# Visual Correlations Part 1

January 4, 2011 2 Comments

I like pictures. In my copious spare time I’m a black-and-white photographer. I use film. I have my own darkroom. Maybe that’s why I think of correlations in terms of diagrams instead of just as numbers.

Correlations are the key to understanding how diversification lowers financial risk, and diversification is one of the simplest and oldest ways that humans have managed risk. By spreading around the chances of bad things happening, you lower your risk. “Don’t put all your eggs in one basket” is obvious, even to children. Most importantly – it works. Financial portfolio management implements this advice by applying investment guidelines like “no more than 5% invested in any one company.” This automatically forces the investments to be spread over at least 20 different companies.

But what happens if those 20 companies’ values/stock prices start rising and falling together – the equivalent of putting your eggs into 20 baskets and then carrying them all down a steep hill? The relevant question then becomes ‘How much do you reduce your risk by spreading the investments across many companies if their stock prices are related?’ The short answer is that it all depends on the correlation between the assets. If two stocks are highly correlated, then putting money into the second stock doesn’t lower the overall risk at all. But if the two stocks are uncorrelated, then you can reduce your risk substantially by investing in the second stock.

It turns out that the math of risk reduction and correlations can be visualized as a bunch of triangles. I particularly like thinking about risk this way because it’s much more intuitive than equations, but I want to assure you that this is not some watered-down oversimplification. The math works out to be exactly the same as the math of triangles, but I’m going to skip the math until a later post. In each of the examples below, we’re going to talk about a two-stock portfolio, but the same things apply no matter how many investments you have. So, suppose you have two stocks in your portfolio and you’ve invested the same amount (say $100) into each. The first stock has a volatility of 18% (about average for US equities) and the second stock has a volatility of 13% (below average). To demonstrate how correlation affects the portfolio’s risk, each of the examples below is different only in the correlation between the stocks. We’ll look at the total portfolio when the correlations are one, zero and negative 0.95. All of my diagrams below are drawn to scale – the lengths and orientations of the arrows correspond to the numbers in the text.

**Correlation = 1.** Figure 1 shows the situation when the two stocks are perfectly correlated. Each investment is represented by a blue arrow, the length of which is a measure of its risk, in dollars (not percent). The red arrow shows the total portfolio risk, which is simply the sum of the individual investments’ risk. If $100 is put into each stock, then the first stock’s risk is $18 (volatility of 18% times the $100) and the second stock’s risk is $13. The portfolio risk is $31, which is 15.5% of the $200 investment, or exactly half way between 18% and 13%. This seems quite intuitive and is pretty straightforward.** **

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** ****Correlation = 0.** What happens if the two stocks are completely uncorrelated? In this case, the picture changes to the one in Figure 2. Two stocks are completely uncorrelated when there is absolutely no relationship between their relative movements – they move completely independently of one another, as if the other doesn’t even exist. The picture is a right triangle, and the total risk is significantly reduced (this red arrow is much shorter than the one in Figure 1). In this case, the two stocks individually still carry 18% risk and 13% risk. But when put into a portfolio together, the total portfolio risk drops to 11%! *What? The portfolio has less risk than either of the two stocks??* That’s exactly right. The total portfolio risk is simply the length of the red arrow, and high-school level geometry gives the Pythagorean formula. With blue sides of $18 and $13, the red side is $22, and $22 of risk out of a $200 investment is 11%. Pretty cool, if you ask me. When correlation is perfect, the risk is 15.5%. When there’s no correlation, it drops to 11%. That’s a risk reduction of almost 30% just from correlations. If that’s impressive, just wait…

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**Correlation → -1.** Negative correlations don’t happen very often in stocks, so they’re a bit unintuitive. It means the two stocks move opposite each other (relative to their own normal movements). When one goes up (relative to its average), the other goes down (relative to its average). So what happens if the two stocks we’ve invested in are negatively correlated? Figure 3 shows the portfolio when the correlation is -0.95 (almost perfectly anti-correlated). In this case, the second arrow is pointing almost backwards against the first. The red arrow, showing the total portfolio risk, is quite short. The math works out, in this case, to a total portfolio risk of only $6.96, out of the $200 investment. That’s only 3.5%. This is why hedging works. If the correlation were exactly -1, the second arrow would double back exactly on the first and reduce the portfolio’s risk to $5, or 2.5%. That’s the smallest risk we can take with equal investments in these stocks.

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**Any Correlation.** So far, we’ve only looked at three extreme cases of correlation, but what’s the picture for any correlation between the stocks? It’s a very specific triangle – where the two legs are the investments and the connector (red arrow in my diagrams) is the total portfolio risk. The angle that the two stock arrows make with each other is determined by the correlation. Figure 4 shows what the triangle for any/all values of correlation. The important thing to note is that the correlation is basically a rotation of one stock’s risk against the other. If the angle is small (correlation close to 1, as in Figure 1), the red arrow is long. If the angle is near 90 degrees (correlation close to 0, as in Figure 2), the red arrow is reduced by up to 29%. But to really lower risk, the angle has to be big (correlation close to -1, as in Figure 3).

I like it when pretty pictures help explain how things works.

Good article. Years ago a risk manager co-worker of mine with a physics background pointed out that modern portfolio theory reduced risk analysis to trigonometry, using these same arguments. Much of the controversy regarding the injection of mathematics into finance circles around how these methods are perhaps too powerful: the inner structure of covariance, i.e., its trigonometry, if you will, gave the Efficient Market Hypothesis much of its glamor and prompted many financial engineers and practitioners to overleverage the model.

In the spirit of a picture being worth a 1000 words, we should visualize how these triangles and vectors change their shapes over a period of market turmoil.

Richard – thanks, and I couldn’t agree more. This is a rather simplistic view of risk, and one that only applies if volailities and correlations remain static. It’s basically like a free-frame photo of something in motion. It doesn’t capture the dynamics. Worse still, as you point out, its mathematical elegance lured many to accept the EMH as reality.

I’ve often thought about how to visualize the triangles/vectors over time and how to visualize each security’s contribution when there are hundreds of them in a portfolio. It’s easy to do for two securities, as I’ve done here, but it becomes something of a challenge as soon as you have four because we quickly run out of dimensions. It’s unfortunately hard for us to visualize even a 4 dimensional vector space.