Portfolio risk and return - Question


I have a qus as the explanation provided in CFA book is not very clear.

Pg 339, Q 32-34

An analyst has made the foll return projections for each of the three possibe outcomes with an equal likelihood of occurence

Asset outcome 1 outcome 2 outcome 3 exp return (all in %)

1 12 0 6 6

2 12 6 0 6

3 0 6 12 6

q. which pair of assets is perfectly negatively correlated

Ans. the book says asset 2 and 3, but I am confused why not 1 and 2 as they also move in the opposite directions at the same magnitude? I think i m missing the concept here. There are 2 more qus but if I can get the concept right then I can answer all.

Can someone help?



It’s because Asset 2 maximizes its return when Asset 3 minimizes it, and vice versa. On the other hand, asset 1 and asset 2 both maximize their returns with outcome 1.

Asset 1 and 2 are positively correlated, because all of the outcomes yield an uneven return, they can’t be perfectly negatively correlated.

The correlation between (12,12) and (0,6) is not even negative.

Rather than asking us what the correct answer is and why, you should calculate the correlation coefficients for each pair of securities; there are only three such pairs, so it’s not a difficult job, and you’ll see for yourself what the answer is and why.

Please post your calculations when you do them.

Thanks a lot. Its clear now. as with 12 percent return on asset 1 only asset 3 has corresponding 0 return and vice versa. so they are perfectly negatively correlated . I was confusing between 6 and 0 but 6 is not the maximum, 12 is.

Thanks :slight_smile:

hello S2000magician

Maybe I am missing something but with the info given can we calculate the correlation coefficents? I couldnt see any calculation in the CFA explanation, and to be honest that was my first thought as to how can I calculate the coefficents? Could you advise?

Thanks a lot

Here’s an article I wrote on covariance and correlation; it includes the calculations: http://financialexamhelp123.com/covariance-and-correlation/.

Calculate the covariances (3) and the standard deviations (3). correlation coefficient for 1,2 = cov(1,2)/(sd1*sd2)… follow this procedure for the others.


Use a process of elimination on your question. Assets 1 and 2 both start with a downward movement, from 12% (can’t be perfectly negatively correlated), where as asset three start lower and moves higher. So, looking more critically, asset 3 and asset 2 move exactly in opposite directions and magnitude (up six and down six, respectively) for each outcome. Assets 1 and 3 move in opposite directions when looking at outcomes 1 and 2, but move in the same direction from outcome 2 to 3 (cant be perfectly negatively correlated). So, we can conclude asset 2 and asset 3 are perfectly negatively correlated.

Thanks a lot to both of you!!

My pleasure.

By the way, I hate the way this question is worded. It isn’t the assets themselves that have negative correlation (or any correlation, for that matter): the assets aren’t quantities. It’s _ the assets’ returns _ that have negative correlation.

This may sound like a mere semantic difference, but this kind of sloppy language (with which the financial industry is rife) leads to sloppy thinking: people – even smart, well-educated people – start to confuse correlation of returns and correlation of prices: they think that if the returns have a strong, negative correlation then the prices will also have a strong, negative correlation. It just ain’t so, but the sloppiness of the language helps ideas like this one fluorish.

Glad to help!

fully agree and I also read that in the link you had sent. I am not as experienced as you but maybe I will learn in the right way :slight_smile:

Even though anamolies exist, more often than not, price converges on return.

I have no idea what this means. As written, it makes no sense.

If the returns of two assets are strongly negatively correlated, then their respective price movements should also behave simillarly.


I just went to Yahoo! Finance and grabbed the two highest-volume stocks listed: BAC and RAD. Over the past 60 months, the correlation of their prices is +0.5623 and the correlation of their returns is +0.3109. Over the past 48 months the numbers are +0.7681 and +0.2632, respectively: they’re diverging. Over the past 340 months (nearly the entire history of BAC), they’re -0.0304 and +0.3111, respectively.

Correlation of prices and correlation of returns don’t have anything to do with each other.

Because value is a function of return, and price tracks value.

But correlation of returns doesn’t track value.

I think you misunderstand how correlation works. Look at the example I posted above, for two stocks that were chosen essentially at random.

How is return calculated in this example, capital gain+income on the stock or net income on the IS? R^2?