INVESTMENTS Factor model of asset returns Suppose that asset returns are driven by a few common factors and diversifiable noise: r=E(r)+6,.+ +u Whe Er is the expected return on asset i f f,f are news on common factors driving all asset returns. f=F-E(F) d b gives how sensitive the return on asset i with respect to news on the k-th factor--is called the loading of asset i on factor fi ui is the idiosyncratic component in asset i's return that is unrelated to other asset returns f, Ji, x, u, have zero means
INVESTMENTS Factor model of asset returns Factor model of asset returns Suppose that asset returns are driven by a few common factors and diversifiable noise: Where is the expected return on asset i; are news on common factors driving all asset returns. gives how sensitive the return on asset i with respect to news on the k-th factor—is called the loading of asset i on factor is the idiosyncratic component in asset i’s return that is unrelated to other asset returns have zero means. i i i iK K ui r = E r + b f + b f + ~ ... ~ ( ) 1 1 Eri K f f f ~ ,..., ~ , ~ 1 2 ( ) ~ k Fk E Fk f = − bik k f ~ u i ~ K ui f f f , ~ ,..., ~ , ~ 1 2
INVESTMENTS exanple Common factors driving asset returns may include gnp interest rates, inflation, etc. Let fit be the news on interest rate. Before a board meeting of the fed, the market expect the Fed not to change the interest rate. After the meeting, Greenspan announces that a There is no change in interest rate---no news"' fint=0 a There is a 4% increase in interest rate--positive surprise 0.25% What should be the sign of the factor loadings on f be for fixed income securities, stocks, commodity futures?
INVESTMENTS example example Common factors driving asset returns may include GNP, interest rates, inflation, etc. Let be the news on interest rate. Before a board meeting of the Fed, the market expect the Fed not to change the interest rate. After the meeting, Greenspan announces that: There is no change in interest rate---”no news” There is a ¼% increase in interest rate—positive surprise What should be the sign of the factor loadings on , be for fixed income securities, stocks, commodity futures? int ~ f 0 ~ fint = 0.25 % ~ fint = int ~ f
INVESTMENTS Properties of factor model The following results provide the building blocks of apt 1. Any diversified portfolio p is exposed only to factor risks E(n)+bn+…b KJK
INVESTMENTS Properties of factor models Properties of factor models The following results provide the building blocks of APT. 1. Any diversified portfolio p is exposed only to factor risks p p p pK K r E r b f b f ~ ... ~ ( ) = + 1 1 +
INVESTMENTS corrIne u a diversified portfolio, that is not exposed to any factor b=b box =0 ) must offer risk-free rat There al ways exists portfolios that are exposed only to the risk of a single factor. pk=Ipx+bpe k Example: suppose two well diversified portfolios, both exposed only to the risk of the first two factors f, f 1=0.2+f1+0.52,2=0.3+2f+15
INVESTMENTS continued continued A diversified portfolio, that is not exposed to any factor risk( ), must offer risk-free rate; There always exists portfolios that are exposed only to the risk of a single factor. Example: suppose two well diversified portfolios, both exposed only to the risk of the first two factors 2 ... 0 1 b p = b p = = bpK = pk pk pk k r r b f ~ = + 1 2 ~ , ~ f f , ~ 0.5 ~ 1 0.2 1 2 r = + f + f , ~ 1.5 ~ 2 0.3 2 1 2 r = + f + f