INTRODUCTION TO FINANCIAL ENGINEERING

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CHAPTER FOUR Index Models and APT
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Problems of Markowitz Portfolio Selection

There are some problems for Markowitz portfolio
selection

• Huge number of estimates of covariance between
  all pairs of available securities
• Vast computing capacity required to resolve an
  optimization quadratic programming for large
  portfolio
• CAPM is a single, static factor model

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Single-Index Models

• A Mini Case

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• Regression Model

Firms or unsystematic factor
Exogenous
Macro or systematic factor
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• Covariance

Unsystematic risk
Systematic risk
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Market Model
The market is at equilibrium
CAPM
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• Can you beat the market?

CML

• The hyperbola through A and M cannot be tangent
  to the efficient frontier
• The point A cannot be located on the efficient
  frontier

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Multi-Index Models
Growth of GDP
Inflation

• The Mini Case

Firms or unsystematic factor
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• Covariance

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More About Arbitrage
A riskless arbitrage opportunity exists if and
only if either

• Two portfolios can be created that have identical
  payoffs in every state but have different costs
  or
• Two portfolios can be created with equal costs,
  but where the first portfolio has at least the
  same payoff as the second in all states, but has
  a higher payoff in at least one state or
• A portfolio can be created with zero cost, but
  which has a non-negative payoff in all states and
  a positive payoff in at least one state.

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• A Mini Case

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• Comparing an equally weighted portfolio of the
  stocks A, B and C with the stock D

The Portfolio
D
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• Expected return and standard deviation and
  correlation between the portfolio and the stock D

The Portfolio
0.94
D
Is there a reskless arbitrage opportunity
?
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• Making arbitrage positions

Investing in A
Investing in B
Investing in C
Short sell D
Net position
0
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Arbitrage Pricing Theory (APT)
Single-Factor APT
Sensitivity of the security is return to the
unexpected change of the macro-economy factor
Pure unsystematic risk
Macro-economy factor the deviation from the
expectation
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• Well-diversified portfolios and the APT

Variance of macro-economy factor
0
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• Well-diversified portfolios and the APT (Cont.)

Two diversified portfolio A and B,

• A Mini Case

Short selling 1 million portfolio B Investing
the amount in portfolio A.
Arbitrage
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Proposition!
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Security Market Line of APT
!
There is an arbitrage opportunity between
portfolios D and C
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• APT for individual securities

It holds almost for all individual securities i
and j
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Multi-Factor APT
Macro-economy factors are the deviations from
their expectations
Diversified portfolios with the following
characteristics

• Factor portfolios

Factor portfolio 1
Factor portfolio 2
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Replicating portfolio Q weight

• Factor portfolios (cont.)

For factor portfolio 1
For factor portfolio 2
Risk-free security
For a diversified portfolio P
For the replicating portfolio Q
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• The replicating portfolio Q is the arbitrage
  portfolio of the diversified portfolio P

Expected return of P
Expected return of Q
Arbitrage opportunity
Long position of Q
Short position of P
Net profit
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• Proposition
• The risk premium for a diversified portfolio
  is the sum of the contributions from all the
  macro-economy factors

Example
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Multi-Factor APT Models
For a portfolio P
For a security i
The extension of Security Market Line
!
It holds almost for all securities in the markets
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Difference Between APT and CAPM
Risk free arbitrage vs. risk/return dominants
Support of equilibrium price relationship
APT
CAPM
When equilibrium is violated
Stronger

• many investors make portfolio changes
• each portfolios change is limited
• the aggregation creates a large volume of buying
  and selling to restore equilibrium

• many investors make portfolio changes
• each portfolios change is limited
• the aggregation creates a large volume of buying
  and selling to restore equilibrium

• implying there exists arbitrage opportunity
• each arbitrageur wants to take as large position
  as possible
• a few arbitrageurs bring the price pressures to
  restore equilibrium

• implying arbitrage opportunity exists
• each arbitrageur wants to take as large position
  as possible
• a few arbitrageurs bring the price pressures to
  restore equilibrium

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Summary of Chapter Four

• Index Models ? Strict Separation of Systematic
  and Unsystematic Risks
• CAPM ? A Special Case of Single-Index Model.
  Whats the Difference?
• How to Beat the Markets?
• The Key of APT Factor Portfolios
• No Arbitrage Equilibrium vs. Risk/Return
  Dominance Arguments ? APT vs. CAPM

APT