Valuing at the Deferral Date Before Discounting

Why Value at the Deferral Date First?

A deferred annuity has two distinct timing questions: what is its value when the payment stream begins, and what is that value at an earlier date? Separating these questions makes the timeline easier to read and reduces the chance of shifting a payment by one period. First value the payments as an ordinary annuity at the appropriate date; then discount that value across the deferral period.

The key is to define the date carefully. For an annuity-immediate, the value one period before the first payment is an annuity value. For an annuity-due, the value at the first payment date is an annuity-due value. These are different focal dates, even when the payment amounts and number of payments are the same.

Definition: Assume a constant effective annual interest rate $i>-1$, with annual periods and level payments. Let $v=1/(1+i)$. A deferred annuity-immediate with $n$ payments of $R$, first paid at time $m+1$, has its payment stream valued at time $m$ as $R a_{\overline{n}|}$. Its value at time 0 is $v^m R a_{\overline{n}|}$. A deferred annuity-due with $n$ payments, first paid at time $m$, is valued at time $m$ as $R\ddot{a}_{\overline{n}|}$, and at time 0 as $v^m R\ddot{a}_{\overline{n}|}$.

Here, $m$ is the number of complete annual periods from time 0 to the date at which the payment stream is valued. For the immediate annuity, the first payment is one period after that date. For the due annuity, the first payment occurs at that date. In both cases, the annuity factor values the payment stream at the specified date; the factor $v^m$ then moves that value back to time 0.

Valuing an Immediate Annuity at the Deferral Date

An annuity-immediate pays at the end of each period. If its first payment is at time $m+1$, its next payments are at times $m+2,\ldots,m+n$. Viewed from time $m$, those payments occur one through $n$ periods later. Thus the value at time $m$ is $R(v+v^2+\cdots+v^n)=R a_{\overline{n}|}$. To obtain the value at time 0, discount this amount $m$ periods.

a_{\overline{n}|}=\frac{1-v^n}{i} \qquad\text{and}\qquad \text{value at time }0=v^m R a_{\overline{n}|}

The formula for $a_{\overline{n}|}$ is used here for $i\ne 0$. If $i=0$, each payment has the same value at every date, so the value at time $m$ is $nR$, and discounting back does not change it. These formulas assume that payments are annual, level, certain, and made at the stated times, and that the same effective annual rate applies over the periods being valued.

Justification: At time $m$, the first immediate payment is one period away, so its value is $Rv$; the second is two periods away, so its value is $Rv^2$, continuing to $Rv^n$. Their sum is $R(v+v^2+\cdots+v^n)=R a_{\overline{n}|}$. A value one period later is multiplied by $v$ to move it one period earlier. Repeating this $m$ times multiplies the time-$m$ value by $v^m$, giving $v^m R a_{\overline{n}|}$ at time 0.

Worked Example: Value an Immediate Annuity at Two Dates

A hypothetical contract pays $1,000 at times 4, 5, and 6. The effective annual interest rate is 10%. Find the value at time 3, one period before the first payment, and the value at time 0.

Method. At time 3, the payments are one, two, and three periods away, so use $a_{\overline{3}|}$. Then discount the time-3 value three periods to time 0. This avoids treating the first payment as if it were due at time 3.

The discount factor is $v=1/1.10=0.909090909$. Therefore, $v^2=0.826446281$, $v^3=0.751314801$, and $a_{\overline{3}|}=(1-v^3)/0.10=2.486851991$.

The value at time 3 is $1{,}000(2.486851991)=\$2{,}486.851991$, or $2,486.85.

Now discount three periods: $v^3(2{,}486.851991)=0.751314801(2{,}486.851991)=\$1{,}868.408708$. The value at time 0 is $1,868.41.

Check. Directly discounting the payments gives $1{,}000(v^4+v^5+v^6)$. The powers are $v^4=0.683013455$, $v^5=0.620921323$, and $v^6=0.564473930$; their sum is 1.868408708. The direct value is therefore also $1,868.41. Amounts are rounded to the nearest cent only at the final step.

Valuing a Due Annuity at the Payment-Start Date

An annuity-due pays at the beginning of each period. If its first payment is at time $m$, the time-$m$ value includes a payment made immediately, followed by payments one, two, and so on, periods later. Accordingly, use $R\ddot{a}_{\overline{n}|}$ at time $m$, not $R a_{\overline{n}|}$. The value at time 0 is that amount discounted $m$ periods.

\ddot{a}_{\overline{n}|}=1+v+v^2+\cdots+v^{n-1} \qquad\text{and}\qquad \text{value at time }0=v^m R\ddot{a}_{\overline{n}|}

The first term is 1 because the first payment is made at the date where the annuity-due is valued. It is not discounted from that date. If a problem instead describes the first payment one period after the proposed deferral date, the stream is immediate at that date, even if the word “deferred” appears in the description.

Worked Example: A Deferred Annuity-Due

A hypothetical fund pays $400 at times 2, 3, 4, and 5. The effective annual interest rate is 5%. Find the value at time 2, when the first payment is made, and the value at time 0.

Method. Since the first payment occurs at time 2, value the stream at that date as a four-payment annuity-due. Then discount the time-2 value two periods. This avoids discounting the payment at time 2 even though it is already due at the valuation date.

Here, $v=1/1.05=20/21=0.952380952$. The powers needed are $v^2=0.907029478$ and $v^3=0.863837598$. Thus $\ddot{a}_{\overline{4}|}=1+v+v^2+v^3=3.723248028$.

The value at time 2 is $400(3.723248028)=\$1{,}489.299211$, or $1,489.30.

Discount two periods to time 0: $v^2(1{,}489.299211)=0.907029478(1{,}489.299211)=\$1{,}350.838287$. The value at time 0 is $1,350.84.

Check. The time-0 payments are at times 2 through 5. Their discount factors sum to $v^2+v^3+v^4+v^5=0.907029478+0.863837598+0.822702474+0.783526166=3.377095716$. Multiplying by $400 gives $1,350.84. Amounts are rounded to the nearest cent only at the final step.

A Reliable Two-Stage Process

Before calculating, identify the annuity type and write down the first payment date. Then select the date at which the annuity factor is to be applied. This short process keeps the within-stream valuation separate from the deferral discount.

Mark the first payment date.
For an immediate annuity, count one period backward from the first payment to get the date where $a_{\overline{n}|}$ applies. For a due annuity, the first payment date itself is where $\ddot{a}_{\overline{n}|}$ applies.

Value the payments at that date.
Multiply the appropriate annuity factor by the payment amount. The factor accounts for the timing of payments relative to this date.

Discount to the requested earlier date.
If the annuity value is at time $m$ and the requested date is time 0, multiply by $v^m$. Use the number of periods between the dates, not the number of payments.

Check the payment dates.
Write at least the first and last payment as powers of $v$ from the requested valuation date. Confirm that the factor-based result gives the same timing.

Worked Example: Check the Deferral Period Against the Payments

A hypothetical benefit pays $600 at times 5, 6, and 7. The effective annual interest rate is 6%. Find its value at time 4 and at time 0.

Method. The first payment is at time 5, so the stream is immediate at time 4. Use $a_{\overline{3}|}$ at time 4, then discount four periods to time 0. The deferral exponent is 4 because the two valuation dates are time 4 and time 0.

The discount factor is $v=1/1.06=0.943396226$. The relevant powers are $v^2=0.889996440$, $v^3=0.839619283$, and $v^4=0.792093663$. Therefore, $a_{\overline{3}|}=v+v^2+v^3=2.673011949$.

At time 4, the value is $600(2.673011949)=\$1{,}603.807169$, or $1,603.81.

At time 0, the value is $v^4(1{,}603.807169)=0.792093663(1{,}603.807169)=\$1{,}270.365498$, or $1,270.37.

Check. Direct discounting uses $600(v^5+v^6+v^7)$. The three factors are approximately 0.747258174, 0.704960541, and 0.665057114; their sum is 2.117275829. The resulting value is $1,270.37. The two-stage calculation and direct payment-by-payment calculation agree. Amounts are rounded to the nearest cent only at the final step.

Common Timing Errors

A frequent error is to use an annuity-due factor at the date one period before an annuity-immediate’s first payment. For example, in the first worked example, the incorrect calculation at time 3 would be $1{,}000\ddot{a}_{\overline{3}|}=1{,}000(1+0.909090909+0.826446281)=\$2{,}735.54$. That calculation incorrectly places a payment at time 3. The actual payments begin at time 4, so the correct value at time 3 is $1{,}000a_{\overline{3}|}=\$2{,}486.85$.

Another error is to discount the time-$m$ annuity value by $v^{m+1}$ for an immediate annuity. The first payment being $m+1$ does not mean the annuity value is at time $m+1$: $a_{\overline{n}|}$ is already valued one period before the first payment, at time $m$. Discount from the actual annuity valuation date to the requested date. Also avoid discounting each payment and then discounting the resulting sum again; that counts the deferral twice.

Payment pattern	Value at the payment-start valuation date	Value at time 0
Immediate: first payment at $m+1$	$R a_{\overline{n}\|}$ at time $m$	$v^m R a_{\overline{n}\|}$
Due: first payment at $m$	$R\ddot{a}_{\overline{n}\|}$ at time $m$	$v^m R\ddot{a}_{\overline{n}\|}$

This two-stage approach is useful whenever a problem asks for a deferred annuity’s value at an earlier date. It makes the annuity’s payment pattern explicit, gives a natural intermediate check, and makes the discount exponent correspond to the distance between two clearly identified dates. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the annuity valuation date before calculating. Round dollar amounts to the nearest cent.

A hypothetical annuity-immediate pays $500 at times 4, 5, and 6. At an effective annual interest rate of 4%, find its value at time 3 and at time 0. (Answer: $1,387.55 at time 3; $1,233.52 at time 0)
A hypothetical annuity-due pays $300 at times 2, 3, and 4. The effective annual interest rate is 25%. Find its value at time 2 and at time 0. (Answer: $732.00 at time 2; $468.48 at time 0)
A hypothetical annuity-immediate pays $700 at times 5, 6, 7, and 8. The effective annual interest rate is 25%. Find its value at time 4 and at time 0. (Answer: $1,653.12 at time 4; $677.12 at time 0)
A hypothetical annuity-immediate pays $250 at times 6, 7, 8, and 9. The effective annual interest rate is 25%. Find its value at time 5 and at time 0. (Answer: $590.40 at time 5; $193.46 at time 0)
A hypothetical annuity-due pays $200 at times 3, 4, 5, 6, and 7. The effective annual interest rate is 25%. Find its value at time 3 and at time 0. (Answer: $672.32 at time 3; $344.23 at time 0)

Payment pattern	Value at the payment-start valuation date	Value at time 0
Immediate: first payment at \(m+1\)	\(R a_{\overline{n}\|}\) at time \(m\)	\(v^m R a_{\overline{n}\|}\)
Due: first payment at \(m\)	\(R\ddot{a}_{\overline{n}\|}\) at time \(m\)	\(v^m R\ddot{a}_{\overline{n}\|}\)

What You'll Learn