Why a Project Needs a Timeline
A project forecast is not just a list of amounts. Each amount belongs to a particular date, and the timing determines how it enters a capital-budgeting analysis. A startup payment belongs at the project start; an operating receipt belongs when it is expected to occur; and a later equipment purchase belongs on its own payment date. Placing any of these in the wrong period can change the analysis even when the nominal amounts are unchanged.
A project cash-flow timeline organizes the relevant incremental cash flows by date. Earlier material on incremental project cash flows distinguishes project-related cash flows from amounts that should be excluded. Here, the task is to place the included amounts correctly. The examples treat the stated amounts as hypothetical and incremental, after-tax cash flows unless specified otherwise. No probability or mortality model is used. Because the focus is timing rather than present value, no interest rate is needed or applied.
Set the Dates Before Adding the Amounts
Start by choosing the project start date and labelling it \(t=0\). Then identify the dates of the cash flows. If the forecast convention puts operating cash flows at each year-end, year 1’s operating cash flow is at \(t=1\), not \(t=0\). A payment made at the beginning of year 3 occurs at \(t=2\) when the project starts at \(t=0\): two full years have elapsed by that date.
A useful schedule separates the components before calculating each period’s net amount. Typical rows include the initial outlay, operating cash flows, later capital spending, and any separately supplied end-of-project flow. Adding components in the same period is appropriate because cash amounts at the same date can be netted. Amounts on different dates should remain separate, even if they occur in the same calendar year.
Set the project start to \(t=0\), and state what each later time index means.
Mark receipts as positive and payments as negative. Use the actual date or the stated annual timing convention.
Sum components at a shared date to obtain that date’s net project cash flow.
Confirm that each amount appears once, at its correct date, and that a closing amount does not replace operating cash flow in the final period.
Why Same-Date Cash Flows Can Be Combined
The timeline records net cash movement at each date. If several project components occur at date \(t\), their net cash flow is their signed sum. This follows from addition of cash amounts: for components \(C_{1,t},C_{2,t},\ldots,C_{k,t}\) occurring at that date, the total amount at \(t\) is their sum.
For example, an operating inflow of $50,000 and a capital payment of $12,000 made on the same date produce a net cash flow of $38,000 at that date. But if the capital payment occurs six months later, it is not part of the same dated cash flow. Keeping the two dates separate preserves the project’s timing information. The annual year-end convention is a modelling assumption, not a claim that every project’s actual receipts and payments occur exactly at year-end.
Worked Example: Initial Outlay, Annual Operations, and Later Capital Spending
A hypothetical four-year project begins at \(t=0\). It requires equipment costing $240,000 and an initial working-capital investment of $18,000, both paid at \(t=0\). After-tax operating cash flow is $82,000 at each year-end, \(t=1\) through \(t=4\). Additional equipment costing $35,000 is paid for at the end of year 2. A separately supplied net end-of-project cash inflow of $38,000 occurs at the end of year 4, in addition to that year’s operating cash flow. All amounts are incremental and after tax; there is no probability model or interest rate in this timeline exercise.
Method. Assign each item to its stated date and add signed amounts only when they occur at the same date. At \(t=0\), both payments are outflows:
At \(t=1\), there is only the first annual operating cash flow, so \(C_1=+\$82{,}000\). At \(t=2\), the $82,000 operating inflow coincides with the $35,000 equipment payment:
At \(t=3\), the operating cash flow is $82,000. At \(t=4\), add the operating cash flow and the separately supplied end-of-project inflow:
The resulting timeline is \(t=0:-\$258{,}000\), \(t=1:+\$82{,}000\), \(t=2:+\$47{,}000\), \(t=3:+\$82{,}000\), and \(t=4:+\$120{,}000\). As a check, the total of the timeline is \(-258+82+47+82+120=73\), in thousands of dollars. This equals \(4(82)+38-35-240-18=73\) thousand. The common mistake avoided here is treating the end-of-project amount as a replacement for year 4 operating cash flow; both occur at \(t=4\) and both belong in its total.
Later Payments and Exact Timing
The phrase “in year 3” can be ambiguous. A payment at the beginning of year 3 occurs at \(t=2\); a payment at the end of year 3 occurs at \(t=3\). If an exercise supplies an exact date, use it rather than silently treating every payment in a year as a year-end amount. If the analysis uses annual periods, state the convention clearly and apply it consistently.
Worked Example: A Payment at the Beginning of Year 3
A hypothetical project starts on July 1 at \(t=0\). It requires an initial capital outlay of $120,000, setup costs of $12,000, and working capital of $8,000, all paid at the start. Operating cash flow is $41,000 at each year-end for four years. An additional working-capital investment of $5,000 is paid at the beginning of year 2. New equipment costing $26,000 is paid for at the beginning of year 3. A separately supplied net end-of-project inflow of $19,000 occurs at the end of year 4, in addition to that year’s operating cash flow. Amounts are hypothetical, incremental, and after tax; no interest rate or probability model is used.
Method. First mark the dates: the start is \(t=0\); the beginning of year 2 is \(t=1\); and the beginning of year 3 is \(t=2\). The year-end operating cash flows occur at \(t=1,2,3,4\). The initial outflows combine at \(t=0\):
At \(t=1\), year 1 operating cash flow and the additional working-capital payment occur at the same date, giving \(C_1=+\$41{,}000-\$5{,}000=+\$36{,}000\). At \(t=2\), year 2 operating cash flow coincides with the beginning-of-year-3 equipment payment, so \(C_2=+\$41{,}000-\$26{,}000=+\$15{,}000\). At \(t=3\), \(C_3=+\$41{,}000\). Finally, at \(t=4\), include both the fourth operating cash flow and the supplied end-of-project inflow:
The timeline is \(-\$140{,}000,\ +\$36{,}000,\ +\$15{,}000,\ +\$41{,}000,\ +\$60{,}000\) at \(t=0,1,2,3,4\), respectively. Its total is \(-140+36+15+41+60=12\) thousand dollars. Independently, the components total \(4(41)+19-140-5-26=12\) thousand dollars. This check confirms that the beginning-of-year-3 equipment payment was placed at \(t=2\), not \(t=3\).
Worked Example: Separating a Midyear Payment from Year-End Cash Flow
A hypothetical three-year project starts on January 1. Its initial outlay is $90,000 at \(t=0\). Operating cash flows are $28,000 at the end of year 1 and $34,000 at the end of each of years 2 and 3. A maintenance payment of $9,000 and replacement equipment costing $16,000 are both paid at the end of year 2. A separately supplied net end-of-project inflow of $22,000 occurs at the end of year 3, in addition to the year 3 operating cash flow. All values are incremental, after tax, and hypothetical. No discounting or probability assumptions are needed.
Method. Keep the project start at \(t=0\) and label each year-end with its elapsed year. The initial outlay gives \(C_0=-\$90{,}000\), and year 1 has \(C_1=+\$28{,}000\). At \(t=2\), both payments coincide with the $34,000 operating cash flow:
At \(t=3\), the operating cash flow and end-of-project inflow combine:
Thus the timeline is \(-\$90{,}000,\ +\$28{,}000,\ +\$9{,}000,\ +\$56{,}000\) at \(t=0,1,2,3\). The total is \(-90+28+9+56=3\) thousand dollars; summing the components gives \(28+34+34+22-90-9-16=3\) thousand dollars. The common mistake avoided is omitting year 2’s operating inflow when netting the two payments, or putting either payment in year 3 merely because it is later capital spending.
Common Timing Errors
A timeline is easier to audit when each component is first listed with its date and sign, then combined into period totals. The following errors are especially consequential:
- Putting the initial outlay at \(t=1\). Under the standard convention, amounts paid when the project begins are at \(t=0\). Year 1 refers to the first period after that start.
- Confusing the beginning and end of a year. Beginning of year 3 is \(t=2\) in a timeline starting at \(t=0\); end of year 3 is \(t=3\).
- Using the wrong sign. An expenditure is negative and a receipt is positive. A net working-capital investment is a cash outflow when paid; use the cash-flow amount and date provided or established for the forecast.
- Omitting ordinary operations in the final period. A separate end-of-project amount is generally added to, not substituted for, the operating cash flow occurring at the same time.
- Combining amounts from different dates. Two cash flows may both be described as occurring “in year 2” but still have distinct dates. Do not combine them unless the stated convention places them at the same time.
- Counting a component twice. If a supplied end-of-project amount is already a net total for that date, do not add its included components again. Read the description of each amount carefully.
Check Your Understanding
For each hypothetical project, construct the timeline using the stated timing convention. Amounts are in dollars, are incremental and after tax, and require no probability model or discount rate.
- A project has an initial outlay of $65,000 at \(t=0\), operating inflows of $22,000 at each year-end for three years, a capital payment of $8,000 at the end of year 1, and a separate end-of-project inflow of $5,000 at the end of year 3. Give the net cash flow at \(t=0,1,2,3\). (Answer: \(-\$65{,}000,\ +\$14{,}000,\ +\$22{,}000,\ +\$27{,}000\).)
- A project starts January 1 with an outflow of $70,000. It generates $30,000 at each of the first three December 31 year-ends. Equipment costing $12,000 is paid for at the beginning of year 2, and an additional net closing inflow of $9,000 occurs at the end of year 3. What is the timeline at \(t=0,1,2,3\)? (Answer: \(-\$70{,}000,\ +\$18{,}000,\ +\$30{,}000,\ +\$39{,}000\).)
- A two-year project begins on January 1 with a $40,000 outflow. It receives $17,000 on each December 31. A $6,000 equipment payment occurs on July 1 of year 2, and a $4,000 closing receipt occurs on December 31 of year 2. State the cash flows by exact time, retaining separate dates where required. (Answer: \(t=0:-\$40{,}000;\ t=1:+\$17{,}000;\ t=1.5:-\$6{,}000;\ t=2:+\$21{,}000\).)
- A four-year project has a $105,000 initial outflow, $26,000 operating inflows at each year-end, a $7,000 payment at the end of year 2, a $15,000 equipment outlay at the end of year 3, and a separate $12,000 closing inflow at the end of year 4. Give the net cash flows at \(t=0,1,2,3,4\). (Answer: \(-\$105{,}000,\ +\$26{,}000,\ +\$19{,}000,\ +\$11{,}000,\ +\$38{,}000\).)
- A project begins with an outflow of $100,000 at \(t=0\), receives $40,000 at each year-end for three years, pays $20,000 for additional equipment at the beginning of year 3, and receives a separate $10,000 closing amount at the end of year 3. Give the timeline at \(t=0,1,2,3\). (Answer: \(-\$100{,}000,\ +\$40{,}000,\ +\$20{,}000,\ +\$50{,}000\).)