Analytical Theory of Investment: Insights from Fischer Black and Real Option Theory

an analytical theory of investment 1

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 theory of real economic activities, instead of financial activities. 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).

Some discussion from Dixit and Pyndick (1994) In quantum electrodynamics the result proves to have immense practical utility. In fact, it underlies Feynman s (1949) diagrammatic technique for summing probabilities over all possible paths of a particle. His approach, developed before dynamic programming and Ito s Lemma had been thought of, was an amazing achievement.

Since the dependent variable --- the probability amplitude --- in quantum electrodynamics is complex valued, the analogy with dynamic programing and contingent claims valuation may not extend beyond the mathematical formalism. If it does, then in addition to all his achievements in physics, Feynman could be claimed as the father of financial economics. (Dixit and Pindyck, 1994, p. 123)

We will show that Feynmans method does work in economics and finance Indeed, there is really no reason why Feynman s method should be restricted to complex valued problems

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 Duration of projects Discount rate Uncertainty 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 = + . dz rdt S

From stochastic process to deterministic equation Most values we observe or sense are the averages of random movements Temperature is the average kinetic energy of individual molecules Air pressure is the average momentum of gas molecules Stock price is the average valuation of investors with widespread opinions

With every stochastic process, there are many corresponding deterministic equations. Which equation we shall chose? The more popular one in economic theory is Fokker-Planck equation.

Quantum mechanics is the averaging process with different weights. So we expect theories of quantum mechanics provide good methods in economics, whose activities mostly involve averaging with weights Among different theories of quantum mechanics, Feynman s method is the most explicit averaging process. Let s try his method.

From Feynman to 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 attended a seminar by Feynman in Cornell Kac realized that it was similar to a problem he was thinking about. He refined it into a precise mathematical formula Feynman-Kac formula is widely used in engineering problems

A correspondence between quantum mechanics and economic activities In quantum mechanics, the averaging factor is action, which Feynman generalized from the least action principle in classical mechanics. In economic activities, the averaging is based on lognormal distribution of underlying factors, discounted over time.

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 2 1 C C C 2 = + 2 2 rS S rC 2 t S S

Comparison with Black-Scholes Equation The only difference is the sign of time direction

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 , S max( ) 0 , K C S

Solution = rT ( ) ( ) C SN d Ke N d 1 2

This formula is identical to Black-Scholes formula in form. But the meanings of the parameters are different.

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 Expected variable cost is lower than the value of product. Otherwise, people would not take on the projects

Some general properties (continued) As fixed cost approaches zero, expected variable cost approaches to the value of the product. No expected positive return without first investing in fixed cost. All these properties are consistent with our intuitive understanding of production processes.

Uncertainty and variable cost (Figure) 1.2 1 Variable cost 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. This suggests that large companies, with high fixed costs, are difficult to engage in innovative activities, which by nature have high uncertainty.

Company size and innovation Innovations predominantly occur in small companies. Innovative companies often start by one or two individuals instead of spin off from large companies. Microsoft, Apple, Yahoo, goggle, Facebook are some examples.

Established large companies often enter new businesses by acquisition. Google acquire Android cell phone operating system, DeepMind, which developed AlphaGo.

Rate of return from investment Then the total value of the products and the total cost of production are and C SQ + ( , ) . K Q K respectively. The return that this producer earns is ( C SQ 1 - K + , ) K Q

NPV of projects the net present value of the project is + = QC ( ) ( ) QS K Q S C K 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 S T K TC TS = + ( ) ( ) C 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. WordPerfect Word Lotus 1-2-3 Excel Netscape Internet Explorer

Exercise 1: Fixed cost and market size A product can be manufactured with two different technologies. The first technology needs two million dollars of fixed investment while the second technology needs five millions of fixed investment. Suppose the unit price of the product is 1 million. A production facility based on either technology will last for fifteen years. The diffusion rate is 50% 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 15 and 30 respectively? Please support your recommendation with calculated rates of return.

Calculations S 1 1 K 2 5 R 0.05 0.05 T 15 15 sigma 0.5 0.5 c 0.676481 0.513506 Market size 15 0.234851 0.180862 30 0.345628 0.470215

Exercise 2: Duration and return A production system has fixed cost of 22, uncertainty level of 40% per annum. Assume the value of annual output is 2, the discount rate is 3% per annum. The project duration can be 50 or 70 years. Calculate the variable costs and net present values of the project for project durations of 50 and 70 years. Explain why it is unsustainable to increase lifespan indefinitely.

Project value with different durations S 1 1 1 K 22 22 22 T 50 60 70 r 4% 4% 4% sigma 30% 30% 30% C 0.54 0.66 0.76 NPV 1.2 -1.7 -5.1

Comments K at 22 represents one needs about 22 years cost before work. Many people graduate from university at 22. Total life span would be 22 +T When T is 50, the total lifespan is 72. When T is 70, the total lifespan is 92.

It explains why individual life does not go on forever. Instead, it is more sustainable for animals to produce offspring. This also determines most businesses fail in the end Parameters could change. With different parameter, answers can be different. We calibrate parameters to be roughly consistent with reality

Discussion Ray Kurzweil from Google, "I believe we will reach a point around 2029 when medical technologies will add one additional year every year to your life expectancy. The lifespan has been increasing over time. Some claim the average lifespan can reach 100, 120 or without limit. What do you think?

Currently, there are a lot of medical research on longevity. How these researches affect the society?

Regression models in statistics and analytical models derived from basic principles What are their differences?

Exercise 3: Investment decision with changing market evaluation 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?

Original project original market conditionnew market condition New project S K R T sigma c Market size 1 1 1.867818219 4.63922371 0.08 20 0.5 0.846537111 0.69589973 0.08 20 45% Return Return 20 30 0.063911253 0.17572294

original project New project with earlier sunken cost new market condition S K R T sigma c Market size 1 1 1.867818219 4.63922371 0.08 20 0.45 0.818501765 0.69589973 0.08 20 0.45 Return Return 20 30 0.13538002 0.09552889

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 always work as expected in economics and biology. One can only hope projects generate positive returns. That is why once successful species can go extinct.