Random Walk Theory

6 Key excerpts on "Random Walk Theory"

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  eBook - ePub

  Investment and Portfolio Management

  A Practical Introduction

  • Ian Pagdin, Michelle Hardy(Authors)
  • 2017(Publication Date)
  • Kogan Page

    (Publisher)

  In this chapter we will not seek to justify, test, or affirm the various theories and models but simply to discuss them and explain their uses and limitations within the modern investment management arena. The reader who finds that their appetite for further knowledge has been awakened will no doubt wish to research the various topics in greater detail and this is certainly encouraged by the authors. Only by researching the various theories and models widely and in more depth can the reader make an informed decision regarding their personal opinions and preferred investment strategies.

  Random Walk Theory

  This theory traces its roots back to the early analysis of stock market price changes over periods of time. Maurice Kendall found to his surprise that he could not identify a predictable pattern in share price changes;4 indeed, the prices appeared to behave in a random manner as they were as likely to rise as they were to fall on any given day. This was of concern to financial economists and analysts of that time, because it seemed to imply irrationality in the markets and cast doubts upon the skills of those involved in dealing in the markets. Closer study, however, indicated that the random nature of share price movements actually pointed to the market being efficient and this gave rise to the Random Walk Theory and the efficient market hypothesis, with its lingering effects upon investment policy and methods. Much of the work initially carried out centred on large quantities of empirical analysis but little in the way of theoretical research.

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  Information Efficiency and Anomalies in Asian Equity Markets

  Theories and evidence

  • Qaiser Munir, Sook Ching Kok, Qaiser Munir, Sook Ching Kok(Authors)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  The random walk hypothesis on the small and medium capitalized segment of the Indian stock market

  Vinod Mishra and Russell Smyth

  1.0 Introduction

  A market in which futures prices cannot be predicted using past historical price data is weak-form market efficient (Fama, 1970). A market that is weak-form efficient is said to be characterized by a random walk. A random walk ‘is a term loosely used in the finance literature to characterize a price series where all subsequent price changes represent random departures from previous prices’ (Malkiel, 2003: 60). The implication is that if a market is characterized by a random walk, an uninformed investor purchasing a diversified portfolio will obtain, on average, a rate of return as good as that realized by an expert (assuming a comparable level of risk).

  Since the early studies of Samuelson (1965) and Fama (1970), a sizeable literature has evolved that tests the efficient market hypothesis (EMH) in financial markets. Hiremath and Kumari (2014: 1) suggest, ‘There is no theory that has attracted volumes of research like the EMH.’ Lim and Brooks (2011) survey much of this literature. Beginning with Chaudhuri and Wu (2003, 2004) there is an increasing number of studies that have tested the EMH in emerging stock markets (see, e.g. Gough and Malik, 2005; Maghyereh, 2005; Phengpis, 2006; Lean and Smyth, 2007; Cooray and Wickremasinghe, 2008; Hasanov and Omay, 2008; Munir, Kok, Furouka and Mansur, 2012; Wang, Zhang and Zhang, 2015).

  One of the most common approaches to test the EMH is to test for a unit root in stock prices. If stock prices contain a unit root, this implies that any shock to stock prices will be permanent and, hence, it is not possible to predict future prices based on movements in past prices. Such a finding would be consistent with stock prices being weak-form efficient or exhibiting a random walk. The alternative hypothesis is that stock prices are mean reverting, implying that prices will revert to their natural mean over time, thus making it possible to forecast future price movements using past data.

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  eBook - ePub

  Financial Valuation and Econometrics

  • Kian Guan Lim(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Chapter 7

  RANDOM WALK APPLICATION: MARKET EFFICIENCY

   

  Key Points of Learning

  Random walk, Rational expectation, Informational efficiency, Equilibrium asset pricing model, Benchmark model, Joint test, Market efficiency, Euler condition, Martingale, Markov process, Strong-form market efficiency, Semi-strong form market efficiency, Weak-form market efficiency, Variance ratio, Long-term memory

  In this chapter, we shall study the mathematical construction of a random walk and how this has been a workhorse for developing the concept of market efficiency in the finance literature. While market efficiency (or informational efficiency, to be more precise about the type of market efficiency discussed here) is a benchmark consideration in an economy with rational agents or investors, it is also important to realise that anomalies and deviations from the benchmark such as in behavioural finance are plausible and possible.

  7.1RANDOM WALK

  Suppose a process Pt +1 follows a random walk:

  where µ is a constant, and et +1 has zero mean and is not correlated with Pt or E(et +1 | Pt ) = 0, and the coefficient of Pt is b = 1. Moreover, et +1 is not correlated with any other economic variables except Pt +1 . The property of et +1 essentially implies that prices at t + 1 are so unpredictable at t, and that very few if at all any investor can claim to beat the market over any reasonable stretch of time. Note that for a random walk, et +1 need not follow any specific probability distribution. It could as well be a normally distributed noise or a discrete Bernoulli random variable.

  The strongest version of a random process as in Eq. (7.1) is where et +1 is i.i.d. and E(et +1 ) = 0. A second strong version is where et +1 is independently, but not necessarily identically, distributed and E(et +1 ) = 0. In particular, this allows variance of et +1 to change over time conditionally. A third weak version is where et +1

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  Modelling Financial Time Series (2nd Edition)

  • Stephen J Taylor(Author)
  • 2007(Publication Date)
  • World Scientific

    (Publisher)

  Testing the random walk hypothesis 157 Expected returns need not be constant even when prices perfectly reflect relevant information. A more general null hypothesis is investigated in Section 6.11 based upon returns having expected values determined by some appropriate equilibrium theory. Tests of the more general hypothesis will be shown to give very similar results to random walk tests. It then follows that some relevant information is reflected in prices during the days following its first impact upon the market. Sections 6.11 and 6.12 show that the apparent inefficient use of informa- tion extends beyond calendar effects and is not caused by market rules restricting prices nor by dealers buying and selling at different prices. Figures given in Section 6.13 suggest the inefficiencies occurred through- out the full period of the long time series. 6.9 It was shown in Chapter 5 that tests based upon the autocorrelation coefficients of returns will be unreliable because the variances of the coefficients exceed lln when the random walk hypothesis is true. Now let T,* and T; respectively denote T* when coefficients rT,x and rT,v are used. A very surprising conclusion is revealed when the tests are evaluated using the coefficients rT,x : for each of the 40 series T,* is less than T;, often by a considerable amount. Table 6.4 shows the values of T,*, T;, and T; - T,*. The number of random walk rejections, at the 5 per cent level for T:, is only 6 out of 15 for the commodity and currency series. A majority of series and lags have rT,v > rr,x; see Table 6.1 for first-lag coefficients. The rejections count for r,,xqn is 14 out of 16 for the stocks series. Considering random variables RT,x and R7,y, it appears true that E[R,,,] < €[R,,,] and this explains the observed conclusion T: < T; . The observation T,* < T; for all series is itself evidence against the random walk hypothesis. Random walks have been simulated to confirm this deduction.

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  A Primer for Unit Root Testing

  • K. Patterson(Author)
  • 2010(Publication Date)
  • Palgrave Macmillan

    (Publisher)

  An Introduction to Random Walks 151 is 45° line, has been superimposed. The following economic examples illustrate this and other points connected with random walks. 5.8 Economic examples In this section we examine some economic time series to see if they exhibit the characteristics of a random walk. 5.8.1 A bilateral exchange rate, UK:US The first series to be considered comprises 7,936 observations on the (nominal) UK:US exchange rate (expressed as the number of units of one £ it takes to buy one US$). The data are daily, excluding weekends and Public Holidays, over the period June 1975 to May 2006. The data are graphed in Figure 5.15 (upper panel). The first point to note is that despite the widespread analysis of nominal exchange rates for random walk behaviour, the limitation of non-negativity rules out a simple ran- dom walk, although it allows an exponential random walk. One could view the lower limit as a ‘reflecting’ barrier in the random walk. The strategy we adopt is to reintroduce ‘Peter’ who gambles on whether the exchange rate will move up, with respective ‘outcomes’ of +1 (up) and –1 (down), and 0 for no change. (In effect, the original sample space has been mapped into a much simpler sample space.) The random variable at each stage in the sequence therefore has outcomes corresponding to a multinomial trial. The sequence so defined is (S t , t = 1, ... , T), whereas the original sequence is denoted (S E,t , t = 1, ... , T). The sequence corresponding to the ‘gamble’ is shown in Figure 5.15 (lower panel). The potential sequences from this formulation do not have the non-negativity limitation of the original series and have a the- oretical mean of zero. They can, then, be considered as being generated Figure 5.14b Scatter graph of S t on S t–1 −10 −5 0 5 10 15 −10 −5 0 5 10 15 20 S t S t−1 152 A Primer for Unit Root Testing by a simple symmetric random walk. Some of the characteristics of this process are summarised in Tables 5.3a and 5.3b.

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  The Economics of Financial Markets

  • Roy E. Bailey(Author)
  • 2005(Publication Date)
  • Cambridge University Press

    (Publisher)

  6 The martingale and random walk models predict that all these autocorrelations equal zero. Given a sample of data, the autocorrelations can be computed, together with formal statistical tests of the null hypothesis that they are equal to zero. As with much empirical work, the statistical results are mixed. 7 The tests do, however, provide at least some evidence that the autocorrelations are non-zero, thus tending to reject the random walk hypothesis in one or other of its forms. That is, the evidence suggests that future asset prices are predictable , albeit with 5 Formally, the expected value of y t + s equals s and the variance of y t + s equals 2 s – the expectation and variance are proportional to the length of the time interval. See Ross (2003, chap. 3) for an introduction to geometric Brownian motion. 6 The autocorrelation coefficient between r t and r t − s is defined as the ratio of cov r t r t − s to the product of the standard deviations of r t and r t − s , respectively. In applied work, the rate of return is often calculated as the difference in the logarithm of prices in adjacent periods – i.e. r t = ln p t − ln p t − 1 . This can be interpreted as either (a) an approximation to the proportional rate of change of the asset’s value, or (b) an exact measure of the continuously compounded rate of change over the interval t − 1 to t . 7 A thorough review of the evidence appears in Campbell, Lo and MacKinlay (1997, chap. 2). 62 The economics of financial markets error, from present and past asset prices. There is evidence that measured rates of return for portfolios of shares are positively autocorrelated over short intervals of time (i.e. high rates of return tend to be followed by high rates of return and vice versa, over periods of several months). For the individual assets that comprise portfolios, the evidence tends to be more supportive of the random walk hypothesis.

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