Corporate Finance: The trade-off theory

Corporate Finance:The trade-off theory

Yossi Spiegel

Recanati School of Business

Corporate Finance 2

The main assumptions

� The timing:

� The entrepreneur wishes to maximize the firm’s value

� X ~ [X0, X1]; dist. function f(X) and CDF F(X)

� The mean earnings are X̂

Period 1 Period 2

The firm is establishedby an entrepreneur

The firm operates -its profit is X

Corporate Finance 3

The dist. function and CDF

X

X

f(X)

f(X)

F(X)

1

Corporate Finance 4

Risky debt� The timing:

� The face value of debt is D

� Debt is paid in full if X ≥ D

� If X < D, the firm goes bankrupt

� Fixed cost of bankruptcy is C < X0 (to avoid uninteresting complications)

� An obvious alternative is proportional costs: C = c(D-X)+

Period 1 Period 2

The firm is establishedby an entrepreneurThe firm issues debt D

The firm operates -its profit is X

Corporate Finance 5

Taxation

� Corporate tax is tc

� Debt is fully deductible

� Interest rate is r (r is the alternative that investors can get for their money)

Corporate Finance 6

All-equity firm: D = 0

( )( ) [ ] ( )444 3444 21

43421

firm in the investing from Payoff

cost eAlternativ

1

0

10 ∫ −=+X

X

c XdFXtXrV

� The value of the firm:

( ) [ ] ( )

( )r

Xt

XdFXtXr

V

c

X

X

c

+

−=

−+

= ∫

1

ˆ1

1

10

1

0

Corporate Finance 7

Riskless debt: D < X0

( ) ( )[ ] ( )

( )( )r

DXt

XdFtDXDXr

DE

c

X

X

c

+

−−=

−−−+

= ∫

1

ˆ1

1

1 1

0

� Equity:

� Debt:

� Total value:

( )r

DDB

+=

1

( ) ( )r

DtXtDBDEDV cc

+

+−=+=

1

ˆ1)()(

Corporate Finance 8

Riskless debt: Implications

� Total value:

� tc = 0 ⇒ Debt is irrelevant ⇒ M&M 1958

� tc > 0 ⇒ The firm will be all-debt ⇒ M&M 1963

� Debt benefits the firm by providing a tax shield

( )r

DtXtDV cc

+

+−=

1

ˆ1)(

Corporate Finance 9

Risky debt: D > X0

Kraus and Litzenberger, JF 1973

( ) ( )[ ] ( )∫ −−−+

=1

1

1X

D

c XdFtDXDXr

DE� Equity:

� Debt:

� Total value:

( ) [ ] ( ) ( )∫∫ ++−

+=

1

01

1

1

1X

D

D

X

XDdFr

XdFCXr

DB

[ ] ( ) ( )[ ] ( )

( ) ( )∫

∫∫

−+

−+

−=

−−+

+−+

=

1

1

0

1

1

1

)(ˆ

1

1

1

1)(

X

D

c

X

D

c

D

X

XdFtDXrr

DCFX

XdFtDXXr

XdFCXr

DV

Corporate Finance 10

First-order conditions

� Simplifying:

� Properties of H(D): H’(D)>0, H(X1)=∞

( ) ( )

( )( ) 011

1

1

)(

1

1)(

1

1

1

)()('

1

=−+

++

−=

++−

++

+−= ∫

DFtrr

DCf

XdFtr

DftDDrr

DCfDV

c

X

D

cc

( )( )C

t

DF

DfDHDFtDCf c

c =−

≡⇒−=)(1

)()(1)(


Hazard rate – Uniform dist.

� F(X)=(X-X0)/(X1-X0)

� f(x) = 1/(X1-X0)

⇒ H’(X)>0

XX

XX

XX

XX

XF

XfDH

−=

−−

−

−=

−≡

1

01

0

01 1

1

1

)(1

)()(


Hazard rate – Exponential dist.

� F(X)=1-e-λx

� f(x) = λe-λx

⇒ H’(X)=0

( ) λλ

λ

λ

=−−

=−

≡−

−

x

x

e

e

XF

XfDH

11)(1

)()(


Illustrating the first-order conditions

X

X1

H(D)

tc/C

D*

tc/C


Illustrating M&M 1958 (C = tc = 0)

X

X1

Xf (X)

Df(x)

D

B(D)

E(D)


Illustrating M&M 1963 (C = 0, tc > 0)

X

X1

Xf (X)

Df(x)

D

B(D)

E(D)

X0

(X-D)(1-tc)f (X)


Illustrating M&M 1963 (C = 0, tc > 0)

X

X1

Xf (X)Df(x)

B(D)

X0

D* = X1


Illustrating K&L 1973 (C > 0, tc > 0)

X

X1

Xf (X)

Df(x)

D

B(D)

E(D)

X0

(X-D)(1-tc)f (X)

Cf(x)


DeAngelo and Masulis, JFE 1980Tax code

X

X1

X

D+SD D+S+G/θtc

(X-D-S)tc

(X-D-S)tc-G

(X-D-S)tC-(X-D-S)θtc


DeAngelo and Masulis, JFE 1980Payoffs

Xtc(X-D-S)-GX-D-tc(X-D-S)+GDX > D+S+G/θtc

Xtc(1-θ)(X-D-S)X-D-tc(X-D-S)+θtc(X-D-S)DD+S < X < D+S+G/θtc

X0X-DDD < X < D+S

X00XX < D

TotalIRSEquity-holdersDebt-holders

State of nature


DeAngelo and Masulis, JFE 1980Equity and debt values

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

( )[ ] ( )

[ ] ( ) ( ) ( )

( ) ( ) ( )∫∫

∫∫

∫

∫∫

++

++

+

+

++

++

+

+

++−−

++

−−+

−−+

=

+−−−−+

+

−−−−−+

+−+

=

1

11

1

/

/

/

/

1

1

1

1

1

1

1

1

1

1

11

1

1

1)(

X

tGSD

tGSD

SD

c

X

SD

c

X

D

X

tGSD

c

tGSD

SD

c

SD

D

c

c

c

c

XGdFr

XdFSDXtr

XdFSDXtr

XdFDXr

XdFGSDXtDXr

XdFSDXtDXr

XdFDXr

DE

θ

θ

θ

θ

θ

θ

( ) ( )∫∫ ++

+=

1

01

1

1

1)(

X

D

D

X

XDdFr

XXdFr

DB


DeAngelo and Masulis, JFE 1980Firm value

( ) ( ) ( )

( ) ( ) ( )∫∫

∫∫

++

++

+

+

++−−

++

−−+

−+

=

1

11

0

/

/

1

1

1

1

1

1

1

1)(

X

tGSD

tGSD

SD

c

X

SD

c

X

X

c

c

XGdFr

XdFSDXtr

XdFSDXtr

XXdFr

DV

θ

θ

θ

( ) ( )∫∫++

++

=+

−+

=ctGSD

SD

c

X

SD

c XdFtr

XdFtr

DV

θ

θ/

01

1

1

1)('

1

( ) ( ) ( )[ ]SDFtGSDFSDF c +−++=+−⇒ θθ /1

Corporate Finance: The trade-off theory

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