 # "perfect negative correlation"

Schweser study guide page 105 from Study Session 12, Book 4: Question 4: Which of the following statements about covariance and correlation is false? … B. If two assets have perfect negative correlation, it is impossible to reduce the portfolio’s overall variance. I understand why this is false–in fact, adding an asset that has negative correlation with another will help diversify the risk, etc. But in the answer to the question, Schweser writes: "If two assets have perfect negative correlation, IT IS POSSIBLE TO REDUCE THE OVERALL RISK TO ZERO. Where did this come from? How do we know this? I understand that it would reduce risk, but how do we know it would reduce it all the way to zero?

I think they say “possible”. It means that the portfolio may be or may not be zero. There is a possible situation that portfolio variance will be zero if two assets in this portfolio have the same risk and same weightage.

Or you weight them proportional to, uh, something or other so that the portfolio has 0 variance (which must be possible).

For example, if asset A goes up by 10%, asset B goes down by 10%, so the risk is zero. All other cases lead to zero as well. Dreary

Or if asset A goes up by 2%, asset B goes down 1%, when asset A goes down 1%, asset B goes up 2%. So, if you give them equal allocation you will make .5 % every day.

If 2 assets A and B have correlation of -1, then there exists a formula such that A = x - yB with y > 0 That means if you have proportions of 1 unit of A for every y units of B , then you end up with: A + yB = (x -yB) + yB = x So you end up with a constant return, with 0 volatility.

Man just a possibility. It is like if you invest EQUAL amount in X and Y . Suppose X goes up by 10% and Y goes down by 10% (Remember perfect -ve). Thats make it RISK = ZERO. remember you got to INVEST EQUAL AMOUNTS.

No YOU DONT. See ChrisMaths answer.

Can you explain the formula in chrismaths post? What are x and y? (proportion of asset A and B in the portfolio?)

Here is an easy way to understand the formula. First, whenever we talk about correlation average returns are not important. It’s dependency of deviations from average that correlation measures. For example, if for any A and B correlation of A and B is the same as correlation of c1*B + c2 where c1 is a positive constant, c2 - any constant. Now what is perfect positive and negative correlations: A- = c*(B-) if c > 0, then A and B are perfectly positively correlated, if c < 0 then they are perfectly negatively correlated. Now we can re-write that equation A - c*B = -c*, if we now define y as (-c) and x as - c*B we will get Chrismaths’ equation: A + y*B = x. I hope that helps.

Oops, forgot to mention that if you assign weight of 1/(1+y) to A and y/(1+y) to B, the portfolio return is going to be constant and equal to x/(1+y).

The portfolio variance (assuming 2 assets) equation is: wA = weighting of asset A wB = weighting of asset B sdA = standard deviation of asset A sdB = standard deviation of asset B p = correlation of A and B portfolio variance = wA^2*sdA^2 + wB^2*sdB^2 + 2*wA*wB*p*sdA*sdB since the sd’s for A and B are going to be positive and correlation =-1, the component (p*sdA*sdB) is going to be negative and it’s always (i think, someone correct me if i’m wrong) possible to get weightings that offset (wA^2*sd^2 + wB^2*sd^2), and therefore get a portfolio variance = 0

If p = -1, then wA^2*sdA^2 + wB^2*sdB^2 + 2*wA*wB*p*sdA*sdB = (wA*sdA - wB*sdB)^2 which can certainly be 0 if wA*sdA = wB*sdB. set wA = 1 - wB, blah, blah,