> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/financial-economics-claims-prices-holders/part-v-firms-as-issuers-of-claims/chapter_22_investment_valuation.md).

# Chapter 22: Investment, Valuation, and M\&A

*Part V: Firms as Issuers of Claims — Financial Economics: Claims, Prices, and Holders*

***

## Opening Episode: The Largest Merger Ever Announced

On Monday, 10 January 2000, America Online and Time Warner announced that they would combine. The deal was the largest ever announced: America Online would issue its own shares to Time Warner's holders at a rate of one and a half AOL shares for each Time Warner share, in a transaction valued at the time at roughly one hundred and sixty-five billion dollars. America Online's shareholders would own about fifty-five percent of the combined company, Time Warner's about forty-five.

The arithmetic behind that split is the first thing to notice. Time Warner owned Warner Bros., HBO, CNN, *Time*, *Sports Illustrated*, and the largest cable systems in the United States, and its annual revenue was several times America Online's. America Online sold dial-up internet access. Yet AOL's equity was worth more than Time Warner's, and it was worth enough more to buy Time Warner outright and keep control. What made that possible was not cash and it was not debt. It was the price of AOL's stock.

The market's immediate verdict was split in the way this chapter will show is typical. Time Warner's shares rose sharply on the announcement, on the order of forty percent — a premium of roughly that size handed to the target's holders in a single session. America Online's shares did not rise. They ended the day close to where they began and drifted lower over the following weeks, and by the time the deal closed on 11 January 2001, after a year of antitrust review, the currency that had paid for it had lost a large part of its value.

Then the accounting caught up. In 2002 AOL Time Warner wrote down the goodwill created by the merger in two installments and reported for the full year a net loss of approximately ninety-nine billion dollars — the largest annual loss reported by an American corporation to that date, and almost entirely a write-down of value the merger had been assumed to create. The company dropped "AOL" from its name in 2003; the two businesses were separated in 2009.

Three threads of this chapter are already visible in that sequence, and it is worth naming them before the machinery starts.

The **announcement return is the market's verdict**, delivered in a day, on a decision whose consequences take a decade. Section 22.5 shows that the pattern here — a large gain to the target, roughly nothing to the acquirer — is the central empirical regularity of the merger literature, and §22.7 asks the harder question of what that verdict actually measures.

**Stock financing is a signal about the currency, not only about the purchase.** A firm that pays with its own shares is issuing equity, and Chapter 12 §12.5 has already established what the market infers from an equity issue. Chapter 23 §23.4 develops the market-timing reading in full, and §22.5 records the fact that stock-financed acquirers do systematically worse.

And **hubris is the behavioral reading**. Chapter 15 §15.6 routed managerial overconfidence here, to sit next to the merger evidence it is trying to explain, and §22.5 takes it up.

Before any of that, though, the firm has to know what a project is worth and what its capital costs. That is where the chapter begins, and it is what Chapter 23 will consume.

***

## 22.1 Capital Budgeting

### Net present value, inside the firm

Chapter 3 §3.1 gave the present-value formula and then issued a warning: present value is an accounting identity that reorganizes prices, not a theory that generates them, and a discount rate typed into a spreadsheet cell launders an assumption into a valuation. Capital budgeting is that identity applied inside the firm. A project generating free cash flows $$\mathrm{FCF}\_t$$ and requiring an outlay $$I\_0$$ today is worth

$$
\mathrm{NPV} = -I\_0 + \sum\_{t=1}^{T}\frac{\mathrm{FCF}\_t}{(1+\mathrm{WACC})^t},
$$

and the rule is to accept every project with positive net present value.

The rule is not arbitrary. It follows from the linearity of the pricing function established in Chapter 3 §3.2: because prices add, the value of the firm is the sum of the values of its projects, and a manager can evaluate each one separately without solving the firm's whole problem. Value additivity is what makes decentralized capital budgeting possible at all, and it is also why the conglomerate discount is a puzzle rather than an accounting artifact.

Two disciplines make the identity honest. Cash flows must be *incremental* — with and without the project, counting working capital and cannibalization of existing products, excluding sunk costs. And flows and discount rate must be denominated consistently: free cash flow to the firm at the weighted average cost of capital, free cash flow to equity at the cost of equity, nominal at nominal.

### The internal rate of return and three traps

The **internal rate of return** is the discount rate that sets net present value to zero. It is popular because it is unit-free, comparable across projects of different sizes, and reportable as a single number to people who do not want to hear about discount rates. It is also wrong often enough to be dangerous, in three distinct ways.

**Scale.** Project A costs 100 and returns 150 in one year: an IRR of 50 percent. Project B costs 1,000 and returns 1,300: an IRR of 30 percent. At a 10 percent cost of capital, A adds 36.36 of value and B adds 181.82. The higher rate of return is on the smaller number, and a firm that can do only one and ranks by IRR gives up 145 of value. Problem 1 shows what repairs the ranking, and why almost nobody bothers.

**Timing.** A short project pays 120 and then 10 on an outlay of 100; a long one pays 10 and then 130. Their IRRs are 27.8 and 19.1 percent. Their net present values cross at 9.09 percent: below that rate the long project is worth more despite the lower IRR, above it the ranking flips. IRR is a summary statistic that has already assumed a reinvestment convention, and the convention is not the firm's cost of capital.

**Multiple roots.** Take cash flows of minus 100 today, plus 230 next year, and minus 132 the year after — an end-of-life cleanup cost produces this shape. Net present value is zero at *both* 10 percent and 20 percent, positive between them, negative outside. The project is worth doing at a 15 percent cost of capital and not at 5 percent, which no single "rate of return" can express. Descartes' rule of signs gives the condition: a stream changing sign more than once may have more than one root.

IRR is a reporting convention; NPV is the decision rule. Where the two disagree, the disagreement is information about the shape of the cash flows.

### The cost of capital in practice

The discount rate for a project financed by both debt and equity is the **weighted average cost of capital**,

$$
\mathrm{WACC} = \frac{E}{E+D}r\_E + \frac{D}{E+D}r\_D(1-\tau\_c),
$$

with $$E$$ and $$D$$ the *market* values of equity and debt, $$\tau\_c$$ the corporate tax rate, and the after-tax treatment of debt reflecting the deductibility of interest — the one piece of Chapter 23's machinery this chapter borrows in advance.

The cost of equity comes from Chapter 4 §4.5's security market line: $$r\_E = r\_f + \beta\_E(\mu\_M - r\_f)$$. Every term in it is a choice, and the choices are where most of the dispersion in practice comes from.

**The riskless rate.** Practice uses a long Treasury yield rather than a bill rate, on the grounds that the project's duration is long. This is defensible and it is not what the one-period CAPM says.

**The equity premium.** Historical realized premia over long US samples run around five to six percentage points; forward-looking estimates implied by dividend and earnings yields typically run lower, nearer four. Chapter 5's equity premium puzzle is the reason the two disagree, and the choice between them moves a high-beta firm's cost of equity by a full percentage point or more.

**Beta.** A regression beta on five years of monthly returns is noisy; practice shrinks it toward one, or unlevers an industry beta and relevers it to the firm. And the beta that belongs in a project's discount rate is the *project's*: a stable utility subsidiary inside a technology company should not be discounted at the parent's beta. Chapter 4 §4.7 adds a warning practice mostly ignores — the empirical security market line is flatter than the CAPM's, so the CAPM overstates the cost of capital for high-beta projects and understates it for low-beta ones.

**Table 22.1: A weighted average cost of capital, assembled**

| Input                            | Value     | Source of the number                       |
| -------------------------------- | --------- | ------------------------------------------ |
| Market value of equity           | 600       | Share price times shares outstanding       |
| Market value of debt             | 400       | Traded bond prices, or book value if close |
| Equity weight $$E/V$$            | 0.60      | Computed                                   |
| Riskless rate $$r\_f$$           | 4.0%      | Long Treasury yield                        |
| Equity premium $$\mu\_M - r\_f$$ | 5.0%      | Long-run historical average, chosen        |
| Equity beta $$\beta\_E$$         | 1.20      | Industry beta, relevered                   |
| Cost of equity $$r\_E$$          | 10.0%     | CAPM: $$4.0 + 1.20\times 5.0$$             |
| Pre-tax cost of debt $$r\_D$$    | 6.0%      | Yield on the firm's own bonds              |
| Corporate tax rate $$\tau\_c$$   | 21%       | Statutory                                  |
| After-tax cost of debt           | 4.74%     | $$6.0\times(1-0.21)$$                      |
| **WACC**                         | **7.90%** | $$0.60\times 10.0 + 0.40\times 4.74$$      |

*Source: Author's calculation. Values in the first two rows are in units of currency; the weights and rates are computed from them.*

![Figure 22.1: The cost of capital and leverage](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-d9da191c2c487bc64dda474cec68d31f3e45f6e3%2Ffig_22_01_cost_of_capital_and_leverage.png?alt=media)

**Figure 22.1: The cost of capital and leverage.** The firm of Table 22.1, with its leverage allowed to vary. The model is calibrated to the table rather than the other way round: the unlevered cost of capital is backed out of the identity WACC = r\_U(1 − tau\_c D/V) at the table's own D/V of 0.40, which puts r\_U at 8.62 percent — and that same r\_U then returns the table's cost of equity, 10.0 percent, through Proposition II, to within a basis point. Without taxes the weighted average is flat and the cost of equity rises exactly enough to keep it so (Chapter 23 §23.1). With the interest deduction the weighted average tilts down. Add a convex cost of distress and it turns, giving the interior optimum the trade-off theory of Chapter 23 §23.2 wants — drawn here at half debt, a placement chosen for the drawing and carrying no empirical content. The horizontal rule on top is the fact that ought to unsettle a reader who has just done the arithmetic: the hurdle rate firms report applying sits several percentage points above any of these curves, and barely moves when rates do. *Source: Author's calculation, calibrated to Table 22.1.*

Figure 22.1 takes that firm and lets its leverage vary, which is the picture Chapter 23 §§23.1-23.2 will build from first principles. Now the survey fact that ought to unsettle anyone who has just done that arithmetic. Graham and Harvey's survey of chief financial officers established that the CAPM is by a wide margin the dominant method firms use to set the cost of equity, so the machinery above is descriptive rather than normative. The subsequent survey literature established something less comfortable: the **hurdle rate** firms actually apply to projects commonly exceeds their own estimated weighted average cost of capital by a substantial margin — several percentage points is a routine finding — and it moves very little when interest rates move. Firms that revised their cost-of-capital estimates downward through the long decline in rates after 2008 largely did not revise their hurdle rates.

The candidate explanations are worth listing because they are not the same claim. Capital rationing makes the hurdle rate a shadow price on a scarce internal budget rather than a cost of capital. Agency: a high hurdle corrects for the optimism of divisional managers who submit the forecasts. Real options: waiting has value, and a high hurdle is a crude way of charging for it (§22.3). And inattention: the number in the policy manual is old. Discussion Question 1 asks which of these the stickiness evidence distinguishes.

***

## 22.2 Valuation

### Discounted cash flow, and where its value actually lives

A discounted-cash-flow valuation of a whole firm has two parts: an explicit forecast over a horizon of five to ten years, and a **terminal value** standing in for everything after it. The terminal value is normally a growing perpetuity on the last forecast flow,

$$
\mathrm{TV}\_T = \frac{\mathrm{FCF}\_T(1+g)}{\mathrm{WACC}-g},
$$

or an exit multiple applied to a terminal-year earnings measure. Chapter 3 §3.1's growing perpetuity is the whole of the formula; the interesting question is how much of the answer it carries.

Table 22.2 answers it for a firm whose free cash flow starts at 100, grows at six percent through a five-year explicit forecast, and is discounted at eight percent.

**Table 22.2: Where a discounted-cash-flow value comes from**

| Terminal growth $$g$$ | Terminal value at year 5 | Its present value | Total firm value | Terminal value as share of total |
| --------------------- | ------------------------ | ----------------- | ---------------- | -------------------------------- |
| 1.5%                  | 1,971                    | 1,342             | 1,788            | 75.0%                            |
| 2.0%                  | 2,146                    | 1,461             | 1,907            | 76.6%                            |
| 2.5%                  | 2,353                    | 1,601             | 2,047            | 78.2%                            |
| 3.0%                  | 2,601                    | 1,770             | 2,216            | 79.9%                            |

*Source: Author's calculation. Free cash flows of 100, 106, 112.36, 119.10 and 126.25 over five years, discounted at a WACC of 8 percent, giving an explicit-horizon present value of 446.1 in every row.*

Figure 22.2 runs the same valuation over the whole grid. Three quarters to four fifths of the answer is the terminal value, and the terminal value is two assumptions. Vary the discount rate instead and the point sharpens: at a WACC of seven percent the same forecast is worth 2,509 and at ten percent it is worth 1,494, a spread of sixty-eight percent produced entirely by three percentage points of discount rate. Extending the explicit forecast to ten years reduces the terminal value's share to about sixty-three percent, which relabels the problem rather than solving it — years six through ten are themselves mostly extrapolation.

![Figure 22.2: Where DCF breaks](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-e4934de5a717cb4bfd88cbe6fd74a2ae7eba6994%2Ffig_22_02_where_dcf_breaks.png?alt=media)

**Figure 22.2: Where DCF breaks.** Panel (a): the whole valuation over the two assumptions that carry it, with Table 22.2's base case outlined. The explicit five-year forecast contributes 446 in every cell; everything else in the grid is the terminal value. Across a plausible range of the two inputs the answer runs from about 1,100 to about 5,400 — a factor of five, from a forecast nobody disputes. Cells where the discount rate is within half a point of the growth rate are left blank, because the perpetuity formula is not meaningful there. Panel (b): the share of the answer the terminal value carries, against terminal growth, at three discount rates and at two forecast horizons. Doubling the explicit forecast to ten years takes the share from about four fifths to about two thirds, which relabels the problem rather than solving it: years six through ten are themselves extrapolation. *Source: Author's calculation from the forecast of §22.2 and Table 22.2.*

Two disciplines keep the terminal value honest. The perpetual growth rate cannot exceed the long-run nominal growth rate of the economy, because a firm growing faster than the economy forever eventually becomes the economy. And growth must be paid for: a firm growing at $$g$$ forever must reinvest, so the terminal free cash flow implied by a given growth rate has to be consistent with a plausible return on new capital. A terminal value built on high growth and no reinvestment is the most common error in the genre.

### Multiples, and what each one assumes

A multiple prices a firm by reference to what the market pays for comparable firms. It is not an alternative to discounting; it is a discounted-cash-flow model with the assumptions compressed into a single ratio and hidden. The useful skill is knowing which assumption each ratio has hidden.

**Table 22.3: What each multiple embeds**

| Multiple                  | Identity behind it                                     | What it assumes about the comparables                                                                            |
| ------------------------- | ------------------------------------------------------ | ---------------------------------------------------------------------------------------------------------------- |
| Price / earnings          | $$P/E\_1 = \text{payout}/(r\_E - g)$$                  | Same growth, same payout ratio, same cost of equity, same leverage and tax position                              |
| Enterprise value / EBITDA | EV over a pre-interest, pre-tax, pre-depreciation flow | Same growth and risk; neutral to capital structure, and to depreciation policy only if capital intensity matches |
| Enterprise value / sales  | EV over revenue                                        | Same growth, risk, *and* margins — the strongest assumption on the list                                          |
| Price / book              | Market-to-book equity                                  | Same return on equity relative to $$r\_E$$; this is Tobin's Q for the equity claim (§22.4)                       |

*Source: Author's construction. The price-earnings identity is the growing-perpetuity formula of Chapter 3 §3.1 divided by next year's earnings.*

Read the first row carefully and the method's fragility is visible. A price-earnings ratio of twenty is the statement that payout divided by $$(r\_E - g)$$ equals twenty — a joint claim about three unobservables. Two firms with identical earnings can justify multiples differing by a factor of three on differences in growth and risk that no screen will surface.

### Where each breaks

Discounted cash flow breaks in the terminal value and in the discount rate, the two inputs nobody observes. Multiples break in the comparable set, and they break circularly: a valuation by multiples is only as good as the market's pricing of the comparables, so the method cannot detect a mispricing common to the industry — exactly the mispricing that matters in a valuation wave (§22.5).

That circularity is where valuation meets Chapter 7's efficiency debate. The dispersion in analysts' published values for the same firm — routinely a factor of two across a large-cap name with a full sell-side following — is not evidence that most analysts are incompetent. It is what Table 22.2's sensitivity looks like when different people make the two assumptions.

***

## 22.3 Real Options

A positive-net-present-value project should be undertaken now. A project with a *negative* net present value today should be abandoned. Both statements are false when the decision can be postponed, because the right to invest later is an option, and an option is worth something.

Three kinds appear in almost every capital budget. The **option to wait** — invest now or observe another year of demand first. The **option to expand** — a small plant today that carries the right to build a large one if the market develops, which is why loss-making pilot projects can be worth their cost. The **option to abandon** — a put on the project, struck at the salvage value.

Chapter 8 §8.3's machinery prices these directly, with one substitution: the underlying is not a traded stock but the present value of the project's cash flows, and the strike is the investment outlay.

Take a project whose cash flows have a present value today of 100, and which can be built for an outlay of 90, now or in one year. The present value will be either 130 or 80 in a year — $$u = 1.30$$, $$d = 0.80$$ — and the riskless gross return is $$R\_f = 1.05$$. Then

$$
\pi^{\ast} = \frac{R\_f - d}{u-d} = \frac{1.05-0.80}{0.50} = 0.50,
$$

and the value of the right to invest in a year is

$$
\frac{0.50\times(130-90) + 0.50\times 0}{1.05} = \frac{20}{1.05} = 19.05.
$$

Investing today has a net present value of 10. Waiting is worth 19.05. The option premium of 9.05 is the value of not committing an irreversible outlay before the bad state is ruled out, and a firm applying the naive rule captures barely half of what the project is worth.

The comparative statics are the reason the section exists, and Figure 22.3 is the whole family. Widen the uncertainty to $$u = 1.50$$, $$d = 0.70$$ and the waiting value rises to 25.00; narrow it to $$u = 1.15$$, $$d = 0.90$$ and it falls to 14.29. Option value rises with uncertainty, and it is worth having only if the commitment is hard to reverse. Dixit and Pindyck's synthesis is exactly this combination: **irreversibility plus uncertainty plus the ability to delay** is what makes real-option value material, and where any one of the three is absent the naive rule is close enough.

![Figure 22.3: Real options](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-374bd524c7c253282f0b2414fbcc8650cb5496a9%2Ffig_22_03_real_options.png?alt=media)

**Figure 22.3: Real options.** The value of the right to invest in a year, against uncertainty, for the project of this section: present value 100, outlay 90, riskless gross return 1.05. The shaded band is the deferral region — where the option is worth more than the ten that investing today delivers, which is everywhere on this axis. Along the curve u and d are centred so that the risk-neutral probability stays at one half, so the only thing moving is uncertainty; the chapter's two variants move d as well and therefore sit slightly off it, which is why they are drawn as open markers. Two things are worth reading off the left edge. Option value rises with uncertainty, which is the comparative static the section exists for. And the premium does not go to zero when uncertainty does: with no uncertainty at all the right is still worth 4.29, the interest saved by deferring the outlay a year — an artifact of the assumption that waiting is free, which is the first of the section's two qualifications and what Problem 4 restores. *Source: Author's calculation from the binomial of §22.3.*

Two honest qualifications. Waiting is not free — a year of delay usually forgoes a year of cash flow, which enters the valuation the way a dividend enters an American call and can make immediate exercise optimal; the example above omits it, and Problem 4 restores it. And competition erodes the option: a right to invest that a rival also holds is a shared option, and shared options are worth less than exclusive ones.

Finally, the organizational fact. Formal real-option pricing is rare in practice: surveys consistently find only a modest minority of firms reporting its use, far behind net present value and payback, and the share has not grown much in twenty years. The reasons given are that the inputs — the volatility of a non-traded present value, above all — are not observable, and that a technique which can rescue any rejected project is a governance hazard. Its durable contribution has been conceptual: it explains why the hurdle-rate wedge of §22.1 is not simply an error, and why firms hold undeveloped land, expiring patents, and idle capacity that no discounted-cash-flow model justifies.

***

## 22.4 Q, Investment, and Asset Prices

### The ratio and the theory

**Tobin's Q** is the market value of the claims on a firm divided by the replacement cost of its capital:

$$
\mathrm{Q} = \frac{E + D}{\text{replacement cost of the firm's assets}}.
$$

*(Notation: this book sets Tobin's Q in roman upper case, because the literature's lower-case italic q collides with Chapter 3 §3.4's state price* $$q\_s$$ *— live in this chapter, since §22.3's risk-neutral probability is built from it. See the notation registry.)*

Tobin's argument was that this ratio is the sufficient statistic for investment. If Q exceeds one, a dollar spent installing capital creates more than a dollar of market value, and the firm should spend it; if Q is below one, the firm should let its capital depreciate or sell it. Investment continues until the ratio is driven back to one.

Stated as a firm's optimization the argument is tighter and gives the modern form. Let $$K\_t$$ be the capital stock and $$I\_t$$ investment, and suppose installing capital carries a convex adjustment cost, so that adding capital quickly costs more per unit than adding it slowly. The firm invests until the marginal cost of an installed unit equals its marginal value:

$$
1 + \frac{\partial(\text{adjustment cost})}{\partial I\_t} = \mathrm{Q}^{\text{marg}},
$$

where $$\mathrm{Q}^{\text{marg}}$$ is the market value of one additional unit of installed capital. With a quadratic adjustment cost this inverts into a linear investment rule: the investment rate $$I\_t/K\_t$$ is proportional to $$\mathrm{Q}^{\text{marg}} - 1$$. Investment is a function of one forward-looking price, and that price is set in the asset market. This is the joint the chapter is built around: the discount rate of §22.1 and the asset prices of Part II are the same object seen from two sides.

### Average versus marginal, and the empirical record

The theory is about marginal Q. What a researcher can compute is **average Q** — the market value of all the firm's claims over the replacement cost of all its capital. Hayashi (1982) established the conditions under which the two coincide: constant returns to scale in both production and adjustment costs, and a price-taking firm. Those conditions are strong. Market power, decreasing returns, or capital that is not homogeneous all drive a wedge, and the wedge is not signed in general.

Measurement adds more. Replacement cost has to be estimated from book values with a perpetual-inventory adjustment; the market value of debt is often unobserved; and the denominator omits intangible capital — brands, software, organizational capital, accumulated research — which has grown from a minor item to a large share of the corporate capital stock. A firm whose capital is mostly intangible therefore has a mechanically high Q, which is much of why measured aggregate Q has trended upward.

The empirical record is accordingly mixed. Regressions of the investment rate on average Q do produce a positive coefficient, but the coefficient is small — implying adjustment costs so large that the capital stock would take decades to adjust — and the fit is poor.

![Figure 22.4: Tobin's Q and investment](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-a082468d44ec36478f419cfc8b385b3de5ed9256%2Ffig_22_04_tobins_q_and_investment.png?alt=media)

**Figure 22.4: Tobin's Q and investment.** The aggregate version of the same failure, from the Financial Accounts. Q — the market value of the sector's claims over the replacement cost of its assets — runs from 0.55 in the late 1940s to 1.51 today, a factor of nearly three, with a trough in the early 1980s and a peak in the late 1960s that the theory says should have produced an investment boom. The investment rate moves between about eleven and fifteen percent of GDP and does not follow: the correlation in levels is 0.10 and in four-quarter changes it is *negative*. A sufficient statistic that explains nothing at the aggregate level is a finding rather than a nuisance, and §22.4's three readings of it are all live. If adjustment costs are the answer they must be enormous. If financing constraints are, the ratio is not sufficient. And if measurement is, then the denominator is the problem — a corporate capital stock that is increasingly intangible is systematically understated at replacement cost, which pushes Q up over time without any change in the incentive to invest, and the upward drift in the blue line is exactly what that would look like. *Source: Financial Accounts of the United States (Z.1), table B.103, and the Bureau of Economic Analysis via FRED; Q is the market value of equity plus total liabilities over total assets at current cost; author's calculations.* Worse for the theory, **cash flow enters those regressions significantly** and stays significant after Q is controlled for, which a sufficient statistic should not permit: nothing else about the firm should forecast its investment. Fazzari, Hubbard and Petersen read that as evidence of financing constraints; Kaplan and Zingales contested both the sorting and the interpretation, and the debate has not closed. A third reading is measurement rather than economics: if average Q is a noisy proxy for marginal Q, its coefficient is attenuated and any correlated variable — cash flow, prominently — absorbs the residual explanatory power. Erickson and Whited's measurement-error-consistent estimators find exactly that pattern, with Q performing far better and cash flow's significance largely disappearing. That relocates the argument rather than settling it, since those estimators are demanding and their results sensitive to assumptions.

### The investment CAPM: what the factors of Chapter 6 look like from the firm's side

Chapter 6 §6.2 promised that the rationalization of the profitability and investment factors would come from the firm's side and would be developed here. This is the promise.

Start from the firm's first-order condition rather than the investor's. The firm invests until the marginal cost of investment equals the present value of the marginal unit's payoff, discounted at the rate investors require. Rearranged, that condition says the **investment return** — the payoff on the marginal unit of capital divided by its marginal cost — must equal the expected return on the firm's securities. This is a pricing equation with no preferences in it: the discount rate appears because the firm faces it, not because a representative agent has one.

Two comparative statics fall straight out, and they are the two factors. Hold expected profitability fixed and raise investment: a firm invests more when the marginal cost of doing so is low relative to the payoff, which is to say when the discount rate applied to that payoff is low. So **high investment implies low expected returns** — the investment factor, conservative-minus-aggressive earning a premium because the aggressive investors are the low-discount-rate firms. Now hold investment fixed and raise expected profitability: if a firm invests the same amount while expecting to earn more from it, the discount rate reconciling the two must be higher. So **high profitability at fixed investment implies high expected returns** — the profitability factor.

Hou, Xue and Zhang built a factor model on precisely these two, alongside the market and size, and showed that it absorbs a long list of documented anomalies, which made the q-factor model a standard benchmark alongside Fama and French's five factors. The two were derived from opposite ends and largely agree.

Read the argument once more for what kind of claim it is. It does not say investors fear investment or love profitability. It says firms optimize against the discount rates investors set, so the cross-section of expected returns leaves an imprint on the cross-section of corporate behavior — and a sort on corporate behavior recovers the discount rates. The profitability and investment factors are what optimal investment looks like from the asset side, which is the strongest available answer to the "which factors?" question the APT left open in Chapter 6 §6.1.

![Figure 22.5: The investment CAPM joint](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-cf94053bc21fef7db5ded7b0d1df510b7868ed15%2Ffig_22_05_investment_capm_joint.png?alt=media)

**Figure 22.5: The investment CAPM joint.** The whole derivation on one page. The firm's first-order condition is at the top, and the first thing to notice is what is missing from it: no utility function, no representative agent, no risk aversion. The discount rate appears because the firm faces it. Below it are the two comparative statics, each holding one of the firm's two choices fixed and varying the other, and each ending in a statement about expected returns that is a factor in Chapter 6. What makes the argument strong is the direction it runs. Every other answer to "which factors?" starts from the investor and asks what she fears; this one starts from the firm's optimization and derives the cross-section of returns as the thing firms optimize *against*. A sort on corporate behaviour then recovers the discount rates, which is why a factor built out of accounting variables can price assets at all. What it does not deliver is the level: like every no-arbitrage argument in this book, it says what must be true of the cross-section given the discount rates, not where the discount rates came from.

At the aggregate level the same ratio is observable in Chapter 2 §2.3's Financial Accounts, and the data exercise plots its history against aggregate business investment.

***

## 22.5 Mergers and Acquisitions

### Waves

Merger activity is not a steady flow. It arrives in **waves** — a burst around the turn of the twentieth century, one in the 1920s, the conglomerate wave of the 1960s, the hostile and bust-up wave of the 1980s, the stock-financed wave of the late 1990s that AOL-Time Warner capped, and further peaks in the mid-2000s, in 2015, and in 2021.

Two explanations compete and both have support. The **industry-shock** account, from Mitchell and Mulherin and developed by Harford, is that waves cluster within industries and follow identifiable shocks — deregulation, a technological change, an input-price move — that make the existing allocation of assets wrong. On this reading waves are the reallocation mechanism doing its job, and Harford adds the condition that the reallocation happens only when capital-market liquidity permits it.

The **valuation** account, from Shleifer and Vishny and from Rhodes-Kropf and Viswanathan, is that waves happen when equity is expensive. An overvalued acquirer buys a less overvalued target with stock, and the transaction is rational for the acquirer's managers even if it creates nothing, because it converts temporarily overvalued paper into real assets. This account predicts what the data show: waves coincide with high market valuations, and stock is the dominant currency precisely at the peaks. It also explains the opening episode without any reference to synergies at all.

### Announcement returns

The event-study evidence is the most replicated result in corporate finance.

**Table 22.4: Abnormal returns around merger announcements, US deals**

| Window                                | Target | Acquirer                                | Combined, value-weighted |
| ------------------------------------- | ------ | --------------------------------------- | ------------------------ |
| Three days around announcement        | \~+16% | \~−0.7% (not distinguishable from zero) | \~+1.8%                  |
| Twenty days before through completion | \~+24% | \~−4%                                   | \~+2%                    |

*Source: Approximate magnitudes as summarized by Andrade, Mitchell and Stafford (2001) for large samples of US deals from 1973 to 1998. Figures are rounded; the acquirer point estimates are small relative to their standard errors in the short window and are not in the long one.*

Three statements the table supports, in descending order of confidence.

**Targets gain, and gain a lot.** A premium in the range of thirty to fifty percent over the pre-announcement price is the norm for a completed deal, and the target's announcement return capitalizes the market's estimate of that premium times the probability the deal closes.

**The combined effect is positive but small.** Weighting the two firms by market value, the merger announcement adds on the order of one to two percent to their combined value. Mergers create value on average. They create much less of it than the premium paid suggests, because most of the premium is a transfer.

**Acquirers, on average, roughly break even or lose slightly** — the **acquirer-loses** regularity, which needs its caveat attached every time it is stated. The mean is a mean: the distribution is wide and roughly centered near zero, many acquirers gain, and the aggregate dollar losses documented for 1998-2001 by Moeller, Schlingemann and Stulz — running into the hundreds of billions of dollars — were driven by a small number of very large deals, not by the typical transaction. Relative size also makes the evidence asymmetric in power: a one percent move in a large acquirer's equity is a larger sum than a sixteen percent move in a target an eighth its size, so the acquirer's return is at once economically decisive and statistically noisy. Problem 6 works the arithmetic.

### Method of payment

Split the sample by how the deal is paid for and the acquirer result sharpens considerably. Cash-financed acquirers do roughly fine at announcement and afterward. **Stock-financed acquirers do worse at announcement, and worse again over the following three to five years** — a result documented by Loughran and Vijh and by Rau and Vermaelen and robust across specifications, though subject to Chapter 12 §12.4's warnings about long-horizon abnormal returns.

The standard reading is Chapter 12 §12.5's, transplanted. Paying with stock is issuing equity, and a manager issues equity when it is dear; the market applies the Myers-Majluf markdown to the acquirer's shares, and the subsequent underperformance is the correction of the overvaluation that motivated the deal. Chapter 23 §23.4 develops the market-timing theory this belongs to, and Chapter 23 §23.3 the Myers-Majluf model the markdown comes from. The competing reading is that stock is a risk-sharing device — the target's holders keep exposure to the combined firm, which is efficient when the acquirer is genuinely uncertain about the target's value — and both are probably operating. The cleanest way to see the difference: risk-sharing predicts stock is used when the *target* is hard to value, timing predicts it is used when the *acquirer* is expensive, and the wave evidence supports the second.

### Managerial hubris

Chapter 15 §15.6 routed managerial overconfidence to this chapter. Here is why it belongs next to Table 22.4.

Roll's hubris hypothesis is the winner's curse of Chapter 12 §12.3 applied to the bidding CEO. Suppose the target's market price is an unbiased estimate of its standalone value, and suppose bidders form their own valuations with error. The bidder who wins is the one whose valuation was highest — which is to say, the one whose error was most positive. Bidding at all requires believing the market's price is wrong; winning the auction is evidence that you were the most wrong. The hypothesis needs no dishonesty and no agency conflict: it needs only managers who do not apply the winner's curse to themselves.

Its predictions are sharp and testable. Acquirer losses should approximately offset target gains; the combined gain should be about zero; and acquirer announcement returns should be worse for managers with more reason to be overconfident. The first two are close to right and not exactly right — the combined effect in Table 22.4 is positive, which says hubris is a component of the total and not all of it.

The third has been tested with the evidence Chapter 15 §15.3 assembles. Malmendier and Tate identified overconfident chief executives by their willingness to hold deep in-the-money options on their own firms far past the point diversification would counsel — a revealed belief that the stock will keep rising. Those executives make more acquisitions, the excess is concentrated in diversifying deals with the weakest strategic rationale, they prefer internal funds to the discipline of an external market, and the reaction to their bids is significantly more negative. Overconfidence is the mechanism behind hubris, and it is measurable.

### Synergies and their base rates

Announced deals come with an announced synergy number, and it is almost always a cost number: duplicate facilities closed, procurement consolidated, headcount reduced. Cost synergies of that kind are realized more reliably than revenue synergies — cross-selling, combined product roadmaps, "platform" arguments — which depend on customer behavior the acquirer does not control and which the post-merger literature finds are met far less often than promised.

The decision-relevant fact is not whether synergies exist but who captures them. The premium capitalizes the acquirer's own estimate of the synergies into the price handed to the target's holders at closing, so a deal that achieves precisely its announced synergies and pays a premium equal to their present value creates value for the target's shareholders and none for the acquirer's. That is the arithmetic behind the acquirer-loses regularity, and it requires no manager to have been wrong about the operations.

> **Box 22.1 — The merger-review clock**
>
> The opening episode compresses a year of antitrust review into a phrase. The machinery behind the phrase is a statutory clock, and because merger arbitrage prices the probability that a deal closes, the clock is what the spread is a term structure of.
>
> The Hart-Scott-Rodino Antitrust Improvements Act of 1976 requires parties to a transaction above a size threshold — adjusted annually — to notify the Federal Trade Commission and the Antitrust Division of the Department of Justice, and then to wait. The initial waiting period is thirty days for most transactions and fifteen for cash tender offers. If neither agency acts, the period expires and the parties may close. Nothing is approved; the clock simply runs out.
>
> The instrument that stops it is the **second request**. Either agency may issue a request for additional documentary material, which suspends the clock until the parties have substantially complied — and substantial compliance on a large transaction means millions of documents and many months. Only after certification does a further short period run, at the end of which the agency must either sue to enjoin or let the deal go. Second requests are rare in absolute terms and concentrated in exactly the transactions the market is unsure about.
>
> Three consequences for §22.7. The distribution of closing dates is not smooth: it has mass at the expiry of the initial period and a long tail beyond a second request. The arbitrageur's position is therefore an exposure to a date as well as to an outcome, and a deal spread that looks generous annualised over three months looks thin annualised over eighteen. And because the agencies' decision is a discrete legal act by an identifiable body, a merger spread is one of the few prices in this book whose fundamental is a probability that a specific committee will do a specific thing.
>
> Chapter 11 §11.6 would call the arbitrageur a liquidity supplier to shareholders who do not want the risk. What he is being paid for is the clock.

***

## 22.6 The Takeover Market as Governance

There is a second reason to buy a company, and it has nothing to do with synergies: the company is being run badly, and the acquirer intends to run it better or break it up. Manne's argument was that a market for corporate control is the sharpest disciplinary device shareholders have. A management team that wastes resources depresses the share price; the depressed price is an invitation; the invitation is accepted by someone who profits from the difference. The threat operates on every management team whether or not any bid arrives.

The 1980s were the exhibit. Hostile tender offers, financed with high-yield debt, targeted firms with the profile Jensen's free-cash-flow theory identified: mature businesses generating cash in industries with poor reinvestment opportunities, whose managers were reinvesting anyway. Oil was the canonical case. Conglomerates assembled in the 1960s were bought and sold in pieces, and Bhagat, Shleifer and Vishny's evidence is that the gains came substantially from reversing that diversification and returning divisions to buyers in their own industries — the strongest empirical case that takeovers do governance work.

The response was defensive and effective. The **poison pill** — a shareholder rights plan that issues cheap shares to everyone except a bidder crossing an ownership threshold, making an unnegotiated purchase prohibitively dilutive — was devised in the early 1980s and validated by the Delaware courts in 1985. State antitakeover statutes followed. The pill alone is not decisive, since a bidder can run a proxy contest, elect a new board, and have the pill redeemed. What makes it decisive is the **staggered board**, which forces a bidder to win two annual meetings a year apart: Bebchuk, Coates and Subramanian's finding is that the two are complements, and that the combination roughly doubles the odds a target stays independent while reducing the premium received by targets that are acquired.

The entrenchment cost is the other side. Gompers, Ishii and Metrick built a governance index from takeover defenses and charter provisions and reported that firms with the strongest shareholder rights outperformed those with the weakest over the 1990s; Bebchuk, Cohen and Ferrell narrowed the result to six provisions that do the work, the entrenchment index. The honest footnote, which the authors themselves supplied, is that the *return* association did not persist out of sample — plausibly because the market learned to price governance once it had been measured — while the association between entrenchment and firm value and operating performance has held up better. State the finding at that strength: defenses are associated with lower value and worse outcomes, not with a standing arbitrage.

Hostile activity declined sharply after 1990 and has never returned to 1980s levels, for four reasons that compound. Defenses became universal and legally secure. The high-yield market that financed the bust-up deals collapsed in 1989-90. Bidders learned that negotiating with a board is cheaper than fighting one, so control transactions continued in friendly form — the opening episode is a friendly deal. And internal governance substitutes arrived: independent boards, equity-based compensation, blockholders, and activist funds that buy a small stake and demand change rather than buying the company. Chapter 24 §24.3 treats those substitutes.

Two modern qualifications belong here. Staggered boards have been largely dismantled at the largest US firms under shareholder pressure, while pills remain available to be adopted overnight if a bidder appears — the "shadow pill," a defense present without being visible in any charter. And approval of a friendly deal now runs through holders whose votes are cast by policy, which §22.7 takes up.

Which is where the opening episode resolves. AOL-Time Warner was not a disciplinary takeover and was never contested. It was a friendly, stock-financed deal at the top of a valuation wave, paid for with a currency the wave had created, approved by both boards and both shareholder bases, and destroyed roughly one hundred billion dollars of recorded value within three years. Nothing in the governance apparatus of §22.6 was designed to stop it, because none of that apparatus asks whether a bid is a good idea. It asks who decides.

> **Box 22.2 — Gulf Oil, 1983-84**
>
> Section 22.6 argues that the takeover market disciplines free cash flow. The argument was made in the 1980s about a specific set of firms, and Gulf Oil is the case that made it concrete.
>
> Gulf was one of the largest oil companies in the world and had the problem the free-cash-flow theory describes: substantial cash generation, a reinvestment programme whose returns the market did not believe, and a share price well below what the reserves in the ground appeared to be worth. In 1983 an investor group led by T. Boone Pickens of Mesa Petroleum accumulated a stake and pressed for a restructuring — a royalty trust that would have distributed the cash flow from producing properties directly to shareholders rather than leaving it inside a corporation to be reinvested.
>
> Gulf's management resisted, and the resistance is the part that matters. The dispute was not about whether the reserves were valuable. It was about who would decide what to do with the cash they generated. Gulf sought a friendly alternative and in March 1984 agreed to be acquired by Standard Oil of California for approximately $13.2 billion, then the largest merger ever completed.
>
> Two readings, and this chapter takes both. On the free-cash-flow reading, the threat worked exactly as Jensen described: a firm with cash and poor investment opportunities was forced to distribute value to shareholders, and the mechanism was the market for corporate control rather than the board. On the reading the same chapter is obliged to record, the acquirer paid a very large premium in a deal whose subsequent returns were mixed, which is §22.5's evidence that the target's shareholders capture most of the gain.
>
> The episode is also the beginning of an era. Figure 22.6's first peak in merger activity is this one, and the leveraged transactions of the later 1980s that Chapter 23's Box 23.2 closes were financed in the market Gulf's era opened.

***

## 22.7 What Does the Market's Verdict at Announcement Actually Measure?

Every claim in §22.5 rests on an abnormal return measured over a few days. Chapter 7 §7.3 explains why that is the sharpest test available: over a three-day window the expected return is a few basis points, so the joint-hypothesis problem shrinks until it cannot account for a sixteen percent move, and the announcement of a merger is a discrete, dated, exogenously timed public event of exactly the kind the method was built for. Merger announcement returns are the best-behaved event study in corporate finance.

They are also, read through Part IV's lens, a price set by a particular and peculiar set of holders. Three complications follow, and each of them changes what the number means rather than whether it is measured correctly.

**The target's post-announcement price is set by merger arbitrageurs.** Within hours of an announcement the target's shareholder base turns over: long-only holders sell into the premium, and the marginal holder becomes a specialized, levered fund holding the target against a short position in the acquirer and earning the residual spread if the deal closes. The target's price is therefore not "the market's estimate of the target's value" but a completion-probability-weighted price quoted by the arbitrage capital of Chapter 18 §18.4 — capital that is redeemable, prime-broker financed, and withdrawn in precisely the states where deals break. Mitchell and Pulvino's characterization is the one to carry: the merger-arbitrage return profile resembles a written put on the market, flat in normal conditions and sharply negative in falling ones, which is a statement that the spread widens when arbitrage capital is impaired. A widening spread is then partly a repricing of deal risk and partly a repricing of the capital that bears it, and the event study cannot tell them apart.

**The vote is cast by mandate.** A merger requires shareholder approval, and a large and growing share of the votes is cast by index funds that cannot sell and vote nearly everything they hold, largely in line with the recommendations of two proxy advisers (Chapter 17 §17.7). The completion probability the market prices is therefore in part a forecast of a published stewardship policy. That is a different object from a forecast of what a dispersed body of owners thinks about the deal, and it is one reason arbitrage spreads on large-cap deals compress so quickly once the advisers report.

**The acquirer's return conflates two announcements.** A stock-financed bid tells the market two things at once: what the acquirer thinks the target is worth, and that the acquirer is willing to issue its own equity at today's price. A fall in the acquirer's stock may price the deal's net present value as negative, or it may be the market revising its estimate of the acquirer's *standalone* value downward on the issuance signal, exactly as Chapter 12 §12.5's seasoned-offering markdown does. The two are observationally similar, and they carry opposite implications: under the first, blocking the deal restores the lost value; under the second, the lost value was never there. Separating them requires variation in the deal that is unrelated to the acquirer's valuation — comparisons of completed with exogenously withdrawn bids are the standard approach — and Chapter 23 §23.4's market-timing evidence is the reason the second channel cannot be assumed away.

The synthesis is the one this part keeps arriving at. An announcement return is a change in the price at which a particular population of constrained holders is willing to hold a claim, measured over the days in which that population is itself turning over. It is an excellent measurement of that. It is not, without further argument, a measurement of whether the merger was a good idea — and the gap between the two is exactly the gap Part IV spent seven chapters describing.

***

## 22.8 Who Holds the Firm's Capital, and What Their Constraints Do to Its Price

This chapter's central price is a discount rate, and §22.1 assembled it out of choices rather than observations. Part IV supplies the question those choices are dodging: a required return is required by somebody, and the somebody is whoever is marginal in the firm's securities.

Read the weighted average that way and every term becomes a holder. The cost of equity is the return demanded by whoever is currently pricing the shares — index funds that cannot sell, an active complex facing redemptions, a household with an embedded tax basis. The cost of debt is the yield at which insurers whose capital charge is nearly flat inside investment grade (Chapter 10 §10.9), pension plans buying duration against their liabilities (Chapter 16 §16.2), and collateralized loan obligations passing coverage tests will hold the firm's paper. None of those is a representative agent, and the consequences are Part IV's: when intermediary capital is impaired the required return rises with no change whatever in the project (Chapter 19 §19.5), and Chapter 4 §4.7's flat empirical security market line is what an investor who cannot borrow does to the price of beta. Table 22.1's tidy 7.9 percent is an estimate of one population's current willingness to hold two claims.

That is also a reading of the hurdle-rate wedge. If the balance sheet financing the project is the firm's own — and Chapter 25 §25.1 reports that internal funds cover most corporate investment in aggregate — then the rate rationing projects is a shadow price on an internal budget rather than a market price, and there is no reason for it to move when market rates do. Of §22.1's four candidate explanations that is the one the stickiness evidence treats most kindly, and it is a holder story: for most firms most of the time, the marginal holder of the firm's capital is the firm.

The third appearance runs the arrow into the real economy. Section 22.4 established that investment is a function of one forward-looking price; Chapter 20 established that such a price is set by demand curves with a slope. A shift in who wants to hold a firm's equity is therefore an investment shock, and a valuation wave is that shock at scale: no synergy had to be real for a hundred and sixty-five billion dollars of assets to change hands in January 2000, only a population of holders willing, that month, to pay it.

That hands the part on. This chapter has valued the firm's assets and priced its capital; the discount rate assembled in §22.1 is the input Chapter 23 §23.1 takes as given when it asks whether the way those assets are financed changes what they are worth, and §22.5's method-of-payment evidence is one of the three exhibits Chapter 23 §23.4 builds the market-timing theory on.

In Chapter 1 §1.2's terms, a discount rate moves on two of the four readings more often than on the first: a constraint binding, when the balance sheets holding the firm's claims are impaired and the required return rises with nothing about the firm having changed, and a model becoming convention, since the rate in the appropriations manual is what the CAPM says it should be because everyone runs the CAPM.

***

## Elsewhere in the Series

* **Corporate treasury operations and the practice of raising capital** — *International Finance*, Chapter 10, which owns the treasury-desk view; this chapter owns the valuation and investment theory that decision rests on.
* **This book**: present value, the linearity that makes value additivity work, and the warning about discount rates — Chapter 3 §§3.1-3.2. The CAPM cost of equity, and the flat-security-market-line warning that biases it — Chapter 4 §§4.5, 4.7. The equity premium behind the second input — Chapter 5. The profitability and investment factors this chapter's §22.4 rationalizes — Chapter 6 §6.2. Event-study method, and the efficiency debate that the multiples circularity runs into — Chapter 7 §§7.1, 7.3, with mechanics in Appendix A. The binomial machinery §22.3 borrows — Chapter 8 §8.3. Aggregate Q from the Financial Accounts — Chapter 2 §2.3. Equity issuance, the seasoned-offering markdown, and stock as acquisition currency — Chapter 12 §§12.1, 12.5. Managerial overconfidence and its measurement — Chapter 15 §15.3, routed here by §15.6. Index-fund voting and proxy advisers — Chapter 17 §17.7. Merger-arbitrage capital and its redemption terms — Chapter 18 §18.4. The holders behind §22.8's discount rate: insurers' capital schedule — Chapter 10 §10.9; liability-driven duration demand — Chapter 16 §16.2; the intermediary stochastic discount factor — Chapter 19 §19.5; and the demand curves whose slope makes a shift in equity demand an investment shock — Chapter 20. Capital structure, the tax shield this chapter borrows for the WACC, payout, and market timing — Chapter 23 §§23.1-23.2, 23.4-23.5. Boards, compensation, blockholders, and activism as the internal substitutes for the takeover threat — Chapter 24 §24.3. How firms actually finance the investment this chapter appraises — Chapter 25 §§25.1-25.3.

***

## Summary

1. **Net present value is Chapter 3's identity applied inside the firm**, and value additivity — a consequence of the linearity of prices — is what makes project-by-project capital budgeting legitimate. Cash flows must be incremental, and flows and discount rates must be denominated consistently.
2. **IRR fails in three specific ways**: it ranks by rate rather than by value (scale), it embeds a reinvestment convention that flips rankings at a crossover rate (timing), and it can have multiple roots when cash flows change sign more than once — a project worth doing at a 15 percent cost of capital and not at 5 percent.
3. **WACC is assembled from choices, not observations.** Table 22.1's 7.9 percent rests on a chosen riskless rate, a chosen equity premium, and an estimated beta, and Chapter 4 §4.7's flat security market line biases the CAPM input for high- and low-beta projects in opposite directions.
4. **Hurdle rates exceed WACC and are sticky.** Surveys find firms applying hurdle rates several percentage points above their own cost-of-capital estimates, and barely revising them as rates move. Capital rationing, agency correction, real-option value, and inattention are the four candidate explanations, and they are not the same claim.
5. **Three quarters or more of a discounted-cash-flow value is the terminal value** (Table 22.2), which is two assumptions; three percentage points of discount rate move the answer by two thirds. Extending the explicit horizon relabels the problem rather than solving it.
6. **Every multiple is a compressed discounted-cash-flow model** (Table 22.3), and the useful skill is knowing which assumption it has hidden. Multiples break circularly: they import whatever error is common to the comparable set, which is the error that matters in a valuation wave.
7. **Real-option value is material when investment is irreversible, uncertain, and postponable.** A project with a naive net present value of 10 is worth 19.05 as an option to wait, and the premium rises with uncertainty. Formal real-option pricing remains rare in practice; its durable contribution is the reason a hurdle-rate wedge is not simply an error.
8. **Tobin's Q is the sufficient statistic the theory wants and average Q is what data give.** Hayashi's conditions for their equality are strong; the denominator omits intangibles; and the empirical record — small coefficients, poor fit, and cash flow entering significantly — is either evidence of financing constraints or an attenuation artifact, with the measurement-error literature favoring the second.
9. **The investment CAPM is Chapter 6's factors read from the firm's side.** Firms invest until the investment return equals the expected return investors require, so high investment implies low expected returns and high profitability at fixed investment implies high ones. That is CMA and RMW derived from corporate optimization rather than from investor preferences.
10. **Merger waves are industry shocks plus available capital, and also valuation waves.** Both accounts have support; the second explains why stock is the dominant currency at market peaks and why AOL-Time Warner needs no synergy story.
11. **Targets gain roughly sixteen percent, acquirers roughly zero to slightly negative, the combined effect roughly one to two percent** (Table 22.4). The acquirer result is a mean over a wide distribution, and the headline aggregate losses of 1998-2001 were driven by a few very large deals. Stock-financed acquirers do worse at announcement and afterward, which is the market-timing reading Chapter 23 §23.4 develops.
12. **Roll's hubris hypothesis is the winner's curse applied to bidders**, and Malmendier and Tate's option-exercise measure of overconfidence supplies its mechanism: overconfident chief executives acquire more, diversify more, and are received worse.
13. **The takeover market is a governance device whose edge was blunted by design.** The 1980s evidence supports the disciplinary reading; the poison pill combined with a staggered board is what defeated it; entrenchment is associated with lower firm value, though the governance-portfolio return result did not survive out of sample. Hostile activity fell for four compounding reasons, and internal substitutes (Chapter 24 §24.3) took over.
14. **An announcement return measures the price at which constrained holders will hold a claim, not the deal's merit.** The target's price is quoted by merger-arbitrage capital, the completion probability is partly a forecast of index-fund voting policy, and a falling acquirer price may price the deal or reveal the acquirer's standalone overvaluation.
15. **A cost of capital belongs to a population of constrained balance sheets rather than to a representative agent.** The cost of equity is whoever is currently pricing the shares and the cost of debt is whoever will hold the paper, so an impaired intermediary raises a firm's discount rate with nothing about the project having changed. The hurdle-rate wedge is what that looks like when the marginal holder of the firm's capital is the firm, and a shift in who wants to hold a firm's equity is an investment shock — which is how a valuation wave reaches the real economy.

***

## Key Terms

* **Net present value (NPV)**: The present value of a project's incremental free cash flows less its outlay, discounted at a rate matching the flows' risk and denomination; positive net present value is the decision rule that value additivity licenses
* **Internal rate of return (IRR)**: The discount rate setting net present value to zero; unreliable for ranking projects of different scale or timing, and non-unique when cash flows change sign more than once
* **Weighted average cost of capital (WACC)**: The market-value-weighted average of the cost of equity and the after-tax cost of debt, and the discount rate for free cash flow to the firm
* **Hurdle rate**: The return a firm actually requires of a project, which surveys find exceeds its estimated weighted average cost of capital by a substantial margin and moves little as interest rates move
* **Terminal value**: The value assigned to all cash flows beyond an explicit forecast horizon, usually as a growing perpetuity or an exit multiple; typically three quarters or more of a discounted-cash-flow valuation
* **Multiple**: A ratio of price or enterprise value to an accounting flow or stock, used to price a firm by reference to comparables; algebraically a discounted-cash-flow model with its assumptions compressed into one number
* **Real option**: The value of managerial flexibility — to wait, expand, or abandon — priced with the machinery of Chapter 8 §8.3, material when investment is irreversible, uncertain, and postponable
* **Tobin's Q**: The market value of the claims on a firm divided by the replacement cost of its capital; the theory says firms invest until marginal Q equals one, and Hayashi (1982) gives the conditions under which average and marginal Q coincide
* **Investment CAPM**: The reading of the profitability and investment factors as consequences of firms' optimal investment against investor-set discount rates, rather than as compensation for investor-side risk
* **Merger wave**: The clustering of acquisition activity in time and within industries, attributed to industry shocks combined with available capital and to episodes of high equity valuation
* **Announcement return**: The abnormal return around a deal's announcement, measured as in Chapter 7 §7.3; large and positive for targets, near zero to slightly negative for acquirers, small and positive for the combination
* **Hubris hypothesis**: Roll's account of acquirer losses as the winner's curse applied to bidding managers — the winning bidder is the one whose valuation error was largest — predicting that acquirer losses roughly offset target gains
* **Poison pill (shareholder rights plan)**: A charter or board-adopted device issuing cheap shares to all holders except a bidder crossing an ownership threshold, making an unnegotiated acquisition prohibitively dilutive; decisive as a defense when combined with a staggered board

***

## Readings

### Required

* Roll, R. (1986). "The Hubris Hypothesis of Corporate Takeovers." *Journal of Business* 59(2): 197-216. *Short, and best read as a piece of reasoning rather than a piece of evidence. Roll shows how far the acquirer-loses fact can be explained with no agency conflict and no irrationality beyond a failure to apply the winner's curse to oneself, and states the testable implication that combined gains should be zero — which is where the data part company with him.*
* Hayashi, F. (1982). "Tobin's Marginal q and Average q: A Neoclassical Interpretation." *Econometrica* 50(1): 213-224. *The paper that turns Tobin's intuition into a theorem with conditions attached. Read it for the conditions — constant returns in production and adjustment costs, and price-taking — because every empirical use of average Q since has been an argument about whether they hold.*

### Recommended

* Andrade, G., M. Mitchell and E. Stafford (2001). "New Evidence and Perspectives on Mergers." *Journal of Economic Perspectives* 15(2): 103-120. *The survey behind Table 22.4 and the best twenty pages on what the merger evidence does and does not establish. Note their care about the acquirer result's dispersion, and their industry-shock account of waves.*
* Hou, K., C. Xue and L. Zhang (2015). "Digesting Anomalies: An Investment Approach." *Review of Financial Studies* 28(3): 650-705. *The q-factor model, and the source of §22.4's argument. Read the derivation of the investment return first and the anomaly tables second; the point is that the factors come out of the firm's first-order condition rather than out of a search over characteristics.*
* Stein, J. C. (2003). "Agency, Information and Corporate Investment." In *Handbook of the Economics of Finance*, Volume 1A. Elsevier. *The survey that connects financing frictions to the investment decision, and the right reading beside §22.4's gap between marginal and average Q: if investment responds to cash flow and to internal politics as well as to Q, the regression's residual has structure.*
* Graham, J. R. and C. R. Harvey (2001). "The Theory and Practice of Corporate Finance: Evidence from the Field." *Journal of Financial Economics* 60(2-3): 187-243. *The survey that established what firms actually do: net present value and internal rate of return dominate, the CAPM dominates for the cost of equity, and small firms behave differently from large ones. The starting point for the hurdle-rate literature §22.1 draws on.*
* Ruback, R. S. (2002). "Capital Cash Flows: A Simple Approach to Valuing Risky Cash Flows." *Financial Management* 31(2): 85-103. *The alternative to §22.1's WACC. Discount cash flows that include the interest tax shield at the unlevered cost of capital, and the discount rate stops depending on a leverage ratio that has to be forecast. Read it as a check on how much of a valuation is riding on the tax shield's discount rate, which Chapter 23 §23.2 shows can halve the shield.*
* Scharfstein, D. S. and J. C. Stein (2000). "The Dark Side of Internal Capital Markets: Divisional Rent-Seeking and Inefficient Investment." *Journal of Finance* 55(6): 2537-2564. *The counterweight §22.1 needs to the assumption that a firm applies one hurdle rate correctly to every project. In a multi-division firm the capital budget is a bargaining outcome, and the direction of the distortion — socialism toward weak divisions — is the opposite of what an efficient internal market would produce.*
* La Porta, R., F. Lopez-de-Silanes, A. Shleifer and R. W. Vishny (2002). "Investor Protection and Corporate Valuation." *Journal of Finance* 57(3): 1147-1170. *Valuation as a function of the legal regime the firm is incorporated in. It routes §22.2's discount rate out to Chapter 25 §25.4 and makes the point that a multiple is not comparable across countries without an adjustment nobody has a clean estimate of.*
* Damodaran, A. (2005). "Marketability and Value: Measuring the Illiquidity Discount." Working paper, Stern School of Business, New York University. *The practitioner's treatment of the marketability discount applied when there is no traded price. It is the missing step between §22.2's multiples and the private-firm valuations Chapter 25 §25.3 and Chapter 18 §18.2 depend on.*
* Dixit, A. K. and R. S. Pindyck (1994). *Investment Under Uncertainty*. Princeton University Press. *The synthesis of the real-options literature, and the source of §22.3's three conditions. The first two chapters carry the whole argument without the continuous-time apparatus and are the right assignment for a masters course.*
* Damodaran, A. "The Promise and Peril of Real Options." Working paper, Stern School of Business, New York University. *More permissive than Dixit-Pindyck about when the option framing applies, and useful for exactly that reason: read it against §22.3's three conditions and mark the places where an option is being claimed for a project whose exclusivity nobody has established.*
* Weston, J. F., M. L. Mitchell and J. H. Mulherin (2004). *Takeovers, Restructuring, and Corporate Governance*. 4th edn. Pearson Prentice Hall. *The practitioner scaffolding behind §§22.5-22.6 — merger waves, the arithmetic of accretion and dilution, defensive tactics, and the deal-structure vocabulary the academic literature assumes its readers already have.*

***

## Discussion Questions

1. **Why does the hurdle rate exceed the cost of capital, and why does it stay put?** Section 22.1 gives four candidate explanations: capital rationing, agency correction for optimistic forecasts, real-option value, and inattention. Take each in turn and state what it predicts about (a) the *size* of the wedge across firms — should it be larger at cash-rich or cash-poor firms, at firms with more or fewer divisions? — and (b) the wedge's *response* to a large fall in interest rates. Which of the four survives the stickiness evidence best? Then argue the opposite case: that the wedge is not a puzzle at all, because the cost-of-capital estimate in the denominator is itself biased, and identify which input of Table 22.1 would have to be wrong, and in which direction.
2. **Does the acquirer-loses fact condemn mergers?** A commentator reads Table 22.4 and concludes that acquisitions are a value-destroying pastime of empire-building managers and should face a higher legal burden. Build the strongest case that the inference does not follow, using at least three of: the distinction between transfer and creation, the dispersion around the acquirer mean, the relative-size arithmetic that makes acquirer returns noisy, the possibility that the counterfactual to a bad acquisition is a worse internal project, and §22.7's revaluation channel. Then build the strongest case that survives your objections, and say what evidence would settle it.
3. **Q's poor fit: theory or measurement?** Section 22.4 offers financing constraints, mismeasured marginal Q, and omitted intangible capital as explanations for why cash flow beats Q in investment regressions. For each, state one testable implication that the other two do not share — think about which firms, which periods, and which industries. Then consider a fourth possibility the section does not raise: that managers do not observe or use their own Q. What would that predict, and how would you distinguish it from the other three?
4. **If waiting is worth so much, why is nobody pricing it?** Section 22.3's option to wait was worth nearly twice the naive net present value, and yet formal real-option valuation is rare. Give three explanations — one about inputs, one about governance, and one about substitutes (what else in the firm's process might already be charging for option value?). Which explanation predicts that real-option methods should be more common in some industries than others, and does the pattern you would expect match the industries where they actually are used?
5. **Defenses: bargaining power or entrenchment?** A poison pill combined with a staggered board roughly doubles a target's chances of staying independent. The board's defense of the arrangement is that it is bargaining power — the ability to refuse a first offer and extract a higher one. The evidence in §22.6 is that defended targets which are nonetheless acquired receive *lower* premiums. Reconcile those two statements, then design the test that would separate the bargaining-power story from the entrenchment story. Why is the fact that most large US firms have dismantled staggered boards while retaining the ability to adopt a pill overnight evidence about which story is right?

***

## Problems

**Problem 1 — Three IRR traps.**

(a) Project A costs 100 today and returns 150 in one year. Project B costs 1,000 and returns 1,300. Compute both internal rates of return and both net present values at a 10 percent cost of capital. Which project should a firm that can undertake only one of them choose, and what does an IRR ranking cost it?

(b) Compute the internal rate of return of the incremental cash flow (B minus A) and show that it repairs the ranking. State in one sentence why this repair is rarely performed in practice.

(c) A short project has cash flows of −100, +120, +10 and a long one −100, +10, +130. Compute both internal rates of return. Find the crossover discount rate at which their net present values are equal, and state which project is preferred above and below it.

(d) A project has cash flows of −100, +230, −132. Show that its net present value is zero at both 10 percent and 20 percent, and determine the sign of net present value at 5 percent, 15 percent, and 30 percent. Write one sentence describing, in plain language, the investment rule this project actually obeys.

(e) State the general condition on a cash-flow stream under which multiple internal rates of return are possible, and give one realistic corporate example of a project with that shape.

**Problem 2 — A cost of capital and a decision.** A firm has 600 of equity at market value and 400 of debt at market value. Its industry beta, relevered to this capital structure, is 1.20. The riskless rate is 4 percent, the equity premium is assumed to be 5 percent, the firm's bonds yield 6 percent, and the corporate tax rate is 21 percent.

(a) Compute the cost of equity, the after-tax cost of debt, and the weighted average cost of capital.

(b) A project requires an outlay of 500 and generates free cash flows of 120, 140, 160, 180 and 200 over five years. Compute its net present value at the weighted average cost of capital, and its internal rate of return.

(c) The firm's capital-appropriations manual specifies a hurdle rate of 15 percent for all projects. Compute the project's net present value at that rate. Does the project pass? Now compute it at 18 percent.

(d) Suppose the project is a regulated utility subsidiary whose asset beta is 0.60 rather than the firm's. Recompute the cost of equity, the weighted average cost of capital (holding the capital structure fixed), and the net present value. State which of §22.1's warnings this part illustrates.

(e) Chapter 4 §4.7 reports that the empirical security market line is flatter than the CAPM's. For the project in (d), state the direction of the resulting bias in the discount rate and its consequence for the firm's mix of safe and risky investment.

**Problem 3 — Terminal value dominance.** A firm's free cash flow next year is 100, growing at 6 percent a year for five years. The weighted average cost of capital is 8 percent and the perpetual growth rate thereafter is 2.5 percent.

(a) Compute the present value of the explicit five-year forecast.

(b) Compute the terminal value at year 5 and its present value today, and express the terminal value as a share of total firm value.

(c) Recompute total firm value for perpetual growth rates of 1.5 percent and 3 percent, and for weighted average costs of capital of 7 percent and 10 percent (holding perpetual growth at 2.5 percent). Report the widest and narrowest values you obtain and their ratio.

(d) Compute the exit multiple of year-5 free cash flow implied by your terminal value in (b). An analyst instead argues for an exit multiple approach and picks a multiple from current trading comparables. State one way in which the two approaches can be made consistent and one way in which the multiple approach imports an assumption the growing perpetuity makes explicit.

(e) A manager notes that the terminal value carries most of the answer and proposes extending the explicit forecast to fifteen years to "reduce the reliance on assumptions." Evaluate the proposal.

**Problem 4 — A real option to wait.** A project's cash flows have a present value today of 100. Building it requires an outlay of 90, payable either today or in one year. In one year the present value of the cash flows will be either 130 or 80. The riskless gross return is 1.05.

(a) Compute the net present value of investing today.

(b) Compute the risk-neutral probability of the up state and the value today of the right to invest in one year. What is the option premium over the naive rule?

(c) Compute the replicating portfolio — the position in the underlying and the riskless borrowing — that manufactures the option's payoff, and verify its cost.

(d) Repeat (b) with $$u = 1.50$$, $$d = 0.70$$ and then with $$u = 1.15$$, $$d = 0.90$$. State the comparative static and connect it to §22.3's three conditions.

(e) Now suppose that building today delivers a cash flow of 12 at the end of the first year that is lost if the firm waits, so that the present value of the cash flows *from year two onward* is $$100 - 12/1.05$$ today and it is this quantity that moves by $$u$$ or $$d$$. Recompute the value of waiting and compare it with the net present value of investing today. Which option-pricing feature of Chapter 8 §8.3.5 does this reproduce?

**Problem 5 — Computing Q.** A firm has 200 million shares outstanding trading at 18. The market value of its debt is 1,400. The replacement cost of its tangible assets, estimated by a perpetual-inventory method, is 4,000. Its book assets are 3,400 and it holds 500 of cash. (All amounts in millions of currency units.)

(a) Compute Tobin's Q.

(b) Recompute using book assets in the denominator instead of replacement cost, and explain the direction of the difference and what drives it.

(c) Recompute excluding cash from both numerator and denominator. Which version is the right one for a statement about the firm's incentive to invest in physical capital, and why?

(d) The firm spends 600 a year on research and development, which it expenses. Suppose capitalizing it at a 20 percent depreciation rate would add 2,400 of intangible capital to the denominator. Recompute Q and state what this does to a cross-sectional regression of investment on Q run across industries of differing research intensity.

(e) Under Hayashi's conditions, marginal Q equals average Q. Name two features of this firm that would break those conditions, and state for each whether marginal Q would then be above or below the average Q you computed in (a).

**Problem 6 — Reading an announcement.** Acquirer A has 500 million shares at 40. Target T has 100 million shares at 25. A announces an all-stock offer of 0.75 A shares per T share. On the announcement, A's shares fall 3 percent and T's rise 16 percent.

(a) Compute both firms' pre-announcement market capitalizations, the offer value per target share at the pre-announcement price, and the premium offered.

(b) Compute the change in each firm's market value and the combined change, in currency units and as a percentage of the two firms' combined pre-announcement value.

(c) Compute the acquirer's loss as a percentage of the *target's* pre-announcement capitalization. Then compute what a one percent move in A's shares is worth as a percentage of T's capitalization. Use the two numbers to explain why acquirer announcement returns are simultaneously economically decisive and statistically noisy.

(d) A commentator concludes that A's management destroyed value and should be replaced. Give the two interpretations of A's decline that §22.7 distinguishes, state what each implies about whether blocking the deal would recover the loss, and describe the research design that separates them.

(e) Two days later the spread between T's price and the offer value narrows sharply, with no news about the businesses. Give two explanations from §22.7 and say what data would distinguish them.

***

## Selected Solutions

*Solutions to Problems 1 and 6 follow. Solutions to the remainder are in the instructor materials.*

**Problem 1.**

(a) A's internal rate of return is $$150/100 - 1 = 50$$ percent; B's is $$1300/1000 - 1 = 30$$ percent. At 10 percent, $$\mathrm{NPV}\_A = -100 + 150/1.10 = 36.36$$ and $$\mathrm{NPV}\_B = -1000 + 1300/1.10 = 181.82$$. The firm should take **B**. Ranking by internal rate of return selects A and costs $$181.82 - 36.36 = 145.45$$ of value. The rate of return is high on a small base, and shareholders are paid in currency, not in percentages.

(b) The incremental stream is $$-900$$ today and $$+1150$$ next year, an internal rate of return of $$1150/900 - 1 = 27.8$$ percent, comfortably above the 10 percent cost of capital, so the extra 900 of investment is worth making and B is preferred. The repair is rarely performed because it requires the analyst to construct a hypothetical project that nobody proposed, and because with more than two mutually exclusive projects the number of pairwise comparisons grows quickly — which is a practical argument for using the rule that does not need repairing.

(c) The short project's internal rate of return solves $$-100 + 120/(1+r) + 10/(1+r)^2 = 0$$, giving $$r = 27.8$$ percent; the long project's gives $$r = 19.1$$ percent. Setting the two net present values equal, the crossover rate is $$1/11 = 9.09$$ percent, where both are worth 18.40. **Below 9.09 percent the long project is worth more** despite its lower internal rate of return, because its cash arrives later and later cash is penalized less at low rates; above 9.09 percent the ranking agrees with the internal rate of return.

(d) Write $$x = 1/(1+r)$$. Then $$\mathrm{NPV} = -100 + 230x - 132x^2$$, whose roots are $$x = 0.9091$$ and $$x = 0.8333$$, that is $$r = 10$$ percent and $$r = 20$$ percent. Evaluating: at 5 percent $$\mathrm{NPV} = -0.68$$; at 15 percent $$\mathrm{NPV} = +0.19$$; at 30 percent $$\mathrm{NPV} = -1.18$$. The rule the project obeys is: **accept if the cost of capital lies between 10 and 20 percent, and reject otherwise** — a rule no single hurdle rate can express, because the project is a bet whose early inflow must be worth more than the late outflow costs, and both sides of that comparison move with the discount rate.

(e) Multiple internal rates of return are possible whenever the cash-flow stream changes sign more than once (Descartes' rule of signs bounds the number of positive roots by the number of sign changes). Realistic examples: a mine or a well with a large end-of-life reclamation obligation; a nuclear plant with decommissioning costs; a project requiring a major mid-life refit.

**Problem 6.**

(a) A's capitalization is $$500 \times 40 = 20{,}000$$ million; T's is $$100 \times 25 = 2{,}500$$ million. The offer is $$0.75 \times 40 = 30$$ per target share, a premium of $$(30-25)/25 = 20$$ percent.

(b) A's shares fall to 38.80, a change of $$-1.20 \times 500 = -600$$ million. T's rise to 29.00, a change of $$+4.00 \times 100 = +400$$ million. The combined change is $$-200$$ million, or $$-200/22{,}500 = -0.89$$ percent of the two firms' combined pre-announcement value. Note that this deal is on the wrong side of Table 22.4's combined average: the market's verdict is that it destroys value.

(c) The acquirer's loss is $$600/2{,}500 = 24$$ percent of the target's entire pre-announcement capitalization. A one percent move in A's shares is 200 million, which is 8 percent of T's capitalization. Both facts follow from A being eight times T's size. Economically, the acquirer's return dominates the aggregate arithmetic — small percentage moves on a large base swamp large percentage moves on a small one, which is why aggregate dollar losses in a merger wave are an acquirer story. Statistically, the same size ratio means the deal is a small perturbation to A's business and its announcement return is buried in A's ordinary daily volatility, so acquirer effects need large samples to detect while a single target's return is unmistakable.

(d) **Deal-value interpretation:** the market judges the deal's net present value to A's shareholders to be about $$-600$$ million — the premium exceeds the synergies. Blocking the deal would recover the loss. **Revaluation interpretation:** by offering stock, A revealed that its managers consider the shares expensive, and the market marks A down on the issuance signal exactly as it marks down a seasoned offering (Chapter 12 §12.5). The 600 million was never there; blocking the deal recovers nothing, and A's shares stay down. The two are observationally identical in the announcement window. The standard design compares deals that completed with deals withdrawn for reasons unrelated to the acquirer's own valuation — a regulatory block, a third-party topping bid, the target's own decision — and asks whether the acquirer's price recovers on withdrawal. Recovery favors the deal-value reading; no recovery favors revaluation.

(e) First, **new information about completion probability**: a regulatory clearance, a target board recommendation, or a proxy adviser's report, any of which raises the probability the arbitrage position pays off and compresses the spread (Chapter 17 §17.7). Second, **a change in the price of arbitrage capital**: merger-arbitrage funds receiving inflows, or an easing of financing terms, bid up target shares without any change in the deal's prospects (Chapter 18 §18.4). To distinguish them, look for whether spreads compressed across *other, unrelated* announced deals on the same days — a deal-specific event moves one spread, a capital-supply shock moves the whole cross-section — and at fund flow and prime-broker margin data over the window.

***

## Data Exercise: Aggregate Q and the Investment It Is Supposed to Explain

Parts A through C run entirely on free data. Part D is the licensed extension.

**Part A — Building aggregate Tobin's Q (free, FRED and Z.1).** The Financial Accounts of the United States report, for the nonfinancial corporate business sector, both the market value of the sector's outstanding equity and the sector's net worth measured with assets at estimated current-cost (replacement) value. Their ratio is the aggregate analogue of §22.4's Q, and Chapter 2 §2.3's data exercise already asked you to note its current and 1980 values.

Work from the Z.1 balance-sheet table for nonfinancial corporate business (the B.103 table in recent releases) rather than from memory of series codes, since FRED's mnemonics for these series have changed across vintages: locate the market value of corporate equities as a liability of the sector, and the sector's net worth at market value, and confirm from the table's own notes which valuation basis each line uses before dividing.

1. Construct the quarterly ratio from the earliest available date to the present and plot it. Mark 1974, 1982, 2000, 2009, and 2021.
2. Construct a second version whose denominator is total assets at current cost rather than net worth, and whose numerator adds the market value of the sector's debt to its equity — the definition in §22.4. Plot both series on one axis. Explain, in two sentences, why they differ and which is closer to the object the theory names.
3. The theory says Q should hover near one. Report the mean of your series and the fraction of quarters above one. Then give two measurement reasons and one economic reason why a persistent level above one is not by itself a refutation.

**Part B — Q against investment (free, FRED).** Retrieve real private nonresidential fixed investment (the standard quarterly national-accounts series on FRED) and a corresponding measure of the nonfinancial corporate capital stock or, more simply, nominal nonresidential investment scaled by the sector's total assets at current cost from Part A.

4. Plot your investment rate against the Q series from Part A on a common time axis. Then plot the scatter, and report the correlation and the slope from a regression of the investment rate on Q.
5. Repeat the regression with Q lagged one quarter and four quarters. Which horizon fits best, and what does that say about adjustment costs?
6. Add the corporate sector's internal funds (undistributed profits plus consumption of fixed capital, both in the Financial Accounts) scaled the same way, as a second regressor. Report what happens to Q's coefficient and to the fit. You have now reproduced, at the aggregate level, the cash-flow-beats-Q result of §22.4; write one paragraph on which of that section's three explanations your aggregate data can and cannot distinguish.
7. Split the sample at 1995 and rerun. State what the intangibles argument of §22.4 predicts for the split and whether your results are consistent with it.

**Part C — The valuation-wave prediction (free).** Using a free aggregate valuation series — Robert Shiller's cyclically adjusted price-earnings data, posted on his Yale page — and any free count of announced US merger activity you can assemble from public sources (annual totals reported by the financial press, or a hand-collected count of deals above a size threshold from public filings):

8. Plot merger activity against the valuation measure. Report the correlation and identify the four largest peaks in activity.
9. For each peak, state from public reporting whether stock or cash was the dominant method of payment. Compare the pattern with §22.5's two competing accounts of waves, and say which the aggregate correlation can and cannot distinguish.

**Part D ★ — Firm-level Q and deal-level returns (WRDS).** With Compustat, CRSP, and SDC Platinum (or its successor):

10. Construct firm-level Q for a broad panel: market equity plus book debt over book assets, and then a second version with the Peters-Taylor intangible-capital adjustment to the denominator. Regress the investment rate on each, with and without cash flow, using firm and year fixed effects, and report all four coefficients. Then repeat with an errors-in-variables estimator and report how much of the cash-flow coefficient survives.
11. Build a deal-level file of US acquisitions over a twenty-year window and compute three-day announcement cumulative abnormal returns for acquirers and targets with the machinery of Appendix A. Reproduce Table 22.4's rows, then split by method of payment and by relative size, and report the acquirer result for each cell with its standard error and the interquartile range — not only the mean.
12. Merge the acquirer's Q from part 10 onto the deal file. Test the valuation-wave prediction directly: are high-Q acquirers more likely to pay with stock, and are their announcement returns worse? Report what you find and state which of §22.5's readings it favors, and what confound you cannot rule out.