Timing Effects in Project Rankings

Why Timing Changes Project Rankings

Two projects can have the same initial cost and the same total future receipts yet create different value at a specified interest rate. The difference is the timing of those receipts: money received sooner has less time to be discounted. A ranking based only on total dollars received can therefore miss a meaningful difference in present value.

The comparison in this tutorial uses net present value (NPV) at a stated annual effective interest rate. It isolates timing by holding the rate fixed and, where appropriate, comparing projects with the same initial outlay and the same total receipts. All examples assume annual cash flows at integer-year times, a constant effective annual rate, and no taxes or other cash flows beyond those stated.

Definition: If $C_t$ is the net cash flow at time $t$, measured in years, and $i$ is the effective annual valuation rate, then the NPV at time 0 is the sum of each cash flow discounted to time 0: $$ \operatorname{NPV}=\sum_{t=0}^{n} C_t v^t \qquad\text{where}\qquad v=\frac{1}{1+i}. $$ A positive cash flow is a receipt and a negative cash flow is an outlay. When two projects have the same outlay at time 0, the difference between their NPVs is the difference between the present values of their future cash flows.

The comparison is conditional on the chosen rate. It does not say that an earlier cash flow is always better in every possible setting: the conclusion here concerns NPV under the stated discount rate and cash-flow assumptions. In particular, the examples below use a positive rate, so future positive receipts are discounted.

The Basic Timing Effect

The simplest timing comparison moves the same positive amount from one date to another. Suppose $K$ dollars is received at the end of year $t+1$, rather than at the end of year $t$. The increase in present value from receiving it one year earlier is the earlier present value minus the later present value.

K v^t-K v^{t+1} =K v^t(1-v) =K v^t\left(\frac{i}{1+i}\right) =K i v^{t+1}.

For $K>0$ and $i>0$, this amount is positive. The conclusion follows because $v=1/(1+i)<1$, so $v^t>v^{t+1}$. More generally, moving $K$ from year $s$ to an earlier year $t$, where $t<s$, increases its present value by $K(v^t-v^s)>0$. This gives a direct way to quantify a timing advantage when the amount being moved is known.

The result applies to a positive receipt. Moving a cost earlier has the opposite effect on NPV: an earlier negative cash flow is more negative in present-value terms. For a full project comparison, discount every signed cash flow rather than applying an “earlier is better” rule without checking whether the cash flow is a receipt or an outlay.

Worked Example: Equal Total Receipts, Different Schedules

Two mutually exclusive projects each require an outlay of $1,000 at time 0. Project A pays $600 at the end of year 1 and $600 at the end of year 2. Project B pays $300 at the end of year 1, $300 at the end of year 2, and $600 at the end of year 3. Both projects have total future receipts of $1,200. Compare their NPVs at an effective annual rate of 10%.

Method. Discount each receipt to time 0 using $v=1/(1+i)$, then subtract the common initial outlay. This method avoids treating equal undiscounted totals as equal present values.

Step 1: Find the discount factors. With $i=0.10$, $$ v=\frac{1}{1.10}=0.9090909,\qquad v^2=\frac{1}{1.10^2}=0.8264463,\qquad v^3=\frac{1}{1.10^3}=0.7513148. $$

Step 2: Calculate Project A’s present value of receipts. $$ 600v+600v^2 =600(0.9090909)+600(0.8264463) =545.4545+495.8678 =1041.3223. $$ Therefore, $$ \operatorname{NPV}_A=-1000+1041.3223=\$41.32. $$

Step 3: Calculate Project B’s present value of receipts. $$ 300v+300v^2+600v^3 =300(0.9090909)+300(0.8264463)+600(0.7513148) $$ $$ =272.7273+247.9339+450.7889 =971.4501. $$ Therefore, $$ \operatorname{NPV}_B=-1000+971.4501=-\$28.55. $$

Step 4: Compare and check. Project A’s NPV exceeds Project B’s by $41.3223-(-28.5499)=\$69.87$, rounded to the nearest cent. The initial costs are equal, and both projects pay $1,200 in total. The ranking comes from discounting the earlier receipts in A more heavily in its favor. The calculation is also consistent with the sign of the result: A has the higher present value of receipts and hence the higher NPV.

When One Payment Is Simply Delayed

A useful controlled comparison keeps the amount and every other cash flow unchanged, but shifts a receipt to a later date. In that setting, the timing difference can be calculated directly using the difference between the two discount factors. This is often quicker than rebuilding and comparing two full cash-flow tables.

Worked Example: Quantifying a One-Year Delay

Two projects each require $1,000 at time 0. Project C pays $1,200 at the end of year 2. Project D pays the same $1,200 at the end of year 3. Find each NPV and quantify the value difference at an effective annual rate of 10%.

Method. Discount the sole receipt in each project. Since D’s receipt is exactly one year later, the direct timing difference is also the present-value difference between a $1,200 receipt in year 2 and one in year 3.

Step 1: Calculate the discount factors. $$ v^2=\frac{1}{1.10^2}=\frac{1}{1.21}=0.8264463,\qquad v^3=\frac{1}{1.10^3}=\frac{1}{1.331}=0.7513148. $$

Step 2: Calculate Project C. $$ \text{PV of receipts for C}=1200(0.8264463)=991.7355, $$ $$ \operatorname{NPV}_C=-1000+991.7355=-\$8.26. $$

Step 3: Calculate Project D. $$ \text{PV of receipts for D}=1200(0.7513148)=901.5778, $$ $$ \operatorname{NPV}_D=-1000+901.5778=-\$98.42. $$

Step 4: Compare and check. Project C’s NPV is higher by $(-8.2645)-(-98.4222)=\$90.16$, rounded to the nearest cent. The direct calculation agrees: $$ 1200(v^2-v^3)=1200(0.8264463-0.7513148)=\$90.16. $$ Both NPVs are negative at this rate, but C is still the higher-NPV choice if the decision is between these two projects. The common mistake to avoid is assuming that a larger total receipt than the outlay guarantees a positive NPV; the receipt must be discounted before it is compared with the outlay.

Schedules That Cross

A simple “one project pays earlier” description is not always available. One project may pay more at first, while the other pays more later. In this case the schedules cross: neither project has at least as much cumulative receipt as the other at every date. The one-period timing result alone cannot determine the ranking. Discount each payment and compare the resulting present values.

Worked Example: A Larger Early Payment Does Not Decide the Ranking

Two projects each require $950 at time 0. Project E pays $700 at the end of year 1, $100 at the end of year 2, and $400 at the end of year 4. Project F pays $400 at the end of year 1, $700 at the end of year 2, and $100 at the end of year 4. Both pay $1,200 in total. Compare their NPVs at an effective annual rate of 10%.

Method. Discount all receipts on both timelines. This avoids inferring the ranking from the first payment alone: E pays more in year 1, but F has paid more cumulatively by the end of year 2.

Step 1: Find the necessary discount factors. $$ v=0.9090909,\qquad v^2=0.8264463,\qquad v^4=\frac{1}{1.10^4}=0.6830135. $$

Step 2: Calculate Project E’s receipt present value. $$ 700v+100v^2+400v^4 =700(0.9090909)+100(0.8264463)+400(0.6830135) $$ $$ =636.3636+82.6446+273.2054=992.2136. $$ Thus, $$ \operatorname{NPV}_E=-950+992.2136=\$42.21. $$

Step 3: Calculate Project F’s receipt present value. $$ 400v+700v^2+100v^4 =400(0.9090909)+700(0.8264463)+100(0.6830135) $$ $$ =363.6364+578.5124+68.3014=1010.4502. $$ Thus, $$ \operatorname{NPV}_F=-950+1010.4502=\$60.45. $$

Step 4: Compare and check. Project F’s NPV is higher by $1010.4502-992.2136=\$18.24$, rounded to the nearest cent. Both have the same initial outlay and total future receipts. E’s larger year-1 payment does not settle the comparison because F’s larger year-2 payment changes the full discounted total. The discounted sums, not a judgment based on one date, give the ranking.

A Reliable Comparison Process

For projects with similar totals, a disciplined calculation helps separate the effect of timing from other differences. The following steps work for equal-total schedules as well as for schedules that do not have identical totals.

Write each timeline.
Record every signed cash flow at its date, including the initial outlay. Do not combine amounts that occur at different times.

Use the stated rate consistently.
For an effective annual rate $i$, use $v=1/(1+i)$ and discount a time-$t$ cash flow by $v^t$.

Calculate each NPV.
Sum the discounted signed cash flows for each project separately.

Compare and interpret.
Subtract the NPVs to quantify the difference, then relate it to the schedules. If one schedule does not consistently pay earlier, do not claim that timing alone has an obvious direction before doing the calculation.

A common trap is to compare total receipts without discounting. In Worked Example 1, the incorrect calculation would be $1200-1000=\$200$ for each project, leading to the false conclusion that both have NPV $200 and are tied. That calculation treats every receipt as if it arrived at time 0. The correction is to discount each receipt at its actual date: the NPVs are $41.32 for A and -$28.55 for B.

Another trap is to use the right discount factor at the wrong time. A receipt in year 3 is multiplied by $v^3$, not $v^2$. Build the timeline before calculating powers of $v$, and keep the stated annual rate paired with years. These checks matter particularly when the cash flows are close enough that a timing or exponent error could change the ranking.

Key takeaway: When the rate is positive, bringing a positive receipt forward increases its present value; delaying it decreases its present value. For full project rankings, discount every signed cash flow at its actual date. Equal undiscounted totals do not imply equal NPVs, and crossing schedules require a complete calculation.

Check Your Understanding

For each comparison, use the stated effective annual rate and report NPVs to the nearest cent.

Projects A and B each cost $450 at time 0. A pays $500 at the end of year 1; B pays $500 at the end of year 2. At 5%, find both NPVs and the NPV advantage. (Answer: A: $26.19; B: $3.51; A’s advantage: $22.68.)
Two projects each cost $600. Project C pays $700 at the end of year 1 and $300 at the end of year 2. Project D pays $300 at the end of year 1 and $700 at the end of year 2. At 8%, which has the higher NPV, and by how much? (Answer: C, by $27.43.)
At 10%, compare a $1,000 receipt at the end of year 2 with the same receipt at the end of year 3. What is the present-value advantage of the earlier receipt? (Answer: $75.13.)
Two projects each cost $820. Project E pays $500 at the end of year 1 and $500 at the end of year 3. Project F pays $200 at the end of year 1 and $800 at the end of year 2. At 10%, find their NPVs and identify the higher-NPV project. (Answer: E: $10.20; F: $22.98; F is higher by $12.77.)
At 8%, $250 of a project’s receipt is moved from the end of year 4 to the end of year 2, with all other cash flows unchanged. By how much does NPV increase? (Answer: $30.58.)

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