Panos Parpas

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Asset Pricing Models

381 Computational Finance

• Panos Parpas

Imperial College London
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Problem Types in Investment Science

• Determining
• correct, arbitrage free price of an asset
• price of a bond, a stock
• the best action in an investment situation
• how to find the best portfolio
• how to devise the optimal strategy for managing
  an investment
• Single period Markowitz model

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Topics Covered

• The Capital Asset Pricing Model (CAPM)
• Single and Multi Factor Models
• CAPM as a Factor Model
• The Arbitrage Pricing Theory (APT)

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M-V model

• investor chooses portfolios on the efficient
  frontier deciding if given portfolio is on
  efficient frontier or not
• no guarantee that a portfolio that was efficient
  ex ante will be efficient ex post
• statistical considerations regarding time period
  over which to estimate which assets to include
  are non-trivial
• not mention implications of m-v optimisation on
  asset pricing
• CAPM describes MV portfolios and provides asset
  pricing

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CAPM Capital Asset Pricing Model

• developed by Sharpe, Lintner and Mossin
• single period asset pricing model
• determines correct price of a risky asset within
  the mean-variance framework
• highlights the difference between systematic
  specific risk

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Assumptions

• All investors
• are mean variance optimisers portfolios on
  efficient frontier
• plan their investments over a single period of
  time
• use the same probability distribution of asset
  returns the same mean, variance, covariance of
  asset returns
• borrow and lend at the risk free rate
• are price-takers investors purchases sales
  do NOT influence price of an asset
• There is no transaction costs and taxes

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Market Portfolio

• Everyone purchases single fund of risky asset,
  borrows (lends)
• at risk-free rate.
• Form a portfolio that is a mix of risk free
  asset and single risky fund
• Mix of the risky asset with risk free asset will
  vary across individuals
• according to their individual tastes for risk
• Seek to avoid risk have high percentage of the
  risk free asset in their portfolio
• More aggressive to risk have a high percentage
  of the risky asset
• What is the fund that everyone purchases?
• This fund is Market Portfolio and defined as
  summation of all assets total invested wealth
  on risky assets
• An asset weight in market portfolio is the
  proportion of that assets total capital value to
  total market capital value capitalization
  weights

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The Capital Market Line (CML)

• Consider single efficient fund of risky assets
  (market portfolio) and a risk free asset (a bond
  matures at the end of investment horizon)
• If a risk free asset does not exist, investor
  would take positions at various points on the
  efficient frontier. Otherwise, efficient set
  consists of straight line called CML.
• Pricing Line prices are adjusted so that
  efficient assets fall on this line
• CML describes all possible mean-variance
  efficient portfolios that are a combination of
  the risk free asset and market portfolio

• Investors take positions on CML by
• buying risk free asset (between M and rf) or
• selling risk free asset (beyond point M) and
• holding the same portfolio of risky assets

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The Capital Market Line

• Equation describes all portfolios on CML
• CML relates the expected rate of return of an
  efficient portfolio to its standard deviation
• The slope the CML is called the price of RISK!
• How much expected rate of return of a portfolio
  must increase if the risk of the portfolio
  increases by one unit?

Expected Value of market rate of return
Standard Deviation of market rate of return
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The Pricing Model

• How does the expected rate of return of an
  individual asset relate to its individual risk?
• If the market portfolio M is efficient, then the
  expected return of an asset i satisfies
• The beta of an asset (risk premium)

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The Pricing Model

• expected excess rate of return of an asset is
  proportional to the expected excess rate of
  return of the market portfolio proportional
  factor is the beta of asset.
• Amount that rate of return is expected to exceed
  risk free rate is proportional the amount that
  market portfolio return is expected to exceed
  risk free rate

describes relationship between risk and expected
return of asset
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Beta of an Asset

• beta of an asset measures the risk of the asset
  with respect to the market portfolio M.
• high beta assets earn higher average return in
  equilibrium because of
• beta of market portfolio average risk of all
  assets

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The Beta of Portfolio

• If the betas of the individual assets are known,
  then the beta of the portfolio is
• This can be shown by using
• rate of return of the portfolio
• covariance

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Systematic and Specific Risk

• CAPM divides total risk of holding risky assets
  into two parts
• systematic (risk of holding the market portfolio)
  and specific risk
• Consider the random rate of return of an asset
  i
• Take expected value and the correlation of the
  rate of return with rM
• The total risk of holding risky asset i is

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Summary CAPM

• The capital market line expected rate of return
  of an efficient portfolio to its standard
  deviation
• The pricing model expected rate of return of an
  individual asset to its risk
• The risk of holding an asset i is

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Beta of the Market

• Average risk of all assets is 1 (beta of the
  market portfolio)
• Beta of market portfolio is used as a reference
  point to measure risk of other assets.
• Assets or portfolios with betas greater than 1
  are above average risk tend to move more than
  market. Example
• If risk free rate is 5 per year and market rises
  by 10 , then assets with a beta of 2 will tend
  to increase by 15.
• If market falls by 10, then assets with a beta
  of 2 will tend to fall by 25 on average.
• Assets or portfolios with betas less than 1 are
  of below average risk tend to move less than
  market.

Capital Market Line
Security Market Line
M
M
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CAPM as a Pricing Formula

• CAPM is a pricing model.
• standard CAPM formula only holds expected rates
  of return
• suppose an asset is purchased at price P and
  later sold at price S.
• rate of return is substituted in CAPM formula

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Discounting Formula in CAPM
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Single-Factor Model

• Consider n assets with rates of return ri for
  i1,2,,n and one factor f which is a random
  quantity such as inflation, interest rate
• Assume that the rates of return and single
  factor are linearly related.
• Errors
• have zero mean
• are uncorrelated with the factor
• are uncorrelated with each other

Factor Loadings
Intercept
Error
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Multi-Factor Model

• Single factor model is extended to have more
  than one factor.
• For two factors f1 and f2 the model can be
  written as
• For k number factors

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How to Select Factors?

• Factors are external to securities
• consumer price index, unemployment rate
• Factors are extracted from known information
  about security returns
• the rate of return on the market portfolio
• Firm characteristics
• price earning ratio, dividend payout ratio
• How to select factors It is part science and
  part art!
• Statistical approach principal component
  analysis
• Economical approach its beta, inflation rate,
  interest rate, industrial production etc.

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The CAPM as a Factor Model

• Special case of a single-factor model f rM

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The CAPM as a Factor Model Example

• Single factor model equation defines a linear fit
  to data
• Imagine several independent observations of
  the rate of return and factor
• Straight line defined by single factor model
  equation is fitted through these points
  such that average value of errors is zero.
• Error is measured by the vertical distance
  from a point to the line

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Arbitrage The law of one price

• Arbitrage relies on a fundamental principle of
  finance the law of one price
• says security must have the same price
  regardless of the means of creating that
  security.
• implies if the payoff of a security can be
  synthetically created by a package of other
  securities, the price of the package and the
  price of the security whose payoff replicates
  must be equal.

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Arbitrage Example

• How can you produce an arbitrage opportunity
  involving securities A, B,C?
• Replicating Portfolio
• combine securities A and B in such a way that
• replicate the payoffs of security C in either
  state
• Let wA and wB be proportions of security A and B
  in portfolio

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Example Continued

• Payoff of the portfolio
• Create a portfolio consisting of A and B that
  will reproduce the payoff of C regardless of the
  state that occurs one year from now.
• Solving equation system, weights are found wB
  0.6 and wA 0.4
• An arbitrage opportunity will exist if the cost
  of this portfolio is different than the cost of
  security C.
• Cost of the portfolio is 0.4 x 70 0.6 x 60
  64 - price of security C is 80. The synthetic
  security is cheap relative to security C.

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Example Continued

• Riskless arbitrage profit is obtained by buying
  A and B in these proportions and shorting
  security C.
• Suppose you have 1m capital to construct this
  arbitrage portfolio.
• Investing 400k in A 400k ? 70
  5714 shares
• Investing 600k in B 600k ? 60
  10,000 shares
• Shorting 1m in C 1m ? 80
  12,500 shares

The outcome of forming an arbitrage portfolio of
1m
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The Arbitrage Pricing Theory

• CAPM is criticised for two assumptions
• The investors are mean-variance optimizers
• The model is single-period
• Stephen Ross developed an alternative model
  based purely on arbitrage arguments
• Published Paper
• The Arbitrage Pricing Theory of Capital Asset
  Pricing, Journal of Economic Theory, Dec 1976.

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APT versus CAPM

• APT is a more general approach to asset pricing
  than CAPM.
• CAPM considers variances and covariance's as
  possible measures of risk while APT allows for a
  number of risk factors.
• APT postulates that a securitys expected return
  is influenced by a variety of factors, as opposed
  to just the single market index of CAPM
• APT in contrast states that return on a security
  is linearly related to factors.
• APT does not specify what factors are, but
  assumes that the relationship between security
  returns and factors is linear.

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Simple Version of APT

• Consider a single factor model.
• Assume that the model holds exactly no error
• The uncertainty comes from the factor f
• APT says that ai and bi are related if there
• is no arbitrage

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Derivation of APT

• Choose another asset j such that
• Form a portfolio from asset i and j with weights
  of w and (1-w)
• Choose w so that the coefficient of factor is
  zero so

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Derivation of APT
ai and bi are not independent
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Arbitrage Pricing Formula

• Once constants are known, the expected rate of
  return of an asset i is determined by the factor
  loading.
• The expected rate of return of asset i
• 
  
• 
  
• 
  CAPM?
  

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CAPM as a consequence of APT

• The factor is the rate of return on the market
• APT is identical to the CAPM with