Answer to Question #184751 in Macroeconomics for john

Question

Consider a Multi-Index Model (MIM) specification for the portfolio return:

𝑟𝑝𝑡 = 𝛼𝑝+𝛽𝑝1𝐹1𝑡 + 𝛽𝑝2𝐹2𝑡 + 𝜀𝑝𝑡

a) Derive the functional form for the variance of 𝑟𝑝𝑡, denoted as 𝜎𝑝2.

b) In deriving 𝜎𝑝2, what are the key assumptions you made under the MIM?

c) If you estimate the above MIM as a regression, and you find that the variance of the residual return 𝜀𝑝𝑡 represents a substantial portion of 𝜎𝑝2 i.e. low 𝑅2 , how

would you interpret this finding? [Hint: More than 1 reason.]

Answer 1

a)

$r_{p𝑡}=\alpha _{p𝑡}+\beta_{p1}F_{1𝑡}+\beta _{p2}F_{2t}+\xi _{p𝑡}$

To remove relation between F₁ and F_2, the coefficient of the following equation can be derived by regression analysis.

$F_2=e_o+e_1F_1+d_i$

where: e_o and e₁ = the regression coefficients

d_i = the random error term

By the assumptions of regression analysis, is uncorrelated with. Therefore:

$F_2=e_o+e_1F_1+d_i$

Which is an index of performance of the sector index without the effect of F₁

defining:

$ˆd_i=F_2-(ˆe_o+ˆe_1F1)$

an index is obtained that is uncorrelated with the market. By solving for F₂ and substituting

$r_{p𝑡}=\alpha _{p𝑡}+\beta_{p1}F_{1𝑡}+\beta _{p2}F_{2t}-\beta_{p2}ˆe_o-\beta _{p2}ˆe_1F_{1t}+\xi_{pt}$

$r_{p𝑡}=(\alpha_{pt}-\beta_{p2}ˆe_o)+(\beta_{p1}-\beta_{p2}ˆe_1)F_1+\beta_{p2}F_2+\xi_{pt}$

b)

By assumption

$E[(F_{1t}+ˆF)(F_{2t}+ˆF_2)]=0$

$E[(F_{1t}+ˆF_1)e_i]=0$

$E[(F_{2t}+ˆF_2)e_i]=0$

and

$E(F_{1t}+ˆF_1)^2=\sigma _1^2$

$E(F_{2t}+ˆF_2)^2=\sigma_2^2$

$E(e_1^2)=\sigma _{ei}^2$

therefore

$\sigma_{pt}^2=\beta_{p1}^2\sigma_1^2+b_{p2}\sigma_2^2+\sigma_{ep}^2$

c)

covariance between security pt and j can be expressed as:

let $pt=i$

$\beta_p=b_i$

$F=I$

$\xi_{pt}=C_i$

substituting the answer becomes

$\beta_{pt1}\beta_{j1}\sigma_1^2+\beta_{pt2}\beta_{j2}\sigma_2^2$