Advanced Corporate Finance

Before you read: write one line guessing what a holder of a ten-year bond loses, in percentage terms, if market yields rise from 1 percent to 5 percent. Keep the line. The third paragraph checks it.

On 31 December 2022, Silicon Valley Bank carried a held-to-maturity securities portfolio at an amortised cost of 91.327 billion dollars. In the same 10-K, in a footnote, it reported the fair value of that portfolio: 76.169 billion. The gap was 15.158 billion dollars. Total shareholders equity was 16.004 billion. Not one of those securities had defaulted, and almost all of them were obligations of the United States government or its agencies. What had happened was that the rate used to discount their cash flows had moved, and the bank had, on that arithmetic, roughly nothing left.

That is a lesson about discounting, not about banking, and it is where this course starts. Everything else in corporate finance is built on the operation performed in that footnote.

Where the fifteen billion went

Take a single instrument of the kind SVB held. A ten-year Treasury note with a face value of 1,000 dollars and a 1 percent coupon pays 5 dollars every six months for twenty periods, then 1,000 at the end. Nothing about those cash flows ever changes. What changes is the rate at which the market discounts them.

In early January 2021 the ten-year constant-maturity yield was 0.93 percent, or 0.465 percent per half-year. Price the note:

P = 5 × [1 - (1.00465)-20] / 0.00465 + 1000 × (1.00465)-20

(1.00465)20 = 1.097225, so (1.00465)-20 = 0.911391. The coupon strip is worth 5 × 19.0557 = 95.28 and the principal 911.39, for a price of 1,006.67.

On 19 October 2023 the same constant-maturity yield touched 4.98 percent, or 2.49 percent per half-year. Same coupons, same principal, same issuer:

(1.0249)20 = 1.635421, so (1.0249)-20 = 0.611462. The coupon strip falls to 5 × 15.6039 = 78.02 and the principal to 611.46, for a price of 689.48.

The holder lost 317.19 on a 1,006.67 investment, or 31.5 percent, without a single missed payment. Check the line you wrote. Most people guess something in the region of 4 percent, reasoning that rates moved 4 points; the answer is roughly eight times that, because the shortfall compounds across twenty periods. Key idea: a bond is a fixed set of cash flows and a variable rate, and the price is the only thing free to move.

The bank's defence, which was true and did not help, was that it intended to hold the securities to maturity and would receive every dollar of par. That is correct and irrelevant: depositors wanted their money in March 2023, not in 2031, and the only way to give it to them was to sell at 76 cents on the dollar.

One identity, written four ways

All of the above is one operation. A dollar at date t is worth 1/(1+r)t today, and the present value of a stream is the sum:

PV = sumt=1..T Ct / (1+r)t

Four special cases collapse that sum into closed form, and every one of them is used later in this course.

Perpetuity. A constant C forever is worth C/r. The British government issued exactly this instrument: the 2.5 percent Consolidated Stock paid 2.50 pounds a year on 100 pounds of nominal stock with no redemption date, and the Treasury only bought back the last undated gilts in 2015. At a 4 percent required return, that 2.50 is worth 2.50/0.04 = 62.50. The perpetuity formula looks like a trick until you notice that the first hundred years carry almost all the value: discounting kills the tail for you.

Annuity. A constant C for T years is a perpetuity starting now minus a perpetuity starting at T. That subtraction is the whole derivation:

PV = C/r - (C/r) × (1+r)-T = (C/r)[1 - (1+r)-T]

Twenty annual payments of 5,000 at 6 percent: (5,000/0.06) × [1 - (1.06)-20] = 83,333.33 × 0.688195 = 57,349.61. If the payments arrive at the start of each year instead of the end, every one of them is discounted one period less, so multiply by 1.06: 60,790.59. That single multiplication is the difference between an ordinary annuity and an annuity due, and it is worth 3,441 dollars here.

Growing perpetuity. If the payment grows at a constant g below r forever, then PV = C1/(r - g), where C1 is next year's payment, not this year's. A share expected to pay 2.00 next year, with a required return of 8 percent and growth of 4 percent, is worth 2.00/0.04 = 50.00. Raise g to 5 percent and the same share is worth 2.00/0.03 = 66.67. One percentage point of assumed growth moved the value by a third. Hold onto that: in Lesson 11 the same formula supplies most of a company's valuation, and the same fragility comes with it.

Growing annuity. Growth for a finite time gives PV = [C1/(r - g)] × [1 - ((1+g)/(1+r))T], which is the workhorse for a wage stream, a lease with an escalator, or a project whose cash flows rise with inflation.

A mortgage, amortised, at two real rates

Weekly averages for the thirty-year fixed mortgage in the United States hit a record low of 2.65 percent in the first week of January 2021 and a two-decade high of 7.79 percent in the last week of October 2023. Take a 400,000 dollar loan at each. The monthly payment inverts the annuity formula:

PMT = P × i / [1 - (1+i)-n], with n = 360 and i the annual rate over twelve.

Same house, same principal, same thirty years. The borrower who signed in October 2023 pays 455,281 dollars more in interest than the one who signed in January 2021. That is what a discount rate is worth.

Now open the schedule for the 7.79 percent loan. Each month, interest is the rate times the outstanding balance and principal is whatever is left of the payment.

Ninety-seven cents of every dollar in month one is rent on money. Run the amortisation schedule forward with the closed form Bn = P(1+i)n - PMT[((1+i)n - 1)/i] and after ten years the balance is 349,396. The borrower has paid 345,184 dollars and retired 50,604 of principal. The point: an amortising loan front-loads interest not by design or malice but because interest is charged on a balance that starts at its maximum, and the arithmetic of that is the same arithmetic as the bond.

Which rate goes in the denominator

Three conversions cause most of the errors in practice, and none of them is difficult.

APR against effective annual rate. A quoted 12 percent annual percentage rate compounded monthly is 1 percent a month, and twelve of those is (1.01)12 - 1 = 12.6825 percent, not 12. Compounded continuously it is e0.12 - 1 = 12.7497 percent. Two lenders quoting 12 percent are not quoting the same price, and the effective annual rate is what makes them comparable.

Nominal against real. The Fisher relation is multiplicative, not additive: 1 + rnominal = (1 + rreal)(1 + inflation). At 5 percent nominal and 3 percent inflation the real rate is 1.05/1.03 - 1 = 1.9417 percent, not 2. The rule that follows is absolute and is broken constantly: discount nominal cash flows at a nominal rate and real cash flows at a real rate. A project whose revenues were forecast in today's money and discounted at a nominal 9 percent has had its inflation removed twice.

Periods must match the payments. The bond above was priced on a semiannual basis because it pays semiannually. Feeding an annual rate into a semiannual schedule is the single most common arithmetic error in a first valuation model.

One further honesty before the course goes further. This course teaches how these numbers are made and where they break. It is not investment advice, and it recommends no security, no company and no trade. When SVB appears here, it is an arithmetic exhibit.

Common misconceptions

• "A bond held to maturity cannot lose money." If nothing defaults, the holder does receive par, and the loss never appears in earnings. The loss is real all the same: it is the difference between what the money did and what it could have done, and it becomes a cash loss the moment the holder needs liquidity, which is exactly the moment rate rises tend to create. SVB's 15.158 billion was disclosed only in a footnote and it still ended the bank.
• "The discount rate is the interest rate." A discount rate is an opportunity cost: the return available on an alternative of similar risk. For a Treasury note that happens to be an interest rate. For a factory it is not, and Lesson 6 spends a whole lesson building one.
• "Ten percent is ten percent." A 10 percent APR compounded daily, a 10 percent effective annual rate and a 10 percent continuously compounded rate are three different prices. Always convert to a common basis before comparing.
• "The annuity formula gives value today." It gives value one period before the first payment. If the first payment arrives at date 3, the formula gives value at date 2 and you must discount that result back two more periods.

What to carry forward

One identity underlies the entire course: value is a cash flow divided by one plus a rate, raised to the number of periods you must wait. From it come the perpetuity C/r, the annuity (C/r)[1 - (1+r)-T], the growing perpetuity C1/(r - g) and the growing annuity. A 1 percent ten-year note worth 1,006.67 at a 0.93 percent yield is worth 689.48 at 4.98 percent, and that 31.5 percent gap, multiplied across a portfolio, is the 15.158 billion in SVB's footnote. A 400,000 dollar mortgage at 7.79 percent costs 635,551 in interest against 180,270 at 2.65 percent. Rates must be converted before they are compared, and nominal cash flows need nominal rates.

The next lesson takes the same discounting machinery and asks what to do with it: given a set of projects and a rate, which do you accept, and why does the rule most managers reach for first give the wrong answer often enough to matter.

Sources

1. SVB Financial Group. (2023). Annual report on Form 10-K for the fiscal year ended December 31, 2022. U.S. Securities and Exchange Commission. sec.gov
2. Board of Governors of the Federal Reserve System. (2023). Review of the Federal Reserve's supervision and regulation of Silicon Valley Bank. federalreserve.gov
3. Federal Reserve Bank of St. Louis. (n.d.). 30-year fixed rate mortgage average in the United States (MORTGAGE30US). FRED. fred.stlouisfed.org
4. Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.), Chapters 2-3. McGraw-Hill Education.
5. Berk, J., & DeMarzo, P. (2019). Corporate Finance (5th ed.), Chapters 4 and 6. Pearson.

A capital committee has two proposals and enough money for one. The kiosk costs 50,000 dollars and returns 80,500 in a year: an internal rate of return of 61.0 percent. The new store costs 500,000 and returns 650,000 in a year: an IRR of 30.0 percent. The memo recommends the kiosk, because 61 beats 30, and the recommendation is wrong. This lesson takes that wrong answer apart, then takes apart two more that fail in different ways.

Failure one: a rate is not an amount

Put both projects through the net present value rule at a 10 percent cost of capital.

The store creates 90,909 dollars of value and the kiosk 23,182. Shareholders cannot spend a percentage. They can spend 67,727 dollars, which is what choosing the store rather than the kiosk adds.

IRR is a rate, and a rate says nothing about the base it is earned on. Sixty-one percent on 50,000 dollars is a smaller gain than thirty percent on 500,000, and no amount of staring at the two percentages reveals that, because the percentages have deliberately thrown the base away.

There is a repair, and it is worth knowing because committees like rates. Compute the IRR of the increment: the extra 450,000 spent and the extra 569,500 received (650,000 minus 80,500). That incremental project returns 569,500/450,000 - 1 = 26.56 percent, comfortably above the 10 percent cost of capital, so the increment is worth taking, so take the store. Incremental IRR and NPV always agree when they are both well defined, which tells you the flaw was never in the arithmetic of IRR but in ranking by it.

The upshot: IRR answers "how fast does this money work?" NPV answers "how much richer am I?" The second question is the one the firm is being paid to answer.

Failure two: two internal rates, both correct

A mine costs 100 million dollars, produces 230 million of net revenue in year one, and then costs 132 million in year two to close the pit and restore the site. Reclamation obligations of this shape are ordinary in mining, in offshore drilling and in nuclear generation. Write the NPV as a function of the rate:

NPV(r) = -100 + 230/(1+r) - 132/(1+r)2

Multiply through by (1+r)2 and set it to zero. With x = 1 + r, that is 100x2 - 230x + 132 = 0, whose roots are x = (230 ± 10)/200, that is 1.20 and 1.10. The project has an IRR of 10 percent and an IRR of 20 percent. Both satisfy the definition exactly.

The NPV profile is a hump. The project adds value only for rates strictly between 10 and 20 percent, which reverses the usual instinct that a lower discount rate is always better news. Descartes' rule of signs tells you when to expect this: a stream can have as many internal rates as it has sign changes. Minus, plus, minus is two changes, so up to two roots. Streams that begin with an outflow and then never turn negative again are called conventional, and they have exactly one IRR.

There is no debate to have here and no judgement to exercise. Ask what the firm's cost of capital actually is. At 12 percent the NPV is -100 + 205.357 - 105.229 = +0.128 million, so the mine is worth opening, barely. The IRRs were never the question.

Failure three: timing, and the reinvestment story you should not tell

Two mutually exclusive projects, each costing 1,000 dollars.

Early has the higher IRR at every cost of capital, because IRR does not depend on the cost of capital at all. Yet at 5 percent, Late is worth 127 dollars more. The two NPV profiles cross where 1,200/(1+r) = 1,400/(1+r)2, that is at r = 1,400/1,200 - 1 = 16.67 percent. Below the crossover the longer project wins; above it the shorter one does. If your cost of capital is 5 percent and you rank by IRR you will systematically choose short projects over valuable long ones.

Textbooks often explain this by saying that IRR assumes cash is reinvested at the IRR while NPV assumes reinvestment at the cost of capital. Be careful with that sentence. Neither rule contains a reinvestment assumption: both are functions of the stated cash flows and nothing else. What is true is narrower and more useful. IRR compresses a stream into one rate, and rates from streams of different length and size cannot be compared without restoring the information that the compression discarded.

The patch that does work is the modified internal rate of return, which states the reinvestment rate explicitly instead of hiding it. Carry every inflow forward to the final date at the cost of capital, then find the rate that grows the outlay into that terminal value. At 5 percent over two years: Early's 1,200 becomes 1,260, and MIRR = (1,260/1,000)1/2 - 1 = 12.25 percent. Late's 1,400 is already at date two, so MIRR = (1,400/1,000)1/2 - 1 = 18.32 percent. MIRR now ranks Late first, agreeing with NPV. It agrees because you told it the rate that NPV was using all along.

Payback, and the one job it still does

The payback period counts the years until cumulative cash flow turns positive. A 1,000 dollar project returning 400 a year pays back in 2.5 years. Discount the flows at 10 percent first and the cumulative present values run 363.64, 694.22, 994.75, then 1,267.96, so discounted payback lands at 3 + 5.25/273.21 = 3.02 years.

Both versions share a fatal property: they are blind after the cutoff. Consider a project returning 600 a year for two years against a 1,000 outlay, followed by a 900 clean-up cost in year three. Payback is 1.67 years, which looks excellent, and the NPV at 10 percent is -1,000 + 545.45 + 495.87 - 676.18 = -634.86. Payback recommended a project that destroys 635 dollars per 1,000 invested. It also ignores the discount rate entirely in its undiscounted form, so a dollar in year four counts the same as a dollar today.

Managers use it anyway, and not only out of laziness. Graham and Harvey surveyed 392 chief financial officers and found roughly 76 percent always or almost always using IRR, roughly 75 percent using NPV, and roughly 57 percent still using payback, with smaller firms and older chief executives leaning on it hardest. Payback is a liquidity and exposure screen: it answers how long the firm's capital is at risk in a country, a technology or a customer relationship it may not be able to forecast past. Read as an answer to that question it is defensible. Read as a value rule it is indefensible.

When the budget itself is fixed

NPV assumes you can raise capital for every positive-value project. Divisions rarely can. Under a hard budget, rank by the profitability index, which is present value of inflows divided by the initial outlay, so it measures value per dollar of the scarce resource. With 100,000 dollars available: project P costs 100,000 with a PV of 130,000 (index 1.30, NPV 30,000); projects Q and R each cost 50,000 with PVs of 72,000 (index 1.44, NPV 22,000 each). Ranking by NPV picks P and creates 30,000. Ranking by index picks Q and R and creates 44,000. Why this matters: the index is the right rule only while the constraint binds, and only when projects are divisible enough to fill the budget without waste; the moment the constraint is relaxed, NPV is right again.

Common misconceptions

• "A high IRR is a good project." A high IRR is a strong return on whatever base the project happens to have, which may be tiny, brief, or both. The 61 percent kiosk creates a quarter of the value of the 30 percent store. Ask for the NPV, and if a rate is wanted for the memo, quote the IRR of the increment.
• "IRR and NPV always agree." For a single conventional project judged accept or reject against a cost of capital, they do agree, always. For ranking mutually exclusive projects, for streams with more than one sign change, and under capital rationing, they routinely disagree.
• "A lower discount rate always raises NPV." True for conventional streams. Untrue for the mine, whose NPV is negative at 0 percent, positive at 15 percent and negative again at 25 percent, because a large late outflow is itself being discounted.
• "Payback is simply wrong and no serious firm uses it." A majority of surveyed CFOs use it, and as a measure of how long capital is exposed it answers a real question. The error is using it to rank value.

Where this leaves us

Accept a project when its NPV at the opportunity cost of capital is positive, and when choosing among alternatives take the highest NPV. IRR fails on scale, where 61 percent on 50,000 loses to 30 percent on 500,000; on sign changes, where minus, plus, minus produced honest IRRs of 10 and 20 percent; and on timing, where two projects crossed over at 16.67 percent. Incremental IRR and MIRR repair the ranking by putting back the information IRR discarded. Payback measures exposure, not value, and recommended a project with an NPV of minus 635. Under a binding budget the profitability index turned 30,000 dollars of value into 44,000.

Bottom line: the rules are only as good as the cash flows fed to them, and so far every cash flow in this lesson has been handed to you. The next lesson builds them, which is where most of the real errors live.

Sources

1. Graham, J. R., & Harvey, C. R. (2001). The theory and practice of corporate finance: Evidence from the field. Journal of Financial Economics, 60(2-3), 187-243.
2. OpenStax. (2022). Payback period method. In Principles of Finance, Chapter 16. Rice University. openstax.org
3. Wikipedia contributors. (n.d.). Internal rate of return. Wikipedia. en.wikipedia.org
4. Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.), Chapter 5. McGraw-Hill Education.
5. Berk, J., & DeMarzo, P. (2019). Corporate Finance (5th ed.), Chapter 7. Pearson.

Table A-1 of IRS Publication 946 says that a five-year asset is deducted at 20.00, 32.00, 19.20, 11.52, 11.52 and 5.76 percent of its cost. Those six numbers are not accounting trivia. At a 21 percent corporate rate they are the government's share of a machine, and on a 12 million dollar bottling line they are worth 1,825,836 dollars in present value. That figure will appear below, alongside three modelling errors that are each worth more.

The rules of Lesson 2 take cash flows as given. Nobody hands them to you. This lesson builds one project's schedule line by line and then prices the mistakes.

The four questions that decide what goes in

An incremental cash flow is the difference between the firm's total cash flow with the project and without it. That definition, applied honestly, settles almost every argument.

• Is it sunk? A sunk cost is unrecoverable whatever you decide, so it is identical with and without the project, so it is not incremental. Our firm spent 600,000 dollars last year on a feasibility study. It is gone in both futures. It does not appear in the schedule, and no amount of institutional embarrassment changes that.
• Is there an opportunity cost? The line occupies a warehouse bay the firm could otherwise let for 250,000 a year. Taking the project forfeits that rent, so 250,000 a year is a genuine cost even though no cheque is written. Assets already owned are not free; they are worth their best alternative use.
• Are there side effects? The new format will take about 500,000 a year of contribution from an existing product line. Cannibalisation is incremental and negative. The honest test is whether the sales would have been lost anyway to a competitor, in which case they belong in the base case, not in the project.
• Is it financing? Interest on the debt raised to fund the line is real cash, and it must not appear here. Financing cost lives in the discount rate. Subtracting interest from the cash flows and then discounting at a cost of capital that already prices debt charges the project twice.

Remember: the numerator holds what the assets do; the denominator holds what the money costs. Keeping them apart is the single most useful discipline in capital budgeting.

The schedule, built

The line costs 12,000,000 installed, runs for five years, and is expected to fetch 2,000,000 on disposal. It adds 9,000,000 of revenue and 4,600,000 of cash operating costs a year. Net working capital runs at 10 percent of revenue, so 900,000 goes out at the start and comes back at the end. The tax rate is 21 percent and the cost of capital is 11 percent.

Start with pre-tax operating cash, after the two adjustments above:

9,000,000 - 4,600,000 - 500,000 - 250,000 = 3,650,000 per year

Depreciation is not a cash flow, but the tax it saves is. The standard form separates the two:

OCF = (Revenue - Costs) × (1 - T) + Depreciation × T

The first term is 3,650,000 times 0.79, or 2,883,500, in every year. The second is the MACRS schedule times 0.21.

Disposal needs care. Five years of MACRS have written off 11,308,800, leaving a tax basis of 691,200. Selling for 2,000,000 produces a taxable gain of 1,308,800, taxed at 21 percent for 274,848, so the after-tax proceeds are 1,725,152. A salvage figure entered gross is one of the commonest errors in a student model and one of the commonest in a real one.

Accept. The project adds about 1.14 million dollars of value, which on a 12.9 million outlay is a thin but genuine margin.

What each mistake is worth

Now run the same model wrong, three ways, and read the damage in dollars rather than in principle.

The ranking is the lesson. Acceleration of depreciation is worth 49,225 dollars here, which is real and which nobody argues about. The side effects are worth 2.19 million, which is forty times as much and which people argue about constantly, because the numbers are estimates and the sponsoring division would rather they were smaller. Why this matters: the disputes in a capital budgeting meeting are almost never about tax tables. They are about what would have happened anyway.

One live complication: since the Tax Cuts and Jobs Act of 2017, US firms have at times been able to expense qualifying equipment immediately rather than depreciate it, and Congress has changed the phase-down schedule more than once. Deducting the whole 12,000,000 in year one on this project raises the NPV to about 1,499,155, a gain of some 358,000 over MACRS. A capital budgeting model must therefore be re-run against the law in force at the time, and a model built two years ago probably has the wrong schedule in it.

Working capital, which people call a wash and is not

The 900,000 goes out at date zero and comes back at date five, so the undiscounted total is zero. The discounted total is not: 900,000 × (0.593451 - 1) = -365,894. Financing five years of inventory and receivables costs the firm 365,894 dollars in present value, which is a third of the project's entire NPV.

Two refinements matter in practice. First, working capital scales with revenue, so a growing project keeps investing in it every year and only recovers the lot at the end; that pattern is what makes fast-growing, profitable companies run out of cash. Second, the recovery at the end is an assumption, not a fact. Inventory is often sold at a discount and some receivables are never collected.

Assets with different lives

Comparing a three-year machine with a five-year machine on total present value is meaningless, because the shorter one leaves you needing another machine. Convert each to an equivalent annual cost: the level annual payment with the same present value as the machine's whole cost stream. At 10 percent, machine A costs 90,000 plus 12,000 a year for three years, a present value of 119,842, and dividing by the three-year annuity factor of 2.486852 gives 48,190 a year. Machine B costs 130,000 plus 9,000 a year for five years, a present value of 164,117, and dividing by 3.790787 gives 43,294 a year. B costs more in total and less per year, so B wins, provided the need is genuinely perpetual and prices are not expected to move.

Common misconceptions

• "Depreciation is a cash outflow, so subtract it." No cash leaves the firm when an asset is depreciated. What is real is the tax the deduction saves, which is why the operating cash flow formula adds back depreciation and then separately adds the shield. A model that subtracts depreciation and forgets to add it back understates this project by more than 1.8 million.
• "We have already spent 600,000, so the project has to earn it back." The 600,000 is unrecoverable in both futures. Including it can only cause you to reject projects that would have made money, which is a second loss on top of the first.
• "Working capital is not a cost because we get it back." You get the nominal amount back later, and later is worth less. Here that timing costs 365,894 dollars.
• "Interest on the project loan is a project cost." It is a financing cost, already inside the discount rate. Counting it in the numerator double counts it and biases the firm against debt-funded projects for no economic reason.

Putting it together

Build the schedule as differences: with the project minus without it. Exclude sunk costs, include opportunity costs and side effects, keep financing out of the numerator, and treat depreciation only through its tax shield. On the bottling line that produced operating cash flows of 3,387,500 to 3,689,900, an after-tax salvage of 1,725,152, a working capital drag of 365,894 in present value, and an NPV of +1,140,844. Charging the sunk study would have cut that by 600,000; omitting cannibalisation and rent would have inflated it by 2,189,821. Where lives differ, compare equivalent annual costs: 48,190 against 43,294.

Every one of those numbers rested on an 11 percent discount rate that arrived by assertion. The next module earns it.

Sources

1. Internal Revenue Service. (2025). Publication 946: How to depreciate property. U.S. Department of the Treasury. irs.gov
2. Internal Revenue Service. (n.d.). New rules and limitations for depreciation and expensing under the Tax Cuts and Jobs Act. irs.gov
3. Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.), Chapter 6. McGraw-Hill Education.
4. Berk, J., & DeMarzo, P. (2019). Corporate Finance (5th ed.), Chapter 8. Pearson.
5. Koller, T., Goedhart, M., & Wessels, D. (2020). Valuation: Measuring and Managing the Value of Companies (7th ed.), Chapters 10-11. John Wiley & Sons.

In March 1952 the Journal of Finance published a fourteen-page article by a Chicago graduate student named Harry Markowitz. Investment practice at the time ran on a simple rule: work out which securities will return the most, and buy those. Markowitz pointed out that the rule cannot be right, because it implies holding a single stock, and nobody sane holds a single stock. The reason they do not is that risk does not add up the way returns do, and the rest of this lesson is the arithmetic of that sentence.

Why risk does not add up

Expected return is linear in weights. Put half your money in each of two assets and your expected return is the average of theirs, always. Variance is not linear, because it carries a cross term:

σp2 = w12σ12 + w22σ22 + 2w1w2ρσ1σ2

Take asset A with an expected return of 12 percent and a standard deviation of 20 percent, and asset B at 8 percent and 15 percent. A fifty-fifty mix returns 10 percent expected, whatever the correlation. Its risk depends entirely on the correlation.

At perfect correlation the risk is the weighted average, 17.5 percent, and diversification does nothing. At the realistic correlation of 0.3 it is 14.19 percent: the same expected return with 3.31 percentage points less risk, obtained by rearranging money rather than by forecasting anything. At minus one the two assets offset almost exactly and the portfolio is nearly riskless. The core of it: diversification is not the free lunch of folklore, it is a discount for buying two things whose bad months do not coincide.

The minimum-variance mix here is not fifty-fifty. Setting the derivative of variance with respect to wA to zero gives wA = (σB2 - ρσAσB)/(σA2 + σB2 - 2ρσAσB), which at ρ = 0.3 works out to 0.0135/0.0445 = 30.34 percent in A. That portfolio has a standard deviation of 13.57 percent and an expected return of 9.21 percent. Plot every possible mix and you trace a curve; the upper half of it, where no other mix offers more return at the same risk, is the efficient frontier.

How far diversification goes, and exactly where it stops

With N equally weighted assets the variance decomposes cleanly:

σp2 = (1/N) × average variance + (1 - 1/N) × average covariance

The first term vanishes as N grows. The second does not. Suppose the average stock has a variance of 0.09, so a 30 percent standard deviation, and the average pair has a covariance of 0.02.

Thirty stocks capture almost the whole benefit: going from thirty to infinity buys 0.80 of a percentage point. And the floor is 14.14 percent, not zero. That floor is the average covariance, the part of every stock's risk that comes from the economy rather than from the company, and no amount of diversification touches it.

The consequence for corporate finance is the one that matters. If the marginal investor holds a diversified portfolio, they will not pay for bearing risk they could have removed for free. Only the undiversifiable part earns a return. A company's own volatility, taken alone, is therefore the wrong input to a discount rate, which is the mistake that ends a lot of first attempts at a WACC.

One portfolio for everybody

James Tobin added a riskless asset in 1958 and the picture simplified sharply. Combine any risky portfolio with borrowing or lending at Rf and you get a straight line in return-versus-risk space. The best line is the one tangent to the frontier, and every investor, whatever their appetite for risk, should hold that same tangency portfolio and vary only how much of it they hold.

Suppose the tangency portfolio returns 9 percent with a standard deviation of 13 percent and the riskless rate is 4 percent. Its Sharpe ratio is (9 - 4)/13 = 0.3846. Put 60 percent in it and 40 percent in bills: the mix returns 0.6(9) + 0.4(4) = 7.0 percent with a standard deviation of 0.6(13) = 7.8 percent, and a Sharpe ratio of 3.0/7.8 = 0.3846, identically. Every point on that line is equally efficient, which is why the ratio, not the return, is the thing to maximise.

If everyone holds the same risky portfolio, and all shares must be held by someone, that portfolio is the market itself, in market-value proportions. That step is the bridge from Markowitz's mathematics to a testable theory of prices.

From the frontier to a price for risk

William Sharpe in 1964, and John Lintner in 1965, asked what an individual security must return in that equilibrium. The argument is short. A portfolio's variance can be written as σp2 = sumi wi Cov(Ri, Rp), so the marginal contribution of asset i to the risk of the market portfolio is its covariance with the market, not its own variance. Scale that covariance by the market's variance and you have beta:

βi = Cov(Ri, Rm) / Var(Rm)

Since every asset must offer the same reward per unit of marginal risk, or investors would rebalance until it did, the capital asset pricing model follows:

E(Ri) = Rf + βi [E(Rm) - Rf]

Read it as a sentence: the return you should require is the price of time plus the quantity of market risk you are taking times the price of market risk. Nothing in it refers to the company's total volatility.

Estimating a beta, and reading the error bars

Beta is a regression slope, so estimate it the way Lesson 2 of any econometrics course does. Twelve monthly returns for a stock and its market index, in percent:

The slope is the covariance over the variance, which on deviations is just the ratio of the two sums:

β = 190/128 = 1.4844, and α = 1.5 - 1.4844(1.0) = +0.016 percent a month.

Fit quality: R2 = 1902/(128 × 553) = 0.510, so market moves explain about half of this stock's month-to-month variation and company-specific news explains the other half. The residual sum of squares is 553 - 1.4844(190) = 270.97, so s2 = 27.10 and

SE(β) = sqrt(27.10/128) = 0.4601, giving t = 3.23 and a 95 percent interval of 1.4844 ± 2.228(0.4601) = [0.46, 2.51].

Stop and look at that interval. Twelve months of data cannot distinguish a defensive stock from one twice as volatile as the market. This is why practitioners use five years of monthly data or two years of weekly, why they estimate betas for industries rather than single firms, and why services report an adjusted beta, typically 0.67β + 0.33, which shrinks the estimate toward one because betas measured high tend to come back down. Here that gives 0.67(1.4844) + 0.33 = 1.324.

Turning the number into a discount rate

With a riskless rate of 4.3 percent and an equity risk premium of 4.6 percent, the raw estimate gives 4.3 + 1.4844(4.6) = 11.13 percent and the adjusted one gives 4.3 + 1.324(4.6) = 10.39 percent. Three quarters of a percentage point of cost of capital rides on a smoothing convention. Lesson 6 sources each of these inputs properly from a filing and a published series; Lesson 5 asks whether beta deserves to be the only term in the formula.

Common misconceptions

• "A risky company needs a high discount rate." Only if the risk is undiversifiable. A biotechnology firm whose value turns on one trial result may have enormous total volatility and a modest beta, because the trial outcome is uncorrelated with the economy. Its shareholders hold it alongside forty other things and are not compensated for the trial risk.
• "Diversification eliminates risk." It eliminates the part that is specific to companies. In the table above the standard deviation falls from 30 percent to a floor of 14.14 percent and stops. Correlations also rise in crises, which shrinks the benefit exactly when it is wanted.
• "A high R-squared means the beta is reliable." R-squared measures how much of the stock's variation the market explains. Reliability of the slope is the standard error, and here an R-squared of 0.51 sits beside an interval running from 0.46 to 2.51.
• "Beta is a property of the company." It is a property of a regression: of the index chosen, the return interval, the sample period and any adjustment applied. Two data providers will hand you different betas for the same firm on the same day, and neither is lying.

What you now know

Portfolio variance carries a covariance term, so a fifty-fifty mix of a 20 percent and a 15 percent asset has a standard deviation of 14.19 percent at a correlation of 0.3 rather than the 17.5 percent average. Across many assets the own-variance term dies at rate one over N and the covariance term does not, leaving a floor of 14.14 percent in the worked case. Adding a riskless asset makes one tangency portfolio optimal for everyone, and in equilibrium that portfolio is the market. An asset's contribution to market risk is its covariance, which normalised is beta, which gives the CAPM. Twelve months of data produced a beta of 1.4844 with a standard error of 0.4601, and a required return of 11.13 percent that falls to 10.39 percent under the usual adjustment.

In short: the model gives you one number and the data give you an interval, and any honest cost of capital carries both. The next lesson deals with the awkward part: for forty years the evidence has said that beta alone does not explain average returns.

Sources

1. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91.
2. Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442.
3. Damodaran, A. (n.d.). Betas by sector. NYU Stern School of Business. pages.stern.nyu.edu
4. French, K. R. (n.d.). Data library. Tuck School of Business, Dartmouth College. mba.tuck.dartmouth.edu
5. Berk, J., & DeMarzo, P. (2019). Corporate Finance (5th ed.), Chapters 10-12. Pearson.

In June 1992 the Journal of Finance carried a paper by Eugene Fama and Kenneth French that sorted every non-financial US stock from 1963 to 1990 into portfolios and asked how much of the difference in their average returns beta explained. The answer was, near enough, none. Two variables that the capital asset pricing model says should be irrelevant did the explaining instead: how small the company was, and how high its book value stood relative to its market value. Thirty years later the model that the paper embarrassed is still the one most corporate finance departments use to set a discount rate. This lesson is about why, and about whether that is defensible.

The position that beta is not enough

Sort stocks into ten groups by beta and their average returns come out almost level. Sort them by market capitalisation and the small groups beat the large ones. Sort them by book-to-market and the high groups beat the low ones. Fama and French's 1993 follow-up turned those two sorts into portfolios that could be used as explanatory variables: SMB, the return on small firms minus big ones, and HML, the return on high book-to-market firms minus low ones. The three-factor model writes an asset's excess return as

Ri - Rf = a + b(Rm - Rf) + s(SMB) + h(HML) + e

and it explains the cross-section of average returns far better than beta alone. Mark Carhart added a momentum factor in 1997. Fama and French themselves added profitability and investment factors in 2015, at which point HML became partly redundant. The literature since has produced hundreds of candidate factors, a fact that is itself part of the argument.

The evidence is not a single study. It is that the size and value sorts produce return differences in most decades, in most developed markets, and on data that predates their discovery.

The position that beta is fine and the factors are noise

Fischer Black, who had helped build the theory, argued in 1993 that the value effect looked like the product of a long and enthusiastic search through a finite dataset. Run enough sorts and some will separate returns by chance. His specific charge, which is testable, is that a real risk factor should keep paying after it is published, and a data-mined one should not.

Richard Roll's 1977 critique cuts deeper and has never been answered. The CAPM's market portfolio is every risky asset there is: equities, bonds, property, private businesses, human capital. Nobody can observe it. Every test of the CAPM therefore uses a stock index as a stand-in, and Roll's critique is that a rejection may say only that the index is not the market. If the model has never been tested, it has never been rejected.

There is also the matter of what the factors mean. Fama and French read them as risk: small, distressed, high book-to-market firms are more exposed to something investors dislike, and are paid for it. Josef Lakonishok, Andrei Shleifer and Robert Vishny read the same numbers in 1994 as mispricing: investors extrapolate poor past growth too far, cheap stocks get too cheap, and the premium is the correction. Both readings fit the returns. They imply opposite advice about whether a company should use a higher discount rate because its shares trade cheaply.

What the disagreement costs, in dollars

Take a small firm with a market beta of 1.10, an SMB loading of 0.60 and an HML loading of 0.45. Use a riskless rate of 4.3 percent, a market premium of 4.6 percent, a size premium of 1.8 percent and a value premium of 2.6 percent.

Two and a quarter percentage points looks academic until you put it in a denominator. Value a perpetual cash flow growing at 2 percent: the CAPM gives a multiple of 1/(0.0936 - 0.02) = 13.59 times and the three-factor model gives 1/(0.1161 - 0.02) = 10.41 times. The same cash flows are worth 23 percent less under the second model. So what?: this is not a debate about elegance. It is a debate that moves the price of a business by a quarter.

What the profession actually does

Graham and Harvey's survey found roughly three quarters of chief financial officers always or almost always using the CAPM to estimate the cost of equity, and only a small minority using a multifactor model. Valuation textbooks used inside banks and consultancies, including Koller, Goedhart and Wessels, recommend the CAPM as the base case while acknowledging the empirical record against it.

The defence is practical rather than scientific, and it is worth stating in its strongest form. A corporate discount rate must be defensible to a board, stable enough that projects approved in March are not rejected in June on the same facts, and computable for a division or a private target that has no traded shares at all. The CAPM needs one estimated parameter. The three-factor model needs three loadings plus three premia, each of which is itself contested, and for a non-traded division you must guess all six. In exchange for that complexity you get a number that is different, not demonstrably better, for the specific job of discounting a corporate project.

The counter is equally strong. Using a model because it is convenient, when its central prediction has failed repeatedly, is exactly the reasoning finance departments criticise everywhere else. Many firms handle the tension by adding a judgemental size premium to a CAPM number for small acquisitions, which is a three-factor model with the arithmetic hidden.

What would settle it

Both camps agree on the test, which is unusual and useful. If the premia are compensation for risk they should persist after publication; if they were discovered by searching, they should shrink once everyone knows about them. David McLean and Jeffrey Pontiff assembled 97 published predictors and found returns roughly 26 percent lower out of sample and about 58 percent lower after publication. That is a large decay and a partial one: the premia weaken rather than vanish, which is what you would expect if some are real risk and some were mining.

The value premium's own recent history reads the same way. Value beat growth for most of the twentieth-century US record and then trailed it badly for roughly the decade after 2007, long enough that the effect was widely declared dead before it returned. A premium that can disappear for thirteen years is not a reliable input to a discount rate on a five-year project, whichever theory of its origin you hold. You can check any of this yourself: the monthly factor series are free, with no account, from Kenneth French's data library.

Common misconceptions

• "Fama and French disproved the CAPM." They showed that beta does not explain the cross-section of average realised returns in their sample. Roll's critique says no such test is decisive, because the market proxy is not the market. What is safe to say is narrower: beta alone is a poor predictor of average returns, and that is enough to matter.
• "A firm should use a higher discount rate because its shares look cheap." That is what a positive HML loading implies, and it means a company whose stock falls must raise its hurdle rate, rejecting projects it accepted last quarter on identical cash flows. Whether that is right depends entirely on whether the value premium is risk or mispricing, which is the unsettled question.
• "More factors means a better model." Each added factor needs a premium estimated from the same finite history, and the estimation error compounds. A five-factor cost of equity has five loadings and five premia, and its confidence interval is wider than the CAPM's, not narrower.
• "Historical average returns are the expected return." The equity premium estimated from realised US returns is high partly because the United States turned out to be the twentieth century's most successful market. Damodaran's implied premium, backed out of current prices and forecast cash flows, is a different and usually lower number.

Summing up

Fama and French found the relation between beta and average return essentially flat over 1963 to 1990, while size and book-to-market sorted returns sharply. Their factor models fit the cross-section better. Black called the value result data mining, Roll pointed out that no test of the CAPM has ever used the real market portfolio, and Lakonishok and colleagues read the same premia as mispricing rather than risk. The choice matters: on one small firm the two models gave 9.36 and 11.61 percent, and a 23 percent difference in what a perpetuity is worth. Practitioners overwhelmingly keep the CAPM, for reasons of estimability and defensibility rather than fit, and then quietly add size premia by hand.

Worth holding on to: when the model you use is known to be wrong in a specific, measured way, the professional obligation is to say so in the memo and to show what the alternative would have given. The next lesson builds a full cost of capital, and every judgement call in it now has a name.

Sources

1. Fama, E. F., & French, K. R. (1992). The cross-section of expected stock returns. The Journal of Finance, 47(2), 427-465.
2. Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56.
3. Roll, R. (1977). A critique of the asset pricing theory's tests. Journal of Financial Economics, 4(2), 129-176.
4. McLean, R. D., & Pontiff, J. (2016). Does academic research destroy stock return predictability? The Journal of Finance, 71(1), 5-32.
5. French, K. R. (n.d.). Fama-French factors. Tuck School of Business, Dartmouth College. mba.tuck.dartmouth.edu
6. Nobel Prize Outreach. (2013). Eugene F. Fama: Facts. The Sveriges Riksbank Prize in Economic Sciences. nobelprize.org

Apple's Form 10-K for fiscal 2023, filed on 3 November 2023, contains every number you need for a cost of capital except two, and both of those are free from public rate series. This lesson builds the number, states the source of each input, and then shows which arguments about it are worth having.

The weighted average cost of capital is the return the firm's providers of capital collectively require, weighted by how much each provides:

WACC = (E/V) × ke + (D/V) × kd × (1 - T)

Step one: the weights, and why book values ruin them

The 10-K cover page reports the aggregate market value of common stock held by non-affiliates as of 31 March 2023: 2,591,165 million dollars. That is the market value of equity, straight from the filing.

Debt takes three lines from the balance sheet and its notes: term debt of 95,281 million classified as non-current, 9,822 million as current, and commercial paper of 5,985 million. Operating lease liabilities of 10,408 million non-current and 1,410 million current are debt in economic substance, since they are fixed obligations that survive a downturn, and since 2019 they sit on the balance sheet rather than in a footnote.

So E/V = 95.47 percent and D/V = 4.53 percent. One of the largest corporate borrowers on earth is, in weight terms, an all-equity company.

Now do it wrong, on book values, which is the single most common error in a first WACC. Apple's book shareholders equity at 30 September 2023 was 62,146 million, giving weights of 33.6 percent equity and 66.4 percent debt. Carry those through and the WACC comes out at 6.39 percent instead of 9.86. Book equity records what shareholders historically paid in less what has been bought back; it is not what they would sell for, and after 77.5 billion dollars of buybacks in one year it is not even close.

Step two: the cost of equity

Three inputs, two of them contested.

Riskless rate. On 29 September 2023, the last trading day of Apple's fiscal year, the ten-year Treasury constant maturity yield was 4.59 percent, from the Federal Reserve's daily series. Use a maturity close to the horizon of the cash flows being discounted; a three-month bill is riskless over three months and nothing else.

Equity risk premium. Take 4.6 percent, which is where implied estimates backed out of index levels and forecast cash flows sat around that date. This is a judgement, not a measurement, and the current figure is on Damodaran's page linked below.

Beta. Take 1.20, in the range published for the computer hardware sector. Lesson 4 showed what the error bars on such a number look like.

ke = 4.59 + 1.20 × 4.6 = 10.11 percent

Step three: the cost of debt, marginal not historical

Two candidates, and the difference between them is instructive. Apple's income statement shows interest expense of 3,933 million for the year against average total debt of about 115,579 million, an effective rate of 3.40 percent. That is the average coupon on bonds issued when rates were far lower, and it tells you what the past cost, not what the next dollar costs.

The marginal rate is what a new Apple bond would have to yield. On 29 September 2023 the ICE BofA index of US AAA corporate effective yields stood at 5.28 percent, and Apple's ratings put it at the top of the scale. Use 5.28.

Tax next. Interest is deductible, so the firm's cost is the after-tax figure. Apple's effective tax rate for fiscal 2023 was 14.7 percent, well below the 21 percent statutory rate because much of its income is earned abroad. Use the effective rate, because it is the rate at which the deduction is actually worth something:

kd(1 - T) = 5.28 × (1 - 0.147) = 4.50 percent

Step four: the answer, and the arguments worth having

WACC = 0.9547 × 10.11 + 0.0453 × 4.50 = 9.65 + 0.20 = 9.86 percent

Now rank the disputes by what they are worth. Swap the marginal cost of debt for the historical 3.40 percent and the WACC falls to 9.78 percent: eight basis points. Drop the operating leases from debt entirely and it moves by less than one basis point. Change beta, though, and the ground shifts.

Beta over a plausible range moves the WACC by 2.19 percentage points. Moving the equity premium from 4.0 to 5.5 percent moves it by 1.72. What matters here: in a firm financed almost entirely by equity, every hour spent negotiating the cost of debt is an hour wasted, and the whole answer lives in two numbers that cannot be measured, only argued for. A WACC memo that does not carry a range is not finished.

The cash question, and the divisional question

Apple held 29,965 million of cash, 31,590 of current marketable securities and 100,544 of non-current ones: 162,099 million in total, more than its entire debt. Net debt is therefore minus 39,193 million, and using net debt in the weights produces a negative weight, which is nonsense in this formula. The convention that works, and the one Koller and colleagues recommend, is to use gross debt in the WACC and to treat excess cash as a non-operating asset added to the value of operations at the end of the valuation. Lesson 11 does exactly that.

A second limit: 9.86 percent is the right rate for a project with the risk of Apple's existing business, and for nothing else. To discount a project in a different industry, find listed pure plays in that industry, unlever their betas to strip out their capital structure using Hamada's relation βU = βL/[1 + (1 - T)(D/E)], average them, and relever at the financing mix the project will carry. For Apple itself, D/E = 0.0474, so βU = 1.20/1.0405 = 1.153: with this little leverage, unlevering barely moves the number, which will not be true of the utilities in Lesson 8.

Common misconceptions

• "Debt is cheaper than equity, so more debt lowers the WACC." Apple's after-tax debt costs 4.50 percent and its equity 10.11, so raising the debt weight looks like free money. It is not, because leverage raises the risk borne by the remaining equity and therefore raises the cost of equity, and past a point raises the cost of debt too. Lesson 7 proves that in the frictionless case the two effects cancel exactly.
• "Use book values, because market values move around." Market values are what capital providers could actually get, which is the definition of what they are giving up. Book weights gave 6.39 percent here against 9.86, a gap large enough to approve a decade of value-destroying projects.
• "Use the interest rate the company is paying." Historical coupons price yesterday's borrowing. The decision at hand is funded at today's rates, which for Apple in 2023 were 188 basis points higher than its average book cost.
• "One WACC per company." A single corporate rate charges a low-risk division too much and a high-risk one too little, which reliably starves the safe business and overfeeds the speculative one. Firms with genuinely different divisions need pure-play betas for each.

The short version

From one filing and two public series: equity of 2,591,165 million and debt of 122,906 million including leases, giving weights of 95.47 and 4.53 percent; a cost of equity of 4.59 plus 1.20 times 4.6, or 10.11 percent; a marginal pre-tax cost of debt of 5.28 percent, which at a 14.7 percent effective tax rate is 4.50 after tax; and a WACC of 9.86 percent. Book weights would have given 6.39. The historical cost of debt would have changed the answer by eight basis points and beta by more than two percentage points across its plausible range.

Remember: a cost of capital is an argument with numbers attached, and the argument that matters is almost always about the cost of equity. Module 3 now asks the question this lesson dodged: does changing the mix of debt and equity change the WACC at all?

Sources

1. Apple Inc. (2023). Annual report on Form 10-K for the fiscal year ended September 30, 2023. U.S. Securities and Exchange Commission. sec.gov
2. Federal Reserve Bank of St. Louis. (n.d.). ICE BofA US corporate AAA effective yield (BAMLC0A1CAAAEY). FRED. fred.stlouisfed.org
3. Damodaran, A. (n.d.). Implied equity risk premiums. NYU Stern School of Business. pages.stern.nyu.edu
4. Koller, T., Goedhart, M., & Wessels, D. (2020). Valuation: Measuring and Managing the Value of Companies (7th ed.), Chapter 15. John Wiley & Sons.
5. Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.), Chapter 9. McGraw-Hill Education.

Two firms own the same machine. It throws off 10 million dollars a year forever, before interest and before tax, and the two firms are identical in every operating respect. Firm U is financed entirely by equity and the market values it at 100 million. Firm L has borrowed 40 million at 5 percent, and its shares trade at a total of 65 million, so the market says Firm L is worth 105 million.

One of those prices is wrong, and you can prove it without knowing anything about machines.

Fifty thousand dollars, risk free, this afternoon

Take one percent of everything. Do four things at once.

• Sell short 1 percent of Firm L's equity. You receive 0.01 × 65,000,000 = 650,000 dollars and take on the obligation to pay whatever those shares earn.
• Borrow 400,000 dollars personally at 5 percent, matching 1 percent of L's debt.
• You now hold 1,050,000 dollars in cash.
• Buy 1 percent of Firm U for 0.01 × 100,000,000 = 1,000,000 dollars, and keep the remaining 50,000.

Check the annual flows. L's equity earns 10 - 2 = 8 million, so your short position costs you 0.01 × 8,000,000 = 80,000. Your personal loan costs 0.05 × 400,000 = 20,000. Total outgoings, 100,000. Your 1 percent of U pays 0.01 × 10,000,000 = 100,000. Net annual flow: zero, in every state of the world, because the machine's output is the machine's output.

You have taken 50,000 dollars out of the market and you owe nothing. So will everyone else, until L's equity falls to 60 million and the two firms are worth the same. That is Modigliani and Miller, Proposition I, and the proof is not an assumption about markets being efficient. It is the observation that an investor can manufacture leverage in a brokerage account, so nobody will pay a company to do it for them.

VL = VU: the value of a firm is set by its assets and by nothing else about how the claims on them are sliced.

Key idea: the pie does not get bigger because you cut it differently. Every real theory of capital structure that follows is an argument about which knife changes the pie.

Where the missing return went

An objection arrives immediately. Debt at 5 percent is cheaper than equity at 10 percent, so surely swapping one for the other must reduce the average cost. Proposition II answers it:

ke = k0 + (k0 - kd)(D/E)

Firm U's assets earn 10 percent, so k0 = 10. Put 40 million of debt against 60 million of equity and the equity now bears the same business risk on a smaller base, plus a fixed 2 million interest bill ahead of it: ke = 10 + (10 - 5)(40/60) = 13.33 percent. Confirm it directly: the equity receives 8 million on a 60 million claim, which is 13.33 percent.

The last column never moves. Cheap debt is exactly offset by the equity it makes expensive, and the weighted average sits where the assets put it. Notice also that the cost of equity is not a preference or a demand: it is an arithmetic consequence of a fixed claim standing in front of a residual one.

The illusion that will not die

Suppose Firm U has 10 million shares at 10 dollars and earns 10 million, so earnings per share are 1.00. It now borrows 40 million and buys back 4 million shares at 10 dollars.

Earnings per share rose by a third and not one shareholder is better off, because the multiple fell by exactly enough to hold the price still. A recapitalisation that raises EPS proves only that debt interest is smaller than the earnings on the shares retired, which is true whenever the borrowing rate is below the earnings yield, which is usually. Any chief executive whose bonus keys off EPS can produce that number on demand, and Lesson 15 deals with what follows from that.

What the tax code changes

Interest is deductible; dividends are not. That single asymmetry breaks the theorem, and Modigliani and Miller published the correction in 1963. Rebuild the example with a 21 percent corporate rate. Unlevered, the firm keeps 10 × 0.79 = 7.9 million a year, so VU = 7.9/0.10 = 79 million.

Borrow 40 million at 5 percent. Interest of 2 million a year shelters 2 million of income, saving 0.21 × 2 = 0.42 million of tax every year. That saving is as safe as the interest payment that generates it, so discount it at the debt rate: 0.42/0.05 = 8.4 million, which is simply Tc × D.

VL = VU + TcD = 79 + 8.4 = 87.4 million

With taxes the cost of equity now follows ke = k0 + (k0 - kd)(1 - T)(D/E), and the WACC falls monotonically. Read the table honestly and it says the firm should borrow every dollar it can, financing itself entirely with debt. No firm does this. Something is missing, and Lesson 8 is about what.

Personal taxes take most of the prize back

Merton Miller returned to the problem in his 1977 presidential address. The corporate deduction is only the first stop. Interest reaches the lender as ordinary income; equity returns reach the shareholder partly as deferred capital gains, taxed later and at a lower rate. The net gain from borrowing a dollar is

1 - (1 - Tc)(1 - Tpe)/(1 - Tpd)

where Tpe is the personal rate on equity income and Tpd the rate on interest. At Tc = 21 percent, Tpe = 15 percent and Tpd = 37 percent, that is 1 - (0.79)(0.85)/(0.63) = 1 - 1.0659 = -0.066. Negative: for an investor in that bracket, corporate borrowing destroys 6.6 cents per dollar of debt rather than creating 21. Change Tpe to 23.8 percent and the figure comes back to roughly minus 1.7 percent, close to a wash.

The upshot: the tax advantage of debt is real but far smaller and far more clientele-dependent than the headline TcD. A firm whose shareholders are pension funds paying no tax at all faces a completely different calculation from one held by high-bracket individuals.

Common misconceptions

• "MM proved capital structure does not matter." They proved it does not matter under stated conditions: no taxes, no bankruptcy costs, no agency conflicts, symmetric information, investors who can borrow on the firm's terms, and an investment policy that does not respond to financing. Every one of those is false somewhere, and the theorem's real function is to tell you exactly which falseness you are relying on.
• "A recapitalisation that raises EPS creates value." The table above raises EPS by 33 percent and leaves the share price at 10.00. Earnings per share ignores the risk borne by the remaining shares, which is the whole of what changed.
• "The tax shield should be discounted at the cost of equity." Its riskiness is the riskiness of the interest payments that create it, so the debt rate is the natural choice, which is what produces the clean result T times D. If the firm might not have enough taxable income to use the deduction, the shield is riskier and worth less, which is a real issue for loss-making firms.
• "Homemade leverage is unrealistic, so the theorem is irrelevant." Individuals do pay more to borrow than investment-grade companies. But the arbitrage only needs enough investors who can trade cheaply, and institutions can. More to the point, the theorem is a benchmark: any claim that a financing change creates value must name which assumption it breaks.

Pulling it together

Two identically productive firms priced at 100 and 105 million offered a riskless 50,000 dollars per one percent stake, and that trade forces their values together. Proposition I says the mix does not change firm value; Proposition II says the cost of equity rises exactly fast enough, from 10 percent unlevered to 30 percent at 80 percent debt, to keep the WACC flat at 10. Earnings per share rose by a third in a recapitalisation that moved the share price not at all. Corporate taxes then add a shield worth TcD, taking the firm from 79 to 87.4 million on 40 million of debt and pushing the WACC down to 9.04 percent, with an implied optimum of all debt. Personal taxes claw most of that back and, on plausible rates, can reverse it.

Two things remain unexplained: why the corner solution of 100 percent debt never happens, and why two firms in different industries, both aware of all of this, choose leverage ratios that differ by a factor of ten. That is the next lesson.

Sources

1. Modigliani, F., & Miller, M. H. (1958). The cost of capital, corporation finance and the theory of investment. The American Economic Review, 48(3), 261-297.
2. Modigliani, F., & Miller, M. H. (1963). Corporate income taxes and the cost of capital: A correction. The American Economic Review, 53(3), 433-443.
3. Miller, M. H. (1977). Debt and taxes. The Journal of Finance, 32(2), 261-275.
4. Wikipedia contributors. (n.d.). Modigliani-Miller theorem. Wikipedia. en.wikipedia.org
5. Wikipedia contributors. (n.d.). Capital structure. Wikipedia. en.wikipedia.org

At 31 December 2023 Duke Energy carried 72,452 million dollars of long-term debt, 2,800 million of current maturities and 4,288 million of short-term borrowings, against an equity market value of 69,081 million reported on the cover of its 10-K. Debt was 53.5 percent of its capital. Four weeks earlier Adobe closed a fiscal year with 3,634 million of debt against 165,530 million of equity: 2.1 percent.

Both finance departments know the arithmetic of Lesson 7. Both face the same 21 percent federal rate and the same bond market. One has borrowed twenty-five times as heavily as the other, relative to size, and neither is behaving irrationally. Two theories explain that gap, they contradict each other on the central question, and both have evidence.

The case for a target: taxes against distress

Trade-off theory says a firm borrows until the marginal tax shield equals the marginal expected cost of financial distress. That gives an interior optimum, a target ratio, and a reason for Duke and Adobe to differ.

Distress costs come in three sizes. Direct legal and professional fees are small: Warner found them around 1 percent of market value for railroads years ahead of filing, and Weiss around 3 percent of assets at filing. Indirect costs are large. Andrade and Kaplan studied highly leveraged transactions that later got into trouble and put the total cost of distress at 10 to 20 percent of firm value: customers who will not sign multi-year contracts with a company that may not exist, suppliers who demand cash, engineers who leave, and assets sold at whatever a forced sale brings.

Then there are the agency costs that appear as soon as default is plausible, and these are worth working rather than describing.

Debt overhang. A firm's assets will be worth 120 next year with probability one half and 60 otherwise. It owes 100. Equity is therefore worth 0.5(20) + 0.5(0) = 10 and the debt 0.5(100) + 0.5(60) = 80. Now offer the firm a project costing 20 today that adds 25 for certain next year: a positive-value project. Fund it with equity and the outcomes become 145 or 85. Equity is now worth 0.5(45) + 0.5(0) = 22.5, a gain of 12.5 against an outlay of 20. Shareholders lose 7.5 and refuse. The debt, meanwhile, goes from 80 to 0.5(100) + 0.5(85) = 92.5. The project's whole 5 of value, plus 7.5 taken from the shareholders, went to the bondholders. This is debt overhang, and it makes distressed firms stop investing exactly when investment would help.

Asset substitution. Same firm, same 100 of debt. It can choose a safe project returning 100 for certain, or a risky one returning 160 or 40 with equal probability. Both have an expected value of 100. Under the safe project the equity is worth nothing and the debt 100. Under the risky one the equity is worth 0.5(60) = 30 and the debt 0.5(100) + 0.5(40) = 70. Shareholders strictly prefer the gamble, and they would still prefer it if its expected value were only 90, at which point the firm is destroying 10 of value to move 30 from lenders to owners. Lenders know this, price it into the coupon in advance, and write covenants against it, which is itself a cost.

Now the theory earns its keep on the two companies. Duke's assets are transmission lines and generating plant: 65 percent of the balance sheet, immovable, and saleable to another utility. Its revenues are set by state regulators, which makes the cash flow that services the debt about as predictable as corporate cash flow gets. Low distress probability plus excellent collateral equals a high target. Adobe's assets are goodwill and software written by people who can leave; net property is 7 percent of the balance sheet, and a lender in liquidation would find very little to sell. High distress cost plus no collateral equals a low target.

The case for no target at all

Stewart Myers and Nicholas Majluf started from a different premise in 1984: managers know more about the firm's prospects than investors do. Suppose a manager believes the shares are worth 60 and the market prices them at 50. Issuing equity at 50 hands value to new shareholders at the expense of existing ones, so a manager acting for existing owners refuses. A manager who believes the shares are worth 40 issues happily. Investors work this out, so an equity issue is read as a signal of overvaluation and the price falls when it is announced.

That prediction has been tested repeatedly and holds: announcements of seasoned equity offerings by industrial firms are associated with share price falls averaging around 3 percent, while straight debt issues move the price by roughly nothing. Debt is less sensitive to what the manager knows, because a bond pays the same in most states of the world.

So the pecking order follows: use retained earnings first, then debt, then equity as a last resort. Crucially, there is no target ratio in this theory at all. Leverage is the accumulated residue of past financing deficits. A firm that has been very profitable pays for everything internally and ends up with almost no debt, not because it wants a low ratio but because it never needed the money.

Adobe earned 5,428 million and holds 7,141 million of cash. It has not been rationing capital; it has simply never had to ask. Why this matters: the two theories make opposite predictions about the most profitable firms. Trade-off says high profits mean high taxable income, a valuable shield and therefore more debt. Pecking order says high profits mean internal funds and therefore less debt. In the data, profitability is one of the most reliable negative predictors of leverage, and that is the trade-off theory's worst problem.

Where each theory breaks

The trade-off theory also has a magnitude problem in the other direction. John Graham estimated the capitalised value of interest tax shields at roughly a tenth of firm value for the typical large firm and concluded that many could roughly double that benefit by borrowing more before the marginal advantage turned down. If a target exists, firms sit persistently below it, and calling that conservatism is a description rather than an explanation.

The pecking order has its own failures. Murray Frank and Vidhan Goyal found that small, high-growth firms, which face the worst information problems and should therefore avoid equity most, are the heaviest issuers of it. The theory also has nothing to say about why utilities cluster near 50 percent and software firms near zero, since industry is not a variable in it. And Malcolm Baker and Jeffrey Wurgler argue for a third story entirely: firms issue equity when the market values them highly and debt when it does not, and observed leverage is the accumulated result of those timing decisions, with no equilibrium of any kind.

What would settle it

There is a clean test. Push a firm away from its ratio with a shock, such as a large share price move that changes market leverage without any decision being taken, and watch. If a target exists, leverage should mean-revert. If leverage is a residue, it should stay where the shock left it.

The answer that keeps coming back is partial adjustment: firms close roughly a third of the gap to their estimated target each year, which is neither the fast reversion a pure trade-off model implies nor the random walk the pecking order implies. Read generously, that is a target plus real costs of moving toward it, with a pecking order governing the choice of instrument in any given year. Read sceptically, it is a model flexible enough to fit anything. Both readings are defensible, and no serious researcher now claims one theory does the whole job.

Common misconceptions

• "Debt is cheap, so more of it is better." Lesson 7 showed the frictionless version of why not. Here is the practical version: Duke's leverage is safe because regulated revenue and physical collateral make distress unlikely and cheap. The same 53 percent at Adobe would put a company whose assets walk out of the building every evening one bad product cycle from an overhang that stops it investing. The cost of debt in the WACC formula prices default risk, not the loss of flexibility, the covenants or the projects not taken.
• "Direct bankruptcy costs are small, so distress does not matter much." The legal fees are indeed a few percent. The 10 to 20 percent figure comes from lost customers, lost staff and forced asset sales, and much of it is incurred well before any filing, by firms that never go bankrupt at all.
• "A firm with no debt is being prudent." It may be. It may also be leaving a tax shield worth a tenth of its value unclaimed, and shielding managers from the discipline of a fixed payment, which is the subject of Lesson 15.
• "The industry average is the target." Industry averages summarise the asset characteristics that drive the target: tangibility, cash flow volatility, growth options. Copying the average without checking that your assets look like the average is copying an answer without the question.

The takeaway

Duke at 53.5 percent debt and Adobe at 2.1 percent are both defensible. Trade-off theory explains the gap through collateral and distress risk: 65 percent tangible assets and regulated revenue against 7 percent tangible assets and 43 percent goodwill. It works the agency costs to numbers, a positive-value project refused because shareholders lose 7.5 while lenders gain 12.5, and a value-destroying gamble preferred because it moves 30 across the same line. Pecking order explains the gap through information: equity issues drop the share price by around 3 percent, so firms use internal funds first, and the most profitable firms end up least levered, which trade-off theory predicts backwards. Adjustment speeds of around a third of the gap per year support a target that firms approach slowly.

In short: ask what a lender could seize, how volatile the cash flow is, and how much the firm already generates internally. Those three answers get you most of the way to a defensible ratio, and no formula gets you the rest. Next: what to do with the cash once it has been earned.

Sources

1. Duke Energy Corporation. (2024). Annual report on Form 10-K for the fiscal year ended December 31, 2023. U.S. Securities and Exchange Commission. sec.gov
2. Adobe Inc. (2024). Annual report on Form 10-K for the fiscal year ended December 1, 2023. U.S. Securities and Exchange Commission. sec.gov
3. Myers, S. C., & Majluf, N. S. (1984). Corporate financing and investment decisions when firms have information that investors do not have. Journal of Financial Economics, 13(2), 187-221.
4. Andrade, G., & Kaplan, S. N. (1998). How costly is financial (not economic) distress? The Journal of Finance, 53(5), 1443-1493.
5. Graham, J. R. (2000). How big are the tax benefits of debt? The Journal of Finance, 55(5), 1901-1941.

Apple's cash flow statement for fiscal 2023 has two payout lines. Repurchases of common stock: 77,550 million dollars. Dividends paid: 15,025 million. Five dollars went back through the market for every one that arrived as a cheque. The company had paid no dividend at all between 1995 and 2012, and its declared dividend per share has risen in careful single-digit steps ever since, from 0.85 in fiscal 2021 to 0.90 to 0.94, while the repurchase line swung from 85,971 to 89,402 and then down 13 percent to 77,550.

Those two patterns, a creeping dividend and a lurching buyback, are not accidents of one company. They are what the two instruments are for.

Why the choice should not matter at all

Merton Miller and Franco Modigliani published the payout version of their argument in 1961. Take a firm worth 100 million, of which 10 million is surplus cash, with 10 million shares outstanding at 10.00 each.

Ten dollars either way. And a shareholder who dislikes the choice can undo it: under the repurchase, someone wanting income sells 1.00 worth of stock; under the dividend, someone wanting no income reinvests it. Homemade dividends do to payout policy what homemade leverage did to capital structure. The point: payout policy cannot create value by itself, so every argument for one method over another has to name a friction.

The four frictions, side by side

Taxes, and the one percent that arrived in 2023

For a taxable investor the two are not equivalent. A dividend is taxed the year it arrives. A repurchase creates a taxable event only for the shareholder who chooses to sell, and even then only on the gain rather than the whole distribution. That deferral is worth real money over a decade, and it is a large part of why buybacks overtook dividends after the 1980s.

The advantage narrowed in 2023. The Inflation Reduction Act of 2022 imposed a 1 percent excise tax on net repurchases by US public companies, reported on IRS Form 7208. On Apple's gross 77,550 million that is roughly 776 million a year, about 0.8 percent of net income: enough to appear in a footnote, not enough to change the decision. A 4 percent rate, which has been proposed more than once, would be a different matter.

The tax argument also runs the other way for a large part of the market. Pension funds, endowments and retirement accounts pay no tax on dividends at all, so for them the deferral is worth nothing.

What a dividend says that a buyback does not

John Lintner interviewed managers in the 1950s and found they did not set dividends from a formula on current earnings. They moved slowly toward a target payout ratio, and they hated cutting. His partial-adjustment model captures it:

Dt = Dt-1 + c × (target payout × EPSt - Dt-1)

With a 40 percent target, an adjustment speed of 0.3 and a dividend of 0.40 last year, earnings per share rising from 1.00 to 1.50 gives 0.40 + 0.3(0.60 - 0.40) = 0.46. A 50 percent jump in earnings produces a 15 percent rise in the dividend. Apple's 5.9 percent and 4.4 percent increases, alongside a 13 percent fall in repurchases in a single year, is that model in a real filing.

The stickiness is what makes the signal informative. Because a cut is expensive in credibility, only a management confident of sustainable cash flow raises the dividend, and the market reads it accordingly. The converse is brutal. General Electric halved its quarterly dividend from 24 cents to 12 in November 2017 and cut it again to a single cent in October 2018, and each announcement was read, correctly, as confirmation that the underlying business was in far worse condition than reported earnings suggested.

A repurchase carries almost no such commitment. Boards authorise a programme and then execute it at whatever pace suits, which makes buybacks the flexible instrument and dividends the promise.

Clienteles, and why the average investor is a fiction

Different shareholders want different things, and firms attract the ones who want what they already do. A retired investor spending from a portfolio prefers a reliable dividend to selling shares each quarter. A high-bracket taxable investor prefers deferral. A tax-exempt endowment is indifferent. This is the clientele effect, and its practical implication is unglamorous but important: the gain from switching policy is usually small, because you mostly swap one set of holders for another, while the cost of surprising your existing holders is immediate.

The argument of the 2010s, with the numbers both sides used

William Lazonick argued in 2014 that S&P 500 firms had spent 54 percent of earnings on buybacks and a further 37 percent on dividends over 2003 to 2012, leaving 9 percent for everything else, and that this starved investment, wages and research. The figures were widely repeated and the policy proposals that followed, including the excise tax, trace back to them.

Jesse Fried and Charles Wang answered on the arithmetic. Gross repurchases overstate what actually leaves the corporate sector, because the same firms issue equity: to employees, in acquisitions, and to the public. Netting issuance against repurchase for S&P 500 firms over 2007 to 2016 cut net shareholder payouts to roughly 41 percent of net income, against a gross figure near 96. They also noted that the S&P 500 is not the economy: cash returned by mature firms is largely reinvested by its recipients in firms that need it, which is what a capital market is for.

Both sides are working from real data and neither number is wrong. The disagreement is about the denominator and about where the money goes next, and it is not settled. What can be said cleanly is that a buyback is not free money, does not create value by raising earnings per share, and is a good decision only when the firm has no project earning above its cost of capital and the shares are not overpriced.

Common misconceptions

• "Buybacks are always manipulation." A repurchase at a fair price is a distribution and nothing more, exactly equivalent to a dividend for a tax-exempt holder. Two real faults sit nearby and should not be confused with the instrument itself: buying above intrinsic value transfers wealth from continuing shareholders to selling ones, and timing repurchases to hit an earnings-per-share target attached to executive pay is a governance failure, which Lesson 15 takes up.
• "A buyback increases shareholder value by raising EPS." The share count falls and the cash falls with it. Lesson 7's recapitalisation raised EPS by a third and moved the price by nothing. If a rising EPS were value, any firm could manufacture it on demand, and many try.
• "Dividends are safer income than selling shares." A dividend reduces the share price by roughly its own amount on the ex-dividend date. Taking 4 percent a year as dividends and taking 4 percent by selling stock are the same withdrawal from the same pot, before tax.
• "A high dividend yield means a healthy company." Yield is dividend over price, and it rises when the price collapses. GE's yield looked excellent shortly before both cuts, for exactly that reason.

What to remember

In frictionless markets a 1.00 dividend and a 10.00 repurchase leave every shareholder with the same 10.00, and homemade dividends let holders override whichever the firm chooses. The frictions decide. Repurchases defer tax for taxable holders and carry no commitment; dividends are taxed on receipt and function as a promise, which is why Lintner's model produces a 15 percent dividend rise from a 50 percent earnings rise, and why Apple's dividend crept up 4.4 percent in a year when its buyback fell 13. The 1 percent excise tax costs Apple roughly 776 million a year and changes nothing. Lazonick's 54 and 37 percent of earnings and Fried and Wang's net 41 percent are both real, and they measure different things.

Bottom line: payout is a residual decision. Fund every project worth more than it costs, then return the rest, and choose the instrument by what you are willing to promise. Module 4 now turns to what the firm is worth in the first place, starting with what its own annual report will and will not tell you.

Sources

1. Apple Inc. (2023). Annual report on Form 10-K for the fiscal year ended September 30, 2023, consolidated statements of cash flows. sec.gov
2. Internal Revenue Service. (n.d.). About Form 7208, excise tax on repurchase of corporate stock. irs.gov
3. Miller, M. H., & Modigliani, F. (1961). Dividend policy, growth, and the valuation of shares. The Journal of Business, 34(4), 411-433.
4. Lintner, J. (1956). Distribution of incomes of corporations among dividends, retained earnings, and taxes. The American Economic Review, 46(2), 97-113.
5. Lazonick, W. (2014). Profits without prosperity. Harvard Business Review, 92(9), 46-55; and Fried, J. M., & Wang, C. C. Y. (2018). Are buybacks really shortchanging investment? Harvard Business Review, 96(2), 88-95.

Netflix's 10-K for 2019 reports revenue of 20,156 million dollars, operating income of 2,604 million and net income of 1,867 million. Turn a few pages to the consolidated statements of cash flows and the top line of the same document reads: net cash used in operating activities, minus 2,887 million.

Both numbers were audited. Both are correct. A reader who stops at the first one concludes that Netflix earned 1.9 billion dollars in 2019, and that conclusion is wrong in the way that matters. This lesson traces exactly where it fails.

The 4,754 million gap, line by line

For most companies operating cash flow exceeds net income, because depreciation is subtracted in the income statement and no cash leaves. Netflix runs 4,754 million the other way. Here is where it goes, using only figures from the filing:

One line does nearly all the work. Netflix capitalises what it spends on films and series, then charges the income statement with amortisation of that library over the years it expects the titles to be watched. In a fast-growing library, cash out runs far ahead of amortisation in, and the difference sits on the balance sheet as an asset instead of in the income statement as a cost. The library grew 4,402 million in one year.

Nothing about that is improper. It is what accrual accounting is for: matching a cost to the period that benefits from it. But it means the income statement measures the cost of content consumed while the business was paying for content acquired, and in 2019 those two numbers differed by more than twice the reported profit.

Where did the cash come from? Also from the filing: long-term debt rose from 10,360 million to 14,759 million, an increase of 4,399 million, against library growth of 4,402. Netflix borrowed almost exactly what it added to the shelf. So what?: a valuation built on 1,867 million of earnings is valuing a company that in cash terms consumed 2,887 million and financed the difference with debt. Those are not the same company.

Why the classification is the whole trick

A manufacturer buying a 4.4 billion dollar factory would show the outflow in investing activities and report healthy positive operating cash flow. Netflix puts content additions in operating activities, which is why its operating line is negative at all. Same economics, different heading, wildly different-looking statement.

The lesson generalises: whenever you compare cash flow across companies, check where each one puts its main recurring investment. The cash flow statement is harder to manage than the income statement, but it is not immune. Classification shifting, selling receivables just before a year end, and stretching payables all improve operating cash flow without improving the business, and all are legal.

Netflix's own solution is a non-GAAP measure, and its 10-K defines it explicitly: free cash flow as operating plus investing cash flows excluding movements in short-term investments. On that basis 2019 was minus 3,274 million. A company that volunteers a worse number than GAAP requires is usually telling you which number it thinks matters.

Nineteen and a half billion dollars that are not liabilities

Turn to the commitments note. At 31 December 2019 Netflix disclosed content obligations of 19,490 million dollars, of which 8,477 million fell due within twelve months. Compare that with total assets of 33,976 million and long-term debt of 14,759 million. The obligations that do not appear as debt on the balance sheet exceed the debt that does.

These are contracts already signed for programming not yet delivered. Under the accounting rules a large part of such commitments is not recognised as a liability until the content is delivered and accepted. They are nonetheless money the company must pay. Any enterprise value that counts the 14,759 million of debt and ignores the 19,490 million of obligations is understating what the equity holder stands behind.

Leases are the historical warning here. Until 2019, operating leases sat entirely in a footnote: an airline could lease its whole fleet and report almost no debt. ASC 842 ended that, and Netflix's first-year figures show the mechanism plainly: a right-of-use asset of 1,532 million and a lease liability of 1,613 million appeared on the balance sheet, being 2,039 million of contracted payments discounted at 5 percent. Those numbers existed before 2019 too. They were simply somewhere the screening software did not look.

A reading order that catches things

An experienced reader does not start at page one. The sequence below is designed so that each step tests the one before it.

1. Cash flow statement first. Three years side by side. Is operating cash flow positive and growing, and how does it compare with net income? A widening gap in either direction is the question the rest of the reading answers.
2. Reconcile the gap. Read the add-backs. Which are genuinely non-cash, such as depreciation and stock compensation, and which are timing that will reverse?
3. Commitments and contingencies note. Purchase obligations, leases, guarantees, litigation. This is where money owed but not yet a liability lives.
4. Segment note. A consolidated margin can hide one excellent business subsidising one failing one.
5. Non-GAAP reconciliation. Item 7 must reconcile every adjusted measure to GAAP. Read what is being excluded and ask whether it recurs. Restructuring charges taken every year for six years are an operating cost with a euphemism.
6. Critical audit matters. Since 2019 the auditor must name the accounts that required the most difficult judgement. It is a free list of where the estimates are softest.
7. Risk factors, compared with last year's. The text is boilerplate; the changes are not. A newly added or newly expanded risk factor was added by lawyers for a reason.

The core of it: the income statement is an opinion supported by estimates, the balance sheet is a snapshot at one date, and the cash flow statement is the closest thing to a fact. Read them in reverse order of how they are printed.

Common misconceptions

• "EBITDA is cash." Netflix's 2019 EBITDA looks strong: operating income of 2,604 million plus depreciation and content amortisation. Actual operating cash flow was minus 2,887 million. EBITDA adds back the amortisation of content while ignoring the cash spent acquiring more of it, ignores working capital entirely, ignores cash taxes and ignores interest on 14.8 billion of debt. It is a useful measure of one thing, operating margin before capital intensity, and it is not cash.
• "A profitable company cannot run out of money." Profit and cash differ by working capital, capital spending and the timing of accruals. Fast-growing profitable firms are the classic insolvency case, because growth consumes cash before it produces it.
• "Off balance sheet means immaterial." Netflix's off-balance-sheet content obligations exceeded its recognised long-term debt. Before 2019 the entire operating lease obligations of the airline and retail industries sat in footnotes.
• "Operating cash flow cannot be manipulated." It is harder to manipulate, which is not the same. Factoring receivables, delaying supplier payments over a year end, and choosing where a recurring outflow is classified all move it without changing anything real.

Recap

Netflix reported 1,867 million of net income and minus 2,887 million of operating cash flow in the same 2019 filing. The 4,754 million gap is almost entirely the 4,402 million growth in a capitalised content library, funded by a 4,399 million increase in long-term debt. Netflix's own free cash flow measure put the year at minus 3,274 million. The commitments note disclosed 19,490 million of content obligations against 14,759 million of recognised debt, and the first year of lease accounting moved 1,613 million from a footnote onto the balance sheet without anything changing in the business.

Why this matters: every input to the model in the next lesson comes from a document like this one, and each of them is a choice somebody made. Read the cash flow statement first, reconcile it to earnings, then go and find what the notes are carrying.

Sources

1. Netflix, Inc. (2020). Annual report on Form 10-K for the fiscal year ended December 31, 2019. U.S. Securities and Exchange Commission. sec.gov
2. U.S. Securities and Exchange Commission. (n.d.). Search EDGAR full-text and company filings. sec.gov
3. Wikipedia contributors. (n.d.). Cash flow statement. Wikipedia. en.wikipedia.org
4. Koller, T., Goedhart, M., & Wessels, D. (2020). Valuation: Measuring and Managing the Value of Companies (7th ed.), Chapters 11-12. John Wiley & Sons.
5. Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.), Chapter 3. John Wiley & Sons.

Coca-Cola's 2023 10-K reports net cash provided by operating activities of 11,599 million dollars, capital spending of 1,852 million, interest expense of 1,527 million and an effective tax rate of 17.4 percent. This lesson turns those four numbers, plus two rates from public series, into a value per share, and then shows how much of that value is assumption rather than arithmetic.

Step one: what the business actually throws off

Free cash flow to the firm is the cash available to everybody who financed the company, before any of it is paid out. Start from operating cash flow, which is already after interest, so add the interest back on an after-tax basis, then subtract the capital spending the business needs to keep running:

FCFF = 11,599 + 1,527 × (1 - 0.174) - 1,852 = 11,599 + 1,261 - 1,852 = 11,008

Two remarks. Interest is added back because it is a payment to a capital provider, and the WACC in the denominator will already price debt; leave it in the numerator and you charge for debt twice, exactly as in Lesson 3. And working capital needs no separate adjustment here, because operating cash flow already contains it: Coca-Cola's working capital consumed 846 million in 2023, and it is inside the 11,599.

The alternative route, starting from operating income and computing NOPAT, gives a materially lower figure for this company, and the reason is worth knowing. A large part of Coca-Cola's earnings comes from equity method stakes in bottlers, which sit below the operating income line. An EBIT-based model quietly values only the concentrate business. Whenever the two routes disagree, find out which cash the second one left behind.

Step two: a forecast somebody could argue with

Five years, with growth stepping down toward the long run. A defensible forecast ties growth to something observable: Coca-Cola's revenue rose from 38,655 million in 2021 to 43,004 in 2022 to 45,754 in 2023, and management guides to mid single digit organic growth.

Step three: the rate, built the way Lesson 6 built Apple's

One detail from the filing is worth pausing on: Coca-Cola's long-term debt has a carrying value of 37,507 million and a fair value of 33,445 million, because much of it was issued at a weighted average coupon of 3.4 percent and now trades in a 5 percent market. That is Lesson 1's bond arithmetic sitting inside a footnote, and it means the company owes less in economic terms than its balance sheet says.

Step four: the assumption that carries the model

Beyond 2028, use a growing perpetuity. With terminal growth of 2.5 percent:

TV2028 = 13,783 × 1.025 / (0.065 - 0.025) = 14,128 / 0.04 = 353,190

PV of TV = 353,190 × 0.729880 = 257,786

Enterprise value is 52,464 + 257,786 = 310,250 million, and 83 percent of it is the terminal value. That proportion is normal, not a flaw in this particular model, and it is the single most important fact about discounted cash flow analysis. What matters here: five years of painstaking forecasting determined a sixth of the answer, and one growth rate determined the rest.

Two disciplines keep terminal value honest. First, terminal growth must not exceed the long-run nominal growth rate of the economy, because a firm growing faster forever eventually becomes the economy. Two and a half percent is roughly consistent with long-run inflation plus modest real growth. Second, cross-check the implied multiple. Year-six free cash flow of 14,128 against a terminal value of 353,190 is 25 times. In EBITDA terms, Coca-Cola's 2023 EBITDA of 12,439 grown at the forecast rates reaches about 15,575 by 2028, so the terminal value implies about 22.7 times EBITDA, against roughly 23.2 times at which the company traded at the time. The terminal assumption is not aggressive relative to today's market; whether today's market is aggressive is a separate question.

Step five: from the firm to the share

The same filing implies a market value of 258,329/4,312 = 59.90 a share. The model sits about 9 percent above it. That is well inside the range the next section produces, which is the point.

Step six: the grid that is the real output

Rebuild the whole model at three discount rates and three terminal growth rates. Every cell is a complete revaluation, not an approximation.

Half a percentage point either way on two inputs nobody can measure produces a range from 51.60 to 87.45, a factor of 1.7. The market price of 59.90 sits comfortably inside it. A model that reports a single number has thrown away its most useful finding: not what the company is worth, but what you would have to believe for it to be worth a given price. Turn the model around and ask what terminal growth justifies 59.90 at a 6.5 percent WACC, and the answer is roughly 2.1 percent. That is a claim about the world that can be argued about.

Common misconceptions

• "The DCF tells you what the company is worth." It tells you what the company is worth conditional on your forecast, your rate and your terminal assumption. Change three inputs by half a point and the answer moves 70 percent. Its real product is a set of conditions.
• "Forecasting ten years instead of five reduces reliance on terminal value." It shifts the reliance rather than removing it, and it does so by making up five more years of forecast whose errors compound. The terminal value share falls arithmetically while the guessing rises.
• "Terminal growth of 4 percent is conservative because the company grew 6 percent last year." Terminal growth applies forever. A firm growing 4 percent in perpetuity in a 4 percent nominal economy keeps a fixed share of it; growing faster than the economy forever is impossible, and the model will happily compute it anyway.
• "If the model exceeds the market price, the market is wrong." Far more often the model is. The productive move is to solve for the input at which the model matches the price, then decide whether you believe that input.

Looking back

Coca-Cola's 2023 free cash flow to the firm was 11,599 plus 1,261 of after-tax interest less 1,852 of capital spending, or 11,008 million. Five years of growth at 5 and then 4 percent discounted at a 6.50 percent WACC gave 52,464 million; a 2.5 percent growing perpetuity gave a terminal value of 353,190 and a present value of 257,786, so enterprise value was 310,250 with 83 percent of it beyond the forecast horizon. Subtracting 42,064 of debt and adding 12,363 of cash gave equity of 280,549, or 65.06 a share against a market-implied 59.90. Across a plausible grid the value ranged from 51.60 to 87.45.

The upshot: a discounted cash flow model is a device for making assumptions explicit and pricing them. The next lesson uses a completely different method, one that lets the market supply the assumptions instead, and asks when that is an improvement.

Sources

1. The Coca-Cola Company. (2024). Annual report on Form 10-K for the fiscal year ended December 31, 2023. U.S. Securities and Exchange Commission. sec.gov
2. Federal Reserve Bank of St. Louis. (n.d.). ICE BofA US corporate A effective yield (BAMLC0A3CAEY). FRED. fred.stlouisfed.org
3. Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.), Chapters 12 and 15. John Wiley & Sons.
4. Koller, T., Goedhart, M., & Wessels, D. (2020). Valuation: Measuring and Managing the Value of Companies (7th ed.), Chapters 8 and 13. John Wiley & Sons.
5. Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.), Chapter 19. McGraw-Hill Education.

Four companies sell non-alcoholic drinks in the United States. In their fiscal 2023 filings they reported enterprise values of 288, 290, 44 and 52 billion dollars, and they traded at 23.2, 19.4, 11.9 and 25.8 times EBITDA. All four figures are computed from the same three lines of each balance sheet and the same two lines of each income statement. This lesson computes them, then explains the spread.

The set, built from the filings

Equity values are the market values reported on each 10-K cover page, all as of mid-2023. Debt is total borrowings; cash includes short-term investments. EBITDA is operating income plus reported depreciation and amortisation.

Reading the spread

Fourteen times earnings against thirty-four, for firms selling substantially the same product to substantially the same customers. Three things explain most of it.

Leverage. Keurig Dr Pepper carries net debt of 12,924 million against 3,731 of EBITDA: 3.5 times. Monster has no debt at all and 3.3 billion of net cash. Interest on that debt is what pushes Keurig's price-earnings ratio below its EV/EBITDA ranking would suggest, because earnings are after interest and EBITDA is not. This is the first reason to prefer enterprise multiples for cross-company work: they compare the business rather than the financing.

Growth. Monster's revenue grew from 5,541 million in 2021 to 7,140 in 2023, about 13.5 percent a year. Coca-Cola grew from 38,655 to 45,754, about 8.8 percent a year, and much of that was price. Keurig grew from 12,683 to 14,814, about 8.1 percent, on falling margins in some years. A multiple prices expected growth, and the market's ranking here is not mysterious.

Capital intensity. PepsiCo's operating margin is 13.1 percent against Coca-Cola's 24.7, because PepsiCo owns bottling and snack manufacturing while Coca-Cola sells concentrate and lets partly independent bottlers own the trucks. Neither is better; they are different businesses wearing similar labels. EV/EBITDA is blind to that difference, since it stops before depreciation, which is precisely where capital intensity shows up.

A multiple is a DCF with the assumptions hidden

Take the constant-growth model from Lesson 1 and divide both sides by earnings. With a payout ratio b, a cost of equity r and perpetual growth g:

P0/E0 = b(1 + g)/(r - g)

At a 60 percent payout, a 7 percent cost of equity and 2.5 percent growth, the fair multiple is 0.6 × 1.025/0.045 = 13.7. So what does a multiple of 26 imply? Solve for g: 0.6(1 + g) = 26(0.07 - g), giving 26.6g = 1.22 and g = 4.59 percent. Perpetual growth of 4.6 percent, in an economy growing 4 percent nominally, forever.

That number may be defensible or it may not. The point is that it was always there. The core of it: using a multiple does not avoid making assumptions; it means somebody else made them and you did not look.

Valuing Coca-Cola the lazy way, and comparing

Take the median EV/EBITDA of its two closest peers, Coca-Cola and PepsiCo, at 21.3 times, and apply it to Coca-Cola's own EBITDA of 12,439:

EV = 21.3 × 12,439 = 264,951, so equity is 264,951 - 42,064 + 12,363 = 235,250, or 235,250/4,312 = 54.55 a share.

Set that beside Lesson 11's discounted cash flow value of 65.06 and the market-implied 59.90. Three methods, three answers, a spread of 19 percent between the highest and the lowest. That is a normal outcome and a useful one: the DCF says the market is slightly pessimistic about long-run growth, the comparables say Coca-Cola is priced at a premium to its own peer group, and both can be true because a premium to peers is exactly what higher expected growth looks like.

The traps, in the order they catch people

1. Choosing the set is the valuation. Include Monster and the median rises; include a slow bottler and it falls. The peer group should be selected on business economics, growth and risk, and written down before the multiples are computed rather than after.
2. Trailing against forward. Every multiple above uses reported figures. Markets price the future, so a firm whose earnings are about to double looks expensive on trailing numbers and cheap on forward ones. Mixing the two across a comparable set is the commonest arithmetic error in the technique.
3. Accounting is not uniform. Capitalised against expensed development costs, differing useful lives, one-off charges that recur, and different lease treatments all move EBITDA without moving the business.
4. EBITDA ignores what it costs to stay in business. A firm needing 6 percent of revenue in maintenance capital spending and one needing 2 percent look identical at the EBITDA line. If capital intensity differs across the set, EV/EBITDA is measuring different things in each column.
5. Circularity. Comparables value a company relative to the market's pricing of similar companies. If the whole sector is mispriced, the method cannot see it, by construction. In 1999 the internet sector's enterprise-value-to-sales multiples were internally consistent and collectively wrong.
6. Broken denominators. A loss-making firm has no meaningful price-earnings ratio, and a negative EBITDA has no meaningful multiple. Analysts then move down to revenue multiples, which are the least informative of all because they say nothing about whether the revenue is profitable.

Common misconceptions

• "A low price-earnings ratio means the shares are cheap." Keurig's 14.4 times sits alongside more leverage, slower growth and lower returns on capital than Monster's 34. A low multiple is a statement that the market expects less, and the question is always whether it expects too little.
• "Multiples are objective and DCFs are subjective." The algebra above converts any multiple into a growth assumption. The difference is transparency: a DCF shows its assumptions in a grid, a multiple hides them inside a peer group whose composition you chose.
• "EV/EBITDA is always better than price-earnings." It is better for comparing businesses with different capital structures, which is most of the time. It is worse when capital intensity varies across the set, and it is unusable for banks, whose debt is raw material rather than financing.
• "Use the average of the comparable set." Use the median, because one outlier moves the mean a long way in a set of four or five, and then explain in one sentence why the subject company should trade above or below it. A valuation that reports a peer median with no such sentence has not made an argument.

Where this leaves us

Coca-Cola at 23.2 times EBITDA, PepsiCo at 19.4, Keurig Dr Pepper at 11.9 and Monster at 25.8 are explained by leverage of 3.5 times net debt to EBITDA at one end and net cash at the other, by revenue growth of 8 percent against 13.5 percent, and by the difference between owning bottling plants and selling concentrate. The constant-growth algebra converts any price-earnings ratio into an implied perpetual growth rate: 26 times, at a 7 percent cost of equity and a 60 percent payout, means 4.6 percent forever. Applying a 21.3 times peer median to Coca-Cola gives 54.55 a share against a discounted cash flow value of 65.06 and a market-implied 59.90.

Why this matters: run both methods, and when they disagree, find the assumption responsible. That disagreement is the most informative output either method produces. Module 5 now turns to what happens when one company decides to buy another at a premium to all of these numbers.

Sources

1. The Coca-Cola Company. (2024). Form 10-K for fiscal 2023. U.S. Securities and Exchange Commission. sec.gov
2. PepsiCo, Inc. (2024). Form 10-K for fiscal 2023. U.S. Securities and Exchange Commission. sec.gov
3. Monster Beverage Corporation. (2024). Form 10-K for fiscal 2023. U.S. Securities and Exchange Commission. sec.gov
4. Liu, J., Nissim, D., & Thomas, J. (2002). Equity valuation using multiples. Journal of Accounting Research, 40(1), 135-172.
5. Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.), Chapters 17-19. John Wiley & Sons.

On 18 January 2022 Microsoft offered 95.00 dollars a share in cash for Activision Blizzard, a transaction it valued at 68.7 billion dollars. Activision had closed the previous trading day at 65.39. The premium was 45 percent, or about 23 billion dollars of value handed to the target's shareholders on the day of the announcement, before a single game had been made differently. The deal closed twenty-one months later, after competition authorities on two continents had been fought to a standstill.

Every acquisition contains the same arithmetic: a premium paid today against benefits promised later. This lesson does that arithmetic three ways.

What a premium obliges you to deliver

Company A is worth 20,000 million, 200 million shares at 100. Target T is worth 5,000 million, 100 million shares at 50. A offers 65 a share in cash.

Premium = (65 - 50) × 100 million = 1,500 million

Management expects cost synergies of 250 million a year before tax, permanently. At a 21 percent tax rate that is 197.5 after tax, and at a 9 percent cost of capital its present value is 197.5/0.09 = 2,194 million.

NPV to the acquirer = 2,194 - 1,500 = +694 million

Invert it and the deal becomes a promise with a number on it. To break even, the acquirer must produce 1,500 × 0.09 = 135 million of after-tax savings a year forever, which is 171 million a year before tax. That figure, not the strategic logic, is what the board should be asked to defend.

Now add the thing every integration plan underestimates. Suppose the savings are real but arrive three years late:

Nothing about the strategy changed. A five-year integration turns a 694 million gain into a loss, purely through discounting. Remember: in acquisitions, the timing of the synergy is roughly as important as its size, and it is the part that is systematically forecast optimistically.

Cash or shares, and who carries the risk

Suppose A pays in stock instead: 6,500 million of shares at 100, or 65 million new shares. If the synergies are real, the combined firm is worth 20,000 + 5,000 + 2,194 = 27,194 over 265 million shares, or 102.62 a share. The target's holders own 65 of those 265 million shares, worth 6,670 million, which is 170 million more than the 6,500 they would have taken in cash.

If the synergies turn out to be zero, the combined firm is worth 25,000 over 265 million shares, or 94.34. The target's holders now have 6,132 million rather than 6,500, and the acquirer's holders have lost 1,132 million rather than 1,500.

A stock deal shares both outcomes. That is why an acquirer confident in its synergies prefers cash, and why the market reads a stock-financed bid as a signal that the acquirer thinks its own shares are dear, which is Myers and Majluf from Lesson 8 arriving in a new setting. The evidence follows the theory: cash acquirers do better on announcement than stock acquirers.

Why the winner is the bidder who was most wrong

Suppose a company is truly worth 5,000 million and five bidders each form an unbiased estimate with a standard deviation of 500. Nobody is foolish; the errors average to zero. But the auction does not select a random bidder. It selects the highest one.

The expected maximum of five draws from a normal distribution sits about 1.163 standard deviations above the mean, so the winning estimate averages 5,000 + 1.163 × 500 = 5,582. The winner overpays by 582 million, or 11.6 percent, on average, while believing they have made an unbiased assessment. With ten bidders the expected maximum rises to 1.539 standard deviations and the overpayment to 769 million.

Read that twice: more competition makes the curse worse, not better. The winner's curse is a property of common-value auctions, and it is defeated only by bidding below your own estimate by an amount that grows with the number of rivals. Richard Roll's hubris hypothesis adds the human layer: chief executives who have just won a contested auction rarely conclude that they were the most mistaken person in the room.

What the evidence says happens

Gregor Andrade, Mark Mitchell and Erik Stafford summarised twenty-five years of US deals: around the announcement, target shareholders gained about 16 percent, acquirer shareholders lost about 0.7 percent, and the combined entity gained about 1.8 percent. The combined gain is positive and small; the split is overwhelmingly to the target.

Michael Moeller, Frederik Schlingemann and René Stulz then examined the 1998 to 2001 wave and found acquiring shareholders had lost about 240 billion dollars, with the damage concentrated in a small number of very large deals by firms whose shares had recently performed well. Take those out and the rest of the period looked ordinary.

The honest summary is not that mergers destroy value. It is that on average the gains are real but modest, they go to the seller, and the distribution has a long and expensive left tail populated by large bids from confident acquirers.

The leveraged buyout, worked

A private equity fund buys a business with EBITDA of 200 million at 9.0 times, an enterprise value of 1,800 million, financed with 1,080 of debt at 9 percent and 720 of its own equity. Depreciation of 40 million equals capital spending, EBITDA grows 5 percent a year, tax is 21 percent, and every dollar of free cash flow repays debt.

Sell at year five at the same 9.0 times: 9.0 × 255.3 = 2,297.7 of enterprise value, less 655.0 of debt, leaves 1,642.7 of equity against the 720 invested.

IRR = (1,642.7/720)1/5 - 1 = 17.9 percent

Where did the 922.7 million of equity gain come from? It decomposes exactly:

• Growth in EBITDA: 9.0 × (255.3 - 200) = 497.7
• Debt repaid out of cash flow: 1,080.0 - 655.0 = 425.0
• Multiple expansion: 0, by construction

Change only the exit multiple and the picture moves sharply: exit at 8.0 times and the IRR is 14.0 percent; exit at 10.0 times and it is 21.4 percent. A leveraged buyout is a bet on operations, on discipline in paying down debt, and on the market being no less generous in five years than it is today, and only the first two are inside anybody's control.

The genre's founding transaction, Kohlberg Kravis Roberts's 25 billion dollar purchase of RJR Nabisco at 109 dollars a share in 1988, is instructive precisely because it was won in a contested auction and returned poorly to its buyer.

Common misconceptions

• "The premium is justified because the target is strategically important." Strategic importance is not a number. The premium is, and it converts into a required annual saving: 171 million a year before tax in the worked case. If nobody in the room will sign up to that figure, the strategic argument has not been made.
• "A competitive auction produces a fair price." In a common-value auction it produces a price above the average estimate, and the more bidders there are the further above. Bidding your honest estimate in a five-way auction loses money on average.
• "Mergers destroy value." The combined announcement effect is positive but small, about 1.8 percent. The finding is about distribution: targets capture the gain, and a minority of very large acquisitions produce very large acquirer losses.
• "An LBO creates value through financial engineering." In the worked case, 54 percent of the equity gain came from EBITDA growth and 46 percent from repaying debt with cash the business generated. Leverage magnifies both the return and the risk; it does not manufacture the cash flow that services it.

What to carry forward

A 1,500 million premium at a 9 percent cost of capital obliges the acquirer to deliver 171 million a year before tax, forever, and a five-year delay in achieving it converts a 694 million gain into a 74 million loss. Paying in stock shares both the upside and the downside with the seller, which is why cash signals confidence. In a five-bidder auction with unbiased estimates the winner overpays by about 11.6 percent, and by more as bidders are added. The evidence gives targets about 16 percent, acquirers about minus 0.7, and the combination about 1.8. And a buyout entered at 9 times with 60 percent debt returned 17.9 percent a year, split 497.7 million from operations and 425.0 million from deleveraging, with everything above that depending on an exit multiple nobody controls.

The upshot: the discipline in a deal is not the model, it is the number the model obliges somebody to promise. The next lesson looks at the value of not committing at all.

Sources

1. Andrade, G., Mitchell, M., & Stafford, E. (2001). New evidence and perspectives on mergers. Journal of Economic Perspectives, 15(2), 103-120.
2. Moeller, S. B., Schlingemann, F. P., & Stulz, R. M. (2005). Wealth destruction on a massive scale? A study of acquiring-firm returns in the recent merger wave. The Journal of Finance, 60(2), 757-782.
3. Roll, R. (1986). The hubris hypothesis of corporate takeovers. The Journal of Business, 59(2), 197-216.
4. Wikipedia contributors. (n.d.). Acquisition of Activision Blizzard by Microsoft. Wikipedia. en.wikipedia.org
5. Wikipedia contributors. (n.d.). Leveraged buyout. Wikipedia. en.wikipedia.org

Before you read: a project has a present value of 100 million and costs 104 million to build. Write down, in one word, whether you reject it. Keep the word.

Its net present value is minus 4 million, so the rule from Lesson 2 says reject. Now add one fact: the firm owns a licence letting it make this decision at any point in the next year, and the project's value a year from now could plausibly be 135 million or 74 million. On that information the licence is worth about 15 million dollars, and the correct action is to acquire it and wait.

Nothing in the NPV rule is wrong. What is wrong is applying it to a decision that does not have to be taken now.

The tree, and the probabilities that are not probabilities

Model the project's value as moving up by a factor of 1.35 or down by 0.7407 each year, with a riskless rate of 5 percent. Do not use your estimate of how likely each is. Use the risk-neutral probability, the weight that makes the underlying itself price correctly:

p = (1 + r - d)/(u - d) = (1.05 - 0.7407)/(1.35 - 0.7407) = 0.3093/0.6093 = 0.5076

Check it works: [0.5076(135) + 0.4924(74.07)]/1.05 = 105.0/1.05 = 100. The tree reproduces today's value, which is what makes it trustworthy for valuing anything else written on the same underlying.

This step is the one people skip. It matters because your own probability estimate would require a risk-adjusted discount rate for an asset whose risk changes at every node, and nobody can supply that. Risk-neutral weighting sidesteps the problem entirely by pricing off an asset whose value you already know.

The option to wait

In one year the project is worth 135 or 74.07 and still costs 104 to build. Build only if it is worth building:

Payoff = max(135 - 104, 0) = 31 in the up state, and max(74.07 - 104, 0) = 0 in the down state.

Value today = [0.5076(31) + 0.4924(0)]/1.05 = 15.74/1.05 = 14.99 million

This is a call option: the project is the underlying, the 104 million construction cost is the strike, and the licence period is the time to expiry. Check the word you wrote down.

Extend the licence to two years and value a two-step tree. The project can be worth 182.25, 100.0 or 54.87 at the end.

Two things to notice. The licence is worth 18.29 million over two years against 14.99 over one: time is the raw material of option value. And at the year-one up node, exercising would bank 31 million while holding is worth 37.83, so the firm waits even though the project is by then comfortably positive. Key idea: a positive NPV is a reason to build only when the alternative of waiting is worth less than the NPV you would collect.

Waiting is not free

The tree above quietly assumed that delaying costs nothing. It usually does cost something. Suppose building now would produce 8 million of cash during the first year, which waiting forfeits. The project value at year one is then measured after that cash has gone: 127 and 66.07 rather than 135 and 74.07.

Value = [0.5076(127 - 104) + 0]/1.05 = 11.68/1.05 = 11.12 million

Waiting cost 3.87 million of option value. This is exactly the role a dividend plays for a call on a stock, and it is why an American call on a dividend-paying stock is sometimes exercised early. In corporate terms, the more cash a project would throw off immediately, the weaker the case for holding the option rather than exercising it.

Two more options on the same tree

The option to abandon. Suppose the firm builds now and can, in one year, sell the assets for 90 million. Its position at year one is max(135, 90) = 135 or max(74.07, 90) = 90.

Value = [0.5076(135) + 0.4924(90)]/1.05 = 112.84/1.05 = 107.47

The right to walk away is worth 107.47 - 100 = 7.47 million. Price it directly as a put option and you get the same figure: 0.4924 × max(90 - 74.07, 0)/1.05 = 0.4924 × 15.93/1.05 = 7.47. That agreement is a useful check on any real options model.

The option to expand. Suppose that in a year the firm could double the project's scale by spending a further 110 million. In the up state that buys another 135 of value for 110, a gain of 25; in the down state it buys 74.07 for 110 and is refused.

Value = 0.5076 × 25/1.05 = 12.09 million

An expansion right is a call struck at the expansion cost. It is why firms accept marginal first plants in new markets, and why the phrase "this project is a platform" is sometimes a real argument and sometimes an excuse.

What the analogy requires

The last row is the one that kills most corporate real option arguments. A patent or a mineral lease genuinely confers the right to decide later. "We could always expand into Germany" confers nothing, because so could everybody else, and a rival's entry is a dividend paid to somebody other than you.

Real options valuation is taught in every finance programme and used by roughly a quarter of surveyed chief financial officers. The gap is not ignorance. It is that the inputs are soft, the technique reliably produces a positive number, and a method that can rescue any rejected project is a governance hazard as much as an analytical tool.

Common misconceptions

• "A negative NPV project must be rejected." A negative NPV means do not build it now. If the decision can be deferred and the outcome is uncertain, the right to build later is a separate asset, worth 14.99 million here on a project whose immediate NPV was minus 4.
• "Volatility is bad, so more of it should lower value." For an obligation, yes. For an option, higher volatility raises value, because the downside is truncated at zero and the upside is not. That is why exploration licences on volatile commodities are valuable and why the same volatility makes the underlying project unattractive.
• "You need the real probability that the project succeeds." The tree uses a risk-neutral weight of 0.5076 derived from the up factor, the down factor and the riskless rate. Your own probability estimate never enters, which is the technique's main advantage and the reason its output feels counter-intuitive.
• "Real option value justifies the project." Only if the option is exclusive, the deferral genuinely available, and the flexibility exercisable in practice. A plant that cannot be closed because of a union agreement or a regulator has no abandonment option, whatever the model says.

Putting it together

On a tree with up and down factors of 1.35 and 0.7407 and a 5 percent riskless rate, the risk-neutral weight is 0.5076 and reproduces the underlying's value of 100 exactly. A one-year right to build for 104 is worth 14.99 million; a two-year right is worth 18.29, and at the year-one up node the firm holds rather than exercises because 37.83 beats 31. Forgoing 8 million of first-year cash flow cuts the one-year option to 11.12. On the same tree, an abandonment right at 90 is worth 7.47 and an expansion right at 110 is worth 12.09, and both can be verified twice by pricing them directly as options.

So what?: ask of every capital proposal not only what it is worth, but what it would be worth to decide later, and then ask who else could take the decision first. The next lesson turns to what happens when the person holding these options is not the person who owns the firm.

Sources

1. Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229-263.
2. Brennan, M. J., & Schwartz, E. S. (1985). Evaluating natural resource investments. The Journal of Business, 58(2), 135-157.
3. Dixit, A. K., & Pindyck, R. S. (1994). Investment under Uncertainty. Princeton University Press.
4. Wikipedia contributors. (n.d.). Binomial options pricing model. Wikipedia. en.wikipedia.org
5. Wikipedia contributors. (n.d.). Real options valuation. Wikipedia. en.wikipedia.org

Enron filed its 10-K for 2000 on 2 April 2001. It reported operating revenues of 100,789 million dollars, up from 40,112 million a year earlier, which made it one of the largest companies in the United States by that measure. Its net income was 979 million, up from 893 million. Its total assets had gone from 33,381 million to 65,503 million in twelve months. Eight months later it filed for bankruptcy.

The interesting question is not how the fraud worked. It is what an ordinary reader could have found in that document, using only Lesson 10's procedure and a calculator. The answer is: nearly everything.

Four ratios from the face of the accounts

Now the line that explains the first row. Against 100,789 million of revenue, the same income statement shows 94,517 million of cost of gas, electricity, metals and other products, leaving a gross margin of 6.2 percent. Enron was booking the full notional value of energy trades as revenue. A broker who buys gas for 94 and sells it for 100 has revenue of 6, not 100, on any economically meaningful measure, and the choice to present it the other way turned a mid-sized margin business into a headline about the seventh-largest company in America.

The second-order reading is harsher still. Assets nearly doubled while earnings rose a tenth. A company that has committed 32 billion dollars of new capital and produced 86 million more of profit is either investing very early in something enormous, or it is not earning a return on what it has bought. What matters here: the question to ask of any spectacular growth story is what happened to return on capital while it was growing.

The note that was there all along

Note 16 of the same filing, headed Related Party Transactions, opens: "In 2000 and 1999, Enron entered into transactions with limited partnerships (the Related Party) whose general partner's managing member is a senior officer of Enron."

That sentence discloses that the chief financial officer was on both sides of transactions with the company that employed him, involving special purpose entities whose non-consolidation depended on thin slices of genuinely independent outside equity. It was printed, audited and filed. The board had waived Enron's own code of ethics to allow the arrangement.

Note 1 disclosed the other structural fact: long-term energy contracts were accounted for using mark-to-market accounting. For a twenty-year contract in a market with no twenty-year price, marking to market means marking to a model, and the model's assumptions were the company's own. Profit could be recognised on the day a contract was signed and revised later, which is why earnings could look smooth while cash did not arrive.

Enron restated its 1997 to 2000 results in November 2001, cutting reported net income by 586 million dollars, and filed for Chapter 11 on 2 December. Its auditor, Arthur Andersen, which earned large consulting fees from the same client, did not survive either.

Why any of this is a finance problem and not just a crime

Michael Jensen and William Meckling framed the modern version in 1976. Once the people who run a firm are not the people who own it, the owner faces costs that do not exist in an owner-managed business: monitoring costs to watch the manager, bonding costs the manager incurs to make promises credible, and a residual loss because no contract closes the gap entirely. Governance is the machinery for keeping that total down, and it is a cost of the corporate form rather than a scandal.

Jensen sharpened it in 1986 with the free cash flow hypothesis: managers with cash and no projects worth the cost of capital will find something to spend it on, because a larger firm brings prestige, pay and safety. Debt, on this view, is a disciplining device. A fixed interest payment removes the discretion that empire building requires, which is a defence of leverage that has nothing to do with tax.

Enron inverted the mechanism. It had persistently negative free cash flow, it borrowed heavily, and the borrowing was structured through entities that kept the obligations off the balance sheet, so the discipline that debt normally imposes never arrived. That is the specific finance failure, and it is separate from the accounting fraud that concealed it.

Paying people to care

The classic answer is to make the manager an owner. Jensen and Murphy measured how much of one in 1990 and found that a chief executive's wealth changed by about 3.25 dollars per 1,000 dollar change in shareholder wealth: an effective ownership stake of about a third of one percent. Their conclusion, that executives were paid like bureaucrats rather than owners, helped drive the option boom of the 1990s.

That correction created its own problems, and they are worth stating precisely rather than as complaint.

• Options are convex. They pay on the upside and stop at zero, so they reward volatility as such. A chief executive holding mostly options prefers a risky strategy to a safe one of equal value, which is Lesson 8's asset substitution problem re-created inside the pay contract.
• Earnings per share targets are manipulable. Lesson 7 showed a buyback raising EPS by a third with no change in shareholder wealth. A bonus keyed to EPS can be earned by financial arithmetic alone.
• Grant dates were gamed. Erik Lie's work on option grants, and the wave of investigations that followed from 2006, found grant dates clustering implausibly at share price lows; Heron and Lie estimated that roughly one grant in seven to top executives between 1996 and 2005 had been backdated or otherwise manipulated.
• Pay is set by people the executive influences. Lucian Bebchuk and Jesse Fried argued that boards and consultants are not arm's length counterparties, so pay structure itself is an agency problem rather than a solution to one.

Enron's executives held large equity stakes and sold heavily into the market while publicly encouraging employees to hold. Alignment of pay with the share price aligns the manager with the price, which is not the same as with the value, and the difference is exactly what accounting discretion can manufacture.

What is left to rely on

No single mechanism works alone, and the useful question is what each one catches.

The Sarbanes-Oxley Act of 2002 made chief executives and chief financial officers personally certify the accounts, required an internal control report with auditor attestation, mandated an independent audit committee with a financial expert, barred most consulting by a company's auditor, and created the Public Company Accounting Oversight Board. It raised the cost of the specific failures of 2001. It did not repeal the agency problem, because nothing can.

Common misconceptions

• "Enron hid everything, so no reader could have known." The related-party structure was in Note 16, mark-to-market accounting was in Note 1, and the ratios above are arithmetic on the face of the statements. What was hidden was the magnitude and the deliberateness. What was disclosed was enough to refuse to own the shares.
• "Tying pay to the share price solves the agency problem." It ties the manager to the price, which reported earnings influence and which accounting discretion can move. It also introduces convexity, so it can buy more risk-taking than the shareholders wanted.
• "Cash on the balance sheet is always good for shareholders." Jensen's argument is that free cash flow without projects invites value-destroying investment, which is why payout policy and leverage are governance instruments and not only financing ones.
• "Independent directors guarantee oversight." Independence is a formal test about employment and contracts. It says nothing about whether directors understood the transactions in front of them, and at Enron they demonstrably did not.

What you now know

Enron's own 10-K reported 100,789 million of revenue against 94,517 million of cost of products, a 6.2 percent gross margin, and 979 million of net income on 65,503 million of assets: a 1.49 percent return on assets, down from 2.68. Assets grew 96 percent and earnings 9.6. Note 16 disclosed that a senior officer was the managing member of the general partner of the entities transacting with the company, and Note 1 disclosed mark-to-market accounting on contracts with no observable market. Jensen and Meckling's monitoring, bonding and residual loss name the problem; Jensen's free cash flow argument explains why debt normally disciplines and why off-balance-sheet structures removed that discipline here; Jensen and Murphy's 3.25 dollars per 1,000 explains the option boom, and the option boom explains the backdating scandal.

Bottom line: governance is a set of partial checks, each of which catches something and misses something else, and the cheapest of them is a reader who computes return on capital before believing a growth story. Module 6 turns to where the capital comes from in the first place.

Sources

1. Enron Corp. (2001). Annual report on Form 10-K for the fiscal year ended December 31, 2000. U.S. Securities and Exchange Commission. sec.gov
2. Jensen, M. C., & Meckling, W. H. (1976). Theory of the firm: Managerial behavior, agency costs and ownership structure. Journal of Financial Economics, 3(4), 305-360.
3. Jensen, M. C. (1986). Agency costs of free cash flow, corporate finance, and takeovers. The American Economic Review, 76(2), 323-329.
4. Jensen, M. C., & Murphy, K. J. (1990). Performance pay and top-management incentives. Journal of Political Economy, 98(2), 225-264.
5. Wikipedia contributors. (n.d.). Enron scandal. Wikipedia. en.wikipedia.org

On 10 December 2020 Airbnb sold 51,323,531 shares of Class A common stock at 68.00 dollars each. The final prospectus sets out the split on its cover: the company offered 50,000,000 shares, selling stockholders offered 1,323,531, and the gross came to 3,490,000,108 dollars. Underwriting discounts and commissions took 1.4416 a share, or 73,988,002 dollars, leaving 3,327,920,000 of proceeds to the company before expenses.

By the close of the first day of trading the stock was up 112.8 percent. Put that figure beside the fee and you have the whole subject of this lesson.

What the fee bought, and what the price gave away

Start with the fee, because the fee is what issuers argue about. Airbnb's gross spread was 1.4416 divided by 68.00, or 2.12 percent. Jay Ritter, who has tracked these numbers since 1980, finds gross spreads on moderate-size US offerings sitting at 7 percent for a quarter of a century. Airbnb, large and badly wanted, got its syndicate to under a third of that.

Now the other number. A 112.8 percent first-day return on a 68.00 offer puts the close just under 145. The standard measure of what that cost is money left on the table: first-day closing price less offer price, times shares sold.

(144.70 - 68.00) × 51,323,531 = about 3,940 million dollars

That is roughly 53 times the underwriting fee, and more than the 3,328 million the company collected. Whether buyers would have absorbed 50,000,000 shares at 100 dollars is exactly what the rest of this lesson asks. Worth holding on to: in a first offering the expensive decision is the offer price, and nearly all the negotiating effort goes into the cheap one.

Two details of the cover page matter later. The underwriters hold an over-allotment option, the greenshoe, letting them buy extra shares at the offer price for thirty days to prop up a weak deal or upsize a strong one. And bookbuilding means the syndicate decides who receives shares: that discretion is the hinge on which two of the four explanations below turn.

Forty-five years of the same result

Ritter's tables cover 9,343 US initial public offerings from 1980 to 2025, excluding penny stocks, trusts and blank-cheque vehicles. The mean first-day return is 19.0 percent.

A discount that survives forty-five years and every kind of market is not an error waiting to be arbitraged. You cannot short a stock that has not traded, and you cannot buy as much of a cheap offering as you want, because the bank rations allocation. Both halves of the correction mechanism are missing.

Four reasons a seller discounts on purpose

Look at the first row again. In Lesson 13 the winner's curse made a bidder pay too much, because the auction selected the most optimistic valuation in the room. Here the same adverse selection runs the other way: the seller discounts the whole issue, because the buyers who turn up for a bad offering are the only ones who turn up for every offering.

The four are not mutually exclusive and the evidence does not cleanly pick one. What it rules out is the comfortable story that the discount buys a guaranteed sale, since the syndicate signs its commitment after the book is built, when the outcome is known.

The mechanisms that were tried instead

If bookbuilding creates the discount, sell the shares some other way. In August 2004 Google ran a modified Dutch auction at 85.00 a share, taking price-and-quantity bids from anyone with a brokerage account. The stock rose 18.04 percent on day one against a 12.3 percent average for the 173 offerings of 2004: better, not transformative, and almost nobody copied it. A quieter variant spread instead. Limit order bookbuilding keeps the bank's discretion but makes every indication of interest carry a reservation price, so the book records what buyers will pay.

Read the two right-hand columns together. Each issuer raised its offer price well above the filing range, and each still left an enormous sum behind. The process cut the fee, the column nobody needed help with, and did nothing to the one that costs billions. A direct listing goes further, skipping the syndicate and the offer price and letting the exchange open on supply and demand, but it raises no money.

The terms that were agreed years before the prospectus

Airbnb reached 2020 having issued six series of redeemable convertible preferred stock since 2009. Those securities are not shares with a nicer name. They are contracts, and their arithmetic decides who gets what in every outcome except the one the press reports.

Take a fully specified case. Founders hold 8,000,000 shares of common. A fund invests 4,000,000 dollars for 2,000,000 shares of preferred at 2.00 a share. Total shares are 10,000,000, the fund owns 20 percent, the post-money valuation is 20,000,000. The liquidation preference is one times the investment, non-participating: on a sale the fund takes either its 4,000,000 back or its 20 percent of the proceeds, whichever is larger.

Three things fall out of that table. The conversion threshold is exactly the post-money valuation, so the preference pays in every outcome below the headline number. A sale at 15,000,000, which anyone would call a decent result, pays the common 1.375 a share against the 2.00 the fund paid. And the founders hold nothing at 4,000,000, though the business sold for a real price.

Now change one word. Make the preference participating and the fund takes its 4,000,000 and then shares what is left. At a 50,000,000 sale it collects 4,000,000 plus 20 percent of 46,000,000, or 13,200,000, and the common falls from 40,000,000 to 36,800,000, which is 4.60 a share instead of 5.00. A two times participating preference takes 8,000,000 and then 20 percent of 42,000,000, leaving the common 4.20. Same percentage, same company, three answers.

Anti-dilution decides what happens when the next round is worse. Suppose the company later sells 5,000,000 shares at 1.00 to raise 5,000,000.

• Full ratchet. The Series A conversion price resets to 1.00, so the original 4,000,000 dollars converts into 4,000,000 shares instead of 2,000,000. Total shares become 17,000,000 and the founders hold 47.1 percent.
• Broad-based weighted average. The conversion price falls in proportion to how much cheap stock was issued: NCP = 2.00 × (10,000,000 + 2,500,000)/(10,000,000 + 5,000,000) = 1.6667, where 2,500,000 is what 5,000,000 dollars would have bought at the old price. The fund converts into 2,400,000 shares, total shares are 15,400,000, and the founders hold 51.9 percent.

One clause, read once, moved 4.8 percent of the company. The point: a venture valuation is a number attached to a contract, and the contract, not the number, sets the payoff in most outcomes that actually occur.

Selling shares when you are already listed

A seasoned equity offering should be the easy case: an observable price, years of filings, analysts with models. Yet Paul Asquith and David Mullins, examining industrial issuers, found share prices falling about 3 percent on the announcement, and Ronald Masulis and Ashok Korwar found the same. Lesson 8 supplied the reason: under Myers and Majluf a management that knows more than the market issues equity when the shares are dear, so the decision to sell is evidence about the seller.

Outside the United States the same money is often raised through a rights issue, which settles who gets the cheap shares by giving every holder the same entitlement. A company with 100,000,000 shares at 10.00, worth 1,000 million, offers one new share for every four held at 6.00, raising 150 million on 25,000,000 shares.

Theoretical ex-rights price = (1,000 + 150)/125 = 9.20

A holder of four shares had 40.00; she pays 6.00 and holds five worth 46.00, so nothing has happened to her. A holder who bins the letter has 36.80, and the 3.20 lost is exactly what the discarded rights were worth, since 9.20 - 6.00 = 3.20 per new share is 0.80 on each share held. The alarming discount is a device for making take-up certain, not a transfer.

Common misconceptions

• "A big first-day jump means the offering was a success." It means the shares sold for less than buyers would pay. Airbnb's jump moved about 3,940 million from the issuer and its selling holders to whoever got an allocation. A deal that closes flat priced correctly.
• "The underwriting fee is the cost of going public." The fee was 74 million, the discount about 3,940 million. Fees are visible, contracted and negotiable; the discount is none of those, which is why it costs fifty times as much.
• "A 20,000,000 dollar valuation means the company is worth 20,000,000." It means somebody paid 4,000,000 for preferred carrying a liquidation preference. The common reaches that figure only at or above it, and is worth nothing at 4,000,000.
• "Underpricing exists because the bank guarantees the sale." The commitment is signed after the book is built, when demand is known. The risk the syndicate carries is measured in hours.

The takeaway

Airbnb sold 51,323,531 shares at 68.00, paid 73,988,002 in fees at a 2.12 percent spread, collected 3,327,920,000, and watched the stock close 112.8 percent higher, leaving about 3,940 million on the table. Across 9,343 offerings since 1980 the mean first-day return is 19.0 percent, and neither Google's 2004 auction nor limit order bookbuilding closed that gap, though both cut the fee. Underneath the prospectus, a one times non-participating preference pays the common 1.375 a share on a 15,000,000 sale and nothing on a 4,000,000 sale, and a full ratchet moves 4.8 percent of a company on one down round. Announcing a seasoned offering costs about 3 percent of the price; a rights issue at 6.00 against a 9.20 ex-rights value costs a subscriber nothing.

In short: every number in this lesson is a price paid for information somebody else did not have. The last lesson puts the whole course on one company, in a world where the cash arrives in currencies the shareholders do not spend.

Sources

1. Airbnb, Inc. (2020). Prospectus filed pursuant to Rule 424(b)(4), 51,323,531 shares of Class A common stock. U.S. Securities and Exchange Commission. sec.gov
2. Ritter, J. R. (2026). Initial public offerings: Updated statistics. Warrington College of Business, University of Florida. site.warrington.ufl.edu
3. Rock, K. (1986). Why new issues are underpriced. Journal of Financial Economics, 15(1-2), 187-212.
4. Loughran, T., & Ritter, J. R. (2002). Why don't issuers get upset about leaving money on the table in IPOs? The Review of Financial Studies, 15(2), 413-444.
5. Asquith, P., & Mullins, D. W. (1986). Equity issues and offering dilution. Journal of Financial Economics, 15(1-2), 61-89.

Apple's fiscal 2023 net sales in Europe were 94,294 million dollars, down 824 million on the year. The 10-K explains the fall in a single clause: weakness in foreign currencies against the US dollar accounted for more than the entire year-over-year decrease. The European business grew. The number shrank.

Every valuation in this course has treated a dollar of cash flow as a dollar. This lesson asks where the cash is earned and what a currency does to it, then puts the whole course on one company and says what the answer is worth.

Where Apple's revenue is actually earned

So 220,725 million, or 57.6 percent of sales, arose outside the Americas, and the Americas segment itself includes Canada and Latin America. Item 7A of the filing states the position: the company is "a net receiver of currencies other than the U.S. dollar". A stronger dollar therefore shrinks reported sales and earnings whatever the underlying business does.

Three exposures, and only one of them changes value

The last row is the one a valuation has to capture and the one no forward contract fixes. The 10-K sets out the mechanism: when foreign currencies weaken, the company either raises international prices, which reduces demand, or holds prices and earns a thinner dollar margin. That is a competitive position, not a settlement date.

The forward rate is arithmetic, not a forecast

Take a spot rate of 1.0500 dollars per euro, a one-year dollar interest rate of 5.00 percent and a one-year euro rate of 3.50 percent. Covered interest parity fixes the one-year forward:

F = S × (1 + rUSD)/(1 + rEUR) = 1.0500 × 1.0500/1.0350 = 1.0652

That is not a prediction about the euro. It is a no-arbitrage price, and if a bank quoted 1.0900 instead, the trade writes itself:

• Borrow 1,000,000 dollars for a year at 5 percent, owing 1,050,000.
• Buy euros at spot: 952,381 euros.
• Deposit them at 3.5 percent: 985,714 euros in a year.
• Sell that amount forward today at 1.0900: 1,074,429 dollars.
• Repay the loan and keep 24,429 dollars, with no capital committed and no price risk.

At the parity forward of 1.0652 the same chain returns exactly 1,050,000 and the profit is zero, which is what the formula means. In practice a small cross-currency basis has persisted since 2008, because the trade consumes bank balance sheet that is no longer free, so the arbitrage narrows the gap without closing it.

What the parity conditions do and do not predict

Work the second row once. With US inflation at 3 percent and euro area inflation at 2 percent, relative parity implies the dollar depreciates about 1.03/1.02 - 1 = 0.98 percent a year. Over one year that is invisible next to exchange-rate volatility ten times larger. Compounded over twenty years it is about 21 percent, which is exactly the horizon a terminal value covers. The core of it: parity conditions are a discipline on the forecast you write down, not a forecasting tool you can trade on.

Whether the firm should hedge at all

Lesson 7's logic transfers without modification. A shareholder who wants less euro exposure can sell euros herself, so a hedge inside the firm changes nothing she could not do more cheaply. Hedging earns its keep only through a friction, and the credible ones are specific: a convex tax schedule, Lesson 8's costs of financial distress, a covenant measured in the reporting currency, and Kenneth Froot, David Scharfstein and Jeremy Stein's argument that a firm hedges to keep internal cash available in the states where its investment opportunities are best.

Apple uses forward and option contracts on certain exposures. Notice what that buys: it fixes the dollar value of contracted flows over a few quarters. It does nothing about what a permanently stronger dollar does to the price of a phone in Frankfurt, and it is that second thing which sits in the valuation below.

Apple, end to end

Step one, the cash. Cash generated by operating activities was 110,543 million and payments for property, plant and equipment were 10,959 million.

Free cash flow = 110,543 - 10,959 = 99,584 million

Step two, the rate. Lesson 6's 9.86 percent, built from a 4.59 percent riskless rate, a beta of 1.20, a 4.6 percent equity premium and a 4.53 percent debt weight.

Step three, a forecast someone can argue with. Growth of 6, 5, 4, 3.5 and 3 percent, fading toward the economy.

Step four, the terminal value. At 2.5 percent perpetual growth, TV = 122,884 × 1.025/(0.0986 - 0.025) = 1,711,361, whose present value is 1,069,409. That single figure is 71.2 percent of the answer.

Step five, from the firm to the share. Enterprise value is 1,502,956. Apple held 162,099 million of cash and securities against 122,906 of debt, so add net cash of 39,193 to reach equity of 1,542,149. The cover page reports 15,552,752,000 shares outstanding on 20 October 2023.

1,542,149/15,553 = 99.16 dollars a share

The same cover page puts the market value of stock held by non-affiliates at 2,591,165 million on 31 March 2023, against the model's 1,542,149. The useful move is not to declare one of them wrong. It is to reverse the model and ask what the market's number requires. Holding the 9.86 percent rate and the five-year forecast fixed and solving for the perpetual growth rate that reproduces 2,591,165 gives 6.02 percent.

Six percent, forever, in nominal terms, from a company already selling 383 billion dollars a year into an economy growing more slowly than that. Either the growth is heroic, or the discount rate is too high, or both inputs are what Lesson 6 said they were: arguments.

Eighty dollars to a hundred and forty, on assumptions every one of which a competent analyst would defend in a meeting. In short: the output of a discounted cash flow model is a range whose width is set by two numbers nobody can measure, and a single point estimate hides that fact rather than resolving it.

What the number does not tell you

• Most of it is the part you cannot check. 71.2 percent of the value sits in a terminal value covering years for which no forecast exists, and one point of perpetual growth moves the share by roughly ten dollars.
• The cash flow flatters itself. Operating cash flow adds back 10,833 million of share-based compensation, a genuine cost to shareholders settled in shares rather than cash. Deduct it and the value falls to about 88.65, some 11 percent lower.
• One growth rate stands in for a geography. Greater China supplied 72,559 million of sales, and much of the manufacturing sits in the same jurisdiction. The model has no line for that; it has a percentage.
• A year of working capital is inside the first number. Payables and receivables swung by billions, and one year's operating cash flow inherits all of it.
• It cannot see what decides the outcome. Whether the next product works, what a regulator does to the margin on Services, whether the buyback pace holds. None of it appears in the arithmetic.

A valuation is a statement about the assumptions you were willing to defend, made auditable by writing them as numbers. That is considerable, and it is not a measurement of a company.

Common misconceptions

• "A DCF tells you what a share is worth." It tells you what your inputs imply. Two analysts with the same filing and defensible disagreements about beta and perpetual growth land 60 dollars apart on Apple, and neither has made an arithmetic error.
• "If the model says 99 and the market says more, the market is wrong." The market's price implies 6.02 percent perpetual growth at your discount rate. That is a hypothesis you can examine, and examining it is more useful than declaring anyone mistaken.
• "Hedging removes the exchange rate from a valuation." Forwards fix contracted flows over a few quarters. Economic exposure, which is what the present value of twenty years of foreign earnings depends on, survives every hedge in the book.
• "A firm should hedge because volatility is bad." Shareholders hedge more cheaply than firms do. Hedging pays only through a named friction: taxes, distress costs, covenants, or keeping cash available for investment.

Looking back

Seventeen lessons, one method. Discount every cash flow that the decision changes, at a rate that reflects its risk, and then say out loud what you assumed. Along the way: a 61 percent internal rate of return that should be refused, a beta of 1.20 with error bars wide enough to matter, a WACC of 9.86 percent whose whole content lives in two contested inputs, Modigliani and Miller proved by an arbitrage that pays 50,000 dollars, a payout record read against Lintner, Netflix profitable while burning 2.9 billion, a binomial tree that prices the right to wait, an LBO returning 17.9 percent from growth and deleveraging alone, Enron's 1.49 percent return on assets printed in its own 10-K, 3.9 billion dollars left on an offering table, and finally 99.16 dollars a share against a market saying 6.02 percent forever.

Bottom line: the arithmetic in this course is not difficult. Sourcing every input, and stating what the answer does not cover, is the difficult part, and it is the part that separates a valuation from a number.

Sources

1. Apple Inc. (2023). Annual report on Form 10-K for the fiscal year ended September 30, 2023. U.S. Securities and Exchange Commission. sec.gov
2. Board of Governors of the Federal Reserve System. (n.d.). Foreign exchange rates, H.10. federalreserve.gov
3. Federal Reserve Bank of St. Louis. (n.d.). U.S. dollars to euro spot exchange rate (DEXUSEU). FRED. fred.stlouisfed.org
4. Damodaran, A. (n.d.). Cost of capital by industry sector. NYU Stern School of Business. pages.stern.nyu.edu
5. Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.), Chapters 27 and 28. McGraw-Hill Education.