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Multi-Factor asset pricing

And more on the homework

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Review item

Define beta.

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Answer

2)(

),(

M

jM

M

Mj

jRVar

RRCov

==

Rate of return on asset j is

Rate of return on the market

portfolio is

jR

MR

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My project A!"

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Regression y # a $ b% $ e

where a and b are constants

y is to be e%plained

% is an e%planatory variable e is a random error term

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For &eta

Rj# a $ bRM$ e

e # idiosyncratic risk 'diversifiable risk(

b # beta a # alpha # sample average advantage

over the market

if statistically significant

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)omponents of risk

Diversifiable risk is uni*ue+

idiosyncratic+ or unsystematic risk

Market risk is systematic or portfoliorisk

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Diversifiable risk

,t is eliminated by buying other assets+

i.e.+

can be diversified away.

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Arbitrage pricing theory

ide-issue Arbitrage is interesting in options+

bonds+ )A/M+ and this course.

0otion 1here are several factors 'inde%es(.

1hey are found by regression analysis.

More notion 2ach factor has its own beta.

Risk unrelated to the factors can be

diversified away+ leaving only systematic risk.

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1he 3-Factor Model

urprise in factors F4+ F5+ 6 +Fk

Ri# 2'Ri( $ i4F4$ i5F5$ 6 $ i3Fk$ i

1he une%pected systematic return is e%plained by

surprise in 7factors.8

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Arbitrage pricing theory is like

)A/M+ 6

Factor risk 'previously market risk(remains even when the portfolio is fullydiversified.

Factor risk is undiversifiable. For any asset+ the betas of factors

measure factor risk.

Re*uired return is linear in the factorbetas.

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1he market rewards the

investor not for bearing diversifiable risk but only for bearing factor 'or market( risk.

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1he market rewards the

investor not for all the risk ' ( of an asset but only for its betas.

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Do low /92 firms contradict

)A/M:

/rice at t # 2arnings at t$49r-g

/rice92arnings # '4$g(9r-g

"ow growth and or high risk imply low/92

;igh risk implies high e%pected return.

1herefore low /92 means+ on average+high return. Doesn

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;ow many assets in a

diversified portfolio: 0ot many. About => well-chosen ones.

tatman ?F@A ept B

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Diversification for an 2*ually

Ceighted /ortfolio

Number of

Securities

Systematicrisk

Total risk2P

Unsystematicrisk

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,nvestors need only two funds.

Figures 4>.+ 4>.E+ and 4>..

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Diversification+ minimum

variance

E(R)

0= 1=

A

&

1=

1= MM

M

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Diversification with a risk-free

asset

E(R)

A#

risk-freeasset

&

M

Diversification+ minimum

variance

E(R)

0= 1=

A

&

1=

1= MM

M

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)apital Market "ine

2%pected returnof portfolio

tandarddeviation of

portfolio

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Argument for the security

market line !nly beta matters A mi% of 1-&ills and the market can

produce any beta. An asset with that beta is no better or

worse than the two-fund counterpart

;ence it has the same return.

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Security Market &ine

E'#ecte returnon security ()

*eta ofsecurity

Rm

Rf

1

M

!

0!+

S!

Security marketline (SM&)

S is o,er,alue!

-ts #rice falls

T is uner,alue!-ts #rice rises

!T!

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Review item

Asset A has a beta of ..

Asset & has a beta of 4.E.

)onsider a portfolio with weights . onasset A and . on asset &.

Chat is the beta of the portfolio:

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Answer

/ortfolio beta is .G.$.G4.E # 4.55.

Cork it out this way

Dev/ # . DevA $ . Dev & 2HDev/GDevMI # . 2HDevAGDevMI $ .

G2HDev&GDevMI.

Divide by 2HDev/ s*uaredI.

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