In financial course today, we were using Harry Markovitz Portfolio Selection Theory. During that , the professor said the following statement "If correlation between A and B is +0.4 and that of B and C is also +0.4 then correlation of A and C cannot be +0.4"
So to prove that his statement is invalid. I came up with following example
(Example starts) Say there are 3 sets of data for X, Y and Z
X Y Z
1 5 10
2 10 20
4 20 40
8 40 80
16 80 160
32 160 320
The correlation between:
1. X and Y is 1
2. Y and Z is 1
3. X and Z is 1
(Example ends)
However, this is only 1 example. I cannot come up with other data sets that would yield all 3 coefficients to be equal.
So I would like to know whether the statement as given by the professor is valid or not? If not then how should I go about proving that it is invalid aside from the example above.
Your professor is certainly wrong. Any valid correlation matrix (where “valid” means positive definite, symmetric, and having ones on the main diagonal) can be the correlation structure of some random variables... just take a multivariate normal with that covariance matrix. It should be easy to check that the $3\times 3$ matrix with ones on the main diagonal and $0.4$ for all off diagonal elements is positive definite.
In fact, it’s a fairly common modeling choice to assume a set of variables all have the same pairwise correlation. It will work for any positive correlation, but there is a limit to how much the negative pairwise correlation can be (above that limit the matrix will cease to be positive definite).