.2500$.)
A four-year temporary annuity-immediate has value $2.5000$, $i=3\%$, and ${}_4p_x=0.64$. Find the corresponding due value to four decimal places. (Answer: $2.9314$.)

Contract	Potential payment dates	Survival conditions	Number of potential payments
Temporary annuity-due, term \(n\)	\(0,1,\ldots,n-1\)	\({}_0p_x,{}_1p_x,\ldots,{}_{n-1}p_x\)	\(n\)
Temporary annuity-immediate, term \(n\)	\(1,2,\ldots,n\)	\({}_1p_x,{}_2p_x,\ldots,{}_np_x\)	\(n\)

What You'll Learn