Varying Annuities at Irregular Payment Dates

Why Payment Dates Must Be Valued Individually

Many varying-annuity formulas are built around payments made at regular intervals. In an irregular stream, the cash flows may instead occur after different lengths of time. A payment at time 1 and one at time 3 cannot be discounted by the same factor simply because each is one item in a list. The relevant quantity is each payment’s actual time from the valuation date.

The reliable method is to write down every cash-flow amount, its date, and the factor that moves value between that date and the valuation date. This approach works whether the amounts are level, increasing, decreasing, or unrelated. It also makes the interest assumptions visible instead of hiding them inside an annuity factor.

Definition: An irregular payment stream is a finite collection of cash flows $C_k$ occurring at times $t_k$, where the intervals between successive payment dates need not be equal. A cash flow is positive when received and negative when paid. To value the stream, pair every $C_k$ with the accumulation or discount factor for its own date.

Discounting Each Payment at Its Own Date

For a fixed effective annual rate $i$, assume fractional-year accumulation follows the convention $(1+i)^t$ for $t$ years. This assumption specifies how to handle dates between integer years; an effective annual rate alone does not always determine a fractional-year convention. If a problem gives a different accumulation rule, use that rule instead.

Under this model, one dollar at time $t_k$ has present value $v^{t_k}$, where $v=1/(1+i)$. Each payment is discounted from its own date to time zero, then the resulting values are added.

\operatorname{PV}_0=\sum_{k=1}^{n} C_k v^{t_k} =\sum_{k=1}^{n}\frac{C_k}{(1+i)^{t_k}}.

The formula does not require equal intervals or a particular pattern in the payment amounts. It does require that each $t_k$ be measured in years from time zero and that the stated interest model apply over the relevant period. If rates vary over time, use the appropriate accumulated factor from time zero to each payment date rather than applying a single constant-rate power; the essential step remains matching each cash flow to its own factor.

Write the dates.
Record every payment time from the valuation date, including fractional years where applicable.

Attach the payment amounts.
Keep the amount and date together; do not replace the dates with payment numbers.

Apply the matching factors.
Discount each payment by the accumulation factor for its own elapsed time.

Add values at one date.
Once every cash flow is expressed at the same valuation date, sum them or set the sum equal to a given value.

Worked Example: Present Value at Unequal Intervals

A fund is scheduled to receive $1,100 at time $0.5$, $1,331 at time $1.5$, and $1,610.51 at time $2.5$. The effective annual rate is 21%, and fractional-year accumulation is defined by $(1.21)^t$. Find the present value at time zero. The payments are hypothetical cash flows.

Method. Pair each payment with its own elapsed time. Since $1.21=1.1^2$, the accumulation factors at times $0.5$, $1.5$, and $2.5$ are $1.1$, $1.331$, and $1.61051$, respectively.

The present value is

\begin{aligned} \operatorname{PV}_0 &=\frac{1{,}100}{(1.21)^{0.5}} +\frac{1{,}331}{(1.21)^{1.5}} +\frac{1{,}610.51}{(1.21)^{2.5}}\\ &=\frac{1{,}100}{1.1} +\frac{1{,}331}{1.331} +\frac{1{,}610.51}{1.61051}\\ &=1{,}000+1{,}000+1{,}000\\ &=\$3{,}000.00. \end{aligned}

Check. Accumulating each $1,000 present value to its corresponding payment date gives $1,100, $1,331, and $1,610.51, exactly the stated payments. The common mistake this method avoids is assigning factors by payment number instead of actual time.

Why the Date-Specific Formula Holds

The formula follows directly from the equivalence of values at different dates. Under the stated model, an amount $P$ at time zero accumulates to $P(1+i)^t$ at time $t$. Therefore, a payment $C$ at time $t$ has the time-zero equivalent obtained by solving $P(1+i)^t=C$ for $P$.

Start with one cash flow.
Let $P$ be its equivalent value at time zero and let its amount at time $t$ be $C$.

Use the accumulation rule.
Equivalence requires $P(1+i)^t=C$.

Solve for present value.
Dividing by the accumulation factor gives $P=C/(1+i)^t=Cv^t$.

Add the separate equivalents.
By additivity of value at the same date, the time-zero value of several cash flows is the sum of their individual time-zero values.

This justification also shows why a regular-annuity factor should not be applied automatically. Such a factor encodes a particular schedule of dates and payments. When the dates differ from that schedule, the factor no longer represents the required set of discount factors. The sum of date-specific present values remains valid.

Solving for an Unknown Payment

A present-value equation can be used to find a missing cash flow. The equation must use the actual date of the unknown payment, just as it does for every known payment. Solve the equation before rounding, then round the payment to the requested monetary precision.

Worked Example: Finding a Payment from a Target Present Value

A proposed stream pays $550 at time $0.5$, $665.50 at time $1.5$, and an unknown amount $X$ at time $2$. The effective annual rate is 21%, with fractional-year accumulation $(1.21)^t$. The stream must have a present value of $1,500 at time zero. Find $X$.

Method. Discount the two known payments using their own dates, then set the full present value equal to the target.

\frac{550}{(1.21)^{0.5}} +\frac{665.50}{(1.21)^{1.5}} +\frac{X}{(1.21)^2} =1{,}500.

Because $(1.21)^{0.5}=1.1$, $(1.21)^{1.5}=1.331$, and $(1.21)^2=1.4641$, the equation becomes

\frac{550}{1.1}+\frac{665.50}{1.331}+\frac{X}{1.4641} =500+500+\frac{X}{1.4641}=1{,}500.

Subtract the known present values:

\frac{X}{1.4641}=500, \qquad X=500(1.4641)=\$732.05.

Check. The unknown payment has present value $732.05/1.4641=500$, so the total is $500+500+500=\$1,500.00$. Treating time $2$ as the third payment period would use the wrong exponent and would not satisfy the stated accumulation model.

Valuing a Stream at a Date Other Than Time Zero

A stream can be valued at any focal date $s$, not only at time zero. Under the same constant-rate and fractional-year assumptions, a payment at time $t_k$ is discounted if $t_k>s$, and accumulated if $t_k<s$. In either case, its value at $s$ is found using the signed time difference $t_k-s$.

Result: At focal date $s$, the value of cash flows $C_k$ occurring at times $t_k$ is $$ \operatorname{V}_s=\sum_{k=1}^{n} C_k(1+i)^{-(t_k-s)}. $$ For a future payment, $t_k-s$ is positive and the payment is discounted. For a past payment, $t_k-s$ is negative, so the factor accumulates that payment forward to $s$.

Worked Example: Moving the Valuation Date

A stream pays $110 at time $1$, $133.10 at time $3$, and $146.41 at time $4$. The effective annual rate is 10%, with fractional-year accumulation $(1.10)^t$. Find the value at time $2$, and check it by moving the result to time zero.

Method. The first payment has already occurred by time $2$, so accumulate it one year. Discount the other two payments one and two years, respectively.

\begin{aligned} \operatorname{V}_2 &=110(1.10)^{-(1-2)} +133.10(1.10)^{-(3-2)} +146.41(1.10)^{-(4-2)}\\ &=110(1.10)+\frac{133.10}{1.10} +\frac{146.41}{(1.10)^2}\\ &=121+121+121\\ &=\$363.00. \end{aligned}

Check at time zero. Discount the time-2 value for two years: $363/(1.10)^2=363/1.21=300$. Directly, the three original payments each have present value $100: $110/1.10=100$, $133.10/1.331=100$, and $146.41/1.4641=100$. Their total is also $300.00, confirming the value-date calculation.

The trap is to discount every cash flow from time zero even when the requested value is at time $2$. At a different focal date, use the time difference from that date; a past cash flow must be accumulated forward.

Common Mistakes and When Annuity Factors Apply

A quick timeline is often more useful than trying to force irregular payments into a familiar formula. Before calculating, confirm that dates are measured from the valuation date and that the interest-rate convention covers fractional periods. Then attach the corresponding factor to each cash flow.

Using payment number as time. The third cash flow is not necessarily three years from the valuation date. Use its stated date.
Ignoring fractional-year conventions. If a payment occurs at $0.5$ years, do not silently assume one full annual period or a particular fractional accumulation rule.
Using one discount factor for the whole stream. Each amount generally has its own elapsed time and therefore its own factor.
Mixing valuation dates. Add cash flows only after expressing them at the same focal date.
Applying an annuity factor outside its schedule. Standard annuity factors require their specified payment timing and spacing. For unequal dates, use the individual terms unless the stream can first be represented exactly by a matching regular schedule.

Common Exam Trap: Replacing Dates With Period Numbers

For the first worked example, the payments occur at times $0.5$, $1.5$, and $2.5$, with effective annual rate 21% and fractional-year accumulation $(1.21)^t$. The correct present value is $3,000.00. A tempting but incorrect calculation treats the payments as if they occur at times 1, 2, and 3:

\frac{1{,}100}{1.21} +\frac{1{,}331}{(1.21)^2} +\frac{1{,}610.51}{(1.21)^3} =909.09+909.09+909.09 =\$2{,}727.27.

The calculation is wrong because the actual times are half-years later than the assumed times. The correct factors are $(1.21)^{0.5}=1.1$, $(1.21)^{1.5}=1.331$, and $(1.21)^{2.5}=1.61051$; dividing the payments by those factors gives $1,000 each and a total of $3,000.00. The exam habit to build is simple: write the date beside every payment before choosing a factor.

Key takeaway: An irregular varying annuity is valued by matching each payment to its actual date, discounting or accumulating it to one chosen focal date, and then adding the resulting values. Under the stated constant-rate model, the exponent is the elapsed time—not the cash flow’s position in the list.

Check Your Understanding

For each problem, use the stated fractional-year convention and value all cash flows at the requested date.

A stream pays $220 at time $1$, $266.20 at time $3$, and $292.82 at time $4$. The effective annual rate is 10%, with accumulation $(1.10)^t$. Find the present value at time zero. (Answer: $600.00)
A stream pays $330 at time $0.5$, $532.40 at time $1.5$, and $585.64 at time $2$. The effective annual rate is 21%, with accumulation $(1.21)^t$. Find the present value. (Answer: $1,100.00)
At an effective annual rate of 10%, with accumulation $(1.10)^t$, a stream pays $242 at time $2$ and $X$ at time $4$. Its present value is $500. Find $X$, to the nearest cent. (Answer: $439.23)
A stream pays $121 at time $1$, $266.20 at time $3$, and $292.82 at time $4$. The effective annual rate is 10%, with accumulation $(1.10)^t$. Find its value at time $2$. (Answer: $617.10)
A stream pays $110 at time $0.5$, $146.41 at time $2$, and $177.1561 at time $3$. The effective annual rate is 21%, with accumulation $(1.21)^t$. Find the present value, rounding money to the nearest cent. (Answer: $300.00)

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