Bus 315 Ch'8 Factor Models and Arbitrage Pricing Theory

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Bus 315 Ch.8 Factor Models and Arbitrage
Pricing Theory
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CAPM downside

• To use CAPM we need to find the market
  portfolio (tangency portfolio) on efficient
  frontier
• To utilize Markowitz portfolio theory, one needs
  too many inputs (for calculating efficient
  frontier). For example, with 50 securities one
  would need 1225 estimates of covariances, 50
  variances and expected returns). With of
  stocks in TSX (1600), one would need to estimate
  1.3 million parameters
• Inputs may not be accurate and thus the
  calculated portfolios may not be efficient.
  Moreover, optimization may lead to nonsensical
  results because of poor estimates(e.g. negative
  variance)

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Returns and systematic risk

• Covariances between returns are caused by some
  common macroeconomic factors (interest rate, GDP
  change etc.)
• Unexpected changes in these factors lead to
  unexpected changes in returns
• We can group all relevant factors in one economic
  indicator (for example, GDP growth)
• Therefore, we can specify the return on any asset
  as
• E(R) is the expected return on the asset at the
  beginning of the period, m is the impact of
  unanticipated macroeconomic events, e is the
  impact of firm specific events

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Single Factor Model

• Different stocks have different sensitivities to
  macroeconomic factors, Fs. These sensitivities
  will be denoted by factor betas.
• F denotes the unanticipated component in factor
  (e.g. unexpected inflation or unanticipated GDP
  growth)
• We can assume that the return on a broad market
  index (TSX) can be a good factor.
• If we use a stock index (TSX) as a factor then
  this model is called a single-index model

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Single Index Model

• With single index model the (excess) return on
  any stock can be decomposed in three parts
• If we rewrite the equation in terms of excess
  returns, it will look like
• Return on an index may be viewed as a sum of
  predicted part (risk-free rate) and unanticipated
  part (market premium)

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Estimating inputs with factor model

• Now, in case of 50 securities we need to
  estimate only 50 expected returns, 50 sensitivity
  betas, 50 firm-specific variances, and market
  variance

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Estimating inputs Example

• Suppose that the index model for stocks A and B
  is estimated with the following results
• Calculate the standard deviation of each stock
  and the covariance between them

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The Single Index Model

• Advantages
• Reduces the number of inputs
• Easier for security analysts to specialize to
  calculate covariance one simply needs to know the
  sensitivity of the stock to a factor. No need for
  inter-industry knowledge.
• Drawback
• The simple separation of factors (macro versus
  micro risk) rules out important risk sources
  (such as industry events)
• Estimation
• Run regression of historical excess returns of a
  security on the market (e.g. SP/TSX composite)
  excess returns
• The regression equation is called Security
  Characteristic Line (SCL)

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Security Characteristic Line
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Interpreting the regression results
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Index Model and Diversification

• If we create a portfolio of n equally weighted
  stocks (w1/n), then the non-systematic part of
  the portfolio will fade away
• To see it, recall that
• The portfolio variance is
• As n increases, the portfolio variance approaches
  the systematic variance

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Risk Reduction with Diversification
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Tracking Portfolios

• Suppose you find a portfolio that you consider to
  be mispriced
• You can replicate the sensitivity to the factor
  by creating a tracking portfolio (140 in TSX
  Index and -40 (borrowing) at risk-free rate)
• Tracking portfolio matches the systematic risk
  (has the same beta)
• If you take long position in portfolio P and
  short the tracking portfolio you can earn alpha
  of 4 irrespective of the market movements.
  Still, the non-systematic risk would be present.

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Multifactor Models

• The single index model uses just one factor as
  explanation for systematic risk
• However, there might be other factors too
• Multifactor models use other factors in addition
  to market return
• Estimate a beta or factor loading for each factor
  using multiple regression
• The way we find what factors to include is purely
  empirical, there is no theory that say which ones
  to include
• Some factor ideas change in expected inflation,
  change in unanticipated inflation, change in
  industrial production (GDP), credit spread, yield
  spread.

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Multifactor Model Equation

• For example, the multifactor (2-factor) model may
  look like this
• Ri E(ri) ßGDP (GDP) ß IR (IR) ei
• Ri Return for security i
• ß GDP Factor sensitivity for GDP
• ß IR Factor sensitivity for Interest Rate
• ei Firm specific events

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Arbitrage Pricing Theory (APT)

• Arbitrage - arises if an investor can construct a
  zero investment portfolio with a sure profit
• Law of one price says that if two assets are the
  same in terms of economic aspects, they should
  cost the same
• Since no investment is required, an investor can
  create large (infinite) positions to secure large
  levels of profit
• In efficient markets, profitable arbitrage
  opportunities will quickly disappear since
  arbitrage is a stronger argument than dominance
  of one security over another
• To get to the APT one needs the following
  assumptions
• Security returns can be described by a factor
  model
• There are enough securities to diversify away
  firm-specific risk
• Arbitrage opportunities are not allowed

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APT and Well-Diversified Portfolios

• For well-diversified portfolios (ones with
  negligible non-systematic risk), this should be
  true
• If the portfolio are well diversified, and have
  the same exposure to the factor risk, they should
  have the same return. Otherwise, arbitrage is
  possible.
• This means that the risk premiums, E(r)-rf,
  should be proportional to the factor betas

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Disequilibrium Example
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APT and CAPM Compared

• APT applies to well diversified portfolios and
  not necessarily to individual stocks
• With APT it is possible for some individual
  stocks to be mispriced - not lie on the SML
• APT is more general in that it gets to an
  expected return and beta relationship without the
  assumption of the market portfolio
• APT can be extended to multifactor models

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A Multifactor APT

• A factor portfolio is a portfolio constructed so
  that it would have a beta equal to one on a given
  factor and zero on any other factor
• These factor portfolios are the benchmark
  portfolios for a multifactor security market line
  for an economy with multiple sources of risk

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Where Should we Look for Factors?

• The multifactor APT gives no guidance on where to
  look for factors
• Chen, Roll and Ross
• Returns a function of several macroeconomic and
  bond market variables instead of market returns
• Fama and French
• Returns a function of size and book-to-market
  value as well as market returns

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Bus 315 Ch.9 Market efficiency
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Efficient Market Hypothesis (EMH)

• Can prices be predicted?
• Do security prices reflect information ?
• Any information that can be used to predict stock
  performance must already be reflected in stock
  prices
• The notion that stocks prices fully reflect all
  available information is called Efficient Market
  Hypothesis (EMH)

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Illustration of Efficient Market Hypothesis

• Suppose there is rule for predicting stock
  prices. The forecast of favourable future prices
  would increase current price as everybody would
  tart buying this stock
• Example suppose that current stock price is
  100, the stock is expected to pay dividend of 3
  next year and have price 107, and according to
  CAPM, this stock return should be 10. If you had
  a model that could predict that future price to
  be 110, what would happen to todays price?

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Market Efficiency and Competition

• As long as analysis of new information generates
  higher return, investors will spend time and
  money to research and uncover new information
• Competition among analysts will make it hard to
  predict future prices
• Stock prices fully and accurately reflect
  publicly available information
• Once information becomes available, market
  participants analyze it
• Competition assures prices reflect information
• Why look at market efficiency
• Implications for business and corporate finance
• Implications for investment

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Random Walk and the EMH

• Random Walk - stock price change is
  unpredictable E(Pt1)E(Pt)et
• Actually stock prices follow a sub-martingale
• Expected price is positive over time
• Positive trend and random around the trend
• Why are price changes random?
• Prices react to information
• Flow of information is random
• Therefore, price changes are random

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Random Walk with Positive Trend
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Forms of the EMH

• Definition of EMH poses two questions
• What is all available information?
• What fully reflect all available information
  means
• Depending on the answers to these questions we
  will have different versions of market
  efficiency.
• Forms of efficiency hypothesis differ by the
  definition of all available information
• Weak Semi-strong Strong
• The phrase prices fully reflect all available
  information means that at market prices
  investments in assets have NPV0.

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Forms of the EMH (2)

• Weak all past market trading information
• Past prices, trading volume, interest rates
• Testing patterns of stock returns (correlation
  etc.)
• Weak form of EMH is supported by the data
• Semi-strong all publicly available information
  regarding the prospects of a firm
• Firm fundamental data, financial statements
• Testing event studies
• Semi-strong form of EMH is supported by the data
• Strong all information relevant to the firm
• All info, including insider info
• Testing
• Strong-form of EMH has mixed evidence

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Types of Stock Analysis

• Technical Analysis - using prices and volume
  information to predict future prices (Ch.16)
• Weak form efficiency technical analysis
• Technical trading rules are not consistently
  profitable (SP 500 index behaviour is similar
  to coin flipping)
• Fundamental Analysis - using economic and
  accounting information to predict stock prices
  (Ch.14-15)
• Semi strong form efficiency fundamental
  analysis

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Implications of EMH

• Trust market prices
• Buying and selling assets are zero NPV
  activities, thus only risk implied return is
  provided
• Market prices give best estimate of securities
  values
• Firms receive fair value for their securities
  when issued
• Information can be extracted from knowing prices
• There are no financial illusions
• Market price reflects value only from an assets
  payoff
• It is not easy to trick the market

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Market Efficiency and Portfolio Management

• Active Management
• Security analysis Timing -gt Not justified
  according to EMH
• Passive Management
• Buy and Hold Index Funds
• Even if the market is efficient a role exists for
  portfolio management
• Diversification (even if markets are efficient,
  diversification is still required)
• Appropriate risk level (different investors will
  require different levels of risk depending on
  investors age
• Tax considerations (some investors prefer capital
  gains to income because of their tax rate)

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Tests of Market Efficiency

• 1) Examine prices and returns over time

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Tests of Market Efficiency (2)

• 2) Returns are adjusted to determine if they are
  abnormal
• Market Model approach
• a. Rt a bRmt et
• (Expected Return)
• b. Excess Return (Actual - Expected)
• et Actual - (at btRmt)

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Tests of Market Efficiency (3)

• 3) Market Model approach
• Cumulate the excess returns over time

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Issues in Examining the Results

• Magnitude Issue
• Difficult to measure abnormal return of say,
  0.01, because of market volatility
• Selection Bias Issue
• Only schemes that couldnt bring abnormal returns
  are reported
• Lucky Event Issue
• Sometimes its just coincidence
• Nice example create 8 newspapers to report
  market direction for 3 years. One paper will
  perfectly predict it by construction!
• Model for adjusting stock returns may be
  misspecified

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What Does the Evidence Show?

• Returns over short horizons
• Very short time horizons small magnitude of
  positive trends
• 3-12 month some evidence of positive momentum
• Filter rules
• Returns over long horizons pronounced negative
  correlation
• Evidence on Reversals
• Winners become losers and vice versa
• Anomalies Exist

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Anomalies (tests of semistrong EMH)

• Small Firm Effect (January Effect)
• Returns for small firms are abnormally hogh in
  January
• Neglected Firm
• Returns on firms with less research are higher
• Book to Market Ratios
• High book-to-market stocks have higher returns
• Post-Earnings Announcement Drift
• Abnormal returns even after announcement

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Explanations of Anomalies

• Risk Premiums or market inefficiencies ?
• Anomalies or data mining ?
• Behavioural explanations
• These are assertions that investors exhibit
  systematic departures from rationality when
  making complicated decisions
• Such departures from rationality cannot be
  exploited because there are limits to arbitrage

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Behavioural Explanations

• Information processing
• Forecasting errors (overweight recent events)
• Overconfidence (90 of drivers in Sweden report
  to be better-than-average)
• Conservatism (update beliefs too slowly)
• Sample size neglect (sample size is not taken
  into account)
• Behavioural biases
• Framing (decision will depend on how choices are
  framed)
• Mental accounting (different mental accounts
  and attitudes toward them)
• Regret avoidance (regret more when decision is
  unconventional)

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Mutual Fund and Professional Managers Performance

• Some evidence of persistent positive and negative
  performance
• Potential measurement error for benchmark returns
• Style changes
• Superstar phenomenon
• Some managers made fortunes with superior
  performance