Finance-Theory · Unit 21 · Video 3 · Interactive Practice

Scale and NPV: Why Firms Budget in Dollars, Not Returns

IKey Formulas

Formula	Name	What you need
$\text{Profit}(x) = B(x) - C(x)$	Total NPV of investing $x$	Present values of benefits $B$ and costs $C$ , both functions of $x$
$\dfrac{d\,\text{Profit}}{dx} = B'(x) - C'(x)$	NPV of the marginal dollar (MB − MC)	How fast benefits and costs grow with $x$
$B'(x^) = C'(x^)$	Profit-maximizing scale: MB = MC	Marginal benefit and marginal cost
$\text{NPV}_{\text{total}} = \text{NPV}_1 + \text{NPV}_2 + \cdots + \text{NPV}_k$	Investing in tranches	The NPV of each tranche at the cost of capital

Key Insight: A small investor cannot move the return, so a theory written in returns works. A firm moves its project's return by the amount it invests, so it judges the project in dollars of NPV and keeps scaling up only while the NPV of each marginal dollar is positive.

IIWho Moves the Return?

An investor's purchase leaves the market's return unchanged; a firm's own investment pushes its division's return down.

Amount invested x ($ millions) 5

IIITranche by Tranche

Each $5 million tranche earns less NPV than the one before; the total peaks at the last positive tranche.

IVMarginal Benefit Equals Marginal Cost

Profit $B(x) - C(x)$ peaks where its slope is zero: $B'(x^*) = C'(x^*)$ .

Amount invested x ($ millions) 6

💡 The curves here are illustrative. The video's point is that no index hands a firm $B'$ and $C'$ ; measuring them well enough to find $x^*$ is the hard part.

VQuiz Questions

Problem 1 · How Many Tranches?

Given: a division can expand in tranches of $5 million, each invested today. Each tranche's only cash flow arrives one year later: $7.085 million for the first tranche, $5.777 million for the second, $5.341 million for the third and $4.905 million for the fourth. The cost of capital is $9\%$ . Find how much the firm should invest.

✅ Correct! The first two tranches add $1.5 million and $0.3 million of NPV; the third would subtract $0.1 million, so the firm stops at $10 million with a total NPV of $1.8 million.

❌ Not quite. The first tranche has the highest return, but the second still adds $0.3 million of NPV. Stopping at the best tranche leaves positive-NPV dollars uninvested.

❌ Not quite. The third tranche pays back more than its $5 million, but only

6.82\%

more, less than the

9\%

cost of capital. Its present value is

5.341 / 1.09 = 4.9

($ millions), so its NPV is −$0.1 million.

❌ Not quite. The total NPV at $20 million is still positive ($1.2 million), but it is lower than the $1.8 million at $10 million: the third and fourth tranches each destroy value. Judge the marginal tranche, not the running total.

❌ Not quite. Discount each tranche's payoff at

9\%

, subtract the $5 million it costs, and invest only in tranches with positive NPV.

Show solution

Present value of each tranche's payoff ($ millions), then its NPV after the $5 million it costs:

\frac{7.085}{1.09} = 6.5, \qquad \frac{5.777}{1.09} = 5.3, \qquad \frac{5.341}{1.09} = 4.9, \qquad \frac{4.905}{1.09} = 4.5

\text{NPV}_1 = 1.5, \qquad \text{NPV}_2 = 0.3, \qquad \text{NPV}_3 = -0.1, \qquad \text{NPV}_4 = -0.5

The NPV of the marginal tranche falls as the firm invests more, and it turns negative with the third tranche. Running totals: $1.5 million at $5 million, $1.8 million at $10 million, $1.7 million at $15 million and $1.2 million at $20 million.

The firm should invest $10 million (two tranches), for a total NPV of $1.8 million.

Problem 2 · The Highest Return Is Not the Best Scale

Given: a division's one-year project returns $r$ on everything invested, but $r$ depends on the scale the firm chooses: $r = 21\%$ if it invests $5 million, $17\%$ at $10 million, $15\%$ at $15 million and $12\%$ at $20 million. The cost of capital is $9\%$ . Find the scale that maximizes NPV.

✅ Correct! At $15 million the NPV is about $0.826 million, more than at any other scale, even though the return is lower than at $5 or $10 million.

❌ Not quite. That is fixating on

r

. The $5 million scale has the highest return, but on so few dollars that its NPV (about $0.550 million) is smaller than at $10 or $15 million.

❌ Not quite. The return falls from

17\%

15\%

when the firm goes from $10 million to $15 million, but the extra $5 million still returns

11\%

, above the

9\%

cost of capital. NPV rises from about $0.734 million to about $0.826 million, so the firm should not stop at $10 million.

❌ Not quite. Every scale earns more than

9\%

, but that is not the test. Going from $15 million to $20 million adds $5 million today and only $5.15 million next year, a

3\%

return on the extra dollars, so NPV falls to about $0.550 million.

❌ Not quite. Compute the dollar NPV at each scale,

\text{NPV} = -x + \dfrac{x(1 + r)}{1.09}

, and compare.

Show solution

For a one-year project, $\text{NPV} = -x + \dfrac{x(1 + r)}{1.09} = \dfrac{x\,(r - 0.09)}{1.09}$ , with $x$ in $ millions:

\frac{5(0.12)}{1.09} \approx 0.550, \qquad \frac{10(0.08)}{1.09} \approx 0.734, \qquad \frac{15(0.06)}{1.09} \approx 0.826, \qquad \frac{20(0.03)}{1.09} \approx 0.550

The marginal view gives the same answer. Going from $10 million to $15 million adds $5 million today and $17.25 - 11.7 = 5.55$ ($ millions) next year, an $11\%$ return on the extra dollars, above $9\%$ . Going from $15 million to $20 million adds $22.4 - 17.25 = 5.15$ , only $3\%$ .

NPV is largest at $15 million. The firm's own investment moves $r$ , so $r$ alone cannot pick the scale.

Problem 3 · Find Where MB = MC

Given: investing $x$ ($ millions) in a division has present value of benefits $B(x) = 2.4x - 0.035x^2$ and present value of costs $C(x) = x$ , both in $ millions.

At what scale is the total NPV largest?

At $x = 30$ , what are the total NPV and the NPV of the marginal dollar?

✅ Correct! MB = MC at $20 million, where the total NPV is $14 million. At $30 million the total is still positive, yet every extra dollar loses 70 cents: the firm has overshot without any loss showing.

❌ Not quite. That is where

B'(x) = 0

, the scale that maximizes benefits alone. Each dollar invested also costs a dollar, so the target is

B'(x) = C'(x) = 1

❌ Not quite. At $40 million,

B(x) = C(x)

and the total NPV is back to zero: that is investing until the project stops being profitable. Either that, or

B'(x)

was taken as

2.4 - 0.035x

; the derivative of

0.035x^2

0.07x

❌ Not quite. Set marginal benefit equal to marginal cost:

B'(x) = 2.4 - 0.07x

and

C'(x) = 1

❌ Not quite.

+0.30

B'(30)

, the marginal benefit alone. The NPV of the marginal dollar is

B'(30) - C'(30) = 0.3 - 1

❌ Not quite. Those are the values at

x^* = 20

, not at

x = 30

. Evaluate

B(30) - C(30)

and

B'(30) - C'(30)

❌ Not quite. Evaluate

B(30) - C(30)

for the total and

B'(30) - C'(30)

for the marginal dollar.

Show solution

Step 1: MB = MC.

B'(x) = 2.4 - 0.07x, \qquad C'(x) = 1 \quad\Rightarrow\quad 2.4 - 0.07x^* = 1 \quad\Rightarrow\quad x^* = \frac{1.4}{0.07} = 20

\text{Profit}(20) = 2.4(20) - 0.035(400) - 20 = 48 - 14 - 20 = 14

Step 2: evaluate at $x = 30$ .

\text{Profit}(30) = 72 - 31.5 - 30 = 10.5, \qquad B'(30) - C'(30) = 2.4 - 2.1 - 1 = -0.7

The best scale is $20 million (total NPV $14 million). At $30 million the total NPV is still $10.5 million, but the marginal dollar loses $0.70: a firm that waits for losses to show (at $40 million) has passed the optimum long before.

Problem 4 · Investor versus Firm

Given: a retiree puts $50,000 into an S&P 500 index fund; a firm puts $150 million into expanding one division. Why does the video judge the firm's decision in dollars of NPV rather than by the rate of return?

✅ Correct! CAPM assumes each investor is small and takes returns as given. A firm pouring capital into one division changes

r

itself, so it must measure value in dollars and stop where the NPV of the marginal dollar reaches zero.

❌ Not quite. Scale works the other way: the more a firm puts into one opportunity, the lower the return on the next dollar. The highest return usually comes with the smallest investment, not the largest NPV.

❌ Not quite. The video still uses the stock market as a guide for computing the project's discount rate. What fails inside the firm is taking the project's return as given.

❌ Not quite. When the return on the whole amount has fallen to the cost of capital, the project as a whole only breaks even (for a one-year project its total NPV is zero), so every bit of NPV earned on the early dollars has been given back. The best scale comes earlier, where the marginal dollar's NPV reaches zero.

❌ Not quite. Ask who can move the rate of return: the retiree or the firm?

Show solution

CAPM is derived for investors who are small: they do not affect prices, so they do not affect returns, and a theory written in returns works for them. $50,000 cannot move the S&P 500.

A firm investing $150 million in one division can change that project's return dramatically, and the return falls as the investment grows. A return computed at one scale therefore says nothing about the best scale. NPV, measured in dollars, does not ignore scale: the firm keeps investing while the NPV of the marginal dollar is positive and stops where marginal benefit equals marginal cost.

The answer is (c). Option (d) describes the point where the total NPV has fallen back to zero, well past the optimum.

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