FIN301 • Chapter 10 — Capital Budgeting

Intro

Capital budgeting evaluates long-term projects by turning future cash flows into today’s value or by finding required returns. For this course, focus on:

NPV (gold standard for value creation)
IRR (rate of return for a project’s cash flows)
Payback (speed of recovery; a screening tool)

Tip: For Excel, cash flows usually go in rows/columns. Make sure signs are right (CF0 < 0, later CFs generally > 0), and that periods are equally spaced.

Slides (PPTX)

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Featured Video — What is Capital Budgeting? (Importance, Methods & Limitations)

Short primer covering definitions, why it matters, common methods, and limitations.

Open on YouTube

Excel Syntax (Quick Cards)

NPV

=NPV(rate, value1, value2, ...)

Values occur at period ends
Add CF0 separately: NPV(...) + CF0

IRR

=IRR(values, [guess])

Array must include CF0 and all cash flows
Guess optional; helps convergence & chooses the root when multiple exist

Payback

Cumulative cash flows until the initial outlay is recovered

Simple and quick for screening
Does not measure total value creation

Calculators

Calculator on this website

Use the full Capital Budgeting Calculator

Need a larger calculator for NPV, IRR, and Payback? Use the calculator here:

Open Capital Budgeting Calculator ↗ Open Calculator I ↗

This page also has a built-in mini calculator right below for quick checks.

Mini Calculator on This Page

Quick check tool: enter CF0 (negative) and later cash flows (comma-separated). Example: -800, 350, 350, 350

Inputs

Discount rate (as %) Cash flows (CSV incl. CF0)

Guess for IRR (%) (blank = 10% Excel default)

Outputs

NPV: —

IRR (by guess): —

Payback (yrs): —

IRR roots (if multiple): —

Manual formulas used

NPV = Σ CF_t / (1+r)^t
IRR: if multiple sign changes, there can be multiple positive roots; we pick the root nearest your Guess (Excel behavior)
Payback = years until cumulative CF ≥ 0 + fraction of final year

In-Class Example — Single Project

WACC = 11%; CFs: CF0 = -800; CF1=350; CF2=350; CF3=350.

Expected (from your notes): NPV ≈ 55.30; IRR ≈ 14.93%; Payback ≈ 2.286 yrs.

Multi-Project Choice — IRR vs NPV

WACC = 7.5%. Mutually exclusive projects S and L:

Year	0	1	2	3	4
S	-1,100	550	600	100	100
L	-2,700	650	725	800	1,400

Reminder: “True value” is measured by NPV. IRR can rank differently when cash flow scale/timing differs.

NPV Profile — Project S vs L (Interactive)

Cash Flows (editable)

Project S

Year	CF

Project L

Year	CF

Controls

WACC (%) Profile Rmax (%)

Tip: Edit any CF cell and the chart refreshes automatically.

S — NPV@WACC: | IRR:

L — NPV@WACC: | IRR:

Crossover r (L–S):

Crossover Rate — What it is and how to get it

Definition (plain English): The crossover rate is the discount rate where two projects’ NPVs are exactly equal—i.e., the point where their NPV profiles cross. Below that rate one project has the higher NPV; above it, the other wins.

How to compute (L − S, then IRR)

Align years for both projects (pad missing years with zeros).
Compute incremental flows = L − S for each year (including CF0).
Compute IRR of that incremental series. That IRR is the crossover rate.

Excel: =IRR(Incremental_Flows_Range)

Current crossover r (L–S): —

Incremental CFs (L − S)

Year	Incremental CF

At the crossover rate, NPV(L) = NPV(S). Ranking flips on either side of this rate.

Excel vs Math

Setup: Put 11% in B1. Cash flows in B3:B6: −800, 350, 350, 350.

A) Easiest — Named Ranges

Select B1 → type Rate in the Name Box → Enter
Select B3 → name CF0
Select B4:B6 → name CFs
Select B3:B6 → name Flow

Formulas (paste anywhere)

=NPV(Rate, CFs)+CF0
=IRR(Flow)

B) Plain Cells (no $)

Tip: Don’t copy these around—just paste once where you want the answers.

Formulas (paste once)

=NPV(B1,B4:B6)+B3
=IRR(B3:B6)

Math (same numbers)

NPV = −800 + 350/1.11 + 350/1.11^2 + 350/1.11^3 ≈ 55.30 IRR ≈ 14.93% (solve NPV(IRR)=0) Payback ≈ 2.286 yrs

Teacher mode (optional): show $-anchored version

$-Anchored (for copy/fill safety)

=NPV($B$1,$B$4:$B$6)+$B$3
=IRR($B$3:$B$6)

Non-Conventional Cash Flows (Multiple IRRs)

Example: CF0 = −90,000; CF1 = 132,000; CF2 = 100,000; CF3 = −150,000. Hurdle (required return) = 15%.

What’s going on (plain English)

IRR solves: $ \mathrm{NPV}(r)=\sum_{t=0}^{n}\frac{CF_t}{(1+r)^t}=0 $.
With sign pattern − + + −, the NPV polynomial can have multiple positive roots ⇒ multiple IRRs (Descartes’ Rule of Signs).
On the NPV profile (NPV vs r), you’ll see the curve cross zero more than once.

Short math derivation

\[ \mathrm{NPV}(r)= -90000 + \frac{132000}{1+r} + \frac{100000}{(1+r)^2} - \frac{150000}{(1+r)^3}=0 \] Multiply by $(1+r)^3$: \[ 90000\,r^3 + 138000\,r^2 - 94000\,r + 8000 = 0 \] Two positive roots here (≈ 10.11% and 42.56%) → two IRRs.

What to do in practice

✅ If NPV at hurdle > 0 → Accept.

⚠️ If IRR conflicts with NPV or you get multiple/no IRRs → trust NPV at the hurdle.
😵 No real/positive IRR? Trust NPV at the hurdle rate.

IRR roots: — and —
NPV@15%: —

Excel IRR “guess” (which root shows up?)

📏 Rule: 🎯 Excel returns the IRR root nearest your guess.

Use a big guess to land on the high root (e.g., 40% → ~42.56%). Use a small guess (e.g., 10%) to land on the low root (~10.11%).

🎯 Guess (%)

—

👉 Default Excel guess is 10% (=IRR(B3:B6)) → tends to return the low root.
😵 If you see #NUM!, try a different guess, check for at least one sign change, and then rely on NPV at the hurdle rate.

NPV Profile — see the two zero-crossings

🔵 NPV(r) curve ➖ NPV = 0 axis 🔴 Zero-crossings (IRR candidates) 🟢 Hurdle line (your r)

Cash flows (CSV, include CF0):

Hurdle r% Explore r%

r=15.0% • NPV(r): —

Method Summary

Approach	Description	Pros	Cons	Suitable Cases	Industry Popularity
Net Present Value (NPV)	Calculates the present value of cash flows, subtracting initial investment	Accounts for time value of money, provides direct measure of profitability	Requires accurate cash flow estimation, sensitive to discount rate assumptions	Long-term projects, capital budgeting	Popular in most industries, especially capital-intensive sectors like manufacturing, energy, and real estate
Internal Rate of Return (IRR)	Finds the discount rate that makes NPV zero	Easy to understand, considers time value of money	Can be misleading with non-conventional cash flows or mutually exclusive projects	Projects with conventional cash flows, early cash inflows	Used across industries, especially finance, healthcare, and tech
Payback Period	Time required to recoup initial investment	Simple, emphasizes liquidity, useful for quick screening	Ignores time value of money, cash flows beyond payback period, and profitability	Short-term projects, high liquidity needs	Common in small businesses and start-ups, used in risk-averse sectors like retail

Quick Quiz

Use the quick quiz below, or open Quiz 2 for more Chapter 10 practice.

Homework (Due with Final)

Question 1

Project with an initial cash outlay of $20,000 with following free cash flows for 5 years.

Year   Cash flows
1      $8,000
2      4,000
3      3,000
4      5,000
5      10,000

1) How much is the payback period (approach one)?   ---- 4 years
2) If the firm has a 10% required rate of return. How much is NPV (approach 2)? -- $2456.74
3) If the firm has a 10% required rate of return. How much is IRR (approach 3)? ---- 14.55%

Question 2

Project with an initial cash outlay of $60,000 with following free cash flows for 5 years.

      Year    FCF
Initial outlay    –60,000
      1          25,000
      2          24,000
      3          13,000
      4          12,000
      5          11,000

The firm has a 15% required rate of return.

Calculate payback period, NPV, and IRR. Analyze your results.
(2.85, $764.27, 15.64%, accept the project)

Question 3: Mutually Exclusive Projects

1) Consider the following cash flows for one-year Project A and B, with required rates of return of 10%. You have limited capital and can invest in one but one project. Which one? (A’s NPV = 72.73, B’s NPV=227.27, so choose B)

§  Initial Outlay: A = -$200; B = -$1,500
§  Inflow:            A = $300; B = $1,900

2) Example: Consider two projects, A and B, with initial outlay of $1,000, cost of capital of 10%, and following cash flows in years 1, 2, and 3:
A: $100                       $200                $2,000
B: $650                       $650                $650

Which project should you choose if they are mutually exclusive? Independent? Crossover rate?

(mutually exclusive: A’s NPV=758.83 > B’s NPV = 616.45, so choose A; Independent, choose all positive NPV, so choose both;
Crossover rate = 21.01%. The calculator does not work. Use IRR in Excel)