\,ll'ltl'-!;ll'!{)¡. ¡'X'l'"ll';:;')¡""!':: nJ"tl~I;' {'\ 1)\,1~v... t .s. ,,-t . L ~ \,... ~ "-'. ._., { '\ 1..)'" , j '- '\.. < ,_ • t i.

Stephen M SchaeferLondon Business School

• More general versions ofthe CAPM predict risk premiathat have \:.'-\<Itt~ the same form as the APT:

'"-. __.,------_._ .._-- ..__ ._-_._----------------~,-~-----

where the fJ's are the sensitivities of returns on asset i tofactors F" F2• FJ etc. and the A's are the factor risk premia-~-------

• Where do these models come from?

" I

hA-D price

w ere: m = --~-s prob(state s)

The challenge is to find the right measure of "bad times," [Le ..) rises in [thestochastic discount factor), so that we can understand high average returnsor low prices as compensation for assets' tendency to pay off poorlyin "bad times," (John Cochrane, Financial Markels and lhe Real Economy)

• Ifwe understand this then we can understand risk premia

(cost of capital)

~-------- -;• f(ey isstl/!: what determines the the sdfs:

• Startil1f.: poiflt is the the states model: relation between riskpremia and the stochastic discount factors (sdj's), ms:

Multi-factor cxtcnsions ofthc CAPM

Multi-factor extcnsions ofthe CAPM

III,'/" ir< I/,//\WI/C¡'\ Ji/,II dri,',' ;1/,'(li1lt' ./oltl/: ".g., badjob market (recession)

,¡' or changes in oPPoJ'tllllities/úr investnlent that makepeople want to save more

• Each ofthese willlead people to consume less now

• The stoc/tastic discountfactor (sdj) is a measure ofthe ,'careitr of available funds and will be relatively~;"i\ in states when:,¡' people have 10'1' 'I¡'{llllt: Le., value of market portfolio

is low

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I________________ J

r.

• In CAPM the value of the market portfolio represents totalwealth in the economy

• Investors care only about wealth (because there is only oneperiod and so they don't worry about investing for laterperiods)

• The s4{therefore depends onlv on wealth (the value ofthemarket portfolio)

• Qne "route" to CAPM:linear relation betweensdfand return on marketportfolio

m =a-bRM

b>Os s

Multi-factor cxtensions orlhe CAPM

• Basics: consumption affected by both H\',IÍfíl and 1I01'-'l','.;ltll

intluences• if 11n",I<

../ '" caJth is lo\\', d<./ othcr factors make investol's save more (311d consume

less)

then consumption will be low and sd{will be high (& vice versa)• The \,It should at each point retlect a level of consumption that

retlects the best trade-off between these two effects,!llili\';I!I"II: consumption should be a good proxy for sdf• consumption low => sdfhigh• consumption high => s4{low

\ilulti-factor extclIsions oflhe CAPM (,

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• As with the conventional CAPM, one route to theconsumption CAPM is via a linear relation between the sdfand (aggregate) consumptionFollowing the samesteps as before we nd:

Aggregate Consumption

',11/ '. 1I/l/¡i{/1I1l

1\.1ulti.f~ctor cxtensions ofthe CAPM

) lil,~/C F[[c{(Jr 1.. \ )l1';U 111pl I1 H: \ lmil:! i:.; ( ',11h i<t;:n1\\ ilh ,HlJ¡íi~t{¡cf()r l'();;t (,)rC~q)il~d :-'1ludcl

,~ '.!.l_~.!.!!.ili!..!.0_~.r::l:,¡' explains risk premia in terms of an asset's beta on (me júClIIr

(C(I I/\'II I11prio 11)

./ just as rs!!\l.tr Lv..:.':J. explains risk premia in terms ofan asset'sbeta on olle./ác/or (the I//arke/)

Level of consumption should be a good measure of"bad times" har:./ When we implement this we have to assume sorne particular

relationship between the sdf and the level of consumption ..¿ Lmll1ple: This may not be linear and using other variables to

explain sensitivity ofthe sdfto consumption leads to a multi-factor

model.Implies that, although consumption CAPM is a single factor model, it ispotentially consistent with a 1l12u,ti-htl<l hl'llllli:lIl"'1

~vt\llti-factor cxtensiol1s orthe CAPM

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