ok , we all know that lower, even negative correlation stocks offer the best diversification benefit, for example:
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)
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)
the above logic correct ? - yes. IF so, 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 ?
If you have a portfolio A with two stocks having correlation of -1, the lowest you can get. You also have a portfolio B with two stocks having a correlation of 0.99.
It makes sense that portfolio A will generally be less risky than B.
If you want to make Portfolio A and B have equal risk, then you need to increase the risk of Portfolio A.
So let’s say you include another asset with a correlation of 0.7 with portfolio A, then overall risk of Portfolio A will increase. If your goal is to ensure that portfolio A’s risk = Portfolio B’s risk, you’ve got to keep increasing the stocks in portfolio A.
can you pls elaborate a bit more ? What’s wrong there ?
as bloodline also pointed out above, if i have a portfolio with 2 stocks and their correlation is -1, then compare with another portfolio with 2 stocks and their correlation is 0.99, then obviously first portfolio risk is lower, due to the diversification benefit.
Jaychou, I think what S2000 is trying to emphasize is that your explanation refers more to price correlations than 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.
Returns correlation, not price correlation, is the appropriate way to measure this association in the context of a portfolio risk.
From the curriculum (vl 6 pg 378): with a higher correlation, the investor would need fewer stocks to obtain the same percentage of minimun possible portfolio variance. So let’s take the example straight from the text: the investor is on a 8% variance portfolio which has 30 stocks with correlation of return among any pair of stock 0.3. now if the corelation among the any pair of stocks is higher at 0.5, he needs less than 30 stocks to produce the same 8% variance portfolio. by explaining it using the formula, i’m perfectly fine. But just think about it logically without referring to the model: hey stock A and B correlation of return used to be 0.3 (stock A return goes up, stock B return likely go up) but now it is 0.5, their return are even more likely to move in the same direction. How does that imply more diversification benefit?
You’re misunderstanding correlation, but that’s probably not your fault: almost nobody teaches the idea properly.
The difference in the size of the return is immaterial.
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:
Look at the definition of covariance:
cov(x, y) = [Σ(x_i – μ_x)(y_i – μ_y)] / n
In this definition, you see that it doesn’t matter whether _x_i is positive or negative, and it doesn’t matter whether _y_i is positive or negative: we’re not multiplying _x_i by _y_i. What matters is whether _x_i is above or below its mean and whether _y_i is above or below its mean.
Suppose that _x_i is always above its mean when _y_i is aboveits mean, and x_i is always below its mean when y_i is belowits mean: we’ll have a positive correlation of returns (even as high as +1.0). But if **μ_x = +5%** and **μ_y = –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 _x_i is always above its mean when _y_i is belowits mean, and x_i is always below its mean when y_i is aboveits mean: we’ll have a negative correlation of returns (even as low as –1.0). But if **μ_x = +5%** and **μ_y = +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:
Where does it say that, and what, exactly does it say? It sounds a bit . . . off.
variance of portfolio = variance of stocks * ( (1- correlation) / n + correlation)
so u see, if n gets real large, the first part will be almost 0, so that will make the variance of the portfolio = variance of stocks * correlation. By using this formular, it is easy to explain when correlation gets larger, n will be smaller for a given portfolio variance.
Both the formulas quoted by magician and jaychou are correct. Jaychou’s formula is a special case where all standard deviations are the same (hence equal to their average). You can easily derive Jaychou’s formula from S2000magician’s formula. Tell me if you need me to.