Corporate Finance:Capital structure and corporate control

Yossi Spiegel

Recanati School of Business

Grossman and Hart, BJE 1980

“Takeover Bids, the Free-Rider Problem, and the Theory of the Corporation’’

Corporate Finance 3

The free rider problem� The timing:

� Consider an individual equityholder with equity participation α

� Let γY be the prob. that the raid succeeds if the equityholder tenders and and γN if he does not tender. The equityholder will tender iff

Period 1 Period 2

The firm is establishedby an entrepreneur and is worth X

A raider appears and can increase valueby R; the raider makes a tender offer X+P

( ) ( )[ ] ( ) ( )[ ]4444 34444 214444 34444 21

t tenderDon'Tender

11 XRXXPX NNYY γγαγγα −++≥−++

Corporate Finance 4

The free rider problem

� Suppose that each individual equityholderis atomistic ⇒ γY = γN= γ

� The condition for tendering:

� The raider’s profit:

� The raider gets nothing and will not ride if the ride requires a cost C!

( ) ( )[ ] ( ) ( )[ ] { {t tenderDon'Tender

t tenderDon'Tender

11 RPXRXXPX ≥⇒−++≥−++444 3444 21444 3444 21γγαγγα

( ) ( ) 0

Paymentpayoffpost Ex

≤−=+−+ PRPXRX4342143421

Corporate Finance 5

Toeholds

� The minimal P to induce atomistic equityholders to tender is R

� If the raider has a fraction β in the firm to begin with then his payoff is

� The takeover will take place iff βR > C, where C is the cost of takeover

( ) ( )( ){

RXRXRX βββ =−−+−+eoverAbsent takover with takePayoff

14444 34444 21

Corporate Finance 6

Dilution

� If the takeover succeeds, the raider can “steal” φ from the firm (φ is implied by the firm’s charter)

� The condition for tendering:

� The raider’s payoff

� The takeover will succeed iff φ > C

( ) ( )[ ] ( ) ( )[ ] { 3214444 34444 21444 3444 21t tenderDon'

Tendert tenderDon'Tender

11 φγφγαγγα −≥⇒−+−+≥−++ RPXRXXPX

( ) CCRXRX −=−−+−+ φφ43421321

Paymentpayoffpost Ex

Corporate Finance 7

Probabilistic C

� Suppose that C ~ [0, ∞) according to F(C)

� The takeover succeeds with prob. F(φ)

� The firm’s value ex ante:

( ) ( )( ) ( )( ) ( ) {φ

φφφφ−

+=−++=R

PFXXFPXFV 1

Corporate Finance 8

The optimal choice of φ

� F.O.C for φ:( ) ( )( ) ( )

{( )( )

( ) 10'

P decreased fromcost Marginal

takeoverofprob increased from

benefit Marginal

=−⇒=−−= φφφ

φφφφ RF

fFRfV

43421

φ

1

φ* R

( )( )

( )φφφ

−RF

f

Stulz, JFE 1988

“Managerial Control of Voting Rights: Financing Policies and the Market for

Corporate Control’’

Corporate Finance 10

The model

� The timing:

� The raider has benefits of control B~[0, ∞)

� To take over the firm the raider needs ½ of the equity

� The raider can try to acquire shares from outsiders. The supply of shares is (1-α)S(p), where S’(p) > 0 (more outsiders submit shares when p is higher)

Stage 0 Stage 2

The firm is established,the entrepreneur keeps equity α and issues 1-αto outsiders

A raider shows up andmay offer p to acquire ½ of the shares

Stage 1

Cash flow X is realized irrespective of the takeover

Corporate Finance 11

Takeover

� To induce a takeover, p needs to be

� p’(α) > 0: α↑ ⇒ p↑

1/2

p

(1-α)S(p)

P(α)

( ) ( ) ( )αα pppS =⇒=−2

11

α↑

Corporate Finance 12

Takeover

� The raider will take over iff B ≥ p(α) ⇒the prob. of takeover is 1-F(p(α))

� The entrepreneur’s payoff:

( ) { ( )( ) ( ) ( )( )( ) ( )[ ]

( )( ) ( ) ( )( )( ) ( )αααα

αααααα

ppFpFBX

ppFXpFBXY

E

E

−−++=

−+−++=

11

11

shares Soldbenefits Privateshares Retained

44444 344444 2143421

Corporate Finance 13

The optimal choice of α� F.O.C for α:

� When α → ½, then (1-α)S(p) = ½ iff S(p) → 1 ⇒ p → ∞ and F(p) → 1:

( ) ( ) ( )[ ] ( )( ) ( )

( ) ( )( )( ) ( ) ( )( )( ) ( )

0

1'11

'1'

effectQuantity shares sold of price on theEffect

takeoverof prob. on theEffect

=

−−−−+

−−=

444 3444 214444 34444 21

44444 344444 21

ααααα

ααααα

ppFppF

ppfpBY E

( ) ( ) ( )( ) ( ) 021'212

2121' <

−=

∞=

ppfp

BY E

321

Corporate Finance 14

The optimal choice of α

� When α = 0:

� If p’(0) > p(0) then y’(0) > 0 so α* > 0

( ) ( )[ ] ( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( )

( )[ ] ( )( ) ( ) ( )( )( ) ( ) ( )( )00'010'00

0010'010'000'

effect Ownershipeffect Price takeoverof prob. on theEffect

pppFppfpB

ppFppFppfpBY

E

E

−−+−=

−−−+−=44 344 21444 3444 214444 34444 21

Israel, JF 1991

“Capital Structure and the Market for Corporate Control: The Defensive Role of Debt Financing”

Corporate Finance 16

The model

� The timing:

� The value of the firm under the entrepreneur is 0

� The value of the firm under the raider is R ~[0, ∞) with mean ER

� If a takeover takes place, R is split between the entrepreneur and the raider in proportions γ and 1-γ

Stage 0 Stage 2

The firm is establishedby an entrepreneur and issues debt with face value D

A raider shows up andmay takeover the firm

Stage 1

Cash flow X is realized and debt is due

Corporate Finance 17

Analysis

� All-equity firm:

� The raider comes, takes over the firm and pays γER

� Leveraged firm:

� The raider takes over the firm iff R > D and pays the entrepreneur γ(R-D)

� If R < D there is no takeover; since the firm is worth 0 under the entrepreneur, debtholdersreceive nothing

Corporate Finance 18

The choice of debt

� The value of the firm:

� The f.o.c for D:

( ) ( ) ( ) ( )4342144 344 21

DebtEquity

∫∫∞∞

+−=DD

RDdFRdFDRDV γ

( ) ( ) ( ) ( )

( ) ( )( ) ( )0

11

'

=

+−−−=

++−= ∫∫∞∞

DDfDF

DDfRdFRdFDVDD

γ

γ

Corporate Finance 19

The choice of debt

� Rewriting f.o.c:

� At the optimum,� D* > 0

� D* < ∞

� Comparative stats:� γ↑ ⇒ D↓

( )( )

( ) γ−=≡−

11

DDHDF

DDf

Corporate Finance 20

Illustrating the first-order conditions

D

DH(D)

1-γ

D*

Corporate Finance 21

Illustrating the model

R1

Rf(R)

Df(R)

D

γRf(R)