Transcript Production Functions - Massachusetts Institute of Technology

Lattice Valuation of Flexibility
Richard de Neufville
Professor of Engineering Systems
and of
Civil and Environmental Engineering
MIT
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 1 of 50
Outline

Uncertainty manifested in evolution of
Outcomes of uncertain process,
— Probabilities associated with these outcomes
— Impacts on system of these uncertain outcomes
—

Integrating Elements of System

Analysis of Value of Flexibility
Principle: a multi-stage decision analysis
— Practice: examples “on” and “in” systems
—

Graphical Illustration of results
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 2 of 50
Manifestations of Uncertainty (1)
Three elements part of valuation of flexibility:
1. The uncertain process that generates a
range of possible outcomes, for example
– Demand or Price of product
– Quantity or Quality of product
– Tax Regime, Environmental Regulations, etc.

Usual to assume that range expands as we
project farther into future
—
Distribution changes over time
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 3 of 50
Manifestations of Uncertainty (2)
2. Probabilities associated with outcomes,
that is, the chance that a state is achieved
•
Usual to assume these probabilities
─ Part of a Diffusion process (as for lattice
projection) , so over time …
─ More chance of extreme outcomes
─ Central outcomes less likely
•
Alternatives possible
─ Example: Probability of any state constant…
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 4 of 50
Manifestations of Uncertainty (3)
The diffusion of probabilities in pictures
PDF @ t = 1...6
1.20
Evolution of uncertainty
12000
10000
1.00
8000
V
0.80
6000
4000
0.60
2000
0.40
0
0
1
2
3
4
5
6
time (years)
0.20
0.00
1
2
3
4
5
6
7
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Graphs from K Konstantinos
Richard de Neufville
Lattice Valuation

Slide 5 of 50
7
Manifestations of Uncertainty (3)
3. Impacts on system: effects of uncertain
outcomes on system performance.
For example, how
—
—
—

Demand or Price impacts profitability
Quantity or Quality determines performance
Tax Regime, Environmental Regulations, etc
alter the efficiency of a system.
Models required to translate uncertain
outcomes into system performance
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 6 of 50
Integration of Elements


Any uncertain outcome …
Influences the performance of the system
The PDF of the uncertain outcomes…
leads to another, different PDF, of system
performances which may be
—
Automatic (no system management), or
Shaped by intelligent control: System
Managers take advantage of flexibility to
adapt system to uncertain environment
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 7 of 50
Example Integration of Elements




The technological system is a copper mine…
The uncertainty concerns price of copper…
Profits depend on price of copper -- but not
linearly, because there are large fixed costs
and variable operating costs…
Operators can use flexibility to shape profits
Close mine if prices low; expand if prices high
— Alter “mine plan” to allocate digging operations
most effectively between exploiting rich deposits
and getting rid of sterile overburden
—
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation
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Slide 8 of 50
Valuation of Flexibility



The question before the system managers:
“What is the value of the flexibility?”
When they answer this, they will know if:
— value of flexibility > cost of acquiring it
— Flexibility should be designed into system
The analytic question is:
“How do we value the flexibility?”
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 9 of 50
Valuation Process (general)





The value of flexibility – of options – is
an Expected Value
… defined like value of information
Decision Analysis of situation without added
flexibility gives “base case” expected value
DA incorporating flexibility gives new EV
Value of Flexibility is the difference
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation
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Slide 10 of 50
Valuation Process (in detail)




Just like a decision analysis!
Lay out the possible states over all periods
With their associated probabilities
From perspective of last period:
—
—
—
knowing the value of the possible results –
calculate the expected value of the best choice
This is the value for the beginning of that period
Repeat process until start of 1st period, which
gives expected value of tree
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation
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Slide 11 of 50
Example situation




A copper mine… producing 5000 tons/year
Control for 6 periods
Revenue/period = 5000 x (price, end of period)
Current price is $2000/ton and we suppose
—
—

The annual $ costs of operating the mine are:
—

Average price increase, v = 5% / year
Standard deviation, σ, is 10%
1,000,000 + 2,200(tons produced)
Discount rate is 12%
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation
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Slide 12 of 50
Monthly Global Price of Copper 1976-2006*
(U.S. Dollar per Lb.)
$4.50
*Data derived from United Nations Conference on Trade and Development.
"Commodity Price Statistics online."
<http://www.unctad.org/Templates/Page.asp?intItemID=1889&lang=1>.
Accessed September 9, 2006 through license of MIT Libraries.
$4.00
$3.50
Oct 09: 2.95
U.S. Dollar ($)
$3.00
Oct 08: 1.45
$2.50
$2.00
$1.50
$1.00
$0.50
$Jan-76
Jan-78
Jan-80
Jan-82
Jan-84
Jan-86
Jan-88
Jan-90
Jan-92
Jan-94
Jan-96
Jan-98
Jan-00
Jan-02
Jan-04
Jan-06
Year
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Richard de Neufville
Lattice Valuation
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Slide 13 of 50
Aerial View
of “Chuqui”,
in Chile.
The mine is
about 1 mile
long.
Other mines
and sites as
marked.
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Lattice Valuation
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Slide 14 of 50
An Open Pit Mine
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Institute of Technology
Source:
www.codelco.com
Richard de Neufville
Lattice Valuation

Slide 15 of 50
Big Trucks !
Source: Briony Hall, BBC News, 10 Dec.
2003 “Monster Trucks Transform Mining”
(in Botswana)
http://news.bbc.co.uk/2/hi/business/32938
89
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Richard de Neufville
Lattice Valuation
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Slide 16 of 50
Blasting
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Source: www.codelco.com
Richard de Neufville
Lattice Valuation

Slide 17 of 50
Moving the rock
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www.codelco.com
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Lattice Valuation
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Slide 18 of 50
Crushing Mill
Source:
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www.codelco.com
Richard de Neufville
Lattice Valuation
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Slide 19 of 50
Smelting
Source:
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Richard de Neufville
Lattice Valuation
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Slide 20 of 50
The Product: copper sheets
Source:
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Richard de Neufville
Lattice Valuation
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Slide 21 of 50
Example situation (repeat)




A copper mine… producing 5000 tons/year
Control for 6 periods
Revenue/period = 5000 x (price, end of period)
Current price is $2000/ton and we suppose
—
—

The annual $ costs of operating the mine are:
—

Average price increase, v = 5% / year
Standard deviation, σ, is 10%
1,000,000 + 2,200(tons produced)
Discount rate is 12%
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 22 of 50
Evolution of Price -- parameters


The historic data enable us to project the
evolution of the copper prices
First we calibrate p, u, d (see lattice slides)
p
0.75
0.5 + 0.5 (ν/σ) (Δt)0.5
u
1.105171
e exp[ (σ) (Δt)0.5]
d
0.904837
1/u
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 23 of 50
Price Evolution –
States and Probabilities
To get (here from binomial lattice.xls):
PROBABILITY LATTICE
1.00
0.75
0.25
0.56
0.38
0.06
0.42
0.42
0.14
0.02
0.32
0.42
0.21
0.05
0.00
0.24
0.40
0.26
0.09
0.01
0.00
0.18
0.36
0.30
0.13
0.03
0.00
0.00
Note: low
values of
poor
results
(1-p) = 1/4
OUTCOME LATTICE
2000
2210
1810
2443
2000
1637
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2700
2210
1810
1482
2984
2443
2000
1637
1341
3297
2700
2210
1810
1482
1213
Richard de Neufville
Lattice Valuation
3644
2984
2443
2000
1637
1341
1098

Slide 24 of 50
Impact of Uncertainty on System

Each state of uncertainty affects system
Here: price of copper in any year affects revenues
Revenue = Tons(price) – (Fixed Cost) – Tons(2200)
= 5000 (price-2200) – 1,000,000
—

=> lattice of net income (losses in red):
—
most entries red, but their probabilities are low
2,000,000
948,291
2,951,626
214,028
2,000,000
3,812,692
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1,498,588
948,291
2,951,626
4,591,818
2,918,247
214,028
2,000,000
3,812,692
5,296,800
4,487,213
1,498,588
948,291
2,951,626
4,591,818
5,934,693
Richard de Neufville
Lattice Valuation
6,221,188
2,918,247
214,028
2,000,000
3,812,692
5,296,800
6,511,884

Slide 25 of 50
How bad is this project?



Quick look at possible outcomes makes
project look terrible…
HOWEVER, PDF is skewed toward success –
probability of losses quite small… (slide 24)
Here is picture of [probability x net income]
which shows contribution to expected value
2,000,000
711,218
737,906
120,391
750,000
238,293
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632,217
400,060
415,072
71,747
923,352 1,064,837 1,107,238
90,293
592,703 1,038,771
421,875 250,038
63,487
178,720 259,420 263,672
20,691
67,263
125,662
5,796
23,277
1,590
Richard de Neufville
Lattice Valuation

Slide 26 of 50
Base Case Value of System

Base Case assumes no flexibility
—

Production is “automatic” -- it continues even if
price low. (Might be required by contract)
Annual revenues = Sum of yearly columns
Annual Expected Revenues
1,449,125
NPV
867,903
254,663
392,359
1,075,023
1,795,294
$398,112
Notes to Calculation of Expected NPV:
1. Annual revenue assumed to depend on end of year
price. Initial, start of year $2000 price is not used
2. Annual expected revenues discounted at 12%
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 27 of 50
Flexibility “on” the System



Assume system operators can close mine
permanently in any year.
What is the value of this flexibility?
As discussed later in course, this flexibility
is a “put option” , “on” the system
“option” because it is “right, not obligation” to
change operations
— “put” because it gets operators out of losses
— “on” system, because it does not change
technology of system
—
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 28 of 50
Decision to Use Flexibility



When to exercise option is NOT OBVIOUS!
Consider evolution of possible revenues…
Should operator close in 1st year because of
possible loss?
Not clear! Good chance of big recovery!!
2,000,000
948,291
2,951,626
214,028
2,000,000
3,812,692
Engineering Systems Analysis for Design
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1,498,588
948,291
2,951,626
4,591,818
2,918,247
214,028
2,000,000
3,812,692
5,296,800
4,487,213
1,498,588
948,291
2,951,626
4,591,818
5,934,693
Richard de Neufville
Lattice Valuation
6,221,188
2,918,247
214,028
2,000,000
3,812,692
5,296,800
6,511,884

Slide 29 of 50
Non-convexity of Feasible Region

Note carefully – in general, the feasible
region is not convex
—

As evolution of system follows upward bending
exponential growth (slide 5, repeated next)
This has an important consequence:
Looking at marginal conditions (also known as
“myopic rule”) is not sufficient
— Distant, longer-run may overtake short-run losses
— See presentation on Dynamic Programming
—
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 30 of 50
Non-convexity of Feasible Region
Evolution of uncertainty
12000
10000
Graph from
Konstantinos
V
8000
6000
4000
2000
0
0
1
2
3
4
5
6
7
time (years)
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation
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Slide 31 of 50
Analysis of Decision to Use Flexibility

To simplify, we restate revenues in millions
2.00
0.95
2.95
0.21
2.00
3.81
1.50
0.95
2.95
4.59
2.92
0.21
2.00
3.81
5.30
4.49
1.50
0.95
2.95
4.59
5.93
6.22
2.92
0.21
2.00
3.81
5.30
6.51
Note: red figures conventionally indicate losses


We now analyze as with decision tree…
For example, suppose we at the end of the
5th year with the worse prices (boxed cell) –
what would our decision be?
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 32 of 50
Decision at a particular state

From this state, prospects for last (6th) year are
losses > closed mine: 5.30 > 1 ; 6.51 > 1
2.00

0.95
2.95
0.21
2.00
3.81
1.50
0.95
2.95
4.59
2.92
0.21
2.00
3.81
5.30
4.49
1.50
0.95
2.95
4.59
5.93
6.22
2.92
0.21
2.00
3.81
5.30
6.51
So, from this state, best choice is exercise
option and “close mine”
— This avoids larger losses and
— … changes the NPV as seen from that state
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation
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Slide 33 of 50
Value seen from “5.93” state

Seen from this state the values change
From
To
5.93

5.30
6.51
5.93
1.00
1.00
Expected PV from “5.93” state (that is, over
last year) then is loss over the last (6th) year,
discounted over one year:
= NPV [p (-1) + (1-p)(-1)] = - 0.89 (at 12%)
Note: fixed costs assumed to be unavoidable
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation
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Slide 34 of 50
Note on discount rate


Analysis here uses a constant discount rate
throughout. This is consistent with standard
practice, as industry and government
organizations typically require (see US
Government mandates shown in lecture on
discount rates).
However, from economic perspective, it is
better to adjust rates to risk content.
—
For example, discount “sure” fixed costs at lower
rate than uncertain, “risky” variable costs
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 35 of 50
Value seen from another state

What if you were in best possible state at
end of 5th year?
9.43
4.49



11.69
6.22
7.39
2.92
You would not close mine
Expected PV for last year is discounted
expectation over possible 6th year states:
= NPV [ p (6.22) + (1-p)(2.92)] = 4.82 (at 12%)
Process can be repeated for each state…
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 36 of 50
Value seen for all states in 5th year


To calculate the value of all states in the next
to last year… we choose the better choice:
“discounted value of maximum of keeping
mine open or exercising option”
= NPV[0.12, Max[EV(mine open), - 1]]
State
PV from 6th
Sum
Thus:
4.49
1.50
0.95
2.95
4.59
5.93
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4.82
2.00
0.30
0.89
0.89
0.89
9.30
3.50
1.25
3.84
5.48
6.83
Richard de Neufville
Lattice Valuation
Note
rounding

Slide 37 of 50
Note on evaluation


From economics perspective, the NPV
analysis should vary the discount rates vary
with amount of uncertainty at different states.
This approach invokes such factors as
Risk-free interest rates
— The identification of an “underlying asset” whose
market behavior acts like the physical asset being
designed (such as Copper for Copper mine)
—

Approach not practical in Engineering Design.
More on Economic approach at end of course.
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation
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Slide 38 of 50
To complete Analysis



We need to repeat process by
… estimating values for end of 4th year
… then of 3rd, 2nd, 1st, until we get to start
0.76

2.42
3.84
6.00
2.89
4.71
8.66
0.07
3.84
5.48
9.93
2.28
2.89
4.71
6.19
9.30
3.50
1.25
3.84
5.48
6.83
In this case, project has an expected profit
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 39 of 50
Strategy Implied by Analysis


Analysis determines at each node if it is better
to close or not. This depends on whether
expected future values > 1.0 cost of closure
Thus it provides strategy about when to use
flexibility (exercise option), in this case:
Strategy for exercise of option to close for example case
O
O
CLOSE
O
CLOSE
CLOSE
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O
O
CLOSE
CLOSE
O
O
O
CLOSE
CLOSE
Richard de Neufville
Lattice Valuation
O
O
O
CLOSE
CLOSE
CLOSE

Slide 40 of 50
Note Use of Dynamic Programming




The process we have just gone through,
to evaluate value of option, is a simplified
form of DYNAMIC PROGRAMMING
DP is especially appropriate for finding
optimum in Non-Convex feasible regions
(like that defined by lattice, see slide 26)
Here, flow is from right to left
Exactly like Decision Analysis! Where
use of Dynamic Programming is Implicit
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Lattice Valuation
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Slide 41 of 50
Used Recurrence formula
Valuation of flexibility calculated from:
= NPV[r, Max[EV(mine open), cost of closing]]
• This transition from one stage to next is
“recurrence formula” (or equivalent analysis)



Compare to Dynamic Programming lecture, in
which we find the optimum for any level K, by
best combination of possible giXi and fS-1 (K)
Where value of “mine open” is immediate
value (giXi) and later stages, fS-1 (K)
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 42 of 50
What is Value of Option?


Value of the option is the increase in expected
value due to flexibility
In this example:
Base Case with option option value
$398,112
$763,158
$1,161,270

Flexibility to close the mine (put option “on”
system) is valuable – it provides ‘insurance’
against bad prices, makes project attractive
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 43 of 50
How does option value change?


A data table shows
variation of option
value
For example, with
“Current Price of
Copper”
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1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
Base Case
398,112
12,632,806
10,185,867
7,738,928
5,291,989
2,845,050
398,112
2,048,827
4,495,766
6,942,705
9,389,644
11,836,582
14,283,521
16,730,460
19,177,399
21,624,338
24,071,277
with option option value
763,158 1,161,270
4,446,161
8,186,644
3,975,140
6,210,727
3,504,119
4,234,809
2,643,176
2,648,813
981,639
1,863,412
763,158
1,161,270
2,704,462
655,635
4,817,839
322,073
7,127,023
184,318
9,491,906
102,262
11,887,531
50,948
14,312,377
28,855
16,746,595
16,135
19,183,247
5,848
21,628,324
3,986
24,073,741
2,464
Richard de Neufville
Lattice Valuation

Slide 44 of 50
How “put” protects against losses
We can plot the sensitivity data to show how
“put” option protects against losses
Mine Value with and without option to close
15,000,000
10,000,000
NPV
5,000,000
0
5,000,000 0
500
1000
1500
2000
2500
3000
10,000,000
15,000,000
current price for copper
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 45 of 50
“Put” Insurance most valuable
when risks greatest
As shown by plot of value of this “put”
Value of Put on Closing Mine
10,000,000
NPV
8,000,000
6,000,000
4,000,000
2,000,000
0
0
500
1000
1500
2000
2500
3000
Current price of copper
Engineering Systems Analysis for Design
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Richard de Neufville
Lattice Valuation

Slide 46 of 50
Flexibility “in” the System

Suppose we design mine with extra vertical
shaft, which enables possible increase in
annual production to 8000 tons
— variable cost to $2400/ton
—

This flexibility is a call option “in” system
“option” because it is “right, not obligation” to
— “call” because it permits profits from high prices
— “in” system, because it changes its technology
—

What is the value of this flexibility?
Engineering Systems Analysis for Design
Massachusetts Institute of Technology
Richard de Neufville
Lattice Valuation

Slide 47 of 50
Valuation Process as before…


Difference is effect of using flexibility
Consider decision at a particular state
2.00

0.95
2.95
0.21
2.00
3.81
1.50
0.95
2.95
4.59
2.92
0.21
2.00
3.81
5.30
4.49
1.50
0.95
2.95
4.59
5.93
6.22
2.92
0.21
2.00
3.81
5.30
6.51
Consider when prices highest (boxed cell)
Note: cells do not reflect cost of new shaft – we
compare initial cost to benefit (EV of flexibility) to
see if worthwhile
Engineering Systems Analysis for Design
Massachusetts Institute of Technology
Richard de Neufville
Lattice Valuation

Slide 48 of 50
At this high state…


Revenues depend on whether operators use
flexibility (exercise option)
4.49
6.22
If NO, then revenues as before:
2.92

If YES, then production and revenues increase,
and pay extra. The net is
4.49
8.95
3.67


In this case, better to take advantage of
flexibility, and its net results go into lattice
The process then continues as with “put”
Engineering Systems Analysis for Design
Massachusetts Institute of Technology
Richard de Neufville
Lattice Valuation

Slide 49 of 50
Summary: 2 Main Ideas

Three elements combine in valuation flexibility
Possible States of an Uncertainty
— Probability this State may occur
— Impact of States on Performance of System
—

Mechanics of process are like decision
analysis,
Difference due to recombinatorial nature of lattice
— Need to focus clearly on effects of using flexibility
and, at each stage, choosing better of choices – to
exercise or not
— Process then repeats “from right to left”
— A form of Dynamic Programming
—
Engineering Systems Analysis for Design
Massachusetts Institute of Technology
Richard de Neufville
Lattice Valuation

Slide 50 of 50
Engineering Systems Analysis for Design
Massachusetts Institute of Technology
Richard de Neufville
Lattice Valuation

Slide 51 of 50
Question about Path Independence

If I understood the question correctly, it is: how can
there be “path independence” for a situation, such as
the state and stage in yellow, when there is the
possibility of management changing the system (by
exercising flexibility) at some previous stage (as in
figure below)?
Strategy for exercise of option to close for example case
O
O
CLOSE
O
CLOSE
CLOSE
Engineering Systems Analysis for Design
Massachusetts Institute of Technology
O
O
CLOSE
CLOSE
O
O
O
CLOSE
CLOSE
Richard de Neufville
Lattice Valuation
O
O
O
CLOSE
CLOSE
CLOSE

Slide 52 of 50
Answer about Path Independence


Path independence is crucial when we are concerned
with the total probability of being in a particular state
and stage. In yellow square, p=0.42 (rounded) is sum
of probabilities of possible paths [=¼(.56) + ¾(.38)]
In process of lattice evaluation, however, we do not
use p=.42. We use probabilities of ¾ and ¼. We focus
on what would happen if we were in that state.
PROBABILITY LATTICE
1.00
0.75
0.25
0.56
0.38
0.06
Engineering Systems Analysis for Design
Massachusetts Institute of Technology
0.42
0.42
0.14
0.02
0.32
0.42
0.21
0.05
0.00
0.24
0.40
0.26
0.09
0.01
0.00
Richard de Neufville
Lattice Valuation
0.18
0.36
0.30
0.13
0.03
0.00
0.00

Slide 53 of 50
Engineering Systems Analysis for Design
Massachusetts Institute of Technology
Richard de Neufville
Lattice Valuation

Slide 54 of 50