Transcript Project investment - University of Northern British Columbia

From option, to
real option to an
analytical theory of
investment
1
• When Black-Scholes theory was initially
developed, it was thought as an arcane
academic theory that only applies to
narrow technical areas. Soon its
application became broader and broader.
At the beginning of the classes, we
explained that most of the financial
activities can be beneficially thought as
options. Today, we will discuss problems in
broader horizons.
Limited liability company as an
option
• As a limited liability company, upward
profit is unlimited while downward loss is
limited to company’s capital. Therefore it is
an call option on its asset. This makes a
LLP more prone to risky businesses,
especially when a company’s capital
comes from other people’s money.
Examples
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S&L scandals
Traders’ pay
industries that are promoted by governments
Subprime lending: Good time, big money; Bad
time, retirement
• CDS selling by large insurance companies
Why LLP become legal entities?
• In the history of human evolution, we cannot
store large amount of wealth or food. Unlimited
upward potential is not very useful for us. On the
other hand, without food for several days may
lead to starvation. Therefore, people are risk
averse. LLP makes people less risk averse.
• Why LLP generally occur in highly developed
societies?
• It is because more resourceful systems can
afford more try and error.
The problem with using NPV to
value project investment
• NPV is the standard method in project
evaluation
• It ignores the irreversibility, ongoing uncertainty,
timing in investment and the context within a
firm.
Real option theory
• Real option theory address these issues.
• However, most articles on this subject either use
stylized numerical examples or adopt a purely
conceptual approach to describing how option
pricing can be used in capital budgeting. Not
clear about the source of advantage.
An analytical theory of project
investment
• Directly investigate the relation among
major factors in economic activities
Main factors in economic activities
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Fixed cost
Variable cost
Discount rate
Uncertainty
Duration of projects
Market size or output capacity
The analytical theory of production
• Living systems need to extract low entropy from the
environment to compensate for continuous
dissipation. It can be represented mathematically
by lognormal processes
•
dS
 rdt  dz .
S
From stochastic process to
deterministic equation
• Most values we observe or sense are the
averages of random movements
• Temperature vs. Velocity of individual
molecules
• Stock price vs. widespread opinions of
different investors
The equation
• If the discount rate of a system is r, from
Feynman-Kac formula, any function of S,
including the variable cost, C, satisfies the
following equation
C
C 1 2 2  2C
 rS
  S
 rC
2
t
S 2
S
A short note on Feynman-Kac
• Feynman developed the path integral
approach in quantum mechanics. It
integrated over probability distributions to
obtain deterministic final results that are
the observable quantities.
• Kac refined it into a precise mathematical
formula
The initial condition
• When the duration of a project is infinitesimal
small, it has only enough time to produce one
piece of product. In this situation, if the fixed cost
is lower than the value of the product, the
variable cost should be the difference between
the value of the product and the fixed cost to
avoid arbitrage opportunity. If the fixed cost is
higher than the value of the product, there
should be no extra variable cost needed for this
product.
The initial condition (continued)
C (S ,0)  max(S  K ,0)
Solution
C  SN(d1 )  Ke
 rT
N (d 2 )
Similarity and difference between
option theory and capital investment
• Both the evolution of share prices and value of
economic commodities are represented by
lognormal processes.
• For a financial option, the strike price at the end
of the contract is known. The problem in option
theory is to estimate the option price when the
strike price, as well as several other parameters,
is given.
• For a business project, irreversible fixed
investment is determined at the beginning of a
project. The problem in project investment is to
estimate variable cost when fixed cost, as well
as other factors, is given.
Some general properties
• Higher fixed costs, lower variable costs
• Longer the duration of a project, higher variable
cost
• Uncertainty increases, variable cost increases.
• Discount rate increases, variable cost increases
• Fixed cost approaches zero, variable cost
approaches to the value of the product.
• Variable cost is lower than the value of product.
• All these properties are consistent with our
intuitive understanding of production processes.
Uncertainty and variable cost
(Figure)
1.2
Variable cost
1
0.8
High volatility
Low volatility
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9 10 11 12 13
Level of fixed cost
Uncertainty and variable cost
• The variable cost of a production mode is
an increasing function of uncertainty. As
fixed costs are increased, variable costs
decrease rapidly in a low uncertainty
environment and decreases slowly in a
high uncertainty environment.
Rate of return from investment
• Then the total value of the products and the
total cost of production are
SQ and C ( K ,  )Q  K .
•
• respectively. The return that this producer
earns is
•
SQ
ln(
)
C ( K ,  )Q  K
Output and return with different
levels of fixed costs (Figure)
Output and return with different
levels of fixed costs
• higher fixed cost investments, which have
lower variable costs in production, need
higher output volume to breakeven.
The fundamental tradeoff
• higher fixed cost investments, which have
lower variable costs in production, need
higher output volume to breakeven.
• The efficiency of high fixed cost systems
and the flexibility of low fixed cost systems
• Market size, level of uncertainty and other
factors determine the level of fixed cost
Implication on capital budgeting
• Microsoft, with high existing assets, can
demand high rate of return. At the same
time, it is very cautious to avoid projects
with high uncertainty.
Uncertainty and duration
• The amount of uncertainty accumulated over
a project is
 T
• Long duration projects require lower
uncertainty
Fixed cost and duration of the
project
4
3
2
Return
1
0
1
2
3
4
5
6
-1
-2
-3
-4
-5
Duration
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8
9
10
One long project
Two short Porhject
• It explains why individual life does not go
on forever. Instead, it is more efficient for
animals to produce offspring. This also
determines most businesses fail in the end
• Question: Currently, there are a lot of
medical research on longevity. How these
researches affect the society?
Fixed cost and discount rate
1.2
change of variable cost
1
0.8
Low fixed cost
High fixed cost
0.6
0.4
0.2
0
0.5
0.45
0.4
0.35
0.3
0.25
0.2
Discount rate
0.15
0.1
0.05
0
• Question: Who will benefit more from low
interest rates? What will happen to
housing pricing in a low interest rate
environment?
Exercise
A product can be manufactured with two different
technologies. The first technology needs ten million dollars
of fixed investment while the second technology needs fifty
millions of fixed investment. Suppose the unit price of the
product is 1 million. A production facility based on either
technology will last for twenty years. The diffusion rate is
60% per annum. The expected rate of change of the price
of the product is 5% per annum. What is the variable cost
for each technology? What technology you will recommend
to your CEO if she estimates market size to be 100 and
300 respectively? Please support your recommendation
with calculated rates of return.
Original motivation
• Biological entities need to extract low
entropy from the environment to
compensate continuous diffusion. This can
be modeled as
dS
 rdt  dz .
S
Universality
• Since this theory is based on the most
fundamental properties of life, we would
expect it offers a universal understanding
of human activities.
Applications
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Projects and firms
Capital structure
Software development: Standard and agile
Product life cycle
Biological evolution
Resource, fertility, economy and stock market
return
• Dynamics of merger and acquisition
Biological evolution
H
Li
He
Be
Na Mg
B
C
N
O
F
Ne
Al
Si
P
S
Cl
Ar
Biological evolution (Continued)
• Carbon is the element of organic compounds.
Except the inert gas Ne, C is at the center of the
period. “Carbon is a particularly mediocre
element, easygoing in the liaisons it forms. … In
chemistry as in life, this unpretentiousness has
rewards, and in its mediocre way carbon has
established itself as king of the Periodic
Kingdom.”
From carbon to silicon
• Carbon’s principle products, living organisms, have struggled over a
few billion years to establish mechanisms for the accumulation and
dispersal of information (an austere distillation and definition of what
we mean by “life”), and silicon has lain in wait. The recent alliance of
two regions, in which carbon-based organisms have developed the
use of silicon-based artifacts for information technology, has resulted
the enslavement of silicon. However, such is the precocity of
carbon’s organisms that they are steadily developing silicon’s latent
powers, and one day silicon may well overturn the suzerainty of its
northern neighbor and assume the dominant role. It certainly has
long term potential, for its metabolism and replication need not be as
messy as carbon’s. Here we may see one of the most subtle
interplays of alliances anywhere in the kingdom, for silicon will not
realize its potential without the burden of development being carried
out by carbon.
From sulfur to oxygen
• Nature discovered that in some respects hydrogen
sulfide (H2S), the analog of water (H2O), can be used by
organisms in much the same way as water is used in the
process of photosynthesis --- as a source of hydrogen.
The great difference to note is that when hydrogen is
removed from a water molecule by a green plant, the
excrement is gaseous oxygen, which then mingles with
the globally distributed atmosphere. However, when
hydrogen is removed from hydrogen sulfide in the
interior of a bacterium, the excrement is sulfur. Sulfur,
being a solid, does not waft away, so the colony of
organisms has to develop a mode of survival based on a
gradually accumulating mound of its own sewage.
The reward and pollution in
economic development
• The transition from sulfur to oxygen was a major
pollution event in biological history, which destroyed
most living systems at that time. This example shows the
importance of dumping high entropy waste in ecological
systems. Those who are able to dump the excrement of
themselves effectively will prosper, often at a cost to
others. This is the same to human beings in economic
development. Many industrial facilities are built by rivers
or near seaside so wastes can be diffused quickly. While
the gains from the industrial output are more
concentrated and harvested by the owners of the
industrial facilities, the wastes are diffused and shared
by many others.
K and r strategies
• The pattern of biological evolution is very similar to the
pattern of the product life cycle. Biological species are
sometimes classified, according to the relative level of
fixed and variable costs, into two categories, the rstrategists and the K-strategists. The fixed costs are low
for the r-strategists. They are usually of small size,
produce abundant offspring and invest very little in each
one. They are the species that prosper in a volatile
environment for low fixed costs make them flexible. But
they cannot compete well with other species in a stable
environment for their marginal costs are high.
K and r strategies
• In contrast, the fixed costs are high for the K-strategists.
They are usually large in size, produce fewer offspring
but invest much more in each one. They are the
conservative species that are able to out-compete the rstrategists in stable environments, for their marginal
costs are low. But they cannot adjust quickly when the
environment changes. Between the extreme r-strategists
such as bacteria and the extreme K-strategists such as
elephants, there lies the r and K continuum