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Corporate Debt Value, Bond Covenants, and Optimal Capital Structure Hayne E. Leland Journal of Finance, 1994

Overview • Develops a structural model for corporate debt, yield spreads, optimal leverage. • Derives closed-form solutions for optimal leverage under various cases. • Has some interesting predictions for junk vs. investment grade bonds. • Though the base model doesn’t explain empirical findings well, some variations of it do come close.

Brief Background • Builds on earlier structural models of Merton (1974) and Black and Cox (1976). • Incorporates effects of taxes, bankruptcy costs, and protective covenants in their model. • Brennan and Schwartz (1978) have a similar idea, but this paper develops an analytical model with closed-form solutions. • Studies optimal leverage, debt pricing, yield spreads, and credit risk issues.

Merton (1974); Black and Cox (1976) models • Common Assumptions • Firm value follows a continuous time diffusion process. • Volatility σ and the risk free rate r are constant over time. • Net payout rate by the firm CF = 0 (Payouts to security holders financed by additional equity issuance) • There are no default costs or tax advantages to debt (so no optimal capital structure – MM world)

• Merton model: • Zero-coupon debt, face value F, maturity T • Default only at T, iff V(T) < F • If default, bond holders get random V(T), equity holders gets zero

Merton (1974); Black and Cox (1976) models • Black & Cox model: • Perpetual debt (no principal repayment), constant coupon rate C • Default at any t, upon first passage of V(t) to default barrier VB • At default, bond holders get nonrandom VB, equity holders gets zero

• Merton, B & C Shortcomings: • Don’t allow for debt that is coupon-paying and has finite maturity • Don’t allow analysis of optimal debt amount/maturity (capital structure) without introduction of taxes, default costs • Have regularly-observed empirical difficulties: • Spreads too low for low-risk and low maturity debt (Jones, Mason, & Rosenfeld (1984), others.)

Model Assumptions • Security value depends on the underlying firm value but time independent. • Face value of debt, once issued, remains static through time. • Firm finances the net cost of the coupon by issuing additional equity. • There exists an asset that pays constant rate of interest r.

Model Parameters • C – coupon • V – current value of assets of the firm • τ – corporate tax rate • α – bankruptcy costs • r – risk free interest rate • σ2 – volatility of asset value • D – value of debt • E – value of equity • v – total value of the firm.

A Generic Model • The asset value V of the firm follows a diffusion process with constant volatility of rate of return: • V is assumed to be unaffected by the financial structure of the firm. • Let F(V,t) be the value of the claim on the firm that pays C continuously when solvent. • The asset’s value must satisfy:

A Generic Model • No closed form solutions for above. • Brennan & Schwartz (1978) use numerical techniques.

• Securities with no explicit time dependence => Ft(V,t)=0. • Then, we have the solution: • Any time-independent claim with equityfinanced payout C must obey equation (4).

A Generic Model • A0, A1, A2 determined by boundary conditions. • Let VB be asset value that triggers bankruptcy, α represent bankruptcy costs. • When bankrupt, αVB is incurred, debtholders get (1α)*VB and equity holders get nothing. • Apply (4) for value of debt D(V) with following boundary conditions:

• We get:

A Generic Model • Debt has two counteracting effects: i. Decrease firm value due to bankruptcy costs BC(V) ii. Increase firm value due to tax benefits TB(V).

• Eq. (4) with appropriate boundary conditions gives closed forms for BC(V) and TB(V). • Now, total value of the firms is given as:

• And equity value is given as:

Specific Cases for VB • Two possible triggers of default are considered in the paper. 1. Unprotected Debt, Endogenous Bankruptcy • Firm chooses VB so as to maximize equity value.

2. Protected Debt, Positive Net Worth Covenant • Bankruptcy when firm value falls below the face value of debt.

• We’ll see the closed form solutions and comparative statistics for each case.

Closed-Form Solutions: Unprotected Debt

Comparative Statics: Unprotected Debt • Table I: These are the comparative statics for an arbitrary coupon C.

Comparative Statics: Unprotected Debt • Most signs as expected. • But when firm is close to bankruptcy (V → VB), some effects reversed (iff τ >0 or α >0). • Since VB is endogenous and VB ↓ if σ ↑ or r ↑ or C ↓ • So, behavior of junk bonds different from investment grade bonds!

• Equity value results (last row) don’t reverse near bankruptcy! • Since bankruptcy costs borne by bondholders.

Comparative Statics: Unprotected Debt; D(V)

Comparative Statics: Unprotected Debt; D(V)

Comparative Statics: Unprotected Debt; D(V)

Comparative Statics: Unprotected Debt; Yield

Comparative Statics: Unprotected Debt; Yield

Comparative Statics: Unprotected Debt; Firm Value

Comparative Statics: Unprotected Debt; Firm Value

Optimal Leverage with Unprotected Debt • At a given asset value V, the coupon C determines the debt level and hence the leverage ratio. • The optimal coupon which maximizes v is:

• Other metrics computed at C* are:

Comparative Statics at Optimal Leverage Ratio, Unprotected Debt • Table II

Comparative Statics at Optimal Leverage Ratio, Unprotected Debt

Comparative Statics at Optimal Leverage Ratio, Unprotected Debt

Now, Protected Debt • Bankruptcy triggered when firm value falls below principal value of debt (D0). • If V0 is asset value when debt is initiated, then D0 is given as: • No closed form solution unless α = 0.

• Plugging VB = D0 gives debt value as a function of V0.

Protected Debt, Comparative Statics

Protected Debt, Comparative Statics

Protected Debt, Comparative Statics

Protected Debt, Comparative Statics • No closed-form solutions for α = 0 • Some differences when compared to unprotected debt.

Comparing with observed values • Empirically Observed: • Leverage in companies with highly rated debt = 40% • Average yield spread of investment grade corporate bonds during 1926-86 = 77 bps • Subtracting 25 bps for call provision premium, avg. yield = 52 bps

• Model parameters: σ2=20%, τ=35%, r=6%, α=50% • Unprotected Debt: Optimal leverage=75%, Yield spread=75 bps, Equity return annual std. dev.=57% • Protected Debt: Optimal leverage=45%, Yield spread=45 bps, Equity return annual std. dev.=34%

Model Extensions • Some assumptions are relaxed and alternatively modeled. 1. No tax shield when asset value falls beyond a point. 2. Firm has net cash outflows after equity financing (so asset value is affected by extent of debt). [Imp] 3. Absolute priority of debtholders not respected.

• When all these are incorporated together: • Unprotected Debt: Optimal leverage=47%, Yield spread=69 bps, Equity return annual std. dev.=36% • Protected Debt: Optimal leverage=32%, Yield spread=52 bps, Equity return annual std. dev.=29%

Protected vs. Unprotected Debt • Asset substitution problem: Equity holders prefer to make firm’s activities riskier to increase equity value at the expense of debt. • But, higher risk benefits equity holders if equity is convex function of V. • This is so with unprotected debt

• With protected debt, equity is concave in V. • Numerically, suppose σ2=20%, τ=35%, r=6%, α=50%

Protected vs. Unprotected Debt • Unprotected debt: optimal C = $6.5, firm value=$128.4, VB=$52.8 • Protected debt: optimal C = $3.26, firm value=$113.3, VB=$50.6 • If asset volatility is changed by managers, then:

• Firm value with unprotected debt and 60% vol. is $111.7 < $113.3.

Summary • Protected & unprotected investment grade bonds behave as expected. • Unprotected junk bonds exhibit different behavior. • Higher risk free rates lead to greater optimal debt level due to tax benefits. • Modified model predicts values close to observed. • Protected debt mitigates agency problems and hence leads to higher firm values. • Equity return volatility changes with firm value.