FIN509 • Chapter 9 — Capital Budgeting (MBA)

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 & MIRR (rates of return; MIRR fixes IRR’s reinvestment assumption)
PI (useful when capital is rationed)
Payback / Discounted Payback (speed of recovery; screening tools)

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 (PPT)

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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 root when multiple exist

MIRR

=MIRR(values, finance_rate, reinvest_rate)

Needs at least one negative and one positive CF
Uses explicit reinvest & finance rates

Profitability Index (PI)

PI = 1 + NPV / |CF0|

Payback

Time to recover initial outlay (undiscounted). Discounted payback uses present values.

Mini Calculators

Inputs

Enter CF0 (negative) and subsequent CFs (comma-separated). Example: -800, 350, 350, 350

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

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

Outputs

NPV: —

IRR (by guess): —

MIRR (fin=reinv=rate): —

PI: —

Payback (yrs): —

Disc. 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)
MIRR: grow positives at reinvest rate, discount negatives at finance rate, then annualize
PI: 1 + NPV / |CF0|
Payback = years until cumulative CF ≥ 0 + fraction of final year
Discounted payback uses PVs

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%; PI ≈ 1.069; Payback ≈ 2.286 yrs; Disc. Payback ≈ 2.72 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)
=MIRR(Flow, Rate, Rate)
=1 + NPV(Rate, CFs)/ABS(CF0)

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)
=MIRR(B3:B6,B1,B1)
=1 + NPV(B1,B4:B6)/ABS(B3)

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) MIRR (r=11%) ≈ 13.50% PI = 1 + 55.30/800 ≈ 1.069 Payback ≈ 2.286 yrs; Discounted Payback ≈ 2.784 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$6)
=MIRR($B$3:$B$6,$B$1,$B$1)
=1 + (NPV($B$1,$B$4:$B$6)+$B$3)/ABS($B$3)

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? Use MIRR and NPV.

IRR roots: — and —
MIRR (15%,15%): — • 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, or switch to MIRR and NPV.

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, assumes reinvestment at IRR	Projects with conventional cash flows, early cash inflows	Used across industries, especially finance, healthcare, and tech
Modified Internal Rate of Return (MIRR)	Similar to IRR, but assumes reinvestment at a specified rate	Corrects IRR’s reinvestment assumption, suitable for comparing different projects	More complex calculation than IRR, still less informative than NPV for absolute profitability	Projects where reinvestment rate is known	Less common, used in finance and advanced corporate finance projects
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
Discounted Payback Period	Time to recoup investment considering time value of money	Considers time value of money, provides a more accurate payback measure than traditional Payback Period	Ignores cash flows after payback, profitability, and project lifetime	High-liquidity projects with cash flow risks	Used in industries with uncertain cash flows like tech and energy
Profitability Index (PI)	Ratio of PV of future cash flows to initial investment	Useful for ranking projects, particularly with capital constraints	Similar drawbacks as NPV for cash flow estimation, not suitable for mutually exclusive projects	Capital rationing scenarios, comparing projects with limited capital	Used in finance, utilities, and large capital-intensive industries

Quick Quiz

Click to check answers.

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%
4) If the firm has a 10% required rate of return. How much is PI (approach 4)? ---- 1.12

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, IRR and PI. Analyze your results.
(2.85, $764.27, 15.64%, 1.013, 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)

Homework Help Videos (Chapter 9)

Resources

Blog — IRR vs MIRR (Why MIRR is often better)

No interactivity here—just a crisp comparison you can reference in class or homework.

The issue in plain English

IRR assumes interim cash flows are reinvested at the IRR itself (often unrealistic), can produce multiple answers (non-conventional cash flows), or no real answer.
MIRR fixes this by using explicit rates: a finance (borrowing) rate for negatives and a reinvestment rate for positives. It produces a single, comparable return.

Definitions (math)

IRR: the rate $ r $ that makes NPV zero.

$ \mathrm{NPV}(r)=\displaystyle\sum_{t=0}^{n}\frac{CF_t}{(1+r)^t}=0 $

MIRR (with finance rate $ r_f $ and reinvest rate $ r_r $):

\[ \mathrm{MIRR}=\left(\frac{\mathrm{FV}_{\text{positives at } r_r}}{-\,\mathrm{PV}_{\text{negatives at } r_f}}\right)^{\!1/n}-1 \]

$\mathrm{PV}_{\text{negatives}}=\displaystyle\sum_{t:CF_t<0}\frac{CF_t}{(1+r_f)^t}$
$\mathrm{FV}_{\text{positives}}=\displaystyle\sum_{t:CF_t>0} CF_t\,(1+r_r)^{\,n-t}$

Excel formulas (no $ anchoring)

=IRR(B3:B6)
=MIRR(B3:B6, finance_rate, reinvest_rate)
  

Tip: For MIRR, a good default is to use your hurdle/WACC for both the finance and reinvest rates (e.g., 10%). Ensure your range has at least one negative and one positive CF.

Worked example (numbers & steps)

Cash flows: CF0 = −1000; CF1 = 400; CF2 = 400; CF3 = 400. Use finance = reinvest = 10%.

IRR
Solve $ \mathrm{NPV}(r)=0 $ for r.
Result: IRR ≈ 9.70%

MIRR
$\mathrm{PV}_{\text{neg}} = 1000$ (only CF0 at t=0)
$\mathrm{FV}_{\text{pos}} = 400(1.1)^2 + 400(1.1)^1 + 400 = 400(1.21+1.10+1)=400\cdot3.31=1324$
$\mathrm{MIRR}=\left(\dfrac{1324}{1000}\right)^{1/3}-1 \approx \mathbf{9.81\%}$

Why MIRR can be higher (or lower) than IRR

If the IRR is below your chosen reinvestment rate, MIRR typically comes out a bit higher (you assumed better reinvestment than IRR’s assumption).
If the IRR is above your reinvestment rate, MIRR will come out lower (more conservative and realistic).

When to use what

NPV remains the gold standard for accept/reject and ranking (value creation at the hurdle/WACC).
MIRR is the most sensible single “rate of return” to report: it uses realistic reinvestment and returns a unique number.
IRR is fine for quick intuition with conventional cash flows—but watch for multiple IRRs or conflicts with NPV.

Bottom line: Use NPV to decide, and use MIRR (with your hurdle as both rates) when you want a single comparable return figure.

Intro

Slides (PPT)

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

Excel Syntax (Quick Cards)

NPV

IRR

MIRR

Profitability Index (PI)

Payback

Mini Calculators

Inputs

Outputs

In-Class Example — Single Project

Multi-Project Choice — IRR vs NPV

NPV Profile — Project S vs L (Interactive)

Cash Flows (editable)

Project S

Project L

Controls

Crossover Rate — What it is and how to get it

How to compute (L − S, then IRR)

Incremental CFs (L − S)

Excel vs Math

A) Easiest — Named Ranges

B) Plain Cells (no $)

Math (same numbers)

Non-Conventional Cash Flows (Multiple IRRs)

What’s going on (plain English)

What to do in practice

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

NPV Profile — see the two zero-crossings

Method Summary

Quick Quiz

Homework (Due with Final)

Question 1

Question 2

Question 3: Mutually Exclusive Projects

Homework Help Videos (Chapter 9)

Q1 Walkthrough

Q2–Q3 Walkthrough

Excel Template Help

Resources

Blog — IRR vs MIRR (Why MIRR is often better)

The issue in plain English

Definitions (math)

Excel formulas (no $ anchoring)

Worked example (numbers & steps)

Why MIRR can be higher (or lower) than IRR

When to use what