Guest Lecture
Risk Analysis of Investment-Based Models
Xiaoji Lin
London School of Economics and FMG
University of Minnesota
November 19, 2010
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
1 / 32
Road Map
1
Dynamic programing and value function iteration
2
Benchmark investment-based model
3
Risk analysis
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
2 / 32
Dynamic Programing: Deterministic Case
V (k )
s.t. k
0
= max0 ff (k, k 0 ) + βV (k 0 )g
i ,k
= (1
δ )k + i
where k’means capital in the next period.
If we de…ne the Bellman operator B (V ) which updates a value function V using
the Bellman equation above, we know the following properties:
1
2
V such that B (V ) = V exists.
V = limt !∞ B t (V 0 ) for any continuous function V 0 . In addition, B t (V 0 )
converges to V monotonically.
In other words, if we supply an initial guess V 0 and keep applying the Bellman
operator, we can asymptotically get to the solution V of the Bellman equation.
Value function iteration is the solution method which uses the properties.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
3 / 32
Algorithm: Deterministic Case
1
2
3
4
5
6
7
8
Set nk number of grid points, upper and lower bounds of k, and tolerance
level ε
Set grid points, usually equi-spaced.
Set an initial value V 0
Update value function and obtain V 1 . For each grid point ki , choose the
optimal kj that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 4. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
4 / 32
Algorithm: Deterministic Case
1
2
3
4
5
6
7
8
Set nk number of grid points, upper and lower bounds of k, and tolerance
level ε
Set grid points, usually equi-spaced.
Set an initial value V 0
Update value function and obtain V 1 . For each grid point ki , choose the
optimal kj that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 4. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
4 / 32
Algorithm: Deterministic Case
1
2
3
4
5
6
7
8
Set nk number of grid points, upper and lower bounds of k, and tolerance
level ε
Set grid points, usually equi-spaced.
Set an initial value V 0
Update value function and obtain V 1 . For each grid point ki , choose the
optimal kj that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 4. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
4 / 32
Algorithm: Deterministic Case
1
2
3
4
5
6
7
8
Set nk number of grid points, upper and lower bounds of k, and tolerance
level ε
Set grid points, usually equi-spaced.
Set an initial value V 0
Update value function and obtain V 1 . For each grid point ki , choose the
optimal kj that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 4. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
4 / 32
Algorithm: Deterministic Case
1
2
3
4
5
6
7
8
Set nk number of grid points, upper and lower bounds of k, and tolerance
level ε
Set grid points, usually equi-spaced.
Set an initial value V 0
Update value function and obtain V 1 . For each grid point ki , choose the
optimal kj that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 4. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
4 / 32
Algorithm: Deterministic Case
1
2
3
4
5
6
7
8
Set nk number of grid points, upper and lower bounds of k, and tolerance
level ε
Set grid points, usually equi-spaced.
Set an initial value V 0
Update value function and obtain V 1 . For each grid point ki , choose the
optimal kj that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 4. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
4 / 32
Algorithm: Deterministic Case
1
2
3
4
5
6
7
8
Set nk number of grid points, upper and lower bounds of k, and tolerance
level ε
Set grid points, usually equi-spaced.
Set an initial value V 0
Update value function and obtain V 1 . For each grid point ki , choose the
optimal kj that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 4. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
4 / 32
Algorithm: Deterministic Case
1
2
3
4
5
6
7
8
Set nk number of grid points, upper and lower bounds of k, and tolerance
level ε
Set grid points, usually equi-spaced.
Set an initial value V 0
Update value function and obtain V 1 . For each grid point ki , choose the
optimal kj that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 4. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
4 / 32
Dynamic Programing: Stochastic Case
V (k, x )
s.t. k
0
Z
= max0 ff (k, k 0 , x ) + β V (k 0 , x 0 )dx 0 jx g
i ,k
= (1
δ )k + i
where x 0 is stochastic. For instance, x 0 is an AR(1) process:
x 0 = (1
ρ)x̄ + ρx + σε0
There are many ways to compute the conditional expectation. Here we will use
quadrature method to approximate the continuous x’process.
1
2
Tauchen (1986) and Tauchen and Hussey (1991) - don’t work well when x 0 is
highly persistent
Rouwenhorst (1995) can handle highly persistent AR (1) process.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
5 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Algorithm: Stochastic Case
1
2
3
4
5
6
7
8
9
Set nk number of grid points for k and nx number for x, upper and lower
bounds of k, and tolerance level ε.
Set grid points for k, usually equi-spaced.
Apply Rouwenhorst (1995) to discretize x to get grid points for x and the
transition matrix.
Set an initial value V 0 . Note it is a two-dimensional object now.
Update conditional expected value and update value function and obtain V 1 .
For each grid point xi and kj , choose the optimal kh that gives highest Vi1,j .
Compare V 0 and V 1 and compute the distance d. If d > ε, update value
function V 0 = V 1 and go back to step 5. If d < ε, then we …nd the
approximated optimal value function.
Check if the bounds of state space is not binding.
Make sure that ε is small enough.
Make sure nk is large enough.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
6 / 32
Tips for Numerical Work: Tony Smith
1
2
3
4
5
6
7
8
Start with the simplest possible model, preferably one with an analytical
solution
Add features incrementally.
Never add another feature until you are con…dent of your current results.
Use the simplest possible methods.
Accuracy is more important than speed or elegance.
When you learn (or develop) a new method, test it on the simplest possible
problem, preferably one with an analytical solution.
Don’t program when you are tired. Don’t program too quickly.
Hamming’s motto: The Goal of Computing is Insight, Not Numbers.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
7 / 32
Tips for Numerical Work: Tony Smith
1
2
3
4
5
6
7
8
Start with the simplest possible model, preferably one with an analytical
solution
Add features incrementally.
Never add another feature until you are con…dent of your current results.
Use the simplest possible methods.
Accuracy is more important than speed or elegance.
When you learn (or develop) a new method, test it on the simplest possible
problem, preferably one with an analytical solution.
Don’t program when you are tired. Don’t program too quickly.
Hamming’s motto: The Goal of Computing is Insight, Not Numbers.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
7 / 32
Tips for Numerical Work: Tony Smith
1
2
3
4
5
6
7
8
Start with the simplest possible model, preferably one with an analytical
solution
Add features incrementally.
Never add another feature until you are con…dent of your current results.
Use the simplest possible methods.
Accuracy is more important than speed or elegance.
When you learn (or develop) a new method, test it on the simplest possible
problem, preferably one with an analytical solution.
Don’t program when you are tired. Don’t program too quickly.
Hamming’s motto: The Goal of Computing is Insight, Not Numbers.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
7 / 32
Tips for Numerical Work: Tony Smith
1
2
3
4
5
6
7
8
Start with the simplest possible model, preferably one with an analytical
solution
Add features incrementally.
Never add another feature until you are con…dent of your current results.
Use the simplest possible methods.
Accuracy is more important than speed or elegance.
When you learn (or develop) a new method, test it on the simplest possible
problem, preferably one with an analytical solution.
Don’t program when you are tired. Don’t program too quickly.
Hamming’s motto: The Goal of Computing is Insight, Not Numbers.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
7 / 32
Tips for Numerical Work: Tony Smith
1
2
3
4
5
6
7
8
Start with the simplest possible model, preferably one with an analytical
solution
Add features incrementally.
Never add another feature until you are con…dent of your current results.
Use the simplest possible methods.
Accuracy is more important than speed or elegance.
When you learn (or develop) a new method, test it on the simplest possible
problem, preferably one with an analytical solution.
Don’t program when you are tired. Don’t program too quickly.
Hamming’s motto: The Goal of Computing is Insight, Not Numbers.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
7 / 32
Tips for Numerical Work: Tony Smith
1
2
3
4
5
6
7
8
Start with the simplest possible model, preferably one with an analytical
solution
Add features incrementally.
Never add another feature until you are con…dent of your current results.
Use the simplest possible methods.
Accuracy is more important than speed or elegance.
When you learn (or develop) a new method, test it on the simplest possible
problem, preferably one with an analytical solution.
Don’t program when you are tired. Don’t program too quickly.
Hamming’s motto: The Goal of Computing is Insight, Not Numbers.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
7 / 32
Tips for Numerical Work: Tony Smith
1
2
3
4
5
6
7
8
Start with the simplest possible model, preferably one with an analytical
solution
Add features incrementally.
Never add another feature until you are con…dent of your current results.
Use the simplest possible methods.
Accuracy is more important than speed or elegance.
When you learn (or develop) a new method, test it on the simplest possible
problem, preferably one with an analytical solution.
Don’t program when you are tired. Don’t program too quickly.
Hamming’s motto: The Goal of Computing is Insight, Not Numbers.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
7 / 32
Tips for Numerical Work: Tony Smith
1
2
3
4
5
6
7
8
Start with the simplest possible model, preferably one with an analytical
solution
Add features incrementally.
Never add another feature until you are con…dent of your current results.
Use the simplest possible methods.
Accuracy is more important than speed or elegance.
When you learn (or develop) a new method, test it on the simplest possible
problem, preferably one with an analytical solution.
Don’t program when you are tired. Don’t program too quickly.
Hamming’s motto: The Goal of Computing is Insight, Not Numbers.
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
7 / 32
Benchmark Model a la Zhang (2005)
Pro…t:
π jt
ktj +1
j
= e xt +zt ktj
= itj + (1
α
f
δ)ktj
SDF:
log Mt +1
γt
= log β + γt (xt xt +1 )
= γ 0 + γ 1 ( xt x )
Adjustment costs:
ctj
8
>
>
>
a+ ktj + b + itj +
>
>
<
c (itj , ktj ) =
0
>
>
>
>
j
j
>
: a kt + b it +
where a > a+ , b
adjustment costs.
Lin
< b + , and c
c+
2
c
2
itj
k tj
itj
k tj
2
2
ktj
if itj > 0
if itj = 0
ktj
if itj < 0
> c + capture the nonconvexities of
Risk Analysis & Investment-Based Models
November 19, 2010
8 / 32
Benchmark Model
Dividend:
dtj
v (ktj , xt , ztj )
=
max
fitj ,k tj +1 g
dtj +
ZZ
π jt
ctj
Mt +1 v (ktj +1 , xt +1 , ztj +1 ) Qx (dxt +1 jxt ) Qz dztj +1 jztj
Risk:
Et [rtj +1 ] = rft + βjt λMt
in which rft 1/Et [M t +1 ] is the real interest rate.
And βjt is risk de…ned as:
βjt
Covt [rtj +1 , Mt +1 ]
Vart [Mt +1 ]
and λMt is the price of risk de…ned as λMt
Lin
Vart [M t +1 ]/Et [M t +1 ].
Risk Analysis & Investment-Based Models
November 19, 2010
9 / 32
Calibration
Parameters
Lin
α
δ
0.60
0.10
z
}|
{
Output
ρx
σx
4
0.014
0.98
z
}|
{
Agg Shock
Risk Analysis & Investment-Based Models
ρz
σz
0.70
0.35
z
}|
{
Idio Shock
November 19, 2010
10 / 32
Frictionless Model
The production function is given by:
j
π jt = e xt +zt ktj
0.6
0
Adjustment costs:
ctj
Lin
8
>
>
>
0ktj + 1itj + 0
>
>
<
c (itj , ktj ) =
0
>
>
>
>
j
j
>
: 0kt + 1it + 0
itj
ktj
itj
ktj
Risk Analysis & Investment-Based Models
2
2
ktj
if itj > 0
if itj = 0
ktj
if itj < 0
November 19, 2010
11 / 32
Frictionless Model
Value Functions
70
low z
mid z
high z
60
v
50
40
30
20
10
0
5
10
15
20
25
30
35
40
45
50
k
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
12 / 32
Frictionless Model
Policy Functions
50
low z
mid z
high z
40
30
20
i
10
0
-10
-20
-30
-40
0
5
10
15
20
25
30
35
40
45
50
k
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
13 / 32
Frictionless Model
Conditional Beta
0.45
low z
mid z
high z
0.4
0.35
β
0.3
0.25
0.2
0.15
0.1
0
5
10
15
20
25
30
35
40
45
50
k
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
14 / 32
Frictionless Model
Quantitative Results: 5 productivity portfolios
Z
Y
Low
-0.68
0.43
2
-0.17
1.19
3
0.00
1.35
4
0.17
2.56
High
0.68
4.32
Lin
I
-1.69
-0.94
-0.69
1.46
7.85
Cost
Div
0.00
2.11
0.00
2.06
0.00
2.04
0.00
0.64
0.00
-3.54
K
2.78
7.32
8.84
13.74
28.25
P
14.66
21.00
22.35
31.58
44.90
Risk Analysis & Investment-Based Models
MB
4.21
2.30
2.18
1.69
1.47
Beta
Ret
0.22
8.89
0.12
6.35
0.10
5.82
0.07
4.75
0.01
3.13
November 19, 2010
15 / 32
Convex Costs and Fixed Costs
Quantitative Results: 5 BM portfolios
Growth
2
3
4
Value
Z
-0.68
-0.17
0.00
0.17
0.68
Lin
Y
0.43
1.19
1.35
2.56
4.32
I
-1.33
-0.97
-0.26
0.91
7.63
Cost
Div
0.00
1.73
0.00
2.08
0.00
1.58
0.00
1.21
0.00
-3.29
K
2.78
7.32
8.84
13.74
28.25
Risk Analysis & Investment-Based Models
P
14.66
21.00
22.35
31.58
44.90
MB
4.21
2.30
2.18
1.69
1.47
Beta
0.22
0.12
0.10
0.07
0.01
November 19, 2010
Ret
8.89
6.34
5.82
4.75
3.13
16 / 32
Convex Costs
The production function is given by:
j
π jt = e xt +zt ktj
α
0
Adjustment costs:
ctj
Lin
8
>
>
>
0ktj + 1itj +
>
>
<
c (itj , ktj ) =
0
>
>
>
>
j
j
>
: 0kt + 1it +
1
2
10
2
itj
k tj
2
itj
k tj
Risk Analysis & Investment-Based Models
ktj
2
if itj > 0
if itj = 0
ktj
if itj < 0
November 19, 2010
17 / 32
Convex Costs
Policy Functions
b + = 1; b =
10;
1.2
low z
mid z
high z
1
0.8
i
0.6
0.4
0.2
0
-0.2
Lin
0
1
2
3
4
5
6
Risk Analysis & Investment-Based
Models
k
7
8
9
10
November 19, 2010
18 / 32
Convex Costs
Policy Functions
b + = 1; b =
10;
7
low z
mid z
high z
6
5
ik
4
3
2
1
0
-1
Lin
0
1
2
3
4
5
6
Risk Analysis & Investment-Based
Models
k
7
8
9
10
November 19, 2010
19 / 32
Convex Costs
Conditional Beta
b + = 1; b =
10;
0.65
low z
mid z
high z
0.6
0.55
β
0.5
0.45
0.4
0.35
0.3
0.25
Lin
0
1
2
3
4
5
6
Risk Analysis & Investment-Based
Models
k
7
8
9
10
November 19, 2010
20 / 32
Convex Costs
Quantitative Results: 5 Productivity portfolios
Z
Y
Low
-0.63
0.30
2
-0.27
0.41
3
0.00
0.51
4
0.27
0.67
High
0.63
0.92
Lin
I
0.11
0.18
0.25
0.35
0.51
Cost
0.03
0.03
0.04
0.05
0.07
Div
0.13
0.17
0.21
0.29
0.44
K
2.29
2.53
2.73
2.99
3.37
P
6.10
6.69
7.14
7.83
8.84
Risk Analysis & Investment-Based Models
MB
2.66
2.63
2.62
2.61
2.60
Beta
Ret
0.31
6.72
0.29
6.38
0.27
6.12
0.26
5.78
0.24
5.36
November 19, 2010
21 / 32
Convex Costs
Quantitative Results: 5 BM portfolios
Growth
2
3
4
Value
Z
-0.05
-0.01
0.02
0.04
0.00
Lin
Y
0.48
0.54
0.58
0.63
0.68
I
0.30
0.30
0.29
0.28
0.24
Cost
Div
0.06
0.11
0.05
0.18
0.05
0.23
0.04
0.30
0.03
0.41
K
2.19
2.50
2.74
3.01
3.46
P
MB
6.62
2.97
7.05
2.75
7.37
2.61
7.73
2.48
8.21
2.29
Risk Analysis & Investment-Based Models
Beta
0.29
0.28
0.27
0.26
0.25
Ret
6.33
6.13
5.99
5.88
5.76
November 19, 2010
22 / 32
Convex Costs and Operational Leverage
The production function is given by:
j
π jt = e xt +zt ktj
0.6
0.07
Adjustment costs:
ctj
Lin
8
>
>
>
0ktj + 1itj +
>
>
<
c (itj , ktj ) =
0
>
>
>
>
j
j
>
: 0kt + 1it +
1
2
10
2
itj
k tj
2
itj
k tj
Risk Analysis & Investment-Based Models
ktj
2
if itj > 0
if itj = 0
ktj
if itj < 0
November 19, 2010
23 / 32
Convex Costs and Operational Leverage
Quantitative Results: 5 Productivity portfolios
Z
Y
Low
-0.63
0.28
2
-0.27
0.38
3
0.00
0.46
4
0.27
0.60
High
0.63
0.83
Lin
I
0.09
0.15
0.21
0.29
0.43
Cost
Div
0.02
0.05
0.03
0.10
0.04
0.14
0.05
0.22
0.07
0.36
K
1.98
2.18
2.35
2.57
2.89
P
3.81
4.32
4.70
5.30
6.18
Risk Analysis & Investment-Based Models
MB
1.79
1.85
1.89
1.95
2.02
Beta
0.29
0.26
0.24
0.22
0.19
Ret
9.33
7.91
7.30
6.61
5.94
November 19, 2010
24 / 32
Convex Costs and Operational Leverage
Quantitative Results: 5 BM portfolios
Growth
2
3
4
Value
Z
0.34
0.13
0.01
-0.13
-0.34
Lin
Y
0.62
0.56
0.53
0.51
0.46
I
0.34
0.27
0.23
0.20
0.13
Cost
0.06
0.05
0.04
0.03
0.02
Div
0.16
0.16
0.16
0.18
0.20
K
2.19
2.29
2.37
2.47
2.65
Risk Analysis & Investment-Based Models
P
5.04
4.96
4.94
4.96
4.97
MB
2.16
1.99
1.89
1.80
1.66
Beta
0.23
0.23
0.23
0.24
0.25
Ret
6.26
6.72
7.19
7.71
9.25
November 19, 2010
25 / 32
Convex Costs and Operational Leverage
Time Series Analysis: Firm Speci…c Productivity
Firm-s pec ifc s hoc k
0.6
Low
High
0.4
0.2
Lev el
0
-0.2
-0.4
-0.6
-0.8
50
100
150
200
250
300
350
400
450
500
Year
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
26 / 32
Convex Costs and Operational Leverage
Time Series Analysis:Output
Y
0.55
Low
High
0.5
0.45
0.4
Lev el
0.35
0.3
0.25
0.2
0.15
0.1
0.05
50
100
150
200
250
300
350
400
450
500
Year
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
27 / 32
Convex Costs and Operational Leverage
Time Series Analysis:IK
IK
0.45
Low
High
0.4
0.35
0.3
lev el
0.25
0.2
0.15
0.1
0.05
0
-0.05
50
100
150
200
250
300
350
400
450
500
Year
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
28 / 32
Convex Costs and Operational Leverage
Time Series Analysis:Adj Costs
Adj Costs
0.14
Low
High
0.12
0.1
Lev el
0.08
0.06
0.04
0.02
0
50
100
150
200
250
300
350
400
450
500
Year
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
29 / 32
Convex Costs and Operational Leverage
Time Series Analysis:Dividend
Dividend
0.5
Low
High
0.4
0.3
Lev el
0.2
0.1
0
-0.1
-0.2
50
100
150
200
250
300
350
400
450
500
Year
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
30 / 32
Convex Costs and Operational Leverage
Time Series Analysis: Conditional Beta
Beta
0.25
Low
Low-High
Zero High Z
0.2
Lev el
0.15
0.1
0.05
0
50
100
150
200
250
300
350
400
450
500
Year
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
31 / 32
Future Work
1
External …nancing: debt vs/& equity, credit risk, capital structure choices
2
Multipal capital goods: labor, intangible capital, etc
3
Additional aggregate risks
4
General equilibrium: aggregate implications
Lin
Risk Analysis & Investment-Based Models
November 19, 2010
32 / 32