Integrated NPV Project-Comparison Workshop

Introduction

A project comparison can look straightforward until the cash flows occur at uneven dates, the investment is spread across several stages, or each project has a different required return. The reliable approach is to put every receipt and outlay on a dated timeline, assign its sign, and discount it using the rate and time period appropriate to that project.

This workshop combines those steps. The focus is not on forecasting cash flows, but on translating a stated forecast into present values and making a consistent selection. A positive NPV means the project adds value relative to its stated discount-rate benchmark; among mutually exclusive alternatives, the highest NPV is the preferred choice, provided accepting no project is also considered when all NPVs are negative.

Definition: Let $C_j$ be the signed net cash flow at time $t_j$, measured in years from the valuation date. A receipt is positive and an outlay is negative. If the effective annual discount rate is $i$, is constant over the project horizon, and fractional years are discounted using fractional powers, then $$ \operatorname{NPV}(i)=\sum_{j=0}^{n} C_j(1+i)^{-t_j} =\sum_{j=0}^{n} C_jv^{t_j}, \qquad v=\frac{1}{1+i}. $$ The cash flow at time zero is not discounted. This formula assumes that the stated rate is appropriate for the project and that all cash flows are measured in consistent monetary units and on the same valuation date.

A Reliable Comparison Procedure

A calculation is only as sound as its timeline. Record the valuation date as time zero, then place each cash flow at its actual date. A staged payment made after six months belongs at $t=0.5$, not at $t=1$ merely because it is the first payment after the initial investment. A receipt at the end of year three belongs at $t=3$, even if no cash flow occurs at year two.

Use one sign convention throughout: investments and other project costs are negative; operating receipts, salvage proceeds, and other benefits are positive. If an amount at a date combines a receipt and an outlay, enter their net amount once, or enter the two signed amounts separately. Either method gives the same NPV.

Set the valuation date and timeline.
Measure each cash-flow date from the same time-zero date, using years if the rate is effective annual.

Assign signs and identify all amounts.
Include the initial outlay, later staged outlays, intermediate receipts, and any terminal receipt or disposal cost.

Choose the applicable discount rate.
Use the rate specified for that project, and do not switch rates partway through its cash-flow stream.

Discount each amount at its own date.
Multiply $C_j$ by $(1+i)^{-t_j}$. Retain the time-zero amount unchanged.

Compare NPVs and test the decision.
For mutually exclusive projects, rank by NPV. If doing nothing is permitted, a project with negative NPV should not be selected over that zero-NPV alternative.

Why Discounting Each Date Works

Under the constant effective annual rate assumption, one dollar invested at time zero accumulates to $(1+i)^t$ dollars by time $t$. Therefore, the time-zero amount equivalent to $C_j$ received or paid at time $t_j$ is the amount that accumulates to $C_j$:

\text{present value}\times(1+i)^{t_j}=C_j \quad\Longrightarrow\quad \text{present value}=C_j(1+i)^{-t_j}.

Present values are additive: amounts at different dates can be translated to the same valuation date and then summed. This gives the NPV formula. The argument also explains why actual dates matter: changing $t_j$ changes the accumulation period and therefore changes the present value. For an effective annual rate, using a fractional exponent for a fraction of a year assumes that the same annual accumulation model applies over that fraction.

For two alternatives $A$ and $B$ evaluated at the same rate, subtracting their NPVs gives the NPV of the incremental cash flows $B-A$. This follows directly from the formula:

\operatorname{NPV}_B(i)-\operatorname{NPV}_A(i) =\sum_j \bigl(C_{B,j}-C_{A,j}\bigr)(1+i)^{-t_j}.

A positive difference means $B$ has greater NPV than $A$ at that rate. If projects have different appropriate discount rates, calculate each NPV using its own stated rate; subtracting those two NPVs can rank the alternatives, but it is not the NPV of a single incremental cash-flow stream discounted at one common rate.

Worked Example: Comparing Two Projects with Staged Outlays

Two mutually exclusive projects have the following cash flows, in dollars. Project A requires $1{,}000$ now and a further $300$ at the end of year one; it pays $900$ at the ends of years two and three. Project B requires $700$ now and $500$ at the end of year one; it pays $700$ at the end of year two and $1{,}000$ at the end of year three. The effective annual discount rate is $10\%$ for either project. Which project has the greater NPV?

Method. Use the same annual effective rate for both projects, discount each dated cash flow, and compare the resulting NPVs. The common mistake to avoid is treating a later staged outlay as though it occurred at time zero.

For Project A, the signed timeline is $-1{,}000$ at $t=0$, $-300$ at $t=1$, $+900$ at $t=2$, and $+900$ at $t=3$. The discount factors are $1$, $1/1.1$, $1/1.1^2$, and $1/1.1^3$. Thus

\begin{aligned} \operatorname{NPV}_A &=-1{,}000-\frac{300}{1.1} +\frac{900}{1.1^2}+\frac{900}{1.1^3}\\ &=-1{,}000-272.727273+743.801653+676.183320\\ &=147.257700. \end{aligned}

For Project B, the signed timeline is $-700$ at $t=0$, $-500$ at $t=1$, $+700$ at $t=2$, and $+1{,}000$ at $t=3$:

\begin{aligned} \operatorname{NPV}_B &=-700-\frac{500}{1.1} +\frac{700}{1.1^2}+\frac{1{,}000}{1.1^3}\\ &=-700-454.545455+578.512397+751.314801\\ &=175.281743. \end{aligned}

Both NPVs are positive, and Project B exceeds Project A by $175.281743-147.257700=28.024043$. Rounded to the nearest cent, the NPVs are $ \$147.26$ and $ \$175.28$, respectively. Select Project B. As a check on the comparison, its incremental cash flows relative to A are $+300$ at time zero, $-200$ at year one, $-200$ at year two, and $+100$ at year three; their NPV is $300-200/1.1-200/1.1^2+100/1.1^3=\$28.02$, consistent with the difference in project NPVs.

Irregular Dates and Terminal Cash Flows

When dates are not equally spaced, do not count the number of payments as the exponent. The exponent is elapsed time from valuation. For example, a cash flow at 2.5 years is discounted by $(1+i)^{-2.5}$, even if it is only the third cash flow on the timeline. The final receipt is treated in the same way as any other dated cash flow: it is included once, with its sign, and discounted from its actual date.

A terminal amount may combine several items, such as a final operating receipt, a sale proceeds amount, and a cleanup cost. If the problem supplies a net terminal amount, use that amount once. If it lists the components separately, preserve each sign and add their discounted values. Do not omit a terminal receipt simply because it occurs after the last regular operating payment.

Worked Example: Uneven Dates and a Half-Year Outlay

At a valuation date, Project A requires $ \$4{,}000$ immediately and $ \$1{,}000$ after six months. It is expected to pay $ \$2{,}500$ at year two and $ \$3{,}000$ at year three. Project B requires $ \$3{,}500$ immediately and $ \$600$ at year one; it pays $ \$2{,}000$ at year two and $ \$3{,}000$ at year three. Both projects are mutually exclusive and use an effective annual discount rate of $8\%$. Compare their NPVs.

Method. Use times in years and discount the half-year outlay at $t=0.5$. The common mistake to avoid is discounting it for a full year because it is the first payment after time zero. The relevant factors, carried at full precision in the calculations, are $(1.08)^{-0.5}=0.962250449$, $(1.08)^{-1}=0.925925926$, $(1.08)^{-2}=0.857338820$, and $(1.08)^{-3}=0.793832241$.

Project A has outlays of $ \$4{,}000$ at $t=0$ and $ \$1{,}000$ at $t=0.5$. Its receipts are $ \$2{,}500$ at $t=2$ and $ \$3{,}000$ at $t=3$:

\begin{aligned} \operatorname{NPV}_A &=-4{,}000-\frac{1{,}000}{1.08^{0.5}} +\frac{2{,}500}{1.08^2}+\frac{3{,}000}{1.08^3}\\ &=-4{,}000-962.250449+2{,}143.347051+2{,}381.496723\\ &=-437.406675. \end{aligned}

For Project B, the time-one outlay has present value $600(0.925925926)=555.555556$. The year-two receipt has present value $2{,}000(0.857338820)=1{,}714.677641$, and the year-three receipt has present value $3{,}000(0.793832241)=2{,}381.496723$. Therefore

\begin{aligned} \operatorname{NPV}_B &=-3{,}500-555.555556+1{,}714.677641+2{,}381.496723\\ &=40.618808. \end{aligned}

To the nearest cent, Project A has NPV $-\$437.41$ and Project B has NPV $ \$40.62$. Project B is preferred and has positive NPV. The half-year timing matters: discounting A's $ \$1{,}000$ outlay for a full year instead would use $1{,}000/1.08=\$925.93$, understating the present value of that cost by about $ \$36.32$. The corrected half-year present value is $ \$962.25$.

Project-Specific Discount Rates

A project-specific discount rate reflects the benchmark specified for that project's cash flows. When the problem supplies different rates for different projects, use each rate consistently for every dated cash flow of its project. Do not apply the lower rate to a project's costs and the higher rate to its receipts, or use one project's rate merely to make the alternatives comparable.

The resulting NPVs answer whether each project adds value relative to its own stated benchmark. For mutually exclusive alternatives, choose the higher NPV, while also checking whether doing nothing is allowed. This comparison relies on the rates being appropriate as given; if the rates or cash-flow assumptions are not appropriate, the arithmetic alone cannot establish a sound decision.

Worked Example: Comparing Projects at Different Required Rates

Two mutually exclusive proposals are being evaluated. Project X has a $ \$5{,}000$ initial outlay, a further $ \$1{,}000$ outlay at year one, and receipts of $ \$3{,}500$ at year two and $ \$4{,}000$ at year three. Its specified effective annual discount rate is $8\%$. Project Y has a $ \$4{,}500$ initial outlay, a further $ \$1{,}500$ outlay at year one, and receipts of $ \$3{,}000$ at year two and $ \$4{,}500$ at year three. Its specified effective annual discount rate is $12\%$. Assume the stated rates are appropriate. Which proposal should be selected?

Method. Calculate each NPV at its own rate, then compare. The common mistake to avoid is using the same rate for both projects just because they are alternatives.

At $8\%$, the factors for years one, two, and three are $0.925925926$, $0.857338820$, and $0.793832241$. Project X's year-one outlay has present value $1{,}000(0.925925926)=\$925.925926$. Its receipts have present values $3{,}500(0.857338820)=\$3{,}000.685871$ and $4{,}000(0.793832241)=\$3{,}175.328964$. Thus

\begin{aligned} \operatorname{NPV}_X &=-5{,}000-925.925926+3{,}000.685871+3{,}175.328964\\ &=250.088909. \end{aligned}

At $12\%$, the factors for years one, two, and three are $0.892857143$, $0.797193878$, and $0.711780248$. Project Y's year-one outlay has present value $1{,}500(0.892857143)=\$1{,}339.285714$. Its receipts have present values $3{,}000(0.797193878)=\$2{,}391.581633$ and $4{,}500(0.711780248)=\$3{,}203.011115$. Therefore

\begin{aligned} \operatorname{NPV}_Y &=-4{,}500-1{,}339.285714+2{,}391.581633+3{,}203.011115\\ &=-244.692966. \end{aligned}

Rounded to the nearest cent, Project X has NPV $ \$250.09$, while Project Y has NPV $-\$244.69$. Select Project X. If the choice were instead between Project Y and doing nothing, doing nothing would be preferable under these assumptions because its NPV is zero, which exceeds Project Y's negative NPV.

Common Checks Before Making the Decision

A careful final review catches many errors that a calculator will not flag. Check the timeline against the wording, particularly phrases such as “immediately,” “after six months,” and “at the end of year three.” Check that every outlay has a negative sign and every receipt a positive sign. Confirm that the rate period and time unit agree: an effective annual rate paired with years uses year-based exponents.

Do not move a staged outlay to time zero. It is discounted from its actual date, just like a receipt.
Do not omit the terminal amount. Include a terminal receipt or cost at the stated final date.
Do not discount a time-zero flow. Its discount factor is $1$.
Do not confuse ranking with acceptance. The highest NPV among projects can still be negative; if the no-project alternative is available, compare it with zero.
Do not apply one project’s rate to another project when distinct project-specific rates are supplied.

These checks support the central comparison principle: a project’s NPV is the sum of its signed cash flows translated to a common valuation date under the stated discounting assumptions. Correct dates and signs are essential because an error in either can change the measured value and, potentially, the decision. This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each question, organize the timeline before evaluating the NPV. Round monetary answers to the nearest cent.

At an effective annual rate of $10\%$, a project requires $ \$1{,}000$ now and $ \$200$ at the end of year one. It pays $ \$700$ at the end of year two and $ \$700$ at the end of year three. Find its NPV. (Answer: $-\$77.39$)
Two mutually exclusive projects use the same effective annual rate of $10\%$. Project A has cash flows $-\$900$ at time zero, $-\$100$ at year one, $+\$600$ at year two, and $+\$600$ at year three. Project B has cash flows $-\$650$ at time zero, $-\$300$ at year one, $+\$500$ at year two, and $+\$650$ at year three. Find both NPVs and state the decision if doing nothing is available. (Answer: NPV of A $=-\$44.25$, NPV of B $=-\$21.15$; choose neither.)
A project has a $ \$2{,}000$ outlay now, a $ \$500$ outlay at year one, a $ \$1{,}200$ receipt at year two, and a $ \$1{,}600$ receipt at year three. At an effective annual rate of $10\%$, calculate its NPV and state whether it adds value relative to that benchmark. (Answer: NPV $=-\$260.71$; it does not add value.)
At an effective annual rate of $10\%$, a project requires $ \$1{,}000$ now and $ \$400$ after six months. It pays $ \$900$ at year two and $ \$900$ at year three. Calculate its NPV using the actual half-year date. (Answer: $ \$38.60$)
A project has cash flows $-\$500$ now, $-\$500$ at year one, $+\$700$ at year two, and $+\$500$ at year three. At an effective annual rate of $10\%$, calculate its NPV and decide whether it should be chosen over doing nothing. (Answer: NPV $=-\$0.38$; choose doing nothing.)

What You'll Learn