Comparing Payment Alternatives at a Common Date

Why Compare at a Common Date?

Two payment schedules can have different amounts and dates, so adding their stated dollar amounts does not necessarily compare their economic values. A dollar received earlier can earn interest before a later comparison date; a dollar due later must be discounted if the comparison date comes first. The practical solution is to value every cash flow in every alternative at one common date, using the same rate assumptions.

This method applies to hypothetical choices such as alternative vendor payment terms, benefit schedules, or contractual receipts. It answers a precise question: which schedule has the greater value at the selected date? The answer depends on the cash flows, the valuation date, and the interest-rate assumption. The method does not by itself account for uncertainty, taxes, fees not included in the cash flows, or differences in risk.

Definition: A common-date value is the value, at one selected date, of all cash flows in a payment alternative. Under a constant effective annual rate $i$, a cash flow $C_j$ at time $t_j$ has value at time $T$ equal to $C_j(1+i)^{T-t_j}$. Use a positive sign for a receipt and a negative sign for a payment. The value of the full schedule is the sum of its cash flows valued at $T$.

The Common-Date Calculation

First identify each cash flow, its date, and its direction. Then select a valuation date that is convenient for comparing the alternatives. A payment before that date is accumulated forward; a payment after it is discounted back. If a cash flow occurs exactly on the common date, its value is unchanged. For annual time measurements and a constant effective annual rate $i$, the calculation is:

V_T=\sum_{j=1}^{n} C_j(1+i)^{T-t_j}

Here, $C_j$ is the signed cash flow at time $t_j$, and $V_T$ is the schedule’s net value at time $T$. A positive exponent accumulates a cash flow forward, while a negative exponent discounts it back. The same rate convention must be used for every alternative. In the examples below, cash flows occur at whole-year dates and the stated rate is effective annually.

For schedules consisting only of costs, it is also possible to compare positive cost amounts at the common date. In that case, the lower common-date cost is preferable. If a schedule includes both receipts and payments, retain their signs and compare the resulting net values. A larger net value is preferable when the alternatives are being evaluated as net receipts.

List each dated cash flow.
Record the amount, date, and whether it is a receipt or payment. Do not combine amounts from different dates before valuing them.

Set the comparison date and rate.
Use the same common date and the same interest-rate assumptions for every alternative being compared.

Move each amount to that date.
Accumulate cash flows before the common date and discount cash flows after it.

Sum and interpret.
Add signed values, compare the alternatives, and state whether a higher or lower value is preferable for the decision being considered.

Why the Method Works

The common-date formula follows from the accumulation rule for a constant effective annual rate. One dollar invested for one year grows to $1+i$ dollars; invested for $m$ years, it grows to $(1+i)^m$ dollars. Therefore, a cash flow at $t_j$ is multiplied by $(1+i)^{T-t_j}$ to express its equivalent value at $T$. If $t_j>T$, the exponent is negative, which gives the corresponding discount factor.

The method also explains why, under one constant rate, changing the common date does not change the ranking. For a schedule with present value $V_0$, its value at time $T$ is:

V_T=\sum_j C_j(1+i)^{T-t_j} =(1+i)^T\sum_j C_j(1+i)^{-t_j} =(1+i)^T V_0

Because $(1+i)^T$ is positive, multiplying every alternative’s value by that same factor preserves the ordering. This result assumes the same constant rate and compatible timing conventions for all alternatives. If rate assumptions differ by date or alternative, one common factor cannot generally be taken outside every schedule; use the applicable rate assumptions consistently instead.

Worked Example: Comparing Two Receipt Schedules

A hypothetical contract offers either Schedule A, which pays $1,000 at the end of year 1 and $600 at the end of year 3, or Schedule B, which pays $700 at the end of year 2 and $1,000 at the end of year 3. Compare the schedules at the end of year 3. Assume certain payments, no fees or taxes, and a constant effective annual interest rate of $i=8\%$. Report final dollar values to the nearest cent.

Method. Accumulate every receipt to year 3. All amounts are positive receipts, so the higher common-date value is preferable.

For Schedule A, the year-1 receipt accumulates for two years, the year-3 receipt needs no adjustment, and the common-date value is:

V_{3,A}=1{,}000(1.08)^{3-1}+600(1.08)^{3-3} =1{,}000(1.1664)+600(1) =1{,}166.40+600.00 =\$1{,}766.40

For Schedule B, the year-2 receipt accumulates for one year and the year-3 receipt remains unchanged:

V_{3,B}=700(1.08)^{3-2}+1{,}000(1.08)^{3-3} =700(1.08)+1{,}000(1) =756.00+1{,}000.00 =\$1{,}756.00

Schedule A has the greater value at year 3 by $\$1{,}766.40-\$1{,}756.00=\$10.40$. As a check, the unadjusted totals are $1,600 for A and $1,700 for B, but those totals do not account for when the receipts occur. The common-date calculation gives a difference of $10.40 in favor of A. The common mistake avoided is choosing B just because its nominal total is larger.

Worked Example: Comparing Payment Costs

A hypothetical buyer can pay $2,500 immediately and another $2,200 at the end of year 1 (Option A), or pay $4,800 at the end of year 2 (Option B). Compare the costs at the end of year 2. Assume the payments are certain, there are no additional charges, and the effective annual interest rate is $i=10\%$. Report final values to the nearest cent.

Method. Accumulate each cost to year 2 and compare the resulting positive cost amounts. The lower common-date cost is preferable.

Option A’s immediate cost accumulates for two years. Its year-1 cost accumulates for one year:

V_{2,A}=2{,}500(1.10)^2+2{,}200(1.10)^1 =2{,}500(1.21)+2{,}200(1.10) =3{,}025+2{,}420 =\$5{,}445.00

Option B is paid at the common date, so it is not adjusted:

V_{2,B}=4{,}800(1.10)^0=\$4{,}800.00

Option B costs $\$5{,}445-\$4{,}800=\$645.00$ less at year 2, so it is the less costly option under the stated assumptions. The nominal totals are $4,700 for A and $4,800 for B, which might suggest A is cheaper. That comparison ignores the time value of the earlier payments. As a check using year 0 instead, A costs $2{,}500+2{,}200/1.10=\$4{,}500.00$, while B costs $4{,}800/1.21=\$3{,}966.94$ to the nearest cent. The year-0 difference is about $533.06; accumulated to year 2, it is $645.00 before rounding. The common mistake avoided is ranking payment schedules by nominal totals rather than by values at a common date.

Worked Example: Comparing Schedules With Receipts and Payments

A hypothetical service agreement has two alternatives, each evaluated through the end of year 3. Under A, the customer receives $1,000 immediately, receives $500 at the end of year 2, and pays a $200 fee at the end of year 1. Under B, the customer pays $100 immediately, receives $900 at the end of year 1, and receives $600 at the end of year 3. Compare net values at year 3. Assume all cash flows are certain, no other charges apply, and the effective annual rate is $i=10\%$. Report final values to the nearest cent.

Method. Assign positive signs to receipts and negative signs to payments, move every amount to year 3, then compare net values. Here the alternative with the larger net value is preferable.

For A, the immediate receipt accumulates for three years, the year-2 receipt for one year, and the year-1 fee for two years:

V_{3,A}=1{,}000(1.10)^3+500(1.10)^1-200(1.10)^2

The individual values are $1{,}000(1.331)=\$1{,}331.00$, $500(1.10)=\$550.00$, and $200(1.21)=\$242.00$. Applying the negative sign to the fee gives:

V_{3,A}=1{,}331+550-242=\$1{,}639.00

For B, the immediate payment accumulates for three years, the year-1 receipt for two years, and the year-3 receipt is already at the common date:

V_{3,B}=-100(1.10)^3+900(1.10)^2+600(1.10)^0 =-133.10+1{,}089.00+600.00 =\$1{,}555.90

Alternative A has the higher net value by $\$1{,}639.00-\$1{,}555.90=\$83.10$. The check is that the year-3 components sum to $1,639.00 for A and $1,555.90 for B, and their difference is $83.10. The common mistake avoided is treating every amount as a receipt or comparing only the schedules’ positive amounts while omitting fees.

Common Traps and Interpretation

The arithmetic is usually straightforward; the main difficulty is keeping dates, signs, and the decision criterion consistent. Before calculating, state the comparison date and whether a larger or smaller value is better. This avoids a correct calculation followed by the wrong recommendation.

Comparing nominal totals. Amounts at different dates are not directly comparable without a rate assumption. Move all amounts to the same date first.
Using the wrong direction. A cash flow before the common date is accumulated; a cash flow after it is discounted. The exponent $T-t_j$ makes the direction explicit.
Reversing the preferred result. For net receipts, a higher value is preferable. For costs only, a lower cost value is preferable.
Dropping payment signs. In a schedule with both receipts and payments, use positive signs for receipts and negative signs for payments before summing.
Changing assumptions between alternatives. The comparison is meaningful only when both schedules are evaluated using the same stated date and rate assumptions.

A common-date comparison is a valuation technique, not a guarantee about which alternative will ultimately be best. If the payments are uncertain, or if important taxes, fees, or risks have not been included, the result is conditional on the simplified cash flows and assumptions in the calculation.

Key takeaway: Compare different payment schedules by expressing every dated cash flow at the same valuation date under the same interest-rate assumptions. Accumulate earlier amounts, discount later amounts, preserve receipt and payment signs, and interpret the result according to whether the alternatives represent net benefits or costs.

Check Your Understanding

For each problem, use the stated effective annual rate and compare values at the specified common date. Round final dollar values to the nearest cent.

At $i=5\%$, Schedule X pays $400 at time 0 and $300 at time 1. Schedule Y pays $750 at time 2. Compare the receipt values at time 2. (Answer: X is worth $756.00 and Y is worth $750.00 at time 2; X is higher by $6.00.)
At $i=8\%$, compare two cost schedules at time 3: A costs $1,000 at time 0 and $500 at time 2; B costs $1,800 at time 3. Which has the lower common-date cost, and by how much? (Answer: A costs $1,799.71 and B costs $1,800.00 at time 3; A is lower by $0.29.)
At $i=6\%$, Schedule A receives $200 at time 1 and $500 at time 4. Schedule B receives $400 at time 2 and $300 at time 3. Compare their values at time 4. (Answer: A is worth $738.20 and B is worth $767.44; B is higher by $29.24.)
At $i=10\%$, compare net values at time 2. Schedule C receives $1,000 at time 0 and pays $300 at time 1. Schedule D receives $700 at time 1 and pays $100 at time 0. Which has the higher net value, and by how much? (Answer: C is worth $880.00 and D is worth $649.00 at time 2; C is higher by $231.00.)
At $i=4\%$, compare receipt schedules at time 3: A receives $250 at time 0 and $250 at time 2; B receives $530 at time 3. Which has the higher common-date value, and by how much? (Answer: A is worth $541.22 and B is worth $530.00 at time 3; A is higher by $11.22.)

What You'll Learn