# PM Question - Benefits of Diversification

I don’t remember the exact wording, but there was a concept checker in Schweser that bothered me. It went something like this… At what level of correlation between two assets would a two-asset portfolio achieve the greatest benefits from diversification? A. +1 B. 0 C. -1 The correct answer was C. I get that a correlation of -1 would absolutely wipe out risks, but wouldn’t it also wipe out any expected return? I mean if stock A is up 15%, and stock B is down 15% then you’ve made nothing, and there is not “benefit” from this type of diversification, you’re just wasting commissions at that point. Maybe I should have read the question as the greatest benefit from a variance perspective, but I just don’t see this as the optimal asset allocation strategy. I would go with corr of zero. Then again, based on todays markets… does diversification really matter?

I says what gives the greatest benefits from diversification. It says nothing about optimal asset allocation strategy. These are two completely different concepts. Why would you go with zero? If you make the (unrealistic) assumption that returns are normal, zero correlation implies independence. What benefits are gained in the form of diversification by combining two independent assets? None.

Yeah but there’s no “benefit” to being perfectly hedged. Might as well sit in cash. Anyone can go long and short on the same stock and have a perfectly neg correlation but it doesn’t provide any benefit of diversification…

lets do it with basic math: you have 2 stocks in your portfolio: stock 1, std.dev1=s1, weight1=w1 stock 2 std.dev2=s2, weight2=w2 port.variance= w1^2*s1^2 + w2^2*s2^2 + 2w1w2s1s2rho for rho=-1 and remembering (a-b)^2=a^2-2ab+b^2 portfolio std.deviation= abs(w1s1-w2s2) first of all this does not imply that portfolio risk =0. for portfolio risk = 0 w1s1-w2s2=0 and for w1+w2=1 this gives you: w2=s1/(s1+s2) w1=1-w2 assuming you are holding 2 stocks long, rho=-1 (play with this a couple of times). stock 1: return 5%, s1=10 stock 2: return 10%,s2=10 x2=10/20=0.5 x1=1-0.5=0.5 return=0.5*5+0.5*10=7.5 so you have 7.5% percent return on the portfolio given portfolio risk=0. as you see this does not mean, you have no returns at all…

Barthezz, What you say makes sense until you build in the return assumptions… If the return on stock 1 is 5%, and the return on stock 2 is 10%, then rho cannot be -1. A correlation of -1 is a perfectly negative correlation, it’s not approximately negative where you could have trading days that stock 1’s return is not perfectly offset by stock 2’s return. Also, I forgot to mention in the original post, but I think the question said something about the portfolio being equally weighted, so that is not a concern here. I just don’t buy that there is any benefit to setting up a portfolio with a correlation of -1. Correlation of -0.9, yes, but not -1.

If two securities have a -1 correlation, it means (in easy terms) when one goes up, the other goes down, not when one goes up 10%, the other goes down 10%. Remember the correlation = COV (X,Y) divided by (std deviation X * std deviation Y) Or the data plot of a perfect negative correlation is a straight line, but does not have to have a slope of -1.

Oh, right… For two variables to be perfectly correlated, the change in one should be able to explain the entire change in the other (i.e. regression R^2=1), but that doesn’t mean it needs to change by the same magnitude. The inverse (for r=-1) should also be true, magnitudes don’t need to be equal. That makes sense. I was really hung up on this for some reason… Thanks guys.

sand is correct. Guys remember this hint: DO NOT OVERTHINK THE QUESTIONS!

Cool! so correlation is about the sign not the magnitude? Is there a way (a tool, a formula, etc) to express the extent/magnitude of the correlation?

The magnitude should show up in the regression coefficient. For example, let’s say the price of exxon mobile stock is perfectly correlated with the price of oil (which it’s not), but that for each one dollar increase in the price of oil, the price of exxon only moves up \$0.50, then your regression equation should look like: P(exxon) = 0.5 * P(oil) + intercept + error R^2 = 1.0 The regression coefficient (0.5) gives you an idea of what the magnitude is.

regression coefficient is not equal to correlation. correlation just gives you an idea about the strength of the linear relationship. i think you got it janky.

props to all of you for having a good thread.