COMM 472: Quantitative Analysis of Financial Decisions

1
COMM 472 Quantitative Analysis of Financial
Decisions

• Part 1
• Fundamental Properties of Returns

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Equity Returns

• Throughout this section, we will make extensive
  use of the dynamic properties of equity returns
• Returns are dynamic in the following sense what
  happens to returns today may, and in fact does,
  effect how returns behave tomorrow.

3
Preliminary Concepts and Notation

• Well focus our attention on equity returns.
• Random returns are the result of one or both of
  the following effects
• Future dividends (usually random),
• Future prices (usually random).

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Preliminary Concepts and Notation

• Note that returns are the sum of two components
• The Dividend Yield
• The Capital Gain

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Preliminary Concepts and Notation

• Note
• This definition is of a gross return (i.e. net
  return gross return - 1).
• The return relates to a price change over some
  specific period of time. That time period can be
  an instant, a day, a month, a year, etc.
  depending on the application. (Well spend a lot
  of time in this course talking about the effect
  of time horizon on returns.)

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Preliminary Concepts and Notation

• Returns can also be stated in their logarithmic
  form
• So that

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Multiperiod Returns

• Returns over long horizons show the effect of
  compounding
• For simple returns (no dividends)
• Long-horizon returns are the product of
  short-horizon returns.

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Multiperiod Returns

• For log returns (no dividends), long-horizon
  returns are the sum of short-horizon returns,
  since

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Multiperiod Returns

• When we choose whether to work with simple or log
  returns we make a tradeoff
• Simple returns are more convenient when
  aggregating returns of stocks in portfolios. (The
  weighted average of simple returns is the simple
  return on the portfolio. This is not true for log
  returns.)
• Log returns are more convenient when aggregating
  returns of a stock or index across time (For
  statistics, it is more convenient to work with
  sums than products. For example, the sum of
  normally distributed log returns is normal
  whereas the product is not.)

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The Distribution of Returns

• Returns are usually random. We can add some
  structure to the return process if we specify a
  distribution for the returns.
• One common specification is that single period
  log returns are normally distributed.
• Is this a reasonable assumption?

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Estimating Moments of Returns

• Random variables can be characterized by their
  moments.
• Typically, we work with the first two moments
  the mean and the variance.

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Other Moments

• Higher moments are sometimes of interest
• These moments are called the skewness and
  kurtosis. For a standard normal distribution
  (mean 0, variance 1) the skewness is 0 and the
  kurtosis is 3.

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Dynamic Portfolio Strategies

• Part 2
• Dynamic Properties of Returns

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Statistical Properties of Long Horizon Returns

• The properties of long-horizon returns can be
  derived from their short-run properties.
• We make use of the following properties of
  expectations and variances

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Case 1 IID Normal Returns

• If 1-period log returns are independent and
  identically normally distributed then the mean
  and variance of the returns distribution grows
  proportionally with the horizon.
• Remember that

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Long-Run Mean Returns

• So
• Notice that
• No one return provides information about any
  other (returns are independent).
• Each return has the same mean (identically
  distributed).

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Long-Run Variance

• Notice that
• Returns are uncorrelated so that no covariance
  terms show up (independent).
• All variances are the same (identically
  distributed).

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IID Return Dynamics

• This model of return dynamics, although very
  simple, has been the mainstay of Dynamic Asset
  Pricing for many years.
• Black-Scholes pricing is based on this model of
  returns.
• Question Does the data support this hypothesis?

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Empirical Properties of Long-Horizon Returns

• Are returns independent?
• Independence has broader implications, but we
  will focus on whether or not returns are
  uncorrelated.
• Two methods of testing
• Directly measure autocorrelations of returns.
• Examine variance ratios.

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Empirical Properties of Long Horizon Returns

• Why do we worry about independence of returns?
• Dependencies, as we will see, have dramatic
  effects on long-run return properties.
• One example time diversification
• If returns from one time period are negatively
  correlated with those from another, then long
  horizon returns will be less risky than would be
  predicted if they were assumed independent.

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Autocorrelation

• Definition
• The autocorrelation coefficient measures the
  correlation between two random variables from a
  time series (eg. two returns on an index).
• The autocorrelation must be specified with
  respect to some lag length (the time between
  measurements of the random returns)

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Autocorrelation

• Well assume throughout that return series are
  covariance-stationary
• One-period returns at all dates have the same
  variance.
• The covariance between returns at different dates
  depends only on the lag (k)

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Estimating Autocorrelations

• Correlation coefficients can be calculated by
  using sample averages

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Testing Significance

• In order to assess the statistical significance
  of these autocorrelations we need to know the
  sampling distributions of the statistics.
• If we assume returns are IID

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Testing for IID Returns with Autocorrelations

• One problem with testing for IID returns using
  autocorrelations is that it is not clear what
  lags to use to test for zero autocorrelation.
• If returns are IID all autocorrelations should be
  zero.
• One solution is to use a statistic that
  summarizes many autocorrelations.

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Portmanteau Statistic

• The Q-statistic simply sums the squares of many
  autocorrelation statistics
• This statistic tests for zero autocorrelation at
  all of m lags, giving power to test against a
  broad variety of alternative hypotheses for
  return dynamics.

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Testing for IID Returns with the Portmanteau
Statistic

• The Q-statistic has a chi-squared distribution
  with m degrees of freedom.
• This distribution can be used to determine,
  statistically, whether or not the statistic is
  significantly different from zero.

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Variance Ratio Statistics

• We saw earlier that if returns are IID, the
  variance of long-horizon returns is proportional
  to the horizon. This result serves as the basis
  for using Variance Ratios to test whether or
  not returns are IID.
• Definition
• The q-horizon variance ratio statistic is the
  ratio of the variance of the q-period return to
  the variance of the 1-period return, divided by q.

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Variance Ratio Statistics

• Note that for IID returns this statistic should
  be identically 1 for all horizons.

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Variance Ratio Statistics

• eg. With two-period IID returns

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Variance Ratio Statistics

• As an alternative, suppose returns are
  autocorrelated

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Variance Ratio Statistics

• This is an important result that has implications
  for portfolio choice
• The investing horizon can have dramatic effects
  on the risk-return relationship.
• In particular, if returns are negatively
  autocorrelated, long horizon investors will face
  less variable equity returns than will short
  horizon investors.

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Testing for IID Returns with Variance Ratio
Statistics

• The null hypothesis is that returns are IID
  normal
• Well work with 2n1 log prices to determine the
  VR(2) statistic

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Testing for IID Returns with Variance Ratio
Statistics

• Note that
• With log returns, the mean return is simply the
  geometric average return (the last log price
  minus the first log price divided by 2n).
• The mean of the two period return is twice the
  one period return.
• Only n two-period returns are used to estimate
  the two-period return variance.

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Testing for IID Returns with Variance Ratio
Statistics

• The variance ratio statistic is normally
  distributed

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Testing for IID Returns with Variance Ratio
Statistics

• The general case

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Testing for IID Returns with Variance Ratio
Statistics

• An improvement (correct bias and overlap)

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Variance Ratio Statistics Distribution of Test
Statistics

• The variance ratio statistics are normally
  distributed