 # Portfolio Mgmt Q: correlation

QUESTION:

Adding a stock to a portfolio will reduce the risk of the portfolio if the correlation coefficient is less than which of the following?

A) 0.00.

B) +1.00.

C) +0.50.

Adding any stock that is not perfectly correlated with the portfolio (+1) will reduce the risk of the portfolio.

The question’s correct answer doesn’t make sense to me. It places no restrictions on risk or return of the securities. For example, what if I had a portfolio (A) of rtn = 10%, std dev = 5% and a separate security (B) of rtn = 3%, std dev = 50%. Suppose corr(A,B) = 0.99. There is no weight of B that could be added to A that would lower A’s std dev, right? I’m assuming no shorting… maybe they aren’t?

If it were true that adding any non perfectly correlated stock to a portfolio would result in a decrease in std dev, that would mean the Optimal Market Portfolio with individual components approaching infinity would have a std dev approaching zero… right? That’s just silly.

What’s more likely is I’m misunderstanding. Please tell me. Thanks!

-F

It doesn’t matter what the standard deviations of the individual stocks are. The optimal portfolio will have a lower standard deviation than either stock in the portfolio. If two stocks aren’t perfectly correlated, you will run into situations where they move in different directions, so your risk is reduced.

Adding an infinite number of stocks eliminates all diversifiable risk, and you’re left with the entire stock market, which fittingly has a standard deviation equal to the market portfolio standard deviation since they’re the same portfolio. This is market risk, and it doesn’t go away.

The answer 0 is definitely wrong because you’re basically saying -0.5 is okay but 0.5 isn’t. In reality, the best way to diversify is by having small absolute correlation values – as close to 0 as possible is best, but everything that isn’t -1 or 1 helps at least a little.

Think about the equation for portfolio variance:

Variance of Portfolio = w1^2*variance1 + w2^2*variance2 + 2*w1*w2*correlation(1,2)*stdev1*stdev2

Focus on the last section of the equation (the only part that includes correlation): 2*w1*w2*correlation(1,2)*stdev1*stdev2

The maximum value (i.e. highest portfolio variance) would occur when correlation is at its maximum (i.e. when correlation = 1). Any correlation coefficient <1 would decrease the overall variance of the portfolio.

The minimum value (i.e. lowest portfolio variance) would occur when the correlation is -1.

When in doubt, reference the equations underlying the questions.

Thank you both for your replies. Recalling the portfolio variance equation formula components helps especially, and I am better understanding the theory of it. But I believe Aaron’s statement above is untrue.

My issue is I still can’t prove this in practice. Consider the following example (similar to what I stated above):

Original Portfolio (A): std dev = 5%, variance = 0.0025

New Stock (B): std dev = 50%, variance = .25

corr(A,B) = 0.9

From your answers above, I would want to add the new stock (B) to my portfolio at some (undetermined) weight because it has a correlation >1 with my original portfolio. Right?

I used Excel Solver to find the minimum variance portfolio of the combination of A and B, and do you know what it said? 0.0025 variance is the minimum, with 100% in (A) and 0% in (B).

If short-selling was not a constraint, 110% (A) and -10% (B) would produce the minimum variance portfolio (0.0006 variance).

This tells me I would NOT want to add Stock B to my portfolio at any (positive) weight.

I used the portfolio std dev formula to look at some varying weights to help confirm:

100% A, 0% B: Std Dev(A) = 0.0500

90% A, 10% B: Std Dev(A,B) = 0.0926

80% A, 20% B: Std Dev(A,B) = 0.1371

So I still believe the original question’s answer is wrong, unless by “risk” they’re talking about things other than std dev and variance.

Although the last part of the equation lowers when corr < 1, the weights also change the other components since you’re adding a stock. If the increase due to weight changes is larger than the decrease due to correlation, the total equation would increase in value.

Maybe what the original question is trying to say is this: If you are forced to add a stock to your portfolio, having a coefficient of less than 1 will decrease the portfolio’s risk relative to if the correlation was 1. I’d agree with that.

Short-selling isn’t a constraint. A portfolio can, and often does, contain negative weights. That’s what optimization is all about. You’re right that it’s impossible to reduce the portfolio risk in all situations if you can only go long, but that’s not an assumption. And, even if you couldn’t short, your answer to the original question is still wrong. The ideal answer is the closest possible to 0 (and if you managed to have -1, that would absolutely reduce your risk since there wouldn’t be any), which includes anything between (-1, 1).

I guess I was assuming the words “Adding a stock” in the question meant adding long, not short positions. But you’re right, in the real world short-selling isn’t a constraint so that was a wrong assumption on my part.

Thanks again.