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In efficient markets prices change randomly. This is often misunderstood and can give rise to what Miller and Upton (2002) call the casino view of the stock market. In this view investors are gamblers whose buying and selling without apparent economic reason gives windfall profits to some and does random damage to others. The alleged casino nature of the stock market is often underlined by comparing graphs of stock price changes with graphs of random numbers, which are indistinguishable from each other. In this chapter we shall see that the opposite is true, that randomly changing prices are the hallmark of properly functioning markets. As we saw in Chapter 2, markets with properly organized price discovery processes aggregate all the information that buyers and sellers possess and that they reveal through their bidding and asking. As a result, prices in such markets will reflect all available information. By consequence, prices will only change if new information becomes available. But new information is random by nature, otherwise it would not be new. Hence, prices have to change randomly in efficient markets.
The concept of market efficiency
Market efficiency, or the efficient market hypothesis, is a deceptively simple concept. But its consequences are profound and not at all easy to understand or accept. We shall look at the concept from different angles and review some of the empirical evidence.
By providing a solid theoretical basis, this book introduces modern finance to readers, including students in science and technology, who already have a good foundation in quantitative skills. It combines the classical, decision-oriented approach and the traditional organization of corporate finance books with a quantitative approach that is particularly well suited to students with backgrounds in engineering and the natural sciences. This combination makes finance much more transparent and accessible than the definition-theorem-proof pattern that is common in mathematics and financial economics. The book's main emphasis is on investments in real assets and the real options attached to them, but it also includes extensive discussion of topics such as portfolio theory, market efficiency, capital structure and derivatives pricing. Finance equips readers as future managers with the financial literacy necessary either to evaluate investment projects themselves or to engage critically with the analysis of financial managers. Supplementary material is available at www.cambridge.org/wijst.
After the groundbreaking contributions by Black, Scholes and Merton in the early 1970s, continuous time finance quickly grew into a large branch of financial economics. The area was strongly invigorated by the enormous proliferation of option trading and it received scientific recognition with the Nobel prize for Merton and Scholes in 1997. Sundaresan (2000) gives an overview of its developments and applications in various areas. Continuous time finance is an area where the mathematical approach can be fully exploited. Given securities prices on complete and arbitrage-free markets, the full force of mathematics can be brought to bear on the problem of how to price new securities. We have already developed the conceptual framework in discrete time, so we can introduce the continuous time techniques with minimal effort. We begin by having another look at the properties of stock returns. We then illustrate how probabilities can be transformed with the simple example of loading a die. Transformation is applied to stock returns, which results, in an equivalent martingale probability measure for stock returns. Pricing can then be done in the by now familiar way of discounting the risk-neutral expectation using the risk-free discount rate. The result is the celebrated Black and Scholes formula.
Preliminaries: stock returns and a die
So far we have mainly used simple, discretely compounded returns. In Black and Scholes’ option pricing we follow individual stocks in continuous time and this makes it necessary to use continuously compounded returns and to detail their properties.
This opening chapter introduces finance as a scientific discipline and outlines its main research tools. We also take a brief look at the book’s central theme, the calculation of value, and the main ways to account for risk in these calculations. To illustrate the differences between finance and the natural sciences we compare results of a Nobel prize-winning financial model with measurements of a NASA space probe.
Finance as a science
What is finance?
Finance studies how people choose between uncertain future values. Finance is part of economics, the social science that investigates how people allocate scarce resources, that have alternative uses, among competing goals. Both scarcity, i.e. insufficient resources to achieve all goals, and possible alternative uses are necessary ingredients of economic problems. Finance studies such problems for alternatives that involve money, risk and time. Financial problems can refer to businesses, in which case we speak of corporate finance, but also to individuals (personal finance), to governments (public finance) and other organizations. Financial choices can be made directly or through agents, such as business managers acting on behalf of stockholders or funds managers acting on behalf of investors. For the most part, we shall study choices made by businesses in financial markets, but the results have a wider validity. As we shall see, financial markets facilitate, simplify and increase the possibilities to choose. Some typical problems we will look at are:
Should company X invest in project A or not?
How should we combine stocks and risk-free borrowing or lending in our investment portfolio?
What is the best way to finance project C?
How can we price or eliminate (hedge) certain risks?
What is the value of flexibility in investment projects?
This chapter summarizes the basic concepts and techniques that are used throughout the other chapters. We first look at the time value of money and common interest rate calculations. We then recapitulate how a firm’s accounting system records and reports financial data about the firm. An example illustrates how these techniques and data can be used for investment decisions. Subsequently, we introduce the economic concepts of utility and risk aversion, and their use in financial decision making. The chapter concludes with a brief look at the role of financial markets, both from a theoretical and practical perspective.
The time value of money
Sources of time value
The time value of money can be summarized in the simple statement that €1 now has a higher value than €1 later. The time value of money springs from two sources: time preference and productive investment opportunities. Time preference, or ‘human impatience’ as the economist Fisher (1930) calls it, is the preference for present rather than future consumption. This is more than just impatience. Some consumption cannot be postponed for very long, for example the necessities of life. For other goods, the time pattern of people’s consumptive needs is almost inversely related to the time pattern of their incomes. People want to buy houses when they are young and starting families, but if they had to accumulate the necessary money by saving, only a few could afford to buy a house before retirement age. Moreover, postponing consumption involves risk. Even if the future money is certain, the beneficiary, or the consumptive opportunity, may no longer be around. As a result, people require a compensation for postponing consumption and are willing to pay a premium to advance it.
Finance has undergone spectacular changes in the last four decades, both as a profession and as a scientific discipline. Before 1973 there were no option exchanges and there was no generally accepted model to price options. Today, the worldwide trade in derivative securities represents a much larger money amount than the global production of goods and services. The famous Black and Scholes option-pricing formula and its descendants are used in financial markets all over the world where an enormous number of derivative securities are traded every day. Professionals in sectors like engineering, telecommunications and manufacturing regularly find that their projects are evaluated with techniques such as real options analysis. Understanding the basic concepts of finance is increasingly becoming a prerequisite for the modern work place.
Many scientific developments in finance are fuelled by the use of quantitative methods; finance draws heavily on mathematics and statistics. This gives students and professionals who are familiar with quantitative techniques an advantage in mastering the principles of finance. As the title suggests, this book gives an introduction to finance in a manner and ‘language’ that are attuned to an audience with quantitative skills. It uses mathematical notations and derivations where appropriate and useful. But the book's main orientation is conceptual rather than mathematical; it explains core financial concepts without formally proving them. Avoiding the definition-theorem-proof pattern that is common in mathematical finance allows the book to use the more natural order of first presenting an insight from financial economics, then demonstrating its empirical relevance and practical applicability, and concluding with a discussion of the necessary assumptions. This ‘reversed order’ reduces the scientific rigour but it greatly enhances the readability for novice students of finance. It also allows the more demanding parts to be skipped or made non-mandatory without loss of coherence.
Modigliani and Miller’s proposition 2 provides a basic relation for the distribution of risk and return over debt and equity holders. In this chapter we will elaborate that relation in more detail. This gives a good illustration of how the interaction between financing and investment decisions can be analyzed. However, the analysis is restricted by the fact that only one market imperfection, taxes, can be incorporated in return rates. The limiting effect of default costs is treated as an exogenous factor.
Risks and discount rates
In the Modigliani and Miller analyses we compared two companies with the same assets that produce the same cash flows. Such companies have the same business risk. Business risk is the uncertainty of the cash flows that are generated by the firm’s assets. It is the risk of, for example, operating a fleet of trucks, or a chemical plant, or a software company. The market price for this risk is the opportunity cost of capital, which is the return that the market offers on investments with the same risk characteristics as the investment we are looking at. If shareholders provide the whole investment sum, so that they bear all the risks and receive the entire cash flow, they will require the opportunity cost of capital as return. Shareholders express their requirements in the price they are willing to pay for the shares – they will set the price such that the expected return on the shares equals the opportunity cost of capital. If there are more categories of investors, financial contracts specify how risk and return are divided over them. The simple, stylized situation we will analyse involves a division of cash flows in a low-risk, low-return part for the debtholders, and a high-risk, high-return part for the equity holders. As we saw in the previous chapter, the additional risk that equity holders accept by giving debtholders a higher priority claim on the firm’s cash flows is called financial risk.
Options are financial contracts that give their holders the right, but not the obligation, to buy or sell something on a future date at a price determined today. The distinction between right and obligation, which gives the holder a choice, is an essential characteristic. Options are derivative securities, they derive their values from the assets to be bought or sold in the future. The use of option-like contracts is very old. The oldest examples go back to Greek antiquity and in the 1600s options on rice were traded in Japan and options on tulips in the Netherlands. However, the world’s first options exchange did not open until 1973 in Chicago (Chicago Board of Options Exchange). In the same year Black and Scholes published their famous option-pricing formula. The trade in standardized options in Europe started in 1978 on the European Options Exchange in Amsterdam and has grown tremendously since then.
In this chapter we will model option prices in discrete time. After discussing the basic characteristics of options, we will lay the foundations of option pricing in state-preference theory. We then look at binomial option pricing with the wonderful Cox–Ross–Rubinstein model.
The models that we have discussed so far are based on market equilibrium or the absence of arbitrage opportunities. In such models, market participants reach consensus about arbitrage-free or market clearing equilibrium prices, which ensure that each participant gets their fair share. For example, in the CAPM all securities are held at their equilibrium market prices. Similarly, in the trade-off theory’s optimal capital structure both equity and debt get their fair share of the firm’s revenue, viz. the market price of the risk they bear. The first cracks in this equilibrium approach appeared in the pecking order theory, where managers with inside information can increase the wealth of the existing shareholders by issuing new debt or equity at too high a price. The cracks widened when we modelled equity as a call option on the firm’s assets. We saw that equity holders can benefit at the expense of debtholders by choosing risky projects rather than safe ones. Such conflicts of interest are much more general. They are studied in agency theory, which looks at the firm as a team of different parties that realize their own interests by cooperating in the firm. That cooperation is governed by a set of contracts that specify how inputs are joined and outputs are distributed. The design of contracts that share risk and return in an optimal way is an important topic in agency theory. This chapter discusses some headlines of agency theory and its influence on the way firms are managed.
Agency theory has its roots in corporate governance theories that go back to the 1930s, but its application in finance is usually associated with Jensen and Meckling’s (1976) pioneering paper. Agency theory is very general – it applies in principle to all situations that involve cooperation. It has a long tradition in political science and labour market economics. We have met several elements of agency theory before; we now summarize them in an agency-theoretic perspective.
As the name suggests, real options have investments in real assets, not financial securities such as stocks and bonds, as their underlying values. Real options analysis is aimed at the valuation of such investments using the option pricing techniques discussed in the previous chapters. Real options analysis is a comparatively new area in finance: the term real options was coined in 1977 by Stewart Myers. But publications on the topic did not appear in significant numbers until the mid 1980s, ten years after the publication of the Black and Scholes model. Real options analysis is rapidly gaining importance and some even predict that it will replace NPV as the central paradigm for investment decisions [Copeland and Antikarov, 2001, p. VI]. As we shall see, real options analysis overcomes the weakness of NPV in valuing flexible projects. The Black and Scholes formula and its descendants allow us to calculate the value of flexibility, that is difficult to price with traditional discounted cash flow methods.
Investment opportunities as options
Before we turn to the valuation of real options, we shall first explore the similarities and differences between real and financial options and investigate what the sources of real option value are.