Asset Return Predictability

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Asset Return Predictability
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Asset Return Predictability

• Chapter 1, CLM
• Introduce notation to a limited extent.
• Discuss the basic assumptions financial
  economists make about returns distributions.
• Review the various forms of the efficient markets
  hypothesis.

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Chapter 2, CLM

• Tests of asset return predictability.
• Various forms of the random walk hypothesis.
• Tests of the random walk hypothesis CJ test,
  runs test, technical trading rules.
• Variance ratio tests (LM 1988).
• Autocorrelations (FF 1988, Richardson 1993).
• Long horizon returns.
• Application Momentum (JT 1993, CK 1998, DT 1985).

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Why Returns?

• The statistical methods you will learn in this
  course will be used primarily to analyze returns
  and the relations between different returns, not
  prices.
• This may seem paradoxical, because one might
  think that asset pricing models would have a lot
  to say about how assets are priced.
• Although it is true that financial economists
  have devised such models,
• dividend discount model
• earnings multiple valuation model
• they work notoriously badly, primarily because
  forecasting both cash flows and future interest
  rates is incredibly difficult.

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The Focus on Returns

• The price formation process is taken as given
  investors have no market power.
• Investment technology is taken as a constant
  returns to scale technology so return is a scale
  free description of the opportunity.
• The focus is then on what stock returns ought to
  look like as a function of
• Risk
• Information flows

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A Technical Issue

• Returns processes are thought to be stationary
  while price processes are not.
• To use the statistical techniques commonly
  applied by economists it is necessary that the
  sample moments converge to the population
  moments. Typically, what is assumed is
  covariance stationarity and ergodicity (or
  perhaps mixing rather than ergodicity).

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Stationarity and Ergodicity

• The material will typically deal with the time
  series properties of returns.
• It is common to compute numbers such as
• expected returns,
• the variances of returns, and
• the covariance between the returns of one asset
  and the returns of another.
• What is required for this to make sense?

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Stationarity and Ergodicity

• These statistics must be well defined in the
  sense that they do not change (except perhaps in
  some pre-specified way) over the course of the
  analysis.
• Covariance stationarity and ergodicity are the
  assumptions commonly employed to ensure this
  basic requirement.

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Covariance Stationarity

• A stochastic process yt is weakly stationary or
  covariance stationary if
• 1. E(yt) is independent of t
• 2. Var(yt) is a finite positive constant,
  independent of t.
• 3. Cov(yt, ys) is a finite function of t-s but
  not of t or s.

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Ergodicity

• Intuitively, this means that values of the
  process sufficiently far apart are uncorrelated.
• Therefore, by averaging a series through time one
  is continually adding new and useful information
  to the average.
• Thus the time series average is an unbiased and
  consistent estimate of the population mean and
  estimates of the variance and autocovariances
  will be consistent.

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Basics

• Simple return from time t-1 to t
• where Pt is the price of the asset at time t.
• Simple Gross Return
• 1Rt

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Basics

• Compound return over k periods

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Annualized Returns

• Often the returns expressed in the popular press
  are annualized. Let k be the number of
  compounding periods and let n be the number of
  compounding periods in a year, so that there are
  Nk/n years of data. The annualized return is
  then defined as the geometric average of the
  returns

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Annualized Returns

• For example, suppose the compounding interval is
  monthly (n 12), the monthly return is 1, and
  there are two years of data (k 24). Then, the
  annualized return would be given by

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Annualized Returns An Approximation

• The annualized return is often approximated using
  the arithmetic mean
• This approximation can be fairly poor.

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Annualized Returns

• Suppose there are instead, 360 compounding
  periods in a year and the return in each is
  1/30.
• Then, the annualized return is actually
• while the approximated return is still 0.12.

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The Alternative Continuous Compounding

• The continuously compounded return, or log return
  ri of an asset is defined as the natural
  logarithm of its gross return
• Why is this called the continuously compounded
  return?

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Continuously Compounded Return

• For some reason, although banks calculate and pay
  interest more frequently, (quarterly, monthly,
  daily, or continuously), it is traditional to
  quote the rate on an annual basis.
• Call that rate Rnom (nominal).
• Then, if there are n compounding periods per
  year, the rate of return per period is Rnom/n.
• And the rate of return per year is (1Rnom/n)n
  1.

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Continuously Compounded Return

• If we take the limit as n goes to infinity, we
  get an annualized return of
• and the balance grows to
• at the end of the year, so that is the
  gross continuously compounded return.

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Continuously Compounded Return

• Taking logs yields r Rnom.
• This is essentially what CLM call the
  continuously compounded return.
• Of course, this is just an approximation, because
  n never goes to infinity but it is usually a very
  good one.

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Why Use Logs

• Conversion of products to sums
• Consider multiperiod return 1Rt(k). Its the
  product of single period returns
• But the log return is

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Why Use Logs

• The continuously compounded return is the sum of
  the continuously compounded single period
  returns.
• This makes some things much easier in modeling
  time series behavior.
• We will see that it is easier to model the
  behavior of sums than of products.
• We will also see that it allows the imposition of
  limited liability in a straightforward way.

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A Slight Problem

• The simple return on a portfolio is the weighted
  average of the simple returns on the individual
  securities in the portfolio
• But

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Example

• Suppose that N2, 1R11.12, 1R21.08, w1 w2
  .5, then 1Rp 1.10.
• The log portfolio return is
• rp ln(1.10) .09531017980
• But
• .5ln(1.12) .5ln(1.08) .09514486322
• When returns are measured over shorter intervals
  of time, the approximation is better.
• Still, it is traditional to use simple returns
  when a cross section of assets is studied but log
  return for time series studies.

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Dividends

• When the asset in question pays periodic
  dividends, the simple net return is
• The simple gross return and the log returns are
  defined from this.

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Excess Returns

• An excess return is the difference between the
  return on an asset of interest and the return on
  a reference asset, R0t.
• Quite often the reference asset is a riskfree
  asset or short term T-bill, maybe a zero beta
  asset.
• The simple excess return on asset i is defined as
  Zit Rit R0t

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Excess Returns

• The log excess return is not the log of the
  excess return, but instead zit rit r0t, the
  difference between the log return on the asset
  and the log return on the reference asset.
• The excess return can be thought of as the payoff
  on an arbitrage portfolio, long asset i and short
  the reference asset so there is no initial
  investment.
• Because the initial investment is zero the return
  is undefined but the payoff will be proportional
  to the excess return.

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• Perhaps the most important characteristic of
  asset returns is their randomness. The return of
  IBM stock over the next month is unknown today,
  and it is largely the explicit modeling of the
  sources and nature of this uncertainty that
  distinguishes financial economics from other
  social scienceswithout uncertainty, much of the
  financial economics literature, both theoretical
  and empirical, would be superfluous.

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Common Distributional Assumptions

• Much of financial econometrics makes the
  assumption of normal distributions, but some care
  must be taken in determining what is normal.
• Normality is appealing because sums of normally
  distributed random variables are normal.
• If simple returns are iid normal what happens?
• First, this violates limited liability.
• Second, multiperiod simple returns then cannot be
  normal since they are products of simple returns.

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Lognormality

• Let single period simple gross returns be
  lognormally distributed so that the continuously
  compounded or log returns are normally
  distributed.
• That is, rit is i.i.d normal with mean µi and
  variance si2.
• Then 1Rit is lognormally distributed and thus
  has a minimum realization of zero.

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Lognormality

• Rit has a mean and variance given by
• The lognormal model is what is generally used.

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Data and Statistics

• This being an empirical course, it is important
  to understand the difficulties associated with
  estimating something as simple as the mean
  return.
• What we focus on now are the properties of
  estimates of
• Means
• Other moments

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Empirical Validation

• In the book you will see empirical properties of
  stock returns.
• They are not really consistent with either the
  simple normal or the lognormal models.
• You need to note the differences and see how
  things might be improved. Our continuing
  discussions will examine these issues.

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Empirical Validation

• Chapter 2 considers the predictability of asset
  returns.
• One question will be Can past return
  realizations tell us anything about expected
  future returns.
• The efficient markets hypothesis (EMH) is an
  important aspect of this discussion.

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The Problem

• Estimating means requires more data than we can
  reasonably expect to get.
• That is, the time series is not likely to be
  stationary for long enough for us to get enough
  precision.
• Luenberger calls this The Blur of History.

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EMH

• Fama (1970)
• A market in which prices always fully reflect
  available information is efficient.
• Malkiel (1992)
• A capital market is fully efficient if it
  correctly reflects all information in determining
  security prices. Formally, the market is said to
  be efficient with respect to some information
  set if security prices would be unaffected by
  revealing that information to all market
  participants. Moreover, efficiency with respect
  to an information set implies that it is
  impossible to make economic profits by trading on
  the basis of the information in that set.

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In an efficient market, prices should be random

• Let the price of a security at time t be given
  by
• Pt EVIt Et V
• The same equation holds one period ahead so that
• Pt1 EVIt1 Et1 V
• The expectation of the price change over the next
  period is
• EtPt1 - Pt EtEt1 V - Et V 0
• because It is contained in It1 , so EtEt1 V
  Et V
• by the law of iterated expectations.

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Discussion

• The second sentence of Malkiels definition
  expands Famas definition and suggests a test for
  efficiency useful in a laboratory.
• The third sentence suggests a way to judge
  efficiency that can be used in empirical work.
• This is what is concentrated on in the finance
  literature.
• Examples mutual fund managers profits if they
  are true economic profits then prices are not
  efficient with respect to their information.
• Difficult to test for good reasons we will
  discuss.

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Versions of Efficiency

• Weak Form
• Information set is the set of historical prices
  (and sometimes volumes).
• Semi-strong Form
• Information set is the set of all publicly
  available information.
• Strong Form
• Information set includes all knowable information.

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Violations of Efficiency

• That technical traders can make money violates
  which form?
• Reading the Wall Street Journal and devising a
  profitable trading strategy violates which form?
• Corporate insiders making profitable trades
  violates which form?
• Question Can markets really be strong-form
  efficient?

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What Does Profitable Mean?

• Were talking about economic profits, adjusting
  for risk and costs.
• Need models of such things, particularly the risk
  adjustment.
• One way of thinking of the tests of efficiency is
  that they are joint tests of efficiency and some
  asset pricing model, or benchmark.
• For example, many benchmarks typically assume
  constant normal returns. This is easier to
  implement, but doesnt have to be right. Hence
  rejections of efficiency could be due to
  rejections of the benchmark.

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The Tests

• Most tests suggest that if the security return
  (beyond the mean) is unforecastable, then market
  efficiency is not rejected.
• With the wrong asset pricing model, we can wind
  up rejecting efficiency. It would be easy to
  find (de-meaned) returns to be forecastable if we
  had the wrong mean.