# Decoupling Volatility and Correlation: the Russo Ratio

May 22, 2014 3 Comments

A few weeks ago, a client asked a deceptively simple question: “You guys tell us how much risk our portfolio has, but can you tell me how much of that risk comes from volatilities and how much comes from correlations?” I sat there a bit dumfounded and finally said “well, that’s a pretty obvious thing to ask. I wonder why I’ve never heard that question before.” In 20 years of professional risk management, that really was the first time I heard that question. Risk attribution is a common thing to analyze, but it’s usually answered in terms of how much risk each part of the portfolio carries: e.g., what fraction of the risk is from stocks, from bonds, etc. or from my technology investments, from my energy sector investoments, and so on. But the basic formulas for calculating portfolio risk take three inputs: the investment amounts into different securities, the volatilities of those securities and the correlations between those investments. It makes a lot of sense to ask how much comes from each part, and I’m quite embarrassed that I never thought about asking it myself. So off we went to solve the problem…

It turns out that the standard portfolio measure of risk, VaR (Value-at-Risk), cannot be easily separated into volatility and correlation but the square of VaR can be. Think of a right triangle: while you can’t relate the length of one side to the lengths of the other sides, you can relate the square of the sides to each other. Similarly, VaR squared can be related to two parts that are additive which we call the Volatility Contribution to Risk (VCR) and the Correlation Contribution to Risk (CCR). Because volatilities can only increase risk (never decrease it), the VCR is always a positive number. But correlations can be negative and decrease risk, making it possible for CCR to also be negative. This makes things a bit tricky when we express these numbers as a percent of total because it’s possible to have the VCR contribute more than 100% to the portfolio’s risk. It’s not uncommon for a portfolio to have, for example, 130% of its risk come from volatilities and -30% come from correlations – that just means that the correlations actively decrease the risk. In the extreme case of a perfectly hedged portfolio, CCR would be equal to -VCR: The correlations exactly cancel the volatilities, leaving no risk. Think of a portfolio that’s long 100 shares of Apple and short 100 shares of Apple for a simple, albeit trivial, example.

Alex Russo, my head of financial engineering, then suggested that the ratio CCR/VCR would be a compact way of conveying this information. He studied a couple of example portfolios to better understand CCR and VCR and we agreed that the ratio captures a lot of info and would be a good way of communicating the degree of diversification. In a meeting with one of our most creative and sharpest clients, we started calling it the Russo Ratio. You can download the full White Paper about this decoupling of volatility and correlation and the new Russo Ratio. One of the more interesting findings in our study is shown in the Figure: although the total risk of the US equity markets varies considerably over the past 10 years, the portion that was caused by volatility and the portion caused by correlation is remarkably stable: about 80% of the total is due to correlations and 20% due to volatitlies throughout the whole period, even though the actual volatility of stocks has been as low at 10% and as high as 90%.

I remain astonished that this approach wasn’t added to the common lexicon of risk management years ago. One rather well known risk manager told me in a recent conversation that he did something very similar at his hedge fund 10 years ago, but that they never let anyone know because they considered it ‘secret sauce’. I just wonder what other (in retrospect) obvious questions aren’t being asked or whose answers aren’t being talked about…

As someone coming from the academia, one criticism I heard all the time about the corporate world was that knowledge wasn’t free and that as a consequence each corporation was reinventing the wheel separately. So that you tell us that some hedge had thought this up and not told anyone is not surprising in the least.

Interesting way of describing that phenomenon as a “criticism” because in academe, knowledge is also used as a currency but the system is set up to eventually share it. You get credit (translate to eventually job offer) for your discoveries, which are often hidden until the formal mechanism of publication allows you to safely reveal your secrets.

In business, knowledge is translated to credit through sales and profit. And that requires a longer delay between discovery and eventually revelation. In some cases, that delay is infinite. Business and the legal system also set up patents and copyright to allow inventors to monetize the discovery and simultaneously reveal its secrets.

I agree that the scientific approach is a better system for participants as a whole. The business approach is better for (some) individuals. Like it or not, that’s the system “99.9%” of the world operates under.

You are not the only one embarrassed by this question. Great insight, thanks for sharing it.