Finance-Theory · Unit 20 · Video 1 · Interactive Practice

One Rule, Many Rates: Capital Budgeting Starts from NPV

IKey Formulas

Formula	Name	What it says
$NPV = CF_0 + \dfrac{CF_1}{1+r_1} + \cdots + \dfrac{CF_T}{(1+r_T)^T}$	The NPV rule	A project's current market value; each cash flow is discounted at the risk-adjusted rate that matches its own risk
$r_1 = \cdots = r_T = r$	Special case: one rate	The early examples, where every cash flow carried the same risk and one market rate discounted every date
$V_{\text{firm} + A + B} = V_{\text{firm}} + NPV_A + NPV_B$	Value additivity	A project adds exactly its NPV to the firm's value, so each project is evaluated standalone; interactions are valued separately
$NPV > 0$ ; all with $NPV_i > 0$ ; the largest $NPV_i$ , if $> 0$	Investment criteria	Single project; independent projects; mutually exclusive projects

Key Insight: The objective is to increase the firm's current market value, and NPV measures exactly what a project adds to it. The rule never changes; the work is in the inputs: the cash flows in the numerators and a rate matched to each cash flow's risk in the denominators.

IIEach Cash Flow at Its Own Rate

Each cash flow is divided by one plus its own date's rate, raised to that date's power.

r₁, % per year 3 r₂, % per year 3.5 r₃, % per year 4

IIIIndependent or Mutually Exclusive

A project adds exactly its NPV to firm value, so which projects to take depends on which combinations are allowed.

The three projects are

💡 Each NPV here is computed on a standalone basis; if two projects interacted, for example by sharing customers, the interaction would be valued separately and added on.

IVQuiz Questions

Problem 1 · NPV with Date-Specific Rates

Given: a project costs $500,000 today and returns $170,000, $175,000 and $190,000 at the ends of years 1, 2 and 3. The risk-adjusted rates matching those three cash flows are $r_1 = 3\%$ , $r_2 = 3.5\%$ and $r_3 = 4\%$ per year. Find the project's NPV.

✅ Correct! The three present values add to about $497,322, which is less than the $500,000 cost, so the NPV is about −$2,678 and the project should be rejected even though its undiscounted cash exceeds its cost.

❌ Not quite. That divides each cash flow by

1 + r_t

only once. The year-

t

cash flow is divided by

(1+r_t)^t

: the year-2 flow by

1.035^2

and the year-3 flow by

1.04^3

❌ Not quite. That discounts every year at

4\%

. Each cash flow has its own rate here: year 1 at

3\%

and year 2 at

3.5\%

; only year 3 uses

4\%

❌ Not quite. $35,000 is the undiscounted total:

-500{,}000 + 170{,}000 + 175{,}000 + 190{,}000

. Future dollars are worth less than dollars today, so every inflow must be discounted before adding.

Show solution

Discount each cash flow at its own date's rate, raised to the power of its date (amounts in dollars, each term rounded to the dollar):

NPV = -500{,}000 + \frac{170{,}000}{1.03} + \frac{175{,}000}{1.035^2} + \frac{190{,}000}{1.04^3}

\approx -500{,}000 + 165{,}049 + 163{,}364 + 168{,}909 = -2{,}678

The NPV is about −$2,678. It is negative, so the project would lower the firm's market value: reject it.

Problem 2 · One Rate for Every Date

Given: a project costs $500,000 today and returns $215,000, $170,000 and $150,000 at the ends of years 1, 2 and 3. The rates matching those cash flows are $r_1 = 3\%$ , $r_2 = 3.5\%$ and $r_3 = 4\%$ . An analyst instead discounts every year at the 3-year rate, $4\%$ , and reports an NPV of about −$2,745.

What is the NPV at the rates that match each cash flow?

What should the firm do?

✅ Correct! At the matched rates the NPV is about +$784, so the project adds value and should be taken. Using one rate for every date overstates the discounting of the years 1 and 2 flows and reverses the decision.

❌ Check the NPV. Discount the year-

t

cash flow by

(1 + r_t)^t

, using that year's own rate.

❌ Check the decision. A single project is taken if and only if its NPV, computed at the rates matching its risks, is positive.

❌ Not quite. That is the analyst's figure, which applies

4\%

to every year. The year-1 and year-2 flows have their own rates,

3\%

and

3.5\%

❌ Not quite. That discounts every year at

3\%

, the year-1 rate. The year-2 flow needs

1.035^2

and the year-3 flow

1.04^3

❌ Not quite. $35,000 is the undiscounted total; every inflow must be discounted to today before adding.

❌ Not quite. The

4\%

figure is not cautious, it is wrong: the year-1 and year-2 cash flows carry the risk priced by

3\%

and

3.5\%

. Proper risk adjustment means each flow at its own rate.

❌ Not quite. The rule is a sign test: any positive NPV, however small, adds that amount to the firm's market value.

Show solution

Matched rates (amounts in dollars, terms rounded to the dollar):

NPV = -500{,}000 + \frac{215{,}000}{1.03} + \frac{170{,}000}{1.035^2} + \frac{150{,}000}{1.04^3}

\approx -500{,}000 + 208{,}738 + 158{,}697 + 133{,}349 = 784

The analyst's single rate:

-500{,}000 + \frac{215{,}000}{1.04} + \frac{170{,}000}{1.04^2} + \frac{150{,}000}{1.04^3} \approx -500{,}000 + 206{,}731 + 157{,}175 + 133{,}349 = -2{,}745

The year-3 term is the same in both; the difference comes entirely from discounting years 1 and 2 at $4\%$ instead of their own lower rates. The correct NPV is about +$784 > 0, so the firm should take the project.

Problem 3 · Three Mutually Exclusive Projects

Given: a firm can build only one of three plants. Rates are $r_1 = 3\%$ , $r_2 = 3.5\%$ and $r_3 = 4\%$ for all three.

Project	Today	Year 1	Year 2	Year 3
A	−$500,000	$220,000	$180,000	$155,000
B	−$440,000	$150,000	$160,000	$175,000
C	−$500,000	$150,000	$190,000	$220,000

Find which project or projects the firm should take.

✅ Correct! The NPVs are about $19,419 (A), $10,567 (B) and $18,577 (C). Only one can be built, so take the largest positive NPV: A.

❌ Not quite. C's undiscounted net cash, $60,000, is the largest, but it arrives late. Discounted at the rates for each date, C is worth about $18,577 against about $19,419 for A.

❌ Not quite. All three NPVs are positive, but only one plant can be built. For mutually exclusive projects, taking every positive-NPV project is the mistake; choose the highest positive NPV.

❌ Not quite. The cost is only one cash flow. B's NPV is about $10,567, the lowest of the three; the criterion is NPV, not the size of the investment.

Show solution

The discount factors are $1/1.03 \approx 0.97087$ , $1/1.035^2 \approx 0.93351$ and $1/1.04^3 \approx 0.88900$ . Present values in dollars, each term rounded (the NPVs come from the unrounded terms):

NPV_A = -500{,}000 + 213{,}592 + 168{,}032 + 137{,}794 \approx 19{,}419

NPV_B = -440{,}000 + 145{,}631 + 149{,}362 + 155{,}574 \approx 10{,}567

NPV_C = -500{,}000 + 145{,}631 + 177{,}367 + 195{,}579 \approx 18{,}577

All three are positive, but the projects are mutually exclusive, so the firm takes the one with the highest NPV, provided it is positive: Project A, adding about $19,419 to firm value. C has more total cash ($60,000 net, against $55,000 for A), but more of it comes in year 3, where discounting is heaviest.

Problem 4 · Value Additivity

Given: a firm's current market value is $8,000,000. It has three independent projects, P, Q and R, with NPVs of $21,500, −$6,300 and $9,800. Separately it must choose at most one of two mutually exclusive projects, S (NPV $14,000) and T (NPV $11,200). Every NPV is computed at that project's own risk-adjusted rates, and no projects interact. Find the firm's market value after it follows the investment criteria.

✅ Correct! Take P and R (every positive NPV among the independent projects), skip Q, and take S (the better of the exclusive pair): $8,000,000 + $21,500 + $9,800 + $14,000 = $8,045,300.

❌ Not quite. That takes both S and T. They are mutually exclusive, so only the one with the higher NPV, S, can be added.

❌ Not quite. That takes every project, including Q, whose NPV of −$6,300 lowers firm value, and both of the exclusive pair.

❌ Not quite. That takes only the single best project. Independent projects do not compete: every one with a positive NPV adds its value, and the exclusive pair contributes its best member too.

Show solution

By value additivity, the firm is worth its current value plus the NPV of every project it takes.

Independent projects: take all with positive NPV, P ($21,500) and R ($9,800); reject Q (−$6,300).

Mutually exclusive pair: take the higher positive NPV, S ($14,000), and not T.

Firm value = $8,000,000 + $21,500 + $9,800 + $14,000 = $8,045,300.

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