AFM 274 · Chapter 17: Capital Structure in a Perfect Market

Exam quick-reference sheet · Modigliani–Miller in a frictionless world

How to use this sheet (last-minute review): Read the Core Intuition boxes once out loud, if you can explain MM I, MM II, and why WACC is flat in your own words, you're 80% there. Then drill the Formulas and Cheat Sheet, re-do the Entrepreneur example from a blank page, and finish with the Active Recall Qs (cover the grey answers). The two Fallacies are almost always tested.

Midterm Must-Know Checklist

State MM I via the Law-of-One-Price / cash-flow-identity argument.
Apply MM II: show r_E rises linearly with D/E.
Argue WACC = r_U is constant; explain the flat line.
Compute value, E, r_E, WACC, unlevered and levered.
Explain homemade leverage & why firm leverage is irrelevant.
Build/read a market-value balance sheet; do a leveraged recap.
Lever & unlever betas (incl. shortcut when β_D=0).
Use net debt (cash = negative debt) for EV & asset beta.
Debunk the two fallacies (EPS, dilution) with numbers.
Know the 6 perfect-market assumptions.

MM Proposition I: the core result

V_L = V_U ⟺ E + D = U = A

Pizza intuition: the assets generate a fixed cash-flow "pizza." Debt & equity just slice it. More slices ≠ more pizza. Proof: (1) debt + equity payoffs = asset payoff in every state; (2) Law of One Price → identical cash flows must have equal price; so E+D = value of assets, whatever the split.

MM Proposition II: cost of equity

r_E = r_U + DE(r_U − r_D)

Debt is paid first, so the same asset-cash-flow swings hit a smaller equity base → equity gets riskier → investors demand more. The rise in r_E exactly offsets putting more weight on cheap debt. No free lunch.

Why WACC is constant (= r_U)

r_wacc = EE+Dr_E + DE+Dr_D

Sub MM II into this and the leverage terms cancel → r_wacc = r_U = r_A. As leverage ↑, both r_E and r_D rise, but more weight shifts to cheaper debt, net effect zero. Discount projects at r_U, not r_E.

Homemade leverage

Investors can borrow/lend on personal account to build any capital structure themselves (e.g., buy unlevered stock on margin). At the same interest rate, this perfectly substitutes for corporate leverage, so nobody pays extra for the firm to do it. ⟹ leverage is value-irrelevant.

Market-value balance sheet

All assets (incl. intangibles) and all securities at current market value; always balances. MV Equity = MV Assets − MV Debt. Key tool for the leveraged recap: borrow $X, buy back $X equity ⟹ assets & debt move together, equity falls by $X, price/share unchanged (zero-NPV swap).

Key Formulas: and When to Use Them

Formula	Gives you / use it when
`V = E + D`	Firm value (the "pizza"). Always market values.
`V_L = V_U`	MM I, value is independent of capital structure.
r_U = EE+Dr_E + DE+Dr_D	Asset / unlevered / pretax-WACC. Same as WACC (no taxes). Discount rate for projects.
r_E = r_U + DE(r_U − r_D)	Levered cost of equity (MM II). After a leverage change.
β_U = EE+Dβ_E + DE+Dβ_D	Unlever a beta (isolate business risk of a comparable).
β_E = β_U + DE(β_U − β_D)	Relever a beta to a target structure. (Twin of MM II.)
β_U = β_E1 + D/E	Shortcut for β_U when β_D = 0.
`r_E = r_f + β_E(r_M − r_f)`	CAPM, estimate `r_E` (or `r_U` with `β_U`).
`Net debt = D − Cash`; EV = E + Net debt	Cash = negative debt. Use net-debt weights for β_U.

⚠ Not this chapter: the tax versions on your formula sheet: V_L = V_U + PV(ITS) and after-tax WACC with (1−t_c), belong to later chapters. In a perfect-markets question there are no taxes, so WACC = r_U.

The pizza (MM I)

Resize the slices (more/less debt), the pizza (firm value) is the same size.

WACC vs. leverage (Fig. 17.1)

Both costs rise, but weight shifts to cheap debt → WACC stays flat at r_U.

Homemade leverage = firm leverage

Firm leverage

Buy levered equity for $500.

Date 1: $875 / $375

Homemade

Buy $1000 unlevered stock = $500 own + $500 borrowed @5%.

Date 1 (after repaying $525): $875 / $375

Identical payoffs ⟹ same price (Law of One Price) ⟹ levered equity = $500.

Worked Example: The Entrepreneur's Project (know cold)

Setup: cost $800 today; pays $1,400 (strong) or $900 (weak), 50/50. Risk-free 5%, risk premium 10% ⟹ r_U = 15%.

Security (Date 0)	Value	Strong $	Weak $	Strong %	Weak %	E[r]
Unlevered equity	1,000	1,400	900	+40%	−10%	15%
Debt (borrow $500 @5%)	500	525	525	+5%	+5%	5%
Levered equity	500	875	375	+75%	−25%	25%
Whole firm	1,000	1,400	900	—	—	15%

NPV = −800 + 1150/1.15 = −800 + 1000 = +$200. Levered equity: E = V_L − D = 1000 − 500 = $500; E[payoff] = ½(875)+½(375)=625 ⟹ r_E = 625/500 − 1 = 25%. Check MM II: 15% + (500/500)(15%−5%) = 25% ✓. Check WACC: ½(25%)+½(5%) = 15% = r_U ✓.

Debt + levered-equity rows sum to the firm row in every column, that cash-flow identity is MM I. Leverage widened equity's range (−10..+40 → −25..+75) and raised its return, but value and WACC didn't budge.

Fallacy 1: "Leverage ↑ EPS, so price ↑"

Wrong: EPS does rise, but leverage also raises r_E (MM II), lowering the P/E by the offsetting amount. Price unchanged.

Levitron: after recap EPS $1.00→$1.10, but r_E 13.33%→14.66%; price = EPS/r_E = 1.10/0.1466 = $7.50 (unchanged). Use EV/EBIT(DA), not P/E, to compare firms.

Fallacy 2: "Issuing equity dilutes value"

Wrong: sold at a fair price, new shares bring in matching assets, more shares and proportionally more assets ⟹ value per share unchanged.

Jet Set Air: $8B (500M×$16) + $1B raised = $9B over 562.5M shares = $16. Any gain/loss comes from the NPV of what you buy, not the issuance. Bonus: "cheap debt lowers WACC" is also false, r_E rises to offset.

Condensed Cheat Sheet perfect markets · no taxes · financing = zero-NPV repackaging of risk

Assumptions (6)

No taxes · no transaction/issuance costs · fairly priced (no arbitrage) · borrow = lend at same rate · no signaling · homemade leverage available.

Core results

V_L = V_U · E + D = U = A · r_wacc = r_U = r_A

MM II + beta twin

r_E = r_U + DE(r_U−r_D)

β_E = β_U + DE(β_U−β_D)

Weighted averages (asset = WACC)

r_U = EE+Dr_E + DE+Dr_D

β_U = EE+Dβ_E + DE+Dβ_D · (β_D=0: β_U=β_E1+D/E)

Multiple securities

Weighted average of all security returns = r_U (add WE+D+Wr_W for a warrant, etc.).

Net debt & recap

Net debt = D − Cash (can be <0); EV = E + Net debt. Leveraged recap (borrow $X, buy back $X): assets ↑X then ↓X, debt ↑X, equity ↓X → price/share unchanged.

As leverage ↑

r_E ↑, r_D ↑ (eventually), β_E ↑, EPS/ROE ↑, but WACC flat at r_U, β_U flat, V flat.

Don't get fooled

EPS↑ ≠ price↑ (r_E rises). Fair-price equity issue ≠ dilution of value. Cheap debt ≠ lower WACC. Discount projects at r_U, not r_E.

One-line WACC proof

Sub MM II into r_wacc = weighted(r_E, r_D) → leverage terms cancel → r_wacc = r_U.

Active Recall: Cover the Grey Answers

1. State MM I + the proof. V_L=V_U. Debt+equity payoffs = asset payoff every state; Law of One Price → equal price; so E+D = asset value regardless of split.

2. Why does r_E rise with leverage (no bankruptcy)? Senior fixed debt concentrates the same asset-cash-flow swings onto less equity → more volatile → higher required return (∝ D/E).

3. E=$600M, D=$400M, r_E=18%, r_D=7%. Find r_U. 0.6(18%)+0.4(7%) = 13.6%.

4. r_U=12%, r_D=6%, D/E=0.5. Find r_E & WACC. r_E=12%+0.5(6%)=15%; WACC=⅔(15%)+⅓(6%)=12%=r_U.

5. β_E=1.2, D/E=0.5, β_D=0. β_U? Relever to D/E=1. β_U=1.2/1.5=0.8; new β_E=0.8+1(0.8)=1.6.

6. Borrow to buy back shares, does the price change? No. Selling $X debt and buying $X equity is a zero-NPV swap; risk changes, value/share doesn't.

7. T/F: issuing equity adds supply, so price falls. False. A fair-price issue brings in matching assets; price/share unchanged. Effects come from new info / NPV of use of funds.

8. Why is "debt is cheaper, so more debt lowers WACC" wrong? Adding debt raises r_E (MM II) by exactly enough to offset the heavier weight on cheap debt: WACC stays at r_U.

AFM 274 Ch. 17 quick sheet · perfect markets, no taxes · all numbers internally verified · tax/distress formulas belong to later chapters.