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# Correlation Coefficient

You can never build a proper portfolio if you ignore the idea of correlation. Because whether and to which degree assets move in sync, or in completely different cycles, is what is going to determine how much you can diversify your portfolio and again, we have not quite defined this concept of diversification just yet. The important thing is that you cannot have a proper portfolio if you ignore this concept of correlation. Now, correlation sometimes, you know, many of the things in finance are referred to with Greek letters. We have already seen Beta and correlation is one of those and the typical letter used to describe correlations is basically Rho (ρ).

What is Correlations?

Correlation is always measured between pairs of variables. It could be two assets, it could be height and weight, and it could be anything. Any two variables can have a correlation and that correlation can actually be estimated.

### Importance of correlation

What does correlation measure?

Correlation measures the strength of the relationship. I can have two variables, which again, could be the return of an asset and the return of another asset or it could be the height and the weight of all the people taking this course, right? In which case you more or less would expect a positive relationship, but that says something about why correlation is so important.

• The sign of the correlation

The sign of correlation could be positive or it could be negative. If it is positive, it means that the two variables tend to move together. So where you would expect taller people to be a little bit heavier and shorter people to be a little bit lighter so in general if you actually look at it over a, a large number of people, you would expect a positive correlation between height and, weight. In the same way, that you would expect a positive correlation in finance between risk and return. So one thing that is important about those correlations is the sign is the correlation positive or is the correlation negative. If you were selling ice cream for example in the end then you actually looked at sales of ice cream and temperature. Well more likely than not there is going to be a negative correlation because the colder is the temperature the less the ice cream you are going to sell. Therefore, a correlation could be positive or correlations could actually be negative.

The other thing that matters in terms correlations is basically the strength of the relationship because its strength relationship can be very loose, i.e. if you know the value of one of the two variables there is not much that you can say about the value of the other variable. It could be very strong and by very strong I mean that if you know the value of one variable, you can make an accurate prediction in terms of what would be the value of the other variable. So it is important that you keep in mind these two dimensions of correlations.

• Correlations measure the sign. Positive or negative?
• Do they tend to move together, or do they tend to move in opposite directions?
• The strength of that relationship. Is there a clear relationship between the two?
• By knowing the value of one variable, can I make an accurate prediction or a very loose prediction about the value of the other variable?

So this correlation coefficient shown in the table measures the sign and the strength of the relationship between these two variables. By measuring the sign and the strength obviously the sign can only be two, it could be positive or could be negative. Of course, it could be zero, too, but that would be a very special case. In terms of the strength, it could be weak or it could be strong. Strong relationship means that if I know the value of one variable, it gives me a very accurate prediction of the other and in the case of a weak relationship if I know the value of one variable; it does not tell me a lot about the variable of the other.

### Range of the Correlation

The range in which correlations fall can go between one on the positive end and minus one on the negative end. Now, on the positive end, the meaning of a correlation equal to one. Well, in terms of strength, it does not get any stronger than that is. If you give me the value of one of the two variables and I could tell you exactly what the value of the other is going to be. Then that is basically a correlation equal to one. That is what sometimes we call it a deterministic relationship if you know the value of one, you can tell exactly what would be the value of the other. In other words, if you have X on one axis and Y on the other axis. The relationship between X and Y is given by a straight line, and all the points would actually fall along the line. So that if I give you the value of one, you could tell me exactly what is the value of the other. That would be a positive relationship, and a relationship equal to one. It does not get any stronger than that. It gives you total accuracy, total predictability. Know the value of one variable; you will know exactly the value of the other.

While, minus one means more or less the same, the only difference is that now, the relationship is negative and so basically we have, if we have X here and Y here, we have a line with a negative slope and all the points along that line would actually fall exactly along that line with a negative slope. Again it remains the case that if I give you the value of one variable then you can tell me exactly what will be the value of the other. In terms of the strength, it doesn’t get any stronger than that. So the two extremes plus one and minus one is as strong as a relationship can be and it is as strong as it can be because then the relationship becomes deterministic. You do not predict more or less what the other variable will be. You actually predict exactly what the other variable would be. In finance, we do not have any relationship with values of one or values of minus one. There are no deterministic relationships in finance. So what matters, is whether we are getting close to one extreme or close to the other. Because that would actually indicate a very strong correlation between the two. But in finance and in mathematics, you have a lot of deterministic relationships. But in finance, we do not have those. But it is still important. Although we do not find in practice, variables, financial variables that are correlated equal to one or correlating equal to minus one and this is the highest possible value and the lowest possible value because the closer we get to those extremes, then the stronger the relationship actually is.

### Correlation Analysis

It is important to keep in mind that although in financial markets we do not expect to find valuables that have a correlation equal to one or equal to minus one. The extreme values of the correlation coefficient, it is important to know the theoretical streams. Because what we really want to know is that the closer, the correlation coefficient gets to one or the closer it gets to minus one, then the stronger is the relationship between the two variables and the more predictability there is going to be between these two variables. Now, on the same token, as we get away from the streams and we are getting closer to zero then the relationship becomes weaker. It means that if we know the value of one variable there is very little that we can tell me about the volume of the other. When we are getting, approaching zero on both from the positive side and from the negative side then we have more than certainty. Basically, our model doesn’t work if I tell you one variable, there’s very little you can tell me about the value of the other variable. Now, here is one important thing. Although in theory, a correlation could be positive or could be negative, it could be all the way or all the way to minus one. For example, if you look across world equity markets, or if you look at individual stocks within a market. If you look at long enough period of time, you are going to find that all these correlations are positive and the reason they are positive is what we call the market factor. That is each company will be affected by a lot of individual factors and in the portfolio sort of diversify way. But, there is going to be someone pulling or something pulling the return. Those global factors, macro factors, pulling the return of all the companies in the same direction or all the market in the same direction. Remember, this is on average and over time. On any given year, some stocks will go up, some stocks will go down within the market but, what really matters is that degree of diversification that we obtain when we put all these assets together.

Now, why is it that it is positive? Well, it depends on that market factor but let me go back one.

#### AM, GM, SD, Beta AND Rho

 Year USA Spain Egypt World 2004 10.7% 29.6% 126.2% 15.8% 2005 5.7% 4.9% 161.6% 11.4% 2006 15.3% 50.2% 17.1% 21.5% 2007 6.0% 24.7% 58.4% 12.2% 2008 -37.1% -40.1% -52.4% -41.8% 2009 27.1% 45.1% 39.7% 35.4% 2010 15.4% -21.1% 12.4% 13.2% 2011 2.0% -11.2% -46.9% -6.9% 2012 16.1% 4.7% 47.1% 16.8% 2013 32.6% 32.3% 8.2% 23.4% AM 9.4% 11.9% 37.2% 10.1% GM 7.6% 7.9% 21.4% 7.7% SD 17.9% 28.1% 64.0% 20.1% Beta W 0.9 1.1 1.5 1.0 Rho W 0.95 0.82 0.48 1.00

Look at last line now, that gives you the correlation between each individual market and the world market so if you look at the 1.00 for the world market that tells you the correlation between a variable and itself is going to be one. It is like plotting the same variable twice and so by definition, the correlation between a variable and itself is always going to be one but look at the other numbers. The US market, very highly correlated with the world market. The Spanish market, pretty highly correlated with the world market and the Egyptian market, much lower correlation with the world market. This should not be surprising. The reason it should not be surprising is that Egypt is an emerging market. Emerging markets tend to be a little bit more isolated from large world capital markets and large world equity markets and so you would expect that small and more isolated markets by definition would have a lower correlation than the large and more integrated market.

Look at the data. Is that all these three correlations are positive. The correlation between the U.S. and the world market, Spain and the world market and Egypt and the world market, all of them are positive. In different degrees, the U.S. is the most highly correlated, Egypt is the least highly correlated but all of them tend to move in the same direction. The fact that all the correlations are positive means that when the world market goes up, these three markets tend to go up too. It does not really; we are not talking about causality here. We are not saying that because the world market goes up, the Spanish or the Egyptian market go up or the other way around. What we are saying is simply that they tend to move together. One more important thing about correlation is that Correlation does not measure causation which means that when we say that two variables have a very large positive correlation or a very large negative correlation, you are not implying anything about which one the determines the other. What you are saying is that they tend to move together in the same direction or the opposite direction and those they are very strongly related. However, we are not saying anything when you calculate a correlation in terms of which one is affecting the other. It does not matter whether x affects y or effects x, the only matter is whether they tend to move together in the opposite directions. Whether their relationship is strong or their relationship is actually much weaker.