Valuing Cash Flows With Unequal Time Gaps

Why Unequal Time Gaps Need Care

A value equation compares cash flows at a common focal date. When payments occur at regular intervals, their dates may be easy to track by counting periods. With unequal gaps, that shortcut can misstate the elapsed time for one or more payments. The reliable approach is to record each cash flow’s date and move it to the focal date using the actual time difference.

This tutorial uses compound interest with an effective annual rate and allows fractional years under the fractional-period convention: a cash flow separated from another date by $t$ years accumulates by the factor $(1+i)^t$. The rate is assumed constant over all dates considered. If the problem specifies a different compounding or fractional-period convention, use that convention instead; the formulas below do not automatically apply to it.

Definition: Let $C_j$ be the signed amount of a cash flow at time $t_j$, measured in years from a chosen time origin. An inflow is positive and an outflow is negative. Under compound interest at effective annual rate $i$, the value of that cash flow at focal time $T$ is $C_j(1+i)^{T-t_j}$. The value of a schedule at $T$ is the sum of the values of all its cash flows. Two schedules are equivalent at $T$ when their values there are equal.

The exponent $T-t_j$ keeps the direction of the calculation explicit. If the cash flow occurs before the focal date, $T-t_j$ is positive and the amount accumulates. If it occurs after the focal date, $T-t_j$ is negative and the amount is discounted. A cash flow at the focal date has exponent zero and is unchanged.

Building the Equation From the Timeline

Before calculating, mark every cash-flow date and the focal date. Then calculate a separate elapsed-time difference for each cash flow. Do not assume the gaps between listed payments are equal, and do not use the number of payments as an exponent.

Choose a focal date.
Use the date named in the problem or choose a convenient date at which to compare the amounts.

Record signed cash flows and dates.
Use positive signs for amounts received and negative signs for amounts paid, or place the two sides of an equivalence equation separately.

Move each amount using its own time difference.
At effective annual rate $i$, a cash flow $C_j$ at $t_j$ has focal-date value $C_j(1+i)^{T-t_j}$.

Add values or set equivalent sides equal.
Keep the dates visible until every exponent has been formed; then solve the resulting equation.

The formula follows directly from compound accumulation. Over one year, an amount $C$ becomes $C(1+i)$; over $n$ years, repeated accumulation gives $C(1+i)^n$. Under the stated fractional-period convention, the same rule applies to an elapsed time $n$ that is not an integer. Thus moving $C_j$ from $t_j$ to $T$ gives $C_j(1+i)^{T-t_j}$. When $t_j>T$, the exponent is negative, so the factor is $1/(1+i)^{t_j-T}$, the corresponding discount factor.

V_T=\sum_j C_j(1+i)^{T-t_j}

Here $V_T$ is the net value at time $T$. To express equivalence between two schedules, either add their signed cash flows into the same sum and set $V_T=0$, or calculate the value of each schedule separately and set those values equal. The two forms are algebraically identical.

Worked Examples

Worked Example: Accumulating a Schedule With Unequal Gaps

A fund receives $400 at time 0 and $700 at time 0.5 years, then pays $300 at time 2.5 years. Find the fund’s net value at time 3 years if the effective annual interest rate is 21%, using compound interest with fractional-year exponents.

Method. Move each signed cash flow to time 3 separately. The gaps between the cash-flow dates are 0.5, 2, and 0.5 years, so there is no single payment interval to use for every amount.

The $400 receipt is 3 years before the focal date. The $700 receipt is $3-0.5=2.5$ years before it. The $300 payment is $3-2.5=0.5$ years before it and is negative:

V_3=400(1.21)^3+700(1.21)^{2.5}-300(1.21)^{0.5}.

The factors are $(1.21)^3=1.771561$, $(1.21)^{2.5}=(1.21)^2(1.21)^{0.5}=1.4641(1.1)=1.61051$, and $(1.21)^{0.5}=1.1$. Therefore the three focal-date values are $400(1.771561)=708.6244$, $700(1.61051)=1{,}127.357$, and $-300(1.1)=-330$.

Adding them gives $V_3=708.6244+1{,}127.357-330=1{,}505.9814$. Rounded to the nearest cent, the net value is $1,505.98.

Check. The payment reduces the total by $330 at time 3, which is greater in magnitude than its original $300 because it is accumulated for half a year. The two receipts accumulate over their respective, longer periods. A common mistake would be to assign both receipts the same exponent because there are two receipts; their actual elapsed times are 3 and 2.5 years.

Worked Example: Solving for a Payment After the Focal Date

A schedule has a payment of $1,000 at time 1 year, a receipt of $900 at time 1.5 years, and an unknown receipt $X$ at time 2.5 years. Find $X$ so the schedule has value zero at time 1. Assume an effective annual rate of 21% and compound interest with fractional-year exponents.

Method. Set the focal-date value of the positive receipts equal to the value of the payment. The $900 receipt is 0.5 years after time 1, while $X$ is 1.5 years after it. Discount each receipt using its own gap:

1000=\frac{900}{(1.21)^{0.5}}+\frac{X}{(1.21)^{1.5}}.

Since $(1.21)^{0.5}=1.1$, the first receipt has value $900/1.1=818.181818\ldots$ at time 1. Also, $(1.21)^{1.5}=1.21(1.1)=1.331$. The equation becomes

1000=818.181818\ldots+\frac{X}{1.331}.

The amount still needed at time 1 is $1000-818.181818\ldots=181.818181\ldots$. Accumulating that amount forward 1.5 years gives

X=181.818181\ldots(1.331)=242.

Thus $X=$$242.00. As a check, $242/1.331=181.818181\ldots$, so the two discounted receipts total $818.181818\ldots+181.818181\ldots=1{,}000$ at the focal date.

Common mistake avoided. Discounting both receipts by the same number of years would ignore their different dates. The correct exponents are $-0.5$ and $-1.5$, not a common exponent based on the number of receipts.

Worked Example: Net Value With Cash Flows on Both Sides of the Focal Date

A contract has a $1,000 receipt at time 0, a $300 payment at time 1, and an $800 receipt at time 4. Find its net value at time 2 if the effective annual interest rate is 10%. Use compound interest with integer-year exponents.

Method. Accumulate the first receipt for 2 years, accumulate the payment for 1 year with a negative sign, and discount the later receipt for 2 years. The respective exponents are $2$, $1$, and $-2$.

V_2=1000(1.1)^2-300(1.1)+800(1.1)^{-2}.

The first value is $1000(1.21)=1{,}210$. The payment contributes $-300(1.1)=-330$. The later receipt contributes $800/1.21=661.15702479\ldots$. Therefore,

V_2=1210-330+661.15702479\ldots=1541.15702479\ldots.

The net value at time 2 is $1,541.16, rounded to the nearest cent. As a check, the positive values total $1{,}210+661.15702479\ldots=1{,}871.15702479\ldots$; subtracting the $330 focal-date value of the payment gives the same net result.

Common mistake avoided. Treating the time-4 receipt as though it occurred at time 2 would leave it undiscounted. It is two years after the focal date, so its value at time 2 must be divided by $(1.1)^2$.

Unequal Gaps and the Interest Model

The actual elapsed time matters even when the listed dates look like a sequence. For example, cash flows at times 0, 1.5, and 4 have gaps of 1.5 and 2.5 years. The middle cash flow is not automatically one “period” from the first or the last. Its exponent comes from its distance to the chosen focal date.

The equation in this tutorial relies on compound interest and the stated fractional-period convention. Under simple interest, the accumulation rule is different: an amount carried for $t$ years uses the simple-interest factor $1+it$, rather than $(1+i)^t$. Do not mix those models in a single value equation. Likewise, if a problem supplies rates that vary by interval, each segment must use the rate applicable to that segment; a single constant-rate power is then not sufficient.

Common Exam Traps

Counting cash flows instead of measuring time. The exponent is the elapsed time between a cash flow and the focal date, not its position in a list.
Using the gap between adjacent payments for every calculation. Each amount is moved directly to the focal date, so its exponent uses its own date.
Discounting an earlier cash flow. A cash flow before the focal date accumulates forward; one after the focal date is discounted back.
Dropping the sign of an outflow. Either subtract its focal-date value or assign it a negative cash-flow amount before summing.
Applying a compound-interest formula under a different model. Confirm the rate and fractional-period convention before forming the equation.

Key takeaway: For a constant effective annual rate $i$ under compound interest, move every signed cash flow $C_j$ at time $t_j$ to focal time $T$ as $C_j(1+i)^{T-t_j}$. Unequal gaps cause no difficulty once each actual time difference is used.

Check Your Understanding

For each problem, use the interest model and focal date stated. Keep the elapsed time for each cash flow explicit.

A fund receives $500 at time 0 and $300 at time 2. Find its value at time 4 at an effective annual rate of 10% under compound interest. (Answer: $1,095.05)
At an effective annual rate of 21%, a schedule receives $250 at time 0.5 and $600 at time 2. Find its value at time 2.5, using fractional-year compound interest. (Answer: $1,026.03)
At an effective annual rate of 10%, a schedule receives $500 at time 1 and an unknown amount $X$ at time 3, and pays $800 at time 2. Find $X$ so the value at time 2 is zero. (Answer: $275.00)
At an effective annual rate of 10%, find the net value at time 3 of a $900 receipt at time 0, a $400 payment at time 2, and a $500 receipt at time 5. (Answer: $1,171.12)
At an effective annual rate of 10%, a schedule receives $600 at time 0 and $400 at time 1.5, then pays $X$ at time 4. Find $X$ so the value at time 2 is zero, using fractional-year compound interest. (Answer: $1,386.08)

What You'll Learn