Correlation diversification and portfolio variance

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pepp

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Statement I: As the average correlation between securities in a portfolio increases, the risk reduction benefits of diversification decrease
Statement II: If stocks B is perfectly correlated with a stock A, then creating a mix of stock A and stock B in portfolio P you can achieve some diversification as opposed to holding a one stock portfolio of stock A.
Is Joker Correct regardign statement I and statement II
A) Yes
B) No about statement I
C) No about statement II
D) No about either
Pls explain your answer.
 
I would say A.
1 is correct. To better diversify your portfolio, you want to have a lower average correlation.
2 sounds correct as well. Because although they are perfectly correlated, there has to be some idiosyncratic risk for each stock that would provide at least some minor diversification benefits.
 
if it just says ‘perfectly correlated’ are we supposed to assume it means +1 and not -1?
And is diversification possible if two stocks has correlation of +1? I thought it was as long as it was less than +1 we would be able to see diversification benefits
 
Diversification benefits is achieved when assets are not perfectly correlated. Practically we know that no asset is perfectly correlated with another… but in theory, and with assumptions of perfect capital markets.. perfectly correlated stocks will not offer diversification benefits (and this is pretty much by definition). Just look at calculation of standard deviation with correlated stocks.
that is my take. but i think it is quite right.
and 1 is correct. so C
 
^^ Assets can also be perfectly negatively correlated, which will mean all kinds of diversification will be possible.
 
My mind is really bent on this topic because on the mock it says:


“As the average correlation between securities in a portfolio increases, the risk reduction
benefits of diversification decrease. Furthermore, as average correlation between securities in a portfolio
rises, the number of securities in the portfolio must be increased in order to achieve the same percentage of
portfolio risk reduction when the average correlation between securities is lower.”

This quote is said to be WRONG.

We all agree from the above responses that the lower your correlation, the better it is for your diversification. So why does that not make sense here?

there is a formula given in the answer (which I was unaware of until now) that adds up, but conceptually I don’t get it.
 
I feel that the first half of the statement is fine and the second half is a bit problematic, but I can’t exactly tell why.
 
Did some reading. So if you’ll go to page 162 of your Schweser book (or 377 of your CFAI book if you enjoy pain), they justify algebraicly that the lowest your standard deviation can go at a large amount of equally weighted stock is going to be your correlation.
So for example let’s say one portfolio has a correlation of .5 and another one with .25. They both start with two stocks and a correlation of .8
The more stocks you add, the more your variance goes down to your correlation. So for the portfolio with a correlation of .5, there is less ground to cover to bring your correlation to for let’s say 110% of your minimum correlation (.8-.55 = .25 ground to cover). Whereas the portfolio with a .25 correlation, there is more ground to cover to get to 110% of that minimum correlation (.8 - .27 = .53 ground to cover). Because there is more ground to cover, there are more stocks that are needed to get to your target percentage for a low correlation portfolio.
The bottom line is that a portfolio with smaller correlation brings your max diversification benefits up (by bringing minimum variance down), but it takes more stocks for it to get there.

That’s how I understand it.
 
“… Furthermore, as average correlation between securities in a portfolio
rises, the number of securities in the portfolio must be increased in order to achieve the same percentage of
portfolio risk reduction”.
It makes sense because when correlation increases, it is as if you have fewer securities, so you need to add more to get any diversification benefit. If your portfolio has many computer networks stocks, which let us assume are all highly correlated, then your portfolio is no good. The statement says: as average correlation between securities in a portfolio rises, the number of securities in the portfolio must be increased. Yep.
 
Dreary - that particular statement is WRONG according to the offical Mock.
Xander086 wrote:
I still don’t quite get (OMG!!!).
Can you correct the statement?
Click to expand...
Just remember that when it states “same portfolio of risk reduction” they mean a percentage of the minimum variance that can be achieved through risk reduction.
The less correlated your portfolio, the lower the minimum variance can potentially be (read: more diversification benefit). If the two portfolios start at the same variance it takes more stocks for the less correlated portfolio to reach its lowest variance because it is simply lower.
So, the way I understand it, the corrected statement would be:
“Furthermore, as average correlation between securities in a portfolio rises, the number of securities in the portfolio needed is less in order to achieve the same percentage of portfolio risk reduction when the average correlation between securities is lower”
 
let me give you an example:
Portfolio A and B. Each has two stocks.
Portfolio A and B both have a portfolio variance of .8
Portfolio A has a correlation of .78
Portfolio B has a correlation of .2
Portfolio A only has to add couple more stocks in order to bring down portfolio variance to 100% of its potential minimum variance: .78.
Portfolio B has to add maybe 50 stocks in order to fully realize its diversification benefits. To reach 100% of its potential minimum variance it has to bring its portfolio variance from .8 to .2!
 
In this rather extensive post http://www.analystforum.com/forums/cfa-forums/cfa-level-ii-forum/91320769, magician says,
“I think the idea is this:
If you have a low average correlation, you get huge diversification benefits, but it takes more stocks to get close to the maximum.
If you have a high average correlation, you get modest diversification benefits, but it takes fewer stocks to get close to the maximum.
So, for example, with a low average correlation, your diversification benefit might be a 50% drop in volatility (compared to having all correlations at +1.0), but you’ll need 75 stocks to get most of that 50% drop. With a high average correlation, your diversificaation benefit might be only a 10% drop in volatility, but you can most of that 10% with only 20 stocks.”
—————————–
My understanding is that if you want to achieve 100% max risk reduction in both situations, looking at the formula, you want portofolio variance to be exactly be average variance * average correlation. When correlation increases, Numerator 1 - ρ will decrease and Denominator n don’t have to be that large anymore to make this component to be close to zero.
variance of portfolio = average variance * { [ (1- average correlation) / n ] + average correlation}
I think I get it now…Thanks a lot, pmond. But the wording is super tricky (WTH is same percentage of portfolio risk reduction?) ! Definitely a trap question!!
 
pmond wrote:
Dreary - that particular statement is WRONG according to the offical Mock.
Click to expand...
I don’t know about the rest of the statement, I just took this statement:
“The statement says: as average correlation between securities in a portfolio rises, the number of securities in the portfolio must be increased”
which makes sense to me.
 
@Dreary I agree with you but the statement is incomplete “As average correlation between securities in a portfolio rises, the number of securities in the portfolio must be increased to maintain the same level of portfolio variance” explained by the formula
σ²port = (avg σ²)/n + (avg cov)(n – 1)/n
The statement in question is “as average correlation between securities in a portfolio rises, the number of securities in the portfolio needed is less in order to achieve the same percentage of portfolio risk reduction when the average correlation between securities is lower”
 
ok, that gets a bit messy, I have no idea what’s right here.
 
Xander086 wrote:
Thanks a lot, pmond. But the wording is super tricky (WTH is same percentage of portfolio risk reduction?) ! Definitely a trap question!!
Click to expand...
Agreed. Thanks for getting me to explain it! Explaining makes you learn it the best. Unfortunately I think this is the only thing that I’ve mastered lol.
 
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