Comparing Projects on a Consistent Basis
Two projects can differ in both the amount invested and the dates on which they return money. One might have a higher percentage return, while another creates more dollars of value. A comparison is meaningful only after specifying the measure, the valuation date, and the rates used. This tutorial compares projects using net present value (NPV), internal rate of return (IRR), and terminal wealth.
For NPV, value every dated cash flow at a common valuation date using the stated discount rate or rates. For IRR, find the rate that makes the project's NPV equal to zero. For terminal wealth, accumulate the signed cash flows to a common future date using the stated reinvestment rates. These measures answer related but distinct questions; they should not be treated as interchangeable without checking their assumptions.
An IRR is not automatically a unique rate. With one initial outflow followed only by inflows, the NPV typically decreases as the discount rate increases, giving at most one economically relevant root. When cash flows change sign more than once, there can be multiple IRRs or no real IRR. NPV at a stated rate remains directly interpretable even in those cases.
Why NPV and Terminal Wealth Can Agree
Suppose a project has cash flows at times \(0,1,\ldots,T\), and the same flat effective annual rate \(i\) is used both to discount to time 0 and to accumulate to time \(T\). The present value of each cash flow, accumulated \(T\) years, becomes that cash flow at its terminal value:
The right-hand side is the terminal wealth. Therefore, for a common horizon and the same flat rate,
Because \((1+i)^T\) is positive, the two measures give the same ranking under these conditions. This equivalence does not say that any terminal wealth calculation must match NPV: a different reinvestment schedule, a different horizon, or inconsistent rate assumptions breaks the stated relationship.
Write each project’s cash flows on a timeline and choose one valuation date for NPV and one common terminal date for terminal wealth.
Use the given discount rate or spot rates for NPV. For terminal wealth, identify the accumulation rate applicable in each period.
Use NPV to compare value added at the valuation date, IRR to compare percentage break-even rates, and terminal wealth to compare net value at the selected future date.
Worked Comparisons
Worked Example: A Higher IRR but a Lower NPV
Two mutually exclusive projects each last one year. Project S requires $100 now and pays $150 at year 1. Project L requires $1,000 now and pays $1,300 at year 1. Use a 10% effective annual discount rate. Compare their NPVs, IRRs, and year-1 terminal wealth. Then assess the incremental investment in L over S.
Method. Calculate both projects at the same 10% rate and on the same dates. Since they are mutually exclusive, NPV measures the dollar value added at the common rate; IRR alone does not account for project scale.
Project S NPV. $$ \operatorname{NPV}_S=-100+\frac{150}{1.10} =-100+136.3636 =\$36.36. $$ Its IRR solves \(-100+150/(1+r)=0\), so \(1+r=150/100=1.5\), giving an IRR of 50%.
Project L NPV. $$ \operatorname{NPV}_L=-1{,}000+\frac{1{,}300}{1.10} =-1{,}000+1{,}181.8182 =\$181.82. $$ Its IRR solves \(-1{,}000+1{,}300/(1+r)=0\), so \(1+r=1.3\), giving an IRR of 30%.
S has the higher IRR, but L has the higher NPV at the stated 10% rate. To compare year-1 terminal wealth using that same rate, accumulate the initial investment to year 1 and add the year-1 receipt: $$ W_{1,S}=-100(1.10)+150=\$40.00,\qquad W_{1,L}=-1{,}000(1.10)+1{,}300=\$200.00. $$ The terminal-wealth difference is \(200-40=\$160.00\), which equals \(1.10(181.8182-36.3636)=\$160.00\), checking the NPV-to-terminal-wealth relationship.
Incremental check. Choosing L instead of S requires an additional $900 at time 0 and produces an additional $1,150 at year 1. Its incremental NPV is \(-900+1{,}150/1.10=\$145.45\), and its incremental IRR is \(1{,}150/900-1=27.78\%\). Since the incremental NPV is positive at 10%, L is preferred if only one project can be selected. The common mistake is to select S solely because its stand-alone IRR is higher.
Worked Example: Different Payment Dates and a Stated Reinvestment Schedule
Project A costs $100 now and pays $120 at year 1. Project B costs $100 now and pays $127 at year 2. Calculate both NPVs at a flat 8% effective annual discount rate and both IRRs. Then calculate terminal wealth at year 2 using a 4% reinvestment rate from year 0 to 1 and a 10% rate from year 1 to 2.
NPV and IRR for A. $$ \operatorname{NPV}_A=-100+\frac{120}{1.08} =-100+111.1111 =\$11.11. $$ Its IRR satisfies \(100=120/(1+r)\), so \(r=120/100-1=20\%\).
NPV and IRR for B. $$ \operatorname{NPV}_B=-100+\frac{127}{(1.08)^2} =-100+\frac{127}{1.1664} \approx -100+108.8820 =\$8.88. $$ Its IRR satisfies \(100=127/(1+r)^2\), so \(1+r=\sqrt{1.27}\) and \(r\approx12.69\%\).
Terminal wealth for A. The initial outflow accumulates for both years, while the year-1 receipt accumulates for one year at 10%: $$ W_{2,A}=-100(1.04)(1.10)+120(1.10) =-114.40+132.00 =\$17.60. $$ Terminal wealth for B. Its only receipt arrives at year 2, so it is not accumulated further: $$ W_{2,B}=-100(1.04)(1.10)+127 =-114.40+127.00 =\$12.60. $$ A ranks higher by NPV, IRR, and terminal wealth in this case. However, the terminal-wealth figures do not follow from multiplying the 8% NPVs by \((1.08)^2\): the terminal calculation uses a different, period-specific reinvestment schedule. The common mistake is to accumulate a year-1 receipt for two years, or to apply the 8% valuation rate to terminal wealth without being asked to do so.
Worked Example: A Project with More Than One Interim Cash Flow
A project costs $500 now, pays $200 at year 1, and pays $400 at year 2. Calculate its NPV at an 8% effective annual discount rate, its IRR, and year-2 terminal wealth using a 6% annual accumulation rate.
NPV. Discount each receipt by its own time: $$ \operatorname{NPV}(8\%)=-500+\frac{200}{1.08}+\frac{400}{(1.08)^2} =-500+185.1852+342.9355 =\$28.12. $$
IRR. Set NPV equal to zero and let \(x=1+r\): $$ -500+\frac{200}{x}+\frac{400}{x^2}=0 \quad\Longrightarrow\quad 500x^2-200x-400=0. $$ The positive solution is $$ x=\frac{200+\sqrt{840{,}000}}{1{,}000} \approx 1.116515, \qquad r\approx11.65\%. $$ The other quadratic root is negative and cannot equal \(1+r\) for a conventional nonnegative annual interest rate.
Terminal wealth. Accumulate the initial outflow for two years and the year-1 receipt for one year, all at 6%: $$ W_2=-500(1.06)^2+200(1.06)+400 =-561.80+212.00+400.00 =\$50.20. $$ The arithmetic checks: \((1.06)^2=1.1236\), so the accumulated initial outflow is \(-\$561.80\). The 6% terminal rate differs from the 8% NPV rate, so the NPV-to-terminal-wealth equivalence does not apply. A common error is to solve the IRR equation with a sign or coefficient error; substituting the reported rate into the NPV equation should give a value close to zero, subject to rounding.
Interpretation and Common Traps
NPV answers, “How much value does this project add at the stated valuation rate?” IRR answers, “What rate makes the project’s NPV zero?” Terminal wealth answers, “What is the net accumulated value at the stated future date under the specified accumulation assumptions?” Each answer is tied to its rate and date assumptions.
- Ranking by stand-alone IRR when scale differs. A small project can have a high percentage return but add fewer dollars of value. For mutually exclusive projects, compare NPVs at a common appropriate rate; incremental analysis can clarify the effect of choosing the larger project.
- Using different horizons without alignment. Terminal values at different dates are not directly comparable unless they are brought to a common date using an explicit accumulation assumption.
- Accumulating every receipt for the full project term. A receipt at year \(t\) accumulates only from \(t\) to the chosen terminal date, not from time 0.
- Assuming IRR is always unique. If project cash flows change sign repeatedly, the IRR equation may have multiple roots. Report the issue rather than treating one root as an unambiguous ranking.
- Mixing valuation and reinvestment rates. A spot or discount rate used for NPV does not automatically specify the realized rate for accumulating interim proceeds. Use the rates stated for each calculation.
Check Your Understanding
For each problem, keep cash-flow timing and rate assumptions explicit.
- Project M costs $200 now and pays $260 at year 1. At a 10% effective annual discount rate, calculate its NPV and IRR, then find year-1 terminal wealth using 10%. (Answer: NPV = $36.36; IRR = 30%; terminal wealth = $40.00.)
- A project costs $300 now, pays $180 at year 1, and pays $180 at year 2. At a 10% effective annual rate, calculate its NPV and year-2 terminal wealth using 10% accumulation. (Answer: NPV = $12.40; terminal wealth = $15.00.)
- Project A costs $1,000 now and pays $600 at each of years 1 and 2. Project B costs $900 now and pays $1,100 at year 2. At a flat 10% discount rate, calculate both NPVs. Then use reinvestment rates of 4% from year 0 to 1 and 7% from year 1 to 2 to calculate each year-2 terminal wealth. (Answer: NPV A = $41.32 and NPV B = $9.09; terminal wealth A = $129.20 and terminal wealth B = $98.48.)
- A project has cash flows of \(-\$100\) at time 0, \(+\$230\) at year 1, and \(-\$132\) at year 2. Find all IRRs by setting \(x=1+r\). (Answer: The two IRRs are 10% and 20%, so there is no unique IRR.)
- A project costs $200 now, pays $100 at year 1, and $150 at year 2. At a flat 5% rate, calculate its NPV and year-2 terminal wealth. Check the relationship between them. (Answer: NPV = $31.29; terminal wealth = $34.50, equal to \(\operatorname{NPV}(5\%)(1.05)^2\) to rounding.)