Stochastic Vol Presentation

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

The Q Theory of Investment with Stochastic Volatility François Gourio Michael Michaux Finance Department The Wharton School University of Pennsylvania

5th December 2008

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Empirical Performance of the Q-theory Investment regressions’ stylized facts for large firms (top quartile of firms sorted by size of the capital stock in 1981) are taken from Eberly, Rebelo, and Vincent (2008).

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Bond Q Philippon (Forth. QJE) uses a Hayashi model with costless default (MM world) and obtains a mapping from bond yields to Tobin’s Q. Using a simple setup, he obtains the following characterization.

Aggregate and firm level investment regressions using numerically constructed bond Q show much improved fit. François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Aggregate Investment Regressions

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Usual Measure of Q and Bond Market’s Q

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Usual Measure of Q and Investment Rate (levels)

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Bond Market’s Q and Investment Rate (levels)

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Results: Aggregate Data

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Firm Level Investment Regressions

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Empirical Performance of the Q-theory Philippon (Forth. QJE)

Results: Firm Level Data

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Step 1 - Contrasting the existing facts Step 2 - Framing the problem in terms of the literature Step 3 - Proposing a different explanation Why Do Standard Models Fails? How is a Model with Stochastic Volatility Different?

Step 1 - Contrasting the existing results

The correlation between I/K and Q is empirically weak. In the standard corporate finance/investment models1 , the correlation between I/K and Q is strong.

This gap between theory and the empirics begs the question why.

1 such as Hayashi (ECO 1982), Gomes (AER 2001), Cooley and Quadrini (AER 2001), and Hennessy and Whited (JF 2005) François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Step 1 - Contrasting the existing facts Step 2 - Framing the problem in terms of the literature Step 3 - Proposing a different explanation Why Do Standard Models Fails? How is a Model with Stochastic Volatility Different?

Step 2 - Framing the problem in terms of the literature Some papers have argued that the data is at fault. The most notable paper is from Erickson and Whited (JPE 2000), where they appeal to mismeasurements of Q using GMM estimation. Some papers have argued that financing frictions are responsible. For example, Hennessy, Levy, and Whited (JFE 2007) develop a Q theory of investment under financing constraints. Eberly et al. (WP 2008) develop a Generalized Hayashi model with Regime Switching and replicates the investment regressions’ stylized facts when adding measurement noise to Q.

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Step 1 - Contrasting the existing facts Step 2 - Framing the problem in terms of the literature Step 3 - Proposing a different explanation Why Do Standard Models Fails? How is a Model with Stochastic Volatility Different?

Step 3 - Proposing a different explanation This research proposes an alternative theoretical explanation of the weak empirical correlation between I/K and Q. The empirical stylized facts are assumed to be true. The data (especially Q) is not mismeasured. The bare bones of the model will be Hayashi 1982. The model in this paper will departs from Hayashi on the real side only in 2 respects: (a) firms exhibit DRS, and (ii) experience stochastic volatility in their productivity shock.

Intuition: The addition of stochastic volatility essentially weakens the correlation between investment and Q. François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Step 1 - Contrasting the existing facts Step 2 - Framing the problem in terms of the literature Step 3 - Proposing a different explanation Why Do Standard Models Fails? How is a Model with Stochastic Volatility Different?

Standard Models The standard models feature (i) a productivity shock z, (ii) capital accumulation k (CRS/DRS, with or without adjustment cost), (iii) debt b (priced or with a collateral constraint/debt capacity), (iv) equity issuance (from costless to infinite cost).

These models look something like that, V (k , b, c, z) = max k 0 ,b0

s.t.



d − λg(−d) + βE max(0, V (k 0 , b0 , c 0 , z 0 ))



d = (1 − τ )ez k α + (1 − δ)k − k 0 + b0 − b(1 + c) + τ (δk + cb), log(z 0 ) = ρ log(z) + σ.

The discount price satisfies the usual Euler equation, 

b0 = βE



1 b0 (1 + (1 − τ )c 0 )1{V (s0 )>0} + θk 0 (1 − 1{V (s0 )>0} ) . 1−τ

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Step 1 - Contrasting the existing facts Step 2 - Framing the problem in terms of the literature Step 3 - Proposing a different explanation Why Do Standard Models Fails? How is a Model with Stochastic Volatility Different?

Intuition

Assume a good shock hits, i.e. z = z HIGH . What happens? 1

MPK increases, which implies that capital investment increases, i.e. I/K ↑.

2

Equity value V increases (as Vz > 0), thus making Q increase, i.e. Q ↑.

3

This effect yields a POSITIVE correlation between I/K and Q.

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Step 1 - Contrasting the existing facts Step 2 - Framing the problem in terms of the literature Step 3 - Proposing a different explanation Why Do Standard Models Fails? How is a Model with Stochastic Volatility Different?

Model with Stochastic Volatility ˆ = (1 + (1 − τ )c)b. Let b ˆ z, σ) = max V (k , b, ˆ0 k 0 ,b

s.t.

h

d − λg(−d) + βE max(0, V (k 0 , bˆ0 , z 0 , σ 0 ))

i

d = (1 − τ )(ez k α − f ) + (1 − δ(1 − τ ))k − k 0 +

ˆ0 b ˆ − b, 1 + (1 − τ )c 0

log(z 0 ) = ρ log(z) + σ 0 , σ is Markov .

The discount price satisfies the usual Euler equation, h

b0 =

βE

1 1−τ

ˆ0 1{V (s0 )>0} + θk 0 (1 − 1{V (s0 )>0} ) b 1+

τ 1−τ



βE 1{V (s0 )>0}



i

.

The implied coupon rate is then, c0 =

1 1−τ

François Gourio and Michael Michaux

 ˆ0



b −1 . b0 The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Step 1 - Contrasting the existing facts Step 2 - Framing the problem in terms of the literature Step 3 - Proposing a different explanation Why Do Standard Models Fails? How is a Model with Stochastic Volatility Different?

Intuition Assume a bad volatility shock hits, i.e. σ = σ HIGH . What happens? Effect on equity: 1

2

The continuation value max(0, V (s0 )) increases, due to the call option feature of equity. This will increase V , and thus Q ↑.

Effect on investment: 1

2

The pricing schedule c 0 (σ 0 ) increases, due to the put option feature of debt. This will (i) decrease debt b and/or (ii) increase yields c 0 . This can potentially reduce capital investment I/K ↓

This effect yields a NEGATIVE correlation between I/K and Q. François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Model with Growth Options and Stochastic Volatility System of PDE’s Region 1 Region 2

Model with Growth Options and Stochastic Volatility Let the productivity (or demand function) be given by, dXt = µdt + σj dwt , Xt

j = L, H.

The Bellman equation is given by, V (X , σj ; cij ) = (1−τ )(Xt Kiα −cij )dt +(1+rdt)−1 E[V (Xt +dXt , σj +λjj 0 dt(σj 0 −σj ); cij 0 )]. Apply Ito’s Lemma, rV =

1 VXX σj2 Xt2 + VX µXt + λjj 0 [V (X , σj 0 ; cij 0 ) − V (X , σj ; cij )] + (1 − τ )(Xt Kiα − cij ). 2

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Model with Growth Options and Stochastic Volatility System of PDE’s Region 1 Region 2

System of PDE’s The system of PDE’s to solve is, 0

=

0

=

1 2 2 H σ X V + µXt VXH − rV H + λH (V L − V H ) + (1 − τ )(Xt Kiα − ciH ), 2 H t XX 1 2 2 L σ X V + µXt VXL − rV L + λL (V H − V L ) + (1 − τ )(Xt Kiα − ciL ). 2 L t XX

The boundary conditions are given by, (Default)

V (XjD , σj ) = 0, ∀j = L, H

(Default)

VX (XjD , σj ) = 0, ∀j = L, H

(Boundedness)

lim

X →∞

V (X , σj ) X

< ∞, ∀j = L, H

We will show later that the default boundary are such that XLD > XHD . Thus we need to partition X into regions over which the system of PDE’s will be solved.

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Model with Growth Options and Stochastic Volatility System of PDE’s Region 1 Region 2

System of PDE’s

The system of PDE’s can be partitioned over 2 regions: On the region XLD ≤ X , 0

=

0

=

1 2 2 H σ X V + µXt VXH − rV H + λH (V L − V H ) + (1 − τ )(Xt Kiα − ciH ), 2 H t XX 1 2 2 L σ X V + µXt VXL − rV L + λL (V H − V L ) + (1 − τ )(Xt Kiα − ciL ). 2 L t XX

On the region XHD ≤ X ≤ XLD , 0

=

1 2 2 H σ X V + µXt VXH − (r + λH )V H + (1 − τ )(Xt Kiα − ciH ). 2 H t XX

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Model with Growth Options and Stochastic Volatility System of PDE’s Region 1 Region 2

Region 1: [XLD , +∞) The system of homogeneous PDE’s to solve is, 0

=

0

=

1 2 2 H σ X V + µXt VXH − rV H + λH (V L − V H ), 2 H t XX 1 2 2 L σ X V + µXt VXL − rV L + λL (V H − V L ). 2 L t XX

Conjecture the solution to be, V H = AH X ν ,

and

V L = AL X ν .

The system of PDE’s translates into a system of algebraic equations, 

0

=

0

=



1 2 σ ν(ν − 1) + µν − r − λH AH X ν + λH AL X ν , 2 H   1 2 σL ν(ν − 1) + µν − r − λL AL X ν + λL AH X ν . 2

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Model with Growth Options and Stochastic Volatility System of PDE’s Region 1 Region 2

Region 1: [XLD , +∞)

These equations must hold for all X , thus we have a system MA = 0, where A ≡ (AH , AL )0 , and, "

M≡

1 2 σ ν(ν 2 H

− 1) + µν − r − λH λL

#

λH 1 2 σ ν(ν − 1) + µν − r − λL L 2

.

This implies det(M) = 0, yielding a 4th order polynomial equation in ν, 

1 2 σ ν(ν − 1) + µν − r − λH 2 H



1 2 σ ν(ν − 1) + µν − r − λL 2 L



− λH λL = 0.

Denote the 2 negative roots by ν1 and ν2 , and the 2 positive roots by ν3 and ν4 .

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

Model with Growth Options and Stochastic Volatility System of PDE’s Region 1 Region 2

Region 2: [XHD , XLD ] The homogeneous PDE to solve is, 0

=

1 2 2 H σ X V + µXt VXH − (r + λH )V H . 2 H t XX

Conjecture the solution to be, V H = CX β . The homogeneous PDE translates into an algebraic equation, 0

=

1 2 2 σ β + 2 H



µ−

1 2 σ 2 H



β − (r + λH ).

The roots of that equation, denoted by β1 and β2 , are given by, {β1 , β2 } = −

µ 1 − 2 2 σH

!

François Gourio and Michael Michaux

±

v u u t

µ 1 − 2 2 σH

!2

+2

r + λH . 2 σH

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

General Solutions Boundary Conditions Solution

General Solutions The mature firm corresponds to the capital level i = 1. The value function for the mature firm can be expressed as follows. On the region XLD ≤ X , (1 − τ ) (1 − τ ) XKiα − ciH , r −µ r (1 − τ ) (1 − τ ) XKiα − ciL . + r −µ r

V (X , σH ; c1H )

=

A1H X ν1 + A2H X ν2 + A3H X ν3 + A4H X ν4 +

V (X , σL ; c1L )

=

A1L X ν1 + A2L X ν2 + A3L X ν3 + A4L X ν4

On the region XHD ≤ X ≤ XLD , V (X , σj ; c1H ) = C1 X β1 + C2 X β2 +

François Gourio and Michael Michaux

(1 − τ ) (1 − τ ) XK α − ciH . r + λH − µ i r + λH

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

General Solutions Boundary Conditions Solution

Boundary Conditions The boundary conditions are, lim

(Boundedness)

X →∞

V (X , σj ) X

< ∞, ∀j = L, H,

(Default)

V (XjD , σj ) = 0, ∀j = L, H

(Default)

VX (XjD , σj ) = 0, ∀j = L, H lim V (X , σH ) = lim V (X , σH ),

(Continuity )

X ↑XLD

(Smooth Pasting)

X ↓XLD

lim VX (X , σH ) = lim VX (X , σH ).

X ↑XLD

X ↓XLD

In addition, the following must hold, AkL = Lk AkH ,

where

Lk =

2 ν (ν − 1) − µν r + λH − 12 σH k k k

François Gourio and Michael Michaux

λH

,

∀k = 1, 2, 3, 4.

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

General Solutions Boundary Conditions Solution

Solution

The boundedness condition implies that A3H = A3L = A4H = A4L = 0. The optimal default conditions are given by, 0

=

0

=

0

=

0

=

(1 − τ ) (1 − τ ) X DK α − c1H , r + λH − µ H i r + λH (1 − τ ) K α, C1 β1 (XHD )β1 −1 + C2 β2 (XHD )β2 −1 + r + λH − µ i

C1 (XHD )β1 + C2 (XHD )β2 +

(1 − τ )XLD K1α

(1 − τ )c1L , r α (1 − τ )K1 A1L ν1 (XLD )ν1 −1 + A2L ν2 (XjD )ν2 −1 + . r −µ

A1L (XLD )ν1 + A2L (XLD )ν2 +

François Gourio and Michael Michaux

r −µ

−

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

General Solutions Boundary Conditions Solution

Solution

The continuity and smooth pasting conditions are given by, 0

=

C1 (XLD )β1 + C2 (XLD )β2 − A1H (XLD )ν1 − A2H (XLD )ν2 λH λH − (1 − τ )XLD K1α + (1 − τ )c1H , (r + λH − µ)(r − µ) r (r + λH )

0

=

C1 β1 (XLD )β1 −1 + C2 β2 (XLD )β2 −1 − A1H ν1 (XLD )ν1 −1 − A2H ν2 (XLD )ν2 −1 λH − (1 − τ )XLD K1α . (r + λH − µ)(r − µ)

The following 2 equations complete the system of 8 equations and 8 unknowns, A1L = L1 A1H ,

François Gourio and Michael Michaux

A2L = L2 A2H .

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

General Solutions Boundary Conditions Solution

System of equations (1 − τ ) (1 − τ ) X DK α − c1H , r + λH − µ H i r + λH (1 − τ ) C1 β1 (XHD )β1 −1 + C2 β2 (XHD )β2 −1 + K α, r + λH − µ i

C1 (XHD )β1 + C2 (XHD )β2 +

0

=

0

=

0

=

0

=

0

=

C1 (XLD )β1 + C2 (XLD )β2 − A1H (XLD )ν1 − A2H (XLD )ν2 λH λH − (1 − τ )XLD K1α + (1 − τ )c1H , (r + λH − µ)(r − µ) r (r + λH )

0

=

C1 β1 (XLD )β1 −1 + C2 β2 (XLD )β2 −1 − A1H ν1 (XLD )ν1 −1 − A2H ν2 (XLD )ν2 −1 λH − (1 − τ )XLD K1α . (r + λH − µ)(r − µ)

(1 − τ )XLD K1α

(1 − τ )c1L , r (1 − τ )K1α , A1H L1 ν1 (XLD )ν1 −1 + A2H L2 ν2 (XLD )ν2 −1 + r −µ

A1H L1 (XLD )ν1 + A2H L2 (XLD )ν2 +

François Gourio and Michael Michaux

r −µ

−

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

General Solutions Boundary Conditions Solution

X The coefficients can be expressed as a function of the default thresholds, 



C1 C2

A1H A2H





=−





=−

(XHD )β1 β1 (XHD )β1 −1

l1 (XLD )ν1 l1 ν1 (XLD )ν1 −1

(XHD )β2 β2 (XHD )β2 −1

−1

∗

l2 (XLD )ν2 l2 ν2 (XLD )ν2 −1

−1

! (1−τ ) (1−τ ) X D K α − r +λ c1H r +λH −µ H i H , (1−τ ) Kα r +λH −µ i 0

∗@

(1−τ )XLD K1α (1−τ )c1L − r −µ r (1−τ )K1α r −µ

0

=

C1 (XLD )β1 + C2 (XLD )β2 − A1H (XLD )ν1 − A2H (XLD )ν2 λH λH − (1 − τ )XLD K1α + (1 − τ )c1H , (r + λH − µ)(r − µ) r (r + λH )

0

=

C1 β1 (XLD )β1 −1 + C2 β2 (XLD )β2 −1 − A1H ν1 (XLD )ν1 −1 − A2H ν2 (XLD )ν2 −1 λH − (1 − τ )XLD K1α . (r + λH − µ)(r − µ)

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

1 A.

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

System of PDE’s General Solutions Boundary Conditions Solution

System of PDE’s Although debt can only be issued when an investment is made, the market value of debt is calculated for each capital level Ki and volatility level σj . The debt value satisfies the Bellman equation, B(X , σj ; ci ) = ci dt + (1 + rdt)−1 E[B(Xt + dXt , σj + λjj 0 dt(σj 0 − σj ); ci )]. Apply Ito’s Lemma, 0=

1 BXX σj2 Xt2 + BX µXt − rB + λjj 0 [B(X , σj 0 ; ci ) − B(X , σj ; ci )] + ci . 2

The system of PDE’s to solve is, 0

=

0

=

1 2 2 H σ X B + µXt BXH − rB H + λH (B L − B H ) + ci , 2 H t XX 1 2 2 L σ X B + µXt BXL − rB L + λL (B H − B L ) + ci . 2 L t XX

Again, we need to partition X into regions over which the system of PDE’s will be solved. François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

System of PDE’s General Solutions Boundary Conditions Solution

System of PDE’s

The system of PDE’s can be partitioned over 2 regions: On the region XLD ≤ X , 0

=

0

=

1 2 2 H σ X B + µXt BXH − rB H + λH (B L − B H ) + ci , 2 H t XX 1 2 2 L σ X B + µXt BXL − rB L + λL (B H − B L ) + ci . 2 L t XX

On the region XHD ≤ X ≤ XLD , 0

=

1 2 2 H σ X B + µXt BXH − (r + λH )B H + ci . 2 H t XX

François Gourio and Michael Michaux

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

System of PDE’s General Solutions Boundary Conditions Solution

General Solutions

The debt value function can be expressed as follows. On the region XLD ≤ X , B(X , σH ; ci )

=

B1H X ν1 + B2H X ν2 + B3H X ν3 + B4H X ν4 +

B(X , σL ; ci )

=

B1L X ν1 + B2L X ν2 + B3L X ν3 + B4L X ν4 +

ci , r

ci . r

On the region XHD ≤ X ≤ XLD , B(X , σH ; ci ) = D1 X β1 + D2 X β2 +

François Gourio and Michael Michaux

ci . r + λH

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

System of PDE’s General Solutions Boundary Conditions Solution

Boundary Conditions The boundary conditions are, (Boundedness) (Default) (Continuity ) (Smooth Pasting)

lim

B(X , σj ; ci )

X →∞

X

B(XjD , σj ; ci ) = ξ

< ∞, ∀j = L, H,

(1 − τ )XjD Kiα

, ∀j = L, H r −µ lim B(X , σH ; ci ) = lim B(X , σH ),

X ↑XLD

X ↓XLD

lim BX (X , σH ; ci ) = lim BX (X , σH ).

X ↑XLD

X ↓XLD

In addition, the following must hold, BkL = Lk BkH ,

where

Lk =

2 ν (ν − 1) − µν r + λH − 12 σH k k k

François Gourio and Michael Michaux

λH

,

∀k = 1, 2, 3, 4.

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

System of PDE’s General Solutions Boundary Conditions Solution

Solution The boundedness condition implies that B3H = B3L = B4H = B4L = 0. The optimal default conditions are given by, ci , r + λH ci + , r

0

=

D1 (XHD )β1 + D2 (XHD )β2 +

0

=

B1L (XLD )ν1 + B2L (XLD )ν2

The continuity and smooth pasting conditions are given by, λH ci , r (r + λH )

0

=

D1 (XLD )β1 + D2 (XLD )β2 − B1H (XLD )ν1 − B2H (XLD )ν2 −

0

=

D1 β1 (XLD )β1 −1 + D2 β2 (XLD )β2 −1 − B1H ν1 (XLD )ν1 −1 − B2H ν2 (XLD )ν2 −1 .

The following 2 equations complete the system of 8 equations and 8 unknowns, B1L = L1 B1H ,

François Gourio and Michael Michaux

B2L = L2 B2H .

The Q Theory of Investment with Stochastic Volatility

Literature Research 101 Model Solution for the Mature Firm (Firm with No Growth Option) Debt Value

System of PDE’s General Solutions Boundary Conditions Solution

System of equations

ci , r + λH ci + B2H L2 (XLD )ν2 + , r

0

=

D1 (XHD )β1 + D2 (XHD )β2 +

0

=

B1H L1 (XLD )ν1

0

=

D1 (XLD )β1 + D2 (XLD )β2 − B1H (XLD )ν1 − B2H (XLD )ν2 −

0

=

D1 β1 (XLD )β1 −1 + D2 β2 (XLD )β2 −1 − B1H ν1 (XLD )ν1 −1 − B2H ν2 (XLD )ν2 −1 .

François Gourio and Michael Michaux

λH ci , r (r + λH )

The Q Theory of Investment with Stochastic Volatility