part 1: q theory and irreversible investment notation

Investment Model q Theory Irreversible Investment Real Options Perfect Competition

Part 1: q Theory and Irreversible Investment

Goal: Endogenize firm characteristics and risk.

Value/growthSizeLeverageNew issues, . . .

This lecture:

q theory of investmentIrreversible investment and real optionsValue and risk under perfect competition


Notation

r = interest rateδ = depreciation rate

Kt = capital stockk0 = initial capital stockIt = investment rate

θ(i , k) = investment costB = BM under risk-neutral probability

dX = µ(X ) dt + σ(X ) dBπ(x , k) = operating cash flowJ(x , k) = market value of firm


Value of Firm

J(k , x) = supI

E∫ ∞

0e−rt [π(Xt ,Kt)− θ(It ,Kt)] dt

subject to

dK = I dt − δK dt ,K0 = k ,X0 = x .


Examples of investment cost θ

Costless adjustment: θ(i , k) = ai for constant a.Quadratic (and linearly homogeneous):θ(i , k) = ai + bi2/k .Purchase price of capital different from resale price:θ(i , k) = ai+ − bi−.Irreversible investment (zero resale price): θ(i , k) = ai+.Irreversible investment with fixed costs:θ(i , k) = ai+ + f1{i>o}.


Operating Cash Flow

Usual model: Assume there is a production function y = kα`β

for α, β > 0 with α+ β ≤ 1. Labor is hired in a perfectlycompetitive market at wage rate w . The industry demand curvehas constant elasticity: p = Xy−1/γ , where X is a GBM.Different versions are obtained from

Constant (α+ β = 1) or decreasing (α+ β < 1) returns toscale .Perfect competition, monopoly, or Cournot oligopoly.

We end up with something like π(x , k) = xkλ.


Linearity/Concavity

Operating cash flow is linear in capital (λ = 1) if there areconstant returns to scale and perfect competition.Operating cash flow is strictly concave in capital (λ < 1)otherwise.


Basic q Theory

Assume θ is linearly homogeneous, so θ(i , k) = kφ(i/k).

If φ is differentiable and strictly convex, then the optimalinvestment-to-capital ratio is a function of the marginalvalue of capital (marginal q).In the quadratic case, the optimal investment-to-capitalratio is an affine function of the marginal value of capital.If π is linear in k , then the marginal value of capital equalsthe average value of capital (marginal q equals average q).In other words, Jk (x , k) = J(x , k)/k .


Proof

HJB Equation:

0 = supi

[π(k , x)− θ(i , k)− rJ(k , x) + Jxµ+ Jk (i − δk) +

12

Jxxσ2].

First-order condition: θi = Jk ⇔ φ′(i/k) = Jk .

Strict convexity implies φ is strictly decreasing, hence invertible, soi/k = (φ′)−1(Jk ).

If φ(y) = ay + by2, then φ′(i/k) = Jk ⇔ i/k = (Jk − a)/(2b).

If π(k , x) = f (x)k , guess J(k , x) = g(x)k and verify. The HJB equationsimplifies to

0 = supi/k

[f (x)− φ(i/k)− rg(x) + µg′(x) + g(x)(i/k − δ) +

12

g′′(x)σ2].

This equation is independent of k . Solve it for g (given boundary conditions).


Nonlinear π

Suppose π(x , k) = xkλ with λ < 1. Then average q is largerthan marginal q.

Suppose π(x , k) = xk − c for a constant c. Then average q isless than marginal q. This is called operating leverage.


Irreversible Investment

Assume θ(i , k) = i+ and π is monotone in k . Let I now denotecumulative investment instead of the investment rate. Then

J(x , k) = supI

E∫ ∞

0e−rt [π(Kt ,Xt) dt − dIt ]

subject to

dK = dI − δK dt ,K0 = k ,X0 = x ,

and subject to I being an increasing process.


Zero Depreciation

.

δ = 0 ⇒ Kt = k + It . Some references in which π dependsdirectly on the control process:

Karatzas, I. and S. E. Shreve, 1984, “Connections between Optimal Stoppingand Singular Stochastic Control I. Monotone Follower problems,” SIAM J. Contr.Opt. 6, 856–877.

Scheinkman, J. A. and T. Zariphopoulou, 2001, “Optimal EnvironmentalManagement in the Presence of Irreversibilities,” J. Econ. Theory 96, 180–207.

Bank, P., 2005, “Optimal Control under a Dynamic Fuel Constraint,” SIAM J.Contr. Opt. 44, 1529–1541.

Back, K. and D. Paulsen, 2009, “Open-Loop Equilibria and Perfect Competition inOption Exercise Games,” Rev. Fin. Stud. 22, 4531–4552.

Aguerrevere, F., 2009, “Real Options, Product Market Competition, and AssetReturns, ”J. Fin. 64, 957–983.

Steg, J.-H., 2012, “Irreversible Investment in Oligopoly,” Finance Stoch. 16,207–224.


Assets in Place and Growth Options

Write π′ = πk and

π(x , k) = π(x , k0) +

∫ k

k0

π′(Xt , `) d` ,

so the objective function is

E∫ ∞

0e−rtπ(Xt , k0) dt + E

∫ ∞0

e−rt

[∫ Kt

k0

π′(Xt , k) dk dt − dIt

].

The first term is the value of assets in place. The maximizedvalue of the second term is the value of growth options.


First-Order Condition

Alternative to dynamic programming when δ = 0.See Bank and Riedel (AAP, 2001), Bank (SIAM J. Contr.Opt., 2005), Steg(FS, 2012).Given K , define a “gradient” D by

Dτ = Eτ∫ ∞τ

e−rtπ′(Xt ,Kt) dt − e−rτ

for all stopping times τ .Given some technical conditions, a necessary andsufficient condition for K to be optimal is that D ≤ 0 and∫ ∞

0Dt dKt = 0 .


Real Options

Rather than choosing the optimal capital stock Kt at eachdate t , we can equivalently choose the optimal date t at whichto invest the unit of capital k for each k ≥ k0.

In this formulation, the value of growth options is the integral ofa continuum of call option values, indexed by the level k ≥ k0 ofthe capital stock.

The equivalence is based on changing the order of integrationand using

τk = inf{t | Kt > k} ,

which is the right-continuous inverse of the path t 7→ Kt .

From the optimal investment times τk , we can recover theoptimal capital stock process as Kt = inf{k | τk > t}.


Real Options cont.

The underlying asset for option k has price

S(x , k) = E[∫ ∞

te−r(u−t)π′(Xu, k) du

∣∣∣∣ Xt = x].

The benefit of investment is that you earn the marginal profit π′

in perpetuity after investment, which has value S(Xt , k) atdate t .

The options are perpetual. The strikes equal 1 (the price ofcapital).

The value of growth options is∫ ∞k0

supτk

E[e−rτk{S(Xτk , k)− 1}

]dk .


Value Matching and Optimal Capital

Define the value of option k :

V (x , k) = supτ

E[

e−rτ{S(Xτ , k)− 1}∣∣ X0 = x

].

When it is optimal to invest, we must have value matching:V (Xt , k) = S(Xt , k)− 1.

In other words, the optimal exercise boundary is{(k , x) | V (Xt , k) = S(Xt , k)− 1}.

Given x , the largest capital stock such that it would beoptimal to invest is

κ(x) def= sup {k | V (x , k) = S(x , k)− 1} .

Hysteresis: The optimal capital stock process is

Kt = k0 ∨ sup0≤s≤t

κ(Xs) .


Real Options and q Theory

The value-matching condition can be expressed in the qlanguage.

The marginal value of investment (marginal q) isS(y , x)− V (y , x).

Investing earns the marginal cash flow π′ in perpetuity butextinguishes the option.Hence, the marginal value of investing is S − V .In fact, a direct calculation shows thatJk (x , k) = S(x , k)− V (x , k).

So, value matching can be expressed as: invest when marginalq equals 1.


Perfect Competition

Assume constant returns to scale and perfect competition, soπ(x , y , k) = h(x , y)k for some h, where y denotes industrycapital (which is exogenous from the point of view of any firm).

The equilibrium condition for perfect competition is thatinvestment occurs as soon as any investment option reachesthe money (no barriers to entry). Because the options are neverstrictly in the money, growth options have zero value.

The value of any firm is the value of its assets in place:

J(x , y , k) = kE[∫ ∞

0e−rt f (Xt ,Yt) dt | Xt = x ,Yt = y

]def= kq(x , y) .


Risk

The return on the firm is its dividend yield plus capital gain:

π dt − dKt + dJJ

.

Because J = Kq and K is continuous with finite variation,

dJJ

=dKK

+dqq.

Because the firm only invests when q = 1 ⇔ J = K , we havedK/K = dK/J, so the return is

π dtJ

+dqq.

If investment were perfectly reversible, industry capital wouldadjust to maintain q = 1, and we would have π/J = r . Withirreversibility, fluctuations in q (the market-to-book ratio) addrisk.


Example

The production function is f (k) = k . The industry output price isPt = XtY

−1/γt , where X is a GBM with coefficients µ and σ,

where µ < r .

Let β denote the positive root (guaranteed to be larger thanone) of the quadratic equation

12σ2β2 +

(µ− 1

2σ2)β = r .

The investment options reach the money when Pt reaches

p∗cdef= (r − µ)

(β

β − 1

).

In equilibrium, P is a GBM reflected at p∗c .


Example cont.

Marginal q = Average q =β

β − 1Pt

p∗c− 1β − 1

(Pt

p∗c

)β.

The stochastic part of dq/q is(1− (Pt/p∗c)β−1

β − (Pt/p∗c)β−1

)βσ dB .

Note that risk decreases as Pt increases towards p∗c , vanishingat Pt = p∗c .

The model suggests several testable implications for the behavior of the conditionalvolatility of returns and investment:

(i) firms’ q should be negatively related to the instantaneous conditional volatilityof returns and should serve as a predictor of return volatility in the near future;

(ii) a similar relation should hold between q and conditional expected returns;(iii) a relatively low investment rate should be contemporaneously associated with a

relatively high volatility of realized returns and should predict high volatility ofreturns.

The third prediction of the model can be related to the empirical findings of Leahyand Whited (1996), who investigate the dependence of the investment rate on theuncertainty faced by the firm. In particular, they regress the rate of investment andthe market-to-book ratio on the predictable component of return variance, which inturn is defined as a one-year-ahead forecast of variance. Leahy and Whited reportthat a 10% increase in the predicted return variance leads to a 1:7% fall in the rate ofinvestment and a 5% decline in q:2 They then conclude that firms respond to anincrease in uncertainty about the future economic environment by reducinginvestment. Note, however, that using the volatility of stock prices as a proxy for

ARTICLE IN PRESS

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.05

0.1

0.15

0.2

0.25

Tobin's q

Con

ditio

nal v

olat

ility

, σR

Fig. 1. Conditional volatility of stock returns. sR denotes the conditional volatility of stock returns of

firms in the second sector. The argument is the average q of firms in the second sector, defined as the ratio

of their market value to the replacement cost of their capital. The subjective time preference rate is

r ¼ 0:05; the expected return and the volatility of the production technology of the first sector are given by

a ¼ 0:07 and s ¼ 0:17; respectively, and the depreciation rate of capital in the second sector is d ¼ 0:05:The preference parameter b is assumed to be small. The three lines correspond to the preference parameter

g of 12(dash), 1 (solid), and 3

2(dash-dot).

2Leahy and Whited (1996) adjust their measures of volatility for financial leverage. This is consistent

with the definitions in the model, where firms are financed entirely by equity.

L. Kogan / Journal of Financial Economics 73 (2004) 411–431422

Kogan, L., 2004, “Asset Prices and Real Investment,” JFE 73, 411–431. The threelines correspond to different levels of risk aversion of the representative investor.


Explanation

The risk comes from the dependence of q on X . When Xchanges, the demand for capital changes.

Suppose the demand for capital increases.

If the supply of capital were perfectly elastic, then thequantity supplied would increase with no change in q.If the supply of capital were perfectly inelastic, then qwould increase with no change in the quantity supplied.

Perfectly reversible investment implies a perfectly elastic supplyof capital. Irreversibility implies that supply is elastic only atq = 1.


Bounded Investment Rate

A simple example of convex adjustment costs is

θ(i , k) =

0 if i ≤ 0 ,i if 0 ≤ i ≤ imax ,

∞ otherwise .

This is equivalent to the constraint 0 ≤ dIt/dt ≤ imax. At theoptimum, the firm will invest at the maximum rate wheneverq > 1.

now q can exceed one and volatility is no longer a monotonic function of q:Qualitatively, the state space can be partitioned into the following three regions:

First region (Low values of q: q51): Firms do not invest and irreversibilityprevents them from disinvesting. Thus, the elasticity of supply is relatively low andstock returns are relatively volatile.

Second region (Intermediate values of q: qC1): Firms are either about to invest,following an increase in q; or are already investing at the maximum possible rate andare about to stop, following a decline in q: The elasticity of supply is relatively highand, as a result, q is not sensitive to shocks and does not contribute much to stockreturns.

Third region (High values of q: qb1): The industry is expanding. Firms areinvesting at the maximum possible rate and are likely to continue investing during anextended period of time. Demand shocks do not immediately change the rate ofentry into the industry, the elasticity of supply is low, and demand shocks are offsetmostly by changes in the output price. Thus, q is relatively volatile and so are stockreturns.

When the rate of investment is allowed to be very high, q rarely exceeds one. Thus,the third regime can be observed only infrequently, during periods of active growthin the industry. In the extreme case of instantaneous adjustment, as in the basic

ARTICLE IN PRESS

0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

Tobin's q

Con

ditio

nal v

olat

ility

, σ R

Fig. 4. Conditional volatility of stock returns, bounded investment rate. sR denotes the conditional

volatility of stock returns of firms in the second sector. The argument is the average q of firms in the second

sector, defined as the ratio of their market value to the replacement cost of their capital. The subjective

time preference rate is r ¼ 0:05; the expected return and the volatility of the production technology of the

first sector are given by a ¼ 0:07 and s ¼ 0:17; respectively, and the depreciation rate of capital in the

second sector is d ¼ 0:05: The maximum investment rate is imax ¼ 0:2: The preference parameter b is

assumed to be small. The three lines correspond to three values of the risk-aversion parameter g: 12(dash), 1

(solid), and 32(dash-dot).

L. Kogan / Journal of Financial Economics 73 (2004) 411–431 425

Kogan, L., 2004, “Asset Prices and Real Investment,” JFE 73, 411–431. We expect

similar figures for more general convex adjustment costs: risks are high when the

adjustment costs are particularly constraining.

part 1: q theory and irreversible investment notation

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