Skipped Payments and Irregular Annuity Starts

Why Payment Dates Must Be Made Explicit

An annuity formula is compact because it assumes a regular pattern of payment dates. But a contract can omit early installments, begin on a date that does not match the usual first-payment date, or leave out one payment while the later installments remain on schedule. In each case, the reliable first step is to write down when each payment actually occurs. The value equation then follows from discounting those payments to the chosen focal date.

This tutorial assumes annual payments and a constant effective annual interest rate unless stated otherwise. Payments are made at the end of each listed year. There is no mortality or other contingency: each stated payment is certain. The same method applies to other regular payment periods if the effective rate per period and the time indices are used consistently.

Definition: A payment stream is specified by its payment amounts and their dates. At effective annual rate $i$, a payment $C_t$ at time $t$ has present value $C_t v^t$, where $v=1/(1+i)$. The present value of a finite stream is the sum of the present values of its payments.

For a level annuity-immediate of $n$ payments $R$, payments occur at times $1,2,\ldots,n$, and its present value is $R a_{\overline{n}|}$. If the first payment is instead at time $k+1$, followed by $n-1$ more annual payments, the stream is the original pattern shifted $k$ years later. Its value at time 0 is $R v^k a_{\overline{n}|}$. The exponent $k$ counts the periods by which the standard annuity-immediate has been deferred; it is not the date of the first payment.

R v^k a_{\overline{n}|} = R v^k(v+v^2+\cdots+v^n) = R(v^{k+1}+v^{k+2}+\cdots+v^{k+n})

This identity also explains the timing convention: a payment at time $k+1$ is discounted $k+1$ years, not $k$ years. For a stream beginning on another integer date $m$, with $n$ payments at times $m,m+1,\ldots,m+n-1$, the value is $R v^{m-1}a_{\overline{n}|}$. Always check the first and last payment dates against the expanded sum.

A Reliable Method for Irregular Streams

When a payment is skipped, do not change the dates of the payments that remain unless the problem explicitly says the schedule shifts. A skipped payment can be handled by subtracting it from a complete scheduled stream. Alternatively, list only the payments that are actually made and discount them one by one. The two methods must give the same value.

Set the focal date.
Identify the date at which the requested value is measured, such as time 0.

Write the actual dates.
List each payment date, amount, and any omitted installment. Do not infer timing from the word “annuity” alone.

Choose a valuation form.
Use a shifted annuity factor when payments are level and regularly spaced; use a dated sum when amounts or dates vary; or start with a complete stream and subtract the omitted payment.

Check the result against the timeline.
Confirm that every included payment is discounted from its actual date and that no omitted payment or later payment has been counted incorrectly.

The subtraction method is a direct consequence of additivity. If the complete scheduled stream has present value $V$, and the omitted payment would have been $C_j$ at time $j$, then removing that payment reduces present value by exactly $C_jv^j$. All other cash flows keep their original dates.

V_{\text{actual}}=V_{\text{complete}}-C_jv^j

Worked Examples

Worked Example: Two Initial Payments Are Skipped

A fund is scheduled to make eight annual payments of $1,200 at the ends of years 1 through 8. The payments at the ends of years 1 and 2 are skipped, and the remaining six payments occur at the ends of years 3 through 8 as scheduled. Find the present value at time 0 if the effective annual interest rate is 4%.

Method. The actual payments are level and regularly spaced, with the first at time 3. This is a six-payment annuity-immediate deferred by two years, so its value is $1{,}200v^2a_{\overline{6}|}$. The scheduled payments are certain; no mortality assumption is involved.

First calculate the discount factor and annuity factor:

v=\frac{1}{1.04}=0.9615384615\ldots,\qquad a_{\overline{6}|}=\frac{1-v^6}{0.04} =\frac{1-(1.04)^{-6}}{0.04} =5.2421369\ldots

The two-year deferral factor is $v^2=(1.04)^{-2}=0.9245562130\ldots$. Thus

\operatorname{PV}_0 =1{,}200(0.9245562130\ldots)(5.2421369\ldots) =1{,}200(4.8466502\ldots) =\$5{,}815.9802\ldots

The present value, rounded to the nearest cent, is $5,815.98. As a timing check, the annuity factor times $v^2$ represents $v^3+v^4+\cdots+v^8$, exactly the six actual payment dates. The common mistake is to use $v^3a_{\overline{6}|}$, which would place the first payment at time 4 instead of time 3.

Worked Example: A Geometric Stream Starts on an Irregular Date

A hypothetical fund pays $110 at the end of year 2, with each subsequent annual payment 10% larger than the preceding one. There are three payments in total, at the ends of years 2, 3, and 4. Find their present value at time 0 at an effective annual interest rate of 10%.

Method. The first payment is two years from time 0, so discount each payment from its actual date. The amounts are $110, $121, and $133.10. Their present values are

\frac{110}{1.10^2}=\$90.9090909\ldots,\qquad \frac{121}{1.10^3}=\$90.9090909\ldots,\qquad \frac{133.10}{1.10^4}=\$90.9090909\ldots

Therefore,

\operatorname{PV}_0 =\frac{110}{1.10^2}+\frac{121}{1.10^3}+\frac{133.10}{1.10^4} =3(90.9090909\ldots) =\$272.7272727\ldots

The value rounded to the nearest cent is $272.73. Each payment has the same present value because the payment growth rate equals the effective annual interest rate: each payment and its discount factor both acquire a factor of 1.10 relative to the previous year. The check is also visible directly: the three discounted terms are equal, and there are exactly three payments. Do not discount the first payment for one year merely because it is the first payment in the list; it occurs at the end of year 2.

Worked Example: One Payment Is Omitted From an Increasing Schedule

A fund is scheduled to pay $300 at the end of year 1, with each scheduled annual payment increasing by $50. The schedule runs through year 6, but the payment at the end of year 4 is omitted. Find the present value of the payments actually made at time 0 if the effective annual interest rate is 5%.

Method. The complete schedule would be $300, $350, $400, $450, $500, and $550 at times 1 through 6. The actual payments are $300 at time 1, $350 at time 2, $400 at time 3, $500 at time 5, and $550 at time 6. The omitted $450 payment remains a missing cash flow at time 4; it does not move the year-5 payment to year 4.

Using $v=1/1.05$, the needed discount factors are $v=0.9523809524\ldots$, $v^2=0.9070294785\ldots$, $v^3=0.8638375985\ldots$, $v^5=0.7835261665\ldots$, and $v^6=0.7462153966\ldots$. Discount the actual payments:

\begin{aligned} \operatorname{PV}_0 &=300v+350v^2+400v^3+500v^5+550v^6\\ &=285.7142857\ldots+317.4603175\ldots+345.5350394\ldots\\ &\quad+391.7630832\ldots+410.4184681\ldots\\ &=\$1{,}750.8911939\ldots \end{aligned}

Thus the present value, rounded to the nearest cent, is $1,750.89. Check by valuing the complete schedule and subtracting the omitted installment:

\operatorname{PV}_{\text{complete}} =\$1{,}750.8911939\ldots+450v^4 =\$1{,}750.8911939\ldots+\$370.2161138\ldots =\$2{,}121.1073077\ldots $$ $$ \operatorname{PV}_{\text{actual}} =\$2{,}121.1073077\ldots-\$370.2161138\ldots =\$1{,}750.8911939\ldots

This agrees with the direct sum. The trap is treating an omitted installment as if it moved all subsequent payments earlier. The dates in the contract schedule remain fixed unless a rescheduling rule is explicitly stated.

Common Timing Errors and Why the Method Matters

An annuity factor is a shortcut for a particular sequence of dates, not a substitute for reading the cash-flow description. When the first payment date is irregular, write the dates before choosing a factor. When one installment is omitted, identify its scheduled amount and date before subtracting it. A correct value equation must preserve both the amount pattern and the actual payment dates.

Using the first-payment date as the deferral exponent. If the first payment is at time 3, a standard annuity-immediate shifted by two periods has first payment at time 3: $v^2a_{\overline{n}|}$. Using $v^3$ shifts it one period too far.
Counting scheduled payments instead of actual payments. If the first two of eight scheduled payments are skipped and the remaining six are paid, the actual stream has six payments, not eight.
Moving later payments after an omission. An omitted payment is a zero cash flow on its scheduled date. It does not normally alter the dates of later installments.
Applying a level annuity factor to varying amounts. A varying stream can be valued by discounting its dated amounts directly. A compact annuity factor is valid only if its payment pattern matches the stream being valued.

These checks matter because a one-period timing error changes every affected discount factor. For exam problems, a clear dated equation is often more reliable than trying to recognize a familiar formula too early. Once the dates and amounts are correct, a standard annuity factor can still make the arithmetic shorter where its assumptions fit.

Key takeaway: Value the cash flows that actually occur, on the dates they actually occur. A skipped initial payment is a deferral only when the remaining payments retain their regular spacing; a single omitted installment is removed at its scheduled date without shifting the rest of the stream.

Check Your Understanding

For each problem, write the payment dates before calculating. Assume all payments are certain, annual, and made at the ends of the stated years.

A fund pays $900 annually at the ends of years 4 through 8. Find the present value at time 0 at an effective annual rate of 6%. (Answer: $3,183.10)
A seven-payment level annuity would pay $250 at the ends of years 1 through 7, but the payment at the end of year 5 is omitted. Find the present value at time 0 at an effective annual rate of 4%. (Answer: $1,295.03)
A fund pays $400 at the end of year 2, and each subsequent annual payment is 5% greater than the previous payment. There are three payments, at the ends of years 2, 3, and 4. Find the present value at time 0 at an effective annual rate of 5%. (Answer: $1,088.44)
A schedule pays $200 at the end of year 2, with each scheduled annual payment increasing by $30. The year-3 payment is omitted, and the payments at the ends of years 4 and 5 are made. Find the present value of the actual payments at time 0 at an effective annual rate of 4%. (Answer: $645.52)

What You'll Learn