Moving Deferred Annuity Values to Any Focal Date

Why the Focal Date Matters

A deferred annuity can be valued at several dates: before its payments begin, at the date immediately before the first payment, or after its final payment. The cash flows do not change when the valuation date changes, but the number of periods used to discount or accumulate them does. A reliable approach is to mark the payment dates and focal date first, then shift a value only across periods for which the same cash flows remain to be valued.

This tutorial considers a level annuity-immediate with payments at regular period ends. Assume a positive effective rate $i$ per period, with the same rate applying throughout. Payments are certain and there are no fees, taxes, or mortality contingencies. The results apply to any regular period length when the rate and dates use that same period.

Definition: Suppose $n$ equal payments of amount $R$ are made at times $m+1,m+2,\ldots,m+n$, where $m$ is a nonnegative integer. The date $m$, immediately before the first payment, is the deferral date. A focal date is the date at which the value of the payment stream is required. Values at different focal dates are equivalent when they represent the same cash flows under the same interest assumptions.

Start With the Deferral-Date Value

At time $m$, the payments are one, two, and up to $n$ periods in the future. The value at that date is $R a_{\overline{n}|}$, where $a_{\overline{n}|}$ is the present-value annuity-immediate factor at rate $i$. To value the stream at an earlier date $f$, discount the deferral-date value by $m-f$ periods. To value it after the final payment, start from the value at time $m+n$ and accumulate.

a_{\overline{n}|}=\sum_{k=1}^{n}v^k,\qquad s_{\overline{n}|}=\sum_{k=0}^{n-1}(1+i)^k,\qquad v=\frac{1}{1+i}

Here $s_{\overline{n}|}$ is the accumulated-value annuity factor at the date of the final payment. It is the value at that date of $n$ payments made at the ends of successive periods. These factors use the same effective rate per period as the timeline.

Result: For payments $R$ at times $m+1,\ldots,m+n$, the value at focal date $f$ is

If $f\leq m$, then $V_f=R a_{\overline{n}|}v^{m-f}$.
If $f\geq m+n$, then $V_f=R s_{\overline{n}|}(1+i)^{f-(m+n)}$.

The first formula discounts from the deferral date to an earlier focal date. The second accumulates from the final payment date to a later focal date.

Why Shifting the Value Works

The shift is valid because each cash flow is being moved by the same number of periods, with the same rate. For a focal date $f\leq m$, the payment at time $m+k$ has value $Rv^{m+k-f}$ at $f$. Its value at the deferral date $m$, $Rv^k$, discounted a further $m-f$ periods, is also $Rv^k v^{m-f}=Rv^{m+k-f}$. Summing over all $k=1,\ldots,n$ gives the earlier-date formula.

R a_{\overline{n}|}v^{m-f} =R\left(\sum_{k=1}^{n}v^k\right)v^{m-f} =R\sum_{k=1}^{n}v^{m+k-f}

For a focal date after the final payment, the same logic uses accumulation instead of discounting. At time $m+n$, a payment made $j$ periods earlier has accumulated by the factor $(1+i)^j$. Moving the whole terminal value forward by $f-(m+n)$ periods multiplies every payment’s value by the same additional accumulation factor.

R s_{\overline{n}|}(1+i)^{f-(m+n)} =R\left(\sum_{j=0}^{n-1}(1+i)^j\right)(1+i)^{f-(m+n)}

These compact shifts apply when the focal date is outside the payment stream: before the first payment or after the last. If the focal date falls within the stream, some payments have already occurred and others are still in the future. The value must then be found by accumulating the earlier payments to the focal date and discounting the later payments back to it; blindly shifting a single annuity value can include the wrong cash flows.

Moving a Value to a Date Before the Payments

Worked Example: Valuing Four Deferred Payments at an Earlier Date

A hypothetical contract pays $800$ at the end of each year from year 5 through year 8. The effective annual interest rate is $i=5\%$. Find the value at time 2. Assume annual periods, certain payments, and a constant rate; there is no mortality assumption.

Method. The first payment is at time 5, so the deferral date is time 4. There are $n=4$ payments. First value the stream at time 4, then discount that value two years to time 2.

a_{\overline{4}|} =v+v^2+v^3+v^4 =0.95238095+0.90702948+0.86383760+0.82270247 =3.54595050

The value at time 4 is

800(3.54595050)=\$2{,}836.76

The focal date is two periods earlier, so $m-f=4-2=2$. With $v^2=0.90702948$, the value at time 2 is

\$2{,}836.76040(0.90702948)=\$2{,}573.02531

Rounded to the nearest cent, the answer is $\$2{,}573.03$. As a check, the payments are three, four, five, and six periods after time 2, so their direct values are $800(v^3+v^4+v^5+v^6)=\$2{,}573.03$. This agrees. The common mistake is to discount from time 4 to time 2 using four periods rather than the two periods between those dates.

Moving a Value to a Date After the Payments

After the final payment, the annuity’s value at that payment date can be accumulated forward. The number of periods is measured from the last payment date, not the deferral date and not the first payment date. This distinction is especially important when a problem gives a focal date several periods after the annuity ends.

Worked Example: Accumulating a Deferred Annuity Beyond Its Final Payment

A hypothetical grant pays $1{,}200$ at the end of each year in years 3, 4, and 5. The effective annual interest rate is $i=10\%$. Find the value at time 7. Assume annual periods, certain payments, and a constant rate; there is no mortality assumption.

Method. There are three payments, and the final payment is at time 5. Find the accumulated value at time 5 using $s_{\overline{3}|}$, then accumulate it two more years to time 7.

s_{\overline{3}|} =1+1.10+1.10^2 =1+1.10+1.21 =3.31

The value at time 5 is $1{,}200(3.31)=\$3{,}972.00$. From time 5 to time 7 is two years, so

\$3{,}972.00(1.10)^2 =\$3{,}972.00(1.21) =\$4{,}806.12

Thus the value at time 7 is $\$4{,}806.12$. Check by accumulating each payment separately: the payments at years 3, 4, and 5 earn interest for four, three, and two years, respectively. Therefore $1{,}200(1.10^4+1.10^3+1.10^2)=1{,}200(1.4641+1.331+1.21)=\$4{,}806.12$. The common mistake is to accumulate the value at time 5 for three periods instead of the two periods from time 5 to time 7.

When the Focal Date Falls Within the Payment Stream

If the focal date is within the payment stream, the outside-the-stream formulas do not apply as a single shift. At an integer focal date, include any payment made exactly on that date at its full amount. Accumulate payments made before the date, and discount payments due after it. This keeps every cash flow on the correct side of the focal date.

Worked Example: Valuing at a Date Within the Payment Stream

A hypothetical scholarship fund pays $500$ at the end of each year in years 2, 3, 4, and 5. The effective annual interest rate is $i=4\%$. Find the value at time 3. Assume annual periods, certain payments, and a constant rate; there is no mortality assumption.

Method. The payment at year 2 is one year before the focal date and accumulates one year. The payment at year 3 is made on the focal date. The payments at years 4 and 5 are discounted one and two years, respectively.

$$ V_3 =500(1.04)+500+500v+500v^2 $$

Using $v=1/1.04=0.96153846$ and $v^2=0.92455621$,

V_3 =500(1.04+1+0.96153846+0.92455621) =500(3.92609467) =\$1{,}963.04734

Rounded to the nearest cent, the value at time 3 is $\$1{,}963.05$. The payment made at time 3 is neither accumulated nor discounted. A common mistake is to treat it as one period in the future or one period in the past; the timeline shows that it is paid at the focal date itself.

A Reliable Valuation Process

A short sequence of checks prevents most focal-date errors. In particular, distinguish the deferral date from the first payment date: for an annuity-immediate, the deferral date is one period before the first payment. Then count periods between the date at which a known annuity value is calculated and the requested focal date.

Mark the dates.
Write down the first and last payment dates, the deferral date, and the requested focal date.

Choose a convenient anchor date.
Use the deferral date for a value before the payment stream, or the last payment date for a value after it.

Count the gap between dates.
Discount when moving to an earlier date and accumulate when moving to a later date. Count only the periods between the anchor and focal dates.

Check the cash-flow boundary.
If the focal date is within the payment stream, separate payments before, on, and after it rather than shifting one whole-annuity value.

Verify the result.
Compare with a direct sum of the discounted or accumulated payments when practical.

Common Focal-Date Traps

The focal date is not just a label attached to an annuity factor. It determines the exponent on every discount or accumulation factor, so a one-period counting error changes the value.

Discounting from the wrong date. If a value is already known at the deferral date, discount only from that date to the requested earlier date.
Accumulating for too many periods. A terminal annuity value is located at the last payment date. Count forward from there.
Using one shift across payments that straddle the focal date. Payments already made and payments still due must be valued separately.
Mixing rate periods and timeline periods. An annual effective rate is used with annual periods; a different payment frequency requires a rate consistent with that period.
Shifting the value without checking what it represents. A value at the deferral date represents all the future payments. If the focal date is after some payments have occurred, those payments should not remain in a value of the remaining stream.

The same timeline method applies whether the focal date is close to the annuity or far away. The amount of the shift changes, but the cash flows, rate assumption, and payment timing do not. This disciplined separation of payment dates and focal dates is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Key takeaway: For a focal date before the first payment, value the annuity at its deferral date and discount across the gap. For a focal date after the last payment, value it at the final payment date and accumulate across the gap. If the date lies within the stream, value each side separately.

Check Your Understanding

For each problem, assume certain level payments, a constant effective annual interest rate, and annual periods. Round money to the nearest cent.

A hypothetical fund pays $600$ at the end of each year from year 4 through year 7. At $i=5\%$, find its value at time 1. (Answer: $\$1{,}929.77$)
A hypothetical scholarship pays $900$ at the end of each year from year 6 through year 9. At $i=4\%$, find its value at time 2. (Answer: $\$2{,}904.27$)
A hypothetical trust pays $750$ at the end of years 2, 3, and 4. At $i=6\%$, find its value at time 6. (Answer: $\$2{,}682.82$)
A hypothetical program pays $400$ at the end of years 1, 2, 3, and 4. At $i=5\%$, find its value at time 2, treating the year-2 payment as occurring at the focal date. (Answer: $\$1{,}563.76$)
A hypothetical endowment pays $1{,}000$ at the end of years 5, 6, and 7. At $i=8\%$, find both its value at time 2 and its value at time 9. (Answer: $\$2{,}209.45$ at time 2; $\$3{,}786.60$ at time 9)

What You'll Learn