Discounting Deferred Payments Conditional on Survival

Why Deferred Annuities Have Two Factors

A deferred life annuity does not begin paying immediately. Its value at issue must account for two distinct effects: the annuitant may not survive to the date payments begin, and money payable at that future date is worth less than money payable now. Keeping these effects separate makes the valuation easier to interpret and helps prevent errors in timing.

This tutorial considers annual payments and a fixed effective annual interest rate. The deferral period is an integer number of years. Payments are made only when the annuitant is alive at the payment date. There is no payment guarantee if death occurs before or during the payment period.

Definition: For a life aged $x$, ${}_mp_x$ is the probability of surviving $m$ years, and ${}_mq_x=1-{}_mp_x$. With effective annual interest rate $i$, the discount factor for $m$ years is $v^m=(1+i)^{-m}$. A deferred life annuity’s deferral factor is $v^m{}_mp_x$: the probability-weighted discount factor to the start of payments.

The factor $v^m{}_mp_x$ is not, by itself, the value of the annuity. It moves the valuation from issue to the deferral date while accounting for survival to that date. The value of the payments after that date must still be included, conditional on being alive then.

Deriving the Deferral Factor

Let the deferral period be $m$ years, and consider a payment of 1 at time $m$, made only if the life is alive at that time. The payment’s present value is $v^m$ if the life survives and 0 otherwise. Its expected present value is therefore the discount factor multiplied by the probability that the payment is made.

E[\text{present value of a payment at time }m] = v^m P(\text{alive at time }m) = v^m{}_mp_x.

For a deferred annuity, each payment has a later payment date and its own survival probability. The deferral factor emerges by factoring the probability of surviving to time $m$ out of every later survival probability.

Derivation: For integer $j\geq 0$, survival probabilities satisfy ${}_{m+j}p_x={}_mp_x\,{}_jp_{x+m}$. Thus a payment at time $m+j$, discounted to issue, has expected present value $v^{m+j}{}_{m+j}p_x=v^m{}_mp_x\,v^j{}_jp_{x+m}$. Summing over payment dates leaves the common factor $v^m{}_mp_x$, multiplied by the annuity value conditional on being alive at age $x+m$.

This factorization assumes the interest rate is deterministic, the payments are annual, and the mortality probabilities refer to the stated life. It does not assume that the life will survive the deferral period. If the contract pays regardless of survival during that period, its cash flows have a different structure and should not be valued by treating every payment as contingent on survival.

Deferred Annuity-Due and Annuity-Immediate Payments

Payment timing determines which conditional annuity value belongs after the deferral factor. For a deferred annuity-due, the first payment is made at time $m$, provided the annuitant is alive then. For a deferred annuity-immediate, the first payment is made one year later, at time $m+1$, provided the annuitant is alive at that later date.

\begin{aligned} \text{Deferred annuity-due:}\quad &v^m{}_mp_x\sum_{j=0}^{n-1}v^j{}_jp_{x+m},\\ \text{Deferred annuity-immediate:}\quad &v^m{}_mp_x\sum_{j=1}^{n}v^j{}_jp_{x+m}. \end{aligned}

Here each annuity makes $n$ annual payments of 1, and the probabilities inside the sums are conditional on survival to the deferral date. The due annuity’s first conditional payment occurs immediately at that date, so its first term is $1$. The immediate annuity’s first conditional payment occurs one year later, so its first term is $v\,{}_1p_{x+m}$. For a whole-life annuity, the corresponding conditional values are $\ddot a_{x+m}$ for due payments and $a_{x+m}$ for immediate payments.

Fix the timing.
Identify the deferral period $m$ and whether the first payment is at time $m$ or $m+1$.

Find survival to the start date.
Use ${}_mp_x$, or calculate it from appropriate survival-function or life-table values.

Discount to the start date.
Calculate $v^m=(1+i)^{-m}$ using the annual interest rate and the number of years.

Value the conditional payments.
Use the annuity value at age $x+m$, with the payment timing and term specified.

Multiply and check.
Multiply the deferral factor by the conditional annuity value and payment amount. Check that the first payment date matches the contract description.

Worked Valuations

Worked Example: Using Survival-Function Values

A hypothetical life aged $x$ has $S_0(x)=0.72$ and $S_0(x+3)=0.54$. A three-year deferred annuity-immediate pays \$1,500 annually for three payments, with conditional value $2.3$ at age $x+3$. The effective annual interest rate is $i=4\%$. Find the expected present value at issue, to the nearest cent.

Method. First calculate the probability of survival from age $x$ to age $x+3$. The survival function values are unconditional from the same origin, so the conditional probability is their ratio:

{}_3p_x=\frac{S_0(x+3)}{S_0(x)} =\frac{0.54}{0.72}=0.75.

The discount factor to the deferral date is:

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

The conditional annuity value is $2.3$, so multiply it by the payment amount and by the deferral factor:

\begin{aligned} \text{Expected present value} &=1500(0.75)(2.3)\frac{1}{1.124864}\\ &=\frac{2587.5}{1.124864}\\ &=\$2{,}300.28. \end{aligned}

The result is rounded to the nearest cent. The method avoids treating $S_0(x+3)=0.54$ as a conditional survival probability: it must be divided by $S_0(x)=0.72$. As a check, the deferral factor is less than 1, so the present value is below the conditional value $1500(2.3)=\$3{,}450$.

Worked Example: A Deferred Annuity-Due with Explicit Conditional Survival

A hypothetical annuity pays \$800 annually for three years, with its first payment at the two-year deferral date. The probability of surviving to that date is ${}_2p_x=0.90$. Conditional on reaching the deferral date, the probabilities of surviving one and two additional years are ${}_1p_{x+2}=0.80$ and ${}_2p_{x+2}=0.60$. The effective annual interest rate is $i=5\%$. Find the expected present value at issue.

Method. Since the first payment is at time 2, this is an annuity-due conditional on survival to age $x+2$. Its three conditional payment contributions per dollar are $1$, $0.80v$, and $0.60v^2$. First calculate $v=1/1.05=20/21$, and then the conditional value:

\begin{aligned} 1+0.80v+0.60v^2 &=1+0.80\left(\frac{20}{21}\right) +0.60\left(\frac{20}{21}\right)^2\\ &=1+\frac{16}{21}+\frac{240}{441}\\ &=\frac{113}{49}=2.306122\ldots. \end{aligned}

The deferral factor is:

v^2{}_2p_x =\left(\frac{20}{21}\right)^2(0.90) =\frac{40}{49}.

Therefore the expected present value is:

800\left(\frac{40}{49}\right)\left(\frac{113}{49}\right) =\frac{3{,}616{,}000}{2{,}401} =\$1{,}506.04.

The result is rounded to the nearest cent. The first conditional payment contributes 1, not $v$, because it is paid at the deferral date. This calculation also checks the payment count: the three contributions correspond to times 2, 3, and 4.

Worked Example: A Deferred Annuity-Immediate from Survival Probabilities

A hypothetical annuity pays \$2,000 at each of times 3 and 4, but only if the annuitant is alive at the payment date. The probability of surviving the two-year deferral period is ${}_2p_x=0.75$. Conditional on surviving to time 2, the probability of surviving one more year is $0.90$, and the probability of surviving two more years is $0.80$. The effective annual interest rate is $i=25\%$. Find the expected present value at issue.

Method. This is a two-payment annuity-immediate deferred two years: its payments are at times 3 and 4. Conditional on survival to time 2, its value per dollar is the sum of the discounted, conditional survival probabilities. With $v=1/1.25=0.8$:

\begin{aligned} {}_1p_{x+2}v+{}_2p_{x+2}v^2 &=(0.90)(0.8)+(0.80)(0.8)^2\\ &=0.72+0.512\\ &=1.232. \end{aligned}

The deferral factor is $v^2{}_2p_x=(0.8)^2(0.75)=0.48$. Thus:

2000(0.48)(1.232) =2000(0.59136) =\$1{,}182.72.

The result is already exact to the cent. As a check, the unconditional probabilities of receiving the payments are $0.75(0.90)=0.675$ at time 3 and $0.75(0.80)=0.60$ at time 4. Direct valuation gives $2000(0.675)(0.8)^3+2000(0.60)(0.8)^4=\$1{,}182.72$, agreeing with the factorized calculation.

Common Mistakes and Why the Factor Matters

The deferral factor is useful because it makes clear which quantities are measured from issue and which are conditional on survival to the start of payments. It also reveals the two most common errors.

Omitting survival to the deferral date. In the first example, using $1500(2.3)/(1.04)^3=\$3{,}067.04$ incorrectly assumes every life reaches age $x+3$. The correction is to multiply by ${}_3p_x=0.75$, giving $1500(2.3)(0.75)/(1.04)^3=\$2{,}300.28$.
Discounting the deferral period twice. If the conditional annuity value is already measured at age $x+m$, apply $v^m{}_mp_x$ once to move it to issue. Do not also discount every conditional payment all the way from issue inside that conditional value.
Starting an immediate annuity one payment too early. A deferred annuity-immediate with an $m$-year deferral first pays at time $m+1$, not at time $m$. The conditional payment terms begin with $v\,{}_1p_{x+m}$.
Using the wrong survival probability. When survival-function values are supplied, ${}_mp_x=S_0(x+m)/S_0(x)$, not simply $S_0(x+m)$, unless $S_0(x)=1$.

Key takeaway: For annual payments contingent on survival, separate the valuation into $v^m{}_mp_x$ to reach the deferral date and the annuity value conditional on being alive at age $x+m$. Then check that the first conditional payment occurs at the correct time. This method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the deferral factor before combining it with the conditional payment value. Round money to the nearest cent.

A hypothetical two-year deferred annuity-due pays \$1,000 for three annual payments. The probability of surviving to the deferral date is $0.80$, the conditional annuity-due value then is $2.5$, and $i=5\%$. Find its expected present value. (Answer: \$1,814.06)
A hypothetical one-year deferred annuity-immediate pays \$500 for two annual payments. Its conditional value at age $x+1$ is $1.4$, ${}_1p_x=0.90$, and $i=10\%$. Find its expected present value. (Answer: \$572.73)
For a life aged $x$, $S_0(x)=0.80$ and $S_0(x+2)=0.60$. A two-year deferred annuity-immediate pays \$1,200 annually, and its conditional value at age $x+2$ is $3.1$. If $i=4\%$, find its expected present value. (Answer: \$2,579.51)
A hypothetical one-year deferred annuity-due pays \$1,500 for three payments. The probability of surviving to time 1 is $0.80$. Conditional on survival to time 1, the probabilities of surviving one and two additional years are $0.90$ and $0.75$. With $i=25\%$, find its expected present value. (Answer: \$2,112.00)
A hypothetical two-year deferred annuity-immediate pays \$1,000 at times 3 and 4. The probability of surviving to time 2 is $0.80$; conditional survival probabilities for one and two more years are $0.75$ and $0.50$. If $i=25\%$, find its expected present value. (Answer: \$471.04)

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