Leland 1994

Corporate Debt Value, Bond Covenants, and Optimal

Capital StructureHayne E. LelandJournal 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 doesnt 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 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: Dont allow for debt that is coupon-paying and has finite

maturity

Dont 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

corporate tax rate

bankruptcy costs

r risk free interest rate

2 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 assets 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 equity-financed payout C must obey equation (4).

A Generic Model

A0, A1, A2 determined by boundary conditions.

Let VB be asset value that triggers bankruptcy, represent bankruptcy costs.

When bankrupt, VB is incurred, debtholders get (1-)*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.

Well 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 >0 or >0). Since VB is endogenous and VB if or r or C

So, behavior of junk bonds different from investment grade bonds!

Equity value results (last row) dont reverse near bankruptcy!

Since bankruptcy costs borne by bondholders.

Comparative Statics: Unprotected Debt; D(V)

Comparative Statics: Unprotected Debt; Yield

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

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 = 0.

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

Protected Debt, Comparative Statics

Protected Debt, Comparative Statics

No closed-form solutions for = 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: 2=20%, =35%, r=6%, =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 firms 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 2=20%, =35%, r=6%, =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.

Documents

Leland 1994