How Cash-Flow Timing Changes NPV

Why Timing Matters

Two projects can generate the same total cash receipts and still have different NPVs. The reason is that a dollar received sooner can be invested or used sooner, so its present value is greater when the required return is positive. Comparing only the undiscounted totals ignores this difference.

The comparison requires cash flows to be placed on a timeline and discounted at the same required return. Assume cash flows are measured in dollars, occur at the stated dates, and are valued using a constant effective annual required return $i$. A positive cash flow is a receipt; a negative cash flow is an outflow. As in the earlier material on NPV cash-flow timelines and discounting project cash flows individually, a cash flow at time $t$ has present value $C_t/(1+i)^t$ when the date is measured in years.

Definition: For cash flows $C_0,C_1,\ldots,C_n$ occurring at times $0,1,\ldots,n$, the net present value at effective annual required return $i$ is $$ \operatorname{NPV}=\sum_{t=0}^{n}\frac{C_t}{(1+i)^t}. $$ The time-zero cash flow is not discounted. This expression assumes the cash-flow dates and the annual rate use consistent time units, and that the same required return is appropriate for the cash flows being compared.

The key comparison is not “which project has the largest total receipts?” but “what is the present value of each dated cash flow?” If two schedules have the same initial cost and the same total positive receipts, but one receives more of those receipts sooner, it will generally have the higher NPV when $i>0$. That is a timing comparison, not a statement that the projects have the same NPV.

The Present-Value Effect of Moving a Cash Flow

Let $C>0$ be a receipt that would otherwise arrive at time $t_2$. Suppose it is moved to an earlier time $t_1$, where $t_1<t_2$, without changing its amount. Its effect on NPV is the present value of the receipt at the earlier date minus its present value at the later date.

\Delta\operatorname{NPV} = \frac{C}{(1+i)^{t_1}}-\frac{C}{(1+i)^{t_2}} = C\left[(1+i)^{-t_1}-(1+i)^{-t_2}\right].

For $i>0$, $1+i>1$. Therefore, increasing the exponent makes $(1+i)^{-t}$ smaller. Since $t_1<t_2$, it follows that $(1+i)^{-t_1}>(1+i)^{-t_2}$. Multiplication by $C>0$ preserves that inequality, so moving the receipt earlier increases NPV. This is the complete reason for the result: the amount is unchanged, but the earlier date has a larger discount factor.

If $i=0$, both discount factors equal 1, and moving the receipt changes neither its present value nor NPV. For a negative cash flow, the conclusion reverses: moving an outflow earlier makes its present value more negative, lowering NPV. Thus, “earlier is better” is not a rule for every cash flow. It applies to positive receipts; the sign of the cash flow matters.

Theorem: If a positive cash flow $C$ is moved from a later date $t_2$ to an earlier date $t_1$, with $t_1<t_2$, then NPV increases when $i>0$, is unchanged when $i=0$, and decreases when $i<0$. For a negative cash flow, the direction of the NPV effect is reversed. The result holds when the amount, all other cash flows, and the applicable discount rate are unchanged.

Comparing Projects with Equal Total Receipts

When two projects have identical total receipts but distribute those receipts differently across time, compare their discounted values date by date. A convenient shortcut is to identify the amounts that have shifted between dates: unchanged cash flows cancel, leaving only the present-value difference caused by timing.

Worked Example: The Same Receipts Arrive in a Different Order

A hypothetical project requires an investment of $100,000 at time 0. Schedule A produces $70,000 at the end of year 1 and $30,000 at the end of year 2. Schedule B produces $30,000 at the end of year 1 and $70,000 at the end of year 2. The annual effective required return is 10%. Which schedule has the higher NPV, and by how much?

Method. Discount each receipt to time 0, include the common initial outflow, and compare the NPVs. Work in thousands of dollars, so the initial outflow is $-100$, and the receipts are 70 and 30.

For Schedule A: $$ \operatorname{NPV}_A =-100+\frac{70}{1.10}+\frac{30}{(1.10)^2} =-100+63.63636\ldots+24.79338\ldots =-11.57\text{ thousand dollars}. $$

For Schedule B: $$ \operatorname{NPV}_B =-100+\frac{30}{1.10}+\frac{70}{(1.10)^2} =-100+27.27272\ldots+57.85123\ldots =-14.88\text{ thousand dollars}. $$

Both schedules have $100,000 in total receipts and the same $100,000 initial investment. Schedule A moves $40,000 of receipts from year 2 to year 1. The difference is therefore $$ 40\left(\frac{1}{1.10}-\frac{1}{(1.10)^2}\right) =3.30578\ldots\text{ thousand dollars} =\$3,305.79. $$ Thus Schedule A has the higher NPV by $3,305.79. Check: $-\$11,570.25-(-\$14,876.03)=\$3,305.78$ using the individually rounded NPV figures; the one-cent difference from the direct comparison is due to rounding the two NPVs separately. Using unrounded values, the direct difference rounds to $3,305.79. The common mistake is to treat the equal $100,000 receipt totals as evidence that the schedules have equal present value.

The direct difference calculation is usually the clearest check. It avoids subtracting separately rounded NPVs and shows exactly which cash flows differ. When reporting to cents, calculate with unrounded values and round the final comparison once.

Worked Comparison of an Earlier and Later Receipt

Worked Example: A Receipt at Year 1 or Year 3

A hypothetical project costs $8,000 at time 0 and will produce a single $10,000 receipt, either at the end of year 1 or at the end of year 3. The effective annual required return is 10%. Calculate both NPVs and the value of receiving the cash earlier.

Method. The time-zero cost is the same under both alternatives. Discount the receipt at its own date; do not discount it for only the number of years between the two alternatives.

If the receipt arrives in year 1: $$ \operatorname{NPV}_{1} =-8{,}000+\frac{10{,}000}{1.10} =-8{,}000+9{,}090.909\ldots =\$1{,}090.91. $$

If the receipt arrives in year 3: $$ \operatorname{NPV}_{3} =-8{,}000+\frac{10{,}000}{(1.10)^3} =-8{,}000+7{,}513.148\ldots =-\$486.85. $$

The increase from receiving the cash in year 1 rather than year 3 is $$ 10{,}000\left(\frac{1}{1.10}-\frac{1}{(1.10)^3}\right) =\$1{,}577.76. $$ The year-1 alternative has the higher NPV. Check: its receipt has a higher present value, and the initial cost is identical, so the NPV difference must equal the receipt present-value difference. A common error is to discount the year-3 receipt for two years because the alternatives are two years apart; NPV discounts from time 0, so this receipt is discounted for three years.

Timing Can Help or Hurt Depending on the Sign

A project may contain both positive and negative cash flows. In that case, it is not enough to say that one schedule has “earlier cash flows.” Identify which signed amounts moved. An earlier receipt raises NPV, while an earlier outflow lowers it. The combined effect is the sum of those two effects.

Worked Example: An Earlier Receipt and a Later Outflow

Two hypothetical schedules each contain a $20,000 receipt and a $20,000 outflow, so each has an undiscounted net total of zero. In Schedule A, the receipt occurs at the end of year 1 and the outflow at the end of year 2. In Schedule B, the outflow occurs at the end of year 1 and the receipt at the end of year 2. The effective annual required return is 10%. Compare their NPVs.

Method. Discount each signed cash flow at its own date. Use thousands of dollars throughout.

For Schedule A: $$ \operatorname{NPV}_A =\frac{20}{1.10}-\frac{20}{(1.10)^2} =18.18181\ldots-16.52892\ldots =\$1{,}652.89. $$

For Schedule B: $$ \operatorname{NPV}_B =-\frac{20}{1.10}+\frac{20}{(1.10)^2} =-18.18181\ldots+16.52892\ldots =-\$1{,}652.89. $$

The NPV difference is $$ \operatorname{NPV}_A-\operatorname{NPV}_B =40\left(\frac{1}{1.10}-\frac{1}{(1.10)^2}\right) =\$3{,}305.79. $$ Schedule A is better by that amount: it receives the positive cash flow earlier and pays the negative cash flow later. Check: both schedules have the same undiscounted net total, yet their NPVs differ because their positive and negative amounts occur at different dates. The common mistake is to move both signed cash flows earlier in the analysis without tracking that an earlier outflow has an adverse, not favorable, effect on NPV.

How to Make a Timing Comparison

A reliable comparison can be organized into a few steps. The purpose is to isolate timing from differences in amount, cost, or discount-rate assumptions.

Put both schedules on a timeline.
Record every signed cash flow at its actual date, including any time-zero investment.

Confirm what is held constant.
Check whether the schedules have the same cash-flow amounts, initial cost, and applicable required return. If some of these differ, do not attribute the entire NPV gap to timing alone.

Discount each cash flow from time 0.
Use its actual date in the exponent and preserve its positive or negative sign.

Compare unrounded NPVs or isolate the shifted amounts.
Either method should produce the same difference before rounding. Round the final dollar result as requested.

Common Timing Traps

Comparing totals instead of present values. Equal undiscounted receipts do not imply equal NPV when dates differ and $i\ne 0$.
Using the interval between two dates as the exponent. A year-3 cash flow is discounted three years from time 0, even if it is being compared with a year-1 cash flow.
Ignoring the cash-flow sign. An earlier positive receipt helps NPV when $i>0$; an earlier negative outflow hurts it.
Changing more than timing. If amounts, investment costs, or required returns differ, a difference in NPV cannot automatically be assigned only to cash-flow timing.
Rounding each line too early. Carry full precision through the calculations, then round the reported final amount. Separately rounded NPVs can produce a one-cent discrepancy in their displayed difference.

For actuarial and corporate-finance work, timing comparisons are useful whenever projected cash receipts or payments arrive on different schedules. The essential discipline is to state the date and sign of each cash flow before discounting it. This tutorial is aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

For each problem, use the stated effective annual required return and calculate from time 0. Report dollar results to the nearest cent.

A project can receive $12,000 at the end of year 1 or year 2. At an effective annual required return of 5%, by how much does receiving it in year 1 increase its present value? (Answer: $544.22.)
A $6,000 receipt is moved from the end of year 4 to the end of year 1. If the effective annual required return is 0%, how much does this timing change affect NPV? (Answer: $0.00.)
Two projects each cost $25,000 at time 0 and produce total receipts of $25,000. Project A receives $15,000 in year 1 and $10,000 in year 2; Project B receives $10,000 in year 1 and $15,000 in year 2. At 10%, which has the higher NPV, and by how much? (Answer: Project A, by $413.22.)
A $9,000 outflow can occur at the end of year 1 or year 3. At an effective annual required return of 8%, by how much does delaying the outflow increase NPV, all else equal? (Answer: $1,188.85.)
At 10%, Schedule A has a $4,000 receipt in year 1 and a $5,000 outflow in year 2. Schedule B has the $5,000 outflow in year 1 and the $4,000 receipt in year 2. Which has the higher NPV, and by how much? (Answer: Schedule A, by $743.80.)

What You'll Learn