Series : Risk measures correlation in finance

In 1990, Harry Markowitz was awarded the Nobel prize for his theory on portfolio choice, where diversification plays a central role. Diversification is made possible through the fact that adding assets together – assets that are not perfectly correlated to each other in terms of potential evolution – allow us to reduce the total variability of a given portfolio. When correlation is lower than 1, it means that a fraction of the variability of an asset can compensate part of the variability of another one, instead of simply adding up.

This concept is crucial because it allows us to quantify the added value of managers that do not “put all their eggs in the same basket”. This has consequences everywhere. For example, if a bank has a better asset diversification, the total dispersion of its asset value will be lower, potentially leading to a lower equity requirement, to act as a buffer. As a simple illustration, and as explained by Markowitz (1976), if we add securities with the same return volatility to a portfolio, but with a correlation of only 25%, with 10 securities, the total variability of the portfolio will be 57% of the volatility of any single asset in it. Still, the effect is very strong with the first ones, but decays sharply as more securities are added to the portfolio, as the table here below shows. And with a high correlation, the capacity to diversify by adding more assets is quickly limited.

Finally, taking the correlation for granted can give us a false sentiment of security about the effective diversification we will benefit from in the future. First, in practice, the correlation is statistically estimated over past data. And nothing guarantees us that the future behaviour won’t be different. Second, we might not be looking at the right correlation. Correlation in good times might well be different from correlation in critical times.

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Systemic phenomena in bad times can make correlations increase substantially, and the expected diversification will vanish away when we need it most. Relying therefore on risk measures that depend on correlation might lead us to the underestimation of exposures in extreme cases. Third, this is “human science” where a better understanding of behavioural economics can explain how our actions can have an the effective correlation, killing the diversity. Let’s illustrate this. In the past, asset allocations tended to praise emerging markets securities because of the diversification power they could add to the portfolio. Indeed, even if those markets were viewed as very volatile, their reduced correlation with our markets made them attractive, since only a small fraction of that volatility would impact the portfolio, being partially cancelled by uncorrelated effects elsewhere. But, as globalization increased, without yet producing the expected benefits of reducing the risk to invest in some of those markets, only correlation increased, killing the diversification benefit, producing a perverse effect (see Goetzmann et al (2001, “Long-Term Global Market Correlations”, NBER). Fourth, the way we compute correlation, tests for the linear dependence between two variables. But there are dependences that are not linear, like: B doesn’t necessarily go up when A goes up, but when A goes down, B surely goes down as well. We can see here that there is a dependence we can describe, but computing correlation might even return a null value in some cases.

Finally, correlation is not causation or causality. We tend to confuse both. The fact that A and B are correlated means their behaviour seems to share some similarities, but we might have found just a statistical relationship in the data. That doesn’t guarantee necessarily there is any direct causal relation between A and B, and if there is one, we don’t know anything about its direction, i.e. that A drives B or that B drives A. Checking for that would require other tests. In the figure above, from Tyler Vigen, undue conclusions could be drawn like “the evolution of arcade revenues encourages more computer scientists to pursue a doctorate” or “having a doctorate make you play more arcade games”. But in fact, there is a trend in both time series and the correlation simply captures that trend. Sometimes, it is due to a hidden variable or reason, driving each of them separately. Here, the discovery of more powerful CPUs might have encouraged the development of more sophisticated and realistic arcade games, whilst encouraging more graduates to do research in IT.

Correlation is a measure used in $