phd course in corporate finance

PhD Course in Corporate Finance

Capital Structure and the Modigliani-Miller Theorem

Ernst Maug1

Revised February 18, 2019

1Professor of Corporate Finance, University of Mannheim; Homepage:http://cf.bwl.uni-mannheim.de, Tel: +49 (621) 181-1952, E-Mail:[email protected]. This note is made publicly available subject to thecondition that any user noti�es the author of its use. Please bring any errorsand omissions to the attention of the author.

Ernst Maug PhD Course in Corporate Finance

Notation

X Operating pro�t of �rm

VU Value of unlevered �rm (= EU)

VL Value of levered �rm (= DL + EL)

DL Debt of levered �rm

EL Equity of levered �rm

α Fraction of shares of unlevered �rm held by some investor

R Interest rate (= 1 + r)


Assumptions

Su�cient conditions:

Borrow and lend at the same risk-free rate R

Perfect capital markets

No taxes

No bankruptcy (costs)

Static model


Suppose an investor buys a share α of the unlevered �rm:Investor pays at time 0: αEU = αVU

Investor receives at time 1: αX

The same payo� can be obtained as follows. Combine two

transactions:

1 Buy α of the shares in the levered �rm.

2 Buy α of the debt in the levered �rm.


Cash �ow consequences if �rm pays risk-free interest at rate R :

Transaction Investment (time 0) Payo� (time 1)

Buy levered equity αEL = α (VL − DL) αDivL = α(X − RDL

)Buy debt αDL αRDL

Total αVL αX

Suppose VL < VU . Then investor buys only securities in levered�rm. Hence VL ≥ VU .

What if VL > VU? What if there is no short-selling of holdingsof levered �rm?

What if X − RDL < 0? How do we square this with limitedliability?


Now suppose investor already holds a position in the levered �rm.Payo�s are:

Transaction Investment (time 0) Payo� (time 1)

Buy levered equity αEL = α (VL − DL) αDiv = α(X − RDL

)


The same payo�s can be obtained as follows, assuming investorscan borrow at the risk-free rate R :

1 Buy fraction α of the equity of the unlevered �rm.

2 Borrow αDL on personal account.

Cash �ows now look as follows:

Transaction Invest (time 0) Payo� (time 1)

Buy unlevered equity αEU = αVU αDivU = αX

Borrow −αDL −αDS = −αRDL

Total α (VU − DL) α(X − RDL

)= αDivL

Suppose VL > VU . Then investor buys only securities in theunlevered �rm. Cash �ow on RHS can be bought for αVL, henceVU ≥ VL. Together with the previous argument (or if short sellingis permitted) this implies that VL = VU .


MM in incomplete markets

Based on Hellwig (1981)

How can we accommodate bankruptcy?

Is it realistic that investors can borrow at the same terms asthe �rm?

How important are short-selling constraints?


MM and intermediaries

Idea: Consider intermediaries (�hedge funds�):

Invest in �rms' stocks and bonds. Borrow, and issue leveredequity in the fund against these investments.

HF can invest in the same proportions as capital structure ofthe �rm (�MM contracts�): funds can undo the capitalstructure of the �rm.

Then HF's costs of borrowing are higher (lower, the same)than those of the �rm if they borrow more (less, the same):law of one price.

Any capital structure of the �rm can be replicated with buyingand short-selling of these contracts: then capital structure ofthe �rm does not matter.

With restrictions on short-selling, capital structure of the �rmcannot always be rebuilt with MM contracts. If not, capitalstructure matters.


MM in a state-preference framework

Based on Kraus and Litzenberger (1973)

Assume a General Equilibrium framework with the followingnotation:

Payo� of �rm X (θ)

Tax rate T

Risk-free rate R

Recovery rate ρ ≤ 1

Payo� to equityholders e (θ)

Payo� to debtholders d (θ)

Valuation function based on state prices E (D) =∫e (θ) dπ (θ)


The case without taxes

The payo� to debtholders and the value of debt is (denote byBC =bankruptcy region, → X (θ) < D; NB =non-bankruptcyregion):

d =

{D if X ≥ D

ρX if X < D⇒ B (D) =

∫NB

Ddπ (θ)+

∫BC

ρX (θ) dπ (θ) ,

where the second part represents the payo� in bankruptcy. Thepayo� to equity holders is:

e =

{X − D if X ≥ D

0 if X < D⇒ E (D) =

∫NB

(X (θ)− D

)dπ (θ) .

The value of the �rm is therefore (drop θ)

V (D) = B(D) + EL =

∫X dπ − (1− ρ)

∫BC

X dπ.


Results without taxes

Proposition

If ρ = 1 then V (D) =∫X dπ, which is independent of D, hence

MM holds.

Remark

This follows without assuming any of the following: (1) debt isriskless; (2) securities fall into homogeneous risk classes; (3)assumptions about risk aversion; (4) assumptions aboutprobabilities. Complete markets are su�cient.


Proposition

If ρ = 1 then V (D) =∫X dπ, which is independent of D, hence

MM holds.

Remark

This follows without assuming any of the following: (1) debt isriskless; (2) securities fall into homogeneous risk classes; (3)assumptions about risk aversion; (4) assumptions aboutprobabilities. Complete markets are su�cient.


The case with taxes

With a positive tax rate T > 0 the valuations for equity and the�rm become (assume D is tax-deductible)

E (D) =

∫NB

(1− T )(X − D

)dπ

V (D) =

∫NB

((1− T ) X + DT

)dπ +

∫BC

ρX dπ

=

∫(1− T ) X dπ −

∫BC

(1− T ) X dπ +

∫NB

DTdπ+∫BC

ρX dπ.

For simplifying the expressions for the BC region, observe that

−(1− T ) + ρ = − (1− ρ) (1− T ) + ρT .


Recall: B(D) =∫BC ρX dπ +

∫NB Ddπ (value of bond).

De�ne: V (0) =∫

(1− T ) X dπ (value of the unlevered �rm).De�ne: BCC =

∫BC (1− T ) (1− ρ) X dπ (bankruptcy costs).

Hence:

V (D) =

∫(1− T ) X dπ︸︷︷︸

V (0)

+

∫NB

DTdπ +

∫BC

ρTXdπ︸︷︷︸TB(D)

−∫BC

(1− T ) (1− ρ) X dπ︸︷︷︸−BCC

=V (0) + TB (D)− BCC .


Result:V (D) = V (0) + TB(D)− BCC .

Remark

This equation can be read as VL = VU + TB−bankruptcy costs.

Remark

(Miller-Modigliani 1963). If bankruptcy costs are zero (ρ = 1),VL = VU + TB .

Exercise

Which conditions need to hold for a unique interior solution for theoptimal debt level D∗ that maximizes V (D)?


Are �rms underlevered?

Idea: Calibrate the parameters of the trade-o�:

Loss given default (LGD) is about 10% - 23% (Andrade andKaplan 1998), hence ρ is between 0.8 and 0.9.

Estimate the cumulative probability of default on a bond. ForBBB bonds this probability is 5.2% (Almeida and Philippon2007).

Multiplying these numbers gives pre-tax expected costs of�nancial distress of about 1% of �rm value.

The net tax bene�ts of debt are 2.6% to 4.8% (Graham 2000).

Marginal tax bene�ts are much larger than marginal costs of�nancial distress. Firms could add 7.5% of �rm value byincreasing leverage (Graham 2000).


Are �rms really underlevered?

Based on Almeida and Philippon (2007)

Argument:

Historical frequencies of default measure the statisticalprobability of default.

The trade-o� model requires that we look at the risk-adjustedprobabilities of default. If

bankruptcies occur in �bad� times

bad times have a higher risk-adjusted probability

⇒ then the conventional method of calibrating costs of

�nancial distress may lead to an understatement.


Idea: Infer risk-neutral probabilities of bankruptcy from bondprices.

Consider 1-period example with bond that has price 1 and (oneplus) yield to maturity Y . The risk-neutral probability of default isq and the recovery rate ρ. Then:

B = 1 =(1− q) + qρ

RY .

Now solve for q:

q =Y − R

Y (1− ρ).

With Y − R = 0.0139, Y = 1.067, ρ = 0.41 we obtain q = 0.022,which contrasts with p = 0.0053 (historical average).


Estimate costs of �nancial distress

Assume:

φ = 0.165 is the fraction of �rm value lost in �nancial distress(midpoint of Andrade/Kaplan range of 10%-23%).

Φ is the present value of all future costs of �nancial distress.

Take an annual model and assume in each year two things canhappen:

1 The �rm goes bankrupt and costs of �nancial distress φ occur

at the end of the year.2 The �rm survives the next year and at the end, �rm value is

reduced by Φ, which re�ects the costs of �nancial distress for

bankruptcies after the next year.


Then Φ obeys:

Φ =qφ+ (1− q) Φ

R.

Solving for Φ gives:

Φ =qφ

R − 1 + q.

The previous parameters then give Φ = 0.046 or 4.6%, more thanfour times the 1%−estimate obtained before!


Risk-neutral vs. historical default probabilities

Use enhanced formulae to account for the term structure of interestrates and credit spreads. Adjust yields for non-default relatedfactors (e.g., liquidity).

Credit rating Historical Risk-neutral

AAA 0.80% 1.65%

AA 0.96% 6.75%

A 1.63% 12.72%

BBB 5.22% 20.88%

BB 21.48% 39.16%

B 46.52% 62.48%


Risk-adjusted costs of �nancial distress

Credit rating Historical Benchmark

AAA 0.25% 0.32%

AA 0.29% 1.84%

A 0.51% 3.83%

BBB 1.40% 4.53%

BB 4.21% 6.81%

B 7.25% 9.54%

Note: The value of 4.53% for BBB-rated bonds is based on the fullmethodology, the 4.6% before is based on back-of-an envelopecalculation using simpli�ed formulae.


The pecking-order theory

Based on Myers and Majluf (1984)

Model: 3 dates, new projects and assets in place; all quantities inpresent value terms

1 Time 0: investment in assets in place.

Symmetric information.

Common knowledge of distributions:

2 Time 1: information about returns on assets in place and newproject become available.

Cash �ow from assets in place at time 2 = a.Cash �ow of new project at time 2 = n , NPV = n− I . Known

only to managers.


3 Assets in place generate cash �ow X1 = 0, commonknowledge. Managers decide to invest or not.

If investment, then spend I = 100 on project, raise money

from outside investors by issuing α shares to new shareholders.

4 Time 2: Payo�s (a, n) become common knowledge. Grosscash �ows of the �rm to investors are:

X2 =

{a + n if investment at time 1a if no investment at time 1

(1)

If there was no investment, X2 goes to old shareholders. Ifmoney was raised to �nance investment, X2 is split betweenold (1− α) and new (α) investors.


Uncertainty and �nancing

Payo�s are unknown to the market at time 1.

There are two types and the market has beliefs µ, which assignprobabilities µ(1), µ(2) to both types. The prior of the marketis that both states are equally probable, i.e.µ(1) = µ(2) = 0.5.

At time 1, the �rm can issue either equity or not invest:

equity: market values �rm at value V and �rm issues α shares

so that αV = I = 100.

no investment: payo� is a from assets in place


Consider example with two types and I = 100:

Type 1 (Pr=1

2) Type 2 (Pr=1

2) Exp. Val.

Assets in place (a) 150 50 100

New Investment (n) 120 110 115

Existing assets (X1) 0 0 0

Total (a + n) 270 160 215

Assume also that managers can only use equity to �nance theinvestment.


Claim

(i) There is no equilibrium in pure strategies of this example wheremanagers invest in both states. (ii) There exists an equilibriumsuch that type 1 managers do not invest and only type 2 managersinvest.

Proof.

(i) Suppose there would exist such an equilibrium. Then managersinvest and raise equity in both states. Hence market beliefscontingent on observing a choice �e� (equity) are µ (1; e) =µ (2; e) = 0.5. Then V = 215 and α = 100/215. The payo�s toold shareholders are:

V old (1, e) = (1− α)X2 =115

215∗ 270 = 144.42

V old (1, 0) = X2 (1, 0) = 150

Then managers of type 1 choose not to invest.


Proof.

(ii) If only managers of type 2 invest, then the market always hasµ (1; e) = 0, µ (2; e) = 1. Then α = 100/160:

V old (1, e) = (1− α)X2 =60

160∗ 270 = 100.25

< X2 (1, 0) = 150

V old (2, e) = (1− α)X2 =60

160∗ 160 = 60

> X2 (2, 0) = 50.

Hence, actions are optimal in the conjectured equilibrium.

Implication: Always prefer debt to equity if information isasymmetric.


phd course in corporate finance

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