Grouping Repeated Payments in an Irregular Stream

Why Group a Repeated Block?

An irregular cash-flow stream may contain a run of equal payments among other payments of different amounts or at different dates. When the equal payments are consecutive and equally spaced, their value can be calculated efficiently with an annuity factor. The key is to value the block at the right date first, then move that value to the requested valuation date.

Grouping is a calculation shortcut, not a change to the cash flows. The block must still represent exactly the same payment amounts and dates as the original stream. In particular, the date immediately before the first payment is one period before that payment—not the valuation date unless the first payment is due one period later.

Definition: A block of $n$ repeated payments consists of $n$ equal cash flows $R$, each separated from the next by the same payment interval. If the effective interest rate per payment interval is $i$, its value one interval before the first payment is $R a_{\overline{n}|}$, where $a_{\overline{n}|}=(1-v^n)/i$ and $v=1/(1+i)$. If that block value is $k$ intervals after the valuation date, its value at the valuation date is $R a_{\overline{n}|}v^k$.

This result assumes a constant effective rate per payment interval over the dates being valued, and that payments occur at the end of each interval. If intervals differ, payments are not equal, or the discount rates differ by date, the ordinary annuity factor does not directly represent the block; value the payments using the applicable factors for their dates. For annual payments with annual effective rate $i$, the rate per interval is $i$. For quarterly payments, use the effective quarterly rate. If an annual effective rate is used with fractional-year timing, state the accumulation convention explicitly.

Locating the Annuity Value Date

Suppose equal payments occur at times $m,m+1,\ldots,m+n-1$, measured in regular payment intervals from time 0. The annuity-immediate value $R a_{\overline{n}|}$ is located at time $m-1$, immediately before the first payment. To obtain the value at time 0, discount this amount for $m-1$ intervals:

PV_0=R a_{\overline{n}|}v^{m-1}.

This placement follows directly from the annuity formula. At time $m-1$, the first payment is one interval away, the next is two intervals away, and the last is $n$ intervals away. Thus its value is $R(v+v^2+\cdots+v^n)=R a_{\overline{n}|}$. Discounting that value by $m-1$ further intervals moves each payment to time 0.

Worked Example: Annual Payments Within a Longer Irregular Stream

A hypothetical project has cash receipts of $700 at the end of year 1, $1,000 at each year-end from year 4 through year 6, and $1,400 at the end of year 8. Find the year-0 value of the three-payment block using a constant effective annual rate of 10%. Do not include the other receipts in the block calculation.

Method. Isolate the equal payments at years 4, 5, and 6. They are consecutive annual payments, so use $i=0.10$ and $v=1/1.10$. The value date for their annuity factor is year 3, one year before the first payment.

Value at year 3. First calculate the three-payment annuity factor:

a_{\overline{3}|}=\frac{1-(1.10)^{-3}}{0.10} =\frac{1-0.751314801}{0.10} =2.486851991.

The block value at year 3 is $1{,}000(2.486851991)=$2{,}486.851991$. Year 3 is three years after the requested year-0 date, so discount by $v^3=(1.10)^{-3}=0.751314801$:

PV_0=1{,}000a_{\overline{3}|}(1.10)^{-3} =1{,}000(2.486851991)(0.751314801) =$1{,}868.4087\ldots

Rounded to the nearest cent, the block value is $1,868.41. To check, discount each payment directly: the year-4, year-5, and year-6 discount factors are respectively 0.683013455, 0.620921323, and 0.564473930. Their sum is 1.868408708, giving $1{,}000(1.868408708)=$1{,}868.41$ to the nearest cent. This check confirms that the block value includes exactly the three stated payments.

Common mistake avoided: discounting the annuity value at year 3 for four years would place it at year −1, one year too early. The correct deferral is three years, because the block value date is year 3.

A Short Derivation

The annuity factor is a geometric sum of discounted payments. At the date one interval before the first payment, the $n$ payments have present values $Rv,Rv^2,\ldots,Rv^n$. Therefore,

R(v+v^2+\cdots+v^n) =R\frac{v(1-v^n)}{1-v} =R\frac{1-v^n}{i} =R a_{\overline{n}|},

because $1-v=i/(1+i)=iv$, so $v/(1-v)=1/i$. If this annuity value is $k$ intervals after the target date, discounting it by $v^k$ gives the target-date value. Equivalently, each original payment is then discounted by $k+1,k+2,\ldots,k+n$ intervals, exactly its actual distance from the target date.

Worked Example: A Quarterly Block Valued Before Its First Payment

Four equal payments of $500 occur at the ends of quarters 3, 4, 5, and 6. Find their value at the end of quarter 1 using an effective quarterly rate of 2%.

Method. The payment interval is one quarter, so use $i=0.02$ per quarter and $v=1/1.02$. The first payment is at quarter 3; the annuity-immediate value date is quarter 2. Discount that block value one quarter to the requested date, quarter 1.

The four-payment annuity factor is

a_{\overline{4}|}=\frac{1-(1.02)^{-4}}{0.02} =\frac{1-0.923845426}{0.02} =3.8077287.

At quarter 2, the block is worth $500(3.8077287)=$1{,}903.8644$. Discount one quarter:

PV_{\text{quarter 1}} =500a_{\overline{4}|}(1.02)^{-1} =500(3.8077287)(0.980392157) =$1{,}866.5337\ldots

The value at the end of quarter 1 is $1,866.53, rounded to the nearest cent. A direct check uses the four payment dates’ distances of 2, 3, 4, and 5 quarters from quarter 1:

500\left[(1.02)^{-2}+(1.02)^{-3}+(1.02)^{-4}+(1.02)^{-5}\right] =500(3.73306735) =$1{,}866.53.

Common mistake avoided: using the annual rate as though it were the quarterly rate would apply the wrong discount period. The stated 2% is effective per quarter, and the date differences in this calculation are counted in quarters.

Payment Intervals Need Not Be One Year

A repeated block can have a regular interval other than one year. The effective rate used in the annuity factor must match that interval. If the problem gives an effective annual rate $i$ and specifies accumulation over fractional years using $(1+i)^t$, then an interval of $h$ years has effective rate $(1+i)^h-1$. This follows because one interval of accumulation is $(1+i)^h=1+i_h$.

Worked Example: Semiannual Payments With a Fractional-Year Convention

Three equal payments of $2,500 occur at months 18, 24, and 30. Find their value at month 12, assuming an effective annual rate of 8% and fractional-year accumulation using $(1.08)^t$.

Method. The payments are six months apart. Under the stated convention, the effective six-month rate is $i_{0.5}=(1.08)^{0.5}-1=0.0392305$, or approximately 3.92305%. Month 12 is one six-month interval before the first payment at month 18, so the value at month 12 is an ordinary three-payment annuity value; no additional deferral is needed.

The three discounts from month 12 are for 0.5, 1, and 1.5 years. Using the specified annual accumulation convention:

PV_{\text{month 12}} =2{,}500\left[(1.08)^{-0.5}+(1.08)^{-1}+(1.08)^{-1.5}\right].

The factors are $0.962250449$, $0.925925926$, and $0.890972638$. Their sum is $2.779149013$, so the unrounded value is $2{,}500(2.779149013)=$6{,}947.8725\ldots$. The answer is $6,947.87, rounded to the nearest cent.

Check. The annuity-factor approach gives the same result: $a_{\overline{3}|}$ at the effective six-month rate is the sum of those three successive half-year discount factors. This confirms both the payment spacing and the valuation date.

Common mistake avoided: treating month 12 as three years before the payment block, or treating the 8% annual rate as an 8% rate every six months, would misstate the timing or the rate. Here the first payment is only half a year after month 12, and the effective half-year rate is derived from the stated annual convention.

When the Shortcut Does—and Does Not—Apply

The annuity shortcut is useful when a block has equal amounts and a constant spacing. The payments outside that block do not change its annuity value, but they are not part of the block. Keeping the block boundaries clear helps prevent an adjacent one-off payment from being counted as a repeated payment.

A block can also be valued at a date other than the target date. First identify the date immediately before its first payment and calculate $R a_{\overline{n}|}$ there. Then count the intervals from that date back to the target and discount accordingly. If the block value date is before the target instead, move the value forward rather than discounting it. In either direction, the value must be stated at a definite date.

If rates vary by payment date, a single $a_{\overline{n}|}$ may not capture the appropriate discounts. Likewise, if a payment is missing from the run or the intervals are not equal, the payments do not form the required ordinary annuity block. In those cases, value the payments at their own dates using the applicable discount factors. A direct sum is also a strong check whenever the block is small.

Key takeaway: Locate the first payment, move back exactly one payment interval to locate the annuity value date, calculate the block there using the effective rate per interval, and then move the block value to the requested date. A direct discount sum should agree.

Check Your Understanding

For each problem, identify the block’s payment interval and valuation date before calculating. Dollar answers are rounded to the nearest cent.

Three equal payments of $1,000 occur at the ends of years 2, 3, and 4. Find their value at year 1 using an effective annual rate of 5%. (Answer: $2,723.25)
Four payments of $750 occur at the ends of quarters 4 through 7. Find their value at the end of quarter 2 using an effective quarterly rate of 1%. (Answer: $2,897.50)
Three payments of $1,800 occur at months 12, 18, and 24. Find their value at month 6 using an effective annual rate of 6% and fractional-year accumulation $(1.06)^t$. (Answer: $5,095.78)
Four payments of $600 occur at the ends of years 6 through 9. Find the block’s value at year 4 using an effective annual rate of 7%. (Answer: $1,899.37)
Three equal payments of $2,000 occur at years 3, 5, and 7. Find their value at year 1 using an effective annual rate of 10%. Explain why the relevant annuity interval is two years, not one year. (Answer: $4,147.87)

What You'll Learn