Capital Asset Pricing Model and Arbitrage Theory

1
Capital Asset Pricing Model and Arbitrage Theory

• Riccardo Colacito

2
Capital Asset Pricing Model (CAPM)

• Equilibrium model that underlies all modern
  financial theory
• Derived using principles of diversification with
  simplified assumptions
• Markowitz, Sharpe, Lintner and Mossin are
  researchers credited with its development

3
Link to Factor Models

• What risk should be priced?
• Idiosyncratic risk no
• Aggregate risk yes
• Only aggregate/macro risk commands a premium

4
Why?

• Because
• idiosyncratic risk can be diversified away
• Macro risk affects all assets and cannot be
  diversified
• Remember our example using factor models?

5
Example two assets
6
What is the portfolio variance?
Systematic Risk
Idiosyncratic Risk
7
Example three assets
8
What is the portfolio variance?
Systematic Risk
Idiosyncratic Risk

• Systematic risk unchanged
• Idiosyncratic risk decreased
• Can you guess what would happen if we had an
  infinite number of assets?

9
Infinite assets

• For a well diversified portfolio
• That is we got rid of any idiosyncratic shock
  and we are left only with systematic risk

10
What lesson did we learn?

• The only source of risk that we are entitled to
  ask a compensation for is aggregate risk.
• Idiosyncratic risk does not entitle to any
  compensation because it can be diversified away.

11
Risk compensation
12
The fundamental equation of the Capital Asset
Pricing Model (CAPM)
Any asset entitles to a risk premium that is
proportional to the risk premium of the market
portfolio. The b of the asset is the coefficient
of proportionality.
13
Questions

• What assumptions need to be satisfied for the
  CAPM to hold?
• Why the market portfolio? What is it anyway?
• Why b? What is b anyway?
• What are the empirical predictions of the CAPM?
• Can we measure it from the data?
• If so, is it empirically accepted?

14
Assumptions

• Individual investors are price takers
• Single-period investment horizon
• Investments are limited to traded financial
  assets
• No taxes, and transaction costs
• Lending and borrowing can be done at same rate
• Information is costless and available to all
  investors
• Investors are rational mean-variance optimizers
• Homogeneous expectations

15
Assumptions in other words!

• Everybody knows how to solve the problem that we
  discussed during the last three classes
• Everybody is forecasting returns, variances and
  correlations in the same way
• if this is the case we are all going to end up
  with same optimal risky portfolio!

16
The Efficient Frontier and the Capital Market Line
17
Some re-labeling

• We call the optimal CAL, the Capital Market Line
• We call the optimal risky portfolio, the Market
  Portfolio
• Remember the separation property?

18
Separation property

• Aka mutual fund theorem
• All investors desire the same portfolio of risky
  assets the optimal risky portfolio or market
  portfolio.

19
This answers Why the market portfolio?

• If everybody is holding this portfolio, then it
  is an excellent candidate to proxy for aggregate
  market risk.
• We are now left with the question of why b is the
  coefficient of proportionality!

20
First things first what is b?

• Remember factor models
• Remember how we computed b

21
b in words

• The more correlated the asset is with market the
  larger b is.
• What does it mean in terms of the CAPM?

22
b and the risk premium

• It means that the higher the correlation with the
  market, the higher the risk premium that an asset
  commands.
• Why?

23
b and the risk premium intuition

• Which asset is more appealing
• An asset whose return goes up when the market
  goes up
• An asset whose return goes down when the market
  goes up

24
Answer

• Everything else equal you prefer an asset that
  co-varies negatively with the market, because it
  gives you an insurance against bad states of the
  world.
• If an asset is perceived as a good asset because
  it provides an hedge against bad states, its
  demand will go up, increasing the price and
  decreasing the expected return.
• How about an example?

25
Example

• Two assets, the market portfolio, a risk free
  rate. Two states of the world.

R1 R2 Rm rf
Boom -2 4 4 .5
Recession 4 -2 -2 .5

• Lets compute bs and required risk premia!

26
Results

• Some computations
• Hence

27
What did we learn?

• Asset 2 doesnt do too much to protect us against
  market fluctuations therefore it must promise
  a higher return to convince us to buy it
• Asset 1 can protect us against market
  fluctuations and therefore we are willing to buy
  it even if its return is low in expectation.

28
What happens to the price of the first security?

• The CAPM says that the expected return of asset 1
  should be 0
• However at the current prices, asset 1 has an
  expected return of 1
• Looks like a great deal!
• Investors will want to buy more of asset 1
• But if its current price goes up, its return will
  go down in expectation
• When does the price increase stop?
• When the expected return at the current prices
  equals the CAPM expected return
• What happens to the current price of security 2?
• Nothing! This security is price correctly!

29
Empirical predictions of the CAPM

• Remember the one factor model
• The intercept should be equal to zero!

30
The Security Market Line and Positive Alpha Stock
31
Alpha (a)

• The abnormal rate of return on a security in
  excess of what would be predicted by an
  equilibrium model such as the CAPM
• Can we test this prediction?

32
Estimating the Index Model

• Using historical data on T-bills, SP 500 and
  individual securities
• Regress risk premiums for individual stocks
  against the risk premiums for the SP 500
• Slope is the beta for the individual stock

33
Characteristic Line for GM
34
Statistical analysis

• In this example we cannot reject that a is equal
  to zero
• However in many cases a is significantly
  different from zero what does it mean?

35
Rejecting the CAPM?

• Not necessarily!
• We may have chosen an imprecise proxy for the
  market risk after all the Market Portfolio is
  not directly observable.
• We may be omitting some risk factors research
  shows that this is possible.

36
Fama-French Research

• Returns are related to factors other than market
  returns
• Size
• Book value relative to market value
• Three factor model better describes returns

37
Why are these factors supposed to help?

• Firms with high ratios of book to market value
  are more likely to be in financial distress
• Small firms are more sensitive to changes in
  business conditions

38
Fama-French applied

• A Fama-French three factors regression for GM
• where

39
Regression Statistics for the Single-index and FF
Three-factor Model
40
Resulting Equilibrium Conditions

• All investors will hold the same portfolio for
  risky assets market portfolio
• Market portfolio contains all securities and the
  proportion of each security is its market value
  as a percentage of total market value
• Risk premium on an individual security is a
  function of its covariance with the market

41
Arbitrage Pricing Theory (APT)

• Arbitrage - arises if an investor can construct a
  zero beta investment portfolio with a return
  greater than the risk-free rate
• If two portfolios are mispriced, the investor
  could buy the low-priced portfolio and sell the
  high-priced portfolio
• In efficient markets, profitable arbitrage
  opportunities will quickly disappear

42
APT and well diversified portfolios

• A well diversified portfolio has no exposure to
  idiosyncratic risk
• Claim aP must equal zero. Why?

43
aP0 to avoid arbitrage opportunities

• Take two well-diversified portfolios
• Pick the following shares of investment
• The resulting portfolio is not subject to any
  risk and provides a non-zero return
• Hence as must be equal to zero

44
Conclusion

• The equilibrium condition is the same as the CAPM
• Only assumption needed is the absence of
  arbitrage opportunities
• Can be extended beyond well diversified portfolios