Cash Flows at Unequal Time Intervals

Why Unequal Time Intervals Need Careful Exponents

A cash-flow stream may contain payments at dates that are not evenly spaced. For example, a fund might receive a contribution after six months, make a distribution one year later, and receive another payment after a further two years. The amounts can be added or compared only after they have been expressed at the same date. The key is to match each exponent to the elapsed time between that cash flow and the valuation date, not to the payment number or the number of gaps in the schedule.

This tutorial assumes one constant interest-rate model over the dates being valued. It focuses on the timing and rate-convention decisions needed when cash flows occur at nonuniform dates. If a contract specifies a day-count rule, compounding rule, or other timing convention, use that rule rather than imposing a different one.

Definition: Let $C_k$ be a signed cash flow at time $t_k$, where time is measured from a stated origin. A positive $C_k$ is a receipt to the party whose value is being calculated; a negative $C_k$ is a payment. A cash-flow stream has unequal time intervals when the differences between consecutive payment times are not all the same. A valuation date is the date to which the cash flows are moved for comparison.

List the actual times first. If payments occur at $t=0.5$, $t=1.25$, and $t=2.5$ years, their successive intervals are $0.75$ years and $1.25$ years. They are not three equally spaced annual payments. A timeline makes this visible and helps prevent the common mistake of assigning exponents by payment order.

Use the Rate Convention to Determine the Exponent

Under a constant effective annual interest rate $i$, the accumulation factor for $t$ years is $(1+i)^t$. For whole-year times, this is ordinary annual compounding. For fractional-year times, using the same formula makes an additional timing assumption: fractional periods follow the power-law extension of the annual accumulation model. In particular, accumulation over $t$ years is $(1+i)^t$ even when $t$ is not an integer. Use this convention when it is specified or when the problem adopts the standard effective-annual-rate model. If the problem instead specifies an actual day-count or a different within-year rule, follow that rule.

A nominal annual rate $i^{(m)}$, convertible $m$ times per year, has periodic rate $i^{(m)}/m$. If payment dates lie on that periodic grid, a cash flow $n$ periods from another date accumulates by $(1+i^{(m)}/m)^n$. The elapsed time $t$, in years, corresponds to $mt$ periods when $mt$ is an integer. For dates between the stated conversion periods, a nominal rate alone does not specify how to handle a fractional period unless the model or contract supplies that rule.

Key rule: Under a constant effective annual rate, a cash flow at $t_k$ has present value at time $0$ of $C_k(1+i)^{-t_k}$. Under a nominal rate convertible $m$ times per year, use the periodic rate $i^{(m)}/m$ and count the actual number of conversion periods between the cash-flow date and valuation date. Do not substitute one convention for the other without converting the rate or establishing an equivalent model.

Moving Unevenly Dated Cash Flows to One Date

A cash flow after the valuation date is discounted to that date; a cash flow before it is accumulated forward. Both operations can be written with one signed exponent. With a constant effective annual rate $i$, the value at date $T$ is

V_T=\sum_k C_k(1+i)^{T-t_k}.

If $T-t_k$ is positive, the cash flow is accumulated forward. If it is negative, the cash flow is discounted back. For nominal interest convertible $m$ times per year, when all relevant dates are on the conversion grid, the corresponding expression is

V_T=\sum_k C_k\left(1+\frac{i^{(m)}}{m}\right)^{m(T-t_k)}.

The second expression requires $m(T-t_k)$ to be an integer under the stated periodic model. For each term, its exponent is the number of periods between that cash flow and the valuation date, with a positive sign for forward accumulation and a negative sign for discounting.

Why the Common-Date Formula Works

The formula follows from the accumulation rule and the equivalence of values at different dates under that rule. Under an effective annual rate $i$, one unit at time $t_k$ becomes $(1+i)^{T-t_k}$ units at time $T$ when $T$ is later. If $T$ is earlier, the same expression has a negative exponent and therefore divides by the accumulation factor. Multiplying by $C_k$ gives the value at $T$ of that individual cash flow. Adding those values is valid because they are now all measured at the same date.

This also explains why unequal intervals cause no special extra operation: each cash flow is moved directly by its own elapsed time to the chosen date. Moving it in stages gives the same result under the constant-rate model, since

(1+i)^{T-s}(1+i)^{s-t_k}=(1+i)^{T-t_k}.

The intermediate date $s$ cancels from the total exponent. This property depends on using the same accumulation model throughout the interval. If the applicable rate or accumulation rule changes with time, a single constant-rate power is not generally adequate.

Write each date.
Measure every payment time and the valuation date from the same origin, in years or in stated conversion periods.

Identify the rate model.
Distinguish an effective annual rate from a nominal rate convertible at a stated frequency. Note any fractional-period or contractual timing condition.

Calculate each elapsed time.
Subtract the cash-flow date from the valuation date. Do not use the payment number or a presumed regular interval.

Move and sum.
Apply the correct accumulation or discount factor to each signed cash flow, then add the resulting values at the common date.

Worked Applications

Worked Example: Present Value with Unequal Annual Gaps

A hypothetical project pays $ \$1{,}000$ at the end of year 1, $ \$1{,}600$ at the end of year 3, and $ \$2{,}000$ at the end of year 5. Find the value at time 0 using an effective annual rate of $8\%$. Assume fractional years, if any, follow the power-law effective annual model. The payment gaps are two years and two years here, but the payment dates are nonconsecutive; the method applies equally when the gaps differ.

Method. Discount each receipt from its actual date to time 0. Here $i=0.08$, and the elapsed times are 1, 3, and 5 years:

V_0=\frac{1{,}000}{1.08^1}+\frac{1{,}600}{1.08^3}+\frac{2{,}000}{1.08^5}.

The accumulation factors are $1.08^1=1.08$, $1.08^3=1.259712$, and $1.08^5=1.4693280768$. The corresponding present values are

\frac{1{,}000}{1.08}=925.9259,\qquad \frac{1{,}600}{1.259712}=1{,}270.1316,\qquad \frac{2{,}000}{1.4693280768}=1{,}361.1664.

Adding the unrounded values gives $V_0=3{,}557.2239$. Rounded to the nearest cent, the present value is $\$3{,}557.22$. As a check, each future receipt has been discounted, so each present value is below its corresponding payment, as expected for a positive interest rate. The mistake to avoid is discounting the year-3 payment by two periods merely because it is the second payment; its date is three years from time 0.

Worked Example: Accumulation Under a Nominal Quarterly Rate

A hypothetical account receives $ \$500$ at $t=0.5$ years, $ \$900$ at $t=1.25$ years, and $ \$1{,}200$ at $t=2.5$ years. Find the accumulated value at $t=2.5$ under a nominal annual rate of $8\%$ convertible quarterly. Assume all dates are on the quarterly conversion grid.

Method. The quarterly rate is $0.08/4=0.02$, or $2\%$. The times correspond to 2, 5, and 10 quarters, respectively; the valuation date is at quarter 10. Thus the cash flows accumulate for $10-2=8$, $10-5=5$, and $10-10=0$ quarters.

V_{2.5}=500(1.02)^8+900(1.02)^5+1{,}200(1.02)^0.

The factors are $(1.02)^8=1.1716593810$, $(1.02)^5=1.1040808032$, and $(1.02)^0=1$. The accumulated amounts are $500(1.1716593810)=585.8297$, $900(1.1040808032)=993.6727$, and $1{,}200(1)=1{,}200.0000$. Their sum is $2{,}779.5024$, so the value at $t=2.5$, rounded to the nearest cent, is $\$2{,}779.50$.

The final cash flow is already at the valuation date and is not accumulated. The unequal intervals are $0.75$ years and $1.25$ years, but the exponents come from the total numbers of quarterly periods to the focal date. A common error is to use an annual exponent such as $2.5-0.5=2$ with the nominal rate $8\%$; the specified model instead requires the quarterly rate and eight quarterly periods.

Worked Example: Avoiding an Effective-Rate Substitution

A hypothetical investor deposits $ \$1{,}000$ at $t=0.5$ years and $ \$1{,}500$ at $t=2$ years. Find the value at $t=3$ under a nominal annual rate of $12\%$ convertible quarterly. All dates lie on the quarterly grid.

Method. The periodic rate is $0.12/4=0.03$, or $3\%$ per quarter. The deposits occur at quarters 2 and 8, and the valuation date is quarter 12. They therefore accumulate for 10 and 4 quarters.

V_3=1{,}000(1.03)^{10}+1{,}500(1.03)^4.

The factors are $(1.03)^{10}=1.3439163793$ and $(1.03)^4=1.12550881$. The accumulated amounts are $1{,}343.9164$ and $1{,}688.2632$. Their sum is $3{,}032.1796$, giving $\$3{,}032.18$ to the nearest cent.

The equivalent effective annual rate for this same periodic model is $(1.03)^4-1=0.12550881$, approximately $12.5509\%$, not $12\%$. Using $12\%$ as though it were an effective annual rate would instead give $1{,}000(1.12)^{2.5}+1{,}500(1.12)$, approximately $\$3{,}007.53$, which is a different model. This illustrates the trap: the stated nominal rate is not itself the annual effective rate.

Common Timing and Rate-Convention Traps

Uneven-date calculations are usually short once the timeline and rate model are clear. Errors tend to occur before the arithmetic, when a candidate interprets a time or rate inconsistently. Check these points before evaluating the sum:

Use elapsed time, not payment order. A payment at $t=3.5$ is 3.5 years from time 0, even if it is the second payment in the stream.
Do not assume regular spacing. Calculate each date difference directly. A gap between payments is not automatically one year or one conversion period.
Keep rate and exponent units together. With nominal quarterly interest, use the quarterly rate and count quarters. With an effective annual rate, use year-based exponents under the stated fractional-year model.
Respect the valuation date. A cash flow later than the focal date is discounted; one earlier than the focal date is accumulated. A cash flow on the focal date has exponent zero.
Check what the rate information guarantees. A periodic nominal rate specifies growth over full conversion periods. For an off-grid date, do not invent a fractional-period rule when the contract or problem has not supplied one.

A useful final check is dimensional and directional: the exponent must represent elapsed periods under the selected rate convention, and positive cash flows moved forward at a positive rate should grow. Cash flows moved backward should shrink. These checks will not replace the calculation, but they often expose a misplaced date or an inconsistent rate before it affects the result.

Check Your Understanding

For each problem, identify the timing convention before calculating. Round monetary answers to the nearest cent.

A fund receives $ \$400$ at $t=1$ and $ \$900$ at $t=4$. Find its present value at $t=0$ using an effective annual rate of $5\%$. Assume the effective annual model applies at all stated dates. (Answer: $\$1{,}121.38$)
An account receives $ \$700$ at $t=0.5$, $ \$500$ at $t=1.75$, and $ \$1{,}000$ at $t=3$. Find the value at $t=3$ using a nominal annual rate of $4\%$ convertible quarterly, with all dates on the quarterly grid. (Answer: $\$2{,}298.74$)
A project pays $ \$300$ at $t=0.5$ and $ \$700$ at $t=2$. Find its value at $t=3$ under an effective annual rate of $9\%$, assuming fractional-year accumulation follows the power-law model. (Answer: $\$1{,}135.12$)
A savings plan deposits $ \$500$ at $t=0.25$ and $ \$800$ at $t=1.5$. Find the accumulated value at $t=2.5$ under a nominal annual rate of $6\%$ convertible quarterly. All dates are on the quarterly grid. (Answer: $\$1{,}420.79$)

What You'll Learn