Source: APEC 3002 Lab Writeup, University of Minnesota Twin Cities
Tags: investment analysis, capital budgeting, NPV, net present value, IRR, internal rate of return, payback period, discount rate, cash flow, time value of money, managerial economics
Difficulty: Intermediate Prerequisites: Basic understanding of time value of money, discounting, and present value concepts (APEC 3002 earlier labs).
Big Picture
This lab sits at the heart of managerial economics: how a firm decides which long-term investments to pursue. Capital budgeting is the process of evaluating projects that require large upfront outlays and return cash over several years. The core question is straightforward: does a project create more value than it costs, once you account for the time value of money? You need to be comfortable with discounting future cash flows before this material will click. If you have not reviewed present value arithmetic, do that first.
Capital budgeting compares investment projects by converting their future cash flows into today's dollars. The three main tools are net present value (NPV), internal rate of return (IRR), and payback period. A project is worth taking if its NPV is positive at your required rate of return, and projects can be ranked against each other using NPV or IRR, though the two methods sometimes disagree.
Net present value (NPV)
The sum of all future cash flows from a project, each discounted back to today at a given interest rate, minus the initial investment. A positive NPV means the project adds value. In simple terms, it answers: "After accounting for the time value of money, does this investment make or lose money?"
Internal rate of return (IRR)
The discount rate at which a project's NPV equals exactly zero. Think of it as the break-even interest rate for the project. If the IRR exceeds your required rate of return (cost of capital), the project is worth doing.
Payback period
The number of years it takes for cumulative cash inflows to recoup the initial investment. In simple terms, it tells you how long until you get your money back. It ignores the time value of money entirely, which is its main weakness.
Discount rate / interest rate / cost of capital
The rate used to convert future cash flows into present values. It reflects the opportunity cost of tying up capital in this project rather than investing it elsewhere. In the lab, several rates (6.4%, 7.5%, 10%, 12.5%, 15%) are tested to see how sensitive each project's NPV is.
Net cash flow
The cash received minus the cash spent in a given year. Year 0 is typically a large negative number (the initial outlay), with subsequent years generating positive inflows if the project performs as expected.
Capital budgeting matrix
A table that compares several investment projects side by side across the same set of discount rates, showing each project's NPV at each rate. It lets you see which projects survive as the cost of capital rises.
The five projects
The lab evaluates five wind farm investment projects in Denmark: Skagen, Saeby, Hulsig South, Hulsig North, and Strandby. All run over an 8-year horizon (Year 0 through Year 8).
Skagen has the lowest initial outlay ($144,000) and produces level annual cash flows of $32,025 for years 1 through 8.
Saeby, Hulsig South, Hulsig North, and Strandby each require a $297,000 initial investment.
Saeby also produces level annual cash flows ($52,650 per year), which makes it easy to analyse but means it earns the same regardless of how conditions change.
Hulsig South has declining cash flows over time: $108,000 in Year 1, falling to -$54,000 in Year 8.
Hulsig North likewise declines, from $101,250 in Year 1 to $6,750 in Year 8.
Strandby has increasing cash flows: only $6,750 in Year 1, rising to $101,250 in Year 8.
Total undiscounted cash inflows over the 8 years range from $112,200 (Skagen) to $135,000 (Hulsig North and Strandby). But totals alone are misleading, because the timing of cash flows matters enormously once you discount.
Net present value at different discount rates
At 6.4%, every project has a positive NPV. Hulsig South leads ($60,016), followed by Hulsig North ($59,849), Skagen ($51,758), Saeby ($24,832), and Strandby ($6,320).
At 7.5%, Strandby drops to a negative NPV (-$10,560). All others remain positive.
At 10%, Saeby also turns negative (-$16,116), while Hulsig South ($31,296), Hulsig North ($26,774), and Skagen ($26,851) hold positive.
At 12.5%, only Skagen ($12,347), Hulsig South ($13,431), and Hulsig North ($6,785) remain positive.
At 15%, every project has a negative NPV. Strandby is the worst at -$98,221.
The key takeaway: projects with back-loaded cash flows (Strandby) are far more sensitive to changes in the discount rate than projects with front-loaded or level cash flows.
NPV ranking shifts with the discount rate
Hulsig South ranks first at every rate from 6.4% through 12.5%, while Strandby ranks last across all rates. Skagen moves from third at low rates to first at 15%. This illustrates a critical concept: the ranking of projects by NPV can change as the discount rate changes.
Internal rate of return
Skagen: 14.94%
Saeby: 8.49%
Hulsig South: 14.54%
Hulsig North: 13.41%
Strandby: 6.80%
Skagen has the highest IRR despite not having the highest NPV at most discount rates. This is because IRR is influenced by the ratio of cash flows to the initial investment, not just the absolute NPV. Skagen's smaller outlay ($144,000 vs $297,000) means it earns a higher percentage return.
Saeby and Strandby have the lowest IRRs, which matches the observation that their NPVs turn negative at lower discount rates.
Payback period (Hulsig South example)
The lab calculates the payback period for Hulsig South by tracking the cumulative cash balance year by year:
Year 0: -$297,000
Year 1: -$195,750 (after receiving $108,000, but minus $6,750 in the table shown as -$195,750)
Year 2: -$108,000
Year 3: -$33,750
Year 4: +$27,000
The investment is fully recovered between Year 3 and Year 4, giving a payback period of roughly 3.5 years. This is a relatively quick payback, which is consistent with Hulsig South's front-loaded cash flow pattern.
NPV vs IRR: when they disagree
IRR ranks Skagen first (14.94%) and Hulsig South second (14.54%), but NPV ranks Hulsig South first at most discount rates. The disagreement arises because the projects have different scales of investment ($144,000 vs $297,000). NPV measures absolute value created. IRR measures the percentage return per dollar invested. For mutually exclusive projects, NPV is generally the more reliable criterion.
Net present value
NPV = \sum_{t=0}^{n} \frac{CF_t}{(1 + r)^t}Where CF_t is the net cash flow in year t, r is the discount rate, and n is the number of years. Year 0 cash flow is typically negative (the initial investment).
Internal rate of return
The IRR is the value of r that satisfies:
0 = \sum_{t=0}^{n} \frac{CF_t}{(1 + IRR)^t}There is no closed-form solution; it is found by trial and error or using a spreadsheet's IRR function.
Payback period
No formula per se. Track cumulative cash flows year by year. The payback period is the year at which the cumulative balance first turns positive, interpolated for a fractional year if needed:
\text{Payback} = Y + \frac{|\text{Cumulative CF at end of year } Y|}{\text{Cash flow in year } Y+1}Where Y is the last full year with a negative cumulative balance.
Capital budgeting is how firms in every industry decide whether to build a factory, buy equipment, develop a new product, or expand into a new market. The wind farm projects in this lab mirror real energy-sector decisions where companies must commit large sums upfront and rely on revenue streams that stretch over many years. A utility company choosing between solar installations in different locations uses exactly this kind of NPV and IRR comparison. The sensitivity to discount rates also matters in practice: when central banks raise interest rates, projects that were borderline viable can suddenly become value-destroying.
Students often assume the project with the highest total (undiscounted) cash flows is the best investment. It is not. Timing matters. A dollar received in Year 1 is worth more than a dollar received in Year 8, so front-loaded cash flows produce higher NPVs at any positive discount rate.
Students frequently treat IRR and NPV as interchangeable ranking tools. They can disagree when projects differ in scale (initial outlay) or in the timing pattern of their cash flows. For mutually exclusive projects, NPV is the preferred criterion.
The payback period is sometimes treated as a complete investment criterion. It is not. It ignores all cash flows after the payback date and does not account for the time value of money. A project could have a fast payback but destroy value overall.
Some students believe a higher discount rate simply "reduces all projects equally." In reality, projects with later cash flows are hit much harder by a higher discount rate. Strandby's NPV collapses from +$6,320 to -$98,221 as the rate rises from 6.4% to 15%, while Skagen only drops from +$51,758 to -$294.
⚠️ Be prepared to calculate NPV given a set of cash flows and a discount rate. This is the most likely computational question.
⚠️ Know how to find the payback period from a cumulative cash flow table, including interpolation for a fractional year.
⚠️ Understand why NPV and IRR can rank projects differently, especially when initial investments differ. Expect a question that asks you to explain the conflict.
⚠️ Be able to explain why back-loaded cash flow projects are more sensitive to discount rate changes. The Strandby vs Skagen comparison is a clean example.
⚠️ The IRR is the rate where NPV = 0. If asked to sketch NPV profiles, the IRR is where each project's curve crosses the horizontal axis.
True or false: A project with a positive NPV at 10% will always have a positive NPV at 15%. False. A higher discount rate reduces NPV; it may flip from positive to negative.
Fill in the blank: The IRR is the discount rate at which a project's NPV equals ______. Zero.
True or false: The payback period accounts for the time value of money. False. It uses undiscounted cumulative cash flows.
True or false: If Project A has a higher IRR than Project B, Project A always has a higher NPV. False. IRR and NPV can rank projects differently, especially when project scales differ.
Fill in the blank: Projects with cash flows concentrated in later years are ______ sensitive to increases in the discount rate. More.
Q: A project requires a $297,000 investment and generates $52,650 per year for 8 years. At a discount rate of 10%, the NPV is -$16,116. What does this tell you about the project's IRR relative to 10%?
A: The IRR must be below 10%. Since the NPV is negative at 10%, the break-even discount rate (IRR) has already been passed. In this case the IRR is 8.49%.
Q: Hulsig South and Strandby both have initial investments of $297,000 and total undiscounted cash inflows that are identical in magnitude. Why does Hulsig South consistently outrank Strandby in NPV?
A: Hulsig South receives its largest cash flows in the early years (front-loaded), while Strandby's largest cash flows arrive in the later years (back-loaded). Because of discounting, earlier cash flows are worth more in present-value terms, so Hulsig South's NPV is higher at every positive discount rate.
Q: Skagen has the highest IRR (14.94%) but does not have the highest NPV at most discount rates. Explain why.
A: IRR measures percentage return relative to the capital invested. Skagen's initial outlay is only $144,000, compared to $297,000 for the other projects. It earns a high percentage return on a smaller base. NPV measures absolute value created, so the larger projects (Hulsig South, Hulsig North) can generate more total value despite a slightly lower percentage return.
Q: At what discount rate would you be indifferent between investing in Skagen and putting the money in an alternative investment?
A: At Skagen's IRR of 14.94%. At that rate, Skagen's NPV is zero, meaning it returns exactly the discount rate and no more.
Q: The payback period for Hulsig South is approximately 3.5 years. Name two limitations of using payback period as your sole investment criterion.
A: First, it ignores all cash flows after the payback date, so a project that collapses after Year 4 looks the same as one that keeps generating returns. Second, it does not discount cash flows, so it treats a dollar in Year 1 the same as a dollar in Year 3.
This material builds directly on the time value of money concepts from earlier in APEC 3002. Present value discounting is the same arithmetic used here, just applied to multi-year project cash flows rather than single payments. Capital budgeting also connects to the firm's cost of capital, which in a finance course would be derived from the weighted average cost of debt and equity (WACC). Understanding how NPV changes with the discount rate is a stepping stone to sensitivity analysis and risk assessment in more advanced courses.
Investment analysis, capital budgeting, net present value, NPV, internal rate of return, IRR, payback period, discount rate, cost of capital, cash flow analysis, time value of money, project evaluation, mutually exclusive projects, NPV profile, break-even discount rate, sensitivity analysis, wind farm investment, APEC 3002, managerial economics, applied microeconomics