Correlation vs. Risk

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keep_running

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One question I am working on right now claims that reduction in correlation between two assets also reduces risk as well.
However, when I look at the covariance equation, it shows that variance and correlation are inversely related. As such, how does correlation reduction also decrease risk when the covariance equation shows the opposite?
 
The equation (for a 2-asset portfolio) should be:
σ²port = w²1σ²1 + w²2σ²2 + 2w1w2σ1σ2ρ(1,2)
So, if ρ decreases, σ²port decreases.
 
But if you look at the covariance formula,
you can see that cov(a,b)/(std a * std b) = correlation (a,b)
how can you explain it in the context of this equation just above?
 
The equation (for a 2-asset portfolio) should be:
σ²port = w²1σ²1 + w²2σ²2 + 2w1w2cov(1,2)
So, if cov decreases, σ²port decreases.
 
How come the relationship is different between your equation and the equation I posted. Correlation seems to have a direction relationship in your equation while having an indirect relationship in my equation with portfolio risk.
 
I don’t see the inverse relationship you mention. What do you see as the inverse relationship?
All I did was to take your equation:
cov(a,b)/(std a * std b) = correlation (a,b)
solve it for covariance:
cov(a, b) = corr(a, b)σaσb
then substitute it into my original equation.
 
Ok. So I made the assumption that Std A * Std B is the portfolio risk, which it is not.
 
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