Deferred Life Annuity Factorization

Why Factor a Deferred Life Annuity?

A deferred life annuity makes no payments during an initial period, then pays while the annuitant is alive. Its value at issue depends on two distinct questions: how likely is the annuitant to survive to the payment period, and what is the annuity worth at that future age? Separating those questions gives a useful valuation method and helps prevent errors in discounting or payment timing.

This tutorial considers discrete annual payments and a fixed effective annual interest rate. The factorization applies to either a whole-life annuity-due or a whole-life annuity-immediate. The distinction is the timing of the first payment after the deferment period.

Definition: An $m$-year deferred whole-life annuity-due on $(x)$ pays 1 at times $m,m+1,m+2,\ldots$, provided the annuitant is alive at each payment time. Its value at time 0 is denoted ${}_{m|}\ddot{a}_x$. An $m$-year deferred whole-life annuity-immediate pays 1 at times $m+1,m+2,\ldots$, provided the annuitant is alive at each payment time; its value is denoted ${}_{m|}a_x$.

Here $m$ is a nonnegative integer number of years. The annuity-due makes its first deferred payment at time $m$; that payment is made only if the annuitant survives to time $m$. The annuity-immediate has its first payment one year later, at time $m+1$. For $m=0$, these definitions reduce to the usual immediate-at-issue life annuity-due and life annuity-immediate.

The Deferred Annuity Factorization

Assume a fixed effective annual interest rate $i$, with $v=(1+i)^{-1}$. Use ${}_mp_x$ for the probability that a life aged $x$ survives $m$ years. The annuity factors $\ddot{a}_{x+m}$ and $a_{x+m}$ are valued at age $x+m$, using the same interest and mortality basis as the deferred annuity.

Theorem: For an $m$-year deferred whole-life annuity-due and an $m$-year deferred whole-life annuity-immediate, $$ {}_{m|}\ddot{a}_x = v^m\,{}_mp_x\,\ddot{a}_{x+m}, \qquad {}_{m|}a_x = v^m\,{}_mp_x\,a_{x+m}. $$ The formulas apply to discrete annual payments and deterministic interest. The annuity-due factor at age $x+m$ includes a payment immediately at time $m$, while the annuity-immediate factor at that age starts one year later.

The three factors have separate meanings. The term ${}_mp_x$ accounts for the chance the annuitant reaches the deferment date. The term $v^m$ discounts a value at time $m$ back to time 0. The age-$x+m$ annuity factor values the future payments conditional on being alive at time $m$.

Why the Factorization Holds

The derivation follows by writing the expected present value as a sum of discounted payments. For a deferred annuity-due, the payment at time $m+j$, for $j=0,1,2,\ldots$, is made if the annuitant survives $m+j$ years. Its expected present value is $v^{m+j}{}_{m+j}p_x$. Conditional on survival for the first $m$ years, the probability of surviving a further $j$ years is ${}_jp_{x+m}$. Thus ${}_{m+j}p_x={}_mp_x\,{}_jp_{x+m}$, and the sum factors as follows.

\begin{aligned} {}_{m|}\ddot{a}_x &=\sum_{j=0}^{\infty}v^{m+j}{}_{m+j}p_x\\ &=v^m{}_mp_x\sum_{j=0}^{\infty}v^j{}_jp_{x+m}\\ &=v^m{}_mp_x\,\ddot{a}_{x+m}. \end{aligned}

For an annuity-immediate, the first possible payment is at time $m+1$, so the corresponding sum starts at $j=1$, not $j=0$. The remaining sum is the annuity-immediate factor at age $x+m$, which gives ${}_{m|}a_x=v^m{}_mp_xa_{x+m}$. This change in the sum’s starting index is exactly the payment-timing distinction between the two formulas.

Worked Applications

A reliable calculation identifies the payment timing first, then uses the survival probability to the deferment date and discounts by exactly $m$ years. If the given information is one-year mortality rather than ${}_mp_x$, multiply the relevant one-year survival probabilities to obtain ${}_mp_x$.

Worked Example: Deferred Annuity-Due With Survival Given

A hypothetical contract provides an annual life annuity-due after a three-year deferment. The effective annual interest rate is $i=4\%$, the probability that a life aged $x$ survives three years is ${}_3p_x=0.90$, and the annuity-due factor at age $x+3$ is $\ddot{a}_{x+3}=7.5$. Find the value at issue per unit annual payment, to four decimal places.

Method. Because the first payment is at time 3 if the annuitant is then alive, this is a deferred annuity-due. Apply ${}_{3|}\ddot{a}_x=v^3{}_3p_x\ddot{a}_{x+3}$.

First calculate the discount factor:

v^3=\frac{1}{(1.04)^3} =\frac{1}{1.124864} \approx 0.88899636.

Next combine survival and the annuity value at age $x+3$:

{}_3p_x\ddot{a}_{x+3} =(0.90)(7.5)=6.75.

Therefore,

{}_{3|}\ddot{a}_x =\frac{(0.90)(7.5)}{(1.04)^3} =\frac{6.75}{1.124864} \approx 6.000725 \approx 6.0007.

The answer is 6.0007 per unit annual payment. As a check, the intermediate value 6.75 is the expected annuity value at time 3 before discounting, so discounting it by three years produces a smaller value at issue. The common mistake avoided is using an inaccurate reciprocal for $(1.04)^3$, which can make the displayed multiplication disagree with the final answer.

Worked Example: Deferred Annuity-Immediate

A hypothetical income contract pays annually in arrears for as long as the annuitant is alive. Payments start one year after a two-year deferment. At issue, $i=3\%$, ${}_2p_x=0.96$, and $a_{x+2}=5.4$. Find the value per unit annual payment, to four decimal places.

Method. Payments start at time 3, so the contract is a two-year deferred annuity-immediate. The factor at age $x+2$ is $a_{x+2}$, not $\ddot{a}_{x+2}$.

The two-year discount factor is

v^2=\frac{1}{(1.03)^2} =\frac{1}{1.0609} \approx 0.9425959.

The value at time 2, conditional on survival to that time, is $a_{x+2}=5.4$. Including the probability of reaching time 2 gives

{}_2p_xa_{x+2}=(0.96)(5.4)=5.184.

Discounting this amount two years to issue:

{}_{2|}a_x =\frac{(0.96)(5.4)}{(1.03)^2} =\frac{5.184}{1.0609} \approx 4.886417 \approx 4.8864.

The value is 4.8864 per unit annual payment. The answer is less than 5.184 because the latter is the survival-adjusted value at time 2, not at issue. The common mistake avoided is discounting for three years just because the first payment occurs at time 3: the attained-age annuity factor already values that payment stream at time 2, so the factorization discounts for the two-year deferment.

Worked Example: Building the Deferment Survival Probability

A hypothetical three-year deferred annuity-due has annual survival probabilities $p_x=0.98$, $p_{x+1}=0.97$, and $p_{x+2}=0.96$. The effective annual interest rate is $i=5\%$, and $\ddot{a}_{x+3}=9.2$. Find the value per unit annual payment at issue, to four decimal places.

Method. Multiply the three consecutive one-year survival probabilities to find ${}_3p_x$. Then use the deferred annuity-due factorization with three years of discounting.

The probability of surviving the three-year deferment is

{}_3p_x =p_xp_{x+1}p_{x+2} =(0.98)(0.97)(0.96) =(0.9506)(0.96) =0.912576.

The discount denominator is

$$ (1.05)^3=1.157625. $$

At time 3, after survival to that date, the annuity-due factor is 9.2. The survival-adjusted value at time 3 is

$$ (0.912576)(9.2)=8.3956992. $$

Thus the issue value is

{}_{3|}\ddot{a}_x =\frac{(0.912576)(9.2)}{1.157625} \approx 7.252521 \approx 7.2525.

The value is 7.2525 per unit annual payment. The survival probability is below 1 and the discount factor is below 1, so the issue value is appropriately below the age-$x+3$ annuity factor of 9.2. The common mistake avoided is using only the final one-year survival probability instead of multiplying survival across all three years.

Timing and Indexing Checks

The factorization is compact, but its terms must match the contract’s timing. In particular, survival is measured from age $x$ to age $x+m$, and the future annuity factor is evaluated at age $x+m$, not at age $x$. The following comparisons are useful when checking a setup.

Contract	First payment	Factor at age $x+m$	Value at issue
Deferred annuity-due	Time $m$, if alive	$\ddot{a}_{x+m}$	$v^m{}_mp_x\ddot{a}_{x+m}$
Deferred annuity-immediate	Time $m+1$, if alive	$a_{x+m}$	$v^m{}_mp_xa_{x+m}$

A frequent timing error is to use $v^{m+1}$ for a deferred annuity-immediate. That is generally wrong when the future factor is $a_{x+m}$, because $a_{x+m}$ already discounts its first payment from time $m+1$ to time $m$. The additional discount from time $m$ to issue is $v^m$, not $v^{m+1}$. Equivalently, one could start from the individual payment at time $m+1$, discount it by $v^{m+1}$, and include its survival probability ${}_{m+1}p_x$; that is a different direct-sum calculation, not the stated factorization.

For a quick check before calculating, write down the first payment date and the date at which the age-indexed annuity factor is valued. For a deferred annuity-due, both are at time $m$. For a deferred annuity-immediate, the factor is valued at time $m$, even though the first payment is at time $m+1$. This distinction helps identify both the right annuity factor and the right discount exponent.

Key takeaway: Value the annuity at the deferment date, multiply by the probability of surviving to that date, and discount back exactly $m$ years. Use an annuity-due factor when payment is possible at time $m$, and an annuity-immediate factor when the first payment is at time $m+1$.

Check Your Understanding

For each problem, identify the appropriate factorization and keep the payment timing explicit. Round numerical values to four decimal places unless another form is requested.

A hypothetical two-year deferred annuity-due has $i=5\%$, ${}_2p_x=0.95$, and $\ddot{a}_{x+2}=6$. Find its issue value per unit annual payment. (Answer: 5.1701)
A hypothetical three-year deferred annuity-immediate has $i=4\%$, ${}_3p_x=0.90$, and $a_{x+3}=4.8$. Find its issue value per unit annual payment. (Answer: 3.8405)
For a hypothetical three-year deferred annuity-due, the one-year survival probabilities are $p_x=0.99$, $p_{x+1}=0.98$, and $p_{x+2}=0.97$. If $i=4\%$ and $\ddot{a}_{x+3}=8$, find the issue value per unit annual payment. (Answer: 6.6930)
A hypothetical two-year deferred annuity has $i=3\%$ and ${}_2p_x=0.92$. At age $x+2$, the annuity-due factor is 7.1 and the annuity-immediate factor is 6.1. Find the issue value for each payment timing. (Answer: annuity-due 6.1570; annuity-immediate 5.2898)
A hypothetical four-year deferred annuity-due has issue value 5.4, $i=4\%$, and $\ddot{a}_{x+4}=7.2$. Find the probability of surviving the four-year deferment. (Answer: ${}_4p_x=0.8774$)

Contract	First payment	Factor at age \(x+m\)	Value at issue
Deferred annuity-due	Time \(m\), if alive	\(\ddot{a}_{x+m}\)	\(v^m{}_mp_x\ddot{a}_{x+m}\)
Deferred annuity-immediate	Time \(m+1\), if alive	\(a_{x+m}\)	\(v^m{}_mp_xa_{x+m}\)

What You'll Learn