1

an analytical theory of

investment

Quote from Fischer Black• I like the beauty and symmetry in Mr. Treynor’s equilibrium

models so much that I started designing them myself. I worked on models in several areas:

•             Monetary theory•             Business cycles•             Options and warrants• For 20 years, I have been struggling to show people the

beauty in these models to pass on knowledge I received from Mr. Treynor.

•  • In monetary theory --- the theory of how money is related to

economic activity --- I am still struggling. In business cycle theory --- the theory of fluctuation in the economy --- I am still struggling. In options and warrants, though, people see the beauty. (p. 93)

• Black had in mind a new foundation of economics.

• After Black-Scholes theory, many efforts were made to develop a general theory of economics.

• The results are called real option theory, to distinguish from financial option theory.

• Most authoritative book, Investment under Uncertainty, by Dixit and Pindyck, 1994

• no analytical results about the key factors in capital investment were obtained. As a result, the real option theory “either use stylized numerical examples or adopt a purely conceptual approach to describing how option pricing can be used in capital budgeting” (Megginson, 1997, p. 292).

Most fundamental properties of life

• First, organisms and organizations, as non-equilibrium systems, need to obtain resources from the environment to compensate for the continuous diffusion of resources required to maintain various functions.

• Second, fixed cost has to be spent before a system can obtain resources from external environment. From the second law of thermodynamics, the diffusion of resource is spontaneous. But the extraction, transformation and storage of resources requires specialized structures. For example, all biological systems require genetic codes and protein structures before they can extract, transform and store external resources. All human beings require substantial amount of investment from parents and the society before they can make a living themselves.

• Third, for an organism or an organization to be viable, the total cost of extracting resources has to be less than the amount of resources extracted, or the total cost of operation has to be less than the total revenue.  Costs include fixed cost and variable cost.

Main factors in economic activities

• 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

•

. dzrdtS

dS

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

rCS

CS

S

CrS

t

C

2

222

2

1

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)

)0,max()0,( KSSC

Solution

)()( 21 dNKedSNC rT

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)

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13

Level of fixed cost

Var

iab

le c

ost

High volatilityLow volatility

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

•

• respectively. The return that this producer earns is

•

.),( and KQKCSQ

1- ),(

or ) ),(

ln( KQKC

SQ

KQKC

SQ

NPV of projects

• the net present value of the project is

• • It is often convenient to represent S as the

value of output from a project over one unit of time. If the project lasts for T units of time, the net present value of the project is

•

KCSQKQCQS )()(

KCSTKTCTS )()(

Output and return with different levels of fixed costs (Figure)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 10 11 12

Output

Low fixed cost

High fixed cost

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.

Exercise 1

A product can be manufactured with two different technologies. The first technology needs ten million dollars of fixed investment while the second technology needs thirty 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 discount rate 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 200 respectively? Please support your recommendation with calculated rates of return.

Calculation

S 1 1

K 10 30

R 0.05 0.05

T 20 20

sigma 0.6 0.6

d1 0.856196 0.446767

d2 -1.82709 -2.23651

c 0.679552 0.532768

Market size

100 0.249036 0.183

200 0.315324 0.3816

Exercise 2

• When the size of a company increases and the business expands, the internal coordination and external marketing becomes more complex. This can be modeled with uncertainty, σ, as an increasing function of the volume of the output.

Assume

• where σ0 is the base level of uncertainty, Q is the volume of output and l > 0 is a coefficient. Assume the product value is 1, fixed cost is 5, discount rate is 10% per annum, duration of project is 10 years. Assume σ0 is 40% per annum and l is 0.005. Calculate the return from the project when market size is 10, 20, 30, 40 and 50.

lQ 0

increasing return initially and then diminishing return.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

10 15 20 25 30 35 40 45 50 55 60 65

Comments

• Increasing return is very difficult to model in standard economic theory.

• But with this production theory, increasing return and decreasing return can be modeled very easily.

Uncertainty and duration

• The amount of uncertainty accumulated over a project is

• Long duration projects require lower uncertainty

T

Exercise 3

• A production system has fixed cost of 5, uncertainty level of 40% per annum. Assume the value of annual output is 1, the discount rate is 6% per annum and project duration is 12 years. Calculate the variable cost and the profit of the project. Another production system has the same parameters except project duration, which is 22 years. Calculate the variable costs and the profits of the two projects. For each project, compare the profit with that of two projects with duration half long while keeping other parameters identical. What conclusion you may obtain?

Projects with durations half long

S 1 1

K 5 5

R 0.06 0.06

T 6 11

sigma 0.4 0.4

d1 -0.7853 -0.05234

d2 -1.7651 -1.37899

c 0.080881 0.262183

Profit 0.514714 3.115984

Projects with reference durations

S 1 1

K 5 5

R 0.06 0.06

T 12 22

sigma 0.4 0.4

d1 0.050924 0.783812

d2 -1.33472 -1.09235

c 0.298872 0.599985

Profit 3.413533 3.800336

Differences in profits with half duration 2.384104 -2.43163

Fixed cost and duration of the project

-5

-4

-3

-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10

Duration

Ret

urn 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

0

0.2

0.4

0.6

0.8

1

1.2

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Discount rate

chan

ge

of

vari

able

co

st

Low fixed costHigh fixed cost

• Question: Who will benefit more from low interest rates? What will happen to housing pricing in a low interest rate environment?

Exercise 4

• Suppose, for a certain product, the initial estimation of uncertainty is 50% per annum and the market size is 20. Assume product value to be 1, discount rate to be 8% per annum and the project will last for 20 years. Find the level of fixed cost so the project will earn highest rate of return. After the project has been built, however, the new estimation of market uncertainty becomes 45% per annum and the market size becomes 30. With this new estimation, what is the new level of fixed cost for a project to earn the highest rate of return? But should we abandon the existing project and build a new project with higher fixed cost, after considering the sunken fixed cost from the existing project?

Comments

• A project will last for a period of time. During that period of time, market condition may change, rendering projects designed for highest return under original estimation of future market condition less profitable.

• That is why optimization does not exist in economics. One can only hope projects generate positive returns.

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 He

Li Be B C N O F Ne

Na Mg 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 r-strategists 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 r-strategists 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