Introduction Asset pricing Implementation

Lecture 7: Arbitrage Pricing andMulti-factor modelsSAPM [Econ F412/FIN F313]

Ramana Sonti

BITS Pilani, Hyderabad Campus

Term II, 2014-15

1/15 APT and multifactor models Ramana Sonti

Introduction Asset pricing Implementation

Agenda

1 IntroductionThe APTMulti-factor models

2 Asset pricingAPT pricing equation

3 ImplementationFactors in the real world

2/15 APT and multifactor models Ramana Sonti

Introduction Asset pricing Implementation

The APT

Introduction to the APT

Multi-factor models proposed as alternatives to the CAPM Allow for multiple risk factors as opposed to the (only) CAPM

market factor

Arbitrage pricing theory A theoretical multi-factor model Asset values determined by the principle of the law of one price, andno arbitrage

Does not assume everyone is optimizing as the CAPM does Only requires that some market participants can arbitrage away any

mispricing, in addition to the usual perfect market assumptions

No arbitrage: No security exists that has a negative price and anon-negative payoff. Implies that: Two securities that have the same payoffs must have the same price No security exists that has a zero price and a strictly positive payoff

in all states

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Introduction Asset pricing Implementation

Multi-factor models

Multi-factor models

Recall that a multi-factor model is written asri = E (ri ) + i,1F1 + + i,KFK + ei Fj represent unanticipated shocks to the the jth factor Any return over and above the expected return on a given security

could be from one of two sources the impact of unanticipated macro events through each of the

factors, in proportion to the securitys factor sensitivities, i,jFj unanticipated idiosyncratic or company specific events, ei

Note that E(Fj) = 0 for all factors j , andE(ei ) = 0 for all securities i

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Introduction Asset pricing Implementation

Multi-factor models

Multi-factor model example

For example, we might write for Infosys stocks next quarter:rI = E (rI ) + I ,1 [GDPG E (GDPG )] + I ,2 [INF E (INF )] + eI Here we have assumed that GDP growth, GDPG , and inflation, INF

are the two macro risk factors affecting all stocks GDPG is expected to be 4%, while INF is expected to be 6% Say Infosys has a sensitivity of 1.0 w.r.t. GDPG and -0.4 w.r.t. INF ,

and an expected return of 6% GDPG actually turns out to be 5%, while INF turns out to be 7% Therefore, the factor model for Infosys stock can be written as

rI = 0.06 + 1.0(5% 4%) 0.4(7% 6%) + eI

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Introduction Asset pricing Implementation

Multi-factor models

Multi-factor models: Variance decomposition Note that factor models are merely convenient ways of representing

asset returns as a systematic part depending on some risk factorsand an unsystematic part

Factor models provide for a useful decomposition of variance For instance, for a two factor model ri = E (ri ) + i,1F1 + i,2F2 + ei ,

we can write

2i = Var (i,1F1 + i,2F2 + ei )

= 2i,121 +

2i,2

22 + 2i,1i,21,2 +

2e,i

More generally, we can write with K factors,

2i =Kj=1

Kk=1

i,ji,kj,k + 2e,i

This looks just like (multiple) regression math from statistics:SST = SSE + SSR

For a well-diversified portfolio, idiosyncratic variance is almost zero6/15 APT and multifactor models Ramana Sonti

Introduction Asset pricing Implementation

Multi-factor models

Risk decomposition: Stock

INVA - Term 4 - 2007 - Indian School of Business

Multi-factor model: Practical application Roll and Ross asset management use following factors

Long term inflation Short term inflation Investor confidence Business cycle

Have researched factors thoroughly and present the following variance decomposition

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