NPV for a Single-Outlay Project

Valuing a Single-Outlay Project

A project with one initial outlay and later benefits has a simple cash-flow structure: an amount is paid at the valuation date, and benefits arrive at specified future dates. The central question is whether the discounted value of those benefits is enough to offset the outlay. A related question is how large a repeated benefit must be for the project to break even at a chosen interest rate.

The timing of each benefit matters. A dollar received sooner has a greater present value than a dollar received later when the effective annual interest rate is positive. Therefore, the benefit amounts should be discounted from their actual dates rather than added as though every dollar arrived immediately.

Definition: Suppose a project requires a single outlay $C$ at time $0$, followed by benefits $B_j$ at times $t_j$, where each $t_j>0$ is measured in years. At effective annual rate $i>-1$, the project’s net present value at time $0$ is

\operatorname{NPV}=-C+\sum_{j=1}^{n}\frac{B_j}{(1+i)^{t_j}}.

The outlay is a negative cash flow; the benefits are positive cash flows. For a nonnegative interest rate, each future benefit is discounted by the factor $v^{t_j}$, where $v=1/(1+i)$.

This model assumes the outlay occurs at time zero, the stated benefits are received in full at their stated dates, and one effective annual rate applies over the valuation period. When a date is a fraction of a year, the examples use annual compounding with accumulation $(1+i)^t$. The model does not itself account for uncertain benefits, taxes, or a different rate for each cash flow; those require additional assumptions.

Why Discounted Benefits Determine NPV

At the valuation date, each future receipt must be translated into an equivalent amount at time zero. A deposit of $1$ at time zero accumulates to $(1+i)^t$ after $t$ years under the stated effective annual rate. Consequently, an amount $B$ payable at time $t$ has present value $B/(1+i)^t$. Since present values of cash flows can be added, the project’s total benefit value is the sum of those discounted amounts. Subtracting the initial outlay gives NPV.

This also gives a direct break-even equation. If every benefit is the same amount $B$, received at times $t_1,\ldots,t_n$, then

\operatorname{NPV}=-C+B\sum_{j=1}^{n}(1+i)^{-t_j}.

The coefficient of $B$ is positive, so NPV increases as $B$ increases. Setting NPV equal to zero therefore gives the unique break-even benefit amount; any larger amount gives positive NPV, and any smaller amount gives negative NPV.

B_{\text{break-even}}= \frac{C}{\displaystyle\sum_{j=1}^{n}(1+i)^{-t_j}}.

If some benefits are fixed and only certain receipts depend on an unknown amount $B$, first subtract the present value of the fixed benefits from the outlay. Then divide the remaining amount by the sum of the discount factors for the receipts that depend on $B$.

Worked Example: NPV From Three Dated Benefits

A hypothetical project requires an outlay of $12,000 now. It is expected to produce benefits of $5,000 at the end of year 1, $6,000 at the end of year 2, and $7,000 at the end of year 3. Find NPV at an effective annual rate of 10%, rounded to the nearest cent.

Method. Discount each benefit separately to time zero, add the present values, and subtract the outlay. This avoids treating later receipts as if they were available at the project start.

The discount factors for years 1, 2, and 3 are:

v=\frac{1}{1.10}=0.9090909,\qquad 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.

The present value of each benefit is:

\begin{aligned} PV_1&=5{,}000(0.9090909)=4{,}545.4545,\\ PV_2&=6{,}000(0.8264463)=4{,}958.6777,\\ PV_3&=7{,}000(0.7513148)=5{,}259.2036. \end{aligned}

Thus the total present value of benefits is $4{,}545.4545+4{,}958.6777+5{,}259.2036=14{,}763.3358$, and

\operatorname{NPV}=14{,}763.3358-12{,}000=2{,}763.3358.

Answer: NPV is $2,763.34. The check is that discounted benefits exceed the outlay by that amount, so the sign is positive.

Finding a Level Break-Even Benefit

When the same benefit is received on several dates, the task is not to compare the outlay with the undiscounted total. Instead, the present value of one unit of benefit is calculated at each date. Those discount factors add to give the present-value contribution of one dollar of the level benefit. Dividing the outlay by that combined factor gives the amount needed at each benefit date.

Worked Example: Minimum Equal Annual Benefit

A hypothetical project costs $19,520 at time zero and pays the same benefit $B$ at the ends of years 1, 2, and 3. Find the minimum $B$ that makes NPV zero at an effective annual rate of 25%.

Method. Set the NPV equal to zero and factor out the common benefit amount. The three payments are at the ends of years 1, 2, and 3, not at time zero.

At 25%, the discount factors are:

v=\frac{1}{1.25}=0.8,\qquad v^2=0.64,\qquad v^3=0.512.

The present-value factor for one dollar paid at each of the three dates is $0.8+0.64+0.512=1.952$. Therefore,

\begin{aligned} 0&=-19{,}520+B(1.952),\\ B&=\frac{19{,}520}{1.952}=10{,}000. \end{aligned}

Answer: The minimum equal benefit is $10,000 at each date. As a check, its present value is $10{,}000(0.8)+10{,}000(0.64)+10{,}000(0.512)=8{,}000+6{,}400+5{,}120=19{,}520$, exactly the outlay.

A quick sign check confirms the threshold: if each benefit were $9,500, the present value would be $9{,}500(1.952)=18{,}544$, giving NPV $18{,}544-19{,}520=-976$. A smaller benefit is not enough to break even.

Combining Fixed and Unknown Benefits

Sometimes some receipts are specified while one common amount remains unknown. In that case, include the present value of the specified receipts first, then solve for the unknown amount using only its own payment dates. This makes clear which benefits contribute to the break-even amount and prevents an unknown receipt from being counted at the wrong time.

Worked Example: Solving for Benefits at Selected Dates

A hypothetical project has a single outlay of $11,296 at time zero. It pays a fixed $5,000 at the end of year 1 and a fixed $4,000 at the end of year 3. It also pays an equal additional benefit $B$ at the ends of years 2 and 4. Find the minimum $B$ for NPV to be zero at an effective annual rate of 25%.

Method. Discount the fixed benefits and the unknown benefits at their respective dates. The fixed receipts reduce the amount of the outlay that must be covered by $B$.

At 25%, the required discount factors are $v=0.8$, $v^2=0.64$, $v^3=0.512$, and $v^4=0.4096$. The fixed benefits have present value

$$ 5{,}000(0.8)+4{,}000(0.512)=4{,}000+2{,}048=6{,}048. $$

The remaining present value needed to cover the outlay is $11{,}296-6{,}048=5{,}248$. The two unknown benefits have a combined discount factor of $0.64+0.4096=1.0496$, so

\begin{aligned} 5{,}248&=B(1.0496),\\ B&=\frac{5{,}248}{1.0496}=5{,}000. \end{aligned}

Answer: The minimum additional benefit is $5,000 at each of years 2 and 4. Checking the full present value gives $4{,}000+5{,}000(0.64)+2{,}048+5{,}000(0.4096)=4{,}000+3{,}200+2{,}048+2{,}048=11{,}296$, equal to the outlay.

Common Mistakes and Checks

A frequent error is to add the future benefits first and compare their undiscounted total with the initial outlay. That calculation ignores the time value of money. For example, with a $10,000 outlay and equal benefits $B$ at the ends of years 1 and 2 at 10%, using $2B=10{,}000$ gives the incorrect answer $B=$5,000$. The correct calculation is

B=\frac{10{,}000}{(1.10)^{-1}+(1.10)^{-2}} =\frac{10{,}000}{0.9090909+0.8264463} =\frac{10{,}000}{1.7355372} \approx 5{,}761.90.

At the incorrect amount of $5,000, the present value of benefits is only $5{,}000(1.7355372)=$8,677.69$, so NPV is $-$1,322.31$, not zero. The correction is to discount each receipt from its own date.

Also check that the discount period matches the cash-flow date. A receipt at the end of year 3 uses $v^3$, not $v^2$, and a receipt at time zero is not discounted. The interest rate and time unit must agree: the examples use an effective annual rate and times measured in years. If the rate is stated on another basis, it must be converted or matched to the period before applying the formula.

For a single outlay with nonnegative benefits and a nonnegative rate, a useful final check is the sign. If the present value of benefits is greater than the outlay, NPV is positive; if it is less, NPV is negative; if the two are equal, NPV is zero. When solving for a level benefit, substituting the answer back into the NPV equation verifies both the arithmetic and the timing.

Key takeaway: For a single-outlay project, discount each dated benefit to the valuation date, sum those present values, and subtract the outlay. To find a level break-even benefit, divide the amount that still needs to be covered by the sum of the discount factors at that benefit’s payment dates.

Check Your Understanding

For each problem, use the stated effective annual rate and value all cash flows at time zero.

A project costs $8,000 and pays an equal benefit at the ends of years 1 and 2. Find the break-even benefit at 10%, rounded to the nearest cent. (Answer: $4,609.52 at each date.)
A project costs $15,000 and pays an equal benefit at the ends of years 1 and 3. Find the break-even benefit at 20%, rounded to the nearest cent. (Answer: $10,622.95 at each date.)
A project costs $10,000 and pays a fixed $4,000 at the end of year 1, plus an equal unknown benefit at the ends of years 2 and 3. Find the break-even unknown benefit at 5%, rounded to the nearest cent. (Answer: $3,495.73 at each date.)
A project costs $12,000 and pays a fixed $3,000 at the end of year 2, plus an equal unknown benefit at the ends of years 1 and 4. Find the break-even unknown benefit at 25%, rounded to the nearest cent. (Answer: $8,333.33 at each date.)
A project costs $8,000 and pays $4,500 at each of the ends of years 1 and 2. Calculate NPV at 8%, rounded to the nearest cent, and state its sign. (Answer: NPV is $24.69, so it is positive.)

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