Payment Dates at the Deferral Boundary

Why the Deferral Boundary Matters

A deferred annuity’s value depends on when its first payment is made. If a deferral period ends at time $m$, a payment at time $m$ is made at the boundary; a payment at time $m+1$ is made one period after the boundary. Those dates differ by one full interest period, so they do not have the same present value.

This distinction is especially important when a problem says that payments begin “after a three-year deferral” or “at the end of the deferral period.” The safest approach is to identify the time of the first payment before selecting an annuity factor. The formulas below assume an effective annual interest rate $i$, annual periods, and valuation at time 0.

Definition: If the deferral period is $m$ years, its boundary is time $m$. A level annuity of $n$ payments made at times $m,m+1,\ldots,m+n-1$ has its first payment at the boundary. A level annuity of $n$ payments made at times $m+1,m+2,\ldots,m+n$ has its first payment one period after the boundary. Here $m$ is a nonnegative integer and $n$ is a positive integer.

The annuity at the boundary is an annuity-due relative to time $m$: its first payment is made at that time. The annuity beginning one period later is an annuity-immediate relative to time $m$: its first payment is made at the end of the first period after that time. These descriptions are about payment timing, not the size of the deferral period.

Two Finite-Annuity Timelines

Let each payment be $R$. The discount factor is $v=1/(1+i)$. The standard present-value factors $a_{\overline{n}|}$ and $\ddot{a}_{\overline{n}|}$ value, respectively, an $n$-payment annuity-immediate and annuity-due at the time from which the payment stream is being valued.

First payment	Payment times	Present value at time 0
At the boundary	$m,m+1,\ldots,m+n-1$	$R v^m\ddot{a}_{\overline{n}\|}$
One period after the boundary	$m+1,m+2,\ldots,m+n$	$R v^m a_{\overline{n}\|}$

The factor $v^m$ discounts a value at time $m$ back to time 0. At time $m$, the boundary stream has payments at times $m$ through $m+n-1$, so it is valued as an annuity-due. The later-starting stream has payments at times $m+1$ through $m+n$, so it is valued as an annuity-immediate.

\begin{aligned} PV_{\text{boundary}}&=R\sum_{k=0}^{n-1}v^{m+k} =Rv^m\ddot{a}_{\overline{n}|},\\ PV_{\text{one period later}}&=R\sum_{k=1}^{n}v^{m+k} =Rv^m a_{\overline{n}|}. \end{aligned}

These expressions apply to the stated payment dates. In particular, “deferred $m$ years” by itself is not enough to decide which expression to use unless the problem’s convention or wording identifies the first payment date. Draw the timeline, or write the payment times explicitly.

Worked Example: Payments at the Boundary or One Year Later

A fund pays \$1,000 annually for three years. The effective annual interest rate is 5%, and the deferral period is two years. Find the present value if the first payment is made (a) at the boundary and (b) one year after the boundary. State the method and check the difference.

Method. List the payment times, then discount each payment using $v^t$, where $t$ is its time in years. This avoids treating the two different first-payment dates as if they were the same annuity.

The discount factor and powers needed are

v=\frac{1}{1.05}=0.95238095,\qquad v^2=0.90702948,\quad v^3=0.86383760,\quad v^4=0.82270247,\quad v^5=0.78352617.

(a) First payment at the boundary. The payments occur at times 2, 3, and 4. Therefore,

PV=1000(v^2+v^3+v^4) =1000(0.90702948+0.86383760+0.82270247) =1000(2.59356955) =\$2{,}593.57.

(b) First payment one year later. The payments occur at times 3, 4, and 5. Therefore,

PV=1000(v^3+v^4+v^5) =1000(0.86383760+0.82270247+0.78352617) =1000(2.47006624) =\$2{,}470.07.

The boundary-starting stream is worth \$123.50 more. As a check, the two three-payment streams are shifted one period apart, so the difference is the value of the payment removed at time 5 minus the value of the payment added at time 2:

1000(v^2-v^5)=1000(0.90702948-0.78352617) =\$123.50.

The check agrees with the difference between the two present values, allowing for rounding to cents. The common mistake avoided here is assuming the difference must equal one payment discounted to the boundary. For a finite stream, shifting all payments also removes the old final payment.

Why the Boundary Formulas Hold

The annuity-factor formulas follow by discounting each payment separately. This also shows exactly where the boundary payment enters.

Value the stream at time $m$.
For a boundary-starting annuity, the first payment is made immediately at time $m$. The remaining payments occur one, two, and up to $n-1$ years later. Its value at time $m$ is $R(1+v+\cdots+v^{n-1})=R\ddot{a}_{\overline{n}|}$.

Discount to time 0.
A value at time $m$ is multiplied by $v^m$ to obtain its present value. Thus the boundary-starting value is $Rv^m\ddot{a}_{\overline{n}|}$.

Repeat for a stream beginning later.
If the first payment is at $m+1$, there is no payment at time $m$. At time $m$, the stream has the timing of an annuity-immediate, with value $Ra_{\overline{n}|}$. Discounting gives $Rv^m a_{\overline{n}|}$.

The same reasoning gives a useful difference formula. The two finite streams contain payments at times $m,\ldots,m+n-1$ and $m+1,\ldots,m+n$. Their common payments at times $m+1,\ldots,m+n-1$ cancel when the present values are subtracted. Only the first payment of the boundary stream and the final payment of the later stream remain.

PV_{\text{boundary}}-PV_{\text{one period later}} =R(v^m-v^{m+n}).

This difference is positive when $i>0$, because $v^m>v^{m+n}$. The boundary stream is worth more: its first payment arrives earlier, although its final payment also arrives earlier. The formula accounts for both timing changes.

Worked Example: Reading “At the End of the Deferral Period”

A hypothetical scholarship fund is valued at an effective annual rate of 5%. It will make six annual payments of \$800, with the first payment at the end of a four-year deferral period. Find the present value. Then calculate the value if the first payment were instead one year after the deferral boundary. Round money to the nearest cent.

Method. Interpret “at the end of a four-year deferral” as a payment at time 4, the boundary. The boundary dates are 4, 5, 6, 7, 8, and 9. If the first payment is one year later, the dates are 5 through 10.

At 5%, the required discount factors are

\begin{aligned} v^4&=0.82270247,& v^5&=0.78352617,& v^6&=0.74621540,\\ v^7&=0.71068133,& v^8&=0.67683936,& v^9&=0.64460892,\\ v^{10}&=0.61391325. \end{aligned}

For the first-payment-at-boundary interpretation, sum the factors at times 4 through 9:

\begin{aligned} \sum_{t=4}^{9}v^t &=0.82270247+0.78352617+0.74621540\\ &\quad+0.71068133+0.67683936+0.64460892\\ &=4.38457365,\\ PV&=800(4.38457365)=\$3{,}507.66. \end{aligned}

For a first payment one year after the boundary, the payment times are 5 through 10:

\begin{aligned} \sum_{t=5}^{10}v^t &=0.78352617+0.74621540+0.71068133\\ &\quad+0.67683936+0.64460892+0.61391325\\ &=4.17578443,\\ PV&=800(4.17578443)=\$3{,}340.63. \end{aligned}

As a direct check, the difference between these two shifted finite streams is the first payment discounted to time 4 less the final payment discounted to time 10:

800(v^4-v^{10}) =800(0.82270247-0.61391325) =\$167.03.

The computed values differ by \$167.03 after rounding. The common mistake avoided is to treat “at the end of the deferral period” as though it meant “one year after the deferral period.” Those instructions put the first payment at different times.

Perpetuities at the Boundary

The same payment-date distinction applies to a perpetuity, an infinite stream of equal payments. Assume $i>0$, so the present-value series converges. If the first payment is at time $m+1$, the stream has payments at $m+1,m+2,\ldots$, and its present value is $Rv^m/i$. If instead the first payment is at the boundary time $m$, the payment at $m$ is included as well as all later payments.

\begin{aligned} PV_{\text{first at }m+1} &=R\sum_{k=1}^{\infty}v^{m+k} =\frac{Rv^m}{i},\\ PV_{\text{first at }m} &=R\sum_{k=0}^{\infty}v^{m+k} =\frac{Rv^m}{1-v} =\frac{Rv^m}{d}. \end{aligned}

Here $d=1-v=i/(1+i)$ is the effective annual discount rate. To justify the formulas, the geometric series satisfies $1+v+v^2+\cdots=1/(1-v)$ when $0<v<1$. Also, $1-v=iv$, so the boundary value can equivalently be written $Rv^{m-1}/i$ when $m\geq 1$. The boundary-starting perpetuity exceeds the later-starting one by exactly $Rv^m$, the present value of the single payment made at time $m$. For a finite annuity, the streams do not have the same final payment date, so the difference is not just the boundary payment.

Worked Example: A Perpetuity Starting at the Boundary

A hypothetical endowment is valued at an effective annual interest rate of 4%. It pays \$1,200 at the end of each year forever, with its first payment made at the end of a five-year deferral period. Find the present value. Compare it with a version whose first payment is one year later.

Method. The boundary is time 5. The first stream pays at times 5, 6, 7, and so on; the later stream pays at times 6, 7, 8, and so on. Since $i=0.04$, the discount rate is $d=0.04/1.04=0.03846154$, and $v^5=(1/1.04)^5=0.82192711$.

For a first payment at time 5, use the perpetuity-due value at time 5 and discount it five years:

PV_{\text{boundary}} =v^5\frac{1200}{d} =0.82192711\left(\frac{1200}{0.03846154}\right) =0.82192711(31{,}200) =\$25{,}644.13.

For a first payment at time 6, use the perpetuity-immediate value at time 5 and discount it to time 0:

PV_{\text{one period later}} =v^5\frac{1200}{i} =0.82192711\left(\frac{1200}{0.04}\right) =0.82192711(30{,}000) =\$24{,}657.81.

The difference is $1200v^5=1200(0.82192711)=\$986.31$, the present value of the payment at time 5. This check works because, after that payment, the remaining perpetuity payments coincide. The common mistake avoided is to use $R/i$ at time 5 even though a payment is also made at time 5; $R/i$ values a stream whose first payment is one period later.

Common Mistakes and a Reliable Check

In a timed problem, the boundary issue can be settled with a short routine: identify the boundary, write the payment dates, and only then select a factor. Do not decide from the word “deferred” alone. For a finite annuity, count the first and last payment dates to ensure there are exactly $n$ payments. For a perpetuity, check whether a payment occurs at the boundary before deciding between $R/i$ and $R/d$ at that time.

Misplacing the first payment. A payment at the end of a four-year deferral is at time 4; a payment one year after the boundary is at time 5.
Using the wrong annuity type. A payment at the valuation date for the annuity is due timing; a payment at the end of the next period is immediate timing.
Forgetting the final payment date. Shifting a finite stream one period later shifts both its first and last payments. Its value difference is $R(v^m-v^{m+n})$, not merely $Rv^m$.
Applying a perpetuity formula at the wrong date. At time $m$, $R/i$ applies to payments at $m+1,m+2,\ldots$; $R/d$ applies to payments at $m,m+1,\ldots$.

A timeline is a practical final check: if the number of dates or the first date on the timeline does not match the problem statement, the annuity factor cannot be correct. This timing discipline is useful whenever a deferred payment stream is valued, and is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the first payment date before calculating. Round monetary answers to the nearest cent.

At an effective annual interest rate of 5%, a four-payment annuity of \$500 has its first payment at the boundary after a two-year deferral. Find its present value. (Answer: \$1,688.55)
At an effective annual interest rate of 8%, four annual payments of \$900 begin one year after a three-year deferral boundary. Find the present value. (Answer: \$2,366.35)
A perpetuity pays \$1,500 annually at an effective annual interest rate of 4%. Its first payment is at time 6, the boundary after a six-year deferral. Find its present value. (Answer: \$30,822.27)
At an effective annual interest rate of 10%, compare two three-payment annuities of \$700. One starts at the boundary at time 2; the other starts one year later. By how much does the boundary-starting annuity exceed the later one in present value? (Answer: \$143.87)
At an effective annual interest rate of 5%, three equal payments are made at times 1, 2, and 3. Their present value is \$2,500. Find the amount of each payment. (Answer: \$918.02)

First payment	Payment times	Present value at time 0
At the boundary	\(m,m+1,\ldots,m+n-1\)	\(R v^m\ddot{a}_{\overline{n}\|}\)
One period after the boundary	\(m+1,m+2,\ldots,m+n\)	\(R v^m a_{\overline{n}\|}\)

What You'll Learn