In mathematical finance, multiple factor models are asset pricing models that can be used to estimate the discount rate for the valuation of financial assets; they may in turn be used to manage portfolio risk. They are generally extensions of the single-factor capital asset pricing model (CAPM).
Model of Rosenberg and Marathe The multifactor equity risk model was first developed by Barr Rosenberg and Vinay Marathe. Initially they proposed a linear model of beta
r ( i , t ) − r ( 0 , t ) = b ( i , t ) [ m ( t ) − r ( 0 , t ) ] + g ( i , t ) {\displaystyle r(i,t)-r(0,t)=b(i,t)[m(t)-r(0,t)]+g(i,t)}
b ( i , t ) = ∑ j X ( i , j , t ) f ( j , t ) + e ( i , t ) {\displaystyle b(i,t)=\sum _{j}X(i,j,t)f(j,t)+e(i,t)}
where r ( i , t ) {\displaystyle r(i,t)} is the return to equity asset i {\displaystyle i} in the period [ t , t + 1 ] {\displaystyle [t,t+1]} , r ( 0 , t ) {\displaystyle r(0,t)} is the risk free return, m ( t ) {\displaystyle m(t)} is the market index return, e ( i , t ) {\displaystyle e(i,t)} is a market residual return and b ( i , t ) {\displaystyle b(i,t)} is a parameter fit by a time series regression over history prior to time t. Then X ( i , j , t ) {\displaystyle X(i,j,t)} are risk exposure values calculated from fundamental and technical data, f ( j , t ) {\displaystyle f(j,t)} are factor returns determined by a cross-sectional regression for each time period and g ( i , t ) {\displaystyle g(i,t)} are the regression residuals. This model was reformulated by Rosenberg et al. into a direct model of asset return,
r ( i , t ) = ∑ j X ( i , j , t ) f ( j , t ) + e ( i , t ) {\displaystyle r(i,t)=\sum _{j}X(i,j,t)f(j,t)+e(i,t)}
Here the factor returns f ( j , t ) {\displaystyle f(j,t)} and specific returns e ( i , t ) {\displaystyle e(i,t)} are fit by a weighted regression over each time period t {\displaystyle t} for a representative asset universe. For instance the model might be fit over the 3000 highest capitalization US common stocks. The primary application of the model is to estimate the asset by asset covariance matrix C {\displaystyle C} of asset returns by the equation
C = X F X t + D {\displaystyle C=XFX^{t}+D}
where F {\displaystyle F} is the covariance matrix of factor returns, and D {\displaystyle D} is a block diagonal matrix of specific returns. The matrix C {\displaystyle C} is then used for Markowitz portfolio construction which involves maximizing the quadratic utility function
u ( h ) = a t h − k h t C h {\displaystyle u(h)=a^{t}h-kh^{t}Ch}
subject to linear constraints on the vector of asset holdings h {\displaystyle h} . Here a is a vector of expected returns and k {\displaystyle k} is a scalar parameter termed the risk aversion.
Modifications by Torre Nicolo G. Torre made a number of improvements to this framework which importantly sharpened the risk control achievable by these means. In Rosenberg's model the risk indices X consisted of industry weights and risk indices. Each asset would be given an exposure to one or more industries, e. g. based on breakdowns of the firms balance sheet or earning statement into industry segments. These industry exposures would sum to 1 for each asset. Thus the model had no explicit market factor but rather the market return was projected on to the industry returns. Torre modified this scheme by introducing an explicit market factor (with unit exposure for each asset.) To keep the model identified by imposed the condition that the industry factor returns sum to zero in each time period. Thus the model is estimated as
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