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.
Marriage as an option
• Marriage is a right, and not an obligation.
Therefore, marriage is an option. But does
labeling marriage as an option enhance our
understanding about marriage?
• An option to wait is more valuable when the
market is more uncertain. Hence marriage, as
an option, is more valuable when the future is
more uncertain. This explains why people
generally delay their marriage in a society, such
as ours, when future is more uncertain.
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
• S&L scandals
• Banking regulation and competition from foreign
countries, e.g. CitiBank in Argentina
• Traders’ pay, such as Nick Leeson from Barings
• industries that are promoted by governments
• Subprime lending: Good time, big money; Bad
time, retirement
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
• It ignores the irreversibility, ongoing uncertainty,
timing in investment and the context within a
firm.
• 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
fixed cost, variable cost, duration of project
and uncertainty.
The evolution of commodity prices
• Lognormal process can represent commodity
prices change
• Variable cost, as a function of commodity
price, satisfy the following equation, when
the discount rate is r
C
C 1 2 2  2C
 rS
  S
 rC
2
t
S 2
S
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. Mathematically speaking, option
theory is a backward problem while the problem in
project investment is a forward problem.
Properties of this analytical
theory
• First, when the fixed cost investment, K, is
higher, the variable cost, C, is lower.
• Second, for the same amount of fixed
investment, when the duration of a project,
T, is longer, the variable cost is higher.
• Third, when uncertainty, , increases, the
variable cost increases.
Properties of this analytical theory
(Continued)
• Fourth, when the fixed cost approaches
zero, the variable cost will approach to the
value of the product.
• Fifth, when the value of the product
approaches zero, the variable cost will
approach zero as well.
• All these properties are consistent with our
intuitive understanding of production
processes.
Uncertainty and variable cost
• The variable cost of a production mode is
an increasing function of uncertainty. As
fixed costs are increased, variable costs,
calculated from (6), decrease rapidly in a
low uncertainty environment and
decreases slowly in a high uncertainty
environment.
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
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
• higher fixed cost investments, which have
lower variable costs in production, need
higher output volume to breakeven.
Output and return with different
levels of fixed costs (Figure)
The fundamental tradeoff
• The efficiency of high fixed cost systems
and the flexibility of low fixed cost systems
• Market size and level of uncertainty
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.
Capital Budgeting
•
•
•
•
•
Fixed cost
Variable cost
Diffusion rate
Market size
Duration of project
Uncertainty and duration
• The amount of uncertainty accumulated over
a project is
 T
• Higher uncertainty situation benefit from
shorter duration
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
7
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?
Explaining real options
• Option to abandon: Lower fixed cost, less
costly to abandon
• Option to switch: Shorter duration of
projects, easy to switch.
• Option to delay: Shorter duration and
lower fixed cost, easy to delay
• Option to grow, invested fixed assets and
variable assets offer new foundation to
grow.
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
•
•
•
•
•
•
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