S2000magician wrote:
jaychou wrote:
S2000magician wrote:When you talk about correlation, do you mean (as most finance people think they do) correlation of returns, or do you mean correlation of prices?
ah, i meant return. im wrong in the concept here?
You’re misunderstanding correlation, but that’s probably not your fault: almost nobody teaches the idea properly.
bloodline wrote:I’m going to take a stab at this.
Jaychou, I think what S2000 is trying to emphasize is that your explanation refers more to price correlationan return correlations.
E.g, stock A closed $5 yesterday and at $6 today –> A one dollar price difference but a 20% return.
stock B closed at $5000 yesterday and at $5001 —> A one dollar price difference but a 0.02%
In price terms, since stock A and B move together by the same amount in the same direction, you might argue that they have a strong positve price correlation. However, if you compare their correlations in terms of returns, then you will see a big difference.
The difference in the size of the return is immaterial.
bloodline wrote:Returns correlation, not price correlation, is the appropriate way to measure this association in the context of a portfolio risk.
Absolutely correct, but if you don’t understand how correlation works, you’re going to come to some very wrong conclusions. Alas, both of you gentlemen have here.
Let’s take the simple one first: the idea that the size of the return matters. Recall the formula for correlation:
ρ(
x,
y) = cov(
x,
y) / (σ
x × σ
y)
Notice that we’re dividing by the standard deviation of
x, and we’re dividing by the standard deviation of
y. If
x has returns that are 10 times the size of
y’s returns, then the covariance of their returns will be 10 times as big, and σ
x will be 10 times as big; they’ll cancel out. The size of the returns doesn’t matter; the only thing that matters is whether they move in the same direction or in opposite directions.
Now, to the heart of my original complaint:
S2000magician wrote:
jaychou wrote:stock A correlation with stock B is -0.8 ( A up, B is very likely to go down - ok great, i have some sort of diversification just with these 2 stocks)
Wrong, wrong, wrong, wrong, wrong!
jaychou wrote:stock C correlation with stock D is 0.8 (C up, D is likely to go up as well - ok i need to add some more stocks to get diversified more because my portfolio with stocks C and D obviously does not offer me diversification benefit)
Wrong, wrong, wrong, wrong, wrong!
Look at the definition of covariance:
cov(
x,
y) = [Σ(
xi – μ
x)(
yi – μ
y)] /
n
In this definition, you see that it doesn’t matter whether
xi is positive or negative, and it doesn’t matter whether
yi is positive or negative: we’re not multiplying
xi by
yi. What matters is whether
xi is
above or below its mean and whether
yi is
above or below its mean.
Suppose that
xi is
always above its mean when
yi is
above its mean, and
xi is
always below its mean when
yi is
below its mean: we’ll have a positive correlation of returns (even as high as +1.0). But if
μx = +5% and
μy = –10%, then
the price of x is going up and the
price of y is going down:
positive correlation of
returns,
negative correlation of
prices.
Suppose that
xi is
always above its mean when
yi is
below its mean, and
xi is
always below its mean when
yi is
above its mean: we’ll have a negative correlation of returns (even as low as –1.0). But if
μx = +5% and
μy = +10%, then
the price of x is going up and the
price of y is going up:
negative correlation of
returns,
positive correlation of
prices.
By the way, remember what I wrote at the top: most people don’t teach correlation correctly, and many, many finance people don’t understand it. (If you read finance literature, you’ll find lots of articles where the authors make exactly the same mistakes as those perpetrated here. That’s sad.)
Now that that’s cleared up, let’s get back to the original question:
jaychou wrote:… can anyone explain to me why in the CFA curriculum, it is stated that lower correlation requires more stocks and higher correlation requires less stocks to achieve the same level of risk ?
Where does it say that, and what, exactly does it say? It sounds a bit … off.
k , we all know that lower, even negative correlation stocks offer the best diversification benefit, for example: