Why Cash-Flow Dates Matter
A cash-flow amount does not determine its present value by itself. Its date matters: a dollar received sooner is discounted for fewer periods and therefore has a larger present value at a positive interest rate. In project calculations, wording such as “at the beginning of each year” or “at the end of each year” specifies the dates, so translating that wording into a timeline is part of the valuation.
The key distinction is that the beginning of year 1 is time 0, while the end of year 1 is time 1. More generally, the beginning of year \(k\) is time \(k-1\), and the end of year \(k\) is time \(k\). The payment amounts may be identical in two schedules, but changing their dates changes NPV.
This formula assumes a consistent rate per period and that each date is correctly expressed in those periods. For example, if \(i\) is an effective annual rate, a cash flow one year from now is discounted by \(v\). Do not use an annual rate as though it were a monthly rate. This tutorial considers integer-period dates; a fractional-period cash flow requires a rate convention appropriate to that fraction.
Putting Beginning and End Dates on a Timeline
A timeline makes the time index visible before any arithmetic begins. The first payment in a beginning-of-year schedule is at time 0, not time 1. A schedule of payments at the end of each year starts at time 1. This one-period difference applies to every payment in the schedule when the entire stream is moved.
| Wording | Time of payment | Present-value factor |
|---|---|---|
| Beginning of year 1 | 0 | 1 |
| End of year 1 | 1 | \(v\) |
| Beginning of year 2 | 1 | \(v\) |
| End of year 2 | 2 | \(v^2\) |
| Beginning of year \(k\) | \(k-1\) | \(v^{k-1}\) |
| End of year \(k\) | \(k\) | \(v^k\) |
Notice that the end of one year and the beginning of the next year are the same instant: the end of year 1 and beginning of year 2 are both time 1. Wording about a period boundary should therefore be converted into a date, not treated as an extra period of waiting.
Worked Example: NPV of Receipts at the Beginning of Each Year
A hypothetical project requires an investment of $1,000 at time 0. It produces receipts of $300, $400, and $500 at the beginning of years 1, 2, and 3, respectively. The effective annual interest rate is 10%. Calculate the NPV. Then quantify the error if the receipts are mistakenly treated as end-of-year payments.
Method. First map each stated date to its time. Beginning of year 1 is time 0, beginning of year 2 is time 1, and beginning of year 3 is time 2. The initial investment is also at time 0.
At \(i=0.10\), \(v=1/1.10=0.9090909\) and \(v^2=1/1.21=0.8264463\). Thus the present value of the receipts is
Subtract the time-zero investment: \(\operatorname{NPV}=1076.8595-1000=76.8595\). The NPV is $76.86, rounded to the nearest cent.
If all three receipts are incorrectly placed at the ends of years 1, 2, and 3, their present value is
That mistaken calculation gives NPV \(978.9632-1000=-21.0368\), or -$21.04. The mistaken NPV is lower by \(76.8595-(-21.0368)=97.8963\), or $97.90. This is a timing error, not a change in the cash-flow amounts.
Common mistake avoided: Discounting the first receipt by one period because it is labelled “year 1.” The beginning of year 1 is time 0, so the $300 receipt is not discounted.
How a One-Period Shift Changes Present Value
Suppose a payment \(C\) is moved from the end of period \(k\) to the beginning of that same period. Its original date is time \(k\); its new date is time \(k-1\). At a positive rate, the new present value is larger because one fewer discount period is applied.
The result follows directly from discounting at the two dates. The earlier value is \(Cv^{k-1}\), and the later value is \(Cv^k\). Their difference is
because \(1/v=1+i\). This derivation also shows why the change is not generally \(iC\): the one-period interest effect applies to the payment’s value at time \(k\), which is \(Cv^k\), not to its undiscounted amount alone.
If every payment in a stream is moved one period earlier, each payment’s present value is multiplied by \(1+i\). This follows because \(Cv^{k-1}= (1+i)Cv^k\). It applies only when the entire stream is shifted by the same one period and there is no separate cash flow—such as an initial investment—that remains fixed at time 0.
Worked Example: Comparing a Beginning-of-Year and End-of-Year Schedule
A hypothetical project costs $1,500 at time 0 and returns $500 at the beginning of each of the next four years. The effective annual rate is 8%. Compare its NPV with the alternative schedule in which the same four receipts occur at the end of years 1 through 4. Calculate the change in NPV when the receipts are moved earlier.
Method. The beginning-of-year receipts occur at times 0, 1, 2, and 3. The end-of-year receipts occur at times 1, 2, 3, and 4. For the calculation, \(v=1/1.08=0.9259259\), \(v^2=0.8573388\), \(v^3=0.7938322\), and \(v^4=0.7350299\).
For the end-of-year schedule, the present value of the receipts is
Its NPV is \(1656.0634-1500=156.0634\), or $156.06.
For the beginning-of-year schedule, the present value of the receipts is
Its NPV is \(1788.5485-1500=288.5485\), or $288.55. The NPV increase from receiving all four receipts one period earlier is \(288.5485-156.0634=132.4851\), or $132.49.
As a check, the entire receipt stream’s present value is multiplied by \(1.08\) when shifted earlier: \(1656.0634(1.08)=1788.5485\). The investment stays at time 0 in both schedules, so it is not multiplied by \(1.08\). The NPV change is therefore the change in receipt value, not 8% of the project’s original NPV.
Common mistake avoided: Applying the one-period shift to the initial $1,500 cost as well as the receipts. The cost date did not change, so its present value remains $1,500.
Shifting a Single Cash Flow
A schedule may contain cash flows at several dates, with only one payment moved. In that case, change the discount factor for that payment alone and leave every other date unchanged. The sign of the cash flow matters: moving a receipt earlier increases NPV, while moving an outflow earlier decreases NPV.
Worked Example: Moving One Receipt One Year Earlier
A hypothetical project requires $1,500 at time 0 and receives $600 at the end of year 1, $700 at the end of year 2, and $800 at the end of year 3. The effective annual rate is 10%. The final $800 receipt is renegotiated to occur at the beginning of year 3 instead. Calculate both NPVs and the change.
Method. In the original schedule, the three receipts occur at times 1, 2, and 3. In the revised schedule, beginning of year 3 is time 2, so only the $800 receipt changes date. At 10%, the relevant factors are \(v=1/1.10\), \(v^2=1/1.21\), and \(v^3=1/1.331\).
The original receipt present value is
The original NPV is \(1725.0187-1500=225.0187\), or $225.02.
In the revised schedule, the $800 is discounted for two periods, not three:
The revised NPV is \(1785.1239-1500=285.1239\), or $285.12. The increase is \(285.1239-225.0187=60.1052\), or $60.11. It also follows from the one-period shift formula: \(0.10(800/1.331)=60.1052\).
Common mistake avoided: Moving the $700 receipt as well as the $800 receipt because both are associated with year 3 in some description. The $700 is explicitly at the end of year 2, which is time 2, and its date is unchanged.
Practical Checks for Timing Calculations
A short timeline is usually more reliable than trying to interpret dates while entering them into a calculator. Before evaluating an NPV, identify the rate period, list each cash flow with its time, and then apply the matching discount factor. A positive NPV or a negative NPV does not confirm that the timing was interpreted correctly; a date check is still needed.
Confirm that the effective rate is expressed per the same period used for the timeline.
Write beginning of year \(k\) as time \(k-1\) and end of year \(k\) as time \(k\).
Record receipts as positive and outflows as negative, without changing dates to simplify the arithmetic.
Apply \(v^t\) to each time-\(t\) amount. If a receipt moves earlier by one period, its present value should increase by \(i\) times its original present value.
The main exam trap is to treat “year 1” as a time index rather than a label for a period. Beginning of year 1 is time 0; end of year 1 is time 1. A second trap is to apply a timing adjustment to the whole NPV when only selected cash flows move. The adjustment belongs to the affected cash flows, while unchanged cash flows retain their original dates.
These timing rules are useful whenever project receipts or costs are stated at period boundaries. They also provide a direct way to compare schedules with identical amounts but different payment dates: value the schedules on one common time-0 basis, then interpret the difference as the effect of the date change. This treatment of beginning- and end-of-period timing is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
Check Your Understanding
For each problem, place every cash flow on a timeline before calculating.
- A project pays $400 at the beginning of each of the next three years. The effective annual rate is 5%. Find the time-0 present value of these receipts. (Answer: $1,143.76)
- A $1,000 receipt is moved from the end of year 2 to the beginning of year 2. The effective annual rate is 6%. By how much does its present value increase? (Answer: $53.40)
- A project costs $1,200 at time 0 and receives $500 at the beginning of year 2 and $600 at the end of year 2. The effective annual rate is 10%. Find the NPV. (Answer: -$249.59)
- Five receipts of $300 each occur at the beginning of years 1 through 5. Compare their present value with that of the same receipts at the ends of years 1 through 5. The effective annual rate is 4%. By how much is the beginning-of-year schedule’s present value larger? (Answer: $53.42)
- A project costs $2,500 at time 0 and receives $1,000 at the end of year 1, $900 at the beginning of year 3, and $800 at the end of year 3. The effective annual rate is 8%. Find the NPV. (Answer: -$167.40)