Discounted Payback Versus NPV

Why Compare Discounted Payback With NPV?

Discounted payback asks when a project’s discounted cash inflows have recovered its initial cost. NPV asks how much value remains after discounting all project cash flows over the full evaluation horizon. Both use the time value of money, but they answer different questions: how long until discounted recovery? and what is the project’s total net present value?

The distinction matters when recovery occurs late, does not occur within a chosen horizon, or depends on a large terminal receipt. A project can meet an NPV test and still fail a manager’s payback deadline. Conversely, a short payback does not by itself show that a project creates the most value. These measures should be interpreted using the same cash-flow assumptions and clearly stated horizons.

Definition: Let $I_0>0$ be an initial investment at time 0, and let $CF_t$ be the project’s net cash flow at the end of year $t$. At a constant effective annual discount rate $i$, the discounted cumulative cash flow through year $k$ is $$ C_k=-I_0+\sum_{t=1}^{k}\frac{CF_t}{(1+i)^t}. $$ The discounted payback period is the first time this cumulative amount reaches or exceeds zero. If it remains below zero through the stated evaluation horizon, discounted payback is not achieved within that horizon.

The NPV over a project horizon of $n$ years is the discounted cumulative cash flow at the end of that horizon:

NPV=-I_0+\sum_{t=1}^{n}\frac{CF_t}{(1+i)^t}=C_n.

These formulas assume that the cash flows are measured at year-end and that the same annual effective rate applies at each date. The cash flows should be net amounts: if a year includes both receipts and costs, combine them before discounting. For a conventional project, the initial outlay is followed only by nonnegative net cash flows. If later cash flows can be negative, the cumulative amount may rise above zero and then fall below it again, so the timing and meaning of “recovery” require care.

Calculating Fractional-Year Discounted Payback

With cash flows recorded only at year-end, recovery occurs at a year-end date if the cumulative amount first becomes nonnegative then. In practice, a fractional-year estimate is often used when the cumulative discounted amount is still negative at year $k$, but would be nonnegative after the next year’s discounted cash flow. The estimate treats that next discounted cash flow as if it accrued evenly during the year.

\text{Estimated discounted payback} = k+\frac{-C_k}{CF_{k+1}/(1+i)^{k+1}}.

Here, $-C_k$ is the discounted amount still unrecovered at year $k$. The denominator is the present value of the next year’s net cash flow. This interpolation is an estimate, not an exact consequence of year-end-only cash-flow timing. If the project truly pays only at year-end, the actual recovery date is the relevant year-end.

Worked Example: Recovery Late in the Project

A hypothetical project requires $I_0=\$1{,}000$ at time 0. It pays $\$300$ at each year-end in years 1, 2, and 3, and $\$800$ at the end of year 4. Use a constant effective annual discount rate of $i=10\%$. Find NPV and estimated fractional-year discounted payback. Use unrounded present values in totals, then round dollar amounts to cents and payback to two decimal places.

Method. Discount each dated cash flow at 10%, find the cumulative discounted amount immediately before the recovery interval, then interpolate within that interval. This avoids treating undiscounted dollars as if they were equivalent across years.

The present values of the four inflows are:

\begin{aligned} PV_1&=\frac{300}{1.10}=272.7273,\\ PV_2&=\frac{300}{1.10^2}=247.9339,\\ PV_3&=\frac{300}{1.10^3}=225.3944,\\ PV_4&=\frac{800}{1.10^4}=546.4108. \end{aligned}

The cumulative discounted inflows through year 3 are $272.7273+247.9339+225.3944=746.0556$. The amount still unrecovered after year 3 is therefore $1{,}000-746.0556=253.9444$. The discounted year-4 inflow is $546.4108$, so the estimated fraction of year 4 required is:

\frac{253.9444}{546.4108}=0.46475\text{ (approximately)}.

Estimated discounted payback is $3+0.46475=3.46475$, or 3.46 years to two decimal places. Using the unrounded discounted inflows, their total is $1{,}292.4664$, so:

NPV=-1{,}000+1{,}292.4664=\$292.47.

Check. The cumulative amount is still negative after year 3, because $-1{,}000+746.0556=-253.9444$, and is positive by the end of year 4, because NPV is positive. Thus recovery occurs during the fourth interval under the interpolation convention. The common mistake avoided here is rounding each present value before summing; the total and NPV are calculated from unrounded values.

When Recovery Is Not Achieved

A negative NPV for a conventional project means that the discounted inflows do not recover the initial cost over the full horizon used in that NPV calculation. Discounted payback is then not achieved within that same horizon. A positive NPV has the opposite implication for a conventional project evaluated over its complete life: discounted payback must occur by the end of that life. This follows because its cumulative discounted cash flow starts below zero and never decreases as nonnegative inflows arrive.

The qualification about the horizon is essential. A decision rule may impose a shorter payback cutoff than the full horizon used for NPV. In that case, positive NPV does not imply recovery by the cutoff: the project may receive enough discounted cash flow later to produce positive NPV overall. The project has not met the cutoff, even though it has recovered the investment over its full evaluation horizon.

Worked Example: No Recovery Within the Full Project Horizon

A hypothetical four-year project costs $\$1{,}000$ at time 0 and pays $\$300$ at each of the four year-ends. The effective annual discount rate is $10\%$. Find the NPV and determine whether discounted payback occurs within four years. Report dollar amounts to cents, using unrounded present values.

Method. Sum the discounted inflows over the entire four-year life and compare that total with the initial cost. This checks recovery at the full horizon rather than assuming that nominal receipts add up to the investment.

The discounted inflows are $272.7273$, $247.9339$, $225.3944$, and $204.9040$ dollars. Using the unrounded values, their total is:

300\left(\frac{1}{1.10}+\frac{1}{1.10^2}+\frac{1}{1.10^3}+\frac{1}{1.10^4}\right) =950.9596.

Thus the NPV is $-1{,}000+950.9596=-\$49.04$. The cumulative discounted inflows are $272.7273$ after year 1, $520.6612$ after year 2, $746.0556$ after year 3, and $950.9596$ after year 4. Each amount is below $\$1{,}000$, so discounted payback is not achieved within the four-year horizon.

Check. The final cumulative discounted cash flow equals the NPV, $-\$49.04$, as it must. The common mistake avoided is summing individually rounded present values: the total is obtained from unrounded amounts, not from adding displayed values rounded to cents.

Why Positive NPV May Miss a Payback Cutoff

The relationship between NPV and recovery can be shown directly. For a conventional project, each discounted inflow is nonnegative, so $C_k$ cannot decrease as $k$ increases. If NPV is positive over the complete horizon $n$, then $C_n>0$. Since $C_0=-I_0<0$, the cumulative amount must have reached or passed zero by the end of the project. Therefore, for a conventional project, positive NPV over the full horizon implies discounted payback by that horizon.

It does not follow that recovery occurs by every shorter deadline. The NPV calculation may include valuable cash flows after that deadline. This can happen, for example, when a project has a large terminal receipt. If an analysis defines payback using operating inflows only but includes terminal proceeds in NPV, it must say so: those are different cash-flow bases, and one measure cannot be used to infer recovery under the other.

Worked Example: Positive NPV but No Recovery by the Cutoff

A hypothetical project requires $\$1{,}000$ initially. It generates $\$100$ at each year-end in years 1 through 4, plus a terminal receipt of $\$1{,}500$ at the end of year 5. The effective annual discount rate is $10\%$. A policy requires discounted payback within four years. Calculate the full five-year NPV and assess the cutoff. Round dollar amounts to cents and the estimated full-life payback to two decimals.

Method. Evaluate the cutoff using cash flows through year 4, but calculate NPV using all five years. This keeps the two horizons explicit and prevents the terminal receipt from being accidentally omitted from NPV or included before it occurs.

The present value of the four annual $\$100$ receipts is:

100\left(\frac{1}{1.10}+\frac{1}{1.10^2}+\frac{1}{1.10^3}+\frac{1}{1.10^4}\right) =316.9865.

The terminal receipt has present value $1{,}500/1.10^5=931.3820$. Therefore the present value of all inflows is $316.9865+931.3820=1{,}248.3685$, and:

NPV=-1{,}000+1{,}248.3685=\$248.37.

At year 4, discounted inflows total only $\$316.9865$, so the unrecovered amount is $1{,}000-316.9865=\$683.0135$. The project therefore fails the four-year discounted-payback cutoff. Its year-5 terminal receipt has a present value of $\$931.3820$, more than the amount still unrecovered. Under the interpolation convention, the fraction of year 5 is $683.0135/931.3820=0.73333$, giving estimated full-life discounted payback of $4.73$ years.

Check. The NPV is positive, and the estimated recovery occurs during year 5, not by the four-year cutoff. This is consistent with the full-life result: positive NPV implies recovery by the end of year 5 for these nonnegative post-investment cash flows, but it does not imply recovery by year 4.

What Each Measure Tells You

NPV measures the net present value created over the stated horizon at the selected discount rate. Discounted payback measures how long it takes for discounted inflows to cover the initial investment, often with a separate maximum acceptable deadline. A payback cutoff is a screening rule, not a substitute for calculating NPV: it deliberately disregards cash flows after the cutoff.

Question	Discounted payback	NPV
What is measured?	Time until discounted recovery	Value of all included discounted cash flows, net of the initial outlay
What happens to cash flows after a cutoff?	They do not help meet a cutoff that has already passed	They are included if they fall within the stated NPV horizon
What does a positive result establish?	Recovery by the date when cumulative discounted cash flow first reaches zero	Present value of included inflows exceeds the initial outlay
What does the measure not establish alone?	Total value created over the complete horizon	Whether recovery occurs before a shorter deadline

A useful comparison begins by writing down the discount rate, cash-flow dates, project horizon, and any payback cutoff. Then calculate both measures from the same underlying cash flows, while keeping the NPV horizon and the cutoff distinct. If a project contains a terminal receipt, state whether it is included in the payback definition. Consistent definitions make the comparison interpretable.

Common Mistakes and Exam Traps

Claiming that positive NPV never implies payback. For a conventional project with only nonnegative future cash flows, positive NPV over the complete horizon does imply discounted recovery by the end of that horizon. It does not guarantee recovery by a shorter cutoff.
Comparing different horizons without saying so. A four-year payback cutoff can fail even when a five-year NPV is positive. Report both periods explicitly.
Using nominal cash flows for discounted payback. Compare the initial outlay with present values of future cash flows, not with their undiscounted total.
Rounding each present value before adding. Small rounding differences can change a reported cumulative amount or NPV. Calculate totals from unrounded values, then round the final result.
Treating fractional-year interpolation as an exact payment date. The fractional estimate assumes the next discounted cash flow accrues evenly within the year. With year-end-only payments, actual recovery occurs at a payment date.
Ignoring the cash-flow basis. If NPV includes a terminal receipt but an operating-payback measure excludes it, say so; those measures do not track the same set of cash flows.

Key takeaway: Discounted payback is a recovery-time measure; NPV is a value measure over a stated horizon. For a conventional project, positive NPV implies discounted recovery by the end of the complete horizon, but it need not satisfy a shorter payback cutoff.

Check Your Understanding

Use year-end cash flows, a constant effective annual discount rate, and unrounded present values in intermediate totals. Round dollar answers to cents and fractional payback to two decimals.

A project costs $\$800$ and pays $\$300$ at the ends of years 1 and 2, then $\$400$ at the end of year 3. At $i=5\%$, calculate NPV and estimated discounted payback. (Answer: NPV = $\$103.36$; discounted payback $\approx 2.70$ years.)
A project costs $\$1{,}200$ and pays $\$250$ at each year-end for four years. At $i=8\%$, determine NPV and whether discounted payback occurs within the four-year life. (Answer: NPV $\approx-\$371.97$; no discounted payback within four years.)
A project costs $\$700$, pays $\$100$ at each of the first three year-ends, and pays $\$900$ at the end of year 4. At $i=10\%$, a policy requires payback within three years. Find the full-horizon NPV and explain whether the cutoff is met. (Answer: NPV $\approx\$163.40$; the three-year cutoff is not met.)
A project costs $\$1{,}000$ and pays $\$200,\ \$250,\ \$300,$ and $\$700$ at the ends of years 1 through 4, respectively. At $i=10\%$, calculate the four-year NPV and estimated discounted payback, and assess a three-year cutoff. (Answer: NPV $\approx\$91.93$; discounted payback $\approx3.81$ years, so the three-year cutoff is not met.)

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