Temporary and Deferred Annuity Premiums

Why Payment Timing Determines the Premium

A life annuity pays a stated amount only while the annuitant is alive, but a temporary annuity can make at most a specified number of payments. A deferred annuity makes no payments during an initial waiting period. Pricing either product therefore requires more than identifying the benefit amount: each possible payment date must be matched to the probability the annuitant is alive on that date.

This tutorial considers net single premiums: the amount paid at issue that equals the expected present value of the annuity benefits, under the stated interest and mortality assumptions. There are no expenses or profit loadings. A temporary annuity’s payments stop at the end of its stated term even if the annuitant survives; a life annuity’s payments stop on death. These conditions are distinct and both matter to the calculation.

Definition: A temporary life annuity pays a benefit for a specified number of scheduled payment dates, provided the annuitant is alive at each date. An annuity-due pays at the beginning of each payment period; an annuity-immediate pays at the end. A deferred annuity has a stated deferment period during which no benefit is paid. In this tutorial, a deferment of $m$ years means the first due payment is at time $m$, while the first immediate payment is at time $m+1$.

Temporary Annuities: Count the Payment Dates

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. For a life aged $x$, ${}_kp_x$ is the probability of surviving $k$ years from age $x$, with ${}_0p_x=1$. Assume annual payments in arrears or in advance as specified, and that a payment is made only if the annuitant is alive on its payment date.

For a level annual benefit of 1, a temporary annuity-due for $n$ years can pay at times $0,1,\ldots,n-1$. A temporary annuity-immediate for $n$ years can pay at times $1,2,\ldots,n$. The corresponding actuarial present values 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.

If each scheduled payment is $B$, multiply the appropriate factor by $B$ to obtain the net single premium. The $k=0$ term for an annuity-due is 1 because the first payment is made immediately, before any future survival condition applies. There is no such term for an annuity-immediate.

Worked Example: A Three-Year Temporary Annuity

A hypothetical insurer offers a three-year temporary annuity-due to a person aged 60. It pays $1,000 at times 0, 1, and 2, provided the person is alive at the payment time. Assume an effective annual interest rate of $i=1/9$, so $v=0.9$, and assume ${}_1p_{60}=0.90$ and ${}_2p_{60}=0.80$. Find the net single premium. Also find the premium for a three-year temporary annuity-immediate with the same benefit, using ${}_3p_{60}=0.70$.

Method. List the due payment dates first, then multiply each payment by its discount factor and probability of survival. The due payment at time 0 is certain.

For the annuity-due, the present values of the three possible payments are:

\begin{aligned} t=0 &: 1000(1)(1)=1000,\\ t=1 &: 1000(0.9)(0.90)=810,\\ t=2 &: 1000(0.9^2)(0.80)=1000(0.81)(0.80)=648. \end{aligned}

Thus the net single premium is $1000+810+648=\$2,458.00$.

For the annuity-immediate, payment dates are 1, 2, and 3, so the present values are:

\begin{aligned} t=1 &: 1000(0.9)(0.90)=810,\\ t=2 &: 1000(0.81)(0.80)=648,\\ t=3 &: 1000(0.9^3)(0.70)=1000(0.729)(0.70)=510.30. \end{aligned}

The immediate premium is $810+648+510.30=\$1,968.30$. The calculations check against the payment schedules: the due contract includes a certain payment at issue, whereas the immediate contract has a third payment one year later and contingent on three-year survival. A common mistake is to use the due payment dates for both contracts.

Deferred Temporary Annuities

A deferment changes the first possible payment date; it does not change the number of payments after payments begin. For a deferment of $m$ years and $n$ annual payments, a deferred temporary annuity-due has payment dates $m,m+1,\ldots,m+n-1$. A deferred temporary annuity-immediate has dates $m+1,m+2,\ldots,m+n$. For a benefit $B$, their net single premiums at issue are:

\begin{aligned} \text{Deferred due:}\quad &B\sum_{k=m}^{m+n-1}v^k\,{}_kp_x,\\ \text{Deferred immediate:}\quad &B\sum_{k=m+1}^{m+n}v^k\,{}_kp_x. \end{aligned}

These formulas use survival probabilities from the original issue age $x$. Equivalently, condition first on survival to the end of the deferment period, then value the annuity at age $x+m$. The probability of reaching that age is ${}_mp_x$, and the $m$-year discount factor is $v^m$.

\begin{aligned} \text{Deferred due premium} &=Bv^m\,{}_mp_x\,\ddot{a}_{x+m:\overline{n}|},\\ \text{Deferred immediate premium} &=Bv^m\,{}_mp_x\,a_{x+m:\overline{n}|}. \end{aligned}

In these expressions, the temporary annuity factors at age $x+m$ use the mortality probabilities conditional on being alive at that age. For instance, for $j\geq 0$, the conditional probability of surviving a further $j$ years is ${}_{m+j}p_x/{}_mp_x$, assuming ${}_mp_x>0$. This is why the factor ${}_mp_x$ appears once, outside the conditional annuity value.

Timing rule: With an $m$-year deferment, the first due payment is at time $m$, not $m+1$. The first immediate payment is at time $m+1$. Always write the dates explicitly before selecting survival probabilities.

Why the Deferred-Annuity Formula Holds

The term-by-term formula follows from the expected present value of each scheduled payment. For a deferred due annuity, the payment at time $k$, where $m\leq k\leq m+n-1$, has present value $Bv^k$ if it is made. It is made exactly when the annuitant is alive at time $k$, an event with probability ${}_kp_x$. The expected present value of that payment is therefore $Bv^k{}_kp_x$. Adding these expected values gives the deferred-due sum.

To obtain the factor form, express each survival probability as ${}_kp_x={}_mp_x({}_kp_x/{}_mp_x)$, and write $k=m+j$. Then:

\begin{aligned} B\sum_{k=m}^{m+n-1}v^k{}_kp_x &=B\sum_{j=0}^{n-1}v^{m+j}{}_mp_x \frac{{}_{m+j}p_x}{{}_mp_x}\\ &=Bv^m{}_mp_x \sum_{j=0}^{n-1}v^j\frac{{}_{m+j}p_x}{{}_mp_x}\\ &=Bv^m{}_mp_x\,\ddot{a}_{x+m:\overline{n}|}. \end{aligned}

The same argument for an immediate annuity uses $j=1,\ldots,n$ instead of $j=0,\ldots,n-1$, giving $Bv^m{}_mp_xa_{x+m:\overline{n}|}$. These relations require the stated annual timing, constant effective annual interest rate, and survival-contingent payments. A different payment frequency or a payment conditional on a different event would require a suitably adjusted cash-flow model.

Worked Example: A Deferred Temporary Annuity-Due

A hypothetical contract pays $1,000 annually for three years as a temporary annuity-due, deferred for two years, to a person aged 50. Assume $i=1/0.95-1$, so $v=0.95$, and assume ${}_2p_{50}=0.90$, ${}_3p_{50}=0.80$, and ${}_4p_{50}=0.70$. Find the net single premium.

Method. A two-year deferment makes the due payment dates 2, 3, and 4. Use the issue-age survival probabilities at those exact times:

\begin{aligned} t=2 &: 1000(0.95^2)(0.90) =1000(0.9025)(0.90)=812.25,\\ t=3 &: 1000(0.95^3)(0.80) =1000(0.857375)(0.80)=685.90,\\ t=4 &: 1000(0.95^4)(0.70) =1000(0.81450625)(0.70)=570.154375. \end{aligned}

The unrounded total is $812.25+685.90+570.154375=\$2,068.304375$, so the net single premium is $2,068.30, rounded to the nearest cent.

A check using the conditional form gives a deferral factor $v^2{}_2p_{50}=0.9025(0.90)=0.81225$. The conditional temporary annuity-due factor at age 52 is:

1+0.95\left(\frac{0.80}{0.90}\right) +0.95^2\left(\frac{0.70}{0.90}\right) =1+0.844444\ldots+0.701944\ldots =2.546388\ldots

Multiplying gives $1000(0.81225)(2.546388\ldots)=\$2,068.30$ to the nearest cent, agreeing with the direct calculation. The common timing error is to start at time 3; that would incorrectly treat a deferred annuity-due as an annuity-immediate.

Worked Example: Deferred Immediate Payments

A hypothetical deferred temporary annuity-immediate provides two annual payments of $1,200 after a three-year deferment. The annuitant is aged 45 at issue. Assume $i=4\%$, $v=1/1.04$, ${}_4p_{45}=0.90$, and ${}_5p_{45}=0.85$. Find the net single premium.

Method. The first immediate payment is one year after the deferment ends, at time 4. The second is at time 5. Thus:

\begin{aligned} v^4&=0.854804\ldots, \qquad v^5=0.821927\ldots,\\ \text{Premium} &=1200\left(v^4{}_4p_{45}+v^5{}_5p_{45}\right)\\ &=1200\left((0.854804\ldots)(0.90) +(0.821927\ldots)(0.85)\right)\\ &=1200(0.769323\ldots+0.698638\ldots)\\ &=1200(1.467961\ldots)=\$1,761.55. \end{aligned}

The result is rounded to the nearest cent. The schedule has two payments, and both occur after the deferment. Using survival to time 3 for the first payment would be wrong: survival must be measured to the payment date, time 4.

Practical Checks and Common Traps

A reliable calculation starts with the contract’s timeline, not with a memorized annuity symbol. After writing the dates, attach to each date its benefit, discount factor, and survival probability. This method is especially useful when a problem combines a deferment with a finite payment term or provides survival probabilities directly rather than a complete life table.

Confusing term and deferment. The deferment determines when payments begin; the term determines how many scheduled payments are possible. An $m$-year deferment followed by $n$ due payments has $n$ dates, beginning at $m$.
Using the wrong survival time. A payment at time $k$ is contingent on survival to time $k$, so its probability is ${}_kp_x$, not the probability of surviving only to the start of the payment period.
Discounting from the wrong date. Every payment is discounted back to issue. A time-$k$ payment has factor $v^k$, even when a conditional annuity factor is used after deferment.
Double-counting survival to the deferment date. In the conditional formula, ${}_mp_x$ is already included outside the annuity factor. The annuity factor then uses conditional survival from age $x+m$.
Including an impossible payment. A temporary annuity has exactly $n$ scheduled dates, whether or not the annuitant survives to all of them. Do not extend its term merely because the annuitant is alive.

A useful final check is to compare the number and dates of the terms in the sum with the contract wording. The due and immediate premiums will generally differ because their payment dates differ; the difference is not just a change in notation. These timing and survival calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the payment dates before calculating. Round dollar answers to the nearest cent.

A two-year temporary annuity-due pays $500 at times 0 and 1 to a person aged 40. Let $i=5\%$ and ${}_1p_{40}=0.90$. Find its net single premium. (Answer: $928.57)
A three-year temporary annuity-immediate pays $800 annually. Let $i=4\%$, ${}_1p_{55}=0.95$, ${}_2p_{55}=0.88$, and ${}_3p_{55}=0.80$. Find its net single premium. (Answer: $1,950.61)
A two-payment temporary annuity-due is deferred for two years and pays $1,000 at each payment date. Let $i=10\%$, ${}_2p_{62}=0.92$, and ${}_3p_{62}=0.86$. Find the premium. (Answer: $1,406.46)
A two-payment temporary annuity-immediate is deferred for one year and pays $1,500 annually. Let $i=5\%$, ${}_2p_{48}=0.97$, and ${}_3p_{48}=0.93$. Find the net single premium. (Answer: $2,524.78)

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