第二章 资本预算与投资决策 — 知识点讲解

1. Core Idea

Time value of money: a dollar in hand today is worth more than a dollar promised at some time in the future — because you can earn interest while you wait, so a dollar today grows into more than a dollar later. Future value (FV) is the amount of money an investment will grow to over some period of time at some given interest rate — the cash value of an investment at some future time. We start with the simplest case: a single-period investment.

2. Single-Period Investing: Where FV Begins

Suppose you invest $100 in a savings account that pays 10 percent interest per year. In one year you have:

$$\$100 \times 1.10 = \$110$$

This $110 equals your $100 principal plus $10 of interest. In general, $1 invested for one period at rate r grows to (1 + r):

$$\text{FV} = C \times (1 + r)$$

3. Multiple Periods and Compounding ("Interest on Interest")

If you leave the money for two years, year-2 interest is computed on $110: $110 × .10 = $11, giving $121 in total. This $121 has four parts:

Compounding is the process of leaving your money and any accumulated interest in an investment for more than one period, thereby reinvesting the interest — earning interest on interest. With simple interest, the interest is not reinvested, so interest is earned each period only on the original principal.

Example 5.1 Interest on Interest: $325 at 14 percent for two years → $325 × 1.14 = $370.50 after year 1, then $370.50 × 1.14 = $422.37 after year 2. Total interest = $97.37; simple interest = $325 × .14 × 2 = $91; the compounding portion = $97.37 − 91 = $6.37 (= year-1 interest $45.50 × .14).

4. The FV Formula and the Future Value Interest Factor (FVIF)

The future value of $1 invested for t periods at a rate of r per period is:

$$\text{FV} = \$1 \times (1+r)^t$$

The term $(1+r)^t$ is called the future value interest factor (FVIF) — the FV factor for $1 at r percent for t periods. Compute the factor, then multiply it by the principal. For example, $100 at 10 percent for five years: factor = 1.10⁵ = 1.6105, so $100 × 1.6105 = $161.05.

FVIF Table (excerpt, Table 5.2):

5. Compound vs. Simple Interest: The $100 Growth Data Card (Table 5.1)

At 10 percent, simple interest stays constant at $10 per year, but the compound interest portion grows every year:

Of the $61.05 total interest over five years, only $50 is simple interest; the other $11.05 comes from compounding — and it grows each year because more and more interest builds up, leaving ever more to compound.

6. The Power of Compounding: Long Horizons and High Rates

• Doubling the rate more than doubles the FV: $1 invested for 10 years grows to about $2.60 at 10 percent but about $6.20 at 20 percent (Figure 5.2).
• Example 5.2: $400 at 12 percent for 3 years → $400 × 1.4049 = $561.97; for 7 years → $400 × 2.2107 = $884.27 — more than doubling your money. Of the $484.27 interest, simple interest is only $336 (7 × $48); compounding contributes $148.27.
• A 200-year miracle: $5 at 6 percent for 200 years — factor 1.06²⁰⁰ = 115,125.90, so FV = $575,629.52. Simple interest totals just $60; virtually everything else comes from reinvesting.
• Example 5.3 How Much for That Island?: Manhattan was bought for about $24 in goods and trinkets in 1626. Invested at 10 percent for roughly 395 years, 1.1³⁹⁵ ≈ 22,393 × 10¹², so the FV is on the order of $537 quadrillion — enough to buy the United States (all of it, cash) with money left over.

Memory aid: "Compounding is the lever of time — slow at first, explosive later."

7. Applications and Tools: Compound Growth and the Financial Calculator

An interest rate is just a compound growth rate, so the same formula answers any fixed-percentage-growth question:

• Employee growth: 10,000 employees growing at 3 percent per year → 10,000 × 1.03⁵ = 10,000 × 1.1593 ≈ 11,593 employees in five years (about 1,593 net new hires).
• Walmart sales: 2019 sales of about $524 billion growing at 15 percent per year → about $1.054 trillion in 2024 (just over twice as large).
• Example 5.4 Dividend Growth (TICO): a $5 dividend increased 4 percent per year for eight years → $5 × 1.04⁸ = $5 × 1.3686 = $6.84.

Financial calculator tips: enter the rate as 10, not .10; enter cash outflows (the PV) with a negative sign; completely clear the calculator before every problem (2nd + CLR TVM on a BA II Plus) — failure to clear is the number-one cause of wrong answers.

Rule of 72: years to double ≈ 72/r%. Example: at 12 percent, 72/12 = 6 years. It is fairly accurate for rates in the 5–20 percent range (it is only an approximation).

8. Three Key Takeaways

1. Compounding means interest on interest — reinvesting interest makes money grow at an ever-faster pace, whereas simple interest pays only on the original principal.
2. FV is driven by three inputs — principal C, rate r, and time t: $\text{FV} = C \times (1+r)^t$, where $(1+r)^t$ is the future value interest factor (FVIF).
3. Compounding's magic shows up over long horizons and high rates — doubling r or t more than doubles the FV; present value, discounting, and DCF valuation all build on this compounding foundation.

1. Core Idea

The time value of money rests on one proposition: a dollar in hand today is worth more than a dollar promised in the future — because you can earn interest while you wait, so a dollar today grows into more than a dollar later.

• Future value (FV) asks: What will money today grow to? — compounding cash flows forward.
• Present value (PV) asks: What is future money worth today? — discounting future cash flows back. Discounting is the reverse of compounding.

Opening example: the Powerball jackpot of $110 million was actually paid $5.5 million per year over 20 years, so the ticket was worth far less than $110 million. The §5.2 counterpart: you need $10,000 in 10 years and can earn 6.5% — you only need to invest $5,327.26 today.

2. Single-Period PV: Where Discounting Starts

How much must you invest today at 10% to get $1 in one year? With $PV \times 1.1 = \$1$:

$$PV = \$1 \times \left[\frac{1}{1+r}\right] = \frac{\$1}{1+r}, \qquad r = 10\% \Rightarrow PV = \$1/1.1 = \$.909$$

Example 5.5 (Textbooks): you need $400 next year and can earn 7%. How much today?

$$PV \times 1.07 = \$400 \quad \Rightarrow \quad PV = \$400 \times (1/1.07) = \$373.83$$

Check: investing $373.83 at 7% for one year produces exactly $400. PV answers: "How much must I invest today to reach a target amount?"

3. Multi-Period PV: Divide by (1 + r) Raised to t

The present value of $1 to be received t periods from now at rate r (Formula 5.2):

$$PV = \$1 \times \left[\frac{1}{(1+r)^t}\right] = \frac{\$1}{(1+r)^t}$$

• Example: $1,000 needed in 2 years at 7%: $1.07^2 = 1.1449$, $PV = \$1{,}000/1.1449 = \$873.44$
• Example 5.6 (Saving Up): a car costs $68,500, purchase in 2 years, r = 9%: $1.09^2 = 1.1881$, $PV = \$68{,}500/1.1881 = \$57{,}655.08$ — you have about $50,000, so you are $7,655 short.

Memory anchor: compounding rolls the snowball forward by multiplying (1 + r); discounting rewinds the tape by dividing — one extra period, one extra division.

4. Discount Factor, Discount Rate, and DCF

The bracketed term $1/(1+r)^t$ goes by several names:

Table 5.3 Present Value Interest Factors (excerpt):

Example: $1,000 in 3 years at 15%: discount factor $1/1.15^3 = 1/1.5209 = .6575$, $PV = \$1{,}000 \times .6575 = \$657.52$ — the present (discounted) value of $1,000 at 15% over three years.

5. How Time and Rate Affect Present Value

• The longer the horizon, the lower the PV: future amounts shrink exponentially and approach zero far enough out.
• The higher the discount rate, the lower the PV: present values and discount rates are inversely related — raising r lowers PV, and vice versa.

Example 5.7 (Deceptive Advertising?): "Come by and we'll give you $100!" — the fine print says a certificate paying $100 in 25 years. At 10%, the factor is $1/1.1^{25} = 1/10.8347 = .0923$, so today's real gift is:

$$.0923 \times \$100 = \$9.23$$

Data card: the same $150,000 at 9% is worth ≈ $58,130 today if paid in 11 years (factor $1/2.5804 = .3875$); but $100 in 25 years at 10% is worth only $9.23. Time and rate jointly compress distant values.

6. The Basic Present Value Equation

The present value factor is the reciprocal of the future value factor (press the "1/x" key on many calculators), linking the two through the basic present value equation (Formula 5.3):

$$PV \times (1+r)^t = FV, \qquad PV = \frac{FV_t}{(1+r)^t} = FV_t \times \left[\frac{1}{(1+r)^t}\right]$$

The equation has only four parts — PV, FV, r, and t: given any three, we can always find the fourth. Everything that follows in the book (bonds, stocks, capital budgeting) builds on this one equation.

Example 5.8 (Evaluating Investments): buy an asset for $335, sell it for $400 in 3 years; the alternative is investing $335 elsewhere at 10%. Both views agree: - By future value: $335 \times 1.1^3 = \$335 \times 1.331 = \$445.89 > \$400$ — the alternative wins; - By present value: $PV = \$400/1.331 = \$300.53 < \$335$ — $400 is worth only $300.53 today, so paying $335 is not a good investment.

7. Applications: Solving for Rate and Time (incl. Rule of 72)

Finding r (single period) — Example 5.9: put up $1,250, get back $1,350 in one year. $1+r = 1{,}350/1{,}250 = 1.08$, so r = 8% (equivalently $100/1{,}250$).

Finding r (multi-period, doubling) — $100 doubles to $200 in 8 years: $(1+r)^8 = 2$; take the 8th root ($2^{.125} \approx 1.09$) to get r = 9%.

Rule of 72: for reasonable rates, the time to double your money ≈ 72/r%. Example: 72/8 = 9, matching the exact solution; accurate in roughly the 5%–20% range. The chapter's opening EE savings bonds: $25 guaranteed to become $50 in 20 years, a minimum return of 72/20 = 3.6%.

Finding t: $25,000 grows to $50,000 at 12%: $1.12^t = 2$, so t = 6.1163 years; the Rule of 72 gives 72/12 = 6 years — an excellent approximation.

Memory aid: "Find any three of PV, FV, r, t to solve for the fourth; for doubling problems, divide 72 by the rate."

8. Key Takeaways

1. PV is today's price of a future cash flow — discounting is compounding in reverse: divide by (1 + r) each period, i.e., by $(1+r)^t$ over t periods.
2. PV moves inversely with both time and rate — longer horizons and higher discount rates shrink PV toward zero; $100 paid in 25 years at 10% is worth just $9.23 today.
3. The basic present value equation PV = FV/(1+r)^t is the master key of corporate finance — with PV, FV, r, and t, knowing any three yields the fourth; both investment evaluation (compare FVs or PVs) and the Rule of 72 follow from it.

1. Core Idea: What Is an Ordinary Annuity?

A series of constant (level) cash flows that occur at the end of each period for a fixed number of periods is called an ordinary annuity. Almost all consumer loans (e.g., car loans), home mortgages, and retirement savings take this form.

Cash flow timing (critical default): unless explicitly told otherwise, assume cash flows occur at the end of each period — every formula, every table, and every financial calculator's default settings rest on this assumption. Cash flows at the beginning of each period (rent, prepaid amounts) form an annuity due (Section 7).

2. Present Value of an Annuity

The present value of an annuity of C dollars per period for t periods at rate r per period: $$PV = C \times \frac{1 - [1/(1+r)^t]}{r} = C \times \frac{1 - \text{PV factor}}{r}$$ The term $\frac{1 - [1/(1+r)^t]}{r}$ is called the annuity present value factor (PVIFA).

Example: an asset pays $500 at the end of each of the next 3 years; required return 10%: - PV factor = $1/1.1^3 = 1/1.331 = .751315$ - Annuity PV factor = $(1 - .751315)/.10 = 2.48685$ - $$PV = $500 \times 2.48685 = $1,243.43$$

Example (6.5): $632 per month for 48 months at 1% per month — how much can you borrow? - Factor = $[1 - (1/1.01^{48})]/.01 = (1 - .6203)/.01 = 37.9740$ - $$PV = $632 \times 37.9740 = $24,000$$

3. Future Value of an Annuity

$$FV = C \times \frac{(1+r)^t - 1}{r} = C \times \frac{\text{FV factor} - 1}{r}$$

Example: contribute $2,000 every year to a retirement account paying 8%; you retire in 30 years: - Factor = $(1.08^{30} - 1)/.08 = (10.0627 - 1)/.08 = 113.2832$ - $$FV = $2,000 \times 113.2832 = $226,566.42$$

Timing trap: the first $2,000 earns interest for 29 years; the last $2,000 is deposited at the end of year 30 and earns no interest at all. Of the t deposits, the first earns t−1 periods of interest, the last earns 0.

4. Finding the Payment

Given PV, r, and t: $C = PV \div \frac{1 - [1/(1+r)^t]}{r}$

Example: borrow $100,000 at 18%, repay in five equal annual payments: - $100,000 = C \times {[1 - (1/1.18^5)]/.18} = C \times [(1 - .4371)/.18] = C \times 3.1272$ - $$C = $100,000/3.1272 = $31,977.78$$

This is exactly how car loan and mortgage payments are computed.

5. Finding the Number of Payments

Example (6.6): $1,000 credit card balance at 1.5% per month, $20 minimum payment per month: - $1,000 = $20 \times [(1 - \text{PV factor})/.015]$ → PV factor = $.25 = 1/1.015^t$ → $1.015^t = 4$ - The problem becomes: how long to quadruple your money at 1.5% per month → $t \approx 93$ months - $$t \approx 93/12 = 7.76 \text{ years}$$

Note: some calculators automatically round t up to the next whole period.

6. Finding the Rate

Example: pay $6,710 today for $1,000 per year for 10 years — what rate is implicit? - $6,710 = $1,000 \times \frac{1 - [1/(1+r)^{10}]}{r}$ → annuity PV factor = 6.71 - Table A.3: the 10-period row at 8% = 6.7101 → r ≈ 8%

Trial and error (no closed-form solution): a $3,000 loan repaid as $1,000 per year for 4 years: | r | Annuity factor | PV | |----|----------------|-----| | 10% | 3.16987 | $3,169.87 (too high → rate too low) | | 12% | — | $3,037.35 | | 13% | — | $2,974.47 (too low) | | ≈12.59% | — | ≈ $3,000 |

PV and discount rates move in opposite directions: the higher the rate, the lower the PV. The answer lies between 12% and 13%, about 12.59%.

7. Annuity PV Factor Table (Table 6.1 excerpt) and Extensions

(At 3 periods and 10%: 2.4869 — exactly the 2.48685 computed in the $500 example)

Memory aid: - The PV factor is "1 minus the present value factor, divided by r"; the FV factor is "the future value factor minus 1, divided by r" — mirror images differing only in where the 1 sits - PV factor $= 1/(1+r)^t$; FV factor $= (1+r)^t$

Annuity due: cash flows occur at the beginning of each period: $$\text{Annuity due value} = \text{Ordinary annuity value} \times (1+r)$$ Example: five $400 payments as an annuity due (10%) = 4-year ordinary annuity $1,267.95 + the extra $400 at Time 0 = $1,667.95.

Perpetuity: level cash flows continue forever: $$PV = C/r$$ Example: $500/.08 = $6,250.

8. Key Takeaways

1. Timing first — ordinary annuities assume end-of-period cash flows; all formulas, factor tables, and calculator defaults rest on this assumption.
2. Two mirror-image formulas — $PV = C \times \frac{1-(1+r)^{-t}}{r}$ and $FV = C \times \frac{(1+r)^t - 1}{r}$; remember only where the "1" goes.
3. Know any four, find the fifth — given any four of PV, FV, C, t, r, solve for the fifth; finding r has no closed-form solution — use a table or trial and error.

1. Three Cash Flow Patterns: It All Starts with Timing

An ordinary annuity is a series of constant, level cash flows that occur at the end of each period — the default assumption of every formula, present value table, and financial calculator. This section extends the baseline in three directions:

Memory anchor: "When in doubt, assume end-of-period." Unless told otherwise, cash flows occur at the end of each period; only "immediate," "today," or "lease upfront" language signals an annuity due.

2. Ordinary Annuity Review: Two Shortcuts, PV and FV

With a level cash flow C per period for t periods at rate r per period:

$$PV = C \times \frac{1 - [1/(1+r)^t]}{r}, \qquad FV = C \times \frac{(1+r)^t - 1}{r}$$

• The bracket $PVIFA(r,t) = {1-[1/(1+r)^t]}/r$ is the annuity present value factor; $[(1+r)^t-1]/r$ is the annuity future value factor (Table 6.1 / Appendix A.3).
• Example: $500/yr for 3 years at 10% → factor 2.48685 → PV = $500 × 2.48685 = $1,243.43
• Example 6.5: $632/month for 48 months at 1% per month → factor 37.9740 → you can borrow $24,000
• Example: $2,000/yr for 30 years at 8% retirement account → factor 113.2832 → FV = $226,566.42

3. Annuity Due: Payments at the Beginning = Value × (1 + r)

An annuity due has cash flows at the beginning of each period; the classic case is an apartment lease, where the first payment is due immediately. Two valuation routes:

• Timeline approach: treat it as a (t−1)-period ordinary annuity plus one extra C at Time 0. Example: five $400 payments at 10% → 4-year ordinary annuity = $1,267.95 plus $400 at Time 0 → $1,667.95.
• Formula approach (one step):

$$\text{Annuity due value} = \text{Ordinary annuity value} \times (1 + r)$$

(Works for both PV and FV — the end-of-period assumption discounts every cash flow one period too many, and multiplying by (1 + r) corrects exactly that.) On a calculator, switch to "begin/due" mode and remember to switch back. Caveat calculator: the same problem — $1.5 million, 30 years, 7.5% — yields $127,006.85 per year as an ordinary annuity, but only $118,145.19 on Calculatoredge, which silently assumes an annuity due — a ~7% gap.

4. Perpetuity: PV = C / r

A perpetuity pays a level cash flow every period forever (also called a consol in Canada and the UK). With infinitely many payments, the only route is a formula:

$$PV = \frac{C}{r}$$

• Example: $500 every year, required return 8% → PV = $500/.08 = $6,250
• Example 6.7 (Fellini preferred stock): preferred stock is a perpetuity — a fixed quarterly dividend forever. The outstanding issue sells for $40/share and pays $1 per quarter → r = $1/$40 = 2.5% per quarter; for Fellini's new issue to sell at $100, the dividend must be $100 × 2.5% = $2.50 per quarter.

5. Growing Annuity: First Payment in One Year, Then Growth at g

A finite stream whose payments grow at a constant rate g each period. Let C be the first cash flow, occurring one year from today:

$$PV = C \times \frac{1 - \left(\frac{1+g}{1+r}\right)^t}{r - g}$$

Lottery example: 20 annual installments, first payment $200,000 one year from now, then growing 5% per year, r = 11%:

$$PV = \$200{,}000 \times \frac{1 - (1.05/1.11)^{20}}{.11 - .05} = \$200{,}000 \times 11.18169 = \mathbf{\$2{,}236{,}337.06}$$

6. Growing Perpetuity: PV = C / (r − g)

Growth forever:

$$PV = \frac{C}{r - g}$$

• The same lottery paid forever: PV = $200,000 × 1/(.11 − .05) = $200,000 × 16.6667 = $3,333,333.33
• Common pitfall: in both growing formulas, C is the cash flow occurring exactly one period from today — not today's, not an "average" one; and we need g < r (the formula breaks down if growth outpaces the discount rate). This is the engine behind stock pricing in the next chapters (the dividend growth model).

7. Formula Summary Card

Memory aid: "End-of-period level flows are annuities; beginning-of-period flows multiply by (1 + r); infinite lives drop the t-term; growth replaces r with r − g."

8. Three Key Takeaways

1. Timing is everything — the default is end-of-period; an annuity due pays one period earlier and is worth the ordinary value times (1 + r); leases and "pay now" insurance premiums are annuities due.
2. Infinite lives are actually the simplest case — perpetuity PV = C/r and growing perpetuity PV = C/(r − g) need no summation; preferred stock is a textbook perpetuity, and the dividend growth (Gordon) model springs directly from the growing perpetuity.
3. In the growing formulas, C is the payment due one year from now — the first cash flow sits exactly one period out, and g must be less than r; the lottery example is worth $2,236,337.06 as a growing annuity and $3,333,333.33 as a growing perpetuity.

1. Core Idea

In reality, most investments involve multiple cash flows, and the amounts are usually unequal — opening a new store means a large outlay today followed by inflows for many years. The previous chapter dealt only with single cash flows; this chapter extends the discounting tools to multiple cash flows.

The key tool is the time line: write each cash flow at the point in time when it actually occurs (today = Time 0, one year from today = Time 1, …). Draw the time line and half the problem is solved — whenever a present or future value problem gives you trouble, draw a time line.

2. Future Value with Multiple Cash Flows: Two Paths

Path 1: Compound forward one period at a time. Roll the balance forward at $(1+r)$ each year, add the new cash flow, and repeat.

Path 2: Compound each cash flow separately and add them up.

$$\mathrm{FV}T = \sum{t} \mathrm{CF}_t \times (1+r)^{T-t}$$

Both paths give the same answer. The critical step is counting the periods each cash flow earns interest — the last cash flow typically earns none (e.g., $2,000 invested at the end of each of five years at 10% grows to $12,210.20; the first flow earns only four years of interest, the last earns none).

Chapter opener: deposit $100 today and another $100 in one year at 8%. Balance at end of Year 1 = $108 + $100 = $208; rolled one more year: $208 × 1.08 = $224.64. Separately: $100 × 1.08² = $116.64 (two years of interest) + $100 × 1.08 = $108 (one year of interest).

Example 6.1: you have $7,000 now and deposit $4,000 at the end of each of the next three years at 8%. Rolling forward: $7,000×1.08+$4,000 = $11,560 → $11,560×1.08+$4,000 = $16,484.80 → $16,484.80×1.08+$4,000 = $21,803.58 (in 3 years); one more year = $21,803.58 × 1.08 = $23,547.87 (in 4 years).

Example 6.2: $100 at end of Year 1, $200 at end of Year 2, $300 at end of Year 3, at 7%. FV in 3 years = $100×1.07² + $200×1.07 + $300 = $114.49 + $214.00 + $300.00 = $628.49, of which interest = $628.49 − $600 = $28.49. Left in place two more years: FV in 5 years = $628.49 × 1.07² = $719.56 (equivalently, $100×1.07⁴ + $200×1.07³ + $300×1.07² = $131.08 + $245.01 + $343.47).

3. Present Value with Multiple Cash Flows: Two Paths

Path 1: Discount back one period at a time. Discount the last cash flow one period, add the next-to-last cash flow, and repeat.

Path 2: Discount each cash flow separately and add them up.

$$\mathrm{PV} = \sum_{t} \frac{\mathrm{CF}_t}{(1+r)^t}$$

Check: to have $1,000 in one year and $2,000 in two years at 9%, you need $1,000/1.09 = $917.43 plus $2,000/1.09² = $1,683.36 today — $2,600.79 in total. Verify: $2,600.79 × 1.09 = $2,834.86; take out $1,000, leaving $1,834.86; $1,834.86 × 1.09 = $2,000 exactly.

Example 6.3: an investment pays $200, $400, $600, and $800 at the end of Years 1–4; you can earn 12%. Discounting one at a time: $200/1.12 = $178.57, $400/1.2544 = $318.88, $600/1.4049 = $427.07, $800/1.5735 = $508.41 — PV = $1,432.93, the most you should be willing to pay.

$1,000 per year for 5 years at 6%: discounting each separately, PV = $4,212.36 (discounting back one period at a time: $1,000/1.06 + $1,000 = $1,943.40 → /1.06 + $1,000 = $2,833.39 → and so on — same result).

The PV of a series of future cash flows is the amount you need today to exactly duplicate those cash flows.

4. Cash Flow Timing: The End-of-Period Default

Unless you are explicitly told otherwise, always assume cash flows occur at the end of each period. Every formula, every table, and every financial calculator default assumes this.

• If cash flows occur at the beginning of each period (e.g., leases, annuities due): compute as an ordinary arrangement, then multiply the whole answer by (1+r)
• On a time line, the first cash flow goes at Time 1 (end of the first period), not Time 0

5. Mixed Cash Flows: Annuity Component + Lump Sum

A mixed cash flow stream combines equal (annuity-form) payments with unequal or lump-sum amounts. Principle: use the annuity shortcut where it applies, discount single cash flows individually, and add everything up.

Strasburg case: a contract stated at $245M, actually seven payments of $23.6M plus $80M in Year 8. Discount each year's salary at 12%:

Example 6.4 (deferred annuity): three $5,000 payments at the end of Years 4, 5, and 6 at 11%. FV at Year 6 = $5,000×1.11² + $5,000×1.11 + $5,000 = $6,160.50 + $5,550 + $5,000 = $16,710.50; discount 6 periods: $16,710.50/1.11⁶ = $8,934.12; check item-by-item: $5,000/1.8704 + $5,000/1.6851 + $5,000/1.5181 = $2,673.20 + $2,967.26 + $3,293.65 = $8,934.11.

Key point: PV and FV can be computed in any order and converted back and forth — as long as the discount rate is consistent and the number of periods is counted correctly, the answer is the same.

6. Calculator and Spreadsheet Tips

• Financial calculator: discount one cash flow at a time — set N and FV for each flow, store each PV in memory and accumulate (no need to write down and rekey numbers); keep I/Y unchanged and only change N and FV. Remember to clear the calculator first!
• Spreadsheet: one row per cash flow, compute each PV with a formula, then SUM the column; the same layout can be reused for different discount-rate scenarios

7. Key Examples Data Card

Memory aid: FV — "roll one year, add one cash flow" or "compound each flow, then add"; PV — "discount each flow back one at a time"; the two paths always converge.

8. Key Takeaways

1. Only two paths exist for multiple cash flows — roll forward/back one period at a time, or handle each cash flow separately and add; count the periods correctly and both work.
2. Timing is everything — cash flows are assumed to occur at period end; beginning-of-period flows require multiplying by (1+r).
3. Stated value ≠ real value — every future cash flow must be discounted back to today before summing: a $245M contract is worth only $140.02M today.

1. Core Idea

A bond is an interest-only loan: the borrower pays interest each period and repays the principal (face value) only at maturity. If Beck Corporation borrows $1,000 for 30 years at a 12 percent market rate, it pays $.12 × $1,000 = $120$ in interest each year and repays $1,000 after 30 years. A bond's cash flows thus have just two parts — the coupon annuity and the lump-sum face value — and the bond's value is the sum of the two present values.

Key Terms:

2. The Bond Value Formula

With face value F, coupon C per period, t periods to maturity, and yield r per period:

$$\text{Bond value} = C \times \frac{1 - 1/(1+r)^t}{r} + \frac{F}{(1+r)^t}$$

Xanth Co. example: 10-year bond, $80 annual coupon, YTM 8%:

• PV of face: $1,000/1.08^{10} = $1,000/2.1589 = $463.19
• PV of coupons: $80 × (1 − 1/1.08^{10})/.08 = $80 × 6.7101 = $536.81
• Bond value = $463.19 + 536.81 = $1,000 (exactly par)

When YTM equals the coupon rate, the bond sells at par — no coincidence, since a bond at $1,000 pays exactly its coupon rate.

3. Discount, Par, and Premium Bonds

One year later the Xanth bond has nine years left. Rates rise to 10%: PV of face = $1,000/1.10^9 = $424.10, PV of coupons = $80 × 5.7590 = $460.72, total = $884.82 ≈ $885 — below face, a discount bond. The $115.18 discount equals the $20 yearly shortfall times 5.7590; the built-in capital gain at maturity compensates for the below-market coupon.

Rates fall to 6%: PV of face = $1,000/1.06^9 = $591.90, PV of coupons = $80 × 6.8017 = $544.14, total = $1,136.03 — above face, a premium bond. The $136.03 premium equals the $20 yearly excess times 6.8017.

Decision rule (compare coupon rate with YTM):

4. Semiannual Coupons

U.S. bonds pay coupons twice a year, and quoted yields work like APRs: rate per period = quoted rate ÷ 2. Example 7.1: 14% coupon (two $70 payments), quoted YTM 16% (8% per six months), 7 years = 14 periods:

$$P = \$70 \times \frac{1 - 1/1.08^{14}}{.08} + \frac{\$1{,}000}{1.08^{14}} = \$70 \times 8.2442 + \$340.46 = \mathbf{\$917.56}$$

The 7% semiannual coupon is below the required 8%, so a price under $1,000 confirms the arithmetic. Effective annual yield = (1 + .08)^2 − 1 = .1664 = 16.64%, higher than the quoted 16%.

Memory aid: "Divide the quoted rate by 2, multiply periods by 2, and convert back with (1 + r/2)² − 1" — work in per-period units, then compound to get the annual yield.

5. Interest Rate Risk

The risk that bond values fluctuate with interest rates is interest rate risk, driven by two factors:

1. All else equal, the longer the maturity, the greater the risk — compounding amplifies the discounting of the face value, so long-term prices are far more rate-sensitive
2. All else equal, the lower the coupon, the greater the risk — low-coupon bonds derive proportionately more value from the face amount

Risk also increases at a decreasing rate: the jump from 1 to 10 years is huge, but from 20 to 30 years is small. The 100-year BellSouth (AT&T) issue, 7.00% coupon, maturing 2095:

6. Finding the Yield to Maturity: Trial and Error

Given price, coupon, and maturity, solve for r. Example: six-year, 8% coupon (annual $80), price $955.14:

$$\$955.14 = \$80 \times \frac{1 - 1/(1+r)^6}{r} + \frac{\$1{,}000}{(1+r)^6}$$

The equation cannot be solved explicitly — use trial and error. Shortcut: a discount bond's YTM exceeds its 8% coupon. Try r = 10%: $80 × 4.3553 + $1,000/1.7716 = $912.89, below the actual price, so 10% is too high; the true yield lies between 8% and 10%, and 9% works exactly — YTM = 9%.

Self-test check: Macrohard's 8% semiannual bond, six years, price $911.37 — trying 6% per period gives $832.32 (too low); the true rate is 5% per period, quoted YTM = 2 × 5% = 10%, EAR = 1.05² − 1 = 10.25%.

7. Current Yield versus Yield to Maturity

$$\text{Current yield} = \frac{\text{Annual coupon}}{\text{Price}}$$

In the example above, current yield = $80/$955.14 = 8.38%, below the 9% YTM — it captures only the coupon income, ignoring the built-in gain from buying at a discount. Example 7.2 confirms the reverse: a premium bond priced at $1,080.42 with $30 semiannual coupons has current yield = $60/$1,080.42 = 5.55% and YTM = 2 × 2.1% = 4.2%, below the current yield — the YTM reflects the built-in loss as the premium amortizes.

Rule: For discount bonds CY < YTM; for premium bonds CY > YTM; at par they are equal.

Bonus — clean vs. dirty prices: quotes are clean prices (net of accrued interest); what you actually pay is the dirty (invoice) price. Example: you pay $1,080, the next $60 semiannual coupon is due in four months, accrued interest = 2/6 × $60 = $20, so the clean price = $1,080 − 20 = $1,060.

8. Three Key Takeaways

1. Price is the sum of two present values — coupons as an annuity plus the face value as a lump sum; given C, t, F, and r you price the bond, and the YTM is simply that discount rate.
2. Prices and interest rates move in opposite directions — rising rates make discount bonds, falling rates make premium bonds; the coupon-vs-YTM comparison decides which, and longer maturities and lower coupons amplify the swings.
3. Quotation conventions matter — semiannual yields are quoted like APRs (divide by 2, double the periods); the YTM is a promised yield, and the current yield covers only the coupon portion (CY < YTM for discounts, CY > YTM for premiums).

1. Core Idea

Interest rate risk is the risk that arises for bond owners from fluctuating interest rates. Even if a bond is default-free, its value still moves in the opposite direction of market interest rates: when rates rise, the present value of the remaining cash flows falls and the bond is worth less; when rates fall, it is worth more.

Why do rates matter so much? The chapter opens with an extreme case: in August 2019, Germany sold 30-year bonds at a yield of −0.11% — investors bought €824M and will get back only €795M in 2050, when more than $15 trillion of government debt carried negative yields. Every wiggle in interest rates is amplified into gains or losses through prices — that is the weight of interest rate risk.

Take the Xanth Co. bond (face value $1,000, $80 annual coupon, 10 years to maturity, YTM = 8% → price exactly $1,000):

$$\text{Bond value} = C \times \frac{1 - 1/(1+r)^t}{r} + \frac{F}{(1+r)^t}$$

2. The Two Determinants of Interest Rate Risk (Textbook Rules)

How sensitive a bond's price is to interest rate changes depends directly on two things (§7.1):

1. All else equal, the longer the time to maturity, the greater the interest rate risk — in Figure 7.2, for 10% coupon bonds, the price-yield line of the 30-year maturity is far steeper than that of the 1-year maturity
2. All else equal, the lower the coupon rate, the greater the interest rate risk — low-coupon bonds derive proportionately more of their value from the face amount paid at maturity

Intuition: a large portion of a long-term bond's value comes from the $1,000 face amount, whose present value is compounded over many years; a high-coupon bond delivers more cash flow early in its life and is therefore less sensitive to the discount rate.

A numerical version of Figure 7.2 (10% coupon bonds, rates rising from 10% to 11%):

The same 1% rate move costs the 30-year bond nearly 10 times as much in price — the numerical source of the two very different slopes in Figure 7.2.

3. Interest Rate Risk Increases at a Decreasing Rate

Like most things in finance, interest rate risk increases with maturity, but at a decreasing rate: the gap between a 1-year and a 10-year bond is huge; the gap between a 20-year and a 30-year bond is fairly small. The textbook's 100-year BellSouth (now AT&T) issue (7.00% coupon, maturing 2095) illustrates the extremes:

Rates first rose and then fell: the price lost 19.66% and then gained 53.03%. Context: Disney's "Sleeping Beauty" 100-year bonds in the 1990s, Rutgers' $330M centennial issue in 2019, and Republic National Bank's 1,000-year bonds.

4. The Interest Rate Risk Premium in the Term Structure (§7.7)

The term structure of interest rates reports nominal rates on default-free, pure discount bonds of all maturities. Its shape reflects three components:

$$\text{Term structure} = \text{Real rate} + \text{Inflation premium} + \text{Interest rate risk premium}$$

The interest rate risk premium increases with maturity — but since interest rate risk itself increases at a decreasing rate, so does the premium. This is why long-term rates usually exceed short-term rates (an upward-sloping yield curve) without diverging without bound. Distinguish two near-synonyms: the term structure is built on pure discount (zero-coupon) bonds, whereas the Treasury yield curve uses coupon bond yields — for practical purposes they are almost the same thing. In total, any bond's yield comprises six components: the real rate plus five premiums (inflation, interest rate risk, default risk, taxability, liquidity).

5. Supplement: Duration — Turning Two Factors into One Number

The textbook treats interest rate risk qualitatively; the quantitative standard is the Macaulay duration — the present-value-weighted average time to each cash flow:

$$D = \frac{\sum_{t=1}^{T} t \times \dfrac{CF_t}{(1+y)^t}}{P}$$

where $P$ is the price and $CF_t$ the cash flow at time $t$. The modified duration:

$$D_{mod} = \frac{D}{1+y}, \qquad \frac{\Delta P}{P} \approx -D_{mod} \times \Delta y$$

Duration captures both textbook factors at once: longer maturity and lower coupon → longer weighted-average payoff time → greater risk. A zero-coupon bond's duration equals its maturity.

Xanth bond worked example (10-year, 8% coupon, YTM 8%):

Duration across maturities (10% coupon, YTM 10%): a 1-year bond has D = 1.00 year; a 30-year bond has D = 10.37 years — duration reproduces the textbook's "longer maturity, greater risk" rule in one number.

Building the Xanth duration (y = 8%):

$D = \$7{,}247 / \$1{,}000 = 7.25$ years. Note: the face value is only 46% of the price yet contributes 64% of the weighted time ($4,632/7,247) — the micro-mechanism behind "low coupon + long maturity → long duration."

Memory aid: "Duration is the time-weighted average of your cash flows; price change ≈ −modified duration × rate change."

6. Supplement: Using Duration to Immunize (Textbook Excel Master It!)

The chapter's Excel Master It! problem introduces duration: a bond portfolio dedicated to a future liability faces reinvestment risk — if rates fall, coupons are reinvested at lower rates. Immunization eliminates this by buying bonds with the same duration as the liability rather than the same maturity: when rates fall, reinvestment income shrinks but bond prices rise, and the two effects offset each other. Moreover, portfolio duration is the weighted average of its bonds' durations, making immunization easy to build — the same logic behind pension funds' liability-driven investing (LDI).

7. Supplement: Convexity — The Curvature of the Price-Yield Curve

Duration is only the tangent (slope) of the price-yield curve — a linear approximation. The curve is convex, calling for a second-order correction:

$$C = \frac{\sum_{t=1}^{T} t(t+1) \times \dfrac{CF_t}{(1+y)^{t+2}}}{P}$$

$$\frac{\Delta P}{P} \approx -D_{mod} \times \Delta y + \frac{1}{2} \times C \times (\Delta y)^2$$

The intuition: a 1% rate cut raises the price more than a 1% rate hike lowers it. Xanth bond evidence:

With convexity = 60.5, the approximation error shrinks from 0.29 to 0.01 percentage points; the gain (7.02%) clearly exceeds the loss (6.42%) — positive convexity favors the holder. Three basic properties (option-free bonds): ① convexity is always positive for fixed-cash-flow bonds; ② the lower the yield level, the greater the convexity; ③ longer maturity and lower coupon mean greater convexity — all pointing the same way as duration and the interest rate risk rules. Memory aid: "Duration is the slope, convexity is the bend; you lose less than you gain."

8. Three Key Takeaways

1. Prices and rates move in opposite directions; risk has two drivers — the longer the maturity and the lower the coupon, the greater the interest rate risk, and risk grows with maturity at a decreasing rate (textbook qualitative rules, §7.1).
2. Duration quantifies interest rate risk — it is the PV-weighted average time to the cash flows (Macaulay); price change ≈ −modified duration × rate change; a zero-coupon bond's duration equals its maturity; immunization matches asset duration to liability duration.
3. Convexity corrects the second-order error — the price-yield curve is convex, so a 1% rate cut gains more than a 1% hike loses; adding the ½C(Δy)² term makes the approximation nearly exact.

Tags: 考点

1. Core Idea

Stock valuation is harder than bond valuation for three reasons: (1) the promised cash flows are unknown in advance (dividends depend on board decisions); (2) common stock has no maturity, so the investment horizon is essentially forever; (3) the required return cannot be easily observed.

The core proposition of the DDM: the current stock price equals the present value of all future dividends. No matter how long you plan to hold the stock, the farther you push the future selling price, the closer its present value gets to zero — leaving only the dividends:

$$P_0 = \frac{D_1}{(1+R)^1} + \frac{D_2}{(1+R)^2} + \frac{D_3}{(1+R)^3} + \cdots$$

where R is the market required return. Zero-dividend "black hole": a firm that never pays any dividend (money goes in, nothing comes out) is worth exactly zero — dividends can be zero for some periods, but they cannot all be zero (e.g., Alphabet pays none currently, not forever).

2. General Formula and the One-Period Case

For a one-year holding period: $P_0 = \dfrac{D_1 + P_1}{1+R}$. Example: expected price in one year $70, dividend at year-end $10, required return 25%:

$$P_0 = \frac{\$10 + \$70}{1.25} = \mathbf{\$64}$$

The one-period formula alone solves nothing (you must still forecast $P_1$), but expanding $P_1$ forward period by period to infinity yields the general result: stock price = PV of all future dividends.

3. Zero Growth and Constant Growth (The Dividend Growth Model)

Case 1: Zero growth. The dividend is constant forever, so the stock is an ordinary perpetuity:

$$P_0 = \frac{D}{R}$$

Paradise Prototyping: constant dividend $10, R = 20% → $P_0 = \$10/.20 = \mathbf{\$50}$.

Case 2: Constant growth. The dividend grows at a constant rate g: $D_t = D_0(1+g)^t$ (Hedless Corp: $D_0 = \$3$, g = 8%, so in five years $D_5 = \$3 \times 1.4693 = \$4.41$). Discounting the growing dividends gives the dividend growth model:

$$P_0 = \frac{D_0 \times (1+g)}{R-g} = \frac{D_1}{R-g}$$

Memory aid: the numerator is always the next dividend $D_1$; the denominator is required return minus growth rate ($R - g$) — don't forget the minus sign.

Textbook example: $D_0 = \$2.30$, R = 13%, g = 5%:

$$P_0 = \frac{\$2.30 \times 1.05}{.13-.05} = \frac{\$2.415}{.08} = \mathbf{\$30.19}$$

Price at any time t: $P_t = \dfrac{D_t(1+g)}{R-g}$. In five years $D_5 = \$2.30 \times 1.2763 = \$2.935$, so $P_5 = \dfrac{\$2.935 \times 1.05}{.08} = \mathbf{\$38.53}$; shortcut: $P_5 = P_0 \times (1+g)^5$ — the stock price grows at the same rate as the dividend.

Gordon Growth Company (Example 8.3): next dividend $D_1 = \$4$, R = 16%, g = 6%:

$P_4 = \$50.50 = \$40 \times 1.06^4 = P_0 \times (1+g)^4$, confirming that the price grows at the dividend growth rate.

Condition for use: g must be less than R. If g > R (or g = R), the PV of dividends keeps growing — the model breaks down and produces meaningless negative or infinite prices. Generalization: the PV of any growing perpetuity is $C_1/(R-g)$ (from Chapter 6).

4. Nonconstant Growth

Procedure: forecast each dividend during the supernormal-growth period, price the stock at the point where constant growth begins (end of period t) with the growth model, then discount everything back to today. The most common mistake is identifying the wrong starting point of the constant-growth phase.

Example: first dividend in 5 years. First dividend $.50 in Year 5, growing at 10% forever thereafter, R = 20%:

$$P_4 = \frac{\$.50}{.20-.10} = \$5; \qquad P_0 = \frac{\$5}{1.20^4} = \frac{\$5}{2.0736} = \mathbf{\$2.41}$$

Textbook example: dividends of $1.00, $2.00, $2.50 over the next three years, then 5% growth forever, R = 10%:

$$P_3 = \frac{\$2.50 \times 1.05}{.10-.05} = \$52.50$$

$$P_0 = \frac{\$1}{1.10} + \frac{\$2}{1.10^2} + \frac{\$2.50}{1.10^3} + \frac{\$52.50}{1.10^3} = \$.91 + 1.65 + 1.88 + 39.44 = \mathbf{\$43.88}$$

Chain Reaction, Inc. (Example 8.4, supernormal growth): 30% growth for 3 years, then 10% forever, $D_0 = \$5M$, R = 20%:

$P_3 = \dfrac{\$10.985 \times 1.10}{.20-.10} = \$120.835M$; $P_0 = \dfrac{6.50}{1.20} + \dfrac{8.45}{1.20^2} + \dfrac{10.985}{1.20^3} + \dfrac{120.835}{1.20^3} = \$5.42+5.87+6.36+69.93 = \mathbf{\$87.57M}$; with, say, 20 million shares, that is $4.38 per share.

5. Two-Stage Growth

The dividend grows at $g_1$ for t years, then at $g_2$ forever:

$$P_0 = \frac{D_1}{R-g_1} \times \left[ 1 - \left( \frac{1+g_1}{1+R} \right)^t \right] + \frac{P_t}{(1+R)^t}, \qquad P_t = \frac{D_{t+1}}{R-g_2} = \frac{D_0 \times (1+g_1)^t \times (1+g_2)}{R-g_2}$$

The first term is the PV of a growing annuity — in Stage 1, $g_1$ may exceed R (the growing-annuity formula is not restricted); the second term is the PV of the price at the start of Stage 2, where $g_2$ must be less than R.

Highfield Company (Example 8.5): g₁ = 20% for 5 years, then g₂ = 4% forever, R = 10%, $D_0 = \$2$:

$$P_5 = \frac{\$2 \times (1.20)^5 \times (1.04)}{.10-.04} = \frac{\$5.18}{.06} = \$86.26$$

$$P_0 = \frac{\$2 \times 1.20}{.10-.20} \times \left[ 1 - \left( \frac{1.20}{1.10} \right)^5 \right] + \frac{\$86.26}{(1.10)^5} = \mathbf{\$66.64}$$

6. Components of the Required Return

Rearranging the constant growth model gives:

$$R = \frac{D_1}{P_0} + g = \text{Dividend yield} + \text{Capital gains yield}$$

Dividend yield = next dividend ÷ current price; capital gains yield = the growth rate g (also the rate at which the stock price grows). Example: stock at $20, next dividend $1, g = 10%:

$$R = \frac{\$1}{\$20} + .10 = .05 + .10 = \mathbf{15\%}$$

Check: $P_1 = \dfrac{\$1 \times 1.10}{.15-.10} = \$22 = \$20 \times 1.10$ — a $1 dividend plus a $2 gain on a $20 investment is 5% + 10% = 15%. P&G (Value Line, March 2020): projected dividend $3.00, price about $120, growth 4% → dividend yield 2.5% + capital gains yield 4% = required return 6.5%.

7. Bonus: Valuation by Multiples (No-Dividend Firms)

Many firms pay no dividends; value them with ratios such as the PE ratio:

$$P_t = \text{Benchmark PE ratio} \times \text{EPS}_t$$

Inactivision example: benchmark PE of 20, trailing four-quarter EPS of $2 → price $20 \times \$2 = \$40$; if next year's EPS is forecast at $2.50, the current $40 price implies a forward PE of 16, and the target price in one year is $20 \times \$2.50 = \$50$. For loss-makers, use the price-sales ratio (typically 0.8–2.0) or EV/EBITDA (Table 8.1: soft drinks PE 24.71 / PS 3.84 / EV/EBITDA 18.48 vs. cable TV PE 23.55 / PS 1.79 / EV/EBITDA 9.12 — ratios vary widely by industry and firm age).

8. Key Takeaways

1. Stock price = PV of all future dividends — the cash flows are dividends, there is no maturity, and R is the market required return.
2. Three solvable special cases — zero growth $P_0 = D/R$; constant growth $P_0 = D_1/(R-g)$ (requires g < R); nonconstant/two-stage growth (discount the supernormal dividends period by period, then discount the price at the start of constant growth).
3. Total return decomposes — $R = D_1/P_0 + g$ = dividend yield + capital gains yield (= g), both recoverable from price and dividend data.

1. Core Idea

Opening anchor: in 2019, Toyota announced plans to invest $13 billion over the next five years to expand plants in Tennessee, Kentucky, West Virginia, Missouri, and Alabama — a classic capital budgeting decision, the strategic asset allocation that defines the firm's operations and products for years to come.

An investment is worth undertaking if it creates value for its owners. Value is created when an investment is worth more in the marketplace than it costs us to acquire — a case of the whole being worth more than the cost of the parts.

Opening example (the "fixer-upper" house): buy a run-down house for $50,000 and spend another $50,000 on painters, plumbers, and materials — total investment $100,000. After the work, the house is worth $120,000:

$$\text{Value created} = \$120{,}000 - \$100{,}000 = \$20{,}000$$

This $20,000 is the value added by management. The real challenge of capital budgeting is to identify ahead of time whether a proposed investment will be worth more, once in place, than it costs.

2. Definition of NPV and the DCF Procedure

The difference between an investment's market value and its cost is called its net present value (NPV) — a measure of how much value is created or added today by undertaking the investment:

$$\text{NPV} = \text{Market value} - \text{Cost}$$

Because the market price of a comparable investment usually cannot be observed (whole fertilizer companies are not routinely bought and sold), we estimate value by discounted cash flow (DCF) valuation: forecast the future cash flows, discount them back to the present, then subtract the cost:

$$\text{NPV} = -\text{Initial investment} + \sum_{t=1}^{T} \frac{CF_t}{(1+r)^t}$$

The capital budgeting process can thus be viewed as a search for investments with positive NPVs — the only way to create value for stockholders.

Decision procedure (the "fixer-upper" method): first look at what comparable, fixed-up properties are selling for in the market; then estimate the cost of buying a property and bringing it to market. If the estimated market value exceeds the estimated total cost, the investment is worth undertaking — with risk, of course, because the estimates may be wrong.

3. The Fertilizer Business: Full Worked Example

Suppose we start a business producing and selling organic fertilizer. The parameters (§9.1 main example):

Cash flow timeline (Figure 9.1, in $000):

The cash flows form an eight-year annuity of $6,000 plus a lump-sum inflow of $2,000 at Year 8:

$$\text{PV} = \$6{,}000 \times \frac{1-(1/1.15^8)}{.15} + \frac{\$2{,}000}{1.15^8}$$

$$= (\$6{,}000 \times 4.4873) + (\$2{,}000/3.0590) = \$26{,}924 + \$654 = \$27{,}578$$

$$\text{NPV} = -\$30{,}000 + \$27{,}578 = -\$2{,}422$$

Negative NPV → not a good investment. With 1,000 shares outstanding, taking the project would reduce value by $\$2{,}422/1{,}000 = \$2.42$ per share.

4. The NPV Rule: Accept–Reject Decision

The net present value rule: an investment should be accepted if its NPV is positive and rejected if it is negative. In the unlikely event that NPV is exactly zero, we are indifferent (economic break-even).

Memory aid: "Positive → go, negative → no, zero → either way" — the accept–reject decision depends only on the sign of NPV.

Concept Question 9.1b: what does it mean to say an investment has an NPV of $1,000? — The investment's market value exceeds its cost by $1,000 today: undertaking it creates $1,000 of value now and increases the total value of the stock by $1,000.

Because the goal of financial management is to increase share value, the only thing we need to know for an accept–reject decision is the sign of the NPV. Note that an NPV like −$2,422 is an estimate — the true NPV could only be known by actually selling the project, so our estimates must be reliable.

5. Example 9.1: New Product Launch

A five-year consumer product project: cash flows of $2,000 in the first two years, $4,000 in the next two, and $5,000 in the last year; it costs $10,000 to begin production; we use a 10% discount rate.

Discounting each flow and summing:

$$\text{PV} = \frac{\$2{,}000}{1.1} + \frac{\$2{,}000}{1.1^2} + \frac{\$4{,}000}{1.1^3} + \frac{\$4{,}000}{1.1^4} + \frac{\$5{,}000}{1.1^5}$$

$$= \$1{,}818 + \$1{,}653 + \$3{,}005 + \$2{,}732 + \$3{,}105 = \$12{,}313$$

$$\text{NPV} = \$12{,}313 - \$10{,}000 = \$2{,}313 > 0 \quad \Rightarrow \quad \textbf{Accept}$$

6. Calculation Notes and the Spreadsheet Trap

Trap (important warning): the NPV function in spreadsheets is actually a PV function — one of the original spreadsheet programs years ago got the definition wrong, and subsequent programs copied it. It discounts every cash flow, including the initial investment at Year 0, so naive use gives a wrong answer. Correct usage: apply the function only to cash flows from Year 1 onward, then add the (negative) initial cost:

$$\text{NPV} = \text{NPV(function applied to Year-1-onward flows)} + (-\text{Initial investment})$$

Memory aid: "Discount the future cash flows first, then subtract today's cost — don't let the function discount the initial outlay too."

The textbook's spreadsheet example reworks Example 9.1 and shows two answers: if the initial −$10,000 is included in the NPV function's discounting range, the result is wrong (the Year-0 flow gets discounted an extra period); only by restricting the function to Years 1–5 and then subtracting $10,000 do we get the correct $2,313.

Self-test (Chapter Review Problem 9.1): an overseas expansion has cash flows of −$200 at Year 0, then $50, $60, $70, and $200 in Years 1–4, at a 10% required return. The discounted flows are $45.45, $49.59, $52.59, and $136.60, cumulating to $284.24 → NPV = $84.24 (for comparison, the ordinary payback is 3.10 years).

Also, NPV is only an estimate: once we have the cash flows and the discount rate, the calculations are mechanical; coming up with those inputs is the hard part (the subject of later chapters).

7. Why NPV Is the Best Investment Criterion

A good criterion must answer two questions: ① Is a particular project a good investment? ② Among several good projects, which one should we take? Only NPV can always provide the correct answer to both (Table 9.7):

• Every alternative — payback, discounted payback, AAR, IRR, MIRR, PI — is flawed in some key way (ignoring time value, arbitrary cutoffs, ignoring post-cutoff cash flows, multiple IRRs, ranking conflicts);
• NPV has no serious flaws and is the preferred decision criterion. Keep two things in mind: ① NPV is always the market value of an asset or project minus its cost; ② the financial manager serves shareholders by identifying and taking positive-NPV projects;
• In practice, firms use multiple criteria alongside NPV: when the signals agree, confidence increases; when they conflict, rely on the NPV and analyze further.

Survey evidence (Table 9.6): in a 1999 survey of 392 CFOs, the percentage who always or almost always used each criterion was: IRR 76%, NPV 75%, payback 57%, discounted payback 29%, AAR 20%, and PI 12%. Historically, in 1959 only 19% of firms used IRR or NPV as their primary technique while 68% used payback or accounting returns; by 1981, IRR or NPV had risen to 82% and become the dominant criterion.

8. Key Takeaways

1. NPV = Market value − Cost — it measures the value created (or destroyed) by an investment today; capital budgeting is a search for positive-NPV projects.
2. The NPV rule: accept if positive, reject if negative, indifferent at zero — accepting a negative-NPV project lowers share value (fertilizer example: −$2.42 per share).
3. NPV is the only criterion without serious flaws — payback, IRR, and the other alternatives each have critical shortcomings; and beware the spreadsheet trap: the NPV function is really a PV function, so discount future flows first, then subtract the initial cost.

1. Core Idea

The internal rate of return (IRR) is the most important alternative to NPV. It seeks a single rate of return that summarizes the merits of a project. The rate is "internal" in the sense that it depends only on the project's own cash flows, not on rates offered elsewhere.

The intuitive example: a project that costs $100 today and pays $110 in one year. For every dollar invested you get $1.10 back, so the return is 10% — that 10% is the project's IRR.

2. Formal Definition and the IRR Rule

Definition of the IRR: the discount rate that makes the NPV of the investment equal to zero.

$$NPV = -\$100 + \frac{\$110}{1+R} = 0$$

$$\frac{\$110}{1+R} = \$100 \;\Rightarrow\; 1+R = 1.1 \;\Rightarrow\; R = 0.10 \;(\text{10%})$$

An NPV of zero means the investment is an economic break-even proposition — value is neither created nor destroyed, and we are indifferent between taking and not taking it.

The IRR Rule:

Accept an investment if the IRR exceeds the required return; reject it otherwise.

3. Computing the IRR on Multi-Period Projects

In general the IRR cannot be solved for directly — use trial and error, by hand or calculator. This is the same problem as finding the unknown rate on an annuity (Ch. 5) or the yield to maturity on a bond (Ch. 7); in both cases you were really finding an IRR.

Example: a project costs $100 and pays $60 per year for two years.

$$NPV = 0 = -\$100 + \frac{\$60}{1+IRR} + \frac{\$60}{(1+IRR)^2}$$

The NPV crosses zero between 10% and 15%; finer trials give IRR ≈ 13.1%. Accept if the required return is below 13.1%, reject if it is above.

The NPV Profile: plot NPV on the y-axis against the discount rate on the x-axis. The smooth curve obtained is the NPV profile, and where it cuts the x-axis the NPV is exactly zero — that point is the IRR. The graph makes the IRR rule and NPV rule geometrically identical.

Memory aid: "The IRR is the discount rate that makes NPV = 0; invest when IRR clears the hurdle, bail when it falls short."

4. When the IRR and NPV Agree

The IRR is sometimes called the discounted cash flow (DCF) return. The two rules give identical accept–reject decisions whenever two conditions hold:

Example 9.4: up-front cost of $435.44; cash flows of $100, $200, $300 in Years 1–3.

The IRR is 15%. With an 18% required return (NPV = −$24.47), reject — the IRR rule (15% < 18%) and the NPV rule agree.

5. Problem 1: Nonconventional Cash Flows and Multiple IRRs

When the cash flows change sign more than once, the question "What's the rate of return?" may have no unambiguous answer.

Strip-mining example: cash flows of −$60, +$155, −$100 (you must spend $100 in Year 2 restoring the terrain).

The NPV profile dips negative, rises positive, then falls again: the NPV is zero at both 25% and 33.33% — there are two IRRs! With a 10% required return, the IRR rule (both IRRs exceed 10%) says accept, yet the NPV at 10% is negative — the project should be rejected. The NPV is positive only for required returns between 25% and 33.33%.

Descartes' Rule of Sign: the maximum number of IRRs equals the number of times the cash flows change sign. Here the signs flip twice (negative → positive → negative), so at most two IRRs — no need to search for a third.

Example 9.5 (no IRR at all): invest $51, receive $100 in one year, pay out $50 in two years. The signs change twice, but trial and error reveals the NPV is negative at every discount rate — the actual number of IRRs can be less than the maximum, even zero. Under no circumstances should this investment be taken.

6. Problem 2: Mutually Exclusive Investments and the Crossover Rate

For mutually exclusive investments (take one or the other, but not both), the highest IRR is not necessarily the best project — compare NPVs.

Investment A vs Investment B (both cost $100; A's cash flows 50/40/40/30, IRR = 24%; B's cash flows 20/40/50/60, IRR = 21%):

A has the higher IRR, but B has greater total cash flow and pays back more slowly: B's NPV is higher at low discount rates. The two NPV profiles cross at about 11.1% (exactly 11.0704%): below 11.1% choose B, above 11.1% choose A. The IRR and NPV rankings conflict below the crossover — go with the higher NPV.

The crossover rate: the discount rate at which the two projects' NPVs are equal. To find it — subtract one project's cash flows from the other's, then compute the IRR of the difference.

Example 9.7: A (−$400, $250, $280) and B (−$500, $320, $340). Moving from A to B requires $100 extra, yielding $70 and $60 extra:

$$NPV(B-A) = 0 = -\$100 + \frac{\$70}{1+IRR} + \frac{\$60}{(1+IRR)^2}$$

The IRR of the difference is 20% — the crossover rate (both projects have NPV = $2.78 at 20%). It makes no difference which way you subtract; IRR(A − B) is the same number.

Investing vs financing cash flows: A (−$100, +$130) and B (+$100, −$130) both have an IRR of 30%, but B has financing-type cash flows (cash in first, out later — e.g., a seminar with fees paid in advance). NPV(B) at 12% = $100 − $130/1.12 = −$16.07. For financing-type projects, accept only if the IRR is below the required return — the IRR here is a rate you are paying, not receiving.

7. The Redeeming Qualities of the IRR

Despite its flaws, the IRR is even more popular in practice than the NPV: people prefer talking about rates of return to dollar values ("Remodeling the clerical wing has a 20 percent return" beats "at 10 percent the NPV is $4,000"). Also, the IRR can be estimated even when the required return is unknown — a project showing a 40% return is almost surely worth taking.

1999 CFO survey (Table 9.6): 76% of CFOs always or almost always use the IRR (average score 3.09/4), versus 75% for NPV and 57% for payback.

MIRR in brief: to fix the multiple-IRR problem, modify the cash flows first and then compute an IRR (discounting / reinvestment / combination approaches — on −$60, +$155, −$100 at a 20% required return they give 19.74%, 19.72%, and 19.87% respectively). But an MIRR depends on an externally supplied rate, so it is no longer truly "internal."

8. Three Key Takeaways

1. The IRR is the discount rate that makes NPV = 0 — the IRR rule (accept when IRR > required return) matches the NPV rule for conventional, independent projects, but NPV remains the ultimate judge.
2. Sign changes create trouble — nonconventional cash flows can produce multiple IRRs, no IRR, or an inverted NPV profile (investing vs. financing); the IRR rule breaks down and only NPV works.
3. For mutually exclusive projects, rank by NPV, not IRR — the highest return is not the highest value; find the crossover rate (the IRR of the difference in cash flows) and choose based on which side of it the required return falls.

1. Core Idea

This lesson answers two intuitive questions: (1) How long until I get my money back? (2) How much value do I create per dollar invested?

• The payback rule measures how long it takes to recover the initial investment. It is simple and intuitive but ignores the time value of money — the textbook classifies it and the discounted payback (§9.2, §9.3) under "payback criteria."
• The profitability index (PI), also called the benefit-cost ratio, equals the present value of future cash flows divided by the initial investment — it measures the value created per dollar invested: "bang for the buck."

Both are closely tied to NPV: PI > 1 is equivalent to NPV > 0, and the discounted payback aligns with NPV in direction.

2. Payback: Definition and Calculation

Payback period = the number of years until the accumulated cash flows equal or exceed the initial cost.

Payback rule: accept the investment if its payback is less than a prespecified cutoff (e.g., two years); reject otherwise.

Fractional-year formula (cash flow received uniformly within the year):

$$\text{Payback} = n + \frac{\text{Amount still to be recovered at end of year } n}{\text{Cash flow in year } n+1}$$

Memory aid: "Whole years + remaining gap ÷ next year's cash flow."

Table 9.1's five odd projects (showing payback can behave strangely):

Key point: Project D has two different payback periods because of the negative Year-3 cash flow — the calculation does not guarantee a single answer; Project E pays back in six months yet is a disaster.

3. Shortcomings of the Payback Rule

Four major flaws:

1. It completely ignores the time value of money — cash flows are added up without discounting;
2. It ignores risk differences — a very risky and a very safe project get the same payback;
3. The cutoff is arbitrary — there is no economic rationale for choosing any particular number;
4. Cash flows after the cutoff are ignored entirely.

The Long vs. Short example (Table 9.2; both cost $250, cutoff two years):

The rule accepts Short and rejects Long — both decisions are wrong (required return 15%):

$$NPV(\text{Short}) = -\$250 + \frac{\$100}{1.15} + \frac{\$200}{1.15^2} = -\$11.81$$

$$NPV(\text{Long}) = -\$250 + \$100 \times \frac{1-(1/1.15^4)}{0.15} = +\$35.50$$

Conclusion: payback is a "break-even" measure in an accounting sense, not an economic one — it does not ask the right question (the impact on share value) and it biases us toward shorter-term projects (against R&D and new projects).

4. Redeeming Qualities and Practice

Despite its flaws, large firms commonly use payback for relatively minor decisions — detailed analysis can cost more than the possible loss from a mistake:

In practice, a firm may require a two-year payback on all investments of less than $10,000, with larger investments getting greater scrutiny. An investment that pays back rapidly and whose benefits extend beyond the cutoff probably has a positive NPV — that is its value as a screen.

5. The Discounted Payback

Discounted payback = the length of time until the sum of the discounted cash flows equals the initial investment. Rule: accept if the discounted payback is less than a prespecified cutoff. It fixes the "ignores time value" flaw.

Textbook example (Table 9.3): cost $300, cash flows $100 per year for 5 years, required return 12.5%:

The ordinary payback is almost exactly 3 years; the discounted payback is 4 years. The discounted payback is break-even in an economic or financial sense — in 4 years the project repays the $300 plus the interest it could have earned elsewhere (in Figure 9.3 the future-value lines cross at exactly four years).

Key property: if a project ever pays back on a discounted basis, it must have a positive NPV — here NPV = $356 − $300 = $56, exactly the value of the cash flows occurring after the discounted payback.

Example 9.3: cost $400, perpetuity of $100 per year, 20% required return → PV = 100/.2 = $500, NPV = $100; ordinary payback 4 years; discounted payback is the number of years that makes the annuity factor equal to 4 → a little less than 9 years.

Yet it is rarely used in practice: it is no simpler to compute than NPV, yet it still relies on an arbitrary cutoff and truncates later cash flows — a compromise that lacks both the payback's simplicity and the NPV's conceptual rigor.

6. The Profitability Index

Definition: present value of future cash flows divided by the initial investment:

$$PI = \frac{\text{PV of future cash flows}}{\text{Initial investment}} = \frac{PV}{C_0}$$

Example: a project costs $200 and its future cash flows are worth $220 → PI = 220/200 = 1.1 (NPV = $20, a desirable investment).

PI rule: accept if PI > 1, reject if PI < 1, indifferent if PI = 1.

Relation to NPV:

$$PI > 1 \iff PV > C_0 \iff NPV > 0$$

Interpretation: a PI of 1.1 means that per dollar invested, $1.10 in value, or $0.10 in NPV, results. For this reason PI is often proposed as a performance measure for government or not-for-profit investments, and when capital is scarce, it makes sense to allocate funds to the projects with the highest PIs.

7. Ranking Problems and Survey Evidence

The mutually exclusive trap: an investment costing $5 with a $10 present value (NPV $5, PI = 2) versus one costing $100 with a $150 present value (NPV $50, PI = 1.5). If they are mutually exclusive, the second is preferred despite its lower PI — the same ranking problem as the IRR: never rank mutually exclusive projects by PI; rank them by NPV.

1999 CFO survey (Table 9.6; 392 CFOs, "always or almost always" used):

Historically, only 19% of firms used IRR or NPV in 1959 (68% used payback or accounting returns); by 1981 the figure had reached 82%. Firms typically use multiple criteria because an estimated NPV can be "soft" — when payback, AAR, and NPV all point the same way ("all systems go"), managers trust the decision more.

8. Three Key Takeaways

1. The PI is the NPV rule in ratio form — PI > 1 ⇔ NPV > 0 and the two usually agree; but for mutually exclusive projects rank by NPV, since the PI (like the IRR) can mislead (the $50-NPV project beats the PI = 2 project).
2. Payback recovers your money but ignores its value — it can accept a value-destroying project (Short, NPV = −$11.81) and reject a value-creating one (Long, NPV = $35.50); the discounted payback fixes time value, and "pays back on a discounted basis ⇒ positive NPV."
3. Every shortcut has an arbitrary cutoff — payback, discounted payback, and PI cannot replace NPV: in practice IRR (76%) and NPV (75%) dominate, payback (57%) still serves as a cheap screen for small decisions, and the PI (12%) is the least used.

1. Mutually Exclusive Investments

If two investments, X and Y, are mutually exclusive, taking one of them means we cannot take the other. For example, with one corner lot, we can build a gas station or an apartment building—but not both. Projects that are not mutually exclusive are said to be independent.

With mutually exclusive projects, the question changes from "Is it worth taking?" to "Which one is best?"—and the answer is simple: the best one is the one with the largest NPV. Can we also say the best one has the highest return? As we show, the answer is no.

2. The Conflict Example: Cash Flows of Investments A and B

Consider two mutually exclusive investments, each costing $100:

IRR(A) = 24%, IRR(B) = 21%. Simple intuition suggests A is better because of its higher return—but simple intuition is not always correct. B has greater total cash flow ($170 vs. $160) but pays back more slowly than A.

3. Which Is Better Depends on the Discount Rate

The NPVs of the two investments at different required returns:

• At a required return of 10%: NPV(B) = 29.79 > NPV(A) = 29.06 → B is better even though A has the higher return; IRR and NPV rankings conflict.
• At a required return of 15%: NPV(A) = 17.18 > NPV(B) = 14.82 → A is better; no conflict.
• The NPV profiles cross at about 11.1% (exactly 11.0704%). At any discount rate below 11.1%, B has the higher NPV; above 11.1%, A does (see Figure 9.8).

4. Why the Conflict Arises

The conflict stems from differences in the timing of cash flows. B's cash flows are back-loaded (year 4: $60 vs. A's $30), making them more sensitive to the discount rate:

• Low discount rate → distant cash flows barely discounted → B, with larger total flows, wins
• High discount rate → distant cash flows heavily discounted → A, with early flows, wins

At a deeper level, the IRR implicitly assumes intermediate cash flows are reinvested at the IRR, while NPV implicitly assumes reinvestment at the required return. Whenever we are comparing investments to determine which is best, looking at IRRs can be misleading—we must look at the relative NPVs to avoid choosing incorrectly.

5. The Crossover Rate (Example 9.7)

The crossover rate is the discount rate that makes the NPVs of two projects equal. Take two mutually exclusive investments:

Moving out of A and into B: invest an extra $100, receive an extra $70 in year 1 and $60 in year 2—a difference stream of −$100, +$70, +$60:

$$\text{NPV}(B-A) = -\$100 + \frac{\$70}{1+R} + \frac{\$60}{(1+R)^2}$$

Setting NPV(B−A) = 0 and solving gives IRR = 20%—this is the crossover rate. Check: at a 20% discount rate, both investments have an NPV of $2.78. General method: subtract one project's cash flows from the other's and compute the IRR of the difference; it makes no difference which one you subtract from which (computing the IRR of A−B gives the same number).

6. Decision Rule and Intuition

We are ultimately interested in creating value for the shareholders, so the option with the higher NPV is preferred, regardless of the relative returns. Intuition: one investment has a 10% return and makes you $100 richer immediately; another has a 20% return and makes you $50 richer. You would rather have $100 than $50, returns notwithstanding.

Related pitfall: financing-type cash flows. Investment B (+$100, −$130, IRR = 30%) has an upward-sloping NPV profile—its IRR is a rate you are paying, not receiving; accept it only if its IRR is lower than your required return (at a 12% required return, NPV(B) = −$16.07, so it should be rejected).

7. Exam Memory Card

Memory aid: "Rank by money, not by rate—highest NPV wins; below the crossover rate, the 'big but slow' project wins; above it, the 'fast and early' project wins."

8. Key Takeaways

1. Mutually exclusive projects must be ranked by NPV, not by IRR — A has a higher IRR (24% vs. 21%), yet B is better whenever the discount rate is below 11.1%.
2. The conflict arises from the timing of cash flows — B has larger total cash flow but pays back slowly and wins at low discount rates; A wins at high rates; the NPV profiles cross at the crossover rate.
3. Find the crossover rate by computing the IRR of the difference in cash flows (Example 9.7: −$100, +$70, +$60 → 20%); when rankings conflict, always take the project with the higher NPV.

1. Core Idea: Which Cash Flows Are "Relevant"?

The most basic — and most error-prone — step in evaluating an investment is determining which cash flows are relevant. Taking on a project changes the firm's overall cash flows today and in the future, so we must look only at the changes it causes in the firm's cash flows and judge whether they add value.

A relevant cash flow = a change in the firm's overall future cash flow that comes about as a direct consequence of the decision to take the project.

2. Definition of Incremental Cash Flows and Its Corollary

Formal definition of incremental cash flows:

$$\text{Incremental cash flows} = \text{Firm's future cash flows with the project} - \text{Firm's future cash flows without the project}$$

The definition has an obvious and important corollary:

Any cash flow that exists regardless of whether the project is undertaken is not relevant.

This corollary is the logical starting point of all identification rules: keep only the cash flows that occur because of the project.

3. The Stand-Alone Principle

Computing the firm's total future cash flows with and without the project would be cumbersome in practice. The stand-alone principle solves this: once we have determined the incremental cash flows, we treat the project as a kind of "minifirm" — with its own future revenues and costs, its own assets, and its own cash flows. We then compare the cash flows of this minifirm to the cost of acquiring it, evaluating the project purely on its own merits, in isolation from any other activities.

4. Sunk Costs: Water Under the Bridge

A sunk cost is a cost already paid, or already incurred as a liability to pay. Today's accept/reject decision cannot change it — the firm must pay it no matter what — so it is clearly irrelevant and must always be excluded.

• Classic case: General Milk Company hired a consultant to evaluate launching a chocolate milk line. The consultant's report omitted the hefty consulting fee as a project cost, and the firm objected. The consultant was right — the fee must be paid whether or not the line is launched. It is a sunk cost.

Memory aid: Sunk costs are water under the bridge — don't let them hijack your decision.

5. Opportunity Costs: Value It by What It's Worth Today, Not What You Paid

An opportunity cost is not an out-of-pocket cash expense; it is the value of the best forgone alternative use of a resource. The common situation: the project uses an asset the firm already owns.

• Classic case: converting an old rustic cotton mill, bought years ago for $100,000, into upmarket condominiums. There is no direct cash outflow for the mill, but it is not "free" — at minimum we could sell it.
• Key question: what price to charge? Not the $100,000 paid (that cost is sunk and irrelevant), but what the mill would sell for today (net of selling costs) — that is the amount given up by using the mill instead of selling it.

Memory aid: Opportunity cost looks at "what it can sell for today," not "what we paid for it."

6. Side Effects: Erosion and Spillovers

Projects frequently have side, or spillover, effects — good and bad — on existing products, and incremental cash flows must include them.

Erosion: the negative impact on the cash flows of an existing product from the introduction of a new product.

Erosion is relevant only when the sales would not otherwise be lost — if future competition would take those sales anyway, do not double-count them.

Beneficial spillover: HP's printers sold for $500–$600 in 1994 but fell below $100 by 2020 — yet HP was unconcerned: the real money is in consumables (ink-jet cartridges, laser toner, special paper), and cheaper printers drive consumable sales.

7. Net Working Capital, Financing Costs, and Other Issues

Net working capital (NWC): besides long-term assets, a project typically requires cash on hand, inventories, and accounts receivable (net of supplier financing). As the project winds down, inventories are sold, receivables are collected, and cash balances are drawn down — the NWC investment closely resembles a loan: supplied at the beginning and recovered toward the end.

Financing costs: we do not include interest paid or any other financing costs such as dividends or principal repaid. Interest is a component of cash flow to creditors, not cash flow from assets; the debt-equity mix is a managerial variable that determines how project cash flow is divided between owners and creditors and must be analyzed separately.

Other issues: - Measure cash flow when it actually occurs, not when it accrues in an accounting sense; - Always use aftertax cash flows — "incremental cash flows" means aftertax incremental cash flows; aftertax cash flow and accounting net income are entirely different things.

Quick-reference table:

8. Three Key Takeaways

1. The definition is the touchstone: incremental cash flows are all changes in the firm's future cash flows that are a direct consequence of the project; any flow that exists without the project is excluded.
2. Three classic pitfalls: exclude sunk costs (water under the bridge); charge opportunity costs at the asset's current market value, not historical cost; adjust for erosion only to the extent sales would not otherwise be lost.
3. Three exclusions: no financing costs (interest), no accrued-but-unrealized items, no pre-tax figures — project analysis always uses aftertax cash flows as they actually occur.

1. Core Idea

The three approaches to operating cash flow — bottom-up, top-down, and tax shield — all measure the same thing: dollars in minus dollars out from operations. They are simply different ways of rearranging the same basic information about sales, costs, depreciation, and taxes. Used correctly, all three always produce the same OCF; no method is inherently better, so use whichever is most convenient for the problem at hand.

2. The Baseline Data and Basic Formula

Baseline example from §10.5:

EBIT and taxes:

$$EBIT = Sales − Costs − Depreciation = \$1{,}500 − 700 − 600 = \$200$$

$$Taxes = EBIT \times T_C = \$200 \times 0.21 = \$42$$

The basic OCF definition (no interest is included in project evaluation):

$$OCF = EBIT + Depreciation − Taxes = 200 + 600 − 42 = \$758$$

Total project cash flow (§10.3): Project cash flow = OCF − Change in NWC − Capital spending.

3. The Bottom-Up Approach

Start at the accountant's bottom line — net income — and add back noncash deductions such as depreciation:

$$OCF = Net\ Income + Depreciation = \$158 + 600 = \$758$$

Shark attractant project (§10.3): net income $21,780 + depreciation $30,000 = $51,780, exactly the OCF we had before.

⚠️ Key caveat: OCF = net income + depreciation is correct only if no interest expense was subtracted in computing net income (interest is a financing cost, not part of operating cash flow).

4. The Top-Down Approach

Start at the top of the income statement with sales and work down, subtracting costs, taxes, and other expenses while leaving out purely noncash items such as depreciation:

$$OCF = Sales − Costs − Taxes = \$1{,}500 − 700 − 42 = \$758$$

Shark attractant project: sales $200,000 − total costs $142,430 − taxes $5,790 = $51,780.

5. The Tax Shield Approach

$$OCF = (Sales − Costs) \times (1 − T) + Depreciation \times T$$

$$= (\$1{,}500 − 700) \times 0.79 + 600 \times 0.21 = \$632 + 126 = \$758$$

This approach views OCF as having two components:

Shark attractant project: ($200,000 − 142,430) × (1 − .21) = $45,480; $30,000 × .21 = $6,300; OCF = $45,480 + 6,300 = $51,780.

6. How the Depreciation Tax Shield Works

Depreciation is a noncash expense, so its only cash flow effect is to reduce taxes. At the 21% corporate tax rate, every dollar of depreciation saves 21 cents in taxes.

Memory aid: "Depreciation × tax rate = tax shield"; Aftertax income + tax shield = OCF

Automation example (§10.6): pretax savings $22,000/year, depreciation = $80,000/5 = $16,000 per year

• Aftertax cost savings: $22,000 × (1 − .21) = $17,380
• Depreciation tax shield: $16,000 × .21 = $3,360
• OCF = $17,380 + 3,360 = $20,740

7. Equivalence and When to Use Each Method

Usage guidance:

• Cost-cutting proposals → the tax shield approach is most intuitive; the two benefits (aftertax savings + tax shield) add directly
• Setting a bid price (§10.6) → first solve for the required OCF = $30,442; then use bottom-up to back out net income $15,442 = OCF − depreciation $15,000; finally solve for sales $128,546, i.e., a bid of $128,546/5 = $25,709 ≈ $26,000 per truck
• With no interest expense the three methods are equivalent — pick whichever is most convenient

8. Key Takeaways

1. Same destination, different routes — the three methods are just different arrangements of the same information; used correctly they always agree ($758 and $51,780 are identical under all three)
2. The tax shield is a hidden cash benefit — depreciation is a noncash expense, yet each $1 of depreciation saves a real 21 cents of taxes at the 21% rate
3. Match the method to the problem — use the tax shield approach for cost-cutting projects and the bottom-up approach to back out a bid price

1. Core Idea

Total project cash flow = Operating Cash Flow − Change in NWC − Capital Spending. Beyond OCF there are two "non-operating" pieces that are easily missed or miscalculated: changes in net working capital (NWC) and the aftertax salvage value at disposal:

• The NWC investment is like a loan — invested at the start, recovered in full at the end;
• Selling an asset whose market value differs from book value creates a tax consequence, so only the aftertax salvage value (never the raw market price) belongs in the cash flow.

2. The NWC Investment: An Internal Loan

Beyond fixed assets, a project needs cash on hand, inventory, and receivables; the balance after the credit supplied by suppliers (A/P) is the NWC:

$$\text{NWC} = \text{Cash} + \text{Inventory} + \text{A/R} - \text{A/P}$$

• An increase in NWC is a cash outflow (funds tied up in the project);
• A decrease in NWC is a cash inflow (funds freed up).

The textbook's mechanical rule: every NWC investment must reappear at some future time with the opposite sign. In the Shark Attractant project, $20,000 invested at Year 0 is recovered at Year 3:

3. NWC Varying with Sales: The Power Mulcher Case (MMCC)

In the MMCC example, NWC is no longer constant but 15% of each year's sales (starting at $20,000). The cash flow in each year equals this year's NWC level minus last year's:

$$NWC_t = 0.15 \times \text{Sales}t, \qquad \Delta NWC_t = NWC_t - NWC{t-1}$$

The final year's $66,000 = $16,500 released during the year + $49,500 of remaining NWC recovered at the end. NWC peaks at $108,000 in Year 3, then declines as sales fall.

4. Why ΔNWC Belongs in the Analysis

Changes in NWC convert accounting revenues and costs into actual cash received and paid:

$$\text{Cash income} = \text{Sales} - \Delta\text{A/R};\quad \text{Cash costs} = \text{Costs} - \Delta\text{A/P}$$

Textbook mini-example: Sales $500, Costs $310 (no depreciation, no taxes), A/R up $30, A/P up $55, NWC down $25:

$$\text{Total CF} = OCF - \Delta NWC = \$190 - (-25) = \$215 = (500 - 30) - (310 - 55)$$

The sign convention is the key: falling NWC releases cash (deducting a negative adds it back); rising NWC absorbs cash (deduct it). Same logic in Example 10.1 (CWT): Sales $998, Costs $734, NWC up $20 → net cash flow = $264 − 20 = $244.

5. Aftertax Salvage Value: Formula and Three Cases

The gain or loss (MV − BV) at disposal is taxed or deductible at the corporate rate T:

$$\text{Aftertax salvage} = MV - T_C \times (MV - BV)$$

When BV = 0 (the common straight-line-to-zero case), the formula simplifies to:

$$\text{Aftertax salvage} = MV \times (1 - T_C)$$

6. Worked Numerical Examples

Case 1: MV > BV (the $12,000 car) — 5-year MACRS, book value $691.20 after Year 5, sold for $3,000: tax = .21 × ($3,000 − 691.20) = $484.85; aftertax proceeds = $3,000 − 484.85 = $2,515.15.

Case 2: MV < BV (Staple Supply, Example 10.2) — $160,000 computer on 5-year MACRS, book value $27,648 after 4 years, sold for $10,000: the loss of $17,648 saves .21 × 17,648 = $3,706 in taxes; total aftertax cash flow = $10,000 + 3,706 = $13,706.

Case 3: BV = 0 (MMCC and cost-cutting) — $800,000 equipment worth $160,000 (20% of cost) in Year 8: aftertax salvage = $160,000 × (1 − .21) = $126,400; the cost-cutting proposal gives $20,000 × .79 = $15,800; Example 10.3 gives $30,000 × .79 = $23,700.

Full version (Project X) — MV $625,000, BV $390,500: aftertax salvage = $625,000 − .21 × ($625,000 − 390,500) = $575,755.

7. The Complete Procedure and Common Pitfalls

Five steps:

1. Compute the book value at the year of sale (MACRS or straight-line);
2. Compute the difference MV − BV (a gain if positive, a deductible loss if negative);
3. Tax = $T_C \times (MV - BV)$;
4. Aftertax salvage = MV − tax;
5. Add it to the final year's total cash flow together with the NWC recovery.

Memory aid: "Sale price − tax = cash in hand. MV above BV pays tax, MV below BV saves tax, BV = 0 means multiply by (1 − T)."

Common pitfalls: - Taxing the difference between sale price and original cost instead of book value (wrong); - Forgetting the NWC recovery in the final year (Shark Attractant: +$20,000 in Year 3); - A project that frees up NWC (Example 10.3: the system releases $45,000) → Year 0 shows an inflow of +$45,000, and the end of the project shows a −$45,000 repayment.

8. Key Takeaways

1. NWC behaves like a loan — invested at the start (outflow), recovered in full at the end; increases are outflows, decreases are inflows.
2. Salvage is taxed on the MV − BV difference — aftertax salvage = MV − T×(MV − BV), simplifying to MV(1 − T) when BV = 0; MV > BV pays tax, MV < BV saves tax.
3. Both items land in the final year — last-year cash flow = OCF + NWC recovery + aftertax salvage; all three pieces are required.

Tags: 难点;考点

1. Core Idea

Depreciation is a noncash deduction — it involves no cash outflow. It affects project cash flows only through one channel: taxes. Depreciation reduces taxable income and therefore lowers the tax bill.

$$\text{Depreciation tax shield} = \text{Depreciation} \times T_C$$

At a corporate tax rate $T_C = 21\%$, every $1 of depreciation saves 21 cents in taxes. Capital budgeting uses tax depreciation (the method allowed by tax law), not accounting depreciation.

2. How Depreciation "Saves" Taxes

Depreciation lowers EBIT first, then lowers taxes:

$$EBIT = Sales - Costs - Depreciation$$

$$\text{Taxes} = EBIT \times T_C$$

Basic example from the text (Sales = $1,500, Costs = $700, Depreciation = $600, $T_C = 21\%$):

Without the $600 depreciation deduction, taxes would be $126 higher — that is the tax shield in a nutshell.

3. The Tax Shield Approach

The third (and most transparent) definition of operating cash flow:

$$OCF = (Sales - Costs) \times (1 - T_C) + \text{Depreciation} \times T_C$$

Two components of the formula:

Plugging in: $OCF = (\$1{,}500 - 700) \times .79 + 600 \times .21 = \$632 + 126 = \$758$

Memory aid: "After-tax revenue + taxes saved by depreciation = operating cash flow."

4. Three OCF Definitions, One Answer

Shark Attractant case (§10.3/10.5): Sales $200{,}000, total costs $142{,}430 (variable $125{,}000 + fixed $17{,}430), depreciation $30{,}000 (= $90{,}000/3).

$$\text{After-tax value} = (\$200{,}000 - 142{,}430) \times (1 - .21) = \$45{,}480$$

$$\text{Tax shield} = \$30{,}000 \times .21 = \$6{,}300$$

$$OCF = \$45{,}480 + 6{,}300 = \$51{,}780 \quad \text{(identical to the other two methods)}$$

5. Tax Depreciation: MACRS

Capital budgeting must use tax depreciation: the Modified Accelerated Cost Recovery System (MACRS). Every asset is assigned to a property class, and yearly depreciation = fixed percentage × asset cost.

MACRS Depreciation Allowances (Table 10.7):

The percentages sum to 100% — the full cost is written off; expected salvage value and economic life are not considered. For a $12,000 car in the 5-year class: Year 1 depreciation $= .20 \times 12{,}000 = \$2{,}400$, Year 2 $= .32 \times 12{,}000 = \$3{,}840$.

Notes: Bonus depreciation was 100% for 2018–2022, dropping 20% per year thereafter to zero after 2026; real property uses straight-line (39 years nonresidential, 27.5 years residential), and land cannot be depreciated.

6. The Tax Shield in Capital Budgeting

Case A: Cost-cutting proposal (§10.6). Equipment costs $80{,}000, saves $22{,}000 pretax per year, 5-year straight-line to zero, $T_C = 21\%$, discount rate 10%.

• Depreciation $= \$80{,}000/5 = \$16{,}000$ per year → shield $= \$16{,}000 \times .21 = \$3{,}360$
• After-tax saving $= \$22{,}000 \times .79 = \$17{,}380$; $OCF = 17{,}380 + 3{,}360 = \$20{,}740$
• After-tax salvage $= \$20{,}000 \times (1 - .21) = \$15{,}800$; NPV = $8,431 > 0 → accept

Case B (Example 10.3): $200{,}000 inventory system, 4-year straight-line, depreciation $50{,}000/yr, shield $= \$10{,}500$, after-tax saving $47{,}400$, $OCF = \$57{,}900$/yr; NPV(16%) = −$4{,}749$, IRR ≈ 14.4% → reject.

Case C (Example 10.4, EAC): the precipitation system's depreciation tax shield ($49{,}875$) exceeds its after-tax operating cost ($7{,}900$), so OCF is positive at +$41{,}975 — a large shield can make OCF positive when operating cost is small relative to purchase price.

Asset disposal: the excess of sale price over book value is "excess depreciation" that must be taxed (recapture). After-tax salvage $= \text{sale price} \times (1 - T_C)$; if book value exceeds sale price, the difference is a tax loss and creates a shield.

7. Data Card

8. Key Takeaways

1. Depreciation is not itself a cash flow — it affects cash flow only through taxes, as a noncash deduction that reduces taxable income.
2. Tax shield $= Dep \times T_C$; the tax shield formula $OCF = (S - C)(1 - T_C) + Dep \times T_C$ is fully equivalent to the top-down and bottom-up definitions.
3. Compute the shield using tax depreciation (MACRS); on disposal, count the after-tax salvage $\times(1-T_C)$ as inflow, with any book-vs-market difference taxed or deductible.

1. Core Idea: Incremental Cash Flows and the Stand-Alone Principle

The first step in any complete project analysis is deciding which cash flows are relevant. Relevant cash flows are incremental cash flows — the changes in the firm's overall future cash flows that occur as a direct consequence of taking the project.

Stand-alone principle: treat the project as a "minifirm" — evaluate it on its own merits by comparing its incremental cash flows with the cost of acquiring it.

Value note: identify cash flows with honesty and look forward — sunk costs must not dictate today's decision; judge projects by incremental contribution, not historical outlay.

2. Case 1: Shark Attractant Project — Pro Forma Statements

Inputs: 50,000 cans/year at $4; variable cost $2.50/can; fixed costs $17,430/year; equipment $90,000 (straight-line to zero over 3 years, ~zero salvage); initial NWC $20,000; tax rate 21%; required return 20%.

Projected income statement (Table 10.1):

Capital requirements (Table 10.2): NWC $20,000 every year; net fixed assets 90,000 → 60,000 → 30,000 → 0 (book values); total investment $110,000 / $80,000 / $50,000 / $20,000.

3. Building Project Cash Flows and Evaluating

Project cash flow = Operating cash flow − Change in NWC − Capital spending:

$$\text{OCF} = \text{EBIT} + \text{Depreciation} - \text{Taxes} = 27{,}570 + 30{,}000 - 5{,}790 = \$51{,}780$$

Memory aid: "NWC investment is like a loan — what you put in at the start comes back at the end (same number, opposite sign)."

$$\text{NPV} = -110{,}000 + \frac{51{,}780}{1.2} + \frac{51{,}780}{1.2^2} + \frac{71{,}780}{1.2^3} = \$10{,}648 > 0 \Rightarrow \text{accept}$$

Other metrics: IRR ≈ 25.8% (> 20%); payback ≈ 2.1 years; AAR = $21,780/$65,000 = 33.51% (average NI ÷ average book investment). The AAR exceeding the IRR is exactly why AAR cannot be interpreted as a rate of return.

4. Case 2: MMCC Power Mulcher Project — Full-Scale Practice

Inputs: 8-year life; price $120 for 3 years then $110; units 3,000→5,000→6,000→6,500→6,000→5,000→4,000→3,000; variable cost $60/unit; fixed costs $25,000/year; equipment $800,000 (7-year MACRS, no bonus depreciation); salvage $160,000 in Year 8; tax 21%; required return 15%.

7-year MACRS depreciation (Table 10.10): 14.29%→$114,320; 24.49%→$195,920; 17.49%→$139,920; 12.49%→$99,920; 8.93%→$71,440; 8.92%→$71,360; 8.93%→$71,440; 4.46%→$35,680.

NWC: $20,000 initially, then 15% of that year's sales (Year 1: 0.15 × $360,000 = $54,000, an extra outflow of $34,000; recovered as sales decline).

OCFs (partial): $146,457 / $258,393 / $294,033 / $257,983 / $232,252 / $192,736…

Aftertax salvage = $160,000 × (1 − .21) = $126,400 (the $160,000 excess of market over zero book value is taxed at 21%).

Conclusion (Table 10.14): NPV(15%) = $146,852 > 0; IRR = 19.86%; payback = 3 + 209,116/258,733 = 3.81 years → accept. Remember: the result is an estimate of NPV — if projections lack confidence, test estimate quality (sensitivity analysis, next chapter).

5. Details: Net Working Capital and Depreciation

Credit-sales adjustment: cash income = sales − increase in A/R; cash costs = costs − increase in A/P. If omitted, total CF = OCF − ΔNWC corrects it automatically (example: sales $500, costs $310, ΔAR +$30, ΔAP +$55 → net cash flow = 190 − (−25) = $215).

MACRS: every asset is assigned to a class (3/5/7-year) that sets its tax life; depreciation = cost × fixed percentage. 5-year class: 20% / 32% / 19.20% / 11.52% / 11.52% / 5.76%, summing to 100%. Expected salvage value and economic life do not affect the deduction.

Aftertax salvage value (key formula):

$$\text{ATSV} = \text{MV} - T_C \times (\text{MV} - \text{BV})$$

Market > book → tax on the recaptured depreciation; book > market → tax loss (e.g., $160,000 equipment → $160,000 × (1 − .21)).

6. Three Equivalent OCF Definitions

With sales $1,500, costs $700, depreciation $600, tax rate 21%: EBIT = $200, taxes = $42:

The depreciation tax shield = D × T = 600 × .21 = $126 (each $1 of depreciation saves 21 cents of tax). All three give $51,780 for the shark attractant project. Which to use? — whichever is most convenient for the problem at hand.

7. Special Cases: Cost-Cutting, Bid Pricing, and EAC

• Cost-cutting: equipment $80,000; pretax savings $22,000/year; 5-year straight-line; salvage $20,000. EBIT = 22,000 − 16,000 = $6,000, taxes = $1,260 → OCF = $20,740; aftertax salvage $15,800. NPV(10%) = $8,431 > 0 → automate.
• Bid pricing: the lowest profitable price is the one that makes NPV = 0. Truck example: required OCF = $78,805 / 2.58873 = $30,442 → annual sales $128,546 → $25,709 per truck ≈ $26,000 bid. Zero-NPV benchmark guards against the winner's curse.
• EAC (unequal lives): Machine A: −$117.36 / 1.7355 = −$67.62/yr; Machine B: −$159.89 / 2.4869 = −$64.30/yr → choose B. Use EAC when lives differ AND service is needed indefinitely (replacement upon wear-out).

8. Key Takeaways

1. Relevance first, numbers second — include only incremental aftertax cash flows: exclude sunk costs and interest, include opportunity costs, erosion, and NWC changes.
2. Assemble cash flows from three pieces — Project CF = OCF (EBIT + Dep − Taxes) − ΔNWC − Capital spending; NWC in at the start, recovered at the end.
3. NPV is the final verdict — accept when NPV > 0 (shark $10,648; MMCC $146,852); IRR/payback are supporting evidence, AAR is not a return; the result is an estimate whose assumptions deserve scrutiny.

1. Core Idea

The NPV rule is the gold standard of capital budgeting: accept a project when its net present value is positive — market value exceeds cost. But NPV cannot be observed; it must be estimated, and estimates can be wrong. This note covers two things: how to compute NPV correctly in Excel and Python (avoiding tool misuse), and how sensitivity analysis assesses the reliability of an NPV estimate — the degree of forecasting risk and which variable is most dangerous.

$$\text{NPV} = -\text{Initial Investment} + \sum_{t=1}^{T}\frac{CF_t}{(1+r)^t}$$

2. Excel Modeling: The NPV Function Trap

The textbook (Example 9.1) issues a classic warning: years ago, one spreadsheet program got the =NPV() function wrong — it is actually a PV function that discounts the first cell as a Year-1 cash flow, and later programs copied the error. Cash flows (5-year life, 10% discount rate): $2,000 in Years 1–2, $4,000 in Years 3–4, $5,000 in Year 5; cost $10,000.

Correct structure: PV = $2,000/1.1 + $2,000/1.1² + $4,000/1.1³ + $4,000/1.1⁴ + $5,000/1.1⁵ = $1,818 + 1,653 + 3,005 + 2,732 + 3,105 = $12,313; NPV = $12,313 − $10,000 = $2,313 > 0, so accept.

Memory aid: "Excel's NPV is a PV — keep Year-0 out of the parentheses; compute the PV first, then subtract the cost."

3. Python Modeling: The NPV Function

In Python, numpy_financial.npv() treats the first cash flow as period 0 (undiscounted) — the opposite of Excel. The safest approach is to write a loop yourself:

def npv(rate, cfs):           # cfs[0] = Year-0 cash flow (negative initial cost)
    return sum(cf / (1 + rate) ** t for t, cf in enumerate(cfs))

npv(0.10, [-10000, 2000, 2000, 4000, 4000, 5000])   # → 2312.7 ≈ $2,313

Whichever tool you use, draw the cash-flow timeline first: CF0 = −10,000 → 2,000 → 2,000 → 4,000 → 4,000 → 5,000. Blindly pressing calculator or computer buttons without understanding what is happening is exactly the real-world danger the textbook warns about.

4. The Fertilizer Example: Full NPV Calculation (§9.1 Main Example)

An 8-year annuity of $6,000 plus a lump sum of $2,000 at Year 8:

$$\text{PV} = \$6{,}000 \times \frac{1-(1/1.15^8)}{.15} + \frac{\$2{,}000}{1.15^8} = (\$6{,}000 \times 4.4873) + (\$2{,}000/3.0590) = \$27{,}578$$

$$\text{NPV} = -\$30{,}000 + \$27{,}578 = -\$2{,}422 < 0 \quad \Rightarrow \text{reject}$$

With 1,000 shares outstanding, the project would reduce share value by $2,422/1,000 = −$2.42 per share. This is the direct meaning of the NPV rule: positive NPV → creates shareholder wealth; negative NPV → destroys it.

5. Sensitivity Analysis: Freeze Everything Except One Variable

Forecasting risk: if cash-flow projections are seriously in error, no amount of careful DCF arithmetic saves you — GIGO (garbage in, garbage out). Sensitivity analysis freezes all variables except one and lets it vary across its plausible range, observing how much NPV moves. If NPV is highly sensitive to small changes in a variable, that variable carries high forecasting risk and deserves the most attention.

Textbook project: cost $200,000, 5-year life, straight-line depreciation to zero, required return 12%, tax rate 21%. Base case: 6,000 units, price $80, variable cost $60/unit, fixed costs $50,000/year.

Base-case income statement: Sales $480,000 − VC $360,000 − FC $50,000 − Depreciation $40,000 = EBIT $30,000 − Taxes (21%) $6,300 = Net income $23,700; OCF = $30,000 + 40,000 − 6,300 = $63,700/year; 5-year annuity factor at 12% = 3.6048:

$$\text{NPV} = -\$200{,}000 + 63{,}700 \times 3.6048 = \$29{,}624$$

6. Results: Unit Sales vs. Fixed Costs

Vary unit sales only (5,500–6,500):

Vary fixed costs only ($45,000–$55,000):

Conclusion: over the given ranges, NPV is far more sensitive to unit sales than to fixed costs — even the fixed-cost worst case keeps a positive NPV, while the sales worst case leaves NPV at only $1,147. Graphically (Figure 11.1): plot NPV on the vertical axis and unit sales on the horizontal; all points fall on a straight line — the steeper the line, the more sensitive NPV is to that variable.

7. Scenario and Simulation Analysis

Scenario analysis: let all variables change at once — everything pessimistic (low sales, low price, high costs) or everything optimistic:

In the worst case the project loses more than half its investment (−$122,732/$200,000). Note that "best/worst" really mean optimistic/pessimistic — absurd extremes (a diet soda that cures the common cold) should not be in the analysis.

Simulation analysis: combine scenario analysis (all variables move, but few values) with sensitivity analysis (one variable, many values): randomly draw unit sales, price, and costs from their ranges, compute NPV, repeat thousands of times, and summarize the average NPV and the percentage of negative NPVs. All three what-if tools share one limitation: they tell you what can happen, not what to do — the decision still requires managerial judgment.

8. Three Key Takeaways

1. Tool trap: Excel's =NPV() is actually a PV function — including the Year-0 cash flow over-discounts it by one period; compute PV of Year-1 onward, then subtract the cost. In Python, writing your own loop is the safest.
2. Sensitivity analysis is "single-variable freezing": the more sensitive NPV is to a variable, the higher its forecasting risk; in the NPV-variable graph, the steeper the line, the greater the risk — here unit sales is far more dangerous than fixed costs.
3. Three escalating what-if tools: sensitivity (one variable) → scenario (all variables, few values) → simulation (random draws, thousands of runs); they expose risk but give no decision rule — final judgment rests with NPV and the manager.