ICAPM and correlations

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Blackou

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Can someone explain further the solution to Q3 of exhibit 10 (page 44 of Reading 15):
Question : what is the correlation between Private Equity and Small/Mid cap?
One is given:
STD(GIM)=7%
STD(Private Equity)=34%
STD(Small/mid cap)=22%
Beta of private Equity with respect to GIM=3.3
Beta of Small/mid cap with respect to GIM=2.06
Solution = BetaPE*BetaSmallCap*STD(GIM)^2 / {STD(Private Equity)*STD(Small/mid cap)}
thks
 
  • BetaPE*BetaSmallCap*STD(GIM)^2
is another way of saying
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC)
as the STD(GIM)^2 cancels out both beta’s denominators
Remember the correlation formula is COV / SD(A) * SD(B)
and since
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC) = COV
then r =
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC)
divided by
  • {STD(Private Equity)*STD(Small/mid cap)}
 
Thanks man, i understand your demonstration but one step is still not obvious for me :
In your line:
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC) = COV
COV in the equation above is covariance(PE,SC)?
Could you explain why covariance(PE,SC)=corr(PE,GIM)*corr(SC,GIM)*SD(PE)*SD(SC)?
Sorry if it is obvious but i am missing something
 
MrSmart wrote:
  • BetaPE*BetaSmallCap*STD(GIM)^2
is another way of saying
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC)
as the STD(GIM)^2 cancels out both beta’s denominators
Remember the correlation formula is COV / SD(A) * SD(B)
and since
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC) = COV
then r =
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC)
divided by
  • {STD(Private Equity)*STD(Small/mid cap)}
Click to expand...
+1
I just was trying to get how you get to the formula of Cov = to all the things in the numerator. I understand it has to do to the 1 factor model of capm and doing the maths
 
Blackou wrote:
Thanks man, i understand your demonstration but one step is still not obvious for me :
In your line:
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC) = COV
COV in the equation above is covariance(PE,SC)?
Could you explain why covariance(PE,SC)=corr(PE,GIM)*corr(SC,GIM)*SD(PE)*SD(SC)?
Sorry if it is obvious but i am missing something
Click to expand...
Well you have for a 2 factor model covariance formula which you can also try to get.. :
Mij=bi1bj1Var(F1)+bi2bj2Var(F2)+(bi1bj2+bi2bj1)Cov(F1,F2)
where Mij is the covariance in the book
If you have a one factor model then:
Mij=bi1bj1Var(F1)
If you have a one factor model then:
Mij=corr(i,market)*sigma(i)/sigma(market)*corr(j,market)*sigma(j)/sigma(market)*sigma(market)^2
Then Mij=corr(i,market)*sigma(i)*corr(j,market)*sigma(j)* [ sigma(Market)^2 / (sigma(Market)*Sigma(Market))]
Then Mij=corr(i,market)*sigma(i)*corr(j,market)*sigma(j)
:)
 
Blackou wrote:
Thanks man, i understand your demonstration but one step is still not obvious for me :
In your line:
  • r(PE,GIM) * r(SC, GIM) * SD(PE) * SD(SC) = COV
COV in the equation above is covariance(PE,SC)?
Could you explain why covariance(PE,SC)=corr(PE,GIM)*corr(SC,GIM)*SD(PE)*SD(SC)?
Sorry if it is obvious but i am missing something
Click to expand...
Yes.
beta i = covariance (i,m) / var(m) = corr i,m * std i / std m
beta j = cov(j, m) / var (m) = = corr j,m * std j / std m
and cov i,j = beta i * beta j * var(m)
Dividing cov i,j by std i and j, to get r(i,j)

You get

r(i,j) = r(i,m) * r(j,m)

If you have two factors, then the covariance of two markets/classes is

B1i*B1J*VAR(1) + B2i*B1J*VAR(2) + (B1i*B2j + B2i*B1j) * COV (1,2)

Since 2 here does not exist, we only have one factor in the model, so you’re only left with the first term, and you know the rest.

And trust me, this is all from the book, I’d stick a fork in my eye before memorizing any of this.
 
Ok! Thanks to the two of you.
Using Mij=bi1bj1Var(F1) it seems more clear
Beta(PE)=bi1 ; Beta(SC)=bi2 , F1= GIM ; Mij=COV(PE,SC)
=> COV(PE,SC)=Beta(PE)*Beta(SC)*Var(GIM)
and since COV(PE,SC)=Corr(PE,SC)*SD(PE)*SD(SC)
hence
Corr(PE,SC)*(SD(PE)*SD(SC)=Beta(PE)*Beta(SC)*Var(GIM)
the result follow
I definitely trust you that the formula B1i*B1J*VAR(1) + B2i*B1J*VAR(2) + (B1i*B2j + B2i*B1j) * COV (1,2) is in the book, i have to work a little more on that one
 
You could also remember that COV(i,j)= B(i)*B(j)*Var(m)
Given that the betas for PE and SC are given, as is the market std. dev., you can solve for the covariance of the two assets.
Once you have the covariance computed, just divided by both assets std. dev.’s to get the correlation.
 
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