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Multi-Factor asset pricing. And more on the homework. Review item. Define beta. Answer. Rate of return on asset j is Rate of return on the market portfolio is . My project: AOL. Regression: y = a + bx + e. where a and b are constants y is to be explained x is an explanatory variable

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multi factor asset pricing

Multi-Factor asset pricing

And more on the homework

review item
Review item
  • Define beta.

Rate of return on asset j is

Rate of return on the market

portfolio is

regression y a bx e
Regression: y = a + bx + e
  • where a and b are constants
  • y is to be explained
  • x is an explanatory variable
  • e is a random error term
for beta
For Beta
  • Rj = a + bRM + e
  • e = idiosyncratic risk (diversifiable risk)
  • b = beta
  • a = alpha = sample average advantage over the market
    • if statistically significant
components of risk
Components of risk
  • Diversifiable risk is unique, idiosyncratic, or unsystematic risk
  • Market risk is systematic or portfolio risk
diversifiable risk
Diversifiable risk
  • It is eliminated by buying other assets, i.e.,
  • can be "diversified away."
arbitrage pricing theory
Arbitrage pricing theory
  • Side-issue: Arbitrage is interesting in options, bonds, CAPM, and this course.
  • Notion: There are several factors (indexes).
  • They are found by regression analysis.
  • More notion: Each factor has its own beta.
  • Risk unrelated to the factors can be diversified away, leaving only systematic risk.
the k factor model
The K-Factor Model

The unexpected systematic return is explained by

surprise in “factors.”

Surprise in factors: F1, F2, … ,Fk

Ri = E(Ri) + bi1F1 + bi2F2 + … + biKFk + ei

arbitrage pricing theory is like capm
Arbitrage pricing theory is like CAPM, …
  • Factor risk (previously market risk) remains even when the portfolio is fully diversified.
  • Factor risk is undiversifiable.
  • For any asset, the betas of factors measure factor risk.
  • Required return is linear in the factor betas.
the market rewards the investor
The market rewards the investor
  • not for bearing diversifiable risk but
  • only for bearing factor (or market) risk.
the market rewards the investor1
The market rewards the investor
  • not for all the risk ( s ) of an asset
  • but only for its betas.
do low p e firms contradict capm
Do low P/E firms contradict CAPM?
  • Price at t = Earnings at t+1/r-g
  • Price/Earnings = (1+g)/r-g
  • Low growth and or high risk imply low P/E
  • High risk implies high expected return.
  • Therefore low P/E means, on average, high return. Doesn’t contradict CAPM.
how many assets in a diversified portfolio
How many assets in a diversified portfolio?
  • Not many.
  • About 30 well-chosen ones.
    • Statman JFQA Sept 87
diversification for an equally weighted portfolio
Diversification for an Equally Weighted Portfolio

Total risk s2




Number of Securities

investors need only two funds
Investors need only two funds.
  • Figures 10.4, 10.5, and 10.6.
capital market line





Capital Market Line

Expected returnof portfolio

Indifference curve

Capital market line







Risk-freerate (Rf )

Standarddeviation ofportfolio’s return.

argument for the security market line
Argument for the security market line
  • Only beta matters
  • A mix of T-Bills and the market can produce any beta.
  • An asset with that beta is no better or worse than the two-fund counterpart
  • Hence it has the same return.
security market line






Security Market Line

Expected returnon security (%)

T is undervalued.

Its price rises

Security market line (SML)






S is overvalued.

Its price falls

Beta ofsecurity


review item1
Review item
  • Asset A has a beta of .8.
  • Asset B has a beta of 1.5.
  • Consider a portfolio with weights .4 on asset A and .6 on asset B.
  • What is the beta of the portfolio?
  • Portfolio beta is .4*.8+.6*1.5 = 1.22.
  • Work it out this way:
  • DevP = .4 DevA + .6 Dev B
  • E[DevP*DevM] = .4 E[DevA*DevM] + .6*E[DevB*DevM].
  • Divide by E[DevP squared].