Question about equations and risk factors

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Hello,

Say I have two risk factors X_1 and X_2. Standard deviation for X_1 is sigma_1 and sigma_2 for X_2. Furthermore, X_1 has a mean of mu_1 and X_2 has a mean of mu_2. Correlation between X_1 and X_2 is rho.

The system is as follows:
X_1=mu_1 + lambda_{11} U_1
X_2=mu_2 + lambda_{21} U_1 + lambda_{22} U_2

U_1 U_2 are standard normals.

My book reads as follows:

"Accordingly:
lambda_{11}=sigma_1 (1)
lambda_{21}^2 + lambda_{22}^2= sigma_2^2 (2)
lambda_{11} lambda_{22}= rho sigma_1 sigma_2 (3)"

I don't understand how they work out lines (2) and (3) from a mathematical point of view.

Can anyone please help?

Thanks in advance,

Julien.

 

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For (2)
Var(X2)=lambda_21^2.Var(U1)+lambda_22^2.Var(U2)+2.lambda_21.lambda_22.cov(U1,U2), Var(U1)=1=Var(U2) and Cov(U1,U2)=0, if I additionally assume they are independent.

For (3), try working the Cov(X1,X2) which u need to equate to rho*sigma1*sigma2

Alan

 

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Hello Alan,
Thanks. I got your reply this morning and I am working through it. I tried to compute the variance using the following formula:
Var(X_2)=E(X^2)-E(X)^2 with E(X)^2 = mu_2^2
I get this:
Var(X_2)=E[(lambda_21 U_1)^2 + 2 (lambda_21 U_1 lambda_22 U_2)+(lambda_22 U_2)^2]
Is it correct? If so, how do I reduce this expression please?
Regards,
Julien.

 

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Julien,

Your method is also correct. The calc. is more.

So, you are left with - lambda_21^2.E(U1^2)+lambda_22^2.E(U2^2)+2.lambda_21.lambda_22.E(U1.U2)
E(U1^2)=Var(U1)+E(U1)^2=1+0^2=1. Similarly for E(U2^2)
If we assume U1 and U2 are independent, then E(U1.U2)=E(U1).E(U2)=0.0=0.

Hope this helps.

Alan.

 

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Wonderful Alan! Thanks a lot. I get it now!
Julien.