Answer to Question #184751 in Macroeconomics for john

Question #184751

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


π‘Ÿπ‘π‘‘ = 𝛼𝑝+𝛽𝑝1𝐹1𝑑 + 𝛽𝑝2𝐹2𝑑 + πœ€π‘π‘‘


a) Derive the functional form for the variance of π‘Ÿπ‘π‘‘, denoted as πœŽπ‘2.


b) In deriving πœŽπ‘2, 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 πœ€π‘π‘‘ represents a substantial portion of πœŽπ‘2 i.e. low 𝑅2 , how

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


1
Expert's answer
2021-04-27T07:12:40-0400

a)

rp𝑑=Ξ±p𝑑+Ξ²p1F1𝑑+Ξ²p2F2t+ΞΎp𝑑r_{p𝑑}=\alpha _{p𝑑}+\beta_{p1}F_{1𝑑}+\beta _{p2}F_{2t}+\xi _{p𝑑}


To remove relation between F1 and F2, the coefficient of the following equation can be derived by regression analysis.

F2=eo+e1F1+diF_2=e_o+e_1F_1+d_i

where: eo and e1 = the regression coefficients

di = the random error term

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

F2=eo+e1F1+diF_2=e_o+e_1F_1+d_i

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

defining:

Λ†di=F2βˆ’(Λ†eo+Λ†e1F1)Λ†d_i=F_2-(Λ†e_o+Λ†e_1F1)


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

rp𝑑=Ξ±p𝑑+Ξ²p1F1𝑑+Ξ²p2F2tβˆ’Ξ²p2Λ†eoβˆ’Ξ²p2Λ†e1F1t+ΞΎptr_{p𝑑}=\alpha _{p𝑑}+\beta_{p1}F_{1𝑑}+\beta _{p2}F_{2t}-\beta_{p2}Λ†e_o-\beta _{p2}Λ†e_1F_{1t}+\xi_{pt}



rp𝑑=(Ξ±ptβˆ’Ξ²p2Λ†eo)+(Ξ²p1βˆ’Ξ²p2Λ†e1)F1+Ξ²p2F2+ΞΎptr_{p𝑑}=(\alpha_{pt}-\beta_{p2}Λ†e_o)+(\beta_{p1}-\beta_{p2}Λ†e_1)F_1+\beta_{p2}F_2+\xi_{pt}


b)

By assumption

E[(F1t+Λ†F)(F2t+Λ†F2)]=0E[(F_{1t}+Λ†F)(F_{2t}+Λ†F_2)]=0

E[(F1t+Λ†F1)ei]=0E[(F_{1t}+Λ†F_1)e_i]=0

E[(F2t+Λ†F2)ei]=0E[(F_{2t}+Λ†F_2)e_i]=0

and


E(F1t+Λ†F1)2=Οƒ12E(F_{1t}+Λ†F_1)^2=\sigma _1^2

E(F2t+Λ†F2)2=Οƒ22E(F_{2t}+Λ†F_2)^2=\sigma_2^2

E(e12)=Οƒei2E(e_1^2)=\sigma _{ei}^2


therefore

Οƒpt2=Ξ²p12Οƒ12+bp2Οƒ22+Οƒep2\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=ipt=i

Ξ²p=bi\beta_p=b_i

F=IF=I

ΞΎpt=Ci\xi_{pt}=C_i





substituting the answer becomes

Ξ²pt1Ξ²j1Οƒ12+Ξ²pt2Ξ²j2Οƒ22\beta_{pt1}\beta_{j1}\sigma_1^2+\beta_{pt2}\beta_{j2}\sigma_2^2


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