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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.
answer
Answer

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

P

Unsystematicrisk

Systematicrisk

Number of Securities

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

.

Y

.

X

Capital Market Line

Expected returnof portfolio

Indifference curve

Capital market line

.

preferred

.

M

.

.

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

.

T

.

S

0.8

Security Market Line

Expected returnon security (%)

T is undervalued.

Its price rises

Security market line (SML)

.

.

M

Rm

Rf

S is overvalued.

Its price falls

Beta ofsecurity

1

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?
answer1
Answer
  • 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].