Finance-Theory · Unit 21 · Video 6 · Interactive Practice

What Firms Use: Survey Evidence and Six Capital Budgeting Rules

IKey Formulas

Formula	Rule	What it asks of you
$\text{NPV} = CF_0 + \displaystyle\sum_{t=1}^{T} \frac{CF_t}{(1 + r_t)^t}$	1 and 5: decide on NPV, with a rate for each date	Take every project with $\text{NPV} > 0$ ; among mutually exclusive ones take the highest NPV; justify each year's $r_t$ by that year's risk
$\text{NPV} = \text{NPV}_{\text{standalone}} + \text{PV}(\text{interaction effects})$	2: value interactions separately	Value the project on its own, then add what it does to the rest of the firm
$CF_t = (R_t - C_t)(1 - \tau) + \tau D_t - I_t$	3: after-tax cash flows, not accounting earnings	Depreciation $D_t$ enters only through its tax shield $\tau D_t$ ; the investment $I_t$ enters when it is paid
$r = r_f + \beta\,(E[R_m] - r_f)$	4: CAPM with the project's beta	The project's own systematic risk sets its discount rate

Key Insight: Rule 6 is to know the alternatives: payback, the profitability index and IRR are widely reported and carry extra information, but the decision rests primarily on NPV.

IIVisualization 1 — Which Rule Do Firms Use?

The answer depends on whom the survey asked and whether it counted any use, the primary method or habitual use.

Survey

IIIVisualization 2 — The Primary Technique, 1959 to 1981

As the primary method, payback fell from 34% to 5.0% while IRR or NPV rose from 19% to 81.8%.

Survey year 1959

IVVisualization 3 — NPV Decides, the Alternatives Come Alongside

For a $20 million project at 7%, IRR and the PI always agree with NPV here; payback can disagree.

CF₁ ($M) 7 CF₂ ($M) 7 CF₃ ($M) 5 CF₄ ($M) 7

💡 IRR and the PI agree with NPV here because the project has one outflow, at $t = 0$ , is the only one under consideration and faces a single $7\%$ rate; between mutually exclusive projects of different scale they can rank the wrong one first.

VQuiz Questions

Problem 1 · Rule 1 with Two Rival Projects

Given: Two mutually exclusive projects, cost of capital $7\%$ for both. Project A costs $10 million today and returns $6 million at $t = 1$ and at $t = 2$ ; its IRR is about $13.1\%$ . Project B costs $30 million today and returns $17.5 million at $t = 1$ and at $t = 2$ ; its IRR is about $10.9\%$ . Which project should the firm take?

✅ Correct! B's NPV of about $1.64 million beats A's $0.85 million; for mutually exclusive projects the rule is the highest NPV, not the highest return.

❌ Not quite. IRR and the PI measure return per dollar and ignore scale. Rule 1 for mutually exclusive projects compares NPVs, and B creates more value.

❌ Not quite. "Take every positive-NPV project" is the rule for independent projects. These are mutually exclusive, so only one can be taken.

❌ Not quite. Both NPVs at

7\%

are positive, and both IRRs exceed

7\%

; at least one project adds value.

❌ Not quite. Compute each NPV at

7\%

and compare.

Show solution

Both projects pay a level amount for two years, so use the two-year annuity factor at $7\%$ (amounts in $ millions):

\frac{1}{1.07} + \frac{1}{1.07^2} \approx 0.934579 + 0.873439 = 1.808018

\text{NPV}_A \approx -10 + 6(1.808018) \approx 0.848

\text{NPV}_B \approx -30 + 17.5(1.808018) \approx 1.640

Both are positive, but the projects are mutually exclusive, so rule 1 says take the one with the higher NPV: Project B.

A has the higher IRR (about $13.1\%$ against $10.9\%$ ) and the higher PI ( $10.848/10 \approx 1.085$ against $31.640/30 \approx 1.055$ ), but both measures are per dollar invested. B earns a smaller margin on three times the capital and adds about $0.79 million more value.

Problem 2 · Rule 2: The Interaction Effect

Given: A firm values a new add-on product line on a standalone basis at $7\%$ and gets an NPV of −$0.60 million. Launching it would also raise the after-tax cash flows of the firm's existing product by $0.25 million a year at $t = 1, 2, 3, 4$ , with the same risk ( $7\%$ ). What is the right decision, and the project's NPV?

✅ Correct! The standalone loss is outweighed by the present value of the interaction, about $0.85 million, so the project adds about $0.25 million.

❌ Not quite. That is only the standalone value. Rule 2 says to value the interaction effects separately and add them in at the end.

❌ Not quite. $0.85 million is the present value of the interaction alone; the standalone NPV of −$0.60 million still has to be added.

❌ Not quite.

4 \times 0.25 = 1.00

adds dollars from different dates. Discount the interaction cash flows at

7\%

before adding them.

❌ Not quite. Add the standalone NPV and the present value of the interaction cash flows.

Show solution

Present value of the interaction effect, a four-year annuity of $0.25 million at $7\%$ :

\text{PV} = 0.25\left(\frac{1}{1.07} + \frac{1}{1.07^2} + \frac{1}{1.07^3} + \frac{1}{1.07^4}\right) \approx 0.25(3.387211) \approx 0.8468

Add it to the standalone value (in $ millions):

\text{NPV} \approx -0.60 + 0.8468 \approx 0.2468

The NPV is about $0.25 million, so accept. Judged on its own the product line would have been rejected; the interaction effect is what makes it worth doing.

Problem 3 · Rules 3 and 4 Together

Given: A project has beta $\beta = 1.2$ ; the risk-free rate is $r_f = 1\%$ and the expected market return is $E[R_m] = 6\%$ . It costs $6 million today, depreciated straight-line to zero over 3 years. Revenue minus cash operating costs is $3 million in each of years 1–3, and the tax rate is $25\%$ (no salvage value, no working capital). The accountants report earnings of $(3 - 2)(1 - 0.25) = 0.75$ ($ millions) a year. What is the project's NPV?

✅ Correct! CAPM gives

7\%

, the after-tax cash flow is $2.75 million a year, and the NPV is about $1.22 million.

❌ Not quite. That discounts accounting earnings. Depreciation is not a cash outflow; the $6 million was already counted at

t = 0

, so add depreciation back:

0.75 + 2 = 2.75

❌ Not quite. That discounts the pre-tax $3 million. Rule 3 calls for after-tax cash flows:

3(1 - 0.25) + 0.25 \times 2 = 2.75

❌ Not quite. That uses

r = r_f + \beta E[R_m] = 8.2\%

. The beta multiplies the market risk premium

E[R_m] - r_f = 5\%

, not the market return.

❌ Not quite. Find the CAPM rate, then the after-tax cash flow, then discount.

Show solution

Rule 4, the discount rate:

r = r_f + \beta\,(E[R_m] - r_f) = 1\% + 1.2(6\% - 1\%) = 7\%

Rule 3, the after-tax cash flow (depreciation $D = 6/3 = 2$ a year):

CF_t = (R_t - C_t)(1 - \tau) + \tau D_t = 3(0.75) + 0.25(2) = 2.25 + 0.50 = 2.75

Equivalently, earnings plus depreciation: $0.75 + 2 = 2.75$ .

Rule 1, the NPV (three-year annuity factor at $7\%$ $\approx 2.624316$ ; amounts in $ millions):

\text{NPV} \approx -6 + 2.75(2.624316) \approx -6 + 7.2169 \approx 1.2169

The NPV is about $1.22 million: accept.

Problem 4 · A Budget Limit

Given: A firm has a budget of $10 million this year (a single limit) and three independent projects, each divisible (any fraction can be taken, with that fraction of its cost and NPV). Costs and NPVs at $7\%$ , in $ millions: A costs $8$ with NPV $2.0$ ; B costs $4$ with NPV $1.4$ ; C costs $5$ with NPV $1.5$ . What is the largest total NPV the budget can buy?

✅ Correct! Ranking by NPV per dollar fills the budget with B, then C, then one-eighth of A:

1.4 + 1.5 + 0.25 = 3.15

❌ Not quite. Starting with the largest NPV (A) spends $8 million to earn $2.0 million, and the last $2 million buys only half of B ($0.7 million): $2.70 million in all. Under a budget limit, what counts is the total NPV the budget buys, so rank by NPV per dollar.

❌ Not quite. B and C cost $9 million and leave $1 million idle. The projects are divisible, so that last $1 million buys one-eighth of A.

❌ Not quite. All three cost $17 million, more than the $10 million budget.

❌ Not quite. Rank the projects by NPV per dollar invested and fill the budget in that order.

Show solution

NPV per dollar invested (equivalently, the profitability index minus one):

\text{B: } \frac{1.4}{4} = 0.35, \qquad \text{C: } \frac{1.5}{5} = 0.30, \qquad \text{A: } \frac{2.0}{8} = 0.25

Fill the $10 million in that order: all of B ($4 million, NPV $1.4$ ), all of C ($5 million, NPV $1.5$ ), and the last $1 million buys $\tfrac{1}{8}$ of A (NPV $\tfrac{1}{8} \times 2.0 = 0.25$ ).

\text{Total NPV} = 1.4 + 1.5 + 0.25 = 3.15

That is $3.15 million, the most the $10 million can buy.

Ranking per dollar works here because the projects are divisible and there is a single budget limit. Without that, the PI's blindness to scale returns, and you must compare the total NPV of the combinations the budget allows. A multi-year budget adds a further link: a dollar committed this year is a dollar later years cannot use.

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