Correlation
We've been doing it wrong this whole time, and the collapse of the global economy in 2008 might have exposed it
By Cal Roff
August 10, 2024
In 2008, something happened that very few saw coming; the burst of a housing bubble of unthinkable proportions slowed the global economy to a halt. Banks incurred losses globally, lending froze, retirement savings vanished and recession loomed. To sum up what went wrong, banks engineered vastly complex webs of financial instruments and executed trading strategies built entirely on the belief that mortgages = money, and lots of it when they are bundled into collatoralised debt obligations (CDOs). But how did this belief form? From assumptions, that's how. Assumptions that the underlying value of a mortgage, which is the mortgage holder's ability to meet their obligations, are generally met. Assumptions that we can objectively assign dollar value to assets dependent on human behaviour.
Felix Salmon[1] writes great article on this following the GFC, where it's suggested the crash stemmed from a culture on Wall street of oversimplifying math in places it shouldn't. Specifically, he's referring to a model called the Gaussian Copula. Consider two random variables X and Y. These could represent the price of two securities, atmospheric pressure in two different locations, the number of times per day dog A and dog B bark. Anything. We define the Gaussian copula as the following function of x and y - observed instances of random variables X and Y respectively:
$$\displaystyle C_{p}(x, y) = \Phi_{\rho}(\Phi^{-1}(F_X(x)), \Phi^{-1}(F_Y(y)))$$
Let's break it down. \( F_{X}:(-\infty, \infty) \rightarrow [0, 1] \) and \( F_Y:(-\infty, \infty) \rightarrow [0, 1] \) are the cumulative distribution functions (CDF) of the random variables. \( F_X(x) \) for instance represents the probability of observing a value less than or equal to x, given a set of assumptions applied to random variable X. Mathematically:
$$\displaystyle F_{X}(x) = P(X \leq x | X \sim D(\theta)) \in [0, 1]$$
Where \( D(\theta) \) is the distribution of X, parametrised by \( \theta \). In practice, we often make assumptions on D by looking at a histogram of X's observations, as they are or transformed, and saying "hey, this looks like such and such distribution", then using maximum likelihood to estimate \( \theta \). In finance the price of a security or CDO is often assumed to follow a log-normal distribution.
Next, \( \Phi^{-1}:[0, 1] \rightarrow (-\infty, \infty) \) represents the inverse of the standard normal distribution CDF. Now we are no longer dealing with a probability, but instead where that probability would exist as an observed data point in a standard normal distribution. And finally, \( \Phi_{\rho}:(-\infty, \infty)^{2} \rightarrow [0, 1] \) is the CDF of the bivariate joint normal distribution, with \( \rho \in [-1, 1]\) the Pearson correlation between outputs \( \Phi^{-1}(F_X(x)) \) and \( \Phi^{-1}(F_y(y)) \). Now because these are strictly increasing functions of x and y, it follows that \( \rho \) here can bee seen as a standardised form Pearson correlation between X and Y. Again, \( \rho \) can be estimated using maximum likelihood.
For those who don't like math or aren't familiar with terms like maximum likelihood, the key takeaway here is that dependence between X and Y under the Gaussian copula model is reducible to a single constant \( \rho \). This is the problem. To explain why, let's say, for example, that company A and company B are both electric vehicle manufacturers in country C. If the government of C passes a bill to subsidise the purchase of new electric vehicles, then we'd likely see the stock price of A and B trend upwards, suggesting positive correlation between them (\( \rho > 0 \)). Now suppose the CEO of company B is exposed in a scandal allowing hush money to be paid to ex-employees who were victims of workplace sexual harassment. The public loses trust, and company A capitalises on the situation seeing their stock price trend upwards and B's go down, suggesting now a negative correlation (\( \rho < 0 \)).
Such an example would be considered extreme, but what kind of an economic collapse would occur under anything other than extreme circumstances? The problem of capturing dependence relies not only on allowing dependence to vary at different regions within the X-Y plane, but also ensuring we have adequate coverage of dependence in the extreme regions. In economic downturns, correlation between assets typically appear stronger. Remaining loyal to the theme of copulas, one might consider the Clayton Copula:
$$\displaystyle C_{\theta}(x, y) = (\text{max}\{ [F_X(x)]^{-\theta} + [F_Y(y)]^{-\theta} - 1, 0\})^{-1/\theta} $$
Let's not get too much into the detail this time. Instead I'll point out that \( \theta \) governs the overall strength of correlation, while allowing that strength to vary locally - within different regions of the X-Y plane. In particular, that strength peaks at the lower limits of X and Y, and deteriorates as one or the other increases. Think of the resulting plot between the original X and Y observations as cone-shaped, whereas for the Gaussian copula, it appears to be more of an elliptical scatter. The key implication is that now we are at a reduced risk of overestimating the value of asset X if correlated asset Y goes pear-shaped. We are also more likely to be conservative in our valuation of asset X if Y sky-rockets. In short, things are less likely to blow up.
The Clayton copula is just one of many alternatives to it's Gaussian counterpart when dependence is assumed to be non-static. In fact, the Clayton copula belongs to a family known as the Archimedean copulas[2], which are characterised by the following form (two-dimensionally):
$$\displaystyle C_{\theta} (x, y) = \psi_{\theta}^{-1}(\psi_{\theta}(F_X(x)) + \psi_{\theta}(F_Y(y)))$$
Where \( \psi_{\theta} : [0,1] \rightarrow [0, \infty) \) is a generator function. For the Clayton copula this is \( (t^{-\theta} - 1)/\theta \). These follow a certain set properties which I won't delve into but instead provide a link in the references if you're keen to do a bit of research. Just like the Clayton copula, the overall strength of dependence between two variables modelled by an Archimedean copula is goverend by \( \theta \), and its variance across different X-Y regions characterised by generator \( \psi \). So if assumptions of extreme lower-tail dependence are not appropriate for the problem at hand, we could transition from the Clayon to the Gumbel copula if there's indication of extreme upper-tail dependence, or the Frank copula when dependence is strongest in the mid X-Y regions and shrinks as we tend to the extremes.
We've only scratched the surface of methods we can turn when the assumption of static dependence between two variables is misleading. There are countless ways to characterise and quantify dependence, beyond copula modelling and even extending to non-monotonic assumptions[3] (for example, X may increase with Y in region 1, decrease with Y in region 2 and then increase with Y again in region 3. Think of this as a snake-like curve in a plot of Y versus X). I will say that Pearson correlation can be helpful when we're after a general idea of how two variables trend against eachother in a straight-line fashion over the long run. I often make use of it in data science projects to get a quick view of which features may be useful in a model. But it's simply impractical when the stakes are high.
To wrap this up, I ask that next time you're curious about how certain things are related, pay some thought to how and when those relations may change. It could save an economy.
Links
[1] Recipe for Disaster: The Formula That Killed Wall Street
[3] Pearson Correlation: Methodology, Limitations & Alternatives
August 10, 2024
In 2008, something happened that very few saw coming; the burst of a housing bubble of unthinkable proportions slowed the global economy to a halt. Banks incurred losses globally, lending froze, retirement savings vanished and recession loomed. To sum up what went wrong, banks engineered vastly complex webs of financial instruments and executed trading strategies built entirely on the belief that mortgages = money, and lots of it when they are bundled into collatoralised debt obligations (CDOs). But how did this belief form? From assumptions, that's how. Assumptions that the underlying value of a mortgage, which is the mortgage holder's ability to meet their obligations, are generally met. Assumptions that we can objectively assign dollar value to assets dependent on human behaviour.
Felix Salmon[1] writes great article on this following the GFC, where it's suggested the crash stemmed from a culture on Wall street of oversimplifying math in places it shouldn't. Specifically, he's referring to a model called the Gaussian Copula. Consider two random variables X and Y. These could represent the price of two securities, atmospheric pressure in two different locations, the number of times per day dog A and dog B bark. Anything. We define the Gaussian copula as the following function of x and y - observed instances of random variables X and Y respectively:
$$\displaystyle C_{p}(x, y) = \Phi_{\rho}(\Phi^{-1}(F_X(x)), \Phi^{-1}(F_Y(y)))$$
Let's break it down. \( F_{X}:(-\infty, \infty) \rightarrow [0, 1] \) and \( F_Y:(-\infty, \infty) \rightarrow [0, 1] \) are the cumulative distribution functions (CDF) of the random variables. \( F_X(x) \) for instance represents the probability of observing a value less than or equal to x, given a set of assumptions applied to random variable X. Mathematically:
$$\displaystyle F_{X}(x) = P(X \leq x | X \sim D(\theta)) \in [0, 1]$$
Where \( D(\theta) \) is the distribution of X, parametrised by \( \theta \). In practice, we often make assumptions on D by looking at a histogram of X's observations, as they are or transformed, and saying "hey, this looks like such and such distribution", then using maximum likelihood to estimate \( \theta \). In finance the price of a security or CDO is often assumed to follow a log-normal distribution.
Next, \( \Phi^{-1}:[0, 1] \rightarrow (-\infty, \infty) \) represents the inverse of the standard normal distribution CDF. Now we are no longer dealing with a probability, but instead where that probability would exist as an observed data point in a standard normal distribution. And finally, \( \Phi_{\rho}:(-\infty, \infty)^{2} \rightarrow [0, 1] \) is the CDF of the bivariate joint normal distribution, with \( \rho \in [-1, 1]\) the Pearson correlation between outputs \( \Phi^{-1}(F_X(x)) \) and \( \Phi^{-1}(F_y(y)) \). Now because these are strictly increasing functions of x and y, it follows that \( \rho \) here can bee seen as a standardised form Pearson correlation between X and Y. Again, \( \rho \) can be estimated using maximum likelihood.
For those who don't like math or aren't familiar with terms like maximum likelihood, the key takeaway here is that dependence between X and Y under the Gaussian copula model is reducible to a single constant \( \rho \). This is the problem. To explain why, let's say, for example, that company A and company B are both electric vehicle manufacturers in country C. If the government of C passes a bill to subsidise the purchase of new electric vehicles, then we'd likely see the stock price of A and B trend upwards, suggesting positive correlation between them (\( \rho > 0 \)). Now suppose the CEO of company B is exposed in a scandal allowing hush money to be paid to ex-employees who were victims of workplace sexual harassment. The public loses trust, and company A capitalises on the situation seeing their stock price trend upwards and B's go down, suggesting now a negative correlation (\( \rho < 0 \)).
Such an example would be considered extreme, but what kind of an economic collapse would occur under anything other than extreme circumstances? The problem of capturing dependence relies not only on allowing dependence to vary at different regions within the X-Y plane, but also ensuring we have adequate coverage of dependence in the extreme regions. In economic downturns, correlation between assets typically appear stronger. Remaining loyal to the theme of copulas, one might consider the Clayton Copula:
$$\displaystyle C_{\theta}(x, y) = (\text{max}\{ [F_X(x)]^{-\theta} + [F_Y(y)]^{-\theta} - 1, 0\})^{-1/\theta} $$
Let's not get too much into the detail this time. Instead I'll point out that \( \theta \) governs the overall strength of correlation, while allowing that strength to vary locally - within different regions of the X-Y plane. In particular, that strength peaks at the lower limits of X and Y, and deteriorates as one or the other increases. Think of the resulting plot between the original X and Y observations as cone-shaped, whereas for the Gaussian copula, it appears to be more of an elliptical scatter. The key implication is that now we are at a reduced risk of overestimating the value of asset X if correlated asset Y goes pear-shaped. We are also more likely to be conservative in our valuation of asset X if Y sky-rockets. In short, things are less likely to blow up.
The Clayton copula is just one of many alternatives to it's Gaussian counterpart when dependence is assumed to be non-static. In fact, the Clayton copula belongs to a family known as the Archimedean copulas[2], which are characterised by the following form (two-dimensionally):
$$\displaystyle C_{\theta} (x, y) = \psi_{\theta}^{-1}(\psi_{\theta}(F_X(x)) + \psi_{\theta}(F_Y(y)))$$
Where \( \psi_{\theta} : [0,1] \rightarrow [0, \infty) \) is a generator function. For the Clayton copula this is \( (t^{-\theta} - 1)/\theta \). These follow a certain set properties which I won't delve into but instead provide a link in the references if you're keen to do a bit of research. Just like the Clayton copula, the overall strength of dependence between two variables modelled by an Archimedean copula is goverend by \( \theta \), and its variance across different X-Y regions characterised by generator \( \psi \). So if assumptions of extreme lower-tail dependence are not appropriate for the problem at hand, we could transition from the Clayon to the Gumbel copula if there's indication of extreme upper-tail dependence, or the Frank copula when dependence is strongest in the mid X-Y regions and shrinks as we tend to the extremes.
We've only scratched the surface of methods we can turn when the assumption of static dependence between two variables is misleading. There are countless ways to characterise and quantify dependence, beyond copula modelling and even extending to non-monotonic assumptions[3] (for example, X may increase with Y in region 1, decrease with Y in region 2 and then increase with Y again in region 3. Think of this as a snake-like curve in a plot of Y versus X). I will say that Pearson correlation can be helpful when we're after a general idea of how two variables trend against eachother in a straight-line fashion over the long run. I often make use of it in data science projects to get a quick view of which features may be useful in a model. But it's simply impractical when the stakes are high.
To wrap this up, I ask that next time you're curious about how certain things are related, pay some thought to how and when those relations may change. It could save an economy.
Links
[1] Recipe for Disaster: The Formula That Killed Wall Street
[3] Pearson Correlation: Methodology, Limitations & Alternatives