Deferred Annuity-Immediate Valuations

Why the First Payment Date Matters

A deferred life annuity-immediate pays only after a specified waiting period, and its payments are made at the ends of payment years. The key valuation decision is therefore the first payment date: after an $n$-year deferment, the first payment is at time $n+1$, not at time $n$. Getting this date right determines both the discount factor and the probability that the life is still alive when payment is due.

For each possible payment, the expected present value is the payment amount multiplied by the probability it is made and by the discount factor back to time 0. This tutorial develops that calculation for whole-life and temporary deferred annuities, with special attention to counting eligible payment dates.

Definition: Consider a life aged $x$ at time 0 and an integer-year deferment of $n$ years, where $n\geq 0$. A deferred annuity-immediate pays at the ends of years $n+1,n+2,\ldots$, provided the life is then alive. If each payment is 1, its expected present value is $$ \sum_{k=n+1}^{\infty} v^k\,{}_kp_x, $$ where $v=1/(1+i)$, $i$ is the effective annual interest rate, and ${}_kp_x$ is the probability that the life aged $x$ survives at least $k$ years. A temporary version with $m$ possible payments has value $\sum_{k=n+1}^{n+m}v^k\,{}_kp_x$.

The payment at time $k$ is contingent on survival to time $k$. Thus the relevant probability is ${}_kp_x$, not merely the probability of surviving the deferment. The whole-life sum includes every eligible future year; a temporary annuity includes only the specified number of eligible payments. If the payment amount is $B$ rather than 1, multiply the unit-payment value by $B$.

Building the Valuation from the Cash Flows

The annuity-immediate convention places payments at the ends of years. An $n$-year deferment means there are no payments during the first $n$ years. The first eligible year-end is $n+1$. For a whole-life annuity, the later payment dates continue annually for as long as the life survives; for a temporary annuity, they stop after the stated number of possible payments.

Mark the payment dates.
For an $n$-year deferment, start at time $n+1$. For $m$ temporary payments, end at time $n+m$.

Attach survival probabilities.
A payment at time $k$ is made only if the life survives $k$ years, so use ${}_kp_x$.

Discount each payment.
With effective annual interest rate $i$, a payment at time $k$ has discount factor $v^k=(1+i)^{-k}$.

Add the expected present values.
Sum $Bv^k\,{}_kp_x$ over the eligible payment dates.

Why the Survival-Weighted Sum Works

Let $I_k$ equal 1 if the life is alive at time $k$, and 0 otherwise. The present value of a unit-payment deferred annuity with $m$ possible payments is the random variable $$ Z=\sum_{k=n+1}^{n+m}v^k I_k. $$ This expression includes a discounted payment at time $k$ precisely when $I_k=1$. Taking expectations and using the fact that the expected value of an indicator is the probability of its event gives

E[Z] =\sum_{k=n+1}^{n+m}v^k E[I_k] =\sum_{k=n+1}^{n+m}v^k\,{}_kp_x.

The sum is valid because expectation distributes over a finite sum; the survival events at different dates do not need to be independent. For a whole-life annuity, the corresponding infinite sum applies when the expected present value is finite. This derivation also explains why each survival probability is tied to its own payment date: the payment at time $k$ is made only if the life is alive at time $k$.

Worked Valuations

Worked Example: Four Payments After a Three-Year Deferment

A hypothetical annuity pays $1,000 at the ends of four consecutive years, starting after a three-year deferment. The life aged $x$ has a 0.90 probability of surviving each year, independently of previous years in the sense that the conditional one-year survival probability is 0.90 at every year. The effective annual interest rate is 5%. Find the expected present value.

Method. This is a temporary deferred annuity-immediate. The three-year deferment leaves possible payment dates at times 4, 5, 6, and 7. Under the stated constant conditional survival model, ${}_kp_x=0.90^k$. Also, $v=1/1.05$, so $0.90v=6/7$.

The survival probabilities at the four payment dates are $$ {}_4p_x=0.9^4=0.6561,\quad {}_5p_x=0.9^5=0.59049,\quad {}_6p_x=0.9^6=0.531441,\quad {}_7p_x=0.9^7=0.4782969. $$ The corresponding survival-weighted discount factors can be calculated as $(0.90v)^k=(6/7)^k$:

\begin{aligned} (6/7)^4&=0.5397751,\\ (6/7)^5&=0.4626644,\\ (6/7)^6&=0.3965695,\\ (6/7)^7&=0.3399167. \end{aligned}

Adding the four factors gives $0.5397751+0.4626644+0.3965695+0.3399167=1.7389257$. Therefore, the expected present value is $1{,}000(1.7389257)=\$1{,}738.93$, rounded to the nearest cent.

Check and common mistake avoided. There are exactly four payments, and the last is at time 7. Starting at time 3 would incorrectly include a payment at the end of the deferment, which is not an annuity-immediate payment under this contract wording.

Worked Example: Whole-Life Payments from Cumulative Survival Probabilities

A hypothetical whole-life annuity for a life aged $x$ has an annual payment of $800 and a two-year deferment. The effective annual interest rate is 4%. The supplied cumulative survival probabilities are ${}_1p_x=0.95$, ${}_2p_x=0.88$, ${}_3p_x=0.72$, ${}_4p_x=0.50$, and ${}_5p_x=0.25$; survival beyond year 5 is impossible. Find the expected present value.

Method. A two-year deferment makes the first payment due at time 3. Payments at times 3, 4, and 5 are possible; there is no payment at time 6 because the life cannot survive to then. Here $v=1/1.04$, and the value is

800\left(v^3\,{}_3p_x+v^4\,{}_4p_x+v^5\,{}_5p_x\right).

Calculate the discount factors: $$ v^3=(1.04)^{-3}=0.8889964,\quad v^4=(1.04)^{-4}=0.8548042,\quad v^5=(1.04)^{-5}=0.8219271. $$

Multiply each by the survival probability for its payment date:

\begin{aligned} v^3\,{}_3p_x&=0.8889964(0.72)=0.6400774,\\ v^4\,{}_4p_x&=0.8548042(0.50)=0.4274021,\\ v^5\,{}_5p_x&=0.8219271(0.25)=0.2054818. \end{aligned}

The sum is $0.6400774+0.4274021+0.2054818=1.2729613$. Multiplying by $800 gives an expected present value of $1,018.37, rounded to the nearest cent.

Check and common mistake avoided. The survival probability at time 3 is ${}_3p_x$, not ${}_2p_x$: the first payment is due one year after the two-year deferment ends. Also, the whole-life label does not create payments after survival becomes impossible.

Worked Example: Three Payments After a One-Year Deferment

A hypothetical contract pays $1,500 at the ends of three consecutive years, beginning after a one-year deferment. The effective annual interest rate is 6%. For a life aged $x$, the cumulative survival probabilities at the relevant dates are ${}_2p_x=0.90$, ${}_3p_x=0.81$, and ${}_4p_x=0.72$. Find the expected present value.

Method. The first payment is at time 2, followed by payments at times 3 and 4. With $v=1/1.06$, calculate each survival-weighted discount factor:

\begin{aligned} v^2\,{}_2p_x&=(1.06)^{-2}(0.90)=0.8899964(0.90)=0.8009968,\\ v^3\,{}_3p_x&=(1.06)^{-3}(0.81)=0.8396193(0.81)=0.6800916,\\ v^4\,{}_4p_x&=(1.06)^{-4}(0.72)=0.7920937(0.72)=0.5703075. \end{aligned}

Their sum is $0.8009968+0.6800916+0.5703075=2.0513959$. Multiplying by the payment amount gives $1{,}500(2.0513959)=\$3{,}077.09$, rounded to the nearest cent.

Check and common mistake avoided. There are three possible payments, not four: the dates are 2, 3, and 4. Using the time-1 survival probability for a payment at time 1 would mistakenly value an annuity-due rather than the stated deferred annuity-immediate.

Common Timing and Valuation Traps

A useful way to audit a deferred-annuity calculation is to write the eligible payment times before inserting any probabilities or discount factors. The deferment determines when the annuity begins; the immediate or due convention determines whether a payment occurs at that boundary. For an annuity-immediate, the first payment is one year after the deferment period.

Key takeaway: For an integer-year deferment of $n$ years, a deferred annuity-immediate has no payment at time $n$. Its first possible payment is at time $n+1$, and the term for that payment is $Bv^{n+1}{}_{n+1}p_x$.

A common incorrect calculation starts a deferred annuity-immediate at time $n$, using the term $Bv^n\,{}_np_x$. That term instead places a payment at the end of the deferment year. The correction is to begin the sum at $n+1$, then use the survival probability and discount factor for each actual payment date. Another error is to count $m$ payments starting at $n+1$ but ending at $n+m+1$; that includes $m+1$ dates. The correct final date for $m$ payments is $n+m$.

These valuations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. In exam calculations, make the timeline explicit whenever the problem uses phrases such as “deferred for $n$ years,” “first payment at the end of year,” or “$m$ annual payments.” Such phrases specify timing, and timing determines the appropriate survival and discount terms.

Check Your Understanding

For each question, identify the payment dates before calculating. Unless stated otherwise, payments are made only if the life is alive on the payment date.

A hypothetical annuity-immediate pays $500 for two years after a two-year deferment. The interest rate is 5%, and ${}_3p_x=0.80$, ${}_4p_x=0.70$. Find its expected present value. (Answer: $633.48)
A life receives two possible annual payments of $200 after a three-year deferment. The interest rate is 0%, and ${}_4p_x=0.90$, ${}_5p_x=0.75$. Find the expected present value. (Answer: $330.00)
A hypothetical annuity-immediate pays $1,000 at times 2, 3, and 4, and no later payments are possible. The effective annual interest rate is 10%, with ${}_2p_x=0.80$, ${}_3p_x=0.50$, and ${}_4p_x=0.20$. Find the expected present value. (Answer: $1,173.42)
A three-year temporary annuity-immediate is deferred for two years. State its possible payment dates and the survival probability attached to its first payment. (Answer: Times 3, 4, and 5; the first payment uses ${}_3p_x$.)
A hypothetical annuity pays $100 for three years after a two-year deferment. The interest rate is 0%, and the life has a conditional annual survival probability of 0.80 in every year. Find the expected present value. (Answer: $124.93)

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