NPV for a Delayed Project Start

Why the Start Date Matters

A project may require a waiting period before its initial investment is made and its operating cash flows begin. The waiting period changes the dates at which those amounts are received or paid, and therefore changes their present values. The key question is whether the entire project cash-flow pattern shifts intact or whether waiting causes some cash flows to change or disappear.

For example, if an investment and every receipt are postponed by two years, the project has the same cash flows relative to its start date, but each is discounted for two additional years when valued today. If a permit expires during the waiting period, however, a receipt that would have occurred after the project began may no longer be available. In that case, the project’s cash-flow pattern has changed; a simple time-shift adjustment is not enough.

Definition: For a valuation at time 0, the net present value (NPV) of cash flows $C_t$ at times $t=0,1,\ldots,n$ is $$ \operatorname{NPV}_0=\sum_{t=0}^{n} C_t v^t,\qquad v=\frac{1}{1+i}, $$ where $i$ is the effective annual interest rate and $C_t$ is positive for a receipt and negative for a payment. In this tutorial, cash flows occur at integer-year times and the same annual effective rate applies throughout.

A project can be described using times relative to its start. Let $C_k$ be its cash flow $k$ years after that start, with $C_0$ usually the initial investment. Thus $C_0$ is negative when the investment is paid at the start. If the project starts $m$ years from today, its cash flow $C_k$ occurs at calendar time $m+k$.

NPV When the Entire Project Is Delayed

Suppose the project starts $m$ years from today and every project cash flow occurs at the same relative time and in the same amount as it would in the original plan. The NPV at the project start date is $$ \operatorname{NPV}_{m}=\sum_{k=0}^{n} C_k v^k. $$ To value that project today, discount each cash flow from its actual calendar time $m+k$:

\operatorname{NPV}_0 =\sum_{k=0}^{n} C_k v^{m+k} =v^m\sum_{k=0}^{n} C_kv^k =v^m\operatorname{NPV}_{m}.

The factor $v^m$ applies to the net project value at its start date, because every component of that project is shifted by the same $m$ years. This does not mean that an investment made today should be discounted: a payment at time 0 stays at time 0. It also does not mean that a receipt can be shifted if its availability depends on a fixed external deadline.

Theorem: If the cash flows $C_0,\ldots,C_n$ are unchanged in amount and relative timing, and the entire pattern occurs $m$ years later, then its NPV at time 0 is $v^m$ times its NPV at the delayed start date. This result assumes a constant effective annual rate and no additional cash flows during the waiting period.

A Reliable Timeline Method

The safest approach is to place every payment and receipt on a calendar timeline before calculating. The following steps keep the start date separate from the cash-flow pattern.

Mark time 0 and the start date.
Write the waiting period $m$ on the timeline and identify whether any costs occur before the project starts.

List project cash flows relative to the start.
Record the initial investment and each later receipt or payment, including its relative year.

Check whether the pattern is intact.
If each cash flow keeps its amount and relative timing, use the uniform-shift formula. If a cash flow is lost or altered, value the actual dated amounts instead.

Discount all amounts to one valuation date.
Use the stated effective annual rate and include any pre-start cash flows at their actual dates.

Worked Examples

Worked Example: An Unchanged Project Delayed Two Years

A hypothetical project requires an investment of $20,000 when it begins, two years from today. It then pays $12,500 at the end of each of its first three operating years. Assume an effective annual interest rate of 25%. Find the project’s NPV today.

Method. First value the project at its start date, time 2. Relative to that date, the investment is at time 0 and the receipts are at times 1, 2, and 3. The discount factor is $v=1/1.25=0.8$.

The NPV at time 2 is

\operatorname{NPV}_{2} =-20{,}000+12{,}500(0.8+0.8^2+0.8^3).

The powers are $0.8^2=0.64$ and $0.8^3=0.512$, so the receipt factor is $0.8+0.64+0.512=1.952$. The present value of the receipts at time 2 is $12{,}500(1.952)=\$24{,}400$. Therefore,

\operatorname{NPV}_{2}=-\$20{,}000+\$24{,}400=\$4{,}400.

The complete project is delayed two years, so discount that net value by $v^2=0.64$:

\operatorname{NPV}_{0}=0.64(\$4{,}400)=\boxed{\$2{,}816.00}.

Check. Directly from today, the investment occurs at time 2 and receipts at times 3, 4, and 5: $-20{,}000(0.64)+12{,}500(0.512+0.4096+0.32768)=-12{,}800+15{,}616=\$2{,}816$. The two methods agree. The common mistake avoided here is discounting the receipts from time 0 but forgetting to discount the initial investment from time 2.

Worked Example: A Planning Cost Before the Delayed Start

A hypothetical project requires a planning payment of $3,000 today. The project itself begins two years from today, when it requires an investment of $10,000. It pays $9,000 at the end of each of the first two operating years. At an effective annual interest rate of 25%, calculate the NPV today.

Method. The planning payment is already at time 0, so it is not part of the project pattern shifted from time 2. At the project start, $v=0.8$, and the local NPV is

\operatorname{NPV}_{2} =-\$10{,}000+\$9{,}000(0.8+0.64) =-\$10{,}000+\$12{,}960 =\$2{,}960.

Discount this project value two years, then include the planning payment at its actual time:

\operatorname{NPV}_{0} =-\$3{,}000+0.64(\$2{,}960) =-\$3{,}000+\$1{,}894.40 =\boxed{-\$1{,}105.60}.

Check. The direct timeline calculation is $-3{,}000-10{,}000(0.64)+9{,}000(0.512+0.4096)$. The project investment has present value $-\$6{,}400$, and the receipts have present value $9{,}000(0.9216)=\$8{,}294.40$. Adding $-\$3{,}000-\$6{,}400+\$8{,}294.40$ gives $-\$1{,}105.60$. The common mistake avoided is multiplying the planning payment by $v^2$, even though it is paid today.

Worked Example: Waiting Changes the Cash Flows

A hypothetical project requires $16,000 immediately and pays $10,000 at the end of each of the next two years. The project authorization expires at the end of year 2. If the project is postponed for one year, the investment is made at year 1 and only one $10,000 receipt, at year 2, can be collected before the authorization expires. At an effective annual interest rate of 25%, compare the original NPV with the NPV under the postponement.

Method. For the original project, $v=0.8$, and the cash flows are $-\$16{,}000$ at time 0 and $\$10{,}000$ at times 1 and 2:

\operatorname{NPV}_{\text{original}} =-\$16{,}000+\$10{,}000(0.8+0.64) =-\$16{,}000+\$14{,}400 =-\$1{,}600.

Under the postponement, the investment occurs at time 1 and the one available receipt occurs at time 2. Thus

\operatorname{NPV}_{\text{postponed}} =-\$16{,}000(0.8)+\$10{,}000(0.64) =-\$12{,}800+\$6{,}400 =\boxed{-\$6{,}400}.

Check. If the entire original pattern could instead be shifted intact by one year, its NPV would be $0.8(-\$1{,}600)=-\$1{,}280$. That shifted pattern would include receipts at years 2 and 3. The authorization rules out the year-3 receipt, so the actual postponed project is not a uniform shift. The common mistake avoided is applying $v$ to the original NPV without checking whether all its cash flows remain available.

Derivation and Interpretation

The time-shift formula follows from the definition of present value, not from a special feature of projects. A cash flow $C_k$ occurring $k$ years after the start date $m$ years from now occurs at time $m+k$. Its present value today is $C_kv^{m+k}$. Since $v^{m+k}=v^m v^k$, summing over the project’s cash flows gives

\sum_{k=0}^{n} C_kv^{m+k} =v^m\sum_{k=0}^{n}C_kv^k =v^m\operatorname{NPV}_{m}.

This derivation also explains an important interpretation: when the entire pattern shifts intact, the delay scales its NPV by a positive factor $v^m$. Since $v^m>0$, the sign of that NPV does not change. A positive NPV at the start date remains positive when valued today, though its value is reduced. A negative NPV remains negative, though its magnitude is reduced. This sign observation applies only when the project pattern and discount-rate assumptions are unchanged.

Common Mistakes and Exam Checks

A delayed start does not automatically mean every cash flow in the problem should be discounted by the same additional number of years. First classify each amount by when it actually occurs.

Shifting only the receipts. If the investment is made at the delayed start, it is delayed too. Discounting only the receipts gives inconsistent dates.
Shifting a cost paid now. A planning or setup cost at time 0 remains at time 0; it is not part of the later project pattern.
Assuming the cash-flow pattern is unchanged. A fixed deadline, expiring authorization, or other condition may remove or alter a later payment or receipt. Build the actual timeline instead of using $v^m\operatorname{NPV}_m$ automatically.
Using the wrong period count. With an annual effective rate, a payment at time $m+k$ is discounted for $m+k$ years, not merely $k$ years when the valuation date is today.

The distinction is aligned with the SOA Exam FM (Financial Mathematics) syllabus: first identify the cash-flow dates, then apply present-value factors consistently. A formula cannot correct a timeline that misstates when an amount is paid or received.

Key takeaway: If every project cash flow is postponed by the same number of years without changing its amount or relative timing, multiply the NPV at the delayed start by $v^m$ to obtain today’s NPV. Keep pre-start cash flows at their actual dates, and revalue the actual timeline whenever waiting changes the project’s cash flows.

Check Your Understanding

For each problem, use the stated effective annual rate and value all amounts at time 0. Assume cash flows occur at the stated year-ends unless otherwise specified.

A project begins one year from today, requires $5,000 at its start, and pays $3,000 at the end of each of its first two operating years. At $i=10\%$, find today’s NPV. (Answer: $187.83)
A project starts two years from today with an $8,000 investment and pays $5,000 at the end of each of its first two operating years. At $i=20\%$, find today’s NPV. (Answer: $-\$250.77$)
A project has a $1,500 planning payment today. It starts one year from today with a $6,000 investment and pays $4,000 at the end of each of its first two operating years. At $i=25\%$, find today’s NPV. (Answer: $-\$1{,}692.00$)
A project’s NPV at its start date is $12,000. It starts two years from today, and all its cash flows are unchanged relative to that date. At $i=10\%$, find today’s NPV. (Answer: $9,917.36)
A project requires $12,000 today and pays $8,000 at the end of each of the next two years. Its authorization expires at the end of year 2. If it is postponed by one year, the investment is made at year 1 and only one $8,000 receipt can be collected at year 2. At $i=25\%$, find the NPV under postponement. (Answer: $-\$4{,}480.00$)

What You'll Learn