Maximum risk reduction

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rf040234

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On Schweser notes book 3 page 209, there’s a table showing the average correlation and the maximum risk reduction benefits. I’ve been reading this over and over again, and still I can’t make a sense of it. Can somebody explain this in a very simple and plain language?
 
variance(p) = avg variance(asset)* [(1 - correlation)/n + correlation]
So as correlation–tendsTo–>0
the term, [(1 - correlation)/n + correlation] –tendsTo–> 1
Hence variance(p) = avg variance(asset)*[1]
Now consider the table
correlation = 0.1, n=90,
[(1 - correlation)/n + correlation] = [0.9/90 + 0.1] = 0.11
Similarly, if,
correlation = 0.5, n=23,
[(1 - correlation)/n + correlation] = [0.5/23 + 0.5] = 0.52
best possible values,
correlation = 0, n = veryLargeNumber (ex: 1000),
[(1 - correlation)/n + correlation] = [1/1000 + 0] = 0.0001
So the multiplier gets smaller and smaller…
Thus as a conclusion it is said that:
variance of the portfolio decreases as correlation decreases and also the number of assets increases to infinity.
variance(p) will become = avg variance(asset) when correlation is 0 and no of assets are infinite.
I think I got you confused even more?
 
actually you’re very clear swaptiongamma. Thanks for clarifying it. But what do the values in the table (e.g. 0.19, 0.14, 0.11, 0.33 etc) mean? Are they the portfolio variance? What are the values called?
 
The values are the numeric solution to square-bracketed term in the formula states below.
variance(p) = avg variance(asset)* [(1 - correlation)/n + correlation]
So [(1 - correlation)/n + correlation] = 0.19 or 0.14 or 0.51 …. depending upon what values u substitute for correlation and n in the equation.
And this is how we have to interpret the terms.
if say we have a portfolio with 10 assets (hence n=10) and average correlation amongst the assets is (correlation = 0.1), substituting in [(1 - correlation)/n + correlation] we get value as [0.19].
and now consider average variance of the assets in the portfolio is 20 (sigma^2(i) = 20))
Then variance of the port = average variance of the assets * [multiplier]
variance of the port = 20 * [0.19]
Thus they are not portfolio variance - but a multiplier that when multiplied to the avg asset variance will give us the port variance.
 
oh!!! i see, this is the multiplier… somehow i just can’t keep my mind straight on these type of things… thanks swaptiongamma!
 
Swaption, what if the co-relation between the assets is -1?
Then the variance of the portofolio will become 0 even if we have only two assets because [(1-cor)/n +cor)] will become 0. So in that case, we don’t need infinite number of assets. am I correct?
 
[(1 - -1)/2 + -1] = 2/2 - 1 = 1-1 = 0 = port variance is zero
This co-incides with the fact that 2 assets which are perfectly negatively correlated (r = -1), then they can provide the best, ideally diversified portfolio.
Say if r(IBM, GLD) = -1. In a volatile market like this -1 would mean, If one goes down 30%, other goes up exaclty by 30%, making the weighted avg variance of port = 0.
Finding such perfectly correlated in great for books, not for real. So at best we can do is - try to take the correlation as low as possible and then add large number of assets in the port to get gr8 diversification.
And the correlation they are talking here is avg asset correlation (correlation-hat).
.. Just my thoughts, and might be way off.
 
-ve correlation only talks about the DIRECTION OF THE MOVEMENT, NOT THE EXTENT OF MOVEMENT.
So with a correlation of -1 - if IBM Went down 30% in above example - GLD would go UP, that is all you can say. Need not go up 30%. Even if it went up, you have reduced the variance of your portfolio, kind of the two assets are not both going into a loss situation. One’s losses offsets the others gains. and you have some diversification benefit, because you have reduced the RISK on your position.
 
guys, you seem to be nailing these concepts..
anyhow, how average variance and correlation are calculated?
 
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