Covariance between two assets in a one-beta model

F

frenchriviera

New member
Joined
Jun 18, 2026
Messages
0
Reaction score
0
Does anybody know how to derive this formula:
cov(i,j) = beta(i)*beta(j)*variance(m)
It was in CFAI Reading 23, p. 50. Somehow, I do not know how to make the connection between cov(i,m), cov(j,m) and cov(i,j). Thanks!
 
cov(i,j) = beta(i)*beta(j)*variance(m) is based on factor model
y(i) = alpha(i) + beta(i)*Market + error(i)
y(j) = alpha(j) + beta(j)*Market + error(j)
cov(y(i),y(j))= //get rid of alphas because they are constants // = beta(i)*beta(j)*Var(m) + Cov(error(i),Market)*beta(i) + Cov(error(j), Market)*beta(j) + Cov(error(i), error(j)) = beta(i)*beta(j)*Var(m) + 0 + 0 + 0 //0 because error(i), error(j) are independent of each other and the market.
 
we know that
cov(i,j) = std(i)*std(j)*corr(i,j)
beta(i) = cov(i,m)/variance(m)
beta(j) = cov(j,m)/variance(m)
corr(i,j) = corr(i,m)*corr(j,m)
here is the proof
cov(i,j) = beta(i)*beta(j)*variance(m)
cov(i,j) = [cov(i,m)/variance(m)] * [cov(j,m)/variance(m)] * variance(m)
cov(i,j) = [std(i)*std(m)*corr(i,m)/variance(m)] * [std(j)*std(m)*corr(j,m)/variance(m)] * variance(m)
cov(i,j) = [std(i)*corr(i,m)/std(m)] * [std(j)*corr(j,m)/std(m)] * variance(m)
cov(i,j) = [std(i)*corr(i,m)*std(j)*corr(j,m)/variance(m)] * variance(m)
cov(i,j) = std(i)*corr(i,m)*std(j)*corr(j,m)
cov(i,j) = std(i)*std(j)*corr(i,j)
 
volkovv Wrote:
——————————————————-
> we know that
> corr(i,j) = corr(i,m)*corr(j,m)
a bit of a leap of faith here, isn’t it?
 
^agree
I should of said, assuming:
corr(i,j) = corr(i,m)*corr(j,m)
 
volkovv Wrote:
——————————————————-
> ^agree
>
> I should of said, assuming:
>
> corr(i,j) = corr(i,m)*corr(j,m)
that’s a pretty strong assumption isn’t it?
 
I’m not 100% sure, but my intuition says that this assumption is valid if the only correlation between I and J is because of their mutual correlations to M (i.e. partial correlation of I and J = 0, when you control for M).
In practice, if I and J are both affected by things, but these things don’t affect the market in general (or don’t affect it as much), then the assumption won’t hold.
 
just read maratikus’ derivation. its spot on.
all you’re doing is
cov(a+bm, c+dm) where a b c d are constants
just solve using the linear transformation formulae for covariance from any intro stat book.
 
bchadwick Wrote:
——————————————————-
> I’m not 100% sure, but my intuition says that this
> assumption is valid if the only correlation
> between I and J is because of their mutual
> correlations to M (i.e. partial correlation of I
> and J = 0, when you control for M).
>
> In practice, if I and J are both affected by
> things, but these things don’t affect the market
> in general (or don’t affect it as much), then the
> assumption won’t hold.
Agreed, the only way this assumption will hold is when movement in the market accounts for the most of the movement in I and J and movements that effect I and J independently without effecting the market is small or 0.
 
You must log in or register to reply here.
Back
Top