## Correlated trades

One major weakness of quantitative trading is that it’s based on the concept of how correlated various instruments and instrument classes are. Today I’m planning to rant about this, thanks to a reader who suggested I should. By the way, I do not suggest that anything in today’s post is new- I’m just providing a public service by explaining this stuff to people who may not know about it.

Correlation between two things indicates how related they are. The maximum is 1 and the minimum is -1; in other words, correlation ignores the scale of the two things and concentrates only on the de-scaled relationship. Uncorrelated things have correlation 0.

All of the major financial models (for example Modern Portfolio Theory) depend crucially on the concept of correlation, and although it’s known that, at a point in time, correlation can be measured in many different ways, and even given a choice, the statistic itself is noisy, most of the the models assume it’s an exact answer and never bother to compute the sensitivity to error. Similar complaints can just as well be made to the statistic “beta”, for example in the CAPM model.

To compute the correlation between two instruments X and Y, we list their returns, defined in a certain way, for a certain amount of time for a given horizon, and then throw those two series into the sample correlation formula. For example we could choose log or percent returns, or even difference returns, and we could look back at 3 months or 30 years, or have an exponential downweighting scheme with a choice of decay (explained in this post), and we could be talking about hourly, daily, or weekly return horizons (or “secondly” if you are a high frequency trader).

All of those choices matter, and you’ll end up with a different answer depending on what you decide. This is essentially never mentioned in basic quantitative modeling texts but (obviously) does matter when you put cash money on the line.

But in some sense the biggest problem is the opposite one. Namely, that people in finance all make the same choices when they compute correlation, which leads to crowded trades.

Think about it. Everyone shares the same information about what the daily closes are on the various things they trade on. Correlation is often computed using log returns, at a daily return horizon, with an exponential decay weighting typically 0.94 or 0.97. People in the industry thus usually agree more or less on the correlation of, say, the S&P and crude.

[I’m going to put aside the issue that, in fact, most people don’t go to the trouble of figuring out time zone problems, which is to say that even though the Asian markets close earlier in the day than the European or U.S. markets, that fact is ignored in computing correlations, say between country indices, and this leads to a systematic miscalculation of that correlation, which I’m sure sufficiently many quantitative traders are busy arbing.]

Why is this general agreement a problem? Because the models, which are widely used, tell you how to diversify, or what have you, based on their presumably perfect correlations. In fact they are especially widely used by money managers, so those guys who move around pension funds (so have $6 trillion to play with in this country and $20 trillion worldwide), with enough money involved that bad assumptions really matter.

It comes down to a herd mentality thing, as well as cascading consequences. This system breaks down at exactly the wrong time, because after everyone has piled into essentially the same trades in the name of diversification, if there is a jolt on the market, those guys will pull back at the same time, liquidating their portfolios, and cause other managers to lose money, which results in that second tier of managers to pull back and liquidate, and it keeps going. In other words, the movements among various instruments become perfectly aligned in these moments of panic, which means their correlation approaches 1 (or perfectly unaligned, so their correlations approach -1).

The same is true of hedge funds. They don’t rely on the CAPM models, because they are by mandate trying to be market neutral, but they certainly rely on a factor-model based risk model, in equities but also in other instrument classes, and that translates into the fact that they tend to think certain trades will offset others because the correlation matrix tells them so.

These hedge fund quants move around from firm to firm, sharing their correlation matrix expertise, which means they all have basically the same model, and since it’s considered to be in the realm of risk management rather than prop trading, and thus unsexy, nobody really spends too much time trying to make it better.

But the end result is the same: just when there’s a huge market jolt, the correlations, which everyone happily computed to be protecting their trades, turn out to be unreliable.

One especially tricky thing about this is that, since correlations are long-term statistics, and can’t be estimated in short order (unless you look at very very small horizons but then you can’t assume those correlations generalize to daily returns), even if “the market is completely correlated” on one day doesn’t mean people abandon their models. Everyone has been trained to believe that correlations need time to bear themselves out.

In this time of enormous political risk, with the Eurozone at risk of toppling daily, I am not sure how anyone can be using the old models which depend on correlations and sleep well at night. I’m pretty sure they still are though.

I think the best argument I’ve heard for why we saw crude futures prices go so extremely high in the summer of 2008 is that, at the time, crude was believed to be uncorrelated to the market, and since the market was going to hell, everyone wanted “exposure” to crude as a hedge against market losses.

What’s a solution to this correlation problem?

One step towards a solution would be to stop trusting models that use greek letters to denote correlation. Seriously, I know that sounds ridiculous, but I’ve noticed a correlation between such models and blind faith (I haven’t computed the error on my internal estimate though).

Another step: anticipate how much overcrowding there is in the system. Assume everyone is relying on the same exact estimates of correlations and betas, take away 3% for good measure, and then anticipate how much reaction there will be the next time the Euroleaders announce a new economic solution and then promptly fail to deliver, causing correlations to spike.

I’m sure there are quants out there who have mastered this model, by the way. That’s what quants do.

At a higher perspective, I’m saying that we need to stop relying on correlations as fixed over time, and start treating them as volatile as prices. We already have markets in volatility; maybe we need markets in correlations. Or maybe they already exist formally and I just don’t know about them.

At an even higher perspective, we should just figure out a better system altogether which doesn’t put people’s pensions at risk.

Correlation is indeed tradable: http://en.wikipedia.org/wiki/Correlation_swap

Great post about the dangers of blindly using correlations from historical data…I realize the best of the best quant are way smarter than I and are not blindly following the data. And yet, there was/is a lot of money moved with limited understanding of the key underlying assumptions. (Marginal Call is a nice Hollywood-style explanation.) You are shining a light on one…where those correlations come from…sure this is not new stuff, but some things are worth repeating.

My more (opinionated) reaction:

1) Diversity of opinion is the KEY…herding should make one worried in any form.

2) Love your distrust of Greek letters in quant models, especially if the person wielding them doesn’t know it’s a Greek letter. Blind faith leads to being blind sided.

3) History does NOT repeat itself…forecasts based on historical relationships may be easy to defend and the best starting point, but if that’s where you stop…you will always be playing catch up

Hi,

first thanks for your posts I really enjoy reading them. Let me comment on this one.

There is a number of trades which give you an exposure to correlation starting from the simplest basket options through correlation swaps both for FX and equity not mentioning CDO. I think the biggest problem is that correlation is a linearization of the dependency and as such is a relatively useless concept. Think of XOR type of problems in Machine Learning.

Imagine, we are in 2006 the DAW is at 14K and interest rates are 5-6% you are a found manager. All your liabilities are quite heavily discounted and they depend on the interest rates, you invested in equity market. Your risk model if you have one assumes a slight (-0.1 ) negative correlation between rates and equity in general. On the other hand you know with a very high degree of probability that if equity markets tanks the rates are going to be slashed to zero this is a conditional correlation of 1. Obviously with zero rates PV of all your liabilities is going to skyrocket. This is obviously what happen a year latter. I did not know a single quant back than who did not know about this, still the hedges where very bespoke and expensive.

The more realistic risk model would assume some sort of regime switching with different correlations for each regime. The problems is that in Q you have too many parameters which cannot be implied in P on the other hand you have to make some assumptions about when the different regimes occurred in past. Once I tried to add a common jump process to rates and equity in order to model similar behavior but couldn’t get it to calibrate in Q.

I posted this on G+ but maybe I’ll put it here too:

I learned from +Constantine Costes via +Cathy O’Neil ‘s blog that there’s a trade called a “correlation swap” where you can make a bet on the measured correlation of two prices! I guess the point is that if you’re a large bank or fund you may have made large bets which are predicated on various instruments you hold moving roughly independently; if they become highly correlated, you lose money, which is to say you are “short correlation.” So you hedge by placing a side bet that the correlation on various instruments will go up.

My question is: who’s on the other side of these trades? Are there funds that specialize in shorting correlation? Or is everyone just trying to get to a place where they’re not exposed to correlation in either direction?

If the latter: surely then they’re exposed to something else, right? Just as funds which hedged away their exposure to gains and losses in the actual asset price ended up short correlation, funds which hedge against correlation increase are exposed to…?

(Other remark: I did not know there was an arXiv category for asset pricing.)

JSE the correlation swaps which are traded currently are not between two prices. These are typically done for full indices like S&P.

Sebastien Bossu paper explains this in details and shows a typical pricing methodology:

http://math.uchicago.edu/~sbossu/CorrelSwapPricing.pdf

You can trade a correlation swap between to FX rates.

This is my favourite blog by a long way.

>figure out a better system altogether which doesn’t put people’s pensions at risk

Society as a whole can’t move wealth into the future through money. The attempt to do so on a massive scale (superannuation/pensions) is a large part of the current problem. [E.g. why is exec remuneration out of control? One reason is that the owners of shares are other companies with policies dictated by their execs.] In a world where total wealth is set to decline for a while (due to the cost of changing infrastructure as the cheap oil runs out) we need to allocate to old people a right to a fraction of total social wealth, not expect them to keep a fixed amount so that everyone else is sharing a smaller part of a smaller pie.

Heck, forget about correlation, we can’t even measure volatility unambiguously, and that’s something that (in a particular realization) is actually traded on in a pretty liquid market. (swaptions, etc.) So much work goes into coming up with a decent term structure of volatility/ volatility grid, and it all depends into how, exactly, you’re defining it. Correlations are just one step higher in complexity.

true.