# Correlations between 3 assets

Hi,

the correlation between asset 1 and asset 2 is negative
the correlation between asset 2 and asset 3 is negative
the correlation between asset 1 and asset 3 should be ______? why?

I can’t figure this out. Any thoughts? Thank you.

Could be almost anything. Positive, negative, zero.

If the first two correlations are, respectively, a and b, then the third correlation has to lie in the range:

\left[ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)},ab + \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}\right]

S2000magician, thank you!

I was looking over a problem in L3 market expectations that said, the correlation between asset 1 and 3 should be positive. It cannot be negative.

It depends on the magnitude of a and b.

In general, they’re wrong.

A little more on this:

If a and b have opposite signs – i.e., one is positive and the other negative – then c can be negative; i.e., the lower limit I gave above, ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}, is negative.

If a = 0 and b ≠ ±1, then c can be negative; i.e., the lower limit I gave above, ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}, is negative.

If a ≠ ±1 and b = 0, then c can be negative; i.e., the lower limit I gave above, ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}, is negative.

If a = 0 and b = ±1, then c cannot be negative; i.e., the lower limit I gave above, ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}, is zero.

If a = ±1 and b = 0, then c cannot be negative; i.e., the lower limit I gave above, ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}, is zero.

Otherwise, a and b have the same sign: either both positive or else both negative.

If a and b satisfy a^2 + b^2 = 1 – i.e., the point \left(a,b\right) is on the unit circle – then c cannot be negative; i.e., the lower limit I gave above, ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}, is zero.

If a and b satisfy a^2 + b^2 < 1 – i.e., the point \left(a,b\right) is inside the unit circle – then c can be negative; i.e., the lower limit I gave above, ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}, is negative.

If a and b satisfy a^2 + b^2 > 1 – i.e., the point \left(a,b\right) is outside the unit circle – then c cannot be negative, nor can it be zero; i.e., the lower limit I gave above, ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}, is positive.

Am I allowed to publish the problem here?

Sure.

I think if you reverse engineer the answer

If Asset 1 goes up, Asset 2 goes down => negative correlation 1,2
If Asset 2 goes down, Asset 3 goes up => negative correlation between 2,3

Therefore Asset 1 is going up and 3 is going up => positive correlation between 1,3

1 Like

Not necessarily.

You have your first two implications pointing in the wrong direction, and when pointing in the proper direction they’re not necessarily true.

This is the problem:

Inconsistency of correlation estimates:
Frequently, the expected correlations b/t asset classes form part of the final expectational data that an analyst needs. if the number of asset classes is n, the analyst will need to estimate (n^2 - n)/2 distinct correlations or the same number of distinct covariances). In doing so, the analyst must be sure that his or her estimates are consistent. for example, consider the correlation matrix for three assets below

Market 1 Market2 Market 3
Market 1 1 -1 -1
Market2 -1 1 -1
Market 3 -1 -1 1

Answer: According the the table above, the estimated correlation b/t each asset and each other asset is -1. These estimates are internally inconsistent and , in fact, not possible. if markets 1 and 2 are perfectly negatively correlated and markets 2 and 3 are as well, then market 1 and 3 should be perfectly positively correlated rather than perfectly negatively correlated.

this makes sense.

Aha!

Not “the correlation between a and b is negative”, but, specifically, “the correlation between a and b is −1”.

It’s so much easier to solve a problem when you’re given all of the information.

Note how this works with the range I gave you; c has to be in the range of:

\left[ab - \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)},ab + \sqrt{\left(1 - a^2\right)\left(1 - b^2\right)}\right]
\left[\left(-1\right)\left(-1\right) - \sqrt{\left(1 - \left(-1\right)^2\right)\left(1 - \left(-1\right)^2\right)},\left(-1\right)\left(-1\right) + \sqrt{\left(1 - \left(-1\right)^2\right)\left(1 - \left(-1\right)^2\right)}\right]
\left[1 - \sqrt{\left(0\right)\left(0\right)},1 + \sqrt{\left(0\right)\left(0\right)}\right]
\left[1,1\right]

So . . . c = 1.

By the way, the test to determine whether a set of correlations is possible is that the determinant of the correlation matrix has to be nonnegative.

Not as written.

Thank you.

My pleasure.