This page intentionally left blank The Economics of Financial Markets The Economics of Financial Markets presents a concise overview of capital markets, suitable for advanced undergraduates and for embarking graduate students in financial economics Following a brief overview of financial markets – their microstructure and the randomness of stock market prices – this textbook explores how the economics of uncertainty can be applied to financial decision making The mean-variance model of portfolio selection is discussed in detail, with analysis extended to the capital asset pricing model (CAPM) Arbitrage plays a pivotal role in finance and is studied in a variety of contexts, including the arbitrage pricing theory (APT) model of asset prices Methods for the empirical evaluation of the CAPM and APT are also discussed, together with the volatility of asset prices, the intertemporal CAPM and the equity premium puzzle An analysis of bond contracts leads into an assessment of theories of the term structure of interest rates Finally, financial derivatives are explored, focusing on futures and options contracts Roy E Bailey is a Reader in Economics at the University of Essex The Economics of Financial Markets Roy E Bailey Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521848275 © R E Bailey 2005 This book is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2005 - - ---- eBook (EBL) --- eBook (EBL) - - ---- hardback --- hardback - - ---- paperback --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate The Theory of Economics does not furnish a body of settled conclusions immediately applicable to policy It is a method rather than a doctrine, an apparatus of the mind, a technique of thinking, which helps its possessor to draw correct conclusions It is not difficult in the sense in which mathematical and scientific techniques are difficult; but the fact that its modes of expression are much less precise than these, renders decidedly difficult the task of conveying it correctly to the minds of learners J M Keynes When you set out for distant Ithaca, fervently wish your journey may be long, – full of adventures and with much to learn C P Cavafy Contents in brief Contents List of Figures Preface page ix xv xvii 1 Asset markets and asset prices Asset market microstructure 33 Predictability of prices and market efficiency 56 Decision making under uncertainty 83 Portfolio selection: the mean-variance model 114 The capital asset pricing model 143 Arbitrage 166 Factor models and the arbitrage pricing theory 183 Empirical appraisal of the CAPM and APT 200 10 Present value relationships and price variability 222 11 Intertemporal choice and the equity premium puzzle 250 12 Bond markets and fixed-interest securities 281 13 Term structure of interest rates 306 14 Futures markets I: fundamentals 336 15 Futures markets II: speculation and hedging 363 16 Futures markets III: applications 393 17 Swap contracts and swap markets 417 18 Options markets I: fundamentals 438 19 Options markets II: price determination 467 20 Options markets III: applications 494 vii Present value relationships and price variability 235 relevant for asset market efficiency only to the extent that the model is allowed to prescribe the criteria for efficiency 10.3 Behavioural finance, noise trading and models of dividend growth The quest for ways to account for asset price volatility has pursued a variety of routes, most of which now shelter under the umbrella of behavioural finance A representative example, mentioned in chapter 1, is the noise trading model The noise trading framework distinguishes between two groups of investors: rational investors (sometimes called smart-money traders) and noise traders Rational investors are assumed to make decisions according to ‘fundamental information’ (presumably, the stream of asset returns) Noise traders, by contrast, act according to whim, fad or fancy, their decisions being made without due regard for commonly accepted investment criteria.11 A consequence is that asset prices tend to reflect the capricious actions of noise traders Expectations of share price increases (or decreases), whether well founded or not, become self-fulfilling It is as if a positive feedback mechanism governs asset prices: price rises stimulate further increases, and conversely Although rational investors are still present in the market, their actions may be swamped by those of noise traders Although attractive at first sight, the distinction between noise traders and rational investors should be treated with caution For, to the extent that the actions of noise traders affect asset prices, rational investors would be sensible to take these actions into account when making their own decisions Even if it is feasible to partition information into ‘fundamental’ and ‘non-fundamental’, rational investors would not ignore the supposedly irrational activities of noise traders deriving from non-fundamental information Thus, rational investors might not appear so rational after all However, despite this reservation, the noise trader approach does succeed in accounting for the impact of investors’ beliefs and actions that appear from some external, objective standpoint to be misguided or, at least, neglectful of relevant information Barsky and De Long (1993) present an explicit model that captures the role of noise traders.12 Their model can be expressed in the simple form pt = 11 12 dt r − gt (10.18) Sometimes noise traders are assumed to include investors who trade for liquidity purposes (for instance, when assets are sold to provide cash for consumption or when assets are purchased as a consequence of income in excess of consumption) The liquidity motive is, of course, not necessarily to be construed as irrational Another important contribution following a similar approach is that of Campbell and Shiller (1989), which focuses on expectations about the ratio of dividends to prices 236 The economics of financial markets where gt is what Barsky and De Long call the ‘permanent’ growth rate in dividends as of date t – essentially, the average dividend growth rate expected from date t onwards Notice the close resemblance between (10.18), pt = dt / r − gt , and (10.9), pt = dt+1 / r − g Apart from a minor technical difference concerning the timing of dividends, the two are identical if there is a fixed dividend growth rate into the infinite future Although gt is constant as of date t (i.e the same for all future time), it is not fixed; investors revise their estimates of dividend growth at each date (presumably as a consequence of the arrival of new information) Barsky and De Long postulate that dt and gt are positively correlated: when dividends change, investors extrapolate the change into the future so that the dividend growth rate changes in the same direction Thus, an increase in the dividend, dt , has a direct effect – via the numerator of pt = dt / r − gt – and an indirect effect – via the growth rate, gt – on the asset price, pt Given that gt responds positively to dt , the share price increases more than proportionately with the increase in dividend Hence, the model accounts for the commonly reported responsiveness (overreaction) of the share price to dividends Barsky and De Long present evidence that supports their model when applied to US data They interpret the evidence as conforming with a rational expectations approach: investors are assumed to act as if information about the past growth of dividends is used to construct forecasts for the future The forecasts are not ‘ex post rational’ in the sense of the previous section, but neither are they irrational expressions of fads or fancies Closely allied with noise trading is the notion of ‘style investing’ (Barberis and Shleifer, 2003) Here, a substantial proportion of investors are supposed to favour particular ‘styles’ – groups of companies – for reasons that have little to with the prospects of the companies paying dividends in the future An example of a style might be the ‘dot.coms’ in the late twentieth century When a style is in fashion, the market values of its companies tend to increase by amounts that appear excessively optimistic This is despite the presence of investors who base their decisions on plausible forecasts of future profits But fashions come and go, so that, over the longer term, average returns tend to be mean-reverting: companies that are in style at one time eventually become inordinately ‘out of style’ – i.e unfashionable A substantial body of evidence suggests that the mechanisms driving aggregate market indexes (e.g the S&P 500) are different from those for individual companies’ shares The prices of individual shares tend to fluctuate in ways more compatible with the NPV relationship than market indexes.13 13 This difference has come to be known as ‘Samuelson’s dictum’ of ‘micro efficiency and macro inefficiency’ (see Samuelson, 1998) Recent evidence is reported by Vuolteenaho (2002) and Jung and Shiller (2002) Present value relationships and price variability 237 For both individual shares and market indexes, price fluctuations may be unpredictable over short periods – but for different reasons The evidence tends to be consistent with companies’ profitability, prospects, etc., dominating fluctuations in individual share prices, with aggregate indexes being more susceptible to the inexplicable vicissitudes of investor sentiment Shiller sums up forcefully (1989, p 8) Returns on speculative assets are nearly unforecastable; this fact is the basis of the most important argument in the oral tradition against a role for mass psychology in speculative markets One form of this argument claims that because real returns are nearly unforecastable, the real price of stocks is close to the intrinsic value, that is, the present value with constant discount rate optimally forecasted future real dividends This argument for the efficient markets hypothesis represents one of the most remarkable errors in the history of economic thought While this view is widely held, not all would agree with Shiller’s claim that ‘mass psychology may well be the dominant cause of movements in the aggregate stock market’ (1989, p 8) The next section reviews the role of mass psychology as a cause of extreme asset price fluctuations 10.4 Extreme asset price fluctuations Asset price volatility sometimes takes the form of spectacular increases in prices followed by equally spectacular collapses Many such historical episodes have been documented, each with its own unique characteristics, some more extraordinary than others Typically they include: (a) a period of manic optimism or frenzy (in which the majority of investors convince themselves that increasing asset prices really are justified by ‘fundamentals’); (b) a crisis of confidence (at the juncture of price increases and declines); (c) blatant fraud (which may instigate the crisis of confidence, or which is blamed, ex post, for the crisis); and (d) intense pessimism accompanied by economic distress (during which the majority opinion is that low prices are justified by ‘fundamentals’ – and, by implication, that the earlier optimism was misplaced) From at least the seventeenth century these phenomena have commonly been called ‘bubbles’, though in the modern literature the word is used in a very specific sense, discussed further below Closely related phenomena emerge from ‘Ponzi schemes’, also considered separately below Yet other incidents are associated with speculative manias or wild bouts of optimism and pessimism in a single market, or a closely aligned set of markets Rather than attempting to construct a taxonomy of all these events, there follows an overview of some of the most notorious historical examples 238 The economics of financial markets 10.4.1 Some examples from history Tulipmania, 1636–7 One of the first recorded speculative manias is that for tulip flower bulbs in the Netherlands in the 1630s Although data are sparse by modern standards, it appears that there was a rapid rise in bulb prices from late 1636 and then a steep decline after February 1637 The magnitude of the rise and subsequent decline remains controversial, because several varieties of tulip bulbs were traded For the commonest varieties of bulbs, the early months of 1637 saw both a frenzied rise and an equally precipitate decline of prices The prices of more unusual, exotic varieties increased somewhat less rapidly over several months and then declined, again somewhat less rapidly and over a longer period of time To the extent that the evidence can be relied upon, it seems that prices for the more exotic bulbs may have responded to a genuine shortage of supply relative to the demand from those who sought to grow the flowers The price fluctuations for common varieties were probably stimulated more by speculative motives – that is, the desire to profit from subsequent price changes.14 The Mississippi and South Sea Bubbles, 1719–20 Two distinct but closely related sequences of extreme price fluctuations in the years up to 1720 provide early examples of speculative booms and busts in the market for shares in joint-stock companies Both involve public share offerings by companies (the Mississippi Company in Paris and the South Sea Company in London), and both involve companies that procured monopoly privileges (from the French and British governments, respectively) in return for taking the responsibility to service government debts The Mississippi bubble preceded that in London and was the culmination of a number of financial experiments promoted by John Law, a famous – or infamous – Scotsman who was a prominent financier in France until he fell into disgrace shortly after the bubble burst John Law established or acquired several companies, of which the Mississippi Company was one Subsequently they merged into a conglomerate, the Compagnie des Indes Much of the stock of these companies was issued in exchange for government debt (the obligations on which were then renegotiated with the government) Also, the companies initiated various commercial and financial ventures The prospects for quick, high profits made the shares popular, the rising prices reflecting their popularity at the same time as embodying the potential gains that made them so attractive in the first place Early in 1720 this self-fulfilling spiral ended suddenly, when some shareholders sought to realize their gains The collapse in prices was rapid 14 See Garber (1990) for a modern analysis that also discusses the role of futures markets in the tulipmania Present value relationships and price variability 239 and sustained despite Law’s claims that the ventures would lead ultimately to a stream of future returns The sequence of events in London mirrored that in Paris The South Sea Company acquired responsibility for significant amounts of government debt on terms that were perceived to be highly favourable for the company’s shareholders An approximately sevenfold increase in the company’s share price occurred during the six months prior to its peak in mid-1720, after which the price fell at an accelerating rate to leave it, in October, roughly where it had started the year Precisely why the price increase was so rapid and why it peaked when it did are matters of debate That the episode took place at all is testimony to the capacity of financial markets to undergo manic bouts of optimism and pessimism accompanied by enormous short-term gains and losses The Wall Street Crash, 1929 In the week following 23 October 1929 the main share price indexes in New York fell by nearly 30 per cent While dramatic enough in itself, even more noteworthy is the fact that the crash marked the beginning of a prolonged decline in prices (of practically all goods and services, not just assets) that lasted for several years Accompanying the price declines were widespread bank failures in the United States and the onset of the Great Depression, a slump that persisted throughout the 1930s in much of the developed world The causes and consequences of the Wall Street Crash remain the subject of lively debate among economic historians The controversy is largely about what linked the crash with the subsequent depression, but there is also a debate about whether the crash was in any sense ‘justified’ in response to the price increases that preceded it Some commentators interpret the crash as a natural outcome of a speculative mania in 1928 and early 1929 Others claim that share prices were not ‘too high’, and were forced down by the monetary authorities and the US government as a deliberate act of policy Whether or not prices were ‘too high’ and the crash ‘justified’ depends, of course, on an underlying model of share price determination While some models are undoubtedly more plausible than others, the available evidence does not favour any one cause of the crash to the exclusion of others Almost certainly it never will The stock market crash, 1987 The most startling feature of the 1987 crash was the fall in share prices of over 20 per cent in New York on a single day, Monday 19 October Repercussions were felt in all the major stock markets around the world, and prices had fallen by nearly a third towards the end of 1987 Thereafter share prices stabilized and began to rise, albeit unsteadily Share prices had increased rapidly for about 240 The economics of financial markets a year before the October crash, so that, with hindsight, the events of 1987 appear as a short-lived boom and bust From the early 1970s an expansion in the trading of financial derivatives (options and futures) had gathered momentum By 1987 the increasing sophistication of these instruments and the prevalence of associated investment strategies (in particular ‘programme trading’) led some observers to blame the crash on their use Other commentators disputed this, placing more emphasis on US trade and budget deficits, and on proposed tax legislation under consideration in the US Congress Although the range of contending causes for the 1987 crash differs from that for 1929, again the evidence does not point unambiguously to any simple explanation of the timing and magnitude of the price fluctuations What is notable is that, by contrast with 1929, the crash of 1987 is not associated with a subsequent recession The financial system continued to function – there was no collapse The stock market bubble, 1999–2000 In the late 1990s the stock prices of companies promoting new information technologies – especially the dot.com Internet companies quoted on NASDAQ15 – began their rapid ascent, even though many had never reported any profits The prices of other shares, especially in New York, had begun their swift ascent several years earlier, in the mid-1990s, marking the onset of ‘irrational exuberance’ (The oft-quoted phrase is attributed to Alan Greenspan, chairman of the US Federal Reserve Board, in a speech on December 1996.) Despite the misgivings of cautious but perceptive analysts (e.g Shiller, 2000), asset prices continued to increase Then, in March 2000, the prices of the Internet stocks fell precipitately By mid-2001 the steep descent of share prices had become widespread, and it continued as the US economy slowed towards recession While stock markets recovered swiftly in the immediate aftermath of the terrorist attacks on 11 September 2001, ‘9/11’, a more sustained collapse of share prices began in mid-2002 Labelled ‘the Great Telecoms Crash’ in The Economist (20 July 2002), the price falls were precipitated by the financial distress that had become apparent among telecommunications companies Many of these companies had accumulated heavy burdens of debt during their boom years Moreover, corporate scandals emerged from the discovery of widespread accounting practices that allowed the overstatement of profits Bankruptcies followed (e.g WorldCom, a large American telecoms firm), together with an atmosphere 15 NASDAQ is an abbreviation for the National Association of Securities Dealers Automated Quotation system, in New York NASDAQ was introduced in 1971 as a development in the operation of the over-the-counter securities market Present value relationships and price variability 241 of distrust, especially following the collapse of Enron, the huge American energy trading corporation 10.4.2 Bubbles The concept of a financial bubble has been given a more formal interpretation in economic research than in the rather imprecise senses used so far This interpretation stems from a recognition that the NPV relationship, pt = i=1 t+i dt+i (10.8), is only one of the solutions to the condition linking prices across time, namely pt = dt+1 + pt+1 / + r (10.2).16 To construct other solutions, suppose that bt bt+1 bt+i is any sequence of numbers satisfying bt+1 = + r bt Now rewrite the NPV relationship as pt = t+i dt+i + bt (10.19) i=1 It can also be checked that (10.19) satisfies pt = dt+1 + pt+1 / + r Hence, because bt is arbitrary, the NPV relationship is not unique Furthermore, it is possible to allow for uncertainty in the usual way, by replacing variables with their expectations, so that equation (10.10) becomes pt = t+i Et dt+i + bt (10.20) i=1 The sequence bt bt+1 bt+i is assumed to satisfy Et bt+1 = + r bt Et bt+2 = + r bt Et bt+i = + r i bt In these extensions of the NPV relationship, the bt term is called the ‘bubble’ The discounted value of the dividend stream is called the ‘fundamental’ value of the asset Viewed in this way, asset prices need not equal the NPV of future payoffs but can become any one of an infinite number of values according to the size of the bubble The ‘bubble’ term captures all the speculative and self-fulfilling aspects of potentially wild asset price changes If, as in most circumstances, asset prices are non-negative pt , then negative bubbles can be ruled out (otherwise, at some finite date, , p < 0) Apart from this restriction, any positive value for the bubble at any date is sufficient to instigate the explosive process The bubbles, as expressed by (10.19) or (10.20), never burst The sequence of price changes goes on for ever This implausible feature can be eliminated by assuming that at each date there is a non-zero probability, say , that the bubble continues and a probability − that it bursts (all subsequent values of the bubble becoming zero, thereafter) 16 It is assumed for simplicity, and without loss of generality, that the interest rate is constant 242 The economics of financial markets Much research has been devoted to theoretical and empirical aspects of bubbles In the theoretical vein the research tends to focus on the circumstances in which bubbles will not occur That is, it seeks to answer the question: what assumptions are sufficient to ensure that all the bt+i values are zero? The required assumptions typically involve investors who optimize over an infinite horizon with perfect foresight (or, at least, know the random process governing the bubble) But the conditions for ruling out bubbles are model-sensitive, without generally applicable conclusions Given the evidence that bubbles, although dramatic when they occur, are isolated incidents, it is a weakness of the approach expressed by (10.19) or (10.20) that it leaves unanswered the question of when the bubble terms are likely to be non-zero Although, in a sense, the size of a bubble is arbitrary (it can start from any positive value), its trajectory, once initiated, is not Empirical work thus concentrates on examining the evidence that time series of prices follow the predicted pattern Because future dividend streams are unobserved, such exercises are fraught with difficulties Careful studies of the data suggest that many phenomena that appear, on first inspection, to be bubbles can be explained by the ‘fundamentals’ term This result can be interpreted in two ways: (a) that bubbles are rare; or, (b) that a more comprehensive theory of bubbles than is expressed by (10.20) is needed to explain the evidence 10.4.3 Ponzi schemes Ponzi schemes are named after one Charles Ponzi (of Boston, Massachusetts), who persuaded investors to participate in the exploitation of foreign exchange rate fluctuations during the unstable period following the First World War Ponzi’s venture ostensibly sought to earn arbitrage profits by trading in international postal coupons Whatever the exact nature of Ponzi’s motives and strategy, the scheme collapsed and Ponzi earned a prison sentence for his ingenuity The Oxford English Dictionary (vol XII, p 101) defines a Ponzi scheme as ‘a form of fraud in which belief in the success of a fictive enterprise is fostered by payment of quick returns to first investors from money invested by others’ The crucial aspect of Ponzi schemes (also known as ‘pyramid schemes’ or ‘chain letters’) in economics is the way in which investors’ gains accumulate from the subsequent contributions of later participants The funds remitted by later investors are used to pay off those who invested earlier The investment may or may not pay a positive stream of dividends, but, if it does, the nature of Ponzi schemes require the dividends to be paid out of the flow of funds from new investors Ponzi schemes are much like bubbles and can be Present value relationships and price variability 243 analysed using expressions such as (10.20), the distinguishing feature being the continuous arrival of new investors prepared to participate in the schemes Pay-as-you-go pension plans organized by governments resemble Ponzi schemes By taxing the younger, working generations, governments offer to pay an increasing flow of pensions to older, retired generations Given (a) the confidence that governments will always have access to the requisite tax-raising powers, (b) the arrival (birth) of successive new generations and (c) a sufficiently strong rate of economic growth, then pay-as-you-go schemes could satisfy everyone concerned – for ever Non-government (private) Ponzi schemes have invariably – to date, at least – ended in collapse, typically in a blaze of recriminations accompanied by the disappearance, castigation or imprisonment of the scheme’s promotor All that is needed for a scheme to fail is a slow down (not even a decline is required) in the arrival rate of new funds As soon as this happens, existing investors, failing to receive their promised returns, take fright, and try to liquidate their assets By design and of necessity, there will be insufficient funds to meet the promised payoffs Despite the inherent fragility of Ponzi schemes, they are commonplace, especially in newly emerging financial systems Witness the popularity of several such schemes following the retreat of communism in the Soviet Union and eastern Europe in the early 1990s Examples include the Caritas scheme in Romania (1992/93), the MMM company in Russia (1995/96), several schemes in Albania (1996) and the ‘Banyumas Mulia Abadi’ company in Indonesia (1999) In most cases the promoters advertise their companies as investing in assets purporting yield a genuine stream of returns The veracity, or otherwise, of their claims is revealed as soon the inflow of funds slows down Even more blatant (in the sense of exploiting pure greed) are the Internet pyramid schemes inviting email subscribers to make a number of small payments in anticipation of massive returns 10.5 Summary The NPV relationship is pervasive in finance (a) It expresses the current price of an asset as the discounted value of its stream of future returns, the discount factors being based on interest rates at which funds can be borrowed or lent (b) If the stream of returns is deterministic (known with certainty), the NPV relationship can be obtained as a consequence of the absence of arbitrage opportunities 244 The economics of financial markets (c) If the stream of returns is uncertain, the validity of the NPV is more fragile and relies on either: (a) expectations based on artificial martingale probabilities (implied by the absence of arbitrage opportunities); or (b) risk-neutral preferences of investors, together with unanimous beliefs about the occurrence of future states of the world A substantial body of empirical evidence suggests that asset prices are more volatile than predicted by the NPV relationship, when investors base their decisions on accurate forecasts of future dividends More successful attempts to model asset price volatility are founded on (a) the existence of ‘noise traders’ – investors who respond to fads and fashions; or (b) imperfect, though not necessarily irrational, forecasts of future dividends Extreme fluctuations in asset prices may be compatible with the NPV relationship, although the usefulness of the relationship in such circumstances (i.e in studying ‘bubbles’) is debatable, because the necessary modifications to the relationship are difficult to construct Further reading The NPV relationship is ubiquitous in finance texts, often being taken for granted Its close relatives, the so-called ‘dividend growth models’ (or ‘dividend discount models’) of share prices, are also commonplace in the literature See, for example, Sharpe, Alexander and Bailey (1999, chap 18) LeRoy (1989) surveys the literature on asset price volatility, a subject treated thoroughly by Shiller in his collection of essays on Market Volatility (1989) Much of the more recent applied work is reviewed carefully by Campbell, Lo and MacKinlay (1997, chap 7) The noise trader approach received early stimulus from Black (1986); an overview is provided by Shleifer and Summers (1990) and a detailed analysis by Shleifer (2000) The literature on extreme asset price fluctuations is large and varied Taking a historical viewpoint, Kindleberger’s Manias, Panics and Crashes (1978) provides an entertaining and perceptive account Less reliable, but equally entertaining, is McKay (1980) More seriously, Garber (1990) offers some important insights.17 Several entries in The New Palgrave Dictionary of Money and Finance (Newman, Milgate and Eatwell, 1992) provide admirably concise surveys from an analytical perspective Of particular interest are the entries on: asset price bubbles; crashes; Ponzi games; rational bubbles; the South Sea Bubble; speculation; the stock market crash of October 1929; and the stock market crash of October 1987 Shiller (2000) provides one of the most carefully researched accounts of the Wall Street boom of the late 1990s Shiller (2002) and LeRoy (2004) explore the determinants of bubbles in modern financial markets 17 For a more detailed study, with the same conclusions, see Garber (2000) Present value relationships and price variability 245 Appendix 10.1: Present values in continuous time If time is measured continuously rather than in discrete unit intervals, the substance of NPV relationships is the same even though appearances differ The rate of return on an asset is now defined as the proportional rate of change in its value at each instant of time – that is, the ratio of its payoff to its market value, v t /p t Given that the payoff is the sum of the asset’s dividend and its capital gain or loss, the instantaneous rate of return, t , at time t (sometimes called the force of interest) is given by t = d t + p˙ t pt (10.21) where v t = d t + p˙ t , and p˙ t denotes the rate of change of the asset’s price with respect to time – i.e its time derivative Equation (10.21) can be rearranged as a linear ordinary differential equation: p˙ t = t p t −d t (10.22) For the mathematical background and method of solution for this sort of equation, see, for example, Sydsæter and Hammond (1995, chap 21) Suppose, provisionally, that t = is constant across time This could be so because the rate of interest is constant and, in the absence of arbitrage opportunities, equal to the rate of return on the asset With t constant, the solution of (10.22) can be written T pt = e− −t d t d + pe− T −t (10.23) where p denotes the value of the asset at the end of its life, time T The integral replaces the summation operator in discrete time and e− −t is the discount factor between today, t, and time Note carefully the distinction between d , the flow of dividends at time , and d , the differential operator defined in conjunction with integration More generally, if t is allowed to vary across time, the solution, (10.23), is replaced with pt = T t e− t d d d + pe− T t d (10.24) d , is analogous to the reciprocal of the product where the discount factor, e− t of the ‘one plus the interest rate’ terms that appear in the discrete time version of the NPV Samuelson (1936–37) merits careful reading for details of this analysis 246 The economics of financial markets If the asset is infinitely lived, T → so-called ‘improper’ integral: pt = e− , then (10.24) is typically replaced by the d t d d t or pt = e− −t d d if is constant (10.25) t In order for (10.25) to hold, notice that (a) the limit of the integral as T → must be well defined (i.e conditions must be placed on the convergence of the present value of the dividend stream), and (b) the solution for p t is not unique unless some condition is imposed to rule out the presence of bubbles In a further generalization, much of modern financial analysis is conducted in continuous time, allowing for random dividends and interest rates This advanced topic is not explored here Appendix 10.2: Infinitely lived assets: constant growth If the dividend stream grows at a constant rate, g, the NPV relationship for an infinitely lived asset takes the form pt = = + g dt+1 dt+1 + g dt+1 + + +··· 1+r 1+r 1+r dt+1 1+r 1+ = dt+1 1+r = dt+1 r −g 1+g 1+g + 1+r 1+r +··· 1+g 1− 1+r if r > g (10.26) If r ≤ g, pt is unbounded – i.e formally undefined Appendix 10.3: The RNVR with multiple time periods In chapter it was shown that the absence of arbitrage opportunities is equivalent to the risk-neutral valuation relationship Applying the notation of this chapter, the RNVR can be summarized as pt = Et∗ vt+1 / + rt+1 , which repeats the relevant expression in chapter with the addition of subscripts to Present value relationships and price variability 247 denote the time dimension implicit in the earlier discussion Recall that the ∗ superscript appears as a reminder that the probabilities underlying the expectation emerge as a consequence of the absence of arbitrage opportunities (and need not correspond to any investor’s beliefs) Substituting the definition of the payoff, vt+1 = dt+1 + pt+1 , pt = Et∗ dt+1 + pt+1 + rt+1 (10.27) Note that Et∗ · has a time subscript: the state probabilities depend on the date at which they are evaluated Equation (10.27) holds for any date, t + s, in the future and hence can be written E∗ dt+s+1 + pt+s+1 pt+s = t+s for s (10.28) + rt+s+1 By letting s take on the values s = , successive values for pt+s from (10.28) can be substituted into (10.27) Repeated application of the law of iterated expectations shows also that Et Et+s · = Et · (see chapter 3, page 79) After making the substitutions using (10.28), and assuming the convergence of the sum of expected discounted returns, (10.27) becomes pt = Et∗ t+1 dt+1 + t+2 dt+2 + · · · + t+s dt+s + · · · (10.29) where t+s = + rt+1 + rt+2 · · · + rt+s −1 , for s 1, so that t+s denotes the discount factor for the time period t to date t + s Note that, if the individual discount factors are all equal, then t+s = + r −s – that is, the common discount factor expressed in terms of a constant interest rate If a behavioural interpretation is given to the condition (10.29), the probabilities underlying the expectation correspond to investors’ (unanimous) beliefs and the ∗ superscript can be omitted It is necessary to remember, however, that, to make sense of the result, an additional assumption about individual behaviour – typically, risk neutrality – must be invoked Furthermore, in the NPV as expressed by equation (10.10), pt = i=1 t+i Et dt+i , the expectations operator is applied on the assumption that the discount factors are non-random – i.e the risk-free interest rates, though not necessarily constant across time, are known with certainty at date t References Barberis, N., and A Shleifer (2003), ‘Style investing’, Journal of Financial Economics, 68(2), pp 161–99 Barsky, R B., and J B De Long (1993), ‘Why does the stock market fluctuate?’, Quarterly Journal of Economics, 108(2), pp 291–311 Black, F (1986), ‘Noise’, Journal of Finance, 41, pp 529–43 Campbell, J Y., A W Lo and A C MacKinlay (1997), The Econometrics of Financial Markets, Princeton, NJ: Princeton University Press Campbell, J Y., and R J Shiller (1989), ‘The dividend-price ratio and expectations of future dividends and discount factors’, Review of Financial Studies, 1(3), pp 195–228 Garber, P M 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(1966), The Collected Scientific Papers of Paul A Samuelsonw, Vol I, Cambridge, MA: MIT Press Sydsæter, K., and P J Hammond (1995), Mathematics for Economic Analysis, Englewood Cliffs, NJ: Prentice Hall International Vuolteenaho, T (2002), ‘What drives firm-level stock returns?’, Journal of Finance, 57(1), pp 233–64 ... xv 37 67 87 10 0 10 4 11 9 11 9 12 0 12 2 12 6 12 8 13 1 13 2 13 7 14 7 15 0 15 2 15 3 15 8 18 5 19 1 208 232 253 289 308 309 xvi 13 .3 14 .1 15 .1 18 .1 18.2 18 .3 18 .4 19 .1 19.2 19 .3 20 .1 20.2 20.3 List of figures... portfolio with a single risky asset References 11 4 11 5 11 8 12 1 12 5 13 1 13 3 13 4 13 5 13 9 14 0 14 1 14 2 Contents xi The capital asset pricing model 6 .1 Assumptions of the CAPM 6.2 Asset market equilibrium... 6 .1: The CAPM in terms of asset prices Appendix 6.2: Linear dependence of j in the CAPM Appendix 6.3: The CAPM when all assets are risky References 14 3 14 4 14 5 14 9 15 1 15 4 15 7 15 9 16 0 16 2 16 2 16 5